plasma-expert/reference/sources/bazelyan-raizer-lightning-physics-2000.txt

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Lightning Physics and 
Lightning Protection 
E M Bazelyan 
Yu P Raker 
and 
IOP 
Institute of Physics Publishing 
Bristol and Philadelphia 
Copyright © 2000 IOP Publishing Ltd.

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IOP Publishing Ltd 2000 
All rights reserved. No part of this publication may be reproduced, stored in a 
retrieval system or transmitted in any form or by any means, electronic, 
mechanical, photocopying, recording or otherwise, without the prior permission of 
the publisher. Multiple copying is permitted in accordance with the terms of 
licences issued by the Copyright Licensing Agency under the terms of its agreement 
with the Committee of Vice-Chancellors and Principals. 
British Library Cataloguing-in-Publication Data 
A catalogue record for this book is available from the British Library. 
ISBN 0 7503 0477 4 
Library of Congress Cataloging-in-Publication Data are available 
Publisher: Nicki Dennis 
Commissioning Editor: John Navas 
Production Editor: Simon Laurenson 
Production Control: Sarah Plenty 
Cover Design: Victoria Le Billon 
Marketing Executive: Colin Fenton 
Published by Institute of Physics Publishing, wholly owned by The Institute of 
Physics, London 
Institute of Physics Publishing, Dirac House, Temple Back, Bristol BS1 6BE, UK 
US Office: Institute of Physics Publishing, The Public Ledger Building, Suite 1035, 
150 South Independence Mall West, Philadelphia, PA 19106, USA 
Typeset in 10/12pt Times by Academic + Technical, Bristol 
Printed in the UK by J W Arrowsmith Ltd, Bristol 
Copyright © 2000 IOP Publishing Ltd.

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Contents 
Preface 
ix 
1 Introduction: lightning, its destructive effects and protection 
1.1 Types of lightning discharge 
1.2 Lightning discharge components 
1.3 Basic stages of a lightning spark 
1.4 Continuous and stepwise leaders 
1.5 Lightning stroke frequency 
1.5.1 Strokes at terrestrial objects 
1.5.2 Human hazard 
1.6.1 A direct lightning stroke 
1.6.2 Induced overvoltage 
1.6.3 Electrostatic induction 
1.6.4 High potential infection 
1.6.5 Current inrush from a spark creeping along the 
earth’s surface 
1.6.6 Are lightning protectors reliable? 
1.7 Lightning as a power supply 
1.8 To those intending to read on 
References 
1.6 Lightning hazards 
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2 The streamer-leader process in a long spark 
27 
2.1 What a lightning researcher should know about a long spark 
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2.2 A long streamer 
32 
2.2.1 The streamer tip as an ionization wave 
32 
2.2.3 
39 
2.2.4 Gas heating in a streamer channel 
42 
2.2.2 Evaluation of streamer parameters 
34 
Current and field in the channel behind the tip 
V 
Copyright © 2000 IOP Publishing Ltd.

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vi 
Contents 
2.2.5 Electron-molecular reactions and plasma decay in 
cold air 
2.2.6 Final streamer length 
2.2.7 Streamer in a uniform field and in the ‘absence’ of 
electrodes 
2.3 The principles of a leader process 
2.3.1 The necessity of gas heating 
2.3.2 The necessity of a streamer accompaniment 
2.3.3 Channel contraction mechanism 
2.3.4 Leader velocity 
2.4 The streamer zone and cover 
2.4.1 Charge and field in a streamer zone 
2.4.2 Streamer frequency and number 
2.4.3 Leader tip current 
2.4.4 Ionization processes in the cover 
2.5.1 Field and the plasma state 
2.5.2 Energy balance and similarity to an arc 
2.6 Voltage for a long spark 
2.7 A negative leader 
References 
2.5 A long leader channel 
3 Available lightning data 
3.1 Atmospheric field during a lightning discharge 
3.2 The leader of the first lightning component 
3.2.1 Positive leaders 
3.2.2 Negative leaders 
3.3 The leaders of subsequent lightning components 
3.4 Lightning leader current 
3.5 Field variation at the leader stage 
3.6 Perspectives of remote measurements 
3.6.1 Effect of the leader shape 
3.6.2 Effect of linear charge distribution 
3.7.1 Neutralization wave velocity 
3.7.2 Current amplitude 
3.7.3 Current impulse shape and time characteristics 
3.7.4 Electromagnetic field 
3.8 Total lightning flash duration and processes in the 
intercomponent pauses 
3.9 Flash charge and normalized energy 
3.10 Lightning temperature and radius 
3.1 1 What can one gain from lightning measurements? 
References 
3.7 Lightning return stroke 
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Contents 
vii 
4 Physical processes in a lightning discharge 
138 
4.1 An ascending positive leader 
138 
4.1.1 The origin 
138 
4.1.2 Leader development and current 
141 
4.1.3 Penetration into the cloud and halt 
144 
4.1.4 Leader branching and sign reversal 
148 
4.2 Lightning excited by an isolated object 
150 
4.2.1 A binary leader 
150 
4.2.2 Binary leader development 
152 
4.3 The descending leader of the first lightning component 
158 
4.3.1 The origin in the clouds 
158 
4.3.2 Negative leader development and potential transport 
161 
4.3.3 The branching effect 
166 
4.3.4 Specificity of a descending positive leader 
168 
4.3.5 A counterleader 
169 
4.4 Return stroke 
171 
4.4.1 The basic mechanism 
171 
4.4.2 Conclusions from explicit solutions to long line 
equations 
175 
4.4.3 Channel transformation in the return stroke 
181 
4.4.4 Return stroke as a channel transformation wave 
185 
4.4.5 Arising problems and approaches to their solution 
190 
4.4.6 The return stroke of a positive lightning 
194 
4.5 Anomalously large current impulses of positive lightnings 
195 
4.6 Stepwise behaviour of a negative leader 
197 
4.6.1 The step formation and parameters 
197 
4.6.2 Energy effects in the leader channel 
199 
4.8 Subsequent components. The problem of a dart leader 
207 
4.8.1 A streamer in a ‘waveguide’? 
207 
4.8.2 The non-linear diffusion wave front 
209 
4.8.3 The possibility of diffusion-to-ionization wave 
transformation 
212 
4.8.4 The ionization wave in a conductive medium 
213 
4.8.5 The dart leader as a streamer in a ‘nonconductive 
waveguide’ 
215 
4.9 Experimental checkup of subsequent component theory 
217 
References 
219 
4.7 The subsequent components. The M-component 
202 
5 Lightning attraction by objects 
222 
5.1 The equidistance principle 
223 
5.2 The electrogeometric method 
226 
5.3 The probability approach to finding the stroke point 
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5.4 Laboratory study of lightning attraction 
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Copyright © 2000 IOP Publishing Ltd.

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..I 
Vlll 
Con tents 
5.5 Extrapolation to lightning 
5.6 On the attraction mechanism of external field 
5.7 How lightning chooses the point of stroke 
5.8 Why are several lightning rods more effective than one? 
5.9 Some technical parameters of lightning protection 
5.9.1 The protection zone 
5.9.2 The protection angle of a grounded wire 
5.10 Protection efficiency versus the object function 
5.11 Lightning attraction by aircraft 
5.12 Are attraction processes controllable? 
5.13 If the lightning misses the object 
References 
6 Dangerous lightning effects on modern structures 
6.1 Induced overvoltage 
6.1.1 ‘Electrostatic’ effects of cloud and lightning charges 
6.1.2 Overvoltage due to lightning magnetic field 
6.2 Lightning stroke at a screened object 
6.2.1 A stroke at the metallic shell of a body 
6.2.2 How lightning finds its way to an underground cable 
6.2.3 Overvoltage on underground cable insulation 
6.2.4 The action of the skin-effect 
6.2.5 The effect of cross section geometry 
6.2.6 Overvoltage in a double wire circuit 
6.2.7 Laboratory tests of objects with metallic sheaths 
6.2.8 Overvoltage in a screened multilayer cable 
6.3 Metallic pipes as a high potential pathway 
6.4 Direct stroke overvoltage 
6.4.1 The behaviour of a grounding electrode at high 
current impulses 
6.4.2 Induction emf in an affected object 
6.4.3 Voltage between the affected and neighbouring 
objects 
6.4.4 Lines with overhead ground-wires 
6.5 Concluding remarks 
References 
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Copyright © 2000 IOP Publishing Ltd.

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Preface 
Today, we know sufficiently much about lightning to feel free from the mystic 
fears of primitive people. We have learned to create protection technologies 
and to make power transmission lines, skyscrapers, ships, aircraft, and space- 
craft less vulnerable to lightning. Yes, the danger is getting less but it still 
exists! With every step of the technical progress, lightning arms itself with 
a new weapon to continue the war by its own rules against the self-confident 
engineer. As we improve our machines and stuff them with electronics in an 
attempt to replace human beings, lightning acts in an ever refined manner. It 
takes us by surprise where we do not expect it, making us feel helpless again 
for some time. 
We do not intend to present in this book a set of universal lightning 
protection rules. Such a task would be as futile as advertising a universal anti- 
biotic lethal to every harmful microbe. The world is changeable, and today’s 
panacea often becomes a useless pill even before the advertising sheet fades. 
Technical progress has so far failed to take lightning unawares. Improvement 
and miniaturization of devices increase our concern about the refined 
destructive behaviour of lightning, but no prophet is able to foresee all of 
its destructive effects. 
We do not plan to discuss in detail all available information on light- 
ning. There are already some excellent books providing all sort of reference 
data, among them the two volumes of Lightning edited by R H Golde and 
Lightning Discharge by M Uman. Our aim is different. We think it important 
to give the reader some clear, up-to-date physical concepts of lightning 
development, which cannot be found in the books referred to. These will 
serve as a basis for the researcher and engineer to judge the properties of 
this tremendous gas discharge phenomenon. Then we shall discuss the 
nature of various hazardous manifestations of lightning, focusing on the 
physical mechanisms of interaction between lightning and an affected 
construction. The results of this consideration will further be used to estimate 
ix 
Copyright © 2000 IOP Publishing Ltd.

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X 
Preface 
the effectiveness of conventional protective measures and to predict technical 
means for their improvement. We give, wherever possible, technical advice 
and recommendations. Our main goal, however, is to help the reader to 
make his own predictions by providing information on the whole arsenal 
of potentionally hazardous effects of lightning on a particular construction. 
We have often witnessed situations when an engineer was trying hard to 
‘impose’ this or that protective device on an operating experimental structure 
which resisted his unnatural efforts. Ideally, the designer must be able to 
foresee all details of the relationship between lightning and the construction 
being designed. It is only in this case that lightning protection can become 
functionally effective and the protective device can be made compatible 
with the construction elements. 
If an engineer is determined to follow this approach, both expedient and 
well-grounded, he will find this book useful. It is a natural extension of our 
previous book Spark Discharge, published by CRC Press in 1997, which dealt 
with streamer-leader breakdown of long gas gaps. The streamer-leader 
process is part of any lightning discharge when a plasma spark closes a 
gigantic air gap. Although the destructive effect of lightning is primarily 
due to the return stroke which follows the leader, it is the leader that 
makes the discharge channel susceptible to it. This is why we give an overview 
of the streamer-leader process, focusing on extrema1 estimations and 
presenting some new ideas. We hope that the second chapter will prove infor- 
mative even for those familiar with our book of 1997. 
Some results of the lightning investigation run in the Krzhizhanovsky 
Power Institute are used in the book. The authors would like to thank 
Dr B N Gorin and Dr A V Shkilev who kindly allowed us to use the originals 
of lightning photographs. We are also grateful to L N Smirnova for 
translation of this book. 
Copyright © 2000 IOP Publishing Ltd.

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Chapter 1 
Introduction: lightning, its 
destructive effects and protection 
If you want to observe lightning, the best thing to do is to visit a special light- 
ning laboratory. Such laboratories exist in all parts of the globe except the 
Antarctic. But you can save on the travel if you just climb onto the roof of 
your own house to give a good field of vision. Better, fetch your camera. 
Even an ordinary picture can show details the unaided human eye often 
misses. You might as well sit back in your favourite armchair, having 
pulled it up to a window, preferably one overlooking an open space. The 
camera can be fixed on the window sill, There is nothing else to do but 
wait for a stormy night. 
There is enough time for the preparations to be made because the storm 
will be approaching slowly. At first, the air will grow still, and it will get 
much darker than it normally is on a summer night. The cloud is not yet visible, 
but its approach can be anticipated from the soundless flashes at the horizon. 
They gradually pull closer, and the brightest of them can already be heard as 
delayed and yet amiable roaring. Ths may go on for a long time. It may 
seem that the cloud has stopped still or turned away, but suddenly the sky is 
ripped open by a fire blade. This is accompanied by a deafening crash, quite 
different from a cannon shot because it takes a much longer time. The first 
lightning discharge is followed by many others falling out of the ripped 
cloud. Some strike the ground while others keep on crossing the sky, competing 
with the first discharge in beauty and spark length. This is the right time to start 
observations: remove the camera shutter and try to take a few pictures. 
1.1 Types of lightning discharge 
The above recommendation to remove the camera shutter should be taken 
literally. Much information on lightning has been obtained from photographs 
taken with a preliminarily opened objective lens. It is important, however, that 
1 
Copyright © 2000 IOP Publishing Ltd.

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2 
Introduction: lightning. its destructive effects and protection 
Figure 1.1. A static photograph of a lightning stroke at the Ostankino Television 
Tower in Moscow. 
no other bright light source should be present within the vision field of the 
camera lens. The film can then be exposed for many minutes until a spark 
finds its way into the frame. After this, the lens should be closed with the shut- 
ter and the camera should be set ready for another shot. Experience has shown 
that at least one third of pictures taken during a good night thunderstorm 
prove successful. 
All lightning discharges can be classified, even without photography, 
into two groups - intercloud discharges and ground strikes. The frequency 
of the former is two or three times higher than that of the latter. An inter- 
cloud spark is never a straight line, but rather has numerous bends and 
branchings. Normally, the spark channel is as long as several kilometres, 
sometimes dozens of kilometres. 
The length of a lightning spark that strikes the ground can be defined 
more exactly. The average cloud altitude in Europe is close to three kilo- 
metres. Spark channels have about the same average length. Of course, 
this parameter is statistically variable, because a discharge from a charged 
cloud centre may start at any altitude up to 10 km and because of a large 
number of spark bends. The latter are observable even with the unaided 
eye. In a photograph, they may look strikingly fanciful (figure 1.1). A photo- 
graph can show another important feature inaccessible to the naked eye - the 
main bright spark reaching the ground has numerous branches which have 
stopped their development at various altitudes. A single branch may have 
a length comparable with that of the principal spark channel (figure 1.2). 
Branches can be conveniently used to define the direction of lightning 
propagation. Like a tree, a lightning spark branches in the direction of 
Copyright © 2000 IOP Publishing Ltd.

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Tipes of lightning discharge 
3 
Figure 1.2. A photograph of a descending lightning with numerous branches. 
growth. In addition to descending sparks outgrowing from a cloud toward 
the ground, there are also ascending sparks starting from a ground construc- 
tion and developing up to a cloud (figure 1.3). Their direction of growth is 
well indicated by branches diverging upward. 
In a flat country, an ascending spark can arise only from a skyscraper or 
a tower of at least 100-200m high, and the number of ascending sparks 
grows with the building height. For example, over 90% of all sparks that 
strike the 530-m high Ostankino Television Tower in Moscow are of the 
ascending type [l]. A similar value was reported for the 410-m high 
Empire State Building in New York City [2]. Buildings of such a height 
can be said to fire lightning sparks up at clouds rather than to be attacked 
by them. In mountainous regions, ascending sparks have been observed 
for much lower buildings. As an illustration, we can cite reports of storm 
observations made on the San Salvatore Mount in Switzerland [3]. The 
receiving tower there was only 70 m high but most of the discharges affecting 
it were of the ascending type. 
Skyscrapers and television towers are, however, quite scarce on the 
Earth. So the researcher has a natural desire to construct, in the right 
place and for a short time, a spark-generating tower of his own. For this, 
a small probe pulling up a thin grounded wire is launched towards a 
Copyright © 2000 IOP Publishing Ltd.

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4 
Introduction: lightning, its destructive effects and protection 
Figure 1.3. A photograph of an ascending lightning. 
storm cloud [4]. When the probe rises to 200-300m above the earth, an 
ascending spark is induced from it. A discharge artificially induced in the 
atmosphere is often referred to as triggered lightning. To raise the chances 
for a successful experiment, the electric field induced by the storm charges at 
the ground surface are measured prior to the launch. The probe is triggered 
when the field strength becomes close to 200 V/cm, which provides spark 
ignition in 60-70% of launches [5]. 
The value 200 V/cm is two orders of magnitude smaller than the thresh- 
old value of E = 30 kV/cm, at which a short air gap with a uniform field is 
broken down under normal atmospheric conditions. Clearly, no spark 
ignition would be possible without the local field enhancement by electric 
charges induced on the probe and the wire. Below, we shall discuss the 
triggered discharge mechanism in more detail. 
A field detector on the Earth’s surface (it might as well be placed on the 
window of your own room) can easily determine the polarity of the charge 
transported by a lightning spark to the ground. The polarity of the spark 
is defined by that of the charge. About 90% of descending sparks occurring 
in Europe during summer storms carry a negative charge, so these are known 
as negative descending sparks. The other descending sparks are positive. The 
Copyright © 2000 IOP Publishing Ltd.

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Lightning discharge components 
5 
proportion of positive sparks has been found to be somewhat larger in 
tropical and subtropical regions, especially in winter, when it may be as 
large as 50%. 
There is no special name for lightning sparks generated by aircraft 
during flights, when they are entirely insulated from the ground. Such dis- 
charges arise fairly frequently. A modern aircraft experiences at least one 
lightning stroke every 3000 flight hours. Almost half of the strokes start 
from the aircraft itself, not from a cloud. This often happens in heap 
rather than clouds carrying a relatively small electric charge. The reason 
for a discharge from a large ground-insulated object is principally the same 
as from a grounded object and is due to the electric field enhancement by 
surface polarization charge. This issue will be discussed after the analysis 
of ascending sparks in section 4.2. 
1.2 Lightning discharge components 
An observer can notice a lightning spark flicker which, sometimes, may 
become quite distinct. Even the first cinematographers knew that the 
human eye could distinguish between two events only if they occurred with 
a time interval longer than 0.1 s. Since lightning flicker is observable, the 
pause between two current impulses must be longer than 0.1 s. 
A current-free pause can be measured quite accurately by exposing a 
moving film to a lightning discharge. With up-to-date lenses and photo- 
graphic materials, one can obtain a good 1 mm resolution of the film, In 
order to displace an image by 1 mm over a time period of 0.1 s, the film 
speed must be about 1 cmjs. It can be achieved by manually moving the 
film keeping the camera lens open (alas, an electrically driven camera is 
unsuitable for this). Then, with some luck, one may get a picture like the 
one in figure 1.4. The spark flashes up and dims out several times. Unless 
the pause is too long, a new flash follows the previous trajectory; otherwise, 
the spark takes a partially or totally new path. 
A lightning spark with several flashes is known as a multicomponent 
spark. One may suggest that the channel of the first component formed in 
unperturbed air differs in its basic characteristics from the subsequent chan- 
nels, if they take exactly the same path through the ionized and heated air. 
The formation of subsequent components is considered in sections 4.7 and 
4.8. Note only that multicomponent sparks are usually negative, both 
ascending and descending. The average number of components is close to 
three, while the maximum number may be as large as thirty. Generally, the 
average duration of a lightning flash is 0.2 s and the maximum duration is 
1-1.5s [6], so it is not surprising that the eye can sometimes distinguish 
between individual components. Positive sparks normally contain only one 
component. 
Copyright © 2000 IOP Publishing Ltd.

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6 
Introduction: lightning, its destructive effects and protection 
Figure 1.4. The image of a multicomponent lightning in a slowly moving film. 
1.3 
Basic stages of a lightning spark 
The affinity of lightning to a spark discharge was demonstrated by Benjamin 
Franklin as far back as the 18th century. Historically, basic spark elements 
were first identified in lightning, and only much later were they observed in 
laboratory sparks. This is easy to understand if one recalls that a lightning 
spark has a much greater length and takes a longer time to develop, so 
that its optical registration does not require the use of sophisticated equip- 
ment with a high space and time resolution. The first streak photographs 
of lightning, taken in the 1930s by a simple camera with a mechanically 
rotated film (Boys camera), are still impressive [7]. They show the principal 
stages of the lightning process - the leader stage and the return stroke. 
The leader stage represents the initiation and growth of a conductive 
plasma channel - a leader - between a cloud and the earth or between 
two clouds. The leader arises in a region where the electric field is strong 
enough to ionize the air by electron impact. However, it mostly propagates 
through a region in which the external field induced by the cloud charge 
does not exceed several hundreds of volts per centimetre. In spite of this it 
does propagate, which means that there is an intensive ionization occurring 
in its tip region, changing the neutral air to a highly conductive plasma. This 
becomes possible because the leader carries its own strong electric field 
induced by the space charge concentrated at the leader tip and transported 
together with it. A rough analogue of the leader field is that of a metallic 
needle connected with a thin wire to a high voltage supply. If the needle is 
sharp enough, the electric field in the vicinity of its tip will be very strong 
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Basic stages of a lightning spark 
7 
even at a relatively low voltage. Imagine now that the needle is falling down 
on to the earth, pulling the wire behind it. The strong field region, in which 
the air molecules become ionized, will move down together with the needle. 
A lightning spark has no wire at its disposal. The function of a conductor 
connecting the leader tip to the starting point of the discharge is performed by 
the leader plasma channel. It takes a fairly long time for a leader to develop - 
up to 0.01 s, which is eternity in the time scale of fast processes involving an 
electric impulse discharge. During this period of time, the leader plasma 
must be maintained highly conductive, and this may become possible only if 
the gas is heated up to an electric arc temperature, i.e. above 5000-6000K. 
The problem of the channel energy balance necessary for the heating and com- 
pensation for losses is a key one in leader theory. It is discussed in chapters 2 
and 4, as applied to various kinds of lightning discharge. 
A leader is an indispensable element of any spark. The initial and all 
subsequent components of a flash begin with a leader process. Although its 
mechanism may vary with the spark polarization, propagation direction 
and the serial number of the component, the process remains essentially 
the same. This is the formation of a highly conductive plasma channel due 
to the local enhancement of the electric field in the leader tip region. 
A return stroke is produced at the moment of contact of a leader with 
the ground or a grounded object. Most often, this is an indirect contact: a 
counterpropagating leader, commonly termed a counterleader, may start 
from an object to meet the first leader channel. The moment of their contact 
initiates a return stroke. During the travel from the cloud to the ground, the 
lightning leader tip carries a high potential comparable with that of the cloud 
at the spark start, the potential difference being equal to the voltage drop in 
the leader channel. After the contact, the tip receives the ground potential 
and its charge flows down to the earth. The same thing happens with the 
other parts of the channel possessing a high potential. This ‘unloading’ pro- 
cess occurs via a charge neutralization wave propagating from the earth up 
through the channel. The wave velocity is comparable with the velocity of 
light and is about 10’ mjs. A high current flows along the channel from the 
wave front towards the earth, carrying away the charge of the unloading 
channel sites. The current amplitude depends on the initial potential distribu- 
tion along the channel and is, on average, about 30 kA, reaching 200-250 kA 
for powerful lightning sparks. The transport of such a high current is accom- 
panied by an intense energy release. Due to this, the channel gas is rapidly 
heated and begins to expand, producing a shock wave. A peal of thunder 
is one of its manifestations. 
The return stroke is the most powerful stage of a lightning discharge 
characterized by a fast current change. The current rise can exceed 
10’ A/s, producing a powerful electromagnetic radiation affecting the 
performance of radio and TV sets. This effect is still appreciable at a distance 
of several dozens of kilometres from the lightning discharge. 
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8 
Introduction: lightning, its destructive effects and protection 
Current impulses of a return stroke accompany all components of a 
descending spark. This means that the leader of every component charges 
the channel as it moves down to the earth, but some of the charge becomes 
neutralized and redistributed at the return stroke stage. Prolonged peals of 
thunder result from the overlap of sound waves generated by the current 
impulses from all subsequent spark components. 
An ascending spark is somewhat different. The leader of the first compo- 
nent starts at a point of zero potential. As the channel travels up, the tip 
potential changes gradually until the leader development ceases somewhere 
deep in the cloud. There is no fast charge variation during this process; as 
a result, the first component has no return stroke. However, all subsequent 
spark components starting from the cloud do develop return strokes and 
behave exactly in the same way as a descending spark. 
Of special interest is the return stroke of an intercloud discharge. Its 
existence is indicated by peals of thunder as loud as those of descending 
sparks. Clearly, an intercloud leader is generated in a charged region of a 
storm cloud, or in a storm cell, and travels towards an oppositely charged 
region. The charged region of a cloud should not be thought of as a con- 
ductive body, something like a plate of a high voltage capacitor. Cloud 
charges are distributed throughout a space with a radius of hundreds of 
metres and are localized on water droplets and ice crystals, known as 
hydrometeorites, having no contact with one another. The formation of a 
return stroke implies that the leader comes in contact with a highly con- 
ductive body of an electrical capacitance comparable with, or even larger 
than, that of the leader. It appears that the role of such a body in an 
intercloud discharge is played by a concurrent spark coming in contact 
with the first one. 
Measurements made at the earth surface have shown that the current 
impulse amplitude of a return stroke decreases, on average, by half for 
about lOP4s. This parameter variation is very large - about an order of 
magnitude around the average value. Current impulses of positively charged 
sparks are usually longer than those of negatively charged ones, and the 
impulses of the first components last longer than those of the subsequent 
ones. 
A return stroke may be followed by a slightly varying current of about 
100 A, which may persist in the spark channel for some fractions of a second. 
At this final stage of continuous current, the spark channel remains electri- 
cally conductive with the temperature approximately the same as in an arc 
discharge. The continuous current stage may follow any lightning compo- 
nent, including the first component of an ascending spark which has no 
return stroke. This stage may be sporadically accompanied by current over- 
shoots with an amplitude up to 1 kA and a duration of about 
s each. 
Then the spark light intensity becomes much higher, producing what is gen- 
erally termed as M-components. 
Copyright © 2000 IOP Publishing Ltd.

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Continuous and stepwise leaders 
9 
1.4 
Continuous and stepwise leaders 
This introductory chapter contains no theory, and this makes the discussion 
of leader details a very complicated task. So we shall mention only its 
principal features which can be registered by a continuously moving film. 
Continuous streak photographs show lightning development in time. One 
needs, however, a certain skill and experience to be able to interpret them 
adequately. Suppose a small light source moves perpendicularly to the 
earth at a constant velocity. It may be a luminant bomb descending with a 
parachute. A film moving horizontally, i.e. in the transverse direction, at a 
constant speed will show a sloping line (figure lS(u)). Given the film speed 
(the display rate), one can easily calculate the light source velocity from 
the line slope. A uniformly propagating vertical channel will leave on a 
film a sloping wedge (figure 1.5(b)) rather than a line. From its slope, too, 
one can find the channel velocity, or its propagation rate. The higher the 
rate of the process in question, the higher must be the display rate in 
streak photography. The highest display rates can be obtained using an 
electron-optical converter, in which an image is converted to an electron 
beam scanned across the screen by an electric field. A conventional photo- 
camera registers the displayed electronic image from the screen onto an 
immobile film. Electron-optical converters have provided much information 
on long sparks, but their application in lightning observations has been 
limited. The main results here have been obtained using mechanical streak 
cameras. We described this technique and analysed streak pictures in our 
book on long sparks [8]. 
Figure 1.6(a) shows the leader of an ascending lightning spark going 
up from the top of a grounded tower in the electric field of a negatively 
charged cloud cell. The leader carries a positive space charge and, therefore, 
it should be referred to as a positive leader. One can clearly see the bright 
trace of the channel tip, which looks like a nearly continuous line. This 
kind of leader is known in literature as a continuous leader. The changing 
trace slope suggests that the leader velocity changes during its propagation. 
These changes are, however, quite smooth, not interrupting the tip travel up 
to the cloud. 
0 
t
o
 
t 
a 
b 
Figure 1.5. The analysis of an image of a vertically descending light source in a 
horizontally moving film (image display in streak photography): (a) point source, 
(b) elongating channel. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 18 ===
10 
Introduction: lightning, its destructive effects and protection 
Figure 1.6. A schematic streak picture of a positive ascending (a) and a negative 
descending (b) lightning leader. 
An essentially different behaviour is exhibited by the leader shown in 
figure 1.6(b). The channel grows in a stepwise manner, covering several 
dozens of metres in each step. Hence, this kind of leader is termed as a 
stepwise leader. The new step in the photograph is especially bright; its 
appearance makes the whole channel behind it also a little brighter. The 
step length varies between 10 and 200m with an average of 30m. The time 
lapse between two steps is 30-90 ps [9]. The stepwise pattern is characteristic 
of negatively charged leaders. Positive leaders, both ascending and descend- 
ing, usually grow in a continuous manner. When averaged over the total time 
of development, the velocity of stepwise and continuous leaders prove nearly 
the same, 105-106m/s, with an average of about 3 x 105m/s. 
If the leader of the next component moves along the hot track of the 
first one, it always develops continuously. The new process, termed a dart 
leader, differs from the first one exclusively in a high leader velocity, about 
(1-4) x 107m/s. It does not change much along its trajectory from the 
cloud to the earth. Streak photographs clearly show the bright head of a 
dart leader, while the channel light intensity is much lower. If the next 
component takes its own path, its leader behaves in the same way as that 
of the first component, i.e. it develops more slowly and often in a stepwise 
pattern. 
Dart leaders have not had a fair share of attention from researchers. 
There is neither theory nor laboratory analogue of this type of gas discharge. 
Still, it is a most fascinating form of discharge developing record high leader 
velocities. The contact of a dart leader with the earth produces the fastest 
current rise, which can reach its amplitude maximum within 
s. This is 
the source of record strong electromagnetic fields which exert one of the 
most hazardous effects on modern equipment. An attempt at a theoretical 
treatment of the dart leader will be made in section 4.8. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 19 ===
Lightning stroke frequency 
11 
1.5 
Lightning stroke frequency 
1.5.1 Strokes at terrestrial objects 
Experience shows that lightning most frequently strikes high objects, 
especially those dominating over an area. In a flat country, it is primarily 
attracted by high single objects like masts, towers, etc. In mountains, even 
low buildings may be affected if they are located on a high hill or on the 
top of a mountain. Common sense suggests that it is easier for an electrical 
discharge, such as lightning, to bridge the shortest gap to the highest 
object in the locality. In Europe, for example, a 30m mast experiences, on 
the average, 0.1 lightning stroke per year (or 1 stroke per 10 years), whereas 
a single lOOm construction attracts 10 times more lightnings. On closer 
inspection, the strong dependence of stroke frequency on the construction 
height does not look trivial. The average altitude of the descending discharge 
origin is about 3 km, so a lOOm height makes up only 3% of the distance 
between the lightning cloud and the earth. Random bendings make the 
total lightning path much longer. One has to suggest, therefore, that the 
near-terrestrial stage of lightning behaviour involves some specific processes 
which predetermine its path here. These processes lead to the attraction of a 
descending leader by high objects. We shall discuss the attraction mechanism 
in chapter 5. 
Scientific observations of lightning show that there is an approximately 
quadratic dependence of the stroke frequency NI on the height h of lumped 
objects (their height is larger than the other dimensions). Extended objects 
of length I ,  such as power transmission lines, show a different dependence, 
NI N hl. This suggests the existence of an equivalent radius of lightning 
attraction, Re, N h. All lightnings displaced from an object horizontally at 
a distance r 6 Re, are attracted by it, the others missing the object. This 
primitive pattern of lightning attraction generally leads to a correct result. 
For estimations, one can use Re! RZ 3h and borrow the stroke frequency 
per unit unperturbed area per unit time, nl, from meteorological observa- 
tions. The latter are used to make up lightning intensity charts. For example, 
the lightning intensity in Europe is nl < 1 per 1 km2 per year for the tundra, 
2-5 for flat areas, and up to 10 for some mountainous regions such as the 
Caucasus. A tower of h = lOOm is characterized by Re, = 0.3 km with 
NI = n17rR& M 1 stroke per year at the average value of nl = 3.5 kmP2 year-’. 
This estimation is meaningful for a flat country and only for not very high 
objects, h < 150m, which do not generate ascending lightnings. 
1.5.2 Human hazard 
It has long been proved that Galvani was wrong suggesting a special ‘animal 
electricity’. A human being is, to lightning, just another sticking object, like a 
tree or a pole, only much shorter. The lightning attraction radius for humans 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 20 ===
12 
Introduction: lightning, its destructive effects and protection 
is as small as 5-6m and the attraction area is less than lop4 km2. If a man 
had stopped alone in the middle of a large field two thousand years ago, 
he might have expected to attract a direct lightning stroke only by the end 
of the third, coming millennium. In actual reality, however, the number of 
lightning victims is large, and direct strokes have nothing to do with this. 
It is known from experience that one should not stay in a forest or hide 
under a high tree in an open space during a thunderstorm. A tree is about 
10 times higher than a man, and a lightning strikes it 100 times more fre- 
quently. When under the tree crown, a man has a real chance to be within 
the zone of the lightning current spread, which is hazardous. 
After a lightning strikes the tree top, its current ZM runs down along its 
stem and roots to spread through the soil. The root network acts as a natural 
grounding electrode. The current induces in the soil an electric field E = pj, 
where p is the soil resistivity and j is the current density. Suppose the current 
spreads through the soil strictly symmetrically. Then the equipotentials will 
represent hemispheres with the diagonal plane on the earth’s surface. The 
current density at distance r from the tree stem is j = IM/(27rr2) the field is 
IMp/(27rr2) and the potential difference between close points r and r + Ar 
is equal to A U  = (ZMp/27r)[r-’ - (Y + AY)-’] 
x E(r)Ar. If a person is stand- 
ing, with his side to the tree, at distance r = 1 m from the tree stem centre and 
the distance between his feet is Ar x 0.3 m, the voltage difference on the soil 
with resistivity p = 200 f2/m will be A U x 220 kV for a moderate lightning of 
ZM = 30 kA. This voltage is applied to the shoe soles and, after a nearly inevi- 
table and fast breakdown, to the person’s body. There is no doubt that the 
person will suffer or, more likely, will be killed - the applied voltage is too 
high. Note that this voltage is proportional to Ar. This means that it is 
more dangerous to stand with one’s feet widely apart than with one’s feet 
pressed tightly together. It is still more dangerous to lie down along the 
radius from the tree, because the distance between the extreme points 
contacting the soil becomes equal to the person’s height. It would be much 
safer to stand still on one foot, like a stork. But it is, of course, easier to give 
advice than to follow it. Incidentally, lightning strikes large animals more 
frequently than humans, also because the distance between their limbs is larger. 
If you have a cottage equipped with a lightning protector, take care that 
no people could approach the grounding rod during a thunderstorm. The 
situation here is similar to the one just described. 
1.6 
Lightning hazards 
1.6.1 A direct lightning stroke 
In the case of a direct lightning stroke, the current flows through the 
conducting elements of the affected object, with the hot channel contacting 
the construction element which has received the stroke. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 21 ===
Ligk tning hazards 
13 
Figure 1.7. Traces of lightning strokes at the steel tip of the Ostankino Television 
Tower in Moscow. 
Thermal effects of lightning are most hazardous at the site of contact of 
a high temperature channel with combustible materials. This often leads to a 
fire which becomes most probable when the continuous current stage has a 
long duration. A return stroke is unlikely to cause a fire even in the case of 
a powerful lightning discharge, because the strong shock wave produced 
blows off the flames and combustion products. In combustible dielectric 
materials a lightning stroke contacts on its way may first be broken down 
by the strong electric field of the leader tip and then, in the return stroke 
and continuous current stages, they may be melted through at the site of 
contact with the hot spark. A burn-through or a burn-off often occurs at 
the point where the spark contacts a metallic surface several millimetres 
thick. The holes and burn-offs are usually of the same size. The photograph 
in figure 1.7 demonstrates the traces of numerous lightning strokes on the 
steel tip of the Ostankino Television Tower. Slight faults cannot disturb 
the mechanical strength of a massive metallic construction. Normally, the 
hazards of burn-offs and fuses are associated with the melted metal in-flow 
into an object which may contain inflammable and explosive materials or 
gas mixtures. Incidentally, not only is a burn-through of a metallic wall 
dangerous but also the local overheating when the temperature of the 
inner metal surface may go up to 700-1000°C. Unfortunately, the surface 
often acts as a lighter. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 22 ===
14 
Introduction: lightning, its destructive effects and protection 
Thermal damage of conductors, through which lightning current flows, 
occurs fairly rarely. It is characteristic of miniature antennas and various 
detectors mounted on the outer construction surfaces. The probability of 
emergency increases if lightning current encounters bolted or riveted joints. 
The electric contact thus formed always has an elevated contact resistance 
which may cause a local overheating. This results in the metal release and 
rivet loosening, disturbing the mechanical strength of the joint. Mobile 
joints (hinges, ball bearings, etc.) are subject to a similar damage. The site 
of a sliding contact becomes locally overheated to produce cavities which 
hamper the motion of mobile parts. In extreme conditions, they may 
become welded. 
Electrodynamic effects of lightning current rarely become hazardous. 
Mechanical stress arising in electrically loaded and closely spaced metallic 
structures or in a single structure with an abruptly changing direction of 
the current is not appreciable and lasts less than 100 ms (it is the attenuation 
time of a current impulse). However, lightning current has been repeatedly 
observed to narrow down thin metallic pipes, to change the tilt of rods and 
to strain thin surfaces. Such effects are not vitally dangerous in themselves 
but, under certain conditions, may lead to an emergency. As an illustration, 
imagine the situation when the lightning-affected pipe is part of an aircraft 
speed control. What will happen if the crew take the readings for granted 
and do not receive corrections from a ground air traffic controller? 
Electrohydraulic effects of lightning are much more hazardous than 
those discussed above. Modern machines have parts made from a variety 
of composite materials. These may include, along with plastics, superthin 
metallic films (both outer and inner), nearly as thin metallic meshes, and min- 
iature conductors monolithic with a dielectric wall. Under the action of light- 
ning current, these metallic parts evaporate, the arising arcs contacting the 
plastic making it decompose and evaporate. A shock wave appears which 
splits and bloats the composite wall. A similar effect arises when a lightning 
spark partially penetrates through a narrow slit between vaporizable plastic 
walls (most plastics possess gas-generating properties). No one questions a 
great future of composite materials, but their peaceful coexistence with light- 
ning is still a challenge to the engineer. 
Direct stroke overvoltage represents a hazardous rise of voltage when 
a lightning current impulse propagates across the construction elements. 
We shall analyse this very dangerous effect of lightning with reference to 
a power transmission line, because engineers first encountered the phenom- 
enon of overvoltage in such lines. Moreover, the problem of electric 
insulation for a transmission line can be stated most clearly. Figure 1.8 
shows schematically a metallic tower with a ground rod (the grounding 
resistance is Rg) 
and a high voltage wire suspended by an insulator string. 
Above the wire, there may be a lightning conductor attached right to the 
tower. It stretches along all the line and is to trap lightning sparks aimed 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 23 ===
Lightning hazards 
15 
Figure 1.8. Lightning current as an overvoltage source on a power transmission line 
during a stroke at the power wire (a) and a grounded tower (b). 
at the line wires. A rigorous solution to this problem is given later in 
this book. Here, we should only like to explain the nature of overvoltages 
phenomenologically. 
Suppose at first that the lightning conductor has proved unreliable, and 
a discharge has struck the wire (figure 1.8(a)). At the point of the stroke, the 
current will branch to produce two identical waves of the amplitude ZM / 2 ,  
where ZM is the lightning current amplitude. The two waves will run towards 
the ends of the line with a velocity nearly equal to vacuum light velocity, 
c = 3 x 10sm/s. Until the end-reflected waves return, the wire potential 
relative to the ground will rise to U, = Z,2/2. 
The wave resistance 
Z = (L1/C1)1/2 
in this expression is defined by the inductance L1 and the 
capacitance C1 per unit wire length; it varies slightly, between 250 and 
350R, with the height and the wire radius. With this wave resistance, the 
average lightning current with an amplitude ZM = 30 kA will raise the wire 
potential up to U, = 3750-5250 kV. The tower potential will practically 
remain unchanged and equal to zero, so the insulation overvoltage will be 
close to the calculated value of U,. 
This will be clear if we compare U, 
with the operating line voltage which does not exceed 1000 kV even in high 
power lines but normally is 250-500 kV. 
In reality, the distance to the line ends I is as large as many dozens of 
kilometres. The time it takes the reflected wave of the opposite sign cutting 
down the overvoltage to arrive back at the stroke point is At = 21/c, or 
many hundreds of microseconds. This time is much longer than the strong 
current duration in the return stroke (loops). For this reason, reflected 
waves, which become strongly attenuated, do not normally have enough 
time to interfere with the process so that the overvoltage acts as long as a 
lightning current impulse. Practically, any lightning stroke at a wire 
represents a real hazard: the insulation will be broken down to produce 
short-circuiting. The power line in that case must be disconnected. 
Suppose now that lightning has struck a tower. More often, this is 
actually not a tower but rather an overhead grounded wire connected to it. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 24 ===
16 
Introduction: lightning, its destructive effects and protection 
The lightning current will flow down the metallic tower to the ground 
electrodes to be dissipated in the earth. Let us take point A at the height 
of the insulator string connection. Due to the lightning current i(t), the 
potential at this point, pA, will differ from the zero potential of the earth 
by the voltage drop in the grounding resistance R, and in the tower induc- 
tance L, between the tower base up to the point A: 
di 
p = R,i+ L,- dt 
However, the power wire potential will practically remain the same (in this 
qualitative description, we ignore all inductances between the power wires, 
tower and grounded wire). The power wire potential is due to the operating 
voltage source of the power line: qw = Uop. Then, the insulator string voltage 
will be 
U = pw - (FA = Uop - R,i - di 
Ls-, 
dt 
Note that the lightning current and operating voltage may have different 
polarities. As a result, the overvoltage U may prove to be the sum of the 
three terms in equation (1.2). 
The inductance component of the overvoltage, L,di/dt, has a short 
lifetime: it acts about as long as the lightning current rises. For a current 
impulse with an average amplitude IM 30kA and an average rise time 
tf = 5 p ,  the inductance voltage at L, 50pH will be about 300kV. The 
resistance component U, at a typical grounding resistance R, = 10R will 
have about the same value but will act an order of magnitude longer, i.e., 
as long as the lightning current flows. For this reason, this component 
makes the principal contribution to the insulation flashover. 
The emergency situation just described is not as bad as a direct stroke at 
a power wire when the same lightning current can induce an order of magni- 
tude higher voltage. The insulation of a ultrahigh voltage line can withstand 
short overvoltages up to 1000- 1500 kV and seldom suffers from lightning 
strokes at a tower or a lightning protection wire. To produce a harmful 
effect, the lightning current must be 3-5 times the average value. Lightning 
strokes of this power do not occur frequently, making up less than 1% of 
all strokes. Quite different is the effect of a direct stroke at a power network 
with an operating voltage of 35 kV and lower. The insulation system will 
suffer equally from a stroke at a power wire or a tower. It is no use protecting 
such a line with grounded wire. 
Insulation flashover due to the tower potential rise is referred to as 
reverse flashover. This name does not imply the definite direction of the 
discharge development but only indicates the direction from which the 
potential rises, i.e., the grounded end of the insulator string rather than 
the power wire. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 25 ===
Lightning hazards 
17 
The above illustration of overvoltages on the line transmission insula- 
tion demonstrates, to some extent, a variety of mechanisms of direct light- 
ning current effects. In actual reality, such mechanisms are much more 
diverse. It is important to remember that in modern technologies, overvol- 
tages are not always measured in hundreds of kilovolts, as for high-voltage 
transmission lines. Short voltage rises of only 100-1OV may be hazardous 
to microelectronic devices. Of special interest in this connection are situa- 
tions when lightning current flows across solid metallic jackets with electric 
circuits inside. These problems are discussed in chapter 6. 
1.6.2 Induced overvoltage 
Induced overvoltage is the most common and dangerous effect of lightning 
on electric circuits of modern technical equipment. This effect is brought 
about by electromagnetic induction. The current flowing through the 
lightning spark and the metallic structures of an affected object generates 
an alternating magnetic field which can induce an induction emf in any of 
the circuits in question. The procedure of estimating induced overvoltages 
is quite simple. If BaV(t) is the magnetic induction averaged over the circuit 
cross section S, the induction emf is expressed as 
When the length of the current conductor inducing the magnetic field is much 
longer than the distance to the circuit, rc, and when the width of the circuit 
normal to the magnetic field is much smaller than rc, we have 
poS di 
27rrc dt 
Eemf 
M - 
- 
where po = 47r x lop7 Him is vacuum magnetic permeability. The order of 
magnitude of the induction emf amplitude is defined as 
where A,, 
is the maximum rate of the current impulse rise equal to 10" A/s 
for the subsequent components of a powerful lightning flash. A circuit of area 
S = 1 m2 located at a distance r, = 10 m from a lightning current conductor 
may become the site of induced overvoltage with an amplitude up to 20 kV. 
This value is only an arbitrary guideline, because induced overvoltage may 
vary with the circuit area, its orientation and distance from the lightning cur- 
rent. Circuits with an area of hundreds and thousands of square metres may 
be created by large industrial metallic constructions and power transmission 
lines. The distance between the circuit and the current flow may also vary 
greatly. For such diverse parameters of a system, the problem will be more com- 
plicated in the case of fast current variations along the spark and in time. It 
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=== PAGE 26 ===
18 
Introduction: lightning, its destructive effects and protection 
cannot then be approached as a quasi-stationary problem, but one must take 
into account the law of current wave propagation along the spark channel 
and the finite velocity of the electromagnetic field in the space between the chan- 
nel and the circuit. Solutions to such problems are illustrated in chapter 6. 
There is another class of situations associated with electromagnetic 
induction in screened volumes. Of special interest is the situation when the 
lightning current i flows across a solid metallic casing and the circuit in 
question is inside it. Unless the casing is circular, an emf-inducing magnetic 
field gradually appears inside the casing. It is remarkable that the time 
variation of the emf is not at all defined by dildt. The magnetic field going 
through the circuit is affected more by the relatively slow current re- 
distribution along the casing perimeter than by the time variation of the light- 
ning current. The problem of pulse induction in aircraft inner circuits or in 
screened multiwire cables ultimately reduces to the problem above. Some 
approaches to its solution will be considered in chapter 6. 
1.6.3 Electrostatic induction 
Benjamin Franklin felt the effect of electrostatic induction when he raised his 
finger up to a lifted wire during a thunderstorm. The electric field of a storm 
cloud had polarized the wire by separating its electric charges. The strong 
electric field of the polarization charge had broken down the air gap between 
the thin wire end and the explorer’s finger and carried the charge through his 
body to the earth. 
Electrostatic induction induces a charge in any grounded conductor or a 
metallic object. Suppose it is a vertical metallic rod of length 1 located in an 
external vertical field Eo. When insulated from the earth, the rod would take 
the potential of the space at its centre, pc = E01/2, which follows from the 
symmetry consideration. The grounded rod potential is zero; hence, the 
external field potential is compensated by the charge q1 induced by this 
field on the rod. The charge can be estimated from the rod capacitance C, 
as q1 = Crpc = E01C,/2. 
The production of the charge q1 implies the existence of current through a 
ground electrode of the object. This is a low current, because it takes several 
seconds for the cloud charge, creating the field Eo, to be formed. As much 
time, At, is necessary for the charge -ql to flow down into the earth, leaving 
behind the induced charge q1 on the conductor. If the field Eo is largely created 
by the leader charge of a close Lightning discharge, the exposure time of the 
induced charge reduces to At x 10-3-10-2 s. But in this case, too, the current 
through the ground electrode is low. For example, at Eo x 1 kV/cm character- 
istic of close discharges, I = 10m, and C,. = lOOpF,t the average current is 
t Approximately, C, = 27rq,l/ In h/r, where r is the rod radius, h is an average distance between 
the rod and the earth (h = l/2), z0 = 8.85 x 
F/m is the vacuum dielectric permittivity. At 
1 = 10m and r = 2cm, C, = 100pF. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 27 ===
Lightning hazards 
19 
ii x qi/At = EolCr/2At x 0.5-0.05 mA. Even if the grounding resistance is as 
high as R, x 10 kR (we deal here with a damaged ground electrode when the 
connection with the grounding circuit is made across the high contact 
resistance of the break), the induced charge current will change the rod poten- 
tial relative to the earth by the value Ap = iiRg = 5-0.5 V. This potential rise 
can be ignored in any situation. 
When a lightning channel reaches the earth and the process of leader 
charge neutralization begins in the return stroke, the field at the earth, Eo, 
rapidly drops to zero, eliminating the charge qi. The same charge now 
flows back through the grounding resistance for a much shorter time, 
about 1 ps, and the current ii increases to about 0.5 A. At the same resistance 
R, x 10 kR the voltage will rise to 5 kV. In practice, it may rise even higher, 
producing a spark breakdown at the site of poor contact. The breakdown 
may become very dangerous if there are explosive gas mixtures nearby, 
since the spark energy is sufficiently high to set a fire. 
There is another mechanism of igniting sparks in an induced charge field, 
which may be hazardous even in the case of perfect grounding of a metallic 
construction. Suppose that a grounded rod of length I and radius r is in the 
leader electric field Eo of a nearby lightning discharge. The charge induced 
on the rod will enhance the field at its top approximately by a factor of I/r. 
With I >> r, this is sufficient to excite a weak counterpropagating leader process. 
Of course, if this leader is only about 10 cm long, it will have no effect on the 
lightning trajectory. Its energy is, however, large enough to ignite an inflam- 
mable gas mixture, if there is any in the vicinity, since the channel temperature 
is close to 5000 K and its lifetime is as long as that of a lightning leader. 
1.6.4 High potential infection 
This unsuitable term has long been used in Russian literature on lightning 
protection. It means that the surface and underground service lines, which 
get into a construction to be protected, may introduce in it a potential 
different from the zero potential of the construction metalwork connected 
to earth connection. This may become possible if a service line is not 
linked to the grounding of the construction but connected or passes close 
to the earth connection of another construction loaded by lightning current 
during a stroke (figure 1.9). This may also be a natural earth connection 
formed at the moment of lightning contact with the earth due to an intense 
ionization in it. If the introduced potential is high, it causes a spark break- 
down between the service line and a nearby metallic structure of the 
object, whose potential is zero owing to the earth connection. The scenario 
of the emergency that follows has been described above. 
To avoid sparking induced by high potential infection, all metallic 
service lines subject to explosion zooms are linked to the earth connection 
of the construction. All metalwork potentials are equalized. The connection, 
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=== PAGE 28 ===
20 
Introduction: lightning, its destructive effects and protection 
Figure 1.9. Schematic input of high potential from remote lightning strokes. 
however, becomes loaded by additional current, which finds its way there 
from a remote lightning stroke, using the service line as a conductor. 
When the earth connection resistance is low and the service line goes through 
the ground with a high resistivity so that the current leakage through the side 
surface is not large, nearly all of the lightning current arrives at the con- 
nection from the stroke site. This situation appears to be somewhat similar 
to a direct lightning stroke. Sometimes, special measures must be taken to 
restrict the infection current. A detailed treatment of the problem of current 
and potential infections will be offered in chapter 6. 
1.6.5 Current inrush from a spark creeping along the earth’s surface 
This phenomenon is familiar to all communications men who have to repair 
communications cables damaged by lightning. The damaged site can be 
detected easily, because it is indicated by a furrow in the ground extending 
far away from the stroke site. A furrow may be as long as several dozens 
of metres, or 100-200m in a high resistivity ground. Such a long gap can 
be bridged by a spark because of the electric field created by the spark current 
spreading out through the ground. The mechanism of spark formation along 
a conducting surface differs from that of a ‘classical’ leader propagating 
through air. A creeping spark can develop in low fields and have a very 
high velocity. 
Underground cables are not the only objects suffering from creeping 
spark current. Similarly, it can find its way to underground service lines 
and to the earth connections of constructions well equipped by lightning 
protectors. But a protector palisade cannot stop lightning. When the conven- 
tional way from the earth surface is blocked, it breaks through from beneath, 
making a bypass in the ground. Lightning thus behaves very much like a 
clever general in ancient times, who ordered his soldiers to make a secret 
underground passageway instead of attacking openly the impregnable 
castle walls. It is reasonable to suggest that the contact of a creeping spark 
with combustible materials is as frequent a cause of a fire as a direct lightning 
stroke. 
The details of the creeping discharge mechanism have been unknown 
until quite recently. They are analysed in chapter 6. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 29 ===
Lightning hazards 
21 
Figure 1.10. This lightning has missed the teletower tip by over 200m. 
1.6.6 Are lightning protectors reliable? 
Lightning protectors are believed to be reliable, since their design has 
changed but little over two and a half centuries. Nevertheless, the photo- 
graph in figure 1.10 makes one question this judgement: the lightning 
struck the Ostankino Television Tower 200 m below its top, i.e., the Tower 
could not protect itself. This is not an exception to the rule. Most descending 
discharges missed the Tower top more or less closely, contrary to what had 
been expected. This is a serious argument against the vulgar explanation of 
the major principle of protector operation that lightning takes a shortcut 
at the final stage of its travel to the earth. There are also other arguments, 
perhaps not as obvious but still convincing. 
Breakdown voltage spread is registered in long gaps even under strictly 
identical conditions. The breakdown probability 9 varies with the pulse 
amplitude of test voltage U (figure 1.11). Deviations from the 50% proba- 
bility voltage, Use%,, are appreciable and may be 10-15% either way. 
Curve 2 in figure 1.11 shows the probability function !F( U )  for a shorter 
gap. In certain voltage ranges, both curves promise breakdown probabilities 
remarkably different from zero. This means that if two different gaps are 
tested simultaneously, there is a chance that any of them (the smaller and 
the larger gap) will be bridged. In general, this situation is similar to that aris- 
ing when a lightning discharge is choosing a point to strike at. It does not 
always take the shortest way to a protector but, instead, may follow a 
longer path in order to attack the protected object. 
For solving the lightning path problem, one has to treat a multielectrode 
system consisting of several elementary gaps. For lightning, all elementary 
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=== PAGE 30 ===
22 
Introduction: lightning, its destructive effects and protection 
Figure 1.11. Distributions of breakdown voltages in air gaps of various lengths with a 
sharply non-uniform electric field. 
gaps have a common high voltage electrode (the leader that has descended to 
a certain altitude), while the zero potential electrodes are formed by the 
earth’s surface with grounded objects and protectors distributed on it. The 
problem of protector effectiveness thus reduces to the calculation of 
breakdown probabilities for the elementary gaps in a multielectrode 
system. The general formulation of this problem is very complex, since the 
spark development in the elementary gap in real conditions cannot be 
taken to be independent. The discharge processes affect one another by 
redistributing their electric fields, which eliminates straightforward use of 
statistical relations describing independent processes. 
We cannot say that the spark discharge theory for a multielectrode 
system has been brought to any stage of completion. But what has been 
done, theoretically and experimentally, allows the formulation of certain 
concepts of the lightning orientation mechanism and the development of 
engineering approaches to estimate the effectiveness of protectors of various 
heights (see chapter 5). 
Investigation of multielectrode systems is also important from another 
point of view: we must find ways of affecting lightning actively. It would 
be reasonable to leave the discussion of this issue for specialized chapters 
of this book, but they will, however, attract the attention of professionals 
only, or of those intending to become professionals. It is not professionals 
but amateurs who, most often, try to invent lightning protectors. They 
have at their disposal a complete set of up-to-date means: lasers, plasma 
jets, corona-forming electrodes for cloud charge exchange, radioactive 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 31 ===
Lightning as a power supply 
23 
sources, high voltage generators stimulating counterpropagating leaders, etc. 
That lightning management has a future has been confirmed by laboratory 
studies on sparks of multimetre length. These experiments and their implica- 
tions will be analysed below, so there is no point in discussing them here. Still, 
it is hard to resist the temptation to make some preliminary comments 
addressed to those who like to invent lightning protection measures. 
When explaining the leader mechanism at the beginning of this chapter, 
we noted that the leader tip carries a strong electric field sufficient for an 
intense air ionization. It is very difficult to act on this field directly, because 
it would be necessary to create charged regions close by, whose charge den- 
sity and amount would be comparable with those in the immediate vicinity of 
the tip. Pre-ionization of the air by radioactive sources is of little use because 
of the low air conductivity after radiative treatment. The initial electron 
density behind the ionization wave front in the leader process is higher 
than 10l2 ~ m - ~ ,  
and in a ‘mature’ leader it is at least an order of magnitude 
higher. These and even much lower densities are inaccessible to radiation at a 
distance of dozens of metres from the radiation source which must present no 
danger to life. The same is true of a gradual charge accumulation due to a 
slow corona formation between special electrodes. Besides, one cannot pre- 
dict the polarity of a particular spark to decide which charge is to be 
pumped into the atmosphere. 
Quite another thing is plasma generation. In principle, we could create a 
plasma channel comparable with the lightning rod height, thus increasing its 
length. A high power laser could, in principle, be used as a plasma source. It 
is clear that it should be a pulse source and the plasma produced should have 
a short lifetime. It must be generated exactly at the right moment, when a 
lightning leader is approaching the dangerous region near the object to be 
protected. This is a new problem associated with synchronization of the 
laser operation and lightning development, giving a new turn to the task of 
lightning protection, which does not at all become easier. 
Finally, we should always bear in mind that most lightning discharges are 
multicomponent. In about half of them, the subsequent components do not 
follow the path of the first component. In fact, these are new discharges 
which would require individual handling. To prepare a laser light source for 
a new operation cycle for a fraction of a second is possible but difficult techni- 
cally. The cost of such protection is anticipated to be close to that of gold. 
It is not our intention to intimidate lightning protection inventors. We 
just want to warn them against excessive enthusiasm. 
1.7 Lightning as a power supply 
The question of whether lightning could serve as a power supply cannot be 
answered positively, no matter how much we wish it to be one. Some authors 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 32 ===
24 
Introduction: lightning, its destructive effects and protection 
of science fiction books force, quite inconsiderately, their characters to har- 
ness lightning in order to use its electric power. Even without this service, 
lightning has done much for people by stimulating their thought. The 
energy of a lightning flash is not very high. The voltage between a cloud 
and the earth can hardly exceed l00MV even in a very powerful storm, 
the transported charge is less than lOOC, and maximum energy release is 
10'OJ. This is equivalent to one ton of trinitrotoluene or 2-4 ordinary 
airborne bombs. A family cottage consumes more power for heating, illumi- 
nation, and other needs over a year. Actually, only a small portion of the 
lightning power is accessible to utilization, while most of it is dissipated in 
the atmosphere. 
Normally, a person lives through 40-50 thunder storm hours during a 
year. All storms send to the earth an average of 4-5 lightning sparks per 
square kilometre of its surface providing a power of less than 1 kW/km2 
per year. In a country of 500 x 400km2, this is about 200MW, which is 
a very small value compared with the electrical power produced by an indus- 
trial country. Just imagine the immense net which would be necessary for 
trapping lightning discharges in order to collect such a meagre power! 
Other natural power sources, such as wind, geothermal waters, and tides, 
are infinitely more powerful than lightning, but they are still not utilized 
much. Clearly, we should not even raise the problem of lightning power 
resources. 
1.8 To those intending to read on 
There will be no more popularized stories about lightning in this book. Nor 
shall we mention ball lightning here. The next chapter will contain a 
thorough analysis of available data and theoretical treatments of the long 
spark, because we believe that without these preliminaries the lightning 
mechanism may not become clear to the reader. Nature has eagerly employed 
standard solutions to its problems, so lightning is quite likely to represent the 
limiting case of the long spark. It would be useful for readers to familiarize 
themselves with our previous book Spark Discharge, because it is totally 
concerned with this phenomenon. But even without it, they will be able to 
find here basic information on long sparks. We have tried to describe their 
general mechanisms and to give predictions as to their extension to air 
gaps of extrema1 length. Even for this reason alone, the next chapter is not 
a summary of the previous book. Lightning is as complicated a phenomenon 
as the long spark and is definitely more diverse. It is a multicomponent 
process. Since its subsequent components sometimes take the path of an 
earlier component, we must consider the effects of temperature and residual 
conductivity in the spark channel on the behaviour of new ionization 
waves. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 33 ===
To those intending to read on 
25 
Even a simple model should not treat a kilometre spark in terms of 
electrical circuits with lump parameters. A lightning spark is a distributed 
system. The time for which the electric field perturbation spreads along the 
sparks is comparable with the duration of some of its fast stages. The 
allowance for the delay can, in some cases, change the whole picture 
radically. This requires new approaches to lightning treatments. Experimen- 
tal data and theoretical ideas concerning the lightning leader and return 
stroke are discussed together. First, there are not many of them. On the 
other hand, we have tried to point out ideological relationships between 
experiment and theory and to offer a more or less consistent physical 
description. 
Spark discharges in a multi-electrode system are the subject of a special 
chapter. We present available data and analyse possible mechanisms of light- 
ning orientation. This is, probably, the most debatable part of the book. 
Field studies of lightning orientation are very difficult to carry out primarily 
because constructions of even 100-200 m high are affected by descending 
discharges only once or twice a year. The observer must have exceptional 
patience and substantial support to be able to reveal statistical regularities 
in lightning trajectories. From field observations, one usually borrows the 
statistics of lightning strokes at objects of various height and, sometimes, 
the statistics of strokes at protected objects, such as power transmission 
lines with overhead grounding wire connections. This material, however, is 
too scarce to build a theory. For this reason, one has to refer to laboratory 
experiments on long sparks generated in 10-15m gaps. No one has ever 
proved (or will ever do so) the geometrical similarity of sparks; therefore, 
experimental data can be extended to lightning only qualitatively. Neverthe- 
less theoretical treatments must be brought to conclusion when one develops 
recommendations on particular protector designs. We analyse the reliability 
of engineering designs, wherever possible. 
The last chapter of the book discusses lightning hazards and protection 
not only in terms of applications. Even the classical theory of atmospheric 
overvoltages in power transmission lines required the solution of com- 
plicated electrophysical problems. Thorough theoretical treatments are 
necessary for the analysis of lightning current effects on internal circuits of 
engineering constructions with metallic casings, on underground cables, 
aircraft, etc. The range of problems to be considered is not limited to 
electromagnetic field theory. We shall also discuss gas discharge mechanisms 
of a spark creeping along a conducting surface, the excitation of leader 
channels in air with the composition and thermodynamic characteristics 
locally changed by hot gas outbursts, and the lightning orientation under 
the action of the superhigh operating voltage of an object. These theoretical 
considerations will not screen our practical recommendations concerning 
effective lightning protection and the application of particular types of 
protectors. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 34 ===
26 
Introduction: lightning, its destructive effects and protection 
References 
[l] Bazelyan E M, Gorin B N and Levitov V I 1978 Physical and Engineering Funda- 
mentals of Lightning Protection (Leningrad: Gidrometeoizdat) p 223 (in Russian) 
[2] McEachron K 1938 Electr. Engin. 57 493 
[3] Berger K and Vogrlsanger E 1966 Bull. SEV 57 No 13 1 
[4] Newman M M, Stahmann J R, Robb J D, Lewis E A et a1 1967 J. Geophys. Res. 
72 4761 
[5] Uman M A 1987 The Lightning Discharge (New York: Academic Press) p 377 
[6] Berger K, Anderson R B and Kroninger H 1975 Electra 41 23 
[7] Schonland B, Malan D and Collens H 1935 Proc. Roy. Soc. London Ser A 152 595 
[8] Bazelyan E M and Raizer Yu P 1997 Spark Discharge (Boca Raton: CRC Press) 
p 294 
191 Schonland B 1956 The Lightning Discharge. Handbuch der Physik 22 (Berlin: 
Springer) p 576 
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=== PAGE 35 ===
Chapter 2 
The streamer-leader process in 
a long spark 
This chapter will deal with the spark discharge in a long air gap. We have 
already mentioned in chapter 1 that this material should not be ignored by 
the reader. But for the long spark, specialists would know much less about 
lightning. Today, high voltage laboratories are able to produce and study 
long sparks of several tens and even hundreds of metres long [l-31. Many 
of the long spark parameters and properties lie close to the lower boundary 
of respective lightning values. Most effects observable in a lightning 
discharge were, sooner or later, reproduced in the laboratory. One exception 
is a multicomponent discharge, but the obstacles lie in the technology rather 
than in the nature of the phenomenon. It would be very costly to instal and 
synchronize several high voltage power generators, making them discharge 
consecutively into the same air gap. 
As for the fine structure of gas-discharge elements, long spark research- 
ers are far ahead of lightning observers. This could not be otherwise, since a 
laboratory discharge can be reproduced as often as necessary, by starting the 
generator at the right moment, within a microsecond fraction accuracy, and 
strictly timing the switching of all fast response detectors. But with lightning, 
the situation is different. It strikes every square kilometre of the earth’s sur- 
face in Europe approximately 2 to 4 times a year. So, even such a high con- 
struction as the Ostankino Television Tower (540m) is struck by lightning 
only 25-30 times a year. Of these, only 2-3 discharges are descending, 
while the others go up to a cloud. Normally, lightning observations have 
to be made from afar, so that many details of the process are lost. The 
gaps in the study of its fine structure must, of necessity, be filled in laboratory 
conditions. 
The long spark theory is far from being completed, and there is no 
adequate computer model of the process. Still, there has lately been some 
progress, primarily owing to laboratory investigations. It would be unwise 
to discard these data and not to try to use them for the description of 
21 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 36 ===
28 
The streamer-leader process in a long spark 
lightning. In this chapter, we shall outline our conception of the basic 
phenomena in a long spark. We shall present some newer data and ideas 
which emerged after the book [4] on the long spark had been published. 
We should like to emphasize again that many details of the spark physics 
are still far from being clear. 
2.1 
What a lightning researcher should know about a 
long spark 
The key point is how a spark channel develops in a weak electric field, by 1-2 
orders of magnitude lower than what is necessary to increase the electron 
density in air (Ei x 30 kV/cm under normal conditions). Naturally, we 
speak of a discharge in a sharply non-uniform field. Near an electrode 
with a small curvature radius (suppose this is a spherical anode of radius 
r, x 1-lOcm), the field is Ea(ra) Ea > E, at the voltage U x 50-500kV. 
This is the site of initiation of a discharge channel. At a distance r = lor, 
from the electrode centre, the channel tip enters the outer gap region, 
where the initial value of E = Ea(r,/r)* is one hundredth of that on the 
electrode. This weak field is incapable of supporting ionization. Nevertheless, 
the channel moves on, changing the neutral gas to a well-ionized plasma. 
There is no other reasonable explanation of this fact except for a local 
enhancement of the electric field at the tip of the developing channel. The 
enhancement is due to the action of the channel’s own charge. Indeed, a con- 
ductive channel having a contact with the anode tends to be charged as much 
as its potential U, relative to the grounded cathode. Current arises in the 
channel, which transports the positive electric charge from the anode (more 
exactly, from the high voltage source, to which the anode is connected). (In 
reality, electrons moving through the channel toward the anode expose low 
mobility positive ions.) Such would be exactly the mechanism of charging a 
metallic rod if it could be pulled out of the anode like a telescopic antenna. 
Then the strongest field region would move through the gap together with 
the rod tip. We can say that a strong electric field wave is propagating through 
a gap, in which ionization occurs and produces a new portion of the plasma 
channel. We can also name it as an ionization wave, and ths term is commonly 
used. 
The wave mechanism of spark formation was suggested as far back as 
the 1930s by L Loeb, J Meek, and H Raether. The channel thus formed 
was termed a streamer (figure 2.1). Experiments showed that the streamer 
velocity could be as high as 107m/s. In lightning, this velocity is demon- 
strated by the dart leader of a subsequent component. Even the mere fact 
that these velocities are comparable justifies our interest in the streamer 
mechanism. It is important to know what determines the streamer velocity 
and how it changes with the tip potential. For this, we have to analyse 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 37 ===
What a lightning researcher should know about a long spark 
29 
grounded 
cathode 
Figure 2.1. A schematic cathode-directed streamer: U, (x), external field potential; 
U ( x ) ,  potential along the conductive streamer axis. 
processes taking place in the streamer tip region where ionization occurs. It is 
necessary to find out how the processes of charged particle production are 
related to electron motion in the electric field, due to which the charged 
region travels through the gap like the crest of a sea wave. 
The specific nature of spark breakdown is not restricted to the ionization 
wave front, because its crucial parameter is the channel tip potential U,. Its 
value may be much smaller than the potential U, of the electrode, from which 
the streamer has started, since the channel conductivity is always finite and 
the voltage drops across it. Therefore, the analysis of streamer propagation 
for a large distance will require a knowledge of the electron density behind 
the wave front and the current along the channel in order to eventually 
calculate the electric field in the travelling streamer and to derive from it 
the voltage drop on the channel. Incidentally, the field and the current 
preset the power losses in the channel. It will become clear below how 
important this parameter is for spark theory. 
The streamer creates a fairly dense plasma. Without this, it would be 
unable to transport an appreciable charge into the gap. A quantitative 
description of the ionization wave propagation provides the initial electron 
density in the channel and defines its initial radius. Behind the wave front, 
the streamer continues to live its own life. A streamer channel may 
expand, through ionization, in the radial electric field of its intrinsic 
charge, provided that the latter grows. The cross section of the current 
flow then becomes larger. The channel continuously loses the majority cur- 
rent carriers - electrons. The rates of electron attachment to electronegative 
particles and electron-ion recombination strongly affect the fate of the 
discharge as a whole. If the air through which a streamer propagates is 
cold and the power input into the channel is unable to increase its tempera- 
ture considerably (by several thousands of degrees), the process of electron 
loss is very fast, since the attachment alone limits the electron average lifetime 
to lop7 s. This is a very small value not only at the scale of lightning but 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 38 ===
30 
also of a laboratory spark, whose development in a long gap takes lop4- 
lop3 s. One must be able to analyse kinetic processes in the channel behind 
the ionization wave front. Without the knowledge of their parameters, one 
will be unable to define the conditions, in which a streamer breakdown in 
air will be possible. 
Here and below, we shall mean by a breakdown the bridging of a gap by 
a channel which, like an electric arc, is described by a falling current-voltage 
characteristic. The channel current is then limited mostly by the resistance of 
the high voltage source. Such a situation in technology is usually called short 
circuiting. 
Current rise without an increase of the gap voltage inevitably suggests a 
considerable heating of the gas in the channel. Due to thermal expansion, the 
molecular density N decreases, thereby increasing the reduced electric field 
E / N  and the ionization rate constant (see [4]). Another consequence of the 
heating is a change in the channel gas composition because of a partial 
dissociation of 02, N2 and H 2 0  molecules and the formation of easily 
ionizable NO molecules. The significance of many reactions of charged 
particle production and loss changes. The importance of electron attachment 
decreases, because negative ions produced in a hot gas rapidly disintegrate to 
set free the captured electrons. The electron-ion recombination rate becomes 
lower. But of greater importance is associative ionization involving 0 and N 
atoms. The reaction is accelerated with temperature rise but it does not 
depend directly on the electric field. This creates prerequisites for a falling 
current-voltage characteristic. 
Clearly, a researcher dealing with long sparks and lightning cannot 
avoid considering the energy balance in the discharge channel, which deter- 
mines the gas temperature. It is here that the final result is most likely to 
depend on the scale of the phenomenon and the initial conditions. In the 
laboratory, a streamer crossing a long gap seldom causes a breakdown 
directly. A streamer propagating through cold air remains cold. It will be 
shown below that the specific energy input into the gas is too small to heat 
it. Even during its flight, the old, long-living portions of a streamer lose 
most of their free electrons. In actual fact, it is not a plasma channel but 
rather its nonconductive trace which crosses a gap. The researcher must pos- 
sess special skills to be able to produce an actual streamer breakdown of a 
cold air gap in laboratory conditions. 
The situation with lightning may be different. Most lightnings are multi- 
component structures. With the next voltage pulse, the ionization wave often 
propagates through the still hot channel of the previous component. It is not 
cold air but quite a different gas with a more favourable chemical composi- 
tion and kinetic properties. Surrounded by cold air, the hot tract shows some 
features of a discharge in a tube with a fixed radius and, hence, with a more 
concentrated energy release. It seems that the mechanism of the phenomenon 
known as a dart leader is directly related to streamer breakdown. One should 
The streamer-leader process in a long spark 
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=== PAGE 39 ===
What a lightning researcher should know ahout a lorig spark 
31 
Figure 2.2. A photograph and a scheme of a positive leader. 
be ready to give a quantitative description or make a computer simulation of 
this process. 
Long gaps of cold air are broken down by the leader mechanism. During 
the leader process, a hot plasma channel (5000- 10 000 K) is travelling 
through the gap. Numerous streamers start at high frequency from the 
leader tip, as from a high voltage electrode, and form a kind of fan. They 
fill up a volume of several cubic metres in front of the tip (figure 2.2). 
This region is known as the streamer zone of a leader, or leader corona, 
by analogy with a streamer corona that may arise from a high voltage elec- 
trode in laboratory conditions. The total current of the streamers supplies 
with energy the leader channel common to the streamers, heating it up. 
The streamer zone is filled up with charges of streamers that are being 
formed and those that have died. As the leader propagates, the zone travels 
through the gap together with its tip. so that the leader channel enters a space 
filled with a space charge, 'pulling' it over like a stocking. A charged leader 
cover is thus formed which holds most of the charge (figure 2.2). It is this 
charge that changes the electric field in the space around a developing 
spark and lightning. It is neutralized on contact of the leader channel with 
the earth, creating a powerful current impulse characteristic of the return 
stroke of a spark. Thus, we can follow a chain of interrelated events, 
which unites the simplest element of a spark (streamer) with the leader 
process possessing a complex structure and behaviour. 
All details of the leader development directly follow from the properties 
of a streamer zone. In lightning, it is entirely inaccessible to observation 
because of the relatively small size and low luminosity. Today, there is no 
other way but to study long sparks in laboratory conditions and to extrapo- 
late the results obtained to extremely long gaps. This primarily concerns a 
stepwise negative leader, whose streamer zone has an exclusively complex 
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=== PAGE 40 ===
32 
The streamer-leader process in a long spark 
structure. It contains streamers of different polarities, starting not only from 
the leader tip but also from the space in front of them. 
The leader channel of a very long spark, let alone of lightning, is its long- 
est element. An appreciable part of the voltage applied to the gap may drop 
on this element. This is why one should know the time variation of the 
channel conductivity. The channel properties mostly depend on the current 
flowing through a given channel cross section. If the current is known, it is 
not particularly important whether it belongs to a long spark or lightning. 
The parameter that changes is the time during which one observes this 
process: for lightning, it is one or two orders of magnitude longer than for 
a spark. By analysing the self-consistent process of leader current production 
in the streamer zone and its effects on plasma heating and conversion in the 
channel, one can derive the conditions for an optimal leader development in 
a gap of a given length. There are reasons to believe that these conditions are 
realized in lightning when it develops in an extremely weak field. Nature 
always strives for perfection, not because it is animated but because optimal 
conditions most often lead to the highest probability of an event. 
To conclude this section, long spark theory is of value in its own right to 
specialists in lightning protection. Lightning current is the cause of the most 
common type of overvoltage in electric circuits. The amplitude of lightning 
overvoltages reaches the megavolt level. In order to design a lightning- 
resistant circuit, one must be able to estimate breakdown voltages in air 
gaps of various lengths and configurations. This can be done only with a 
clear understanding of the long spark mechanism. 
2.2 
A long streamer 
2.2.1 
Let us consider a well developed ‘classical’ streamer, which has started from 
a high voltage anode and is travelling towards a grounded cathode. The 
main ionization process occurs in the strong field region near the streamer 
tip. We shall focus on this region. The front portion of a streamer is 
shown schematically in figure 2.3 together with a qualitative axial distribu- 
tion of the longitudinal field E, electron density ne, and a difference between 
the densities of positive ions and electrons, or the density of the space charge 
p = e(n+ - ne) (the time is too short for negative ions to be formed). 
The strong field near the tip is created mostly by its own charge. In front 
of the tip where the space charge is small, the field decreases approximately as 
E = E,(T~/Y)’, which is characteristic of a charged sphere of radius Y,. 
Here, 
E, is the maximum streamer field at the tip front point. In fact, the radius at 
which the field is maximum should be termed the tip radius Y,. 
It approxi- 
mately coincides with the initial radius of the cylindrical channel extending 
behind the tip. The front portion of a conventionally hemispherical tip 
The streamer tip as an ionization wave 
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=== PAGE 41 ===
A long streamer 
33 
Figure 2.3. Schematic representation of the front portion of a cathode-directed 
streamer and qualitative distributions of the electron density ne, the density difference 
n, - ne (space charge), and longitudinal field E along the axis. 
should be called the ionization wave front. The streamer tip charge is primarily 
concentrated in the region behind the wave front. The field there becomes low, 
dropping to a value E, in the channel, small as compared with E,. The lines of 
force going radially away from the tip in front of it become straight lines inside 
the tip and align axially along the streamer channel. 
Let us mentally subdivide the continuous process of streamer develop- 
ment into stages. The strong field region in front of the tip is the site of 
ionization of air molecules by electron impact. The initial seed electrons 
necessary for this process are generated by the streamer in advance. Their 
production is due to the emission of quanta, accompanying the ionization 
process because of electronic excitation of molecules. In our case, highly 
excited N2 molecules are active so that the quanta emitted by them ionize 
the O2 molecules, whose ionization potential is lower than that of N2. The 
radiation is actively absorbed, but its intensity is high enough to provide 
an initial electron density M~ of about 105-106~m-3 at a distance of 0.1- 
0.2cm from the tip. Each of these electrons gains energy from the strong 
field, giving rise to an electron avalanche. Since the number of avalanches 
developing simultaneously is very large, they fill up the space in front of 
the streamer tip to form a new plasma region. Owing to the electron outflow 
towards the channel body, the positive space charge of the plasma becomes 
exposed. Simultaneously, electrons that have advanced from the ahead 
region neutralize the positive charge of the ‘old’ tip which turns to a new 
channel portion, thereby elongating the streamer. 
The gas in the wave front region must be highly ionized for the electron- 
ion separation to produce an appreciable charge capable of creating a strong 
ionizing field in front of the newly formed tip. For this reason, the region of 
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=== PAGE 42 ===
34 
The streamer-leader process in a long spark 
concentrated tip charge is somewhat shifted towards the channel body 
relative to the intensive ionization site (figure 2.3). Normally, the electric 
field is pushed out of a good plasma conductor, and the space charge (if 
the conductor is charged) quickly concentrates near its surface as a ‘surface’ 
charge. The plasma of a fast streamer (‘fast’ in the sense that will be specified 
below) possesses a fairly high conductivity, and these properties apply to 
such a streamer. Therefore, the region of strong field and space charge in 
the tip looks like a thin layer, as is shown in figure 2.3. 
If the streamer length is I >> r,, its velocity and the tip parameters change 
little during the time the tip travels a distance of its several radii. T h s  means 
that, depending on the time t and the axial coordinate x, all parameters 
are the functions of the type E(x. t) = E ( x  - V,t), and what is shown in 
figure 2.3 moves to the right as a whole, without noticeable distortions. The 
picture changes only as the streamer velocity changes relatively slowly. This 
kind of process represents a wave, in ths case a wave of strong field and 
ionization. The external parameter determining the wave characteristics (its 
velocity V,, maximum field E,, 
tip radius r,, electron density behind the 
wave n,) is the tip potential U,. It is indeed an external characteristic of the 
tip, although it partly depends on the properties of the wave itself. The poten- 
tial U, is equal to the anode potential U, minus the voltage drop on the 
streamer channel. The channel properties, however, are initially determined 
by the ionization wave parameters, so that the problem of streamer develop- 
ment is, strictly speaking, just one problem. Still, it can be approximately 
subdivided into two parts: the ionization wave problem and the problems of 
voltage drop and current in the channel. Both parts will be related by the 
dependencies of V,( Ut) and current il at the channel front on velocity V,. 
2.2.2 Evaluation of streamer parameters 
The formulas to be derived in this and subsequent sections of this chapter do 
not claim high accuracy. The streamer and leader problems are very complex, 
and a rigorous solution can be obtained only by numerical computation. But 
a simplified analytical treatment may also be useful because it provides an 
understanding of basic laws and relations among the process parameters. 
In other words, one is able to get a general idea of the physics of the phenom- 
enon under study and to estimate the order of values of its characteristics. 
Let us consider a fast streamer, whose velocity is much higher than the 
electron drift velocity in the wave. Streamers are fast in many situations of 
practical interest. The calculation of electron production can ignore the 
slight drift of electrons from a given site in space for the short time the 
ionization wave passes through it. In this case, the ionization kinetics 
along the streamer axis is described by the following simple equations: 
= exp vidt =exp 
s 
n C  
3 
= yn,. 
- 
at 
a0 
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=== PAGE 43 ===
A long streamer 
35 
Figure 2.4. Ionization frequency of air molecules by electron impact under normal 
conditions (from the data on ionization coefficient cy and electron drift velocity V, 
in [l 11). 
where vi = vi(E) is the frequency of electron ionization of molecules. Its time 
integral has been transformed to the integral over the coordinate x along 
the wave axis, according to the equality dx = V, dt corresponding to the 
coordinate system moving together with the wave. Due to the sharp increase 
of the ionization frequency with field (figure 2.4), the region where the field is 
not much less than its maximum makes the largest contribution to the 
electron production. This region in the wave is of the same order of 
magnitude as the tip radius (figure 2.3). So we can write the approximate 
expressions for the integral (2.1) and streamer velocity: 
This type of formula was first suggested by Loeb [5] and has been used since 
that time, in this or modified form, in all streamer theories [6-lo]. The 
velocity of a fast streamer is weakly related to the initial no and final n, 
electron densities and is determined only by the maximum field E, and the 
tip radius r,. 
The quantities E, and r, which determine V, are not independent. They 
are interrelated by the tip potential U,. For an isolated conductive sphere with a 
uniformly distributed surface charge Q', we have U = r,E, 
= Q'/~TTTE~Y,, 
where E~ = ( 3 6 ~  
x lo")-' 
FZ 8.85 x 10-12F/m is vacuum permittivity. A 
streamer looks more like a cylinder with a hemispherical rounded end (see 
figure 2.3). We can show [4] that in a long perfect conductor of this shape, 
one half of the potential at the hemisphere centre is created by charges 
concentrated on the hemisphere surface and the other half by those on the 
cylinder surface, so that the tip charge is Q = ~ T T T E ~ Y , U ~ .  
The field at the tip 
front point is, to good approximation, only one half of that in an isolated 
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=== PAGE 44 ===
36 
The streamer-leader process in a long spark 
sphere with the same potential, or 
0; = 2E,r,. 
The tip charge moves because of the electron drift under the field action. 
The electron density in the wave plasma and the respective plasma conduc- 
tivity must provide the charge transport with the same velocity as that of 
the wave. This permits estimation of the plasma density in the streamer 
just behind its tip. With the same assumptions as those in (2.2), the electron 
density in the strong field region on the streamer axis increases as 
ne M no exp (vimt) for the time At M rm/Vs. During this period of time, the 
electron density rises to its final value nc M no exp (vi,&) 
, and the electron 
drift towards the channel with velocity V, = peE, (where pe is electron 
mobility taken, for simplicity, to be constant) exposes the charge which 
creates the field E, in the region of the new streamer tip. 
The electron charge that flows through a unit cross section normal to the 
axis in the wave front region over time At is 
PeEmnc 
vim 
At 
q = ep,E,noSo 
exp(6,t) dt = 
~ 
It leaves behind it a positive charge of the same surface density q. It is this 
charge that creates the field E,. We shall see soon that the effective thickness 
of a positively charged layer is Ax << r,. In electrostatics, the field of such a 
layer at the conductor surface is equal to E, M q / q  (ths equality is 
absolutely exact for the surface charge of a perfect conductor). By substituting 
q from (2.4), we get an estimate for the density behind the ionization wave: 
n, M Eovim/epCL,. 
The plasma density n, is not related directly to the streamer velocity and is 
essentially determined by the maximum field value which defines the ioniza- 
tion frequency. 
Let us make sure that the tip charge is indeed concentrated in the thin layer. 
The effective time for the charge to be formed in the layer approximately is 
1 
(At - t )  exp(vi,r) dt 
exp(q,t) dt = vi;. 
The space charge layer of thickness Ax moves to a new site at velocity 
Ax/& 
which is equal to the streamer velocity V,, since the ionization 
wave moves as a whole. Hence, using (2.2), we obtain 
Ax = VsAt, M V,/q, M r,/ln(n,/no). 
(2.5) 
The final plasma density is many orders of magnitude larger than the initial 
density no, so that the logarithm in (2.5) is a large value. Therefore, we have 
Ax << r,. 
The formulas derived here claim for nothing more than an illustration of 
functional relations among streamer parameters. Numerical factors allowing 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 45 ===
A long streamer 
37 
the transition from an order of magnitude to a specific value have been deliber- 
ately ignored. This is justified because we have simplified all initial conditions 
and the derivation procedure in order to reveal the physics of the phenomenon 
in question. The significance of a formula will increase if it is ‘equipped’ with 
numerical factors, even approximate ones. Since we know the origin of these 
factors, we can partly judge about the theory validity and meaningfully 
compare the analytical results with computations and measurements. 
A more rigorous treatment of a fast ionization wave, using one- 
dimensional equations consistent with the streamer physical model [4,9], 
yields the expressions 
(2.6) 
nm 
= ln- 
nC - 
n, = 
q m r m  
v, = 
(2k - 1) ln(n,/no) ’ 
kePe ’ 
n m  
no 
where k is the power index from the approximate formula vi(E) N Ek and n, is 
the electron density in the wave front at the point of maximum field (the 
density is an order of magnitude smaller than the maximum achievable 
density n,). In the field range characteristic of an air streamer, k = 2.5. 
The issue of the streamer tip radius or maximum field represents the 
most complicated and least convincing point in streamer theory. It is likely 
that their values are established under the action of a self-regulation mechan- 
ism related to the proportionality V, = q(E,) and to the rapidly increasing 
(at first) and then slowly growing dependence of q on E (figure 2.4). If, at 
constant tip potential, the tip radius turns out to be too small and the field 
E, respectively too high, corresponding to the slow growth of vi(E), the 
channel front end will not only move forward quickly but it will expand as 
fast under the action of a strong transverse field. The value of r, will rise 
while that of E,, according to (2.3), will fall. 
Suppose, on the contrary, that the radius Y, 
is too large and the field E, 
is too low, corresponding to a rapid growth of vi(E). Any slight plasma pro- 
trusion in the tip front will locally enhance the field. The ionization rate will 
greatly increase there, and the protrusion will run forward as a channel of a 
smaller radius. Some qualitative considerations of this kind were suggested in 
an old work of Cravath and Loeb [12], but these authors discussed a lightning 
channel obeying other mechanisms, because lightning develops via the leader 
process. These ideas were used in [6,7] to formulate an approximate streamer 
theory. A semi-qualitative criterion was suggested for choosing the maxi- 
mum field feasible in the streamer tip. According to [6,7], E, corresponds 
to the saturation point or bending in the function vi(E). This criterion was 
refined in [13] by establishing a quantitative relation of E, to the slope of 
the q ( E )  function and to the charge and normal field distributions over 
the streamer tip surface. It was shown that the field E, at the tip front 
point is established such that the normal field on its lateral surface 
corresponds to the point of transition from the rapid to the slow growth of 
the ?(E) function (figure 2.4). 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 46 ===
38 
The streamer-leader process in a long spark 
The streamer’s choice of maximum field was manifested during a 
numerical simulation of short fast streamers within the framework of a 
complete two-dimensional formulation [ 14- 161. The mechanism of auto- 
matic establishment of E, was demonstrated in [13] in the calculation of 
long streamer development with arbitary initial conditions and considerably 
simplified equations (see also [4 Suppl.]). No one has been able yet to 
calculate a long streamer within a complete two-dimensional model. We 
can conclude from these results that the field at the front point of an air 
streamer propagating at atmospheric pressure and room temperature 
seems to be E, M 150-170kV/cm. The tip radius varies with the tip poten- 
tial, approximately satisfying (2.3). 
We shall give a numerical example as an illustration. At E, = 170 k V / m  
in air (vim M 1.1 x 10 s , pe M 270 cm2/V s) and r, = 0.1 cm (corresponding 
to U, = 34 kV), the streamer velocity for no M lo6 cmP3 from formula (2.6) is 
close to 1.7 x lo6 mjs and the electron density in the newly born channel 
portion is n, M 9 x 1013 ~ m - ~ .  
Within 20% accuracy, these values coincide 
with the results of integration of unreduced equations in the one-dimensional 
model illustrated in figure 2.3 [lo]. They show not more than a 2- or 3-fold 
disagreement with numerical simulations of streamers in a two-dimensional 
model developed by different workers. This type of computation is extremely 
complicated and not particularly advanced. So the simple formulas (2.6) are 
useful since they can provide rough working estimations. They are also 
applicable to a gas of lower density, in whch similarity laws are operative. 
Since v, M N f ( E / N ) ,  where N is the number of molecules per 1 cm3, f is a 
function of the type given in figure 2.4, and pe M N-‘, we have 
11 -1 
E, - N ,  
r, - u,/N. 
v, U,. 
n, 
N ~ .  
(2.7) 
The streamer velocity is independent of N and n, of the tip potential U,. 
The latter fact opens up a tempting but yet unused possibility to test 
experimentally the theoretical concept of the maximum tip field E, being 
constant. For this, it is sufficient to measure the electron density right 
behind the tip of a fast streamer, in which U, = U, - EJ rapidly decreases 
with the channel length 1 (Eav is the average channel field). The constant data 
on n, will indicate the constant values of E,. On the other hand, the decrease 
in electron density with decreasing streamer velocity could become a strong 
argument to support the hypothesis of constant tip radius r,, which still has 
advocates. 
From (2.6) and (2.3), the velocity of a fast streamer is proportional to its 
tip potential, because its radius is proportional to the potential at 
E, = const. The velocity V, becomes lower than the electron drift velocity 
V,(E,) 
RZ 4 x lo5 mjs at U, M 5-8 kV. At lower voltages, the streamer 
moves slower than the drift electrons, so that the formulas become invalid. 
The analysis of equations presented in [lo] shows that the streamer velocity 
drops with further decrease in U, at V, < Vem. The electron density at the 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 47 ===
A long streamer 
39 
wave front decreases, all electrons are drawn out of the tip, and space charge 
fills it up. However, the final plasma density behind the ionization wave does 
not decrease. The tip radius becomes very small, and the streamer stops at 
low U,. This result agrees with experiments: no streamers with a velocity 
less than (1.5-2) x lo5 mjs have ever been observed in air under normal 
conditions. 
2.2.3 
As the streamer develops, its channel is under high potential which changes 
from the anode potential U, at the starting point to a certain value U, at the 
front end, close to the tip potential U, (the difference between U, and U, of 
about E,Ax << U, is due to a small potential drop in the tip). The channel 
is electrically charged, since the potential at any point x along it is higher 
than the unperturbed potential of space Uo(x) created by electrode charges 
in the absence of streamer. Current must flow through the channel to 
supply charge to the new portions of the growing streamer. When setting our- 
selves the task of estimating this current and the current through the external 
circuit (this is the streamer current to be measured), we must first find the 
channel charge, for it is the time variation of this charge that produces the 
current. Suppose a streamer has started from an anode of small radius P , .  
Let us examine the stage when the streamer length becomes 1 >> ra (I is 
much larger than the channel radius r). We can then neglect the time 
variation of the anode charge, because its capacitance is small, and take 
the external current to be close to the current i, entering the channel through 
its base at the anode. Besides, a streamer conductor can be regarded as being 
solitary, and the unperturbed potential U, far from the anode can be ignored. 
Assume first that the channel is a perfect conductor. From a well-known 
electrostatics formula, the capacitance of a long solitary conductor is 
C = 27r~~l/ln(l/~). 
Its charge is Q = CU, because a perfect conductor is 
under only potential U .  Introduce now the concept of capacitance per unit 
length of the conductor, C1, which is frequently used in electro- and radio- 
engineering to analyse long lines. The average capacitance per unit length 
Current and field in the channel behind the tip 
C 
I 
ln(I/r) 
ln(I/r) 
27rq - 5.56 x lo-” 
c1 =-=-- 
has a nearly constant value which only slightly varies with I and r. Calcu- 
lations show that the local capacitance C1 (x) practically coincides with the 
average value from (2.8) along the whole length of a long conductor, 
except for its portions lying close to its ends. But even at the ends, the 
local capacitance is less than twice the average value. This, however, does 
not concern capacitances of the free ends which are much larger (see below). 
As an approximation justifiable by calculations, we shall use the capaci- 
tance per unit length from (2.8) and apply it to a real streamer channel. If a 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 48 ===
40 
The streamer-leader process in a long spark 
channel possesses a finite conductivity, then it must have a longitudinal 
potential gradient and U = U ( x ) ,  when current flows through it. The 
charge per unit channel length has the form 
27r&O[U(X) - U()] 
Wllr) 
.(x) 
% C,[U(x) - Uo(x)] 
= 
which allows for the fact that the local charge of a channel raises its potential 
relative to unperturbed potential. Uo(x). 
A similar refinement should be 
introduced in formula (2.3), which generally looks like 
U, - UO(l) = 2Emrm. 
(2.10) 
as well as in (2.7). If the channel radius r varies along its length, a character- 
istic value may be substituted into (2.9), because the capacitance varies with r 
only logarithmically. 
Now turn to channel current. When a channel elongates by dl, its new 
portion acquires charge ~ [ d l ;  
index E will denote parameters of the front 
channel end, x = 1. This charge is supplied directly by local current ir over 
time dt = dl/V,. Therefore, at any stage of streamer development, we have 
(2.11) 
The current at the tip is defined mainly by the tip potential and streamer 
velocity. At the anode, the current is 
/ 
1 
0 
(2.12) 
s o  
1, . = dt, 
dQ 
Q = J’ ~ ( x )  
dx = 
C1 [U(x) - Uo(x)] 
dx 
where, Q is the total channel charge. Strictly, Q should be supplemented by 
the tip charge Q, = 2 7 r ~ ~ r ~ [ U ,  
- U,([)], but it is relatively small in a long 
streamer. 
Currents i, and ii at the opposite ends of a streamer channel do not 
generally coincide. Of course, their values may be very close or differ con- 
siderably, depending on particular conditions. For example, if the electrode 
voltage is raised during the streamer development, the potential and charge 
distributed along the channel increase. Some of the anode current is used to 
supply an additional charge to the old channel portions, so that only the 
remaining current reaches its front end: i, > il. But if the electrode voltage 
is decreased, the ‘excess’ charge of the old channel goes back to the supply 
through the anode surface, so that the current decreases nearer to the 
anode (positive current is created by charges moving away from the 
anode): i, < if. 
A long streamer can develop at constant voltage when the electric field in 
the channel, E ( x .  t), does not vary much with time. The potential at any point 
of the existing channel U ( x )  = U, - Jt E(x) dx and ~ ( x )  
vary slightly with 
time, which means that current does not branch off on the way from the 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 49 ===
A long streamer 
41 
anode to the channel tip. In this case, the anode current is close to the end 
current defined by (2.11) including potential U, which may be much lower 
than U,. Many experiments have shown that the average channel field 
must exceed a certain minimum value of about 5 kV/cm for air in normal 
conditions (see sections 2.2.6 and 2.2.7) to be able to support a long positive 
streamer. For instance, if U, = 600 kV at the anode and the streamer length 
is 1 
1 m, nearly all voltage drops in the channel and U, << U,, but currents 
i, and il still do not differ much. 
To get a general idea about the orders of magnitudes, let us consider a 
variant which seems quite realistic and is manifested by some calculations 
(section 2.2.6). This is the variant with constant applied voltage and 
slowly varying average field in the channel, when the current along it 
changes little and, therefore, can be evaluated from (2.1 1). For example, at 
I = 1 m, r = 0.1 cm, V, = 1.7 x lo6 m/s, and Ul = U, 
34 kV, as in the 
illustration in section 2.2.2 (with U, > 500 kV), we have ln(l/r) = 6.9, 
C1 = 8 x 10-12F/m, r1 = 2.7 x lOP7C/m, and i, = i, = 0.46A. Streamer 
currents of this order of magnitude (as well as much higher or much lower 
currents) have been registered in many experiments. These values can also 
be obtained from calculations with the account of possible streamer velocities 
from lo5 to lo7 m/s in air, which have been found in some experiments to be 
even higher [4]. 
In a simple model of potential and current evolution in a developing 
streamer channel; the latter can be represented as a line with distributed 
parameters: the capacitance C1 and resistivity R1 = (xr2epene)-' per unit 
length. The electron density n,(x. t )  should be calculated in terms of the 
plasma decay kinetics (see section 2.2.5). Estimations show that self-induction 
effects are not essential in streamer development [4]. Then, the process is 
described by the following equations for current and voltage balance: 
(2.13) 
dU 
. 
-- = zR1, 
7 = C1(U - Uo). 
dr 
di 
-+--0, 
at 
dx 
d X  
A boundary condition in the set of equations (2.13) at x = 1 is the equality 
4 = c1 [U1 - UO(4l Vs 
(2.14) 
equivalent to (2.1 1). Formula (2.12) automatically follows from (2.13) and 
(2.14). Another boundary condition may be the setting of anode potential, 
since U(0, t )  = Ua(t). Equations (2.13) and (2.14) will be used in the next 
section to evaluate the heating of a streamer channel. Illustrations of 
streamer development calculations will be given in sections 2.2.6 and 2.2.7 
after a discussion of the plasma decay mechanism. A complete set of 
equations for a long line, generalized by taking self-induction into account, 
will be applied in section 4.4 to the treatment of a lightning return stroke. 
Equality (2.11) allows evaluation of longitudinal field E, in the channel 
behind the streamer tip, where the electron density is still as high as that 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 50 ===
42 
The streamer-leader process in a long spark 
created by an ionization wave. The current behind the tip is conduction 
current il = 7rrfnencpeEc. By equating this expression to (2.11) and using 
(2.6) and (2.10) with U, = U,, we find 
For a 1 m streamer, the product of logarithms in the denominator of (2.15) is 
close to 100. Therefore, the field in the front end of a streamer channel in 
normal density air is E, M 4.2 kV/cm (E, = 170 kV/cm from section 2.2.2). 
Within the theory accuracy, this value does not contradict the average 
measured channel field of 5kV;cm necessary to support the streamer. 
There is no ionization in such a weak field, therefore electrons are lost in 
attachment and electron-ion recombination processes. 
Current il near the channel end is lower than that of the tip adjacent to 
the channel, because the charge per unit tip length Tt = QJr, is larger than T 
in the channel. This is a typical consequence of end effects for long conduc- 
tors, well-known from electrostatics. The surface charge density at the free 
end of a conductor is much higher than on its lateral surface. In our 
simple model, in which a channel tip has been replaced by a hemisphere of 
radius Y, 
and charge Q, written after formula (2.12), the average charge 
per unit length is T, M 2 7 r ~ ~ [ U ~  
- UO(l)]. 
It is In (l/rm) times larger than T, 
at the channel end (see (2.9)). The tip current i, much exceeds il. This does 
not affect the total charge balance, because the charge Q of a long channel 
is much larger than the tip charge Q,. 
Note that current perturbation in the tip region has a local character. It 
cannot be detected by current registration from the anode side. The streamer 
here makes use of its own resources - the charge of the ‘old’ tip has moved on 
into the gap with the elongating streamer. It is the charge overflow that 
creates current it. If a current detector were placed at the site of a newly 
born portion of the channel, it would register current i M it for a very 
short period of time At = rm/Vs M lop9 s; then the current would decrease 
to il and evolve as the solution of equations (2.13) and (2.14) indicates. 
2.2.4 Gas heating in a streamer channel 
A streamer process is accompanied by current flow and, hence, by Joule heat 
release. As was mentioned above, the viability of a plasma channel depends 
primarily on temperature, so this issue is of principal importance. The initial 
heating of a given gas volume occurs when a streamer tip with its high current 
and field passes through it. As the channel develops, the gas is heated further 
by streamer current flowing through it. Let us evaluate both components of 
released energy. 
The energy released in 1 cm3 per second is j E  = aE2, where j = aE is the 
current density and a is the plasma conductivity in a given site in a given 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 51 ===
A long streamer 
43 
moment of time. The energy released in 1 cm3 as a result of ionization wave 
passage is 
W = aE dt = aE2 dx/Vs 
s 2  s 
(2.16) 
where the integrals are formally taken from --x to +x 
but actually over the 
ionization wave region. The principal contribution to energy release is made 
by a thin layer behind the wave front where the electron density and field are 
high. The integral of (2.16) was found rigorously to be ~ ~ E i / 2 ,  
using 
equations for this wave region [4]. This value has the physical meaning of 
electrical energy density at maximum field. The contribution of the region 
before the wave front is In (nm/no) times, or an order of magnitude, smaller 
than this value. Although the field there is as high as that behind the front, 
the electron density is of the order of n, and the conductivity 0 is 
In (nm/no) times smaller (section 2.2.2). Therefore, 
W x eoEk/2 x 2.6 x 
J/cm3 
(2.17) 
where the numerical value corresponds to E, = 170 kV. 
The fact that the density of energy release in a gas is of the same order of 
magnitude as the energy density of the electric field is quite consistent with 
electricity theory. When a capacitor with capacitance C is charged through 
resistance R to voltage U of a constant voltage supply, half of the work 
QU = CU2 done by the supply is stored by the capacitor as electrical 
energy, and the other half is dissipated due to resistance, irrespective of its 
value. The value of R determines only the characteristic time of the capacitor 
charging, RC. Something like this is valid for the case in question but, of 
course, without both energies being rigorously equal to each other, because 
this situation is much more complicated. Indeed, according to the results 
of section 2.2.2, the tip capacitance is C, = Q / U ,  % 27re0rm, volume 
V, x 4rrLI3, and field E, E Ut/rm, so that the energy dissipation per unit 
tip volume is W FZ CtU:/2Vt z eOEk (we have ignored the unessential 
term Uo(l)). 
Joule heat is released directly in a current carrier gas, or an electron gas. 
Then electrons give off their energy to molecules in collisions. An appreciable 
portion of electron energy (even most of it in a certain range of E I N )  is used 
for the excitation of slowly relaxing vibrations of nitrogen molecules. Some 
energy is used for ionization and electron excitation of molecules, about 
U’ = 100 eV per pair of charged particles produced, i.e., n , ~ ’  = 
J/cm3 at 
n, FZ 1014 cmP3. But even without the account of these ‘losses’, the gas tempera- 
ture rise in the wave front region appears to be negligible: AT < Wlcv = 3 K. 
Here, cv = qkBN = 8.6 x loP4 J/(cm3/K) is the heat capacity of cold air and 
kB is the Boltzmann constant. 
Let us see what subsequent gas heating can provide by the moment it 
is somewhere in the middle of a long streamer channel. We multiply the 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 52 ===
44 
The streamer-leader process in a long spark 
second equation of (2.13) by i and integrate over the whole channel length, 
assuming, for simplicity, that constant voltage U, is applied to the anode. 
After taking a by-part integral in the left side of the equation, we substitute 
ailax from (2.13) and i(1) i, from (2.14), followed by simple transforma- 
tions. As a result: we have 
dt I 
1 
U,i, = 
i RI dx + - - 
dx + [y 
- Ci UIUO(/)] 
Vs 
(2.18) 
So 
which describes the power balance in the system; here, Uo(1) is unperturbed 
potential of the external field at the streamer tip point x = 1. The input power 
Uai, into a discharge gap is used for Joule heat release in the channel (the first 
term on the right), for increasing the electric energy stored in its capacitance 
(the second term), and for the creation of new capacitance due to the channel 
elongation (the third term). Joule heat associated with the ionization wave is 
not represented here. The field burst and the tip impulse current that make up 
W calculated above are absent from equations (2.13) and (2.18). Having 
integrated equality (2.18) over the period of time from the moment of 
channel initiation to the moment t the channel has acquired length I ,  we 
get the equation for the energy balance in the system at the moment t :  
where charge Q is given by (2.12). The energy input into the channel, U,Q, is 
used to create capacity (the last term on the right), partly stored in this 
capacity (the integral) and partly dissipated (&Is). 
The braces ( )t indicate 
the time averaging of the process. In case of a long channel, much of the 
applied voltage drops across its length, so the tip potential U, is small 
most of the time, as compared with average channel potential U,, of about 
U,. Then, the last term in (2.19) can be neglected. 
If we compare the left side of (2.19) with the substituted expression for Q 
from (2.12) and the integral in the right side of (2.19), we can conclude that 
the difference between these values cannot be much smaller than their own 
values but rather have the same order of magnitude. Therefore, the energy 
KdlS dissipated in the channel is equal, in order of magnitude, to the gained 
electrical energy, which is in agreement with a similar situation discussed 
above. 
The average energy dissipated per unit channel length is W,, x CI Uiv/2 
and the average energy contributed per unit channel volume is 
(2.20) 
where rav is the average channel radius. With the formation of every new 
portion of the channel, its radius was approximately proportional to the 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 53 ===
A long streamer 
45 
tip potential owing to the fact that the maximum tip field remained approxi- 
mately constant. So we have Uav/r,, M E,. Substituting this expression and 
(2.8) into (2.20), we find 
One can see that subsequent heating of the channel gas adds little to the initial 
heating by an ionization wave passing through the particular channel site. 
To conclude, gas heating due to streamer development is negligible if the 
gap voltage remains constant. Higher voltage does not change the situation 
because the energy dissipated in the channel grows in proportion with the 
channel cross section and the air volume to be heated. Specific heating 
remains unchanged, since it is determined by a more or less fixed volume 
density of electric energy. 
2.2.5 Electron-molecular reactions and plasma decay in cold air 
Electron loss in cold air is due to attachment to oxygen molecules and 
dissociative recombination. The main attachment mechanism in dry air at 
moderate fields is a three-body process 
0 2  + e  + 0 2  + 0; + 0 2 ,  
cm6/s, 
(2.21) 
k,, = (4.7 - 0.257) x 
y = E / N  x 10'6V.cm2 
where kat is the rate constant at T = 300K. In higher fields, the dominant 
process is dissociative attachment O2 + e -+ 0- + 0 with the rate constant 
-9.42 - 12.717 
-10.21 - 5.7,'~ 
at y < 9 
at 7 > 9. 
log k, = 
(2.22) 
In not excessively high fields of E < 70 kV/cm at 1 atm, air is ionized at the 
rate constant ki = q / N  
logki = -8.31 - 12.7,'~ at y < 26. 
(2.23) 
Since the rate of electron loss through attachment is proportional to electron 
density y1, and that through recombination is proportional to y12, the latter is 
unimportant at the beginning of ionization. The equality ki = k, valid at 
y M 12 determines the minimum field mentioned above, which is necessary 
to initiate the growth of electron density in unperturbed air; Ei E 30 kV at 
p = 1 atm and room temperature. 
Oxygen molecules possessing a lower ionization potential than N2 are 
ionized in fields not much exceeding the ionization threshold. Electrons 
recombine with 0; at the rate constant 0, usually termed a recombination 
coefficient: 
(2.24) 
0; + e  -+ 0 + 0, 
/? M 2.7 x 10-7(300/Te)'/2 cm3/s 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 54 ===
46 
The streamer-leader process in a long spark 
where T, is electron temperature in Kelvin degrees. However, complex ions 
are more effective with respect to electron-ion recombination. The most 
important ions in dry air are 0; ions, while in atmosphere saturated by 
water vapour, as in thunderstorm rain, H30+(H20)3 cluster ions are more 
important. For these, the recombination coefficients 
0; + e  + O2 + 02. 
D = 1.4 x 10-6(300/T,)'/2 cm3/s 
(2.25) 
H30+(H20)3 + e  ---f H + 4H20, 
B 
6.5 x 10-6(300/T)'12 cm3/s 
(2.26) 
Complex 0; ions are formed from simple ions in the conversion 
k = 2.4 x 10-30(300/Te)'/2 cm6/s. (2.27) 
Chains of hydration reactions lead to the production of H30'(H20)3 ions. A 
typical chain looks like this: 
are an order of magnitude larger than for simple ions. 
reaction 
0; + 0 2  + 0 2  --f 0; + 02, 
0; + H20 -+ O;(H20) + 0 2 >  
Oi(H20) + H20 + H30' + O H  + 0 2 ?  
H30+ -t H20 + (M) + H30+(H20) + (M), 
k = 1.5 x lop9 cm3/s 
k = 3.0 x lo-'' cm3/s 
k = 3.1 x 
cm3/s 
H30+(H20) + H20 -t (M) + H30+(H20)2 + (M). 
H3O+(H20)2 -t H20 i- 
(M) -+ H30+(H20)3 -t (M), k = 2.6 x 
k = 2.7 x lop9 cm3/s 
cm3/s 
(2.28) 
(M is any molecule, k correspond top = 1 atm, T = 300 K); here, a hydrated 
ion replaces a 0; ion. 
Another, similar chain begins with the production of an H20+ ion in 
ionization of water molecules by electron impact. Then comes the conversion 
reaction 
H20+ + H 2 0  -+ H30+ + OH, 
k = 1.7 x lop9 cm3/s 
producing an H30+ ion, followed by the reaction chain of the type (2.28). 
0; ion, this is the reaction 
The production of complex ions is accompanied by their decay. For an 
0; + 0 2  + 0; i- 
0 2  -t 0 2 ,  
k = 3.3 x 10p6(300/T)4exp (-504O/T) cm3/s. 
(2.29) 
It is greatly accelerated by gas heating, but in cold air the reaction effect 
is negligible. The same is true of other complex ions, including hydrated ions. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 55 ===
A long streamer 
41 
In a cold streamer channel, simple positive ions turn to complex 
ions very quickly, for the time T,, 
x 10-8-10-7s. It is these ions that 
determine the rate of electron-ion recombination in cold air, except for a 
very short initial stage with t 6 r,,,,. 
If the ionization rate is too low and if the detachment-decay of negative 
ions is slow, as in a cold streamer channel, the plasma decay is described by 
the equation 
2 
3 
= -vane - pn, 
dt 
(2.30) 
where va is electron attachment frequency. Its solution at initial electron 
density equal to the plasma density behind the ionization wave, n,, is 
(2.31) 
where the time is counted from the moment the streamer tip passes through a 
particular point of space. 
Accordingto(2.15), wehaveE x 4.2kV/cmandE/N x 1 . 7 ~  
10-16V/cm 
for a streamer channel just behind the tip at p = 1 atm. The electron 
attachment frequency from (2.21) is va x 1.2 x lo7 sC1 and the characteristic 
attachment time is 7, = v;’ x 0.8 x 
Over this time, most simple 0; 
ions in dry air turn to complex 0; ions. Electrons recombine with them 
with the coefficient ,8 x 2.2 x lop7 cm3/s corresponding to electron tempera- 
ture Te x 1 eV = 1.16 x lo4 K at the above value of E/N. The initial electron 
density n, x 1014 cmP3 is so high that the parameter @nc/va x 2 determining 
the relative contributions of recombination and attachment is larger than 
unity. This means that at an early decay stage with t < ra x lop7 s, electrons 
are lost primarily due to recombination, with attachment playing a lesser role. 
Later, at t > 2ra x 2 x lO-’s, the electron density decreases exponentially, as 
is inherent in attachment, but as if starting from a lower initial value 
nl = nJ(1 + @ n c ~ a )  
x 0.3n,; ne x nl exp (-vat). 
The plasma conductivity decreases by two orders of magnitude, as 
compared with the initial value, over t x 3 x 
s. At the streamer velocity 
V, x lo6 m/s, this occurs at a distance of 30 cm behind the tip. A micro- 
second later, the conductivity drops by six orders of magnitude. The streamer 
plasma in cold humid air decays still faster because of a several times higher 
rate of recombination with hydrated ions and due to the appearance of an 
additional attachment source involving water molecules. These estimations 
indicate a low streamer viability. It is only very fast streamers supported 
by megavolt voltages that are capable of elongating to 1 x 1 m in cold air 
without losing much of their galvanic connection with the original electrode. 
This is supported by experiments with a single streamer and a powerful 
streamer corona [4]. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 56 ===
48 
The streamer-leader process in a long spark 
Note that a streamer plasma has a longer lifetime in inert gases, where 
attachment is absent and recombination is much slower. This makes it 
possible to heat the plasma channel by flowing current for a longer time 
after the streamer bridges the gap (the estimations of section 2.2.4 do not 
extend to these conditions). Such a process sometimes leads to a streamer 
(leader-free) gap breakdown [ 171. Still, the formulation of the streamer 
breakdown problem is justified for hot air and is related to lightning (see 
section 4.8 about dart leader). 
2.2.6 Final streamer length 
When a streamer starts from the smaller electrode (anode) of radius Y,, to 
which high voltage U, >> Elya is applied, it propagates in a rapidly decreasing 
external field. It is first accelerated but then slows down after it leaves the 
region of length Y, where it senses a direct anode influence. If the voltage is 
too low, the streamer may stop in the gap, without reaching the opposite 
electrode (say, a grounded plane placed at a distance d). The higher is U,, 
the longer is the distance the streamer can cover; at a sufficiently high voltage, 
it bridges the gap. In order to estimate the sizes of the streamer zone and 
leader cover in a long spark or lightning - a task important for their 
theory - we need a criterion that would allow estimation of maximum 
streamer length under different propagation conditions. No direct measure- 
ments of this kind have been made for single long streamers in air, because 
there is always a burst of numerous streamers. This, however, is quite 
another matter (see below). So we shall use indirect experimental results 
and invoke physical considerations, theory, and calculations. 
It has been established experimentally that streamers comprising a 
streamer burst are able to cross an interelectrode gap of length d only if the 
relation E,, = U,/d exceeds a certain critical value E,, which varies with the 
kind of gas and its state. Under normal conditions in air, ths critical value 
is E,, e 4.5-5kV in a wide range of d = 0.1-10m. The data spread does 
not exceed the measurement error. Bazelyan and Goryunov [18] recommend 
the value E,, = 4.65 kVjcm for positive streamer, averaged over various 
measurements. Therefore, the voltage necessary for a streamer to bridge a 
gap of length d is U,,, 
= E,,d or more. For example, a gap of 1 m length 
requires about 500 kV (Ec, e 10 kV/cm for negative streamer in air). 
At the moment of crossing a gap, all voltage U, is applied to the 
streamer, so Ea" is also the average field in the streamer. If a gap is long 
enough, E,, can be identified with the average channel field. Indeed, in criti- 
cal conditions with E,, = Ecr, a streamer crosses a gap at its limit parameters. 
It approaches the opposite electrode at its lowest velocity corresponding to 
the minimum excess of the tip potential U, e U ,  over the external potential, 
AUl = U, - Uo(d) = 5-8 kV, below which the streamer practically stops. In 
the case of a grounded electrode, Uo(d) = 0. If a gap is so long (say, 1 m) that 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 57 ===
A long streamer 
49 
AU, = U, << U,, nearly all applied voltage drops in the channel. Therefore, 
E,, can be treated as the lowest field limit, at which a streamer is still capable 
of propagating.? 
This interpretation remains valid when a streamer does not bridge a gap 
but stops somewhere on the way. Indeed, according to (2.6) and (2.10), the 
streamer velocity, and hence its ability to move on, is determined only by 
the tip potential excess over the external potential, A U ,  = U, - Uo(I), and 
is independent of the latter. No matter where a long streamer stops, we 
shall have A U ,  << U,, though the external potential value at this point, 
Uo(lmax), may be high. Generally the average field limit in the channel and 
the streamer length at the moment it stops, I, 
are interrelated as 
In order to be able to use these relations in practice, we must know not only 
the easily registered gap voltage U, but also potential Uo(Imax) inaccessible to 
measurement. In most cases, it is hard to estimate even by calculation. For a 
particular streamer, the external field is determined, in addition to the anode 
charge, by the whole combination of charges that have emerged in the gap 
and its vicinity. Especially important is the charge of all other streamers 
that were formed together with the one under study. Consequently, the 
field Uo(x), in which the streamer is moving, represents a self-consistent 
field. An outburst of hundreds of streamer branches is characteristic of air; 
they fill up a space comparable with l,,,. 
It is this maximum length, rather 
than the small anode radius, that will determine the external field fall 
along the gap length. Estimations of a self-consistent field Uo(x) involve 
considerable difficulties and errors (we shall come back to this when 
evaluating the size of the streamer zone of a leader). So in reality, critical 
field E,, can be evaluated only from experimental data that relate to a 
situation with streamers bridging a discharge gap. Then, the potential 
Uo(I,,,) 
is known reliably because it coincides with the potential of the 
electrode, usually the grounded one: Uo(lmax) = Uo(d) = 0. But if it is 
known that Uo(lmax) << U,, as in the case of a long streamer moving in a 
sharply non-uniform field, the criterion of (2.32) for a definite streamer 
length will be extremely simplified: I, 
x U,/Ecr. 
The existence of critical field E,, has a rather clear physical meaning. The 
reason for the appearance of a minimum average field in a channel is its finite 
t The average quantities E,, and E,, describe, to some extent, the actual field strengths in the 
channel even when the external field is extremely non-uniform, changing by several orders of 
magnitude along the gap far from the conductor. For example, if we close the gap with such a 
thin wire that short-circuiting current does not change the electrode voltage, a short time 
later, after the current along the wire is equalized, the actual gap field will become constant 
along its length and exactly equal to Eav. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 58 ===
50 
The streamer-leader process in a long spark 
resistance. A channel must conduct current necessary to support the motion 
of the streamer tip. This is the current which supplies charge to a new portion 
of the channel produced at its tip front. The nature of the streamer process is 
such that the current il just behind the tip is proportional to its velocity (see 
formulas (2.1 1) and (2.14)). Local field E, necessary to support this current is 
defined by (2.15) derived from the channel resistivity per unit length for a still 
dense plasma. The value of E, in (2.15) slightly depends on varying streamer 
parameters, such as length, tip radius, and velocity, and is largely determined 
by the gap gas composition, which predetermines maximum field E, at the 
tip and the slope of the v,(E) curve (the latter was taken into account in 
(2.15) by the k parameter). The calculated value of E, = 4.2kV/cm for air 
appeared to be surprisingly close to the measured value E,, = 4.65 kV/cm. 
One should not give too much importance to this coincidence of the 
values, but the agreement in the order of magnitude is definitely not 
accidental. 
Because of the plasma decay and conductivity decrease, the current 
support in other channel portions may require a stronger field than E,. 
For this reason, decay processes appreciably affect the value of Ecr, as is 
indicated by experiments. An important mechanism of electron loss in cold 
air in a relatively low field E,, is the attachment in three-body collisions 
(section 2.2.5). Here, the attachment frequency is v, M N 2 ,  so the conven- 
tional similarity principle E N N for field E,, is violated: the reduced field 
E,,/N does not remain constant and the value of E,, decreases more rapidly 
than density N [19]. When a streamer propagates through heated air, the 
critical field becomes lower not only due to a lower density but as a result 
of a direct temperature action. This was found from measurements at various 
p and T ,  up to 900 K [ 19,201. The reason is clear: on gas heating, the action of 
attachment and recombination becomes weaker (section 2.2.5). In electro- 
positive gases, in which there is no attachment, the value of E,, is lower 
than in cold air, other things being equal. For instance, in nonpurified 
nitrogen with an oxygen admixture up to 2%, the field is E,, M 1.5kV/cm 
at p = 1 atm. In inert gases where the attachment is absent and the recombi- 
nation has a lower rate than in molecular gases, E,, is much lower, about 
0.5 kV/cm [21,22]. 
The channel field does not vary much in time along its length because of 
the compensation due to countereffects. On the one hand, the conductivity in 
an old channel portion is lower than in a new one because of the plasma 
decay. On the other, that old portion was produced by a faster ionization 
wave at a higher tip potential corresponding to a larger channel cross section. 
As a result, the resistivity per unit length R1 = ( ~ r ~ e p ~ n , ) - ~  
does not vary 
much along the channel. Of course, it grows in time because of electron 
loss, but at the same time, the streamer velocity decreases together with the 
channel current. For this reason, the time variation of the channel field 
E(x, t )  = RI (x, 
t)i(x. t )  is much slower than that of any of the cofactors. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 59 ===
A long streamer 
51 
We shall illustrate this by giving a particular analytical solution to the 
set of equations (2.13), (2.14), (2.6), (2.10), and (2.31), which is not far 
from the actual result (see below). Assume the channel field and capacitance 
per unit length to be constant, with the current along the channel being the 
same: E(x: t )  = const, Cl(x, t )  = const, and i(x, t )  = i(t). Neglect potential 
Uo(x), 
inessential to a long streamer in a sharply non-uniform field, and 
suppose that the plasma at point x decays exponentially starting from the 
moment t, of its production (as is inherent in attachment without recombi- 
nation). We shall have 
U ( X )  = U, - E X ,  
Ui = U, -El, 
Vs = AU,, 
i = C,V,U, = CIAU: 
where the nearly constant coefficient A is, according to (2.6) and (2. lo), equal 
to 
= const. 
vim 
A =  2(2k - l)Em In (n,/no) 
(2.33) 
The integration of dl = V, dt yields I x lmax[l - exp (-AEt)], I, 
= Ua/E, 
t, = t(1) at x = 1. But the requirement R1 (x, t )  = RI ( t )  involved in the initial 
assumptions can be met only for one value of the channel field: 
E = va/2A = (AUt)min/2Vem~a 
x 1.2kV/cm. Here, V,, 
4 x 105m/s is 
electron drift velocity at maximum tip field E,, 
(Aut),, x 8 kV and 
r, = vi' x 0.85 x lop7 s is the attachment time. The relation for E can be 
interpreted as follows. The potential difference (A U,),,, 
necessary to 
provide the minimum streamer velocity Vsmin x V, 
must be gained in 
field E along the plasma decay length V,,T,. 
A similar treatment of the 
streamer process will be offered in the next section when discussing the 
streamer motion in a uniform field. The order of magnitude of the 'critical' 
field E is correct. Therefore, the assumptions underlying the particular 
solution are not meaningless, so the solution illustrates the main idea. 
The existence of a critical field has been confirmed quantitatively by 
numerical models of the streamer process. Let us discuss the calculations 
obtained from a simple, evident model. We mean the above set of equations 
(2.13) and (2.14) supplemented by expressions (2.6), (2. lo), and (2.33), which 
define the streamer velocity and local channel radius, together with (2.3 1) for 
the plasma decay, in which the time is counted off from the moment t, of its 
production at point x. The streamer development in air from a spherical 
anode of radius ra = 5 cm at U, = 500 kV is demonstrated in figure 2.5.1 
The calculations were made with 7, = 0.85 x 
s and recombination 
coefficient /3 = 2 x 
cm3/s. The general tendency in the behaviour of 
principal parameters is quite consistent with the qualitative picture above. 
t The numerical simulation was made in cooperation with M N Shneider. This type of equation, 
but with a constant channel radius, was solved in [23]. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 60 ===
52 
5M) 
400 
2 300 
s 2 200 
v 
- 
m 
.- 
L 
' E  
0 
100 
0 
The streamer-leader process in a long spark 
2 -  23.37 ns 
3 -  
53.5ns 
4 -  IOOns 
5 -  
310ns 
20 
40 
60 
80 
1 
x ,  cm 
I 
U t- 
4 
0.1 
0 
20 
40 
60 
80 
1 
x ,  cm 
0 
x ,  cm 
Figure 2.5. Streamer propagation in air from a spherical anode of 5cm radius at 
500 kV. The distributions of potential U ,  current I ,  field E, and electron density ne 
along the channel at various moments of time until the streamer stops. 
Note that the nonmonotonic character of some current distributions when 
the potential at a given point x grows with time is associated with a slight 
time decrease in capacitance (2.8) of the elongating channel. The streamer 
acquires its maximum velocity lo7 mjs very soon, over 
s; then it steadily 
decelerates and stops at I, 
= 0.94 m. 
It was found from (2.32), with the account of U,,(lmax) and (AUt),in, 
that the actual average field in the channel at the moment of zero velocity 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 61 ===
A long streamer 
53 
was E,, = 4.9 kV/cm. From a simplified criterion, it was found to be 
E,, = U,/I,,, 
= 5.3 kV/cm. The agreement with the experimental value of 
4.65 kV/cm is quite satisfactory. Calculation with U, = 250 kV yielded 
I, 
= 0.39m and at U, = 750 kV it was 1.42m. Equation (2.32) is satisfied 
at the same E,, = 4.9 kV/cm, i.e., the constant value of the average channel 
field is confirmed by calculations of the final stage of a streamer process 
for various streamer lengths. t When one takes into account only recombina- 
tion, a streamer elongates much more, to 1.25 m; with no account of electron 
losses, it elongates to 3 m  (Ec, = 1.7kV/cm), as the qualitative picture 
suggests. 
2.2.7 
So far, we have dealt with a streamer which starts from a high-voltage elec- 
trode, to which it remains galvanically connected, and is supplied by current 
from a voltage source through the electrode. Such are typical experimental 
designs and, partly, conditions in the streamer zone of a positive leader, in 
which the leader channel and its tip possessing a high positive potential act 
as the electrode. However, a situation may arise when a streamer is initiated 
in the body of a gas gap, where the external field is sufficiently high. This kind 
of streamer develops without a galvanic connection with a high-voltage 
source. Such streamers seem to be present in negative leaders. Note that 
lightning propagating from a cloud down to the earth most often carries a 
negative charge, while that going up from an object on the earth is positive. 
In some situations, a streamer may take its origin from the electrode vicinity 
and begin its travel being connected to it, but later it may break off because of 
the plasma decay in an old channel portion. If the external field is still strong 
at some gap space length, the streamer will move on, having ‘forgotten’ about 
its former connection with the electrode. This behaviour is characteristic of 
the streamer zone of a leader. 
Consider a simple case when there is a uniform electric field Eo at some 
distance from the electrode and a fairly long conductor of length 1 and 
vector Eo along the x-axis. This may be a metallic rod in laboratory 
conditions, or a plane or rocket going up to charged clouds, or a dense 
plasma entity created in this way or other. The conductor is polarized by 
the external field to form a charged dipole. The vectors of the dipole and 
external fields are summed. The total field E,,, 
in the body of a perfect 
conductor drops to zero, since an ideal conductor is always equipotential. 
In the symmetry condition, all its points take the potential of the external 
unperturbed field at the middle point of the conductor. Sometimes, the 
field in the conductor body is said to be pushed out into the external 
Streamer in a uniform field and in the ‘absence’ of electrodes 
t The calculation using a simplified formula without the account of Uo(lmax) at l,, 
= 0.39m 
gives an error: 5.7 instead of 4.9 kV cm. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 62 ===
54 
The streamer-leader process in a long spark 
Figure 2.6. The potential distribution along a conductive rod in a uniform electric 
field. Broken line, the potential in the absence of a conductor. 
space. The dipole charge enhances the field (E,,, > Eo) at the ends of the 
polarized body (figure 2.6). 
The problem of field redistribution by polarization charge can be solved 
rigorously by numerical methods for any geometry, but simplified evalua- 
tions are also possible. In the close vicinity of a charged dipole ‘tip’ of 
radius Y << 1, the longitudinal field varies nearly in the same way as the 
field of a sphere of identical radius. Therefore, the external field perturbation 
by polarization charges is attenuated at a distance Y from each of the two 
conductor ends. Let us take the conductor middle point to be the coordinate 
origin. The end potentials of a polarized conductor differ from that of 
an unperturbed one, U. = -Eox at the same points by AU x Eol/2. The 
absolute strengths of the total field at the conductor ends rise to 
E, x AU/r x Eol/2r, and the field increases with increasing l/r. This 
estimate fits fairly well the numerical evaluation in [24]. 
At 1 >> r, ionization processes and streamers may arise at the ends of a 
polarized conductor even at a relatively low external field Eo (figure 2.7(a)). 
Ionization waves run in both directions, leaving plasma channels behind, in 
much the same way as with a streamer starting from a high-voltage electrode. 
If their velocity V, appreciably exceeds the electron drift velocity, the condi- 
tions of streamer travel from the positive and negative ends will not differ 
much. The total charge of developing streamers is zero at any moment of 
time. This could not be otherwise, because none of them is connected with 
the electrode and, through it, with the high-voltage source. Charges do not 
escape the gap but are only redistributed by the streamer current. A streamer 
Figure 2.7. Excitation of streamers of both signs from the ends of a conductor in a 
uniform field. The charge distributions per unit channel length are shown schemati- 
cally: (a) with active plasma, (b) with plasma decay in the older channel portions. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 63 ===
A long streamer 
55 
travelling along the vector Eo is charged positively, while a counterpropa- 
gating streamer acquires a negative charge. Naturally, the current and the 
charge separation occur due to electron drift. As in a streamer starting 
from an electrode, the work done for plasma production and charge separa- 
tion is done at the expense of the power source creating external field Eo. If it 
is a conventional high-voltage source, current flowing through the streamer 
during its propagation is due to the variation in the charge induced on the 
electrode surfaces when the value and distribution of polarization charges 
in the gap bulk change. This current takes away some of the source power 
which is eventually used for the streamer development. 
As streamers develop, the total length of a polarized conductor 
increases, increasing the potential difference AU pushed out of the plasma 
channels. On the other hand, the plasma in the old, central channel decays, 
so that the charge overflow from one half of the conductor to the other 
becomes more difficult. Finally, a streamer cannot elongate any more, 
because the gain in length at the tip is lost due to the plasma decay at the 
‘tail’. What one observes now is a pair of detached plasma sections of limited 
length going away in both directions. At the front ends, they have a limited 
potential difference AU with respect to the external potential. The process is 
stabilized. It may probably go on until there is the external field. 
We should like to emphasize that this issue is of principal importance. A 
streamer needs the external field for charge redistribution in the created 
plasma, i.e., behind the tip but not in front. The streamer creates its own 
field that contributes to the gas ionization in the tip region, while the channel 
field provided by an external source is necessary to support the current to the 
tip, without which the streamer could not move on. 
When the streamer plasma in the old channel near the starting point has 
decayed completely, two galvanically disconnected streamers continue to 
move in opposite directions. Now the polarization effects of each of the 
conductive sections are added to the earlier polarization effects of the 
whole channel. As a result, there are four charged regions with alternating 
polarity (figure 2.7(b)). Nevertheless, there are still only two ionization 
waves moving only forward, away from the channel centre, trying to elongate 
the streamer. No return ionization waves arise in its central portion which 
has lost conductivity because of a smooth charge distribution towards the 
ends. The charge is ‘smeared’ along more or less extended ‘semiconducting’ 
regions and cannot create a sufficiently strong field to initiate ionization. 
High-voltage engineers are familiar with this phenomenon. They can some- 
times decrease an electric field, alternating in time, at the site of its local 
rise between a sharp metallic edge and an insulator, by coating the dielectric 
at their boundary with semiconducting material. 
The parameters of fast ‘electrodeless’ streamers are described by the 
same formulas (2.6), (2.10), and (2.14). The role of U, - U. here is played 
by the quantity AU x Eo1/2, where 1 is the length of a channel section with 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 64 ===
56 
The streamer-leader process in a long spark 
preserved conductivity. Here, A U also represents the excess of the streamer tip 
potential over the external one. In particular, as the length I increases and the 
field at the conductor ends becomes as high as E, M 150-170kV/cm for 
normal density air (section 2.2.2), the growth of E, ceases. As I and A U  
increase further, the streamer tip radius Y, 
not E,, 
increases because 
A U z E,Y,. 
This is accounted for by the self-regulation mechanism discussed 
in section 2.2.2. 
For the understanding of the streamer process in the streamer zone of 
a leader, where the field is nearly uniform and the leader channel acts as a 
high voltage ‘electrode’, it is essential that the ‘electrode’ and ‘electrodeless’ 
situations should be strictly equivalent, provided that the positive and the 
negative streamers are identical and the external field is uniform. Let us 
mentally cut, at the centre, a plasma conductor developing in both directions 
from this centre and discard, say, the negatively charged half. Let us now 
replace it by a plane anode under zero potential and assume a negative 
potential to be applied to a remote plane cathode. A cathode-directed 
streamer produced at the anode by a local inhomogeneity, whose field 
initially supported ionization, will be identical to a positive streamer in the 
electrodeless case. Indeed, in both cases, the conductor potential U coincides 
with the external potential, U(0) = Uo(0). The charge pumped from the 
negative half into the positive one will now be supplied by the source current 
from the anode. Here, the principle of mirror reflection in a perfectly 
conducting plane, well-known from electrodynamics, reveals itself in every 
detail. According to this principle, the distributions of charge, current and 
field in half-space do not change if the plane is replaced by the mirror 
reflection of half-space charges. 
These considerations were used in the calculations and representation of 
results on streamer development in a uniform field U, = -Eox from the 
point x = 0 towards lower potential (figures 2.8 and 2.9).t The solution 
was derived from the same set of equations (2.13), (2.14), (2.6), (2.10), 
(2.31) and the same plasma decay characteristics as in section 2.2.6. The 
calculations show that a streamer does not develop if the external field is 
lower than a certain minimum value. The values of Eomm do not differ 
much from the critical channel field E,, which determines the streamer 
length in a non-uniform field calculated with (2.32): Eo 
M 7.7 kV/cm. One 
may probably use for estimations the experimental value E,, x 5 kV/cm as a 
realistic Eo 
(section 2.4.1). If the uniform external field slightly exceeds 
the minimum, the excess tip potential is small, the streamer velocity is low, 
and the channel field is close to the unperturbed external field Eomn. This 
situation is illustrated in figure 2.8. 
however, the tip potential is much 
higher than the external potential, and the streamer develops a high velocity 
If Eo is appreciably higher than Eo 
t Numerical simulation was made in cooperation with M N Shneider. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 65 ===
A long streamer 
57 
-3001 
' ' . 
' . ' .  ' ' " 
1 
0 
5 
IO 
15 
20 
25 
30 
35 
40 
x, cm 
x, cm 
x, cm 
'
2
 
0 
0 
5 
10 
15 
20 
25 
30 
35 
x, cm 
0 
Figure 2.8. An air streamer in a uniform field Eo =7.7 kV/cm, slightly exceeding the 
critical minimum, with calculated distributions of potential U ,  current I ,  field E and 
electron density ne. Dashed line, applied field potential counted from the streamer 
origin. The oppositely charged streamer running in the opposite direction is not 
shown. 
(figure 2.9). The current in well-conducting portions at the tip is high but 
decreases towards the channel centre. Owing to the high current, much 
positive charge is pumped into the tip region even at an early stage; in the 
tail, however, where the conductivity has decreased, the current is low. As 
a result, positive charge pumped out of it is not reconstructed; moreover, 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 66 ===
58 
The streamer-leader process in a long spark 
---I_. 
--_ 
5 
-800 
3 - 
482.1 ns 
4 -  576.6ns 
5 -  653.1 ns 
-1000 0 
20 
40 
60 
80 
100 
h 
. 4  
\ 
x, cm 
2.5 7 
x, cm 
x, cm 
x, cm 
Figure 2.9. An air streamer in a uniform field Eo =10 kV/cm with the charge distribu- 
tion 7 instead of the ne curves similar to those in Figure 2.8. 
the tail becomes negatively charged. The calculated charge distributions in 
figure 2.9 were found to be exactly as those represented schematically in 
the right-hand side of figure 2.7(b) in terms of double polarization of the 
whole channel and each of its conducting sections individually. At 
Eo zz Eo min and low current, the polarization effect of the conducting section 
is very weak (figure 2.8). 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 67 ===
The principles of a leader process 
59 
2.3 
The principles of a leader process 
This section is a key one for the understanding of a long spark and the first 
lightning component. We shall try to answer the question why a simple struc- 
tureless plasma channel has no chance to acquire a considerable length in cold 
air of atmospheric pressure. The reader will see what is necessary for a spark to 
become long and have a long lifetime and how Nature realizes this possibility. 
2.3.1 The necessity of gas heating 
Section 2.2 dealt with the development of a simple plasma channel - a 
streamer - which has no additional structural details. The theoretical con- 
siderations concerning the streamer process are, in general, supported by 
experiment, indicating that the streamer gas is cold and the channel field is 
too low for ionization to occur. In these conditions, the plasma produced in 
the tip by an ionization wave decays later. Electrons are lost due to recombi- 
nation (it exists in any gas) and attachment inherent in air as an electronegative 
gas. Losing its conductivity and, hence, the possibility to use current from an 
external source, a streamer eventually stops its development, unless it encoun- 
ters a strong field on its way (section 2.2.7). Sometimes, the streamer lifetime 
can be made longer by a steady voltage rise, but this possibility is, naturally, 
limited. Even at a megavolt voltage of a laboratory generator, an air streamer 
can become only several metres long. Voltages of a few dozens of megavolts 
inducing lightning discharges are, at best, capable of increasing the streamer 
length to several tens of metres but not to the kilometre scale characteristic 
of lightning. At high altitudes, however, the air density is low and a streamer 
may cover a longer distance. This probably accounts for vertical red sprites 
above powerful storm clouds dozens of kilometres above the earth’s surface 
[25], which were found to travel downwards. 
The only way of preventing or, at least, slowing down air plasma decay 
in a low electric field is by increasing the gas temperature in the channel to 
several thousands of Kelvin degrees and, eventually, to 5000-6000 K or 
more. In a hot gas, electron loss through attachment is compensated by 
accelerated detachment reactions, and recombination slows down. The 
mechanism of associative ionization comes into action, and electron 
impact ionization is enhanced because the gas density decreases on heating. 
These processes make it possible for a plasma channel to support itself, or, at 
least, to approach this condition, in a relatively low field. A hot spark looks 
like a hot arc or a glow discharge column after contraction [26]. We shall not 
discuss here the details of these processes (for this, see section 2.5). It suffices 
to take for granted the statement that gas heating does maintain plasma 
conductivity, making the spark viable. 
It follows from section 2.2 that an increase of potential U at the streamer 
front does not contribute to gas heating. Total energy release grows as U2, 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 68 ===
60 
The streamer-leader process in a long spark 
but the streamer cross section 7rrk also increases as U2. So the released energy 
density proportional to ( U / r m ) 2 ,  which defines the heating, remains low. To 
increase the channel temperature considerably, it is necessary to accumulate 
a much higher energy in a much narrower plasma column. For this, the func- 
tional relation providing the low U / r m  ratio must be violated. This is 
impossible in a primary ionization wave but becomes possible in a differently 
organized channel development. Let us try to approach this problem by 
considering the final result and estimate voltages and plasma channel radii, 
at which the gas temperature would become sufficiently high. This can be 
done in terms of the general energy considerations discussed in section 2.2.4. 
Suppose there is a charged space with characteristic size R in the front 
region of a developing plasma channel. Its capacitance is C x moR with 
C1 x x0 
per unit length along the channel axis. If the tip potential is U ,  
the energy dissipated per unit length of a new portion of the system including 
the channel and the space being charged is C1 U2/2, provided the spark devel- 
ops steadily. Since the capacitance per unit length of a new system portion is 
independent of its radius R or of any other geometrical dimension, we are 
free to assume any nature, size, and volume of this charged space. (The 
specific capacitance of the central portion of a long system does vary with 
total length and radius but only logarithmically, as is clear from formula 
(2.8)) The dissipated energy includes all expenditures for the creation of a 
new channel portion and space charge. Attribution of this energy to the 
various expenditures is a special problem which requires details of the 
process to be specified. But we can estimate the upper limit of air mass 
that can be heated to the necessary temperature, say, to T = 5000K. For 
this, let us assume that all energy has been used heating an air column of 
initial radius ro. This will be an estimate of the upper radius limit. This 
temperature will lead to considerable thermal gas expansion, because a hot 
channel, as will be shown later, develops much more slowly than a cold 
streamer channel. Current must have enough time to heat the gas, because 
it is eventually the released Joule heat of current that does the heating. If 
the heating rate is not high enough, pressure in the gas space is equalized, 
so that the gas of a thin channel becomes less dense. The air heat capacity 
does not remain constant within a wide temperature range, so energy calcu- 
lation should be made in terms of specific enthalpy h( T , p ) .  Therefore, the 
expression to a maximum radius rOmax of a cold air column that can be 
heated to temperature T is 
7rrim,,p0h(T) x T & ~ U ~ / ~ .  
(2.34) 
Here, h( T )  is specific enthalpy for air at p = 1 atm and po is its density at 
p = 1 atm and To = 300K. 
With tip potential U, = 1 MV and T = 5000 K, when h(5) = 12 kJ/g, an 
air column that can be heated must have an initial radius less than 
yomax = 0.054cm. The maximum radius due to thermal expansion will be 
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=== PAGE 69 ===
The principles of a leader process 
61 
less than rmax = ro 
[po/p( 5 ) ]  'I2 M 0.26 cm, where p( 5) is the air density at 
T = 5000 K and pressure 1 atm. A channel of this thickness has been observed 
in laboratory spark leaders. At U, M 100 MV, characteristic of very powerful 
lightning, the radius estimated from formula (2.34) must be two orders of 
magnitude larger. A lightning leader, however, has a temperature higher 
than 5000K and h - T2 approximately (h(10) = 48kJ/g), so that the 
radius does not grow as much as U, and remains as small as several centi- 
metres. It may seem surprising, but a leader channel is thinner than a streamer 
channel at the same tip potential (their radii are established due to different 
reasons: a leader radius follows the heating conditions, while a streamer 
radius is such that the lateral field is too low for intensive ionization). 
2.3.2 The necessity of a streamer accompaniment 
The existence of a long spark and lightning are due to two main mechanisms, 
even in the presence of a very high voltage source. One is the mechanism of 
current contraction in a thin channel which can practically be heated. The 
other is the attenuation of a very strong radial field that arises at the lateral sur- 
face of a very thin conducting channel under a very high potential relative to the 
earth. We shall begin with the second mechanism, because it opens the way 
for the first one. In reality, the tremendous value of U / r  M 10-100 MV/cm is 
not the field scale near the channel tip of radius r. Nor does the value of 
U/[rln (Ilr)] l-lOMV/cm, which is somewhat less, determine field E, at 
the lateral surface of a channel of length I behind the tip, as could be 
suggested from formula (2.9) and the Gaussian theorem, E, = 7/(27rq,r). 
This would be valid only for such a simple structureless channel as a 
streamer, but its lateral field cannot maintain a high strength for a long 
time. Lateral ionization expansion would immediately increase the channel 
radius. On the other hand, a channel cannot be heated to the necessary 
high temperature unless its radius is small. This is the reason why a single 
simple channel cannot be heated. 
A long-living spark requiring megavolt voltages will inevitably have a 
complex structure. The reader has, no doubt, guessed that we mean the 
streamer zone in front of a leader tip and its production - the leader cover 
representing a thick charged envelope around the channel (figure 2.2). The 
space charge of a streamer zone and leader cover, having the same sign as 
that of the channel potential, greatly reduces the field at the channel surface. 
Roughly, owing to the field redistribution by space charge, the huge potential 
U now drops across a much longer length R of the streamer zone and the 
charge cover radius, rather than across a length nearly as short as the channel 
radius r. In this case, the field scale is a moderate magnitude U / R  but not 
U / r ,  because even a laboratory spark has R of about a metre long. 
Indeed, the radius of a streamer zone and, hence, of a leader cover is 
defined by the maximum distance streamers may cover when they travel 
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=== PAGE 70 ===
62 
The streamer-leader process in a long spark 
away from the leader tip. We already know (sections 2.2.6 and 2.2.7) that the 
average field necessary for streamer development in atmospheric air must be at 
least E,, x 5 kV/cm. Since streamers stop at the end of the streamer zone, the 
voltage drop along the zone length R is about AU, x E,,R (cf. formula (2.32)). 
About as high voltage drops outside the streamer zone, because the field there, 
i.e., in a zero-charge space, drops from about E,, to zero, as for a solitary 
sphere of radius R. Hence, we have U x 2AUs and R = U/2Ec,. At 
U x 1 MV, the streamer zone radius is R x 1 m, in agreement with laboratory 
measurements. 
It follows from both calculations and measurements that the current, 
field, electron density, and conductivity of a heated leader channel are 
generally comparable with respective parameters of a fast streamer. If they 
are somewhat larger, the difference is not orders of magnitude. So the heating 
time to achieve a much higher gas temperature must be much longer. This 
explains why a leader propagates much more slowly than a fast ionization 
wave. 
The capacitance per unit length of a leader system (the channel plus a 
charged cover) will be described by the same formula (2.8) if I is substituted 
by leader length L and the conducting channel radius Y by cover radius R, the 
actual radius of a charged volume. This follows directly from electrostatics. 
Similarly, the current iL at the leader channel front is related to the tip 
potential U and leader velocity VL by the same expression (2.1 l)t 
(2.35) 
For a laboratory leader of length L x 10m and R M 1 m, the logarithmic 
values are several times smaller, while the linear capacitance is larger than 
in a streamer with Y x lo-’ cm. 
The linear capacitance of a conventionally semispherical streamer zone 
is C1 M 27reo, like the capacitance of a streamer tip. The tip current flowing 
into the streamer zone 
it = 27r&O U v, 
(2.36) 
is by a factor of In (LIR) higher than iL, again like in a streamer. But since the 
leader logarithm is closer to unity, currents iL and it do not differ as much as 
for a streamer. If the current along a leader channel does not vary much, as in a 
fairly short leader at constant voltage, the tip current will not differ much from 
experimental current i in the external circuit. A typical laboratory leader 
has i x iL x it M 1 A, U x 1 MV, and from (2.36) VL M 2x lo4 m/s whch is 
close to numerous measurements, in which VL x (1-2.2) x lo4 mjs [27,28]. 
Formula (2.36) or (2.35) permits the estimation of any of the three parameters 
t Here and below, the external field potential Uo(x) 
is omitted for brevity. It is indeed small in 
laboratory leaders normally observed in a sharply non-uniform field. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 71 ===
The principles of a leader process 
63 
- i, U ,  or VL - from the other two. This is especially useful in studying 
lightning leaders when actual data are very scarce. 
Thus, a key condition for a long-term spark development is the formation 
of a thick space-charge cover around it, having the same sign as the channel 
potential. The charge reduces the field on the channel surface, depriving the 
channel of its ability to expand due to ionization. It is only a channel with a 
small cross section that can preserve the ability to be heated. A charge cover 
also contributes somewhat to the linear leader capacitance, because it is 
now determined by the much larger cover radius R rather than by the small 
channel radius r. An increase in linear capacitance is accompanied by an 
increase in the energy input into the channel. 
If Nature were a living being and decided to make a spark or lightning 
travel as large a distance as possible, it would do this by organizing the 
streamer zone and charge cover. In actual reality, everything happens 
automatically: the huge voltages that create long sparks produce numerous 
streamers at the front end (figure 2.10). 
This reminds us of a high voltage electrode creating, under suitable 
conditions, a multiplicity of streamer corona elements. This kind of corona 
can be registered in laboratory experiments. 
Currents of all streamers starting from a leader tip are summed up, 
heating the spark channel. This total current charges the region in front of 
the tip, neutralizing the charge of the old tip, and when a new tip is 
formed, the spark elongates by a length of about the tip length, as in a 
Figure 2.10. Photograph of a positive leader in a rod-plane gap of 9 m length at 2 MV; 
the electronic shutter was closed at the moment of contact of the streamer zone and 
the plane. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 72 ===
64 
The streamer-leader process in a long spark 
single streamer. Part of the streamer zone appears to be behind the tip, 
transforming to a new cover for the newly born leader portion. But this 
does not decrease the streamer zone length, because meanwhile the zone 
has moved forwards together with the tip. Note only that if there are 
many streamers they are very close to one another, and they travel in a 
self-consistent field close to the critical field (sections 2.2.6 and 2.2.7). Such 
streamers are slow and have a low current [4], so that the leader current is, 
indeed, a sum of numerous low streamer currents. 
2.3.3 Channel contraction mechanism 
The mechanism of current contraction in the front region of a leader channel 
is not quite clear, especially quantitatively. One may assume the existence of 
ionization-thermal instability. This effect looks like the one leading to glow 
discharge contraction [26], but it has its own specificity [4]. The instability is 
associated with the dependence of electron impact ionization frequency on 
field and molecular number density: v,(E, N )  = N f ( E / N ) ,  where f ( E / N )  
is a rapidly rising function at small E / N  (figure 2.4). This is the ionization 
component of the instability. Its thermal component is due to the fact that 
the gas pressure p rapidly equalizes in small volumes at a moderate heating 
rate. With p N NT = const, a more heated site proves less dense, and the 
reduced field E/ N ,  determining the ionization frequency, increases there. 
As was mentioned above, numerous streamers start from the front end 
of a developing leader. The frequency of streamer emission has been shown 
experimentally to exceed lo9 sC1 at a typical laboratory spark current of 1 A 
[29]. Younger streamers have not lost their conductivity yet. The streamers at 
the leader tip are so close to each other that they form a continuous conduct- 
ing channel of radius rSum. Current it flows along ths and the initial leader 
channel. It is external current relative to the tip, because it is created by the 
whole combination of charges exposed and displaced by the streamer zone 
bulk. This current is practically independent of the tip conductivity. In 
terms of electric circuit theory, the streamer zone acts as a current source 
(an electric power generator with an inner resistance R + m) relative to the 
leader tip. Its actual value is very large: R M AU,/i, M U/2it, where AU, is 
the voltage drop across the streamer zone. At U M 1 MV and i x 1 A, the 
value is R 
ZZ 0.5MR. No matter what happens to the leader tip or its short 
front portion, the current there does not change. What changes is the electric 
field, because it depends on the conductivity and radius of the region loaded by 
current (in a glow discharge, the field is fixed and the current can vary during 
the instability development). 
Suppose the current density, whose average cross section value is 
j = it(7rr;,J1, 
has increased, for some reason or other, in a thin current 
column of radius ro << r,,,. 
Then the released energy density j E  and gas 
temperature T will also increase. The gas density N will become smaller 
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=== PAGE 73 ===
The principles of a leader process 
65 
and EIN larger due to thermal expansion. As the ionization frequency is a 
steep function of reduced field, it will grow much faster than E / N .  So, the 
electron density ne and conductivity a N n,/N will rise. As a result of this 
long chain of cause-effect relationships, the current density j = aE in the 
fluctuation region will become still larger, etc. The process may begin with 
any link in the chain. In any case, the current density in a particular fluctua- 
tion region will be rising without limit until all current it accumulates there. 
At the initial stage of instability development, the perturbed current density 
does not exceed much an average value. But as current concentrates within a 
small cross section mi, the gas heating rate there rises sharply. The instability 
now develops very quickly, acquiring an explosion-like character. The 
acceleration effect is manifested better in a thinner column with high density 
current. 
A perturbation region, however, cannot be infinitely thin, and this sets 
a limit to the rate of instability development. The matter is that non- 
uniformities of electron density ne are dispersed by diffusion, which is 
ambipolar at very high density values. The characteristic time for perturba- 
tion dispersion is Tamb = ri/4Da, where D, = p+Te is an ambipolar diffusion 
coefficient (p+ is ion mobility and T, is electron temperature in volts). In 
addition to charge diffusion, non-uniformity dispersion is due to heat 
conduction with a characteristic time Tth = r i / 4 ~ ,  
where K is thermal diffusiv- 
ity. The former mechanism appears to be more effective in initial air plasma, 
since D, x 4 cm2/s (p+ x 2 cm2/s 
e V, T, GZ 2 eV) is an order of magnitude 
larger than K x 0.3 cm2/s. If a non-uniformity takes less time for dispersion 
than for development, i.e., if Tamb is smaller than the instability lifetime qns, 
the latter is suppressed at its origin. 
The scale for qns is the characteristic time of, say, gas temperature 
doubling in a perturbed plasma column, as compared with initial tempera- 
ture To. This time is pocpTo/jE, where j E  is the power of Joule heat release 
and cp is specific heat at constant pressure with the account of thermal 
expansion. But this is not all. The higher the instability development rate 
is, the greater is the steepness of the ionization frequency dependence on 
reduced field E / N  - ET, i.e., on gas temperature. For instance, if a 10% 
increase in T raises the ionization rate by 20%, the instability will, generally, 
double its rate, as compared with a 10% increase in the ionization rate. This 
circumstance brings the factor Ci 
d In vild In ( E / N )  into the theoretical 
formula for qns [26], which characterizes the q ( E / N )  function steepness. 
This yields the following expression to be used for estimations: 
(2.37) 
For calculations, we shall take laboratory leader current i x 1 A and conductiv- 
ity a x lop2 (R cm)-’ corresponding to the electron density ne GZ 1014 cmp3 of 
air ionization by a streamer zone; Ci = 2.5. Suppose the current density in a 
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=== PAGE 74 ===
66 
perturbation region is j x 40A/cm2, i.e., somewhat higher than the average 
value of 30A/cm2 along a channel with the initial radius taken to be 
rsum = 0.1 cm. We shall obtain T,,~ 
x 1O--6 s. From the condition Ta,b 
2 -qTins, 
under which the instability has a chance to develop further, we find that the 
initial radius of a column with accumulated leader current must exceed 
ro min M 3 x lO-3 cm. Taking into account the upper limit ro max x 
5 x 1O-2 cm derived from energy considerations, we conclude that a probable 
leader radius prior to thermal expansion is about ro - lop2 cm. For details, the 
reader is referred to [4], but reservation should be made concerning the result 
accuracy, which cannot be too high in the present state of the art. 
2.3.4 
Leader velocity 
Streamers generated at the leader channel front cover a distance of several 
metres and stop. As was mentioned in section 2.3.2, such streamers are 
weak and their propagation is slow; their velocity is close to its low limit 
of V, M 105m/s, which means that their lifetime is R/Vs x lO-5 s. This 
time is so long that the streamer plasma decays considerably. Only young 
streamers, whose lifetime is about the electron attachment time ra x lO-’s 
(section 2.2.5), can preserve good conductivity. A young streamer length is 
It M Vs7a M 1 cm. A dense fan of such plasma conductors starts from the 
channel front. It is this young streamer fan that seems to be registered in 
photographs as a bright spot with a radius of r, - 1 cm in order of magnitude 
(figure 2.11) and is generally considered as a leader tip. 
The streamer-leader process in a long spark 
Figure 2.11. An instantaneous photograph (0.1 ps exposure) of the tip region of a 
leader. 
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=== PAGE 75 ===
The streamer zone and cover 
67 
This suggestion is supported by the fact that the radius of a leader 
travelling through air pre-heated to 900K is r, RZ lOcm [20]. Indeed, the 
plasma decay slows down and the values of r, and It become higher. 
Thus, a necessary condition for leader propagation is the tip region 
contraction to a very small radius. This results from instability development, 
taking a time of about qns. Over this time, all short young streamers 
supplying the leader with current transform to the leader channel. Therefore, 
over the time of the process providing a steady propagation of the leader tip, 
the latter must cover a distance of about its size, i.e., a young streamer length. 
Only in this case can a new front region be formed to replace the old one. 
Hence, the leader velocity can be evaluated from the respective parameters as 
In the absence of attachment or if its rate is low, the role of 7, is performed by 
the time of another plasma decay process - recombination. The evaluation 
with (2.36) gives a correct order of magnitude for the velocity of laboratory 
leaders: at rins 
N lop6 s and It - 1 cm, we obtain V, N 104m/s. We should 
like to note that these qualitative and, probably, questionable considerations 
have not yet been substantiated by a more rigorous treatment. 
Some of the above problems of the leader process will be discussed in 
more detail in the subsequent sections of this chapter and further. Here, 
our aim was only to give a general idea of the propagation of a long spark 
and, presumably, of the first lightning component. A reader interested 
exclusively in lightning hazards may find this information sufficient. 
2.4 
The streamer zone and cover 
We have shown above that a streamer zone plays the key role in a leader 
process. It is here that a space charge cover is formed which stabilizes the 
leader channel, preventing its ionization expansion which would otherwise 
exclude plasma heating. A streamer zone is the site of current generation 
for heating the leader, providing its long life. In this section, we shall deal, 
in some detail, with processes occurring in the streamer zone and leader 
cover, defining the priorities in the causative relationships among leader 
parameters. We shall show how the process of streamer generation from a 
leader tip becomes automatic. 
2.4.1 
The tip of a long leader possesses a very high potential: U, N 1 MV for 
laboratory sparks and -10 MV or, probably, more for lightning. Streamers 
are continuously produced in a leader tip, which means that the field at its 
surface Et exceeds the ionization threshold Ei z 30 kV/cm (under normal 
Charge and field in a streamer zone 
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=== PAGE 76 ===
68 
The streamer-leader process in a long spark 
conditions). This excess cannot be very large, otherwise a streamer flux would 
become too intensive. The excessive charge of the same sign as U, introduced 
into the space would create a much stronger reverse field which would reduce 
Et to a level close to E,. Therefore, the field E, does not exceed much E, and 
has the same order of magnitude. An automatic field stabilization is inherent 
in any continuous threshold process of charge generation by an electrode, for 
example, in a steady-state corona. Measurements have shown that the field 
near a corona-forming electrode is stabilized with high accuracy and does 
not respond to voltage rise across the gap; what changes is the corona 
intensity, i.e., its current. A leader tip, too, is the site of corona formation, 
with an intensity high enough to support a quasi-stationary state in the tip 
and streamer zone, corresponding to potential U,. The field at the corona 
electrode E,, is shown by stationary corona experiments to be by a factor 
of 1.5 higher than E,, if the electrode radius is about the leader tip radius 
r -  lcm. 
At E, = E,, = 50 kV/cm and Y, = 1 cm, the leader tip charge q, = 
4mO~:Et 
% 5 x lO-'C 
is capable of creating only a small portion of 
E,r, = 50kV of an actually megavolt potential U,. The main potential 
source is, therefore, the space charge of the streamer zone and cover 
surrounding the tip. But the value of U, is primarily determined by character- 
istics external relative to the tip. This is the electrode (anode) potential minus 
the voltage drop across the leader channel. Consequently, the charge Q, and 
the size R of a streamer zone, as well as respective cover parameters, are 
established such that they correspond to the proper potential U,. The 
mechanism by which a leader 'chooses' the values of Q, and R are directly 
related to streamer properties. There are many streamers present in the 
zone at every moment of time. They are emitted by the tip at a high frequency 
(see below), have different lengths at any given moment and are at different 
stages of evolution, with their charges filling up the zone space. Every single 
streamer moves in a self-consistent field created by the whole combination of 
streamers. The contribution of the leader tip itself (or of its channel) to the 
total field has just been shown to be small. One exception is the region 
around the tip with a size of its radius. 
There are experimental and theoretical grounds to believe that the field 
strength in the streamer zone, except for the tip vicinity, is more or less 
constant and close to the minimum at which streamers can grow. This is 
indicated by measurements of streamer velocity, which does not vary along 
the streamer zone. (Attempts to measure a single streamer in the tip 
region, where the streamer density is high, have so far failed.) Experiments 
show that until the streamer zone of a laboratory leader touches the opposite 
electrode, streamers move slowly, at a nearly limit velocity of about lo5 mjs. 
This is possible only in a uniform field close to Eo mn (section 2.2.7). Streamers 
can travel for such a distance R, at whch the field Eo mn still exists, but they 
stop on entering the region with E < E,, ml,. 
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=== PAGE 77 ===
The streamer zone and cover 
69 
Suppose, for simplicity, that the streamer zone is a hemisphere with the 
centre in the leader tip. The hemisphere changes to a cylindrical cover of the 
same radius R with the same order of the space-charge density. A thin con- 
ducting leader channel goes along the cylinder axis as far as the hemisphere 
centre. When evaluating the zone parameters, one should take into account 
the cover charge at the leader end, which also affects the zone field. We can 
do this simply by connecting the hemisphere, simulating a streamer zone, to 
another hemisphere by mentally cutting it out of the cover space. Let us 
assume that there is a uniform radial field E = Eomi, in the sphere. As was 
mentioned in section 2.2.7, the theoretical limit of Eo 
is close to the experi- 
mental critical average field in the streamer channel, below which a streamer 
cannot propagate. For air, therefore, we have E x E,, x 5 kV/cm. A uni- 
form field in sphere geometry corresponds to the space charge density 
p = 2 ~ & / r .  If the leader tip is far from the earth and grounded electrodes, 
its potential is 
The sphere charge Q and its surface potential UR are 
(2.39) 
(2.40) 
U R  = U, - E R  = Ut/2. 
For example, for Ut = 1.5 MV, we have R = l S m ,  the charge of a hemi- 
spherical streamer zone equal to Q, = Q/2 = 6.2 x 
c. The leader tip 
charge qt = Q(E,/E)(r,/R)2 - lOP3Q is indeed negligible, as compared 
with the zone charge. Its physical role, however, is very important: the 
high field it creates near the tip, Et > Ei >> E,,, is capable of generating 
streamers. 
As the streamer zone approaches the grounded plane, its length 
increases because its boundary potential UR decreases under the action of 
charge of opposite polarity induced in the earth. Now it is most of the voltage 
U,, rather than its half, which drops across the streamer space. At the 
moment of streamer contact with the 'earth', potential UR = 0 and the 
zone length L, = U,/E is doubled relative to the value of R from formula 
(2.39). This is clearly seen in streak pictures of a laboratory spark (figure 
2.12). It is at the moment of streamer contact with a grounded plane that 
the critical field E,, x 5 kV/cm was registered experimentally. The measure- 
ments make sense only for short (compared with the interelectrode distance) 
leaders, when the voltage drop across the channel could be neglected. By 
equating the potentials of the anode U, and of the tip, we can write: 
E,, = U,/L, 
U,/L,. 
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=== PAGE 78 ===
70 
The streamer-leader process in a long spark 
Figure 2.12. Streak photograph of the positive leader channel bottom in an air rod- 
plane gap of 12 m in length. The streamer zone is seen to elongate when approaching 
the plane cathode. 
2.4.2 Streamer frequency and number 
The number of streamers present in a streamer zone at every moment of time 
is N, = Q,/q,, where q, is the average charge of a streamer. Both charges 
were measured experimentally [29,30], the first from the integral of con- 
duction current through the anode for short leaders with as yet small cover 
charge and the second from the integral of current through a cathode 
measurement cell with such a small radius that only one streamer could 
touch its surface (with good luck). After a successful contact, the charge of 
a conductive streamer section flew into the cathode and through an integrat- 
ing circuit. The charge averaged over many registrations was found to be 
q, = 5 x lo-’’ C. For the illustration mentioned in the previous section of 
this chapter, we find that the number of streamers in a streamer zone of 
length 1.5m and charge Q, = 6.2 x lO-5 C is N, M 1.2 x lo5. Similar data 
for qs can be derived from the calculations presented in figure 2.8 or from 
a simple theoretical treatment, we shall just perform. 
A streamer produced by a leader tip crosses the streamer zone over the 
time t, = R/ V, min M lop5 s, which is by two orders of magnitude larger than 
the attachment time 7, M lO-’s. Therefore, all electrons are lost from older 
streamer portions comparable in length with R. Conductivity is preserved 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 79 ===
The streamer zone and cover 
71 
only along the length I, - V, m i n ~ a  - 1 cm behind the streamer tip. During 
the motion of the streamer tip, the charge of the older streamer portions 
flows into the new ones located closer to the tip before conductivity turns 
to zero. Under steady-state conditions, when the tip goes far away from 
the start, the charged portion of length I, moves together with the tip, 
supporting by its charge the minimum excess of tip potential over external 
potential, (AU,),, 
x 5 kV, necessary for the streamer propagation (section 
2.2.7). The conductor of length I, and radius r, M 0.1 cm carries the charge 
(2.41) 
As for the charge accumulated in the streamer tip 
qst z 2 r ~ O ( A U ! ) ~ i ~ r ~  
M mO(AU,)iin/Em M 5 x lo-" C, 
which was not taken into account in the calculation or (2.39), it is by one 
order less than that distributed along the channel. At N, M lo5, the average 
interstreamer space is R/N;I3, i.e., about several centimetres. With this large 
separation of streamer tips, the streamers can really be considered solitary 
and propagating in an average self-consistent field. 
When a streamer reaches the end of the streamer zone, it stops because it 
enters a field lower than Eo ,in. Since the streamer zone approaches this field 
fairly slowly, at leader velocity VL an order of magnitude lower than 
streamer velocity V, 
(for laboratory streamers), the streamer loses its 
conductivity entirely. The ions of its space charge are gradually repelled 
(and diffuse), reducing the field near the charge trace, so that the streamer 
trace becomes lifeless 'forever'. The streamer zone still passes by for a time 
tL = R/ V, N lop4 s, after which the immobile charged trace, which is now 
behind the leader tip, becomes a cover component. Viable streamers fly 
across the streamer zone over time t, = R/Vsmin - lop5 s, an order of 
magnitude shorter. Therefore, if the frequency of streamer production is 
v,, the number of viable streamers in a streamer zone is N I  - v,t,, while 
the number of non-viable traces, practically coinciding with the total 
number of charged streamer portions, is N - v,tL. Hence, the streamer 
generation frequency is v, - N / t L  - N V L / R  - lo9 s-'. 
2.4.3 Leader tip current 
The streamer production frequency is directly related to the leader tip current: 
it = qp,. With 9, = lop9 C and v, = lo9 s-l, we get it x 1 A, a value typical 
for laboratory leaders in the initial stage while the streamer zone has not yet 
reached the cathode. T h s  current value has been registered in many experi- 
ments [27-291, and the relation v, = iL/qs has been confirmed by direct 
measurements. The streamers were counted by piece from current impulses 
in small cathode measurement cells after the streamer zone boundary had 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 80 ===
72 
The streames-leader process in a long spark 
touched its surface [29]. The counts were integrated over the area. Measure- 
ments 
made 
at 
different 
currents 
showed 
that 
the 
relation 
iL/v, = qs RZ 5 x 10-”C remained constant. It is consistent with measure- 
ments and the estimations of average streamer charge presented here. 
The formula for leader tip current can be given in a conventional form of 
the type i = 7rr eneve = T ~ V ~ ,  
when current is expressed as the number of 
charge carriers (electrons) per unit current column length T, and electron 
velocity V,. This formula can also be changed to the phenomenological 
expression (2.36) describing the result of the current process without 
indicating the nature of carriers. Although the current carriers in this case 
are electrons, we can also speak of ‘macroscopic’ carrier-moving charged 
streamer sections. With what we mentioned at the end of section 2.3.2 and 
formula (2.36), we have 
2 
(2.42) 
where T, = Q J R  RZ m O U t  is the linear charge in a streamer zone and 
T~ = q,N1/R is the linear charge of ‘macroscopic’ carriers. The three 
expressions for current are equivalent to one another but reflect different 
aspects of the current process. The second expression in the chain of 
equalities (2.42) indicates the current origin whle the latter shows the actual 
process of charge transport; the penultimate expression is phenomenological 
and describes the result of travel of the streamer zone as a whole. 
The mechanism of leader current production just described is valid until 
a streamer zone touches the electrode of opposite polarity (the earth or a 
grounded object in the case of lightning). Then the situation changes 
radically. In the final jump, the charges of all streamers ‘hitting’ the electrode 
leave the gap through its surface. Nonviable streamers are no longer 
produced, and they possess an ever-decreasing portion of total streamer 
zone charge. The last expression in (2.42) is valid in this case, too, 
it = T~ V, mln, but the value of T~ is no longer equal to the portion VL/ V, mln 
of the total streamer zone charge Q,. In the limit, when the leader tip reaches 
the electrode, all zone charge will be provided by moving streamers, so the 
current will be it = T,V,. The linear charge of the streamer zone in the final 
jump remains the same in the order of magnitude as in the initial stage. Of 
course, the streamer zone has now a different shape - it is elongated, looking 
more like a cylinder than a hemisphere. But still, its longitudinal L, and 
transverse R, dimensions are comparable, and the value of ln(L,/R,) 
which appears in the denominator of the expression for linear capacitance 
in cylindrical geometry (2.8) and (2.35) is about unity. Therefore, leader cur- 
rents after the transition to the final jump and before it are related as Vs/ VL, 
the ratio being above 10. 
Moreover, the streamer velocity rapidly increases as the streamer zone is 
reduced. The potential difference between the leader tip and the grounded 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 81 ===
The streamer zone and cover 
1 3  
electrode remains the same, U,, whereas the length L, becomes shorter. The 
average field in the streamer zone E = U,/L, rises, together with streamer 
velocity (section 2.2.7) and leader current it N V,. The final jump current 
was shown by laboratory leader experiments to rise by a factor of tens or 
hundreds, from about 1 A to 102-103 A. This fast current rise lasting for 
several microseconds is a prelude to a still higher current of the return 
stroke. The latter begins when the leader channel reaches the electrode. 
The current rise of the final jump is stimulated by the fact that fast streamers 
cross a shorter streamer zone much faster than before, so that the zone 
plasma is unable to decay as much, preserving the streamer conductivity. 
At the end of the final jump, the leader channel appears to be linked to the 
opposite electrode by numerous streamer filaments with current (for details, 
see [4]). 
2.4.4 Ionization processes in the cover 
A leader cover contains a large number of non-viable charged traces of 
earlier streamers. They were produced and developed when a streamer 
zone was passing through this site. The axial field in the cover is very low, 
much lower than in the streamer zone. No ionization can occur in it, so it 
is of no interest to us. What is important is the radial field created by the 
leader surface charge and all cover charges, as in a streamer zone, the only 
difference being in geometry. In contrast to a channel cover, however, a 
leader tip with a streamer zone is formed as a self-consistent system from 
the very beginning. The tip pumps into the zone as much charge as necessary 
to maintain the tip field at the level Ei, providing the production of the 
necessary number of streamers. A leader channel ‘inherits’ a ready-made 
cover. The charge amount and distribution in the former quasi-spherical 
zone is unsuitable for the cylindrical geometry and channel potentials U ( x )  
different from U,. But the inherited charge of dead streamer traces is 
invariable, which means that there must be a mechanism to make the 
channel-cover system self-consistent and controllable, since the potential 
distribution U ( x )  and linear leader capacitance vary in time. 
We mentioned in section 2.3.2 that the ‘intrinsic’ charge of a conducting 
channel of length L and radius rL x 0.1 cm would create at its surface a huge 
radial 
field 
Er, = U / [ r L  In ( L / r L ) ]  = 1 MV/cm 
at 
channel 
potential 
U x 1 MV. This critical situation is unfeasible owing to the presence of a 
cover. The cover charge induces in the conductor an opposite charge which 
is to be subtracted from the intrinsic surface charge. As a result, the field E, 
created by the resultant charge has a moderate value. It is hard to imagine, 
however, that the inherited charge will be as large as is necessary for confining 
the resultant surface field in the narrow range -Eln < E, < Ein with 
Ei, = 50 kV/cm << El, x 1 MV/cm. No doubt, the cover charge will turn out 
to be either too large or too small. In the first case, the channel will be charged 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 82 ===
I4 
The streamer-leader process in a long spark 
negatively (if the leader is positive), the field around it will become negative, its 
module exceeding Ei,. A negative corona will be excited and introduce into the 
cover as much charge as is necessary to reduce IE,I to Ein. This situation 
becomes feasible when the gap voltage is constant or decreases, since then 
the channel ‘enters’ the cover with a too-large charge. Indeed, suppose the 
charge of a streamer hemisphere of radius R is distributed as p = 2 ~ ~ E / r  
(section 2.4.1), creating potential U, = 2ER (2.39) in its centre. The cover 
inherits a charge of the same radial density distribution p = 2 ~ ~ E / r  
= 
~ E ~ U , / R ~  
and amount r’ = 27reoUt per unit length. At point x far enough 
from the channel ends, L >> R, this charge will create potential 
2 ~ r p  
dr dz 
r’ 
L e  
L e  
1,2 M -In- 
= U, In- R 
(2.43) 
1 
U’(x) 
= - 
5” J’ 
YTEO 
o o [r2+ ( z  - x) ] 
2
~
~
0
 
R 
where e is the natural logarithmic base. The potential U’ is by the logarithmic 
factor larger than the actual value of U,. The excess cover charge which has 
created excessive potential is U’ - U, and must be compensated by intro- 
ducing a charge of opposite sign. 
The other situation, when the inherited charge is too small, is usually 
feasible if the gap voltage rises appreciably during the leader evolution. 
The channel potential U ( x )  at a given point increases, and the cover must 
be charged up. 
It is this mechanism of direct or reverse corona display by a ‘wire’, such 
as a leader channel, which leads to a self-consistent channel-cover system. 
The system is controlled and corrected automatically but not very quickly. 
It is sensitive to the slightest variation in potential distribution along the 
channel due to the field Ei, being too small compared with El., of the ‘intrin- 
sic’ channel charge. A slight effect on the cover is sufficient to change the field 
value, and even its direction, at the channel surface. The cover of a develop- 
ing leader with a reverse corona acquires a double-layer structure: outside is 
the charge inherited from the streamer zone and inside is the new charge of 
opposite sign, introduced by the corona. For example, at U, = 1.5 MV, 
R = 1.6m, and L = 10m, the linear cover capacitance from (2.43) is 
C1 = r’/U’ = 2 x lo-” Fjm. It is defined by the same formula (2.8) but 
with the effective radius Reff varying with the radial charge distribution; for 
p M 1/r,  re^ = R /  e and for p(r) = const,  re^ = R/eli2, etc. With U = U,, 
the corrected steady state charge in the cover is rL = C1 U = 3 x lo-’ Cjm. 
Since the values of C1 and rL gradually decrease with the leader length, 
the surface field must support a negative corona; hence, E, M -50 kV/cm. 
The actual channel charge (intrinsic charge minus induced charge) is found 
to be rLc = ~ T E ~ E ,  
= -8.2 x IO-’ Cjm << rL. Therefore, it is easy to control 
the value of rLc and even to reverse its sign. 
We have deliberately considered the mechanism of cover-leader self- 
regulation in so much detail, because a reverse corona neutralizes the 
cover charge in a laboratory spark and lightning during the return stroke, 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 83 ===
A long leader channel 
75 
when the channel potential becomes equal to the earth’s zero potential (see 
section 4.4). 
2.5 
A long leader channel 
All ionization processes responsible for the leader development are localized in 
the streamer zone, leader tip and a short channel section behind it. In the latter, 
gas heating is completed and a quasi-stationary state characteristic of a long 
spark is established. In t h s  sense, the rest of the channel plays a minor role, 
simply connecting the operating part of the leader to a high-voltage source. 
High potential and current vital to the ionization and energy supply are 
transmitted through the channel. But how much voltage reaches the leader 
tip depends on the channel conductivity which, in turn, is determined by the 
channel state. For this reason, what is going on in a developing channel is 
as important to the leader process as the mechanisms described above. 
2.5.1 
There are no direct experimental data on the state of a lightning leader 
channel. Therefore, of special value is the information derived from laboratory 
spark experiments, since it can serve as a starting point in lightning treatments. 
Here we present some values derived from experimental data [27] with a 
minimum number of assumptions. Streak photographs were taken continu- 
ously of a leader propagating from a rod anode to a grounded plane. Pulses 
of voltage U, with the microsecond risetime were applied to gaps of various 
length d. By measuring the streamer zone length L, in the photographs at 
the moment the zone touched the grounded electrode and assuming the aver- 
age zone field to be E,, = 4.65 kV/cm (section 2.4), one can find the leader tip 
potential U, = E,,L, and evaluate the average field in the leader channel as 
EL = (Uo - U,)/L, where L = d - L, is the channel length (table 2.1). 
The accuracy of EL evaluation, however, is not high, because one 
calculates a small difference between large values. Besides, measurements 
of the streamer zone length at the moment of contact with the cathode 
contain errors as large as those of the channel length. When determining 
the latter from streak pictures, one can hardly take into account all channel 
Field and the plasma state 
Table 2.1. Leader parameter derived from experimental data. 
5 
1.3 
2.3 
2.7 
1.1 
750 
10 
1.9 
3.2 
6.8 
1.5 
590 
15 
2.2 
3.6 
11.4 
1.7 
440 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 84 ===
76 
The streamer-leader process in a long spark 
bending, which may increase the length by 20-30%. Finally, the accuracy of 
E,, is not as high as is necessary for such a delicate operation. Experimental 
researchers know that the streamer zone field varies with air pressure, 
humidity, and temperature but they do not know the respective corrections. 
Nevertheless, the data in table 2.1 demonstrate a decrease in the average field 
with increasing channel length. This is also evident from experiments with 
superlong sparks. A voltage of 3-5 MV is sufficient to create a spark 100 m 
long or longer. The tip potential necessary for the development of a streamer 
zone of several metres in length is 1-2 MV, as is clear from table 2.1, there- 
fore the average field in such a long channel will be as low as 200-250 V/cm. 
These values are more applicable to older, remote channel sections which 
have acquired a quasi-stationary state, but the fields close to the tip are much 
higher. This follows from many experiments indicating a regular increase in 
average field with decreasing leader length. More explicitly this was shown 
by supershort spark experiments, when the channel length was only a few 
dozens of centimetres [31]. The field in a supershort leader and, therefore, 
the field at the respective distance from the tip of a long spark may be 2- 
4 kV/cm. At the site of ionization-thermal instability, where current is 
accumulated within a thin column, the field was found (section 2.3) to be 
20 kV/cm [4]. But far from the tip, it is nearly two orders of magnitude lower. 
At a typical experimental leader current of i x 1 A, its velocity is 
V, M 1.5-2cm/p, and the lifetime of a leader section at a distance of 3 m  
from the tip is at least 150 ps. This time is long enough for the relaxation 
processes in the channel to be nearly completed and for a nearly steady- 
state to be established. The thermal expansion of the channel, very fast at 
the beginning, is also completed by that time. Measurements made in a 
10m gap between a cone anode and a grounded plane [28] (voltage 1.6- 
1.8 MV, average current about 1 A, and average leader velocity 2 cm/ps) 
showed that the channel expansion rate was l00m/s at first but loops 
later the rate dropped to 2m/s. Measurements made by the shadow tech- 
nique showed the average thermal expansion radius to be rL = 0.1 cm. 
According to spectroscopic measurements, the temperature of a channel 
which has reached the gap middle is 5000-6000 K. Some other experimental 
data on laboratory leaders can be found in [4, 27, 28, 31, 321. 
The air ionization mechanism changes radically at temperatures T M 3000- 
6000K and relatively low reduced fields (for example, at E = 450V/cm, 
T = 5000K, and p = 1 atm, we have E / N  = 3 x 
V/cm2).t In cold air, 
t We should like to warn against the commonly used postulate that the field in a leader channel 
has a constant value of E / N  
8 x 10-'6VcmZ. Some authors use it for the calculation of 
breakdown voltages in air gaps, including long ones. At T = SO00 K, we have E = 1.15 kV/cm, 
which disagrees with experimental data for more or less long leaders (table 2.1) and contradicts 
the physics. The underlying implicit suggestion is that the only consequence of air heating is a 
change in its density. We shall show that this is not the case. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 85 ===
A long leader channel 
71 
it is ionization of O2 molecules by electrons gaining energy in a strong field, but 
at the above value of E / N ,  the ionization rate of unexcited oxygen and nitro- 
gen molecules and atoms by electron impact is negligible. Electrons are mostly 
produced in the associative ionization reaction 
N + 0 + 2.8eV + NO' + e. 
(2.44) 
Due to a low ionization potential of NO (9.3eV), the reaction requires a 
small activation energy and occurs at a large rate constant. Recent data 
[33] give 
T [K] 
(2.45) 
Direct NO ionization by electron impact may compete with associative 
ionization (2.44) but at an electron temperature higher than T, M lo4 K. 
Both estimations and kinetic calculations [34] show that thermodynami- 
cally equilibrium concentrations of N, 0, NO and electrons at T M 4000- 
6000 K (table 2.2) are established for 20-50 ps. 
During this time, a leader elongates only by 20-100cm, i.e., the process 
of establishing a thermodynamic equilibrium in the channel seems to occur 
concurrently with the transitional process of channel formation and heating 
to a quasi-stationary state. Although the electron density does not practically 
differ from the density due to streamer generation, ne M 10'4cm-3, the 
ionization degree at T = 5000, n,/N M 3.3 x lo-', is an order higher than 
that in the streamers, n,/N M 4 x lop6. Therefore, intensive ionization 
occurs during the evolution of ionization-thermal instability and subsequent 
heating to the final temperature. 
Thus, a long laboratory leader channel can be subdivided into two 
unequal parts. First, there is a relatively short (about 1 m) transitional por- 
tion just behind the tip where the gas is gradually heated and additionally 
ionized. This is accompanied by a change in the plasma density and con- 
ductivity. Second, there is the rest of the channel heated to 5000-6000K, 
which has reached a quasi-stationary state. The suggestion of an equilibrium 
electron density in this part of the channel generally leads to a correct value 
of its radius, Y x 0.13 cm, close to the measured thermal radius [28]. It can be 
k,,, = 2.59 x 10-'7T'.43 exp (-31 140/T) cm3 s-'. 
Table 2.2. Equilibrium air composition at p = 1 atm. 
T, K 
4000 
4500 
5000 
5500 
6000 
N ,  1 0 ' ~ c m - ~  
1.79 
1.60 
1.48 
1.35 
1.27 
ne, 1013 cm-3 
0.63 
1.70 
4.90 
11.2 
21.4 
N ~ ,  
10" cm-3 
4.70 
4.90 
4.60 
4.35 
3.81 
N ~ ,  
1 0 ' ~  
cm-3 
0.25 
1.15 
3.61 
9.92 
20.6 
N
~
~
,
 
1 0 ' ~  
cm-3 
1.62 
4.54 
2.73 
1.67 
1.03 
ne/N, lop5 
0.35 
1.06 
3.31 
8.30 
16.8 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 86 ===
78 
derived from the relation for current i = 7rr2en,p,E if we take electron 
mobility to be pe w 1.5 x 1022N-’ cm2(V.s)-‘, i w 1 A, E M 250V/cm and 
if we use the values of ne and N from table 2.2, corresponding to T = 5000 K. 
In reality, with the T and E / N  values characteristic of remote channel 
portions, the electron temperature T, may differ considerably from the gas 
temperature: T, may be as high as l0000K at T = 5000K. This slightly 
shifts the quasi-stationary values of ne relative to the thermodynamic equili- 
brium values corresponding to T (as in table 2.2). Stationary n, corresponds 
to the equality of forward and reverse reaction rates in (2.44). The forward 
reaction rate is independent of T,, while the reverse reaction rate at 
T, x 104K is proportional to TL3’*. Hence, the stationary value of II, 
will 
be larger than that in table 2.2 by a factor of (Te/T)3/4 
As for the less heated, recent channel sections, the difference between 
the electron and gas temperatures is greater. The reduced field E,” 
and 
Te must be higher in the unheated channel to provide impact ionization, 
since there is no other source of electron production. At temperatures 
T < 2500K, this is O2 ionization by electron impact. As the channel is 
heated further, NO ionization requiring lower T, and E,” 
begins to domi- 
nate, and only at T > 4000-4500K requiring a still lower field does the 
reaction of (2.44) become important. Clearly, the channel field cannot 
follow the condition E,” 
= const because of the change of ionization 
reactions with different energy thresholds. Calculations show [34] that the 
value of E / N  drops from 55 to 1.5Td with heating from 1000 to 6000K 
(figure 2.13). 
The streamer-leader process in a long spark 
2. 
1 o4 
0
1
2
3
4
.
5
6
 
Temperature, kK 
Figure 2.13. Parameters of the initial leader channel right behind the tip as a function 
of the gas temperature (model calculations of [34]; 1 Td = lo-’’ V . cm2). 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 87 ===
A long leader channel 
I9 
2.5.2 Energy balance and similarity to an arc 
The older leader portions are similar to an arc in atmospheric air. Current 
1 A and temperature 5000 K correspond approximately to the minimum 
arc values and the field equal to 200-250V/cm is only by a factor of two 
or three stronger than that in a low current arc. So it is natural to look at 
the leader channel as an arc analogue. 
All characteristics of a long stationary arc at atmospheric pressure when 
the plasma is usually quasi-stationary (maximum temperature T, along the 
axis, longitudinal field E, current channel radius ro) are defined only by one 
‘external’ parameter, normally, by current i. Joule heat released in the current 
channel is carried out by heat conduction. Radiation is essential only for very 
intensive arcs when the channel temperature exceeds 1 1 000- 12 000 K. The 
further fate of the energy depends on the arc cooling providing its steady- 
state. Heat can be removed via heat conduction through the cooled walls of 
the tube containing the arc. It can be carried away by the cooling gas flow 
or due to a natural convection if the arc burns in a free atmosphere. Definite 
relations among E, T ,  and ro with current i are obtained if the heat release 
mechanism is known. None of the above mechanisms (convection does not 
seem to have enough time to develop) are operative in a leader channel. One 
may suggest that heat is carried farther away from the channel via heat con- 
duction, gradually heating an ever increasing air volume. Strictly, this is not 
a steady state process, so it is not a simple matter to find all relations. 
However, the state of the channel itself is close to a stationary one. This is 
due to a small and definite temperature variation in the current channel owing 
to an exceptionally strong dependence of equilibrium plasma conductivity on 
its temperature. So one can find the relation between the leader channel 
temperature T, and the power PI = iE released per unit length. Using the 
available experimental values for T and i, one can find E to see that a fairly 
low field is sufficient to support plasma in a well developed leader channel. 
The electron density in an equilibrium plasma is ne E exp (-Zeff/2kT), 
where Zeff is an effective ‘ionization potential’ of the gas. The relation with 
an actual ionization potential of atoms is strictly valid for a homogeneous 
gas (the Saha equation). For the temperature range T E 4000-6000K, we 
have, in accordance with table 2.2, Zeff = 8.1 eV and Ien/K = 94000K, 
which is close to INo = 9.3 eV. Since Ieff/2kT E 10, the conductivity 0 
N ne 
is strongly temperature dependent and decreases with radius much more 
than the temperature. Therefore, we can use the concept of a current channel 
with a more or less fixed boundary - the radius ro (figure 2.14). By denoting 
the channel boundary temperature as Ti and bearing in mind that the 
temperature variation in the channel is A T  = T, - Ti << T,, we can write 
an approximate expression for the channel energy balance: 
(2.46) 
A T  
PI = -27rroX, (g ) M 47rr0X, - 
= 47rX,AT 
10 
YO 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 88 ===
80 
The streamer-leader process in a long spark 
.T 
Figure 2.14. Schematic arc channel with nearly the same distributions of T and o at 
the axis of the ‘old’ portion of a leader channel. 
where A, 
is heat conductivity at T = T,. The refined factor 4 appears instead 
of 2 if the heat conduction equation for a uniform distribution of heat 
sources is integrated over the range 0 < r < yo. 
An arbitrary channel boundary should be set such as to allow an adequate 
current through the channel. The current should not be too low, because 
another channel will appear ‘outside’; it should not be too high either, because 
a current-free periphery will arise inside the current channel. Assume, for the 
sake of definiteness, that the channel boundary conductivity cr( Ti) is by a 
factor of e less than the axial value cr( T,). With the exponential dependence 
of cr(T), this approximately yields AT x 2kTi/Ief. By substituting this 
expression into (2.46), we find the desired relation: 
1 12 
T, = (*PI) 
. 
(2.47) 
k T i  
Ieff 
87rA,k 
PI = iE 
87rAm-, 
Expression (2.47) does not contain the radius ro, its account requires a 
consideration of the channel environment [26]. The channel temperature 
T, grows more slowly with power than Pi12 because the air heat conductivity 
rises rapidly with temperature in the range of interest. At T, = 5000K, 
A, 
= 0.02 W/cm K; so we have PI x 130 W/cm. For current i x 1, for 
which this temperature seems to be characteristic, E x 130 V/cm. The 
values of E and PI are only by a factor of two smaller than those found 
from experimental evaluations of the field in older leader channel portions. 
It is possible that the channel instability and the necessity to heat an increas- 
ing air volume require a higher power and a higher field at the given current. 
This problem remains unsolved and deserves close study. When applied to 
arcs, formula (2.47) gives a fairly good agreement with experiments. 
These considerations of the thermal balance of a leader channel permit 
establishing the ‘current-voltage’ characteristic (CVC) E(i) that we shall 
need in the next section. Heat flow from the channel grows with temperature 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 89 ===
Voltage for a long spark 
81 
but not very rapidly. But the conductivity 0 and current i = .irr2uE would 
grow very quickly if the field remained constant. As fast would be the 
growth of energy release PI = iE, which would set the system out of heat 
balance. The balance is maintained because the field drops with rising 
current, while the power and temperature do not change much. In an ideal 
case with T, = const, which is close to the low temperature conditions for 
a leader and low current arc, we have E x i-' from (2.47). The arc CVC is 
indeed a descending curve, though its goes down somewhat more slowly 
because T, and P1 slightly rise with current [26]. 
A similarity between the states and CVC curves for quasi-stationary 
leaders and arcs was established in model experiments [4]. Sparks 7cm long 
were generated in air between rod electrodes. The circuit parameters were 
chosen such that the channel current was stabilized at a level characteristic 
of a long laboratory spark at the moment of gap bridging. The stabilizing 
mode lasted several milliseconds. During this time, a quasi-stationary state 
was established under energy supply conditions close to those of the leader 
process. The CVC thus measured is approximated by the expression 
E = 32 + 52/i V/cm, 
i [A]. 
(2.48) 
The obtained field appears to be lower than in a leader (84 V/cm at 1 A) and 
closer to the arc field. 
2.6 Voltage for a long spark 
The problem of minimum voltage, at which a spark can develop to a certain 
length is of primary importance for high-voltage technology. This quantity 
characterizes the electric strength of an air gap, since its bridging by a 
leader results in a breakdown. This problem also applies to lightning, because 
it is interesting to know the minimum cloud potential at which a lightning 
discharge is possible. 
Experiments show that the leader process has a threshold character. An 
initial leader cannot survive in normal air at gap voltages less than 300- 
400 kV. A leader can only be formed at low voltages in a short gap when it 
develops as a final jump from the very beginning. Then streamers immedi- 
ately reach the opposite electrode, and the energy supply mode differs 
from that of the initial stage, with the streamer zone isolated from the 
grounded electrode. The reason for a threshold is easy to understand in 
terms of the discussion in section 2.2.3 and formula (2.34). A leader channel 
has a minimum possible radius. The radius of a cold air column, in which 
current can accumulate, is ro > lOW2cm. A thinner current channel is 
immediately enlarged by ambipolar diffusion. To heat a column of such 
initial radius to 5000K, the leader tip potential from (2.34) must be at 
least 200 kV. If we consider the inevitable energy expenditure for ionization 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 90 ===
82 
The streamer-leader process in a long spark 
and gas excitation in the streamer zone, this value will increase by, at least, a 
factor of 1.5 [4]. 
Therefore, the tip potential in the initial leader stage will be 
several hundreds of kilovolts even under favourable conditions. 
The voltage U. applied to the gap drops across the leader channel and is 
partly transported to the tip. The general formula is 
U0 = EL + U, 
(2.49) 
where E is the average field in a leader channel of length L. We showed in 
section 2.5 that in a long channel, most of which is in a quasi-stationary 
state, E is a more or less definite value varying with current i. The channel 
field decreases with increasing current. But current growth requires that 
the tip potential determining the leader velocity and current i = C1 U, VL 
should be raised. At a fixed length L, the Uo(i, 
L )  function has a minimum, 
since it is the sum of a falling component and a component rising with i. 
Minimum voltage U 0 ~ * ( L )  
corresponds to current iOpt(L) optimal for a 
leader of length L. It is hardly possible, in the present state of the art, to 
find the Uomin(L) 
function theoretically. We shall try to define its character 
using semi-empirical data. 
Many experimental physicists have measured the leader velocity varia- 
tion with applied voltage U,. Much work has been done on short leaders 
because one can neglect the voltage drop across the channel, assuming 
U, = U,. With the account of this approximate equality, Bazelyan and 
Razhansky [35] suggested an empirical formula: VL z uU:’~, where 
a E 1.5 x lo3 V-1/2 cm s-l. Physically, the velocity increase with voltage 
looks quite natural (though this variation is not strong). We also know 
that the tip current is defined by (2.36). This gives the relation 
U, = Ai2l3(VL 
N i1/3) with A = ( C ~ U ) - ~ / ~  
= ( 2 7 r ~ ~ a ) - ~ / ~ .  
Let us use the 
analogy between a well developed leader and an arc and take the CVC 
E = b/i typical for a low current arc. Let us put b = 300 V A/cm for numerical 
calculations and ignore the difference between the tip and channel currents. 
We shall then get U, = Lb/i + AiZi3 and after differentiation 
iopt = (3Lb/2A)3i5, 
(2.50) 
If a leader develops under optimal conditions, the applied voltage is 
shared by the tip and the channel in comparable proportions. The mode 
with a low tip potential close to the limit admissible from the energy criteria, 
is unprofitable for a long leader, because it corresponds to low current lead- 
ing to a considerable voltage drop across the channel. The long spark param- 
eters in table 2.3 found from (2.50) with semi-empirical constants are quite 
reasonable: these orders of magnitude for current, voltage, and velocity 
meet the requirements on the optimal experimental conditions for long 
leader development. Besides, the experiment requires a nonlinear, slow 
dependence of minimum breakdown voltage on the gap length. It is generally 
known that increasing the length of a multi-metre gap is not a particularly 
U. mln = A3/5(3bL/2)215 
= $ U, opt 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 91 ===
A negative leader 
83 
Table 2.3. Long spark parameters. 
50 
3.3 
2.0 
1.1 
2.1 
260 
100 
4.3 
2.6 
1.3 
2.4 
170 
3000 
17 
10 
17 
4.1 
22 
effective way of raising its electrical strength. This is a key challenge to those 
working in high-voltage technology. 
The results of extrapolation of formula (2.50) to a lightning leader 
(L = 3 km) also lie within reasonable limits. 
What is the rate of gap voltage rise necessary for the optimal mode of 
spark development? Clearly, the gap voltage must be raised as the spark 
length becomes longer according to (2.50), where L is an instantaneous 
leader length. The existence of an optimal mode of spark development has 
been confirmed experimentally [36-381. It has been shown that for a 
breakdown to occur at minimum voltage, the pulse risetime tf must increase 
with the gap length d. The authors of [39] recommend the following empirical 
formula for the evaluation of an optimal risetime: 
lfopt 
50d [PSI, 
d [ml 
(2.51) 
Generally, optimal voltage impulses have a fairly slow risetime. Their values 
vary between 100 and 250ps in modern power transmission lines with the 
insulator string length of 2-5m. We shall return to this issue in chapter 3, 
when considering the diversity of time parameters of lightning current 
impulses. The minimum electric strength of an air gap with a sharply non- 
uniform field can be found from the formulas [4] 
[kV], 
d < 15m 
3400 
1 + 8/d 
u50% min = - 
(2.52) 
U,,,, 
min = 1440 + 55d [kV], 
15 < d < 30 m 
2.7 
A negative leader 
Most lightnings carry a negative charge to the earth because they are ‘anode- 
directed’ discharges. It is always more difficult to break down a medium- 
length gap between a negative electrode and a grounded plane. A negative 
leader requires a higher voltage. The difference between leaders of different 
polarities is due to the streamer zone structure, while their channels and 
voltage drop across them are quite similar. Indeed, a gap of about lOOm 
long, in which an appreciable part of voltage drops across the channel, is 
bridged by leaders of both signs at about the same voltages [2,3]. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 92 ===
84 
The streamer-leader process in a long spark 
The streamer zone formation in a negative leader requires a higher tip 
potential for the same reason as a single anode-directed streamer needs a 
higher voltage for its development. Fast streamers with the velocity V, 
much higher than the electron drift velocity V, do not exhibit much difference 
associated with polarity. But streamers in a leader streamer zone are slow: 
V, x V,. It is of great importance whether the components of electron 
velocity relative to the streamer tip are summed, V, + V,, as in a cathode- 
directed streamer, or subtracted, V, - V,, as in an anode-directed one. In 
the former case, electrons produced in front of the tip move towards it, 
and the ionization occurs in a strong field near the tip. In the latter, electrons 
tend to ‘run ahead’ of the moving tip and spend most of their time in a lower 
field, so that the ionization occurs under less unfavourable conditions. 
The fact that negative streamers generally require a higher field and 
voltage has been supported by many experiments. They show that the 
average critical field, which defines the maximum streamer length in formula 
(2.32), is twice as high for an anode-directed streamer as for a cathode- 
directed one: E,, x 10 kV/cm against E,, x 5 kV/cm. We shall illustrate 
this with figure 2.1 5 for a gap of length d = 3 m between a sphere of radius 
ro = 50cm and a grounded plane. The streamers stopped, having covered 
the distance I,,, 
at negative sphere potential U, = 1.5 MV (the unperturbed 
potential at the stop with the account of the sphere charge reflection in the 
plane is Uo(Zmax) x 0.25 U,). Under these conditions, cathode-directed 
streamers practically cross the whole gap. 
The propagation mechanism and streamer zone structure of a negative 
leader are much more complicated than those of a positive leader and are still 
Figure 2.15. Anode-directed streamers from a spherical cathode of 50 cm radius at a 
negative voltage impulse of 1.8 MV and a 50 ps front duration. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 93 ===
A negative leader 
85 
poorly understood. In the 1930s, when Schonland started his famous studies 
of lightning [41], a negative leader was found to have a discrete character of 
elongation, so it was termed stepwise. Streak photographs exhibit a series of 
flashes, indicating that the leader propagates in a stepwise manner. Later, a 
similar process was found in a long negative leader produced in laboratory 
conditions 142,431. With every step, a negative leader elongates by dozens 
of centimetres, or by several metres in superlong gaps [3]; steps of a hundred 
metres have been registered in negative lightning discharges. Every step of a 
laboratory leader is accompanied by a detectable current overshoot which 
quickly vanishes during the time between two steps. 
Without going into theoretical explanations of this mechanism, based 
on an unverified hypothesis, let us see what information can be derived 
from streak photographs of the process, made during laboratory experiments 
[44]. These are naturally more informative than streak photographs of a step- 
wise lightning leader. It is seen from figures 2.16 and 2.17 that in the intervals 
between the steps, the tip of a negative leader slowly and continuously moves 
on together with its streamer zone made up of anode-directed streamers. The 
main events occur near the external boundary of the negative streamer zone. 
It seems that a plasma body elongated along the field arises there and is 
polarized by the field (compare with the discussion in section 2.2.7). The 
positive plasma dipole end directed towards the main leader tip serves as a 
starting point for cathode-directed streamers. They move towards the tip, 
thus elongating the conducting portion of the channel and enhancing the 
negative field at its end directed to the anode. Almost at the same time, the 
plasma body generates an anode-directed streamer. T h s  nearly mystic 
picture of streamer production in the gap space is clearly seen in a streak 
photograph in figure 2.18. Nothing like this has ever been observed with a 
positive leader. 
Figure 2.16. A schematic streak picture of a negative stepped laboratory leader: (1,2) 
secondary cathode- and anode-directed streamers from the gap interior; (3) secondary 
volume leader channel; (4) main negative leader channel; (5) its tip; (6) plasma body; 
(7), (8) tip of secondary positive and negative leader (9) leader flash concluding step 
development. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 94 ===
86 
The streamer-leader process in a long spark 
Figure 2.17. A streak photograph of the initial stage in a negative laboratory leader. 
Marking numbers correspond to figure 2.16. 
The polarized plasma section becomes the starting point not only of 
streamers but also of secondary leaders which follow them. They are 
known as volume leaders. A positive cathode-directed volume leader grows 
intensively. Normally, its streamer zone almost immediately reaches the 
main negative leader, so it looks as if the secondary positive leader develops 
Figure 2.18. The origin of anode-directed (1) and cathode-directed (2) streamers from 
the gap interior; (3) initial flash of a negative corona (static photograph) which trigger 
a streak photograph regime; (4) arisen negative leader. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 95 ===
A negative leader 
87 
in the final jump mode, i.e., very quickly. The negative volume leader moves 
towards the anode somewhat more slowly. When the tips of the main 
negative and of the positive volume leaders come into contact, they form a 
common conducting channel, giving rise to the process of partial charge 
neutralization and redistribution. As a result, the former volume leader 
acquires a potential close that of the main negative leader tip. This process 
looks like a miniature return stroke of lightning, accompanied by a rapidly 
rising and just as rapidly falling current impulse in the channel and external 
circuit. The intensity of the channel emission increases for a short time. It is 
hard to say what exactly stimulates this increase - the short temperature rise 
or the ionization in the channel cover: which changes the cover charge, 
thereby getting ready for a potential redistribution along the channel (see 
section 2.4.4). The negative portion of the plasma dipole turns to a new 
negative tip of the main leader. This is the mechanism of step formation 
and stepwise elongation of the main channel. Then the story is repeated. 
The picture just described gives no ground to draw the conclusion about 
a stepwise character of negative leader development. The motion of a nega- 
tive leader is continuous, but secondary positive volume leaders, also contin- 
uous, produce a stepwise effect. Discrete is the final result of their ‘secret 
activity’, but only if the observer is equipped with imperfect optical instru- 
ments. In other words, what is generally known as a step is an instant 
result of a long continuous leader process. As for gap bridging by a main 
negative leader, one should bear in mind that most of the channel is created 
by auxiliary agents - by a succession of positive volume leaders. 
This picture has been reconstructed from streak photographs. But we 
still do not know how polarized plasma dipoles are formed far ahead of 
the main leader tip. Their appearance is hardly a result of our imagination. 
Steps can be produced deliberately by making a volume leader start from a 
desired site in the gap. For this, it suffices to place there a metallic rod several 
centimetres long (figure 2.19). A series of rods placed in different sites of a 
gap will create a regular sequence of volume leaders. The work [45] describes 
an experiment with a negative leader 200 m long. Its perfectly straight trajec- 
tory was predetermined by seed rods suspended by insulation threads at a 
distance of 2-3m from each other. A volume leader started from a rod 
when it was approached by the negative streamer zone boundary of the 
main leader. Clearly, the rods are polarized by the streamer zone field to 
serve as seed dipoles instead of natural (hypothetical) plasma dipoles. 
There are many hypotheses concerning the stepwise leader mechanism, 
but they are so imperfect, lacking strength, and, sometimes, even absurd that 
we shall not discuss them here. We are not ready today to suggest an alter- 
native model either. Additional special-purpose experiments could 
certainly stimulate the theory of this complicated and challenging phenom- 
enon. It would be desirable to take shot-by-shot pictures of a negative 
leader tip region with a short exposure. A sequence of such pictures would 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 96 ===
88 
The streamer-leader process in a long spark 
Figure 2.19. An artificially induced step: (1) initial flash of a negative corona from a 
spherical cathode; (2, 3) cathode- and anode-directed leaders from a metallic rod 
2.5 cm long, placed in the gap interior; (4) leader flash concluding the step develop- 
ment; (5) new streamer corona flash from the tip of the elongating channel. 
form a film more accessible to unambiguous interpretation than continuous 
streak photographs with confusing overlaps of many details. 
References 
[l] Lupeiko A V, Miroshnizenko V P et a1 1984 Proc. II All-Union Conf Phys. of 
[2] Baikov A P, Bogdanov 0 V, Gayvoronsky A S et a1 1998 Elektrichestvo 10 60 
[3] Gayvoronsky A S and Ovsyannikov A G 1992 Proc. 9th Intern. Conf on Atmosph. 
[4] Bazelyan E M and Raizer Yu P 1997 Spark Discharge (Boca Raton, New York: 
[5] Loeb L B 1965 Science (Washington D.C.) 148 1417 
[6] D’aykonov M I and Kachorovsky V Yu 1988 Zh. Eksp. Teor. Fiz. 94 32 
[7] D’aykonov M I and Kachorovsky V Yu 1989 Zh. Eksp. Teor. Fiz. 95 1850 
[8] Shveigert V A 1990 Teplofz. Vys. Temperatur 28 1056 
[9] Bazelyan E M and Raizer Yu P 1997 Teplofiz. Vys. Temperatur 35 181 (Engl. 
[lo] Raizer Yu P and Simakov A N 1996 Piz. Plazmy 22 668 (Engl. transl.: 1996 
[ll] Dutton J A 1975 J. Phys. Chem. Rex Data 4 577 
[12] Cravith A M and Loeb L B 1935 Physics (N.Y.) 6 125 
[13] Raizer Yu P and Simakov A N 1998 Piz. Plazmy 24 700 (Engl. transl.: 1996 
Electrical Breakdown of Gases (Tartu: TGU) p 254 (in Russian) 
Electricity 3 (St Peterburg: A.I. Voeikov Main Geophys. Observ.) p 792 
CRC Press) p 294 
transl.: 1997 High Temperature 35) 
Plasma Phys. Rep. 22 603) 
Plasma Phys. Rep. 24 700) 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 97 ===
References 
89 
[14] Vitello P A, Penetrante B M and Bardsley J N 1994 Phys. Rev. E 49 5574 
[15] Babaeva N Yu and Naidis G V 1996 J. Phys. D: Appl. Phys. 29 2423 
[16] Kulikovsy A A 1997 J. Phys. D: Appl. Phys. 30 441 
[17] Aleksandrov N L, Bazelyan E M, Dyatko N A and Kochetov I V 1998 Fiz. 
Plazmy 24 587 (Engl. transl. 1998 Plasma Phys. Rep. 24 541) 
[18] Bazelyan E N and Goryunov A Yu 1986 Elektrichestvo 11 27 
[19] Aleksandrov N L and Bazelyan E M 1998 J. Phys. D: Appl. Phys. 29 2873 
[20] Aleksandrov D S, Bazelyan E M and Bekzhanov B I 1984 Izv. Akad. Nauk 
[21] Bazelyan E M, Goryunov A Yu and Goncharov V A 1985 Izv. Akad. Nauk 
[22] Aleksandrov N L and Bazelyan E M 1999 J. Phys. D: Appl. Phys. 32 2636 
[23] Gayvoronsky A S and Razhansky I M 1986 Zh. Tekh. Fiz. 56 11 10 
[24] Kolechizky E C 1983 Electric Field Calculation for High- Voltage Equipment 
[25] Raizer Yu P, Milikh G M, Shneider M Nand Novakovsky S.V. 1998 J. Phys. D: 
[26] Raizer Yu P 1991 Gas Discharge Physics (Berlin: Springer) p449 
[27] Gorin B N and Schkilev A V 1974 Elektrichestvo 2 29 
[28] ‘Positive Discharges in Air gaps at Las Renardieres - 1975’ 1977 Electra 53 31 
[29] Bazelyan E M 1982 Izv. Akad. Nauk SSSR. Energetika i transport 3 82 
[30] Bazelyan E M 1966 Zh. Tekh. Fiz. 36 365 
[31] Bazelyan E M, Levitov V I and Ponizovsky A Z 1979 Proc. 111 Inter. Symp. on 
[32] Meek J M and Craggs J D (eds) 1978 Electrical Breakdown of Gases (New York: 
[33] Makarov V N 1996 Zh. Prikl. Mekh. Tekhn. Fiz. 37 69 
[34] Aleksandrov N L, Bazelyan E M, Dyatko N A and Kochetov I.V. 1997 J. Phys. 
D: Appl. Phys. 30 1616 
[35] Bazelyan E M and Razhansky I M 1988 Air Spark Discharge (Novosibirsk: 
Nauka) p 164 (in Russian) 
[36] Stekolnikov I S, Brago E Nand Bazelyan E M 1960 Dokl. Akad. Nauk SSSR 133 
550 
[37] Stekolnikov I S, Brago E N  and Bazelyan E M 1962 Con$ Gas Discharges and the 
Electricity Supply Industry (Leatherhead, England) p 139 
[38] Bazelyan E M, Brago E N and Stekolnikov I S 1962 Zh. Tekh. Fiz. 32 993 
[39] Barnes H and Winters D 1981 IEEE Trans. Pas-90 1579 
[40] Gallet G and Leroy J 1973 IEEE Conf. Paper C73-408-2 
1411 Schonland B 1956 The Lightning Discharge. Handbuch der Physik 22 (Berlin: 
1421 Stekolnikov I S and Shkilev A B 1962 Dokl. Akad. Nauk SSSR 145 182 
[43] Stekolnikov I S and Shkilev A B 1963 Dokl. Akad. Nauk SSSR 145 1085; 1962 
1441 Gorin B N and Shkilev A V 1976 Elektrichestvo 6 31 
[45] Anisimov E I, Bogdanov 0 P, Gayvoronsky A S et a1 1988 Elektrichestvo 11 55 
SSSR. Energetika i transport 2 120 
SSSR. Energetika i transport 2 154 
(Moscow: Energoatomizdat) p 167 (in Russian) 
Appl. Phys. 31 3255 
High Voltage Engin. (Milan) Rep. 51.09 p 1 
Wiley) 
Springer) p 576 
Intern. Con$ (Montreux) p 466 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 98 ===
Chapter 3 
Available lightning data 
Scientific observations of lightning were started over a century ago. Much 
factual information has accumulated about this natural phenomenon since 
that time. Most of it, however, has been obtained by remote observational 
techniques which can reveal only external manifestations of lightning. This 
is not the researchers’ fault. Even a long laboratory spark keeps the 
experimenter at a respectful distance: there have been single and mostly 
unsuccessful attempts to study the leader interior and the ionization region 
in front of its tip. No attempts of this kind have yet been made with lightning. 
Nevertheless, the accumulated material is being analysed and systematized, 
so that our knowledge about atmospheric electricity is gradually expanding. 
A number of carefully written books has made the results of field studies 
of lightning accessible to specialists. Among them, of great interest is the 
recent book by Uman [l] and the co-authored work edited by Golde [2]. 
The reader will find there nearly all available data on lightning, so there is 
no need to discuss them in this book. We have set ourselves a different 
task - to select the few data available on the lightning discharge mechanism 
and to try to build its theory. In addition, we shall make a detailed analysis of 
lightning characteristics important from the practical point of view. The 
nature of hazardous effects of atmospheric electricity on industrial objects 
will be considered in much detail and lightning protection principles will 
be offered. 
This task cannot be solved completely, because many lightning param- 
eters have never been measured or, more often, even estimated in order of 
magnitude. One hope is a method similar to the identical text analysis used 
in cryptography to read a text written in a dead language. If there is at 
least part of the text written in an accessible, better, related, language, the 
task is not considered hopeless. With patience and ingenuity, the researcher 
has a chance if he compares these texts carefully. In this respect, we expect 
much from long spark studies. Clearly, a spark and lightning are phenomena 
90 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 99 ===
Atmospheric field during a lightning discharge 
91 
of different scales, but it is also clear that both have a common nature. For 
this reason, we shall often compare the parameters of lightning with those of 
a long spark. We should like to emphasize that this will be a comparison 
rather than a direct extrapolation, because there is no complete analogy 
between the two phenomena. 
3.1 Atmospheric field during a lightning discharge 
There is no strict answer to this physically ambiguous question. It is neces- 
sary to specify what part of the space between the cloud and the earth is 
meant. One thing is clear - the electric field at the lightning start must be 
high enough to increase the electron density by impact ionization. This 
value is Ei = 30 kV/cm for normal density air and about 20 kV/cm at an 
altitude of 3 km (the average altitude for lightning generation in Europe). 
Such a strong field has never been measured in a storm cloud. The maximum 
values were recorded by rocket probing of clouds (10kV/cm, Winn et al, 
1974 [7]) and during the flight of a specially equipped aeroplane laboratory 
(12kV/cm). The value obtained by Gunn [4] in 1948 during his flight on a 
plane around a storm cloud was about 3.5 kV/cm. The values between 1.4 
and 8 kV/cm were obtained from some similar measurements [3-91. It is 
hard to judge about the accuracy of these measurements, especially those 
made in strong fields, because parts of the field detector or the carrier- 
plane parts close to it can produce a corona discharge. In any case, the 
corona space charge will not allow the strength in the region being measured 
to go beyond a threshold value (for details, see [20]). There are reasons to 
believe, however, that a corona on hydrometeorites (water droplets, snow 
flakes, ice crystals) keeps the field at a level below Ei in the whole of the 
cloud, If this is indeed so, a field can be enhanced above Ei only in a small 
volume for a short time, say, as a result of eddy concentration of charged 
hydrometeors. This enhancement will be reduced to zero by a corona for 
less than a second. The experimenter has no chance to guess where the 
field may be locally enhanced to be able to introduce a probe detector there. 
Theoretically, it is also important to know the average gap field capable 
of supporting a lightning leader. The field decreases in the charge-free space 
from the cloud towards the earth. At the earth, the storm field was found to 
be 10-200 V/cm. Such a low field did not prevent the lightning development. 
Lightnings were deliberately produced in numerous experiments described 
by Uman [l, 10-161. A rocket was launched from the earth, pulling behind 
it a thin grounded wire. A lightning leader was excited at 200-300m above 
the earth’s surface. The near-surface field during a successful launching 
was usually 60- 100 V/cm. 
Strictly, measurements made at two points, at the earth and in the cloud, 
are insufficient for an accurate evaluation of an average electric field. The 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 100 ===
92 
Available lightning data 
km 
6- 
4- 
2- 
0 I
Figure 3.1. The ‘dipole’ model of the charge distribution in a storm cloud. 
space between the cloud and the earth should be scanned, and this must be 
done for some fractions of a second, just before a lightning discharge, in 
the vicinity of its anticipated trajectory. Unfortunately, such attempts have 
not been quite successful. More successful were measurements made at 
points on the earth’s surface separated at a distance of hundreds and 
thousands of metres [17-191. These have been used to reconstruct the 
charge distribution within a storm cloud, invoking the results of direct 
cloud probing. The reconstruction procedure and its possible errors are 
discussed in [20]. Generally, with simultaneous field measurements made at 
n points, one can write a closed set of equations for the same number of 
parameters of charged regions. Its solution provides the parameters, for 
example, the average space charge densities in pre-delineated regions. 
Most often, the number of points is too small, so the results obtained only 
permit the construction of simplified models with point charges. Very common 
is the dipole model with a negative charge at 3-5 km above the earth with the 
same value of the positive charge raised at a double altitude. Sometimes, a 
small positive point charge is added to them, which is placed at a distance 
by 1-2 km closer to the earth than the negative charge. All point charges are 
assumed to be located along the same vertical line (figure 3.1). 
The concept of a cloud filled by charged layers of different signs is 
based on probe measurements of charge polarity in hydrometeors. In this 
respect, this model raises no doubt. But as for the field distribution, the 
measurement error is too large, especially for the space in the cloud between 
two point charges. Luckily, the descending lightning trajectory lies mostly 
outside of the cloud, in the air free from charged particles. For this part of 
the trajectory, the average field evaluation in terms of a simple model 
makes sense. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 101 ===
Atmospheric field during a lightning discharge 
93 
We shall illustrate the procedure of deriving information from such 
measurements. Suppose we have at our disposal the values of field El at 
the earth's surface just under an anticipated charged centre of a storm 
cloud, as well as the values for field E2 at the lower cloud boundary (also 
under the charged centre) measured during a plane flight around the 
cloud. The altitude of the lower boundary, h, is also known. Assume the 
centre of the main lower charge q to be above the lower cloud boundary 
at an unknown distance r. If we ignore the effects of the remote upper 
charge and of the additional charge lying under the main one, we can 
write 
(3.1) 
4 
E2 = - 
q
+
 
2q 
El = 
47r&o(h + Y)2 ' 
47reor2 
47r&o(2h + r)2 ' 
The factor 2 in El and the second term in E2 are due to the action of charge 
induced in the earth's conducting plane (mirror reflection). Since in reality 
El << E2, it is natural to suggest that r << h. Then, with the second term in 
E2 ignored, we find 
q = 27r~o(h + r)2E1, 
r M ah, 
a = (E1/2E2)1'2. 
(3.2) 
Substituting the values above, i.e., El = lOOV/cm, E2 = 3000V/cm, and 
h = 3 km, we shall find q = 6.3 C, a = 0.13, and r M 390m. The average 
field in the region between the lower cloud boundary and the earth, 
equal to the lower boundary potential p2 M q / 4 7 r ~ ~ r  
divided by the cloud 
distance from the earth, h, Ea" 
(ElE2/2)112 
M 390V/cm. Allowance for 
the second term in E2 (the effect of mirror charge reflection by the earth) 
can hardly be justified, because our model did not take into account the 
effect of the upper charge of opposite sign. This charge is closer to the 
cloud edge than that reflected by the earth and has, therefore, a greater 
effect on E2, Its consideration, however, simple though it may seem, 
would require field measurement at another point of space and another 
equation for finding a new unknown - the altitude of the upper charge 
centre of the dipole. 
A larger scale correction would, probably, be necessary to account for 
the effect, on the near-earth field, of the space charge induced by coronas 
from pointed grounded objects (tree branches, high grass, various buildings, 
etc.) [21]. Estimations made just at the earth's surface show that this charge 
reduces the actual field of a storm cloud by half. So one cannot say that 
lightning moves in an unusually low electric field. These are just the values 
at which superlong sparks are excited in laboratory conditions (see 
chapter 2). Therefore, there is no need to invent a special propagation 
mechanism for lightning, different from that of a long laboratory spark, if 
we deal with average electric fields capable of breaking down the cloud- 
earth gap. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 102 ===
94 
Available lightning data 
3.2 The leader of the first lightning component 
The leader of the first component of lightning develops in unperturbed air, so 
only this leader behaviour should be compared directly with laboratory data 
on long spark leaders. The comparison can be carried out along two lines. We 
can first compare the leader structure in lightning and a spark and, second, 
their quantitative parameters, primarily velocities. 
3.2.1 Positive leaders 
Streak pictures of a positive leader are easy to interpret. So we shall begin 
with positive lightnings, though their occurrence is not frequent. Many 
books and papers refer to the successful streak photographs of a descending 
positive leader taken by Berger and Fogelsanger in 1966 [22]. Its schematic 
diagram is reproduced in figure 3.2. The leader became accessible to photo- 
graphy at 1900m above the earth's surface. It moved down in a continuous 
mode, without an appreciable intensity variation. The average leader velocity 
over the registration time was 1.9 x lo6 m/s, increasing somewhat as the 
leader tip approached the earth. Such a leader is much faster than a long 
laboratory spark, whose velocity is 50-100 times lower at minimum break- 
down voltage before the streamer zone contacts a grounded electrode. 
In the streak picture, the leader tip looks much brighter than its channel, 
but no signs of a streamer zone can be identified. We cannot say how the 
original negative looks, but in the published photograph the resolution 
threshold is hardly less than 50m. It is a very large value for a streamer 
leadp channel 
b- 
1.5 ms ,-A 
L - return 
E stroke 
Figure 3.2. Schematic streak picture of a positive descending lightning leader regis- 
tered on the San Salvatore Mount in Switzerland [22]. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 103 ===
The leader of the first lightning component 
95 
Figure 3.3. Schematic streak picture [22] of a positive ascending leader from a 70-m 
tower on the San Salvatore Mount. 
zone. With the data of section 2.4.1, we can calculate the leader tip potential 
U,, at which the streamer zone will exceed at least twice the length of about 
loom, a threshold value for the measuring equipment. For the average 
streamer zone field E, = 5 kVjcm, the potential derived from (2.39) is 
U, % 100 MV. Such a high value cannot be typical of lightning with average 
parameters. There is another circumstance preventing streamer zone registra- 
tions - different radiation wavelengths of a hot leader channel and a cold 
streamer zone. Violet and ultraviolet radiation from streamers is dissipated 
by water vapour and rain droplets in the air much more than long wavelength 
radiation characteristic of a mature channel. At a distance of about a kilo- 
metre between the lightning and the registration site (closer distances are 
practically unfeasible), a streamer zone may become quite invisible to the 
observer's equipment. Note that this is totally true of optical registrations 
of an ascending positive leader. 
A schematic streak picture of an ascending positive leader, based on 18 
successful registrations [22], is shown in figure 3.3. All lightnings started from 
a 70-m tower on the San Salvatore Mount near the Lake Lugano. The leader 
does not exhibit specific features that would distinguish it from a long labora- 
tory spark. On the whole, it developed in a continuous mode with irregular 
short-term enhancements of the channel intensity. Normally, they did not 
accelerate the leader development. Something like this has been observed 
in a long laboratory spark. The streak picture in figure 3.4 demonstrates 
this with reference to a positive leader in a sphere-plane gap 9 m  long. But 
this phenomenon has nothing to do with a stepwise elongation of a negative 
leader channel. 
The velocity of an ascending positive leader near the starting point is 
close to that of a laboratory spark, about 2 x 104m/s. From some data 
[22], it was in the range of (4-8) x 104m/s for a channel length of 40- 
100m; but when the leader tip was at a height of 500-1 150m, it increased 
by nearly an order of magnitude, to 105-106 mjs. 
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=== PAGE 104 ===
96 
Available lightning data 
Figure 3.4. A streak photograph of the initial stage of a positive leader in a 9-m rod- 
plane gap, displaying short flashes of the channel. 
3.2.2 Negative leaders 
Negative lightnings occur more frequently than positive, and their registra- 
tion is more common. The main distinguishing feature of the negative 
leader of the first lightning component is its stepwise character. The leader 
tip leaves a discontinuous trace in streak pictures which look like a movie 
film (figure 3.5). One can sometimes find such pictures in a sports magazine 
illustrating the successive steps in a sportsman’s performance. The bright 
flash of the tip and the channel right behind it are followed by a dead zone 
with practically zero intensity. This is followed by another flash showing 
that the tip has moved on for several dozens of metres. Such negative 
leader behaviour was observed by Schonland and his group as far back as 
the 1930s [24,25]. According to their registrations, the average pause between 
the steps was close to 60ps, with a spread from 30 to loops, and the step 
Figure 3.5. Schematic streak picture of a descending negative leader. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 105 ===
The leader of the first lightning component 
91 
I 
I 
I 
I 
I 
1 
b 
20 
25 
Yp 
0 
5 
10 
15 
Figure 3.6. Typical integral velocity distributions for descending leaders of the CY- and 
P-types (from [24, 251). 
length varied between 10 and 200 m with the average value being 30 m. The 
duration of a step is likely to be within several microseconds. The available 
streak photographs are not good enough to identify the details of a step. 
In any case, it is hard to decide whether it is similar to a step of the long 
negative spark described in section 2.7. 
A stepwise negative leader approaches the earth at an average velocity 
of 105-106m/s. Two descending leader types can be identified in terms 
of their velocity: slow a-leaders and fast P-leaders. The former travel 
at their step-averaged velocity; it varies with the discharge in the range of 
(1-8) x105m/s with the average value of 3 x 105m/s. The respective P- 
leader parameters are 3-4 times higher. This can be seen in figure 3.6 showing 
the integral velocity distributions described in [24,25]. Usually, P-leaders are 
more branched and their steps are longer. They abruptly slow down when 
they approach to the earth, after which they behave as a-leaders. 
An ascending negative leader also has characteristic steps. Most of the 
13 registered leaders ascending from a 70-m tower on the San Salvatore 
Mount [22] were identified as a-leaders. They have relatively short steps 
(5-18 m) and a velocity of (1.1-4.5) x lo5 mjs. Two of the discharges were 
referred to @-leaders because their velocity was (0.8-2.2) x lo6 m/s and the 
step length up to 130m. On the whole, ascending and descending stepwise 
leaders do not show significant differences. 
The registrations of ascending discharges from the San Salvatore Mount 
provide direct evidence for the existence of a streamer zone in a lightning 
leader. Registrations made at a sufficiently close distance, which became 
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=== PAGE 106 ===
98 
Available lightning data 
Figure 3.7. Schematic diagram of the initial development of a leader ascending from 
a 70-m tower on the San Salvatore Mount, as viewed from close distance streak 
photographs. 
possible due to the tower top being the only starting point, show streamer 
flashes arising at the moment a new step begins. Streamers were initiated 
not only from the tip of the main channel but also from its branches 
(figure 3.7). 
3.3 
The leaders of subsequent lightning components 
Leaders of lightning components following the first one are known as dart 
leaders because of the absence of branches. The streak photograph in 
figure 3.8 shows the trace of only one bright tip looking like a sketch of an 
arrow or dart. A dart leader follows the channel of the previous lightning 
component with a velocity up to 4 x lo7 mjs. Averaging over many registra- 
tions gives the value (1-2) x 107m/s, with the minimum values being an 
order of magnitude less than the maximum one [23,25]. The dart leader 
velocity does not vary much on the way from the cloud to the earth. 
Figure 3.8. A schematic streak picture of a dart leader. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 107 ===
The leaders of subsequent lightning components 
99 
Figure 3.9. A streak photograph of a well-branched lightning striking the Ostankino 
Television Tower; the components along the branches A and B are formed at different 
moments of time. 
Similar results have recently been obtained from 23 streak photographs 
of fairly good quality showing dart leaders taken in Florida by a camera with 
a time resolution 0.5 ps [26]. The average velocity of a dart leader varied from 
5 x lo6 to 2.5 x lo7 m/s in some registrations; it was (1.6-1.8) x lo7 m/s for 
three typical pictures presented in the publication. 
It is clear that a dart leader somehow makes use of the previous channel 
with a different temperature, gas density and composition. There are several 
indications to this. First, there is a tendency for the dart leader velocity to 
decrease with increasing duration of the interleader pause. This is because 
the gas in the trace channel is gradually cooled to return to the original 
condition. If a pause lasts longer, the subsequent component may take its 
own way. Figure 3.9 shows a lightning discharge which struck the Ostankino 
Television Tower in Moscow. Some of its initial components followed a 
common channel but then the discharge trajectory changed. Naturally, 
there is nothing like a dart leader in unperturbed air - the leader of each 
next component develops in a step-wise manner. 
Second, it has been found in triggered lightning investigations [ 15,271 
that a dart leader requires a current-free pause for its development. The 
long-term current, which supports the channel conductivity in the period 
between two subsequent components, must entirely cease to allow the 
channel to partly lose its conductivity. Only then will the channel be ready 
to serve as a duct for a dart leader. But if a new charged cloud cell is involved 
and raises the potential of the channel with current, an M-component is 
produced instead of a dart leader. Its distinctive feature is a higher intensity 
of the existing channel lacking a well defined tip (figure 3.10). The absence of 
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=== PAGE 108 ===
100 
Available lightning data 
Figure 3.10. A streak photograph of a well-branched lightning with M-components. 
a clear luminosity front is a serious obstacle to the measurement of its 
velocity. Investigators often point to a nearly simultaneous increase of the 
light intensity in the whole channel. This suggests either an almost sub- 
light velocity of the leader or an exceptionally smeared boundary of its 
front. In contrast to a dart leader, an M-component is never followed by a 
distinct return stroke with a high (10-100 kA) rapidly rising current impulse. 
Both the external view and photometric data obtained from streak 
photographs reveal clearly a dart leader tip. Some authors [26] made an 
attempt to measure the time variation of the leader light intensity. Although 
the measurements were performed near the time resolution limit, they indi- 
cate that the light pulse front at the registration point rises for 0.5-1 ps 
and then is stabilized for 2-6 ps. Therefore, with the dart leader velocity of 
1.5 x lo7 m/s, the extension of the front rise is 7.5-15m with the full pulse 
front length of 35-105m. It can be mentioned, for comparison, that the 
M-component has a pulse front, if any, of a kilometre length. 
It is important for the theory of dart leaders that they always move from 
a cloud down to the earth. This means that the voltage source that excites 
them is 'connected' to the trace channel of the previous component right in 
the cloud. The direction of the previous leader does not matter much because 
their channels are equally suitable for the development of a dart leader. 
3.4 
Lightning leader current 
We can only guess about the values of descending leader currents or estimate 
them from indirect data. We shall make such estimations in section 3.5, using 
leader charge data, also obtained indirectly. Ascending leader currents are 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 109 ===
Lightning leader current 
101 
Figure 3.11. A schematic oscillogram of the leader current in an ascending lightning. 
not difficult to measure, and there have been many measurements of this 
kind. Normally, a current detector is mounted on top of a tower dominating 
the locality [28-301. The current impulse of an ascending leader, registered on 
an oscillogram, lasts for about 0.1 s, corresponding to the time of ascending 
leader development. The current nearly always rises in time (figure 3.11). The 
current supplies an elongating leader with charges. Physically, these charges 
are induced by the electric field of a cloud. When a leader approaches a cloud, 
going through an increasingly higher field, the linear density of induced 
charge T increases. Besides, the leader goes up with an increasing velocity 
V,, reducing the time for the charge supply. A combination of these factors 
raises the current i = 7VL. At the moment an ascending leader starts its 
travel, its current is lower than 10A, whereas at the end of the travel, it 
may rise to 200-600 A, with an average value of about 100 A. Sometimes, 
just before the leader begins its continuous elongation, impulses with an 
amplitude of several amperes may arise against the background of a milli- 
ampere corona current. 
Current oscillograms of an ascending leader triggered from a thin wire 
elevated by a small rocket to 100-300m [13,31] give a similar picture. 
They show the same slowly rising impulse with an amplitude of 100-200A 
and duration 50-looms. It has no overshoots at the front, even if the 
leader goes up in a stepwise mode. 
There are no reasons to suggest any principal difference between average 
currents of ascending and descending leaders. In both cases, the leader is 
supplied by charges induced by the electric field of a storm cloud, and the 
leader lifetimes are comparable because they move at approximately the 
same velocity. 
Qualitatively, the current variation of the first component leader is 
similar to that of a laboratory spark. When the gap voltage is raised 
slowly, one can observe initial leader flashes at the high-voltage electrode, 
followed by distinct current impulses [32]. As for long spark steps, they 
practically do not change the current at the leader base. It has been shown 
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=== PAGE 110 ===
102 
Available lightning data 
[20] that this is to be expected if the charge perturbation region is separated 
from the registration point by an extended channel section with high 
resistivity and distributed capacitance (section 4.4). The perturbation wave 
travelling along the channel towards the detector is attenuated. Of course, 
the current of a laboratory spark rarely exceeds a few amperes, but such a 
difference is predictable. it follows from the expression for the current 
i =  TV, cited above. A lightning leader has an order higher velocity 
V, and, at least, an order larger linear charge T (due to the voltage being 
10-20 times higher). All in all, this increases the current to within the 
anticipated two orders of magnitude. 
Now one can judge about the current of a dart leader. There are no 
direct registrations of this current. One exception was an attempt at its 
measurement in a triggered lightning just before its contact with the earth. 
This is principally possible since the point of contact is known exactly - 
this is the point of wire fixation to the earth. The wire evaporates completely, 
having passed the current of the first lightning component. Using the still hot 
trace channel, a dart leader follows the path of the wire. A current detector 
can be placed at the wire grounding site. 
It is much more difficult to interpret the recorded oscillograms, because 
it is unclear at what moment of time the development of a dart leader stops 
and the return stroke with the high current begins. Nevertheless, the 
published current measurements vary from 0.1 to 6kA with the average 
value of 1.7 kA [33]. The lower limit of the range is more typical for the 
first component (this may be the next component, too, but after a long 
current-free pause, when the previous trace channel has nearly totally 
decayed). The value of several kiloamperes seems reasonable, since the 
velocity of a dart leader is 30-50 times higher than that of the first 
component. 
3.5 
Field variation at the leader stage 
The subdivision of experimental data between this and the previous section is 
somewhat arbitrary. Electric field measurements provide information about 
leader charge, while charge and current are related by leader velocity. On the 
whole, this is a general problem. If the observations were arranged properly 
and the data analysis was made carefully, relatively simple field measure- 
ments can add much to our knowledge of electrical parameters of lightning. 
The knowledge of the field itself is rarely of importance, probably, except in 
some applied problems of lightning protection of low voltage circuits. For 
this reason, it is not the measurements but, rather, methodological 
approaches to their treatment which are significant. So we shall begin with 
these approaches and the general principles underlying a treatment of most 
lightning stages. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 111 ===
Field variation at the leader stage 
103 
- 
Figure 3.12. Measurement of fast variations in the electric field during the lightning 
development. 
Suppose the electric field at an observation point near the earth’s surface 
is Eo(0) at the moment of the lightning start. When the discharge is com- 
pleted, the field takes a new value Eo(tm). The field has changed by the 
value AEo = Eo(tm) - Eo(0) over the time t,. 
When the measurement time 
is relatively short, it is more convenient to register the field change rather 
than to measure its values. This is usually done with electrostatic antennas, 
i.e., metallic conductors (normally, flat) grounded through a reservoir capa- 
citor C (figure 3.12). If the capacitor and the measurement circuits connected 
to it have an infinitely high leakage resistance RI, 
the capacitor voltage, at 
any moment of time, is 
(3.3) 
where Sa is the area of a flat antenna and qc is the charge induced on it. If RI 
is finite (which is always the case due to the input resistance of the circuit 
taking voltage readings from the capacitor), the use of (3.3 ) requires the con- 
dition RIC >> t,, which can be easily met for the lightning duration ~ 1 0 - *  
s 
but becomes problematic for a time interval of several minutes between two 
flashes. 
An accurate measurement of the field change AEo(t,) requires the 
necessary time constant of the measurement circuit RIC and the account of 
effects of external field variation in the atmosphere, Eo, by making allowance 
for local effects. (The antenna may be raised above the earth, say, mounted 
on a building roof, so that the field there will be higher than on the earth. On 
the other hand, a nearby high construction may reduce the field, acting as an 
electrostatic screen.) The field value obtained is not particularly informative. 
In order to get information about the lightning discharge, we have to make 
certain assumptions concerning the distribution of charges which have 
changed the field. 
Let us begin with a simple illustration. Suppose a lightning leader 
passing from a spherical volume has changed the charge of only one sign 
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=== PAGE 112 ===
104 
Available lightning data 
in a storm cloud cell. If there are other changes in the sphere charge during 
the leader travel, they are assumed to have been completely neutralized later, 
at the return stroke stage. If this assumption is correct, the measured value of 
AEo(tm) can give an idea about the quantity of charge transported by the 
leader from the cloud to the earth: 
(3.4) 
27r~o(H~ 
+ R2)3/2AEo(tm) 
H 
AQM = 
Here, H is the altitude of the charged storm centre and R is its radial dis- 
placement relative to the registration point. Both parameters should be 
measured by an independent method or simultaneous field registrations at 
two more points should be made at given distances from the first one. This 
will provide additional equations for the unknown values of H and R. 
Such an unambiguous treatment results from the simple model we have 
chosen, which contains no geometrical parameter except for the distance to 
the charge. However, a slightly more complicated, dipole model deprives the 
measurement treatment of this advantage. Still, electric field measurements 
have always been attractive to lightning researchers owing to their 
simplicity. Interest in such measurements increased with the application of 
lightning triggering by small rockets raising a grounded wire to 150-300 m 
above the earth’s surface (triggered lightning). The first component of such 
lightning is genuinely artificial, but then the first trace channel is used by 
practically natural dart leaders travelling to the earth. Their point of contact 
with the earth is predetermined, so field detectors can be placed at any dis- 
tance from the leader. This registration system is quite sensitive and capable 
of responding to the linear charge density not far from the leader tip when it 
approaches the earth. 
To illustrate our analysis, we shall use the field measurements described 
in [34,35]. The authors of this work kindly made them available to us after 
their discussion at the IXth International Conference on Atmospheric Elec- 
tricity, held in St. Petersburg in 1992. The files contained detailed records of 
electric fields, taken during the flight of dart leaders, and of their return 
stroke currents. Detectors were placed at the distance of R = 500m and 
30m from the contact point. Regretfully, the recordings at these distances 
were not simultaneous but made in different years. Their comparison is 
still possible because the fields were recorded at the same time as the 
return stroke currents. By sorting out identical current oscillograms, one 
can select lightning discharges with about the same leader tip potentials. 
This provides close values of leader velocity and linear charge density in 
the charge cover not too far from the leader tip. Some representative oscillo- 
grams of AE(t)/AEmax normalized by their amplitudes are shown in 
figure 3.13. They correspond to discharges with really close currents in the 
return strokes (IM 
= 6 kA at point R = 500 m and 7 kA at point 30 m). The 
amplitude values of field variation AE,,, 
over the time of the dart leader 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 113 ===
Field variation at the leader stage 
105 
w 
0 
100 
200 
300 
400 
500 
I
0.8 
1.0- 
0.8 - 
0.6 - 
0.4 - 
10 
20 
30 
40 
Time, ps 
Figure 3.13. Oscillograms taken in Florida, USA [35], from the vertical field com- 
ponent during the development of the dart leader in the subsequent component of 
a triggered lightning. The detectors were positioned at 30 and 500 m from the point 
of strike; the pulses are related to their maximum amplitudes. 
travel were 6.9 V/cm and 120 V/cm, respectively. Note that the measure- 
ments in [35] result in AEm,,/IM M const at every point. There is no 
geometrical similarity between the pulses AE( t)/AEmax at the different 
points. On the contrary, there is a sharp difference in the rates of strength 
rise, as a dart leader was approaching the earth. The field increase in the 
range (0.5-1.0)AEm,, took Atl12 = 76ps for point R = 500m and only 
5 ps for point R = 30 m. 
These data will be treated in terms of a simple model, in which a dart 
leader is represented as a uniformly charged axis with linear charge density 
rL. Naturally, the real cover radius R, can be ignored in the field calculation 
at a distance R. We shall show below that field calculations can only take into 
account the charge distribution along a relatively short length behind the tip, 
comparable with R. This will justify the assumption of rL being constant, 
because it actually refers to a short length of about R near the tip. Therefore, 
the field change due to the leader charge at point R at the earth, with the 
allowance for its mirror reflection by the earth, is described as 
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=== PAGE 114 ===
106 
Available lightning data 
where h is the height of the leader tip from the earth at the moment of regis- 
tration and H is the height of the leader base. The field change is maximum 
when the tip contacts the earth, and for R << H this gives 
It indeed follows from (3.6) that only the charge distribution along a short 
length comparable with R is important for field evaluation. (For example, 
at H > 5R, the error of the model with rL = const will be less than 20% 
for any charge distribution, unless TL grows rapidly from the tip toward 
the base; but there is no reason for this, because the channel field E, is 
weak and the cloud potential does not vary much.) 
Formula (3.6) allows charge density evaluation with a good accuracy, 
since the leader is strictly vertical at the earth - it reproduces the path of 
the rocket taking up the wire which has evaporated. The value calculated 
from the measurements at point R = 30m appears to be unexpectedly 
small: rL x 2 x lop5 Cjm. Nearly as much charge is transported by long 
laboratory sparks (section 2.4). The potential of a lightning leader tip, 
U,, does not seem to be much larger than that of a laboratory spark. 
According to (2.8) and (2.35), the linear leader capacitance is 
C1 FZ 2mo/ln (H/RL) x (2-10) x 
F/m, even with indefinite leader 
radius RL. From this, we have U, x rL/C1 x 2-10MV. 
The velocity of a dart leader proves to be very high. For its evaluation, we 
shall use the measured value of Atlp, which is 5 ps for R = 30 m. Formula (3.5) 
gives AE = AE,,,/2 
at h = J?;R. Hence, the average velocity along a path of 
length h FZ 50m at the earth’s surface is VL FZ f i R / A t 1 / 2  = lo7 m/s, quite 
consistent with direct measurements. It should be emphasized that this velocity 
refers to the perfectly vertical path at the earth’s surface, so it is the true velocity. 
Similar evaluations can be made with the measurements at the far point 
R = 500 m but with a lower reliability, since the parameter averaging is to be 
made over a leader length of about lo3 m with an unknown path. Neverthe- 
less, the values of T~ = 2.3 x 
Cjm and VL = 1.15 x lo7 mjs are found to 
be close to those above. It will be shown in the next section that an indefinite 
trajectory may produce an error much larger than the obtained difference in 
the values of TL and VL. So the dart leader of triggered lightning with the 
definite path at the earth is a lucky exception. 
Another illustration of AE(t), cited in [35], characterizes a more power- 
ful dart leader. The current amplitude in the return stroke was as high as 
40 kA. The maximum field change was found to be AE,,, 
= SlOVjcm, 
i.e., a little more than a value proportional to current, while the characteristic 
time of the process, Atlp, decreased to 1.8 ps. Calculations similar to those 
described 
above 
give 
T~ x 1.35 x 10-4C/m, 
U, =20-30MV, 
and 
VL = 2.9 x 107m/s, thereby supporting the hypothesis of a direct, though 
not very strong, dependence of the leader velocity on the tip potential. For 
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=== PAGE 115 ===
Perspectives of remote measurements 
107 
the calculated values of linear charge and velocity at the earth, the leader cur- 
rent is found to be iL = T~ VL = 3.9 kA, only an order of magnitude lower 
than the current amplitude in the return stroke. 
3.6 Perspectives of remote measurements 
What we described in the previous section is a very favourable situation, in 
which the point of leader contact with the earth is fixed and its final path 
is strictly vertical, at least, at a length of 150-300m above the earth. One 
should not expect such favourable conditions for natural lightnings, espe- 
cially for their first components. Still, one should take quietly and with 
some scepticism the idea of indirect remote measurements of lightning 
parameters. The experimeter resorts to them because, otherwise, his life 
would turn out too short to bring his experiment to a conclusion. Recon- 
struction of an electromagnetic field source from strength measurements 
made at definite points is an incorrect solution to a fairly common problem 
of electrodynamics in various areas of science and technology. Lightning is 
not an exception to the rule. We shall consider critically the treatments of 
results obtained from solutions to such problems and discuss inverse electro- 
static problems, as applied to the lightning leader. 
Generally, the density of space charge p(x, y. z )  between some boundary 
surfaces can be found if the electric field in the whole confined volume is 
known. Experimentally, this means simultaneous field measurements at an 
infinitely large number of points, which is practically unfeasible. A well 
organized service for field lightning observation has, at best, several synchro- 
nized field detectors. A theoretical treatment of the field records always 
suggests an a priori construction of a simplified field source model. The 
inverse problem can be solved if the number of unknown parameters in 
this model does not exceed the number of registration points. What follows 
is quite obvious. One writes down a set of equations with the measurements 
on the right and the expression for field at a given point (derived from the 
model with yet unknown charge parameters) on the left. The solution defines 
the parameters as rigorously as the measurements permit. One should always 
remember, however, what has been found from the equations, since these are 
parameters of a speculative model rather than a real phenomenon. How 
much they coincide is not a matter of accuracy of measurements or calcula- 
tions but that of the model adequacy ‘to the phenomenon under study. Most 
often, it is here that possible errors originate. 
3.6.1 
Without claiming a general analysis, we shall consider a special but fre- 
quently used model of near-earth field variation at a large distance from a 
Effect of the leader shape 
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=== PAGE 116 ===
108 
Available lightning data 
I
I
 
<” 0.00- 
w 
U z 
WO -0.02 - 
e 
N 
-0.04- 
-0.06- . 
Figure 3.14. Electric field variation at the earth, evaluated for a large distance from 
the vertical channel of a descending leader. 
descending lightning leader. In this model, the leader is represented by a thin 
vertical uniformly charged thread with a linear charge density rL, and the 
storm cell, from which the leader started, is taken to be so small that it is 
replaced by a point charge Q at height H. The value of Q may be unknown, 
because the analysis uses the time variation of the field rather than its 
absolute value [l]. 
The field at point R near the earth (figure 3.14(a)) varies in time for two 
reasons. The absolute field decreases because of the charge reduction in the 
storm cell by the charge AQ = TLL carried away by the leader on its way 
to the earth.t The second component is due to the charge accumulation on 
the leader of length L; as the leader moves on, this charge goes down, 
enhancing the field at the earth. As a result, for the change of the vertical 
field component at moment t with L = VLt, where VL is average leader 
velocity, we have 
(3.7) 
AQH 
+‘“I 
(H-x)dx 
AE(L) = - 
27rq,(H2 + R2)312 2% 
o [(H - x)’ + R2I3/’ ’ 
This expression takes into account the doubled field associated with the 
earth-induced charge. The evaluation of the integral gives the known 
expression 
- 
LH 
}. (3.8) 
1 
1 
AE(L) = - 
[(H - L)’ + R2I1/’ - (H’ + R’)”’ (H‘ + R2)3/2 
t This component may be ignored in field measurements at a small distance from the leader, as 
described in section 3.5. 
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=== PAGE 117 ===
Perspectives of remote measurements 
109 
When analysing this expression, an experimenter will find it difficult to avoid 
a temptation. In the range of R/H < 1.4, the function AE(L) has an 
extremum (figure 3.14(b)) which can be easily recorded by an oscillogram. 
What remains to be done is to find the height H a t  which the lightning started 
(it may be taken to be equal to the average height of the storm front), to 
measure the distance between the observation point and the leader (e.g., 
from the thunder peal delay time, since the sound velocity is known and 
the point of sound wave excitation is near the earth's surface), and to find 
the moment of time when the leader tip descended to the height H - L, 
by calculation, from (3.8), the leader length L, corresponding to the 
extrema1 point. At least one more moment of time is registered exactly by 
the oscillogram - the moment of the leader contact with the earth, giving 
rise to the return stroke. The oscillogram indicates this moment by a 
field strength overshoot. The time interval At in figure 3.14(b) defines the 
leader average velocity along the path length H - L, at the earth: 
VL M ( H  - L,)/At. 
Note that one of the boundaries of the measured 
length might also be found from the moment of sign reversal of the field 
being registered. It follows from the analysis of (3.8) that the curve AE(L) 
intercepts the abscissa if the registration point lies at a distance R z (0.8- 
1.4)H from the vertical path axis. Technically, the reference point is easier 
to find than the extremum. 
It is known that the appetite comes with eating. If one substitutes the 
geometrical parameters used and the measured values of AE(L) into (3.8), 
one can find the average linear charge density T ~ .  
Together with the velocity, 
this provides the average current in the leader for the final period of time 
At: iL M T ~ F ' ~ .  
The calculation of charge density was replaced in [36] by 
graphical differentiation of the oscillogram E( t )  at the point corresponding 
to the moment of leader contact with the earth; this, however, gave a 
low accuracy. Therefore, the electric field registration only at one point on 
the earth's surface seemed to be sufficient to evaluate one of the least 
accessible parameters - the leader current in a descending lightning 
discharge. 
Let us now try to assess this situation without considering the measure- 
ment errors. Obviously, the main error is associated with finding the starting 
point of a lightning spark and the distance to it. A common 10% error in 
measurements gives much larger errors in evaluations of leader velocity 
and current. This always happens when one deals with the difference of 
two comparable parameters. We shall focus on errors of the model itself. 
The problem of leader branching effects will be ignored. After all, one can 
always consider a dart leader which has no branches. The representation 
of a real leader as a vertical axis is quite another matter. Any photograph 
shows numerous bendings of a lightning trajectory, so a straight vertical 
leader is nothing more than a speculative mathematical concept. To assess 
its implications, let us make another step and consider a tilted straight 
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=== PAGE 118 ===
110 
Available lightning data 
0.054 
L/H 
Figure 3.15. Electric field variation at the earth for a leader deviating from its vertical 
path; RIH = 0.7. 
leader. Suppose a leader moves in a vertical plane passing through a field 
detector and is tilted towards it by the angle Q relative to the vertical line 
(figure 3.15). Then, instead of (3.8), we have 
where a = R - H tan a. The result of numerical integration of (3.9) is 
presented in figure 3.15 for R = 0.7H. Even a slight tilt from the vertical 
line (a zz 20") entails a nearly three-fold (two-fold for R / H  = 0.7) change 
in the pulse amplitude AE(L) and in the parameters usually derived from 
field measurements at the earth's surface. The value of L corresponding to 
the extremum AE(L) depends only slightly on the tilt, but the signal ampli- 
tude variation is sufficient to make one treat the derived leader parameters 
only as estimations of orders of magnitude. It is quite another matter if the 
leader shape is recorded simultaneously with electric field registration at 
two points. The leader trajectory can then be reconstructed in space quite 
accurately, and a computer processing of records can completely eliminate 
this type of error. 
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=== PAGE 119 ===
Perspectiws of remote measurements 
111 
3.6.2 
The model of a leader with a uniform charge distribution and the concept of a 
storm cell as an electrode with a capacitor battery supplying conduction 
current to the leader are extremely far from reality. A storm cloud does 
not look like a giant capacitor plate, to which a lightning leader is connected 
galvanically during its motion to the ‘plate’ of opposite sign, i.e., to the earth. 
In actual reality, the cloud charge is concentrated on hydrometers which do 
not contact one another and their assemblage does not possess the properties 
of a metallic electrode. A better analogy would be that of an electrode-free 
spark, rather than of a spark starting from a high voltage electrode of a 
laboratory generator. 
To illustrate this, consider a small metallic rod suspended along the 
field vector in an inter-electrode gap, where the field is supported by a 
high-voltage generator. The rod has no contact with the generator poles. 
Two sparks of opposite sign are excited simultaneously at the rod ends, 
i.e., in the region of enhanced local field (figure 3.16). The charges appearing 
on the sparks must be regarded as polarization charges. This is the way a 
Effect of linear charge distribution 
Figure 3.16. Streak photographs of a simultaneous development of a positive and a 
negative leader from the ends of a metallic rod of 50 cm in length in a uniform electric 
field: (1) rod; (2,3) channel and streamer zone of a positive leader; (4,5) 
the same for a 
negative leader. 
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=== PAGE 120 ===
112 
Available lightning data 
metallic conductor is polarized when it is introduced into an electric field. 
Similarly, a charged storm cloud possessing no conductivity is only a field 
source in the space extending to the earth. A plasma conductor arising in 
this way or other is polarized in the field and grows, being supplied by polar- 
ization current. This system is definitely not a perpetuum mobile. Its energy 
source is the electric energy of the cloud field. As the leader develops, this 
energy decreases, in accordance with the conservation law. If the external 
field and the conductor are homogeneous, the linear density of polarization 
charge is equal to zero exactly at the conductor centre and its absolute value 
rises towards the ends of different polarities. As long as the conductor has no 
contact with the high-voltage generator terminals, its total charge, naturally, 
remains equal to zero. The latter is also valid when the field and conductor 
are inhomogeneous. Using numerical methods, one can find the polarization 
charge distribution for any electric field. The distributions presented in 
figure 3.17 have been found by the equivalent charge method [37]. This 
method is simple and convenient for long conductors, like those used to 
simulate lightning leaders. 
Numerical computations show that a uniform field in a perfectly con- 
ducting rod creates a polarization charge 7 rising almost strictly linearly 
from the rod centre towards its ends. The ends are an exception, because 
Figure 3.17. Polarization charge distribution along a straight conductor (a leader 
system) in the cloud dipole field, with allowance for a dipole reflection in the conduct- 
ing earth. 
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=== PAGE 121 ===
Perspectives of remote measurements 
113 
the charge density here (along a length of the rod radius) rises rapidly. The 
charge distribution at the rod base can be approximated as T ( X )  = ax, 
where the coordinate origin x is at the rod centre. The reader will soon see 
that the contribution of the end charges fq is small at larger distances, if 
the rod length 2d is much greater than its radius r. 
For simplicity, let us assume that the charge is concentrated on the outer 
rod surface, as is the case when it has a conductivity, and that the potential 
will be calculated along the longitudinal axis. The rod centre will be taken as 
the zero point of the external field Eo potential. The potential p at the point x 
is a sum of potentials created by the external field, -Eox, the end charges, pq, 
and the charges distributed along the rod, pr: 
1 
(d - x )  + [(a - x12 + r2] '12 
- [(d + x12 + r2] 1/2 + x In 
-(d + x )  + [(d + X)* + r 2 p 2  
(3.10) 
Here, the last approximate expression refers to the rod sites lying far from its 
ends, Id f 
X I  >> r. Here, the term pq can be neglected, and we shall approxi- 
mately have pT - Eox = 0. With the actual charge distribution ~ ( x )  
and the 
end charges providing pq, the rigorous equality p ( x )  = 0 should be valid 
along the whole rod length. By relating the approximate equality to the 
centres of the semi-axes x = fd/2, we find 
(3.11) 
The potential at the rod ends must be calculated with the account of 
their higher charges. Assuming this charge to be concentrated along the 
end circumference, the potential at the centre of the end plane (at the 
points x = fd on the axis) can be described as 
(3.12) 
The potential pr must now be calculated from the unsimplified formula 
(3.10). It follows from (3.12) that the end charge is approximately equal to 
q x 27r~orEod and by a factor of 
ad2 
d 
2q 
2r In (&/er) 
K=-NN 
(3.13) 
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=== PAGE 122 ===
114 
Available lightning data 
smaller than the charge distributed over each half (it is an order of magnitude 
smaller for d / r  z 100). Therefore, the account of localized tip charge may be 
necessary only for the calculation of electric field in the region close to the tip, 
at a distance less than 10r from it, In a remote region, where measurements 
are usually made, such a subtlety is unnecessary - it is sufficient to consider 
only the charge distribution along the leader channel. Clearly, this is not a 
uniform distribution, taken for granted by some researchers. 
It is time to look at the shape of a field strength pulse at the earth, deter- 
mined by the charge of a linearly polarized vertical axis with charge 
~ ( x )  
= f a x  per unit length. It is defined by the algebraic sum of terms 
from the positively and negatively charged semi-axes and is equal to 
AE(L) =r a x(H - X) dx 
-L 2 r E o  [(H - x)’ + R2I3/’ 
where L are the lengths of leader sections which have moved away from the 
starting point to the earth and upwards. Integration with (3.1 1) gives 
AE(L) = 
L 
L 
In (fiL/er) 
[(H - L)’ + R2]‘/’ - [(H + L)’ + R2I1/’ 
- In 
[H + (H’ + R2)”’]2 
Eo 
[ 
{ H  - L + [(H - L)* + R2I1/’}{H + L + [(H + L)’ + R2]’/’J 
(3.14) 
Here, H is the height of the leader start, r is its radius, and R is the 
distance between the leader axis and the observation point. It can be 
shown that the function AE(L) of (3.14) rises smoothly with L and has 
no extrema. 
The linear charge distribution assumed in the above illustration is, of 
course, another speculation (section 4.3). Moreover, a leader goes up and 
down non-uniformly, and the field in the earth-cloud gap is far from 
being uniform: its strength decreases towards the earth. This limits the 
linear charge growth from the start downward. The finite channel conductiv- 
ity exhibits similar behaviour, reducing the tip potential. So it is impossible to 
find the actual charge distribution exactly without knowing these param- 
eters. Thus, a processing of field oscillograms can give nothing more than 
what they actually show. The field at a point is an integral effect of the 
whole combination of charges created or transported by a given moment 
of time. It is probably worth speculating about registrations but one 
should assess the results soberly, considering all possible variants and 
insuring oneself whenever possible. The best insurance is, of course, to 
increase the number of registration points and parameters determined by 
independent methods. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 123 ===
Lightning return stroke 
115 
3.7 Lightning return stroke 
All lightning hazards are associated with the return stroke, and this accounts 
for the great effort of investigators to learn as much as possible about this 
discharge stage. It has been established that the contact of a descending 
lightning leader with the earth or a grounding electrode produces a return 
wave of current and voltage. It travels up along the leader channel, partially 
neutralizing and redistributing the charge accumulated during the leader 
development (figure 3.18). The travel is accompanied by an increased light 
intensity of the channel, especially at the wave front. At the earth, the 
wave front intensity acquires its maximum over 3-4ps [31]. As the wave 
goes up to the cloud, the wave intensity steepness and amplitude decrease 
many-fold, indicating a considerable decay. Judging by streak pictures, the 
region of a high light intensity at the wave front extends to 25-1 10m. The 
whole wave travel takes 30-5Ops. This time is especially convenient for 
electron-optical 
methods of streak photography. However, available 
attempts to use such methods can hardly be considered successful. A serious 
obstacle is the exact synchronization of a streak camera and lightning contact 
with the earth. Although there are many synchronization methods, they have 
no simple technical solutions and are seldom used in lightning experiments. 
Continuous (e.g. sinusoidal) electron streak photography has not justified 
hopes. Basic results on return stroke velocities have been obtained using 
cameras with a mechanical image processing, which do not need synchroni- 
zation (Boyce camera). 
Figure 3.18. Scheme of the return stroke propagation after the contact of a descend- 
ing leader with the earth (at moment t = 0). A leader brings potential U < 0; ZM is 
return stroke current. 
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=== PAGE 124 ===
116 
Available lightning data 
1.0 
0.8 
.- 
O 0.6 
3 
& 0.4 
- 
cd 
D 
0.2 
3.7.1 Neutralization wave velocity 
The measurements made half a century ago [25,39] and those performed 
recently [40] indicate a high velocity of a return current-voltage wave. The 
minimum measured values are close to (1.5-2) x lo7 mjs and the maximum 
ones are an order of magnitude higher, reaching 0.5-0.8 of light speed c. A 
velocity comparable with light speed does not mean that we deal with 
relativistic particles or purely electromagnetic perturbations. The wave 
velocity is the phase velocity of the process. 
There are not so many successful optical registrations of the return stroke, 
the number of really good ones being about 100. Most of the available data 
concern subsequent lightning components. This is natural because every 
successfully registered discharge includes the return strokes of several compo- 
nents. The wave velocities of subsequent components are somewhat higher 
than those of the first ones. According to [40], the first component has an aver- 
age velocity V, x 9.6 x lo7 mjs while the subsequent ones are a factor of 1.25 
higher. Similar data are cited by other authors for subsequent components of 
lightning discharges triggered from a grounded wire elevated by a rocket. 
To illustrate the statistical velocity spread in individual measurements 
and in those made by different researchers, figure 3.19 shows integral 
distribution curves for the data of [25] and [40]. The first and subsequent 
- 
- 
- 
- 
- 
0.01 - . * 
.
I
 
1 
0.05 
0.1 
0.2 
0.5 
1 
v,ic 
Figure 3.19. Velocity distribution of the lightning return stroke: (1) averaged over the 
visible channel length [25]; (2) averaged over 1.3 km above the earth [40]. 
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=== PAGE 125 ===
Lightning return stroke 
117 
components were not separated. Within a 50% probability, there is a 2-fold 
difference between the velocities. Earlier measurements generally give lower 
return stroke velocities. The point is that most measurements performed 
during the 1980s were two-dimensional, usually providing higher velocities, 
whereas the earlier data had allowed conclusions only about the vertical 
component of velocity. Moreover, the application of improved optics and 
photographic materials, as well as higher relative motion rates of the 
image and film, improved the time resolution of streak photographs. As a 
result, the velocity value obtained in the 1980s was more accurate and 
higher because the measurements were averaged over the initial stroke 
length of about 1 km at the earth’s surface, where the wave moves 1.5-2 
times faster, rather than over the whole stroke length. 
All measurements show that the return stroke velocity gradually 
decreases and that the velocity V, drops abruptly when the wave front 
passes through the point of leader branching. The latter fact suggest a certain 
relation between the stroke velocity and the current transported by the wave: 
at the branching point, the current is divided among the branches, so the 
velocity becomes lower. The knowledge of this relation could improve the 
calculation accuracy of overvoltages in electrical circuits during lightning dis- 
charges. Unfortunately, the available data are insufficient to allow finding 
this relation reliably. Simultaneous registrations of current and velocity 
have been made only for return strokes of subsequent components of 
triggered lightnings but they cannot provide a representative statistics. 
With reference to [12,41], there is note in [l] about a satisfactory agreement 
between these registrations and Lundholm’s semi-empirical formula 
V,/c = (1 + 40/1M)-1’2, where lM is a return stroke current amplitude 
expressed in kA (see section 3.7.2). The lack of factual data is sometimes 
compensated by a superposition of distribution statistics. It is assumed 
that the values of current and velocity characterized by an equal probability 
correspond to each other. There are no serious arguments in favour of this 
operation but it is used for the lack of a better method. 
3.7.2 Current amplitude 
The current amplitude is an important lightning parameter. Most hazards of 
lightning are associated, directly or indirectly, with stroke currents, whose 
registration has taken much time and effort. Very few of them were made 
by direct methods, using a shunt and a Rogovski belt [28,29,42-461. Still 
fewer direct measurements were made by equipment with a wide dynamic 
range, which can register both powerful impulses with an amplitude to 
200 kA and low currents of a few hundreds of amperes, which are equally 
important for the understanding of the lightning physics. 
A large number of measurements have been made by magnetic detec- 
tors. Such a detector represents a rod several centimetres in length, made 
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=== PAGE 126 ===
118 
Available lightning data 
from magnetically hard steel. Preliminarily demagnetized rod detectors were 
placed at a fixed distance from a conductor aimed at leading lightning current 
to the earth. This could be a grounding lead of a lightning conductor or a 
metallic tower of a power transmission line. With the appearance of lightning 
current, the detector proves to be within the range of its magnetic action and 
becomes magnetized. One measures the residual steel magnetization and 
calculates the current by solving the inverse problem. The advantages of 
this method are its simplicity and low cost. Usually, magnetic detectors are 
installed by the thousand to obtain the necessary statistics. However, they 
can yield nothing else but a current impulse amplitude. Of course, by mark- 
ing the ends of the detector, one can also determine the direction of current 
and attribute it to the lightning type (positive or negative). The accuracy of 
current measurements is very low for several reasons. 
First, there are few objects with a simple system of current spread over 
metallic constructions. A single conductor would be ideal in this respect, 
because it excludes current branching. In reality, lightning current is distribu- 
ted among many conductors, the distribution pattern being unpredictable 
since it varies with temporal parameters of the impulse. We shall illustrate 
this situation with reference to a simple system consisting of two parallel induc- 
tively connected branches with their own inductances L1 and L2, mutual 
inductance M ,  and resistances R1 and R2. Suppose a rectangular current 
impulse I with a short risetime is applied to the system. The current distribu- 
tion between the two branches is described as 
Initial currents ilo and i20 at the stage when current I(t) is stabilized to Z 
are generated over a very short time equal to the I risetime. The branch 
currents, therefore, rise from zero very quickly. The reactive components of 
voltage drop -di/dt produced by them are much larger than the ohmic 
ones -i 
that can be neglected for the time being. Hence, we have 
i1o/i20 = (L2 - M ) / ( L 1  - M ) ,  and the initial current, say, in the first branch 
is ilo = Z(L2 - M ) / ( L 1  + L2 - 2 M ) .  When the transitional process, whose 
duration is defined by the time constant At = (Li 
+ L2 - 2M)/(RI + R2), 
is over, currents ilx = IR2/(R1 + R2) and i2x = ilxR1/R2 are established 
in the circuits. The durations of lightning currents are usually comparable 
with the time constant At. Therefore, a magnetic detector placed in one of 
the branches will register a current intermediate between the initial and estab- 
lished values having a maximum amplitude, since the residual magnetization 
of the rod contains information only about the maximum magnetic field of 
current. For this reason, one can calibrate a magnetodetector for deriving a 
full current amplitude only if the impulse shape is known. This cannot be 
done in a real experiment, so one has to resort to a rough estimation of current 
distribution over metallic constructions and use it in data processing. 
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=== PAGE 127 ===
Lightning return stroke 
119 
Second, the operating range of the rod magnetization curve is not large, 
and the transition from a linear to saturation region may produce additional 
errors in data processing. To avoid saturation, the magnetodetector is placed 
far from the conductor, which creates difficulties in data processing of light- 
nings with low current and low magnetic field. Besides, when the distance 
between a conductor and a detector is large, the magnetic field effects of 
other metallic elements with current are hard to take into account. So a 
100% error does not seem too high for magnetodetectors, even when several 
detectors are placed at different distances from a current conductor. Their 
records provide sufficient material for engineering estimations or for a 
qualitative comparison of storm intensity in different regions, but they are 
insufficient for theory. Organization of direct registrations takes much time 
and effort. There are no more than a hundred successful registrations 
made over a decade. Let us see what information can be derived from them. 
Current impulse amplitudes vary widely, from 2-3 to 200-250 kA. 
Some magnetodetector measurements give even 300-400 kA, but these 
amplitudes seem doubtful. According to [42,46], the integral amplitude dis- 
tributions for the first and subsequent lightning components obey the so- 
called lognormal law, in which it is current logarithms, rather than currents 
themselves, that meet the normal distribution criterion. The probability of 
lightning with a current larger than ZM, is defined as 
(3.16) 
where (lg I)av 
is an average decimal logarithm of the currents measured and 
olg is the mean square deviation of their logarithms. This approximation 
cannot be considered accurate. The relative deviation of the value of (3.16) 
from the real one may be several tens percent; it may be even more for prac- 
tically important current ranges. Nevertheless, lognormal distributions allow 
measurement comparison and serve as a guide to engineering estimations. 
For example, about 200 current oscillograms for lightnings that struck the 
70m tower on the San Salvatore Mount in Switzerland [42] satisfactorily 
obey the lognormal law with (lgZ)av = 1.475 and olg = 0.265 for the first 
component currents of a negative lightning discharge. This means that the 
50% current value is estimated to be 30 kA; 95% of lightnings must have 
currents exceeding 4kA and 5% of lightnings 80kA. The probability of 
higher currents rapidly decreases: 100 kA is expected in 2% of cases and 
200kA in less than 0.1% of cases (figure 3.20). It should be emphasized 
again that the distribution boundaries must be treated with caution. The 
curve shape in the low current range strongly depends on the sensitivity of 
the measuring instruments used (its left-hand limit is usually taken to be 
1-3 kA in distribution plots). There are few measurements in the high current 
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=== PAGE 128 ===
120 
Available lightning data 
1 
0.01 
I
'
I
.
I
.
I
.
1
 
50 
100 
150 
200 
250 
Lightning current, kA 
Figure 3.20. Lognormal distributions of return stroke currents: A, the first com- 
ponent of a negative lightning with (lg I)," = 1.475 and clg = 0.265; B, subsequent 
components with (lgZ),v = 1.1 and olg = 0.3; C, positive lightnings with 
(lgZ)," = 1.54 and clg = 0.7. 
range: it is considered as good luck if they provide a reliable order of magni- 
tude. Note that negative lightning currents above 200 kA have never been 
registered reliably. 
The approximation of data on subsequent lightning components in [42] 
gives a much lower integral probability for high currents. A lognormal distri- 
bution can be satisfactorily described by (lg& = 1.1 and clg = 0.3. The 
calculated 50% current is 12.5kA, 5% current is only 39kA, and the 
chance for a subsequent component to exceed lOOkA is close to 0.1% 
(figure 3.20). 
The statistics for positive lightnings, whose number is about 10% of the 
total registrations, is less representative. All descending positive lightnings 
are one-component. The integral current distribution for them has a large 
spread. The probabilities of both low and high currents are larger without 
an essential change of the 50% value. The 50% value is close to 35 kA, 
i.e., it is nearly the same as for the first component of negative lightning. 
An approximate description of the lognormal distribution of positive cur- 
rents in [42] can be made with (lgI)av = 1.54 and clg = 0.7 (figure 3.20). 
Positive high current lightnings are more frequent than negative ones. A 
5% probability corresponds to 250 kA, and 100 kA can be expected with a 
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=== PAGE 129 ===
Lightning return stroke 
121 
20% probability. Among 26 successful registrations of positive lightnings in 
[43], only one showed 300 kA current. It seems likely that many positive light- 
nings were, under the observation conditions [43], ascending ones, which 
may account for the large spread. Such lightnings have practically no 
return stroke, and the equipment seems to have registered the relatively 
low leader current of the final development stage. These data were used to 
derive the integral distribution extending to the low current region. 
The great importance of lightning current statistics to applied lightning 
protection necessitated a unification of theoretical distribution curves. Other- 
wise, engineers would have been unable to compare the frequency of harmful 
lightning effects and protection efficiency. This work is being done within the 
frame of the CIGRE (Conference Internationale des Grands Reseaux 
Electrique a haute tension) - an operating international conference on 
high-voltage networks. Data on current from all over the globe are collected 
and analysed. However, there is no unified approach to these data: different 
data are discarded for different reasons, so that the distributions obtained 
differ markedly. For example, a report submitted to [47] compares two log- 
normal laws with (lg&" = 1.4 and 1.477al, = 0.39 and 0.32. The latter is 
preferable for power transmission lines, since the measurements for objects 
higher than 60 m were excluded from this derivation (power transmission 
lines are usually lower). In his book, Uman [l] gives a table of lightning 
currents mostly based on the measurements of [42]. 
Attention to details is inevitable, since slight corrections in parameter 
distributions may cause manifold changes in the calculated probabilities of 
currents above 100 kA, especially important in lightning protection of 
important objects. Both theory and applications suffer from a lack of light- 
ning current measurements. We shall list here some key issues to be discussed 
in more detail below. 
We have mentioned the importance of an object's height. It has been 
known since Benjamin Franklin's experiments that hgh constructions 
attract more lightnings. It seems likely that the process of attraction depends 
on the potential of a descending leader. If this is so, the statistics of descend- 
ing leader currents for objects of various height may prove different: there 
will be a kind of lightning separation, In this case, a comparison of reliable 
current statistics for various objects could help resolve the much debated 
problem of lightning-object interaction mechanism. 
Of interest in this connection is the following fact. In the case of a very 
high construction, many first components are of the ascending type having 
no return stroke. But the first component is followed by subsequent descend- 
ing components, whose average stroke currents are lower than in subsequent 
components affecting low buildings. This suggests an involvement of high 
ground constructions in the formation of storm clouds. It appears that an 
ascending leader starts from a high construction before the cloud has 
matured. Its charge and potential are, therefore, lower. This accounts for 
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=== PAGE 130 ===
122 
Available lightning data 
the fact that subsequent components discharging an immature cloud will 
transport lower stroke currents than a mature cloud. 
Finally, an important issue is the effect of grounding resistance of objects 
on lightning current. This may provide information on the resistance of 
lightning itself. This resistance is to be introduced in equivalent circuits, 
when calculating overvoltages affecting various electrical circuits. This 
problem is still much debated: some investigators suggest the substitution of 
a lightning channel by a current source with an ‘infinite’ resistance, others 
ascribe to the channel the wave resistance of a common wire (about 300 0). 
It would not be hard to solve this problem if we had at our disposal reliable cur- 
rent statistics for objects of various height but different grounding resistances. 
No such statistics exist yet. To speed up the work in this area and to reduce its 
cost, various remote registration techniques are being employed. They register 
electromagnetic fields and coordinates of points where the lightning strikes (ide- 
ally, the lightning trajectory), followed by the solution of the inverse problem 
for the field source, i.e., lightning current (see section 3.7.4). 
There is also an increasing number of direct current registrations from 
lightnings triggered from a wire lifted by a rocket to the height of 150- 
250m. The first component of a triggered lightning (ascending leader) has 
no return stroke; therefore, one deals only with subsequent components. A 
comparison of such registrations with natural lightning currents was made 
in Alabama, USA [15]. The statistics were not particularly representative 
(45 measurements), so no principal differences were revealed. The lognormal 
distribution of currents corresponded to the parameters (lg& = 1.08 and 
olg = 0.28, nearly the same as those obtained in Switzerland for subsequent 
components of natural lightnings [42]. We should like to warn the reader 
against a possible overestimation of this coincidence. The comparison 
involved measurements from geographical points separated by large dis- 
tances, whereas the global variation of lightning parameters still remains 
unclear. A more important thing is that the lightnings studied in [42] 
cannot be regarded as totally natural. They struck a 70-m tower on a 
mountain elevated at 600m above the earth’s surface close to a lake. The 
conditions here are more similar to those of lightning triggering than to its 
natural development in a flat country. Lightning parameters are known to 
differ with altitude: currents registered by magnetodetectors at an altitude 
of 1-2km were two times lower than in a flat country, for less than 50% 
probabilities [48]. 
3.7.3 
Records of lightning current impulses look more like abstractionists’ 
pictures - they are so diverse and fanciful. The conventional approximation 
of a impulse by two exponents Z ( t )  = Io[exp (-at) - exp (-@)I, 
which is 
suggested in various guides to equipment testing lightning resistance, is 
Current impulse shape and time characteristics 
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=== PAGE 131 ===
Lightning return stroke 
123 
Figure 3.21. A schematic oscillogram of a current impulse in the first component of a 
negative lightning. 
intended for currents of laboratory sources simulating lightning, rather than 
for natural lightning. Let us try to identify the main features of the time 
variation of current, essential for the understanding of the return stroke 
mechanism and applications. 
The most reliable data have been obtained for first component currents 
of a negative lightning. This current is easy to register, since the impulse front 
takes several microseconds and an oscillographic record reproduces it in 
detail. A sketch of a current impulse averaged over many oscillograms is 
shown in figure 3.21 in two time scales. Note the concave shape of the 
front. An expression of the type 1 - exp (-pt) looks least suitable for its 
description. The first current peak is often followed by a higher one, and 
evaluation of the impulse risetime tf is associated with some reservations. 
For example, [42] measured the time of current rise from 2kA, a value 
close to the resolution threshold, to the first maximum Z,. 
In this case, 
about 50% of negative lightnings had the risetime of the first component 
over 5.5 ps, 5% exceeded 18 ys, and another 5% less than 1.8 1s. The knowl- 
edge of the risetime allows calculation of the average impulse slope 
AI 
= Z,/tf, 
However, the calculation of electromagnetic fields of lightning 
and the evaluation of possible hazards require a maximum slope 
AI,,, 
= (dZ/dt),,,, 
rather than an average one. The error in evaluations 
of this parameter from current oscillograms may be very large, because 
one has to replace the tangent to the Z ( t )  curve by a secant. Nevertheless, 
this operation has a sense for a fairly long impulse of the first component. 
The integral distribution of the values, like the current itself, is described 
by the lognormal law with the parameters (lgZ)," = 1.1 and glg = 0.255, if 
the slope is expressed in kA/ps. It results in 12 kA/ps for 50% current, and 
the slope exceeds 33 kA/ys with a 5% probability. 
To describe the electromagnetic effect of lightning current, let us find 
the induced emf U, in a frame of area S = 1 m2 placed at distance D = 1 m 
from the channel or a grounding conductor, when the first component 
current flows through it (the frame is in a plane normal to the current 
magnetic field). Even for a moderate steepness AI 
= 33 kA/ps, we have 
U, = p0AI maxS(2~D)-' 
= 6.6 kV, where p0 = 47r x lo-' Hjm is vacuum 
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=== PAGE 132 ===
124 
Available lightning data 
magnetic permeability. The role of a frame can be performed by any metallic 
structure within the construction affected by lightning-wires, wall fittings, rails, 
metallic stripes touching each other, etc. At the site of a poor contact, induced 
emf will produce a spark, much more effective than that in an electric lighter. 
This is dangerous because the spark may come in contact with an explosive gas 
mixture. 
Sometimes, lightning current behaves in a dual way, creating the induc- 
tion emf and voltage at the resistance of the grounding electrode U, = IR. It 
is important, therefore, to have knowledge about the relation between the 
current amplitude and maximum slope. Although both parameters obey 
the same lognormal law, no correlation has been found between them. 
This is bad for engineering applications, for one has to calculate the 
probabilities of each current with a whole set of possible slopes. 
There have been attempts at a more detailed description of the current 
impulse front. They were initiated by the CIGRE mentioned above to 
handle hazardous effects of lightning on power transmission lines. A set of 
additional parameters has been suggested to reduce errors in current oscillo- 
gram processing and some regions of the impulse front have been described 
quantitatively. This is illustrated in figure 3.22 and requires no comment. 
Some of the results were cited in 1471. The processing technique used did 
not lead to considerable data refinement, since the 50% maximum slope 
AI,,, 
= 12kA/ps is only two times larger than the 50% average slope 
A ~ o %  
= Z5oyO/tf5o~ = 30/5.5 = 5.5 kA/ps. But the factor of 2 is essential to 
the electrical strength of ultrahigh voltage insulation. 
I- 
Eo - 
Figure 3.22. The distribution of current impulse parameters in the return stroke, 
based on oscillograms. 
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=== PAGE 133 ===
Lightning return stroke 
125 
Current impulses of subsequent components have a shorter risetime. In 
the work cited above [42], 9 < 1.1 ps for a 50% probability and 0.2 ps for a 
5% probability. The latter should be treated with caution because this value 
is close to the resolution limit of the measuring equipment. The impulse front 
in subsequent components is likely to rise faster. It is mentioned in [l] with 
reference to other publications that in many digital registrations the 
current could rise to a maximum during the first detector reading (for 0.2 ps). 
The maximum slope of a impulse front in subsequent components 
obeys, in the first approximation, the lognormal law: (lg AI ,,,),, 
= 1.6 
and glg = 0.35. The value of AI 
exceeds 40 kA/ps with a 50% probability 
and is larger than 120 kA/ps with a 5% probability. When affected by such a 
steep impulse, the amplitude of induced voltage would exceed 25 kV in the 
above example of a frame. 
Current of positive lightnings rises slowly. In 5% of cases, the front 
duration was 9 > 200ps. With these impulses, the electric strength of air 
gaps of several metres in length is close to a minimum (section 2.6, formula 
(2.51)). The voltage with tr 
200ps is much more dangerous than a 
‘common’ lightning overvoltage impulse with a risetime of several micro- 
seconds. Minimum breakdown voltage in air gaps with a sharply non- 
uniform field (see formula (2.52)) is about 1.5 times lower than in a standard 
lightning overvoltage impulse of 1.2/50 ps (in accordance with the con- 
ventional way of presenting time characteristics of a impulse, 1.2 is the 
risetime and 50 is the impulse duration at 0.5 amplitude, all in ps). 
The duration of a current impulse is as important for lightning protec- 
tion practice as the risetime. Impulse duration is usually characterized as a 
time span between its beginning and the moment its amplitude decreases 
by half, Since current is related to the neutralization wave travelling along 
the channel, the impulse duration t, is comparable with the time of the 
wave travel. If its velocity is V, E 108m/s and the average channel length 
is 3km, the value of tp will be several tens of microseconds. A similar 
value is derived from experimental data. The impulse duration in the first 
component of a negative lightning is above 30, 75 and 200 ps for the prob- 
abilities 95, 50 and 5%, respectively. For subsequent components, the 
impulse is much shorter: 6, 32 and 140 ps for the same probabilities. Positive 
lightnings must be longer because most of the positive charge of a storm 
cloud is located 2-3km higher than the negative charge. Indeed, tp is 
above 230ps with a 50% probability. The shortest durations for positive 
lightnings are the same as for the first component of a negative one. ‘Anom- 
alously’ long impulses stand out against this background - about 5% of posi- 
tive currents decreased to half the amplitude for 2000 ps. 
Today, we know nothing about the nature of superlong positive 
impulses. One thing is clear: they are unrelated to the wave processes in 
the lightning channel. One may suggest that hydrometeor charge is accumu- 
lated and descends to the earth due to an ionization process in the positively 
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=== PAGE 134 ===
126 
Available lightning data 
charged region of a cloud. But we can only speculate about the nature of this 
process producing final current of 100 kA and ask why it is manifested only in 
positive lightnings. 
3.7.4 Electromagnetic field 
Electromagnetic field of lightning is familiar to those leaving a TV or radio 
set on during a thunderstorm. Sound and video noises inform about a 
storm long before it actually begins. Lightning was the first natural radio 
station used by the founders of radio engineering for testing their receivers. 
The lightning detector designed by A S Popov in 1885 is still Russia’s 
national pride. For many years meteorologists surveyed approaching 
storm fronts by registering so-called atmospherics - pulses of electro- 
magnetic radiation from lightning discharges occurring hundreds of 
kilometres away. In the late 1950s, much interest in atmospherics was due 
to the nuclear weapon race: suspiciously similar to radiation pulses from 
nuclear explosions, they interfered with the diagnostics of the latter. 
It is clear from the foregoing that in a return stroke the charge accumu- 
lated by a leader cover varies and is redistributed rapidly along the channel, 
producing variation of the static component of the electric field. Charge 
variation occurs simultaneously with the propagation of a current wave 
along the channel, inducing a magnetic field. The induction emf varying in 
time gives rise to an induction component of the electric field. Finally, 
variation in the current dipole moment (a leader channel can be regarded 
as a dipole, with the account of its mirror reflection by the earth) gives rise 
to an electromagnetic wave producing a radiation component of the electric 
field with a concurrent magnetic radiation component. There is another mag- 
netic component - a magnetostatic one proportional directly to current. 
It is common practice to distinguish between the near and far regions of 
electromagnetic radiation. In the near region, static field components may be 
dominant: the electric component, damped in proportion to the cubic 
distance Y to the dipole centre, and the magnetic component, varying with 
distance as F2. These can be neglected for the far region, because they are 
much smaller than the radiation components E ,  H cz Y-’. Now, after these 
preliminary remarks, we shall turn to experimental data showing how 
much the shape of a registered pulse varies with distance between a lightning 
discharge and a field detector. 
The shapes of return stroke radiation pulses are shown schematically in 
figure 3.23 for the near and far regions. At large distances, where the static 
components of magnetic and electric fields are nearly completely damped, 
the pulses E( t )  and H (  t )  become geometrically similar. Both are bipolar 
and have a high front slope, a well defined initial maximum and several 
smaller ones along the slowly falling pulse slope, producing the effect of 
damping oscillations. Note that the oscillation period is smaller than the 
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=== PAGE 135 ===
Lightning return stroke 
127 
Figure 3.23. Schematic oscillograms of electromagnetic pulses of lightning in the near 
(top) and far (bottom) zones at the distances 2 km (top) and 100 km (bottom). 
double time of the wave run along the channel. After passing the zero point, 
the pulse part opposite in sign rises and then decreases with nearly the same 
rate; its amplitude is 2-3 times smaller that the first ‘half period’. 
The inverse proportionality of radiation components to the distance 
from the radiation source was the reason why measurements are presented 
in the above form: they are normalized to the basic distance Ybas = 100 km 
as E:,, 
= 10-5Em,,r with r in metres. For the first lightning component, 
the average values of the initial pulse peak of the vertical component, 
EA,,, lie within 5-10 V/cm [49-541 (compare: radio receivers detect well 
signals of lmV/m in the medium bandrange). The electric component of 
subsequent lightning components is 1.5-2 times smaller. The spread of 
measurements is as large as that of lightning currents. The standard deviation 
oE is in the range 35-70% for the first lightning component and 30-80% for 
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=== PAGE 136 ===
128 
Available lightning data 
subsequent ones. The horizontal component of magnetic field strength 
= 
(po/~O)-'/2E~ax 
varies 
respectively. 
Magnetic 
induction 
B,,, = poHmax 
is about lo-* T at a distance of 100 km from the lightning. 
The radiation pulse of the first lightning component rises to the initial 
peak with an increasing rate. In oscillogram processing, the risetime is 
arbitrarily subdivided into two components: the initial slow one of 3-5 ps 
duration and the final fast one taking 1-0.1 ps. The standard deviation is 
also large here: 30-40% of the average value for the slow front and 
about 50% for the fast front. In the final stage, the signal rises for about 
0.5-1.0E~,,. With some reservations, a subdivision into a slow and fast 
component can be also made for radiation pulse of the return stroke of 
subsequent lightning components. But it would be more correct to consider 
that the rise to the initial peak occurs quickly there, for 0.15-0.6 ps. Note that 
the risetimes for the first and subsequent components are close to those of 
their current impulses in a return stroke. 
The moment of sign reversal for radiation pulses of the first components 
is delayed, relative to the onset of a return stroke, by 50ps in temperate 
latitudes [54] and by 90 ps in the tropics [52]. The sign reversal for subsequent 
components occurs by a factor of 1.3- 1.5 earlier. The time for maximum field 
to be established after the sign reversal is of the same order of magnitude as 
that prior to the reversal. 
The radiation components E and H are, naturally, present in the near 
region, too, but they are much smaller than the static component. One 
exception is the initial moments of time. The initial peaks in oscillograms 
E(t) and H(t) should be attributed to radiation, since the static field 
components did not have enough time to reveal themselves. The monotonic 
rise of electric field over 20-50ps, the time long enough for the radiation 
component to be damped, is nearly totally due to electrostatic effect. The 
induced electrostatic field is quite powerful, because the charge accumulated 
by the stepwise leader of the first component or by the dart leader of 
subsequent components is neutralized during the return stroke. For 
example, the electrostatic field changes by several kV/m at the distance of 
1 km from the channel lightning during the first 50ps (for the subsequent 
component, the signal is 2-3 times lower than for the first one ); a slower 
field rise may continue for about 100 ps. All in all, the field of the first light- 
ning component is an order of magnitude higher than the initial radiation 
rise. With increasing distance r to 15-20km, the radiation component 
becomes dominant over the others, and the initial radiation peak becomes 
an absolute maximum of the registered signal. 
The magnetostatic component in the near region is not so important. 
Still, at a distance of 1 km, it contributes as much to the signal as the 
radiation component (figure 3.23). The magnetic induction here is as high 
as lop5 T. The absolute magnetic field maximum is achieved later than the 
stroke current peak registered at the earth's surface. This is clear because 
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=== PAGE 137 ===
Total lightning flash duration and processes in the intercomponent pauses 
129 
the magnetostatic component is proportional not only to the current but to 
the conductor length. The length increases as a neutralization wave travels 
from the earth up to the cloud. For the same reason, the times for the first 
and subsequent components do not differ much. The duration of pulse 
B(t) in the near region is comparable with that of current inducing a 
magnetic field. 
3.8 Total lightning flash duration and processes in the 
intercomponent pauses 
A descending negative lightning flash has on average two or three compo- 
nents, each terminated by a more or less powerful current impulse of the 
return stroke. The average number of components in an ascending lightning 
is four. The maximum number of components in a lightning flash may be as 
large as 30. The pauses between the components At,,, 
vary from several 
milliseconds to hundreds of milliseconds. With a 50% probability, their 
duration exceeds 33 ms; the integral distribution curve is described by the 
lognormal law with the parameters (lg At,,,)av 
= 1.52 and olg = 0.4, at 
At,,, 
[ms]. The total flash duration varies with the number of components. 
Negative one-component flashes are the shortest ones, since their current 
often ceases right after the return stroke, for less than a millisecond. An 
ascending one-component positive flash can pass current for a longer time, 
0.5 s, in spite of the absence of a return stroke. Of course, this is a low current, 
less than 1 kA. The average flash duration is close to 0.1-0.2s and the 
maximum is 1.5s. These large times are discernible by the naked eye, so 
lightning flickering is not a physiological by-product of vision but a physical 
reality. 
Intercomponent pauses take most of the flash time. They cannot be said 
to be current-free. A lightning leader is supplied by current nearly all the 
time, and this current is high enough to support plasma in a state close to 
that of a steady-state arc. Current of an intercomponent pause is referred 
to as continuous current, which is a fairly ambiguous term. Average 
continuous current varies between 100 and 200A. Nearly as high current 
supplies an arc in a conventional welding set used for cutting metal sheets 
or for welding thick pipes. Most thermal effects of lightning are associated 
with its continuous current, rather than with return stroke impulses which 
are more powerful but shorter. The hghest continuous current measured 
[55] was 580A. Continuous current usually slowly decreases with time. In a 
one-component ascending lightning having no return stroke, the contact of 
the leader with the cloud is terminated by charge overflow from the cloud to 
the earth as a decreasing continuous current of about the same value. 
Cloud discharging by continuous current can be easily registered by an elec- 
tric field detector. Field varies monotonically, as long as current flows through 
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=== PAGE 138 ===
130 
Available lightning data 
the channel. These are appreciable changes, since current of lOOA extracts, 
from a cloud, charge AQ x 1OC over the time 0.1 s. The field on the earth 
right under a cloud changes by the value A E  = AQ/(27qH2) M 200V/cm if 
the height of the charged cell centre is H = 3 km; at distance r = 10 km from 
the lightning axis, A E  = AQH/[2q,(H2 + Y~)~'~] zz 5V/cm. Similar values 
were registered during observations. 
Continuous current flow is accompanied by slowly rising and as slowly 
decreasing current impulses with an amplitude up to 1 kA. These are M- 
components of lightning. The risetime of a typical M-component is about 
0.5ms, an average impulse duration (on the level 0.5) is twice as much, an 
average amplitude is 100-200 A, although M-components with current up 
to 750 A have also been registered [56,57]. Pulsed current rise is always 
accompanied by an increase in light emission intensity of the whole channel, 
from the cloud down to the earth. Streak photographs (even taken slowly) do 
not show the propagation of a well defined emission wave front similar, say, 
to the tip of a dart leader. It seems as if most of the channel flares up 
simultaneously, although excitation, no doubt, propagates down from a 
cloud with a high velocity, (2.7-4)x lo7 mjs (from measurements of [58]). 
Two M-components were identified in [58] as ascending ones. In later 
measurements, the existence of ascending processes were questioned, because 
there were no clear physical reasons for the appearance of an inducing 
perturbation at the earth's surface. 
Variations in current and electric field of M-components were registered 
in triggered lightning flashes at a short distance from the channel (r = 30 m) 
[57]. The field variation of a vertical component at the earth is shown in figure 
3.24. The pulse A E  rises to its maximum 70ps earlier than the current 
impulse. The field rises and decreases at nearly the same rate. The pulse 
component of field perturbation is nearly completely damped while the 
current still has a high amplitude. 
2?0 
I 
400 
600 
b 
0.5 I- -- 
1.5 
loO1 
E, kV/m 
m 
,e- t, PJ- 
Figure 3.24. Superimposed schematic oscillograms of M-component electric field and 
current at the earth [57]. 
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=== PAGE 139 ===
Flash charge and normalized energy 
131 
The number of M-components in a flash may even be larger than that of 
subsequent components, but they are of little interest to lightning protection 
practice - their charge and current are too low. Theoretically, however, these 
components are of great interest, because they seem to contain information 
on unobservable processes occurring in storm clouds. It is quite likely that 
these processes give rise to a dart leader with a return stroke or to a 
stroke-free M-component. Some authors [27] believe that an M-component 
is always formed against the background of continuous current, whereas a 
necessary prerequisite for a dart leader is a current-free pause, during 
which the grounded lightning channel partly loses its conductivity. This is 
a very important detail shedding light on processes occurring in a storm 
cloud after a grounded plasma channel of the first lightning component 
has penetrated it. The transport of the earth’s zero potential to a cloud by 
a conducting channel, resulting in a rapid increase in the cloud electric 
field in the vicinity of the channel top, is a powerful stimulus for gas discharge 
processes there (for details, see sections 4.7 and 4.8). 
3.9 
Flash charge and normalized energy 
During intercomponent pauses, charge is transported from a cloud to the 
earth by both powerful return stroke impulses and continuous current, the 
latter being much lower but longer-living. The contributions of these currents 
to the total charge effect are comparable. With a 50% probability, the stroke 
charge transported by the first component of a negative flash is over 4.5C, 
while 5% of flashes transport over 20C and another 5% less than 1.1C 
[42]. The lognormal law described above is suitable for an approximate repre- 
sentation of the integral distribution curve with the values (lg 
= 0.653 
and olg = 0.4. The return strokes of subsequent components have, for the 
same probabilities, five times smaller charges due to their shorter duration 
and lower currents. The largest spread of charge measurements is character- 
istic of positive lightning, in agreement with the diversity of their shape and 
duration. Positive pulse charges exceed 16C with a 50% probability, 150C 
with a 5% probability, and are less than 2C with a 5% probability. These 
seem to be positive lightning with no return stroke. For the description of 
integral charge distribution for positive pulses, the lognormal parameters 
may be taken to be (lg Q)av = 1.2 and rlg = 0.6. 
We have already mentioned that the charge of a lightning flash is always 
larger than the sum of charges transported by the return strokes of the first 
and subsequent components, since a substantial contribution to the total 
charge is made by continuous current. The total negative flash charge exceeds 
7.5C with a 50% probability, 40C with a 5% probability, and is nearly the 
same as the first negative pulse charge in the least powerful flashes. The 
total positive charge is appreciably larger - with 95%, 50% and 5% 
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=== PAGE 140 ===
132 
Available lightning data 
probabilities, it exceeds, respectively, 20, 80 and 350C. One cannot say that 
the charge transported by a flash is very large. For comparison, even a very 
large lightning charge of 350C flows through the arc of a conventional 
welding unit for 3-5 s. 
Charge transport is accompanied by energy release. An average negative 
flash with a charge Q = 1OC and gap voltage 50 MV dissipates about QU = 
5 x 10sJ, which is equal to the energy released by a 100 kg trinitrotoluene 
explosion. While most energy is released within the lightning trace, the 
problem of energy release and heating of metal constructions is of much inter- 
est. Normally, the resistance of metallic conductors and that of a grounding 
electrode are much less than the equivalent resistance of a lightning channel 
RI = U / I M  (IM is the impulse amplitude of a return stroke); RI = 1 kR if 
U x 50 MV and ZM = 50 kA. Therefore, lightning can be regarded as a current 
source, assuming that current IM is independent of the object’s resistance. Any 
conductor with lightning current flow releases the energy 
K = R 1; i2 dt 
(K/R)R. 
KIR = 1; i2 dt 
proportional to the conductor resistance R. For practical calculations, data 
on ‘normalized’ energies K / R  characterizing lightning only are published. 
According to [42], 95%, 50% and 5% probabilities correspond to the 
measured values exceeding 2.5 x lo4, 6.5 x IO5 and 1.5 x 107A2s for 
positive flashes and 6.0 x lo3, 5.5 x IO4 and 5.5 x 105A2s for negative 
flashes, respectively. For subsequent components of negative flashes, the 
respective values are an order of magnitude smaller and do not contribute 
much to the total energy release. To get an idea about thermal potency of 
lightning, evaluate the heat of a steel conductor with a cross section of 
S = 1 cm’. With resistivity p = lop5 R cm, the energy density released by a 
powerful positive flash (K/R = 1.5 x lo7 A’s) is (K/R)(p/S2) 
= 150 J/cm3, 
with the conductor temperature increasing by 40°C. Owing to Joule heat, a 
lightning flash is capable of burning down only a very thin conductor with a 
cross section less than 0.1 cm’. In many cases, however, heating just by several 
hundred degrees may become hazardous. 
3.10 
Lightning temperature and radius 
Plasma temperature is usually measured by spectroscopic methods. Light- 
ning spectroscopy is a hundred years old, and it was used even before 
photography and field-current measurements. Reviews of spectroscopic 
results can be found in Uman’s books [l, 591 together with extensive 
references. However, direct data on lightning plasma are still very scarce. 
Lightning spectra, naturally, contain lines of molecular and atomic oxygen 
and nitrogen, as well as singly charged ions N2, argon, cyane and some 
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=== PAGE 141 ===
Lightning temperature and radius 
133 
other impurities. No doubly charged ions have been detected, indicating that 
the temperature does not exceed 30 000 K. Measurements of time resolved 
NI1 (N') line intensities show that the return stroke temperature reaches 
30 000 K for the first 10 ps [59,62] and drops to 20 000 K in 20 ps. Average 
temperatures are estimated to be about 25 000 K. These results are obtained 
assuming that a plasma channel is optically transparent and that the 
excitation of atoms in the plasma is equilibrium (of the Boltzmann type). 
The estimations justify this assumption. 
Electron densities found from the Stark broadening of the Ha lines 
are 1 0 ' * ~ m - ~  
for the first 5ps of the stroke life. Under thermodynamic 
equilibrium conditions at T = 30 000 K, this value of ne corresponds to 
the pressure of 8atm [63]. About lops later, ne decreases to lO''~m-~, 
corresponding to the pressure drop down to the atmospheric pressure. 
Then the value of ne remains unchanged over the time of the NII line 
registration. This does not seem strange. Equilibrium electron density in 
air at p = const = 1 atm changes only slightly in a wide temperature 
range 15 000-30 000 K, remaining about 1017 ~ m - ~ .  
As the channel cools 
down, the ionization degree x = ne/N certainly decreases, but when the 
pressure reaches the atmospheric value, the gas density N rises simul- 
taneously. For this reason, ne = x N  does not change much. High intensity 
radiation is observed for about loops (from 40 to lOOOps). The first 
peak is often followed by another one several hundreds of microseconds 
later. 
Spectroscopic measurements were mostly made during a return stroke, 
but some authors [64] managed to register the spectrum of a 2-m portion of a 
stepwise leader. The leader tip temperature calculated from the N 11 lines lies 
within 20 000-35 000 K. The diameter of the radiation region is less than 
35 cm. More accurate evaluations are unavailable. It seems unlikely that 
this temperature is characteristic of the whole leader channel. Rather, the 
experiment registered a short temperature rise during a powerful step 
which was akin to a miniature return stroke (section 2.7). The step-induced 
perturbation involving part of the channel region is most likely to be 
damped rapidly along the leader length. 
It is not only the plasma dynamics but the channel radius, too, which 
still remains enigmatic. In making evaluations of the radius, one usually 
relies on photographs. But in this case, it is very important to agree on the 
kind of radius being evaluated. This may be the radius of the channel, 
through which current flows during the leader and stroke stages. Clearly, 
such a radius will include the best conducting and, hence, the hottest core 
of the plasma channel. Or, one can follow another approach. When solving 
the problem of electric field variation during the lightning development, one 
has to deal with the radius of the leader cover where most of the space charge 
is concentrated. This is the charge radius of lightning. Therefore, each time 
we speak of radius, we must define exactly what we mean. 
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=== PAGE 142 ===
134 
Available lightning data 
Here, we shall use the concept of channel radius as applied to the region 
where the lightning current is accumulated and the concept of cover radius to 
the region where most of the space charge is concentrated. The former can.be 
determined, to some extent, by using optical methods, although this is a com- 
plicated task. With reference to the optical measurements [65], one usually 
deals with radii of several centimetres. This resolution is accessible to 
modern cameras at a distance of about a kilometre, but the cameras must 
have the highest resolution possible. Anyway, we have never heard about 
the application of such perfect optical equipment in lightning research. 
In addition to using special-purpose optics, the experimentalist must 
match perfectly the sensitivity of photographic materials and exposures. 
A longer exposure produces a halo, increasing the actual radius. Unless 
special measures are taken, the error may be very large, especially for 
flashes with a high light intensity. For some reasons, the optical radius of 
a lightning channel may exceed manifold the thermal radius. Such an 
effect was observed in studies of spark leaders in laboratory conditions 
[66]. Registration of the thermal radius appears problematic even for trig- 
gered lightning, with a fixed point of contact with the earth. For natural 
lightning, this task is much more complicated. As for the cover radius, 
there is no reliable technique for its registration at all. So lightning radius 
measurements cannot provide unquestionable data, and the researcher is 
to rely on theoretical evaluations only. 
3.1 1 What can one gain from lightning measurements? 
It was not our task to review all experimental studies on lightning: this has 
been well done in [l, 591. We believe that the latest experimental data will 
be presented in a new Uman book now in preparation. But the basic facts 
have been discussed here, and we can now ask ourselves whether the 
available data are sufficient to build lightning theory and to check it by 
experiment. 
The situation with lightning is somewhat similar to that for a long 
laboratory spark, i.e., experiments give mainly external parameters of a dis- 
charge. In the laboratory, these are velocities of the major structural elements 
(streamers and leaders), their initiating voltages, currents, transported 
charges, and, possibly, some other characteristics Sometimes, we have 
some information on channel radii, or on the time variation of radii, or 
scarce data on plasma parameters. But that is all. 
The arsenal of lightning researchers is much smaller. First, they have no 
information about the voltage in the cloud-earth gap at the lightning start, 
and there are no data on the initial distribution of electric field. Both literally 
and figuratively, the bulk of a storm cloud, where a descending leader 
originates, is obscure. Measurements made at the earth’s surface cannot 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 143 ===
What can one gain from lightning measurements? 
135 
help much, because the number of registration points is too small, so it is 
impossible to reconstruct the initial field distribution along the whole 
lightning path. 
The fine structure of a lightning flash is not clear either. Observations 
give no information about the size of the streamer zone in a lightning 
leader, and even the existence of such a zone is largely speculative. Nor do 
we know the origin and structure of volume leaders, which are responsible 
for the stepwise pattern of a negative leader, at least, observable in laboratory 
conditions. There is no information on the gas state in the track of a 
preceding component, when a dart leader travels along it. The only dart 
leader parameter that has been measured is its velocity. What has just been 
listed refers primarily to the return stroke. It appears that space charge 
neutralization - the basic process occurring in it - is related to the fast 
radial propagation of streamers away from the channel. This is the way 
the cover charge is supposed to change. But there are no experimental data 
on this process, nor can we hope to obtain any in the near future. 
Most available findings concern lightning currents and transported 
charges. As in a laboratory spark, lightning currents are usually registered 
at the earth’s surface, so we have data on leader currents for ascending dis- 
charges only. There are no direct measurements of currents for descending or 
dart leaders, the latter fact being especially disappointing. There are more or 
less detailed descriptions of currents for return strokes, but the measurements 
made at one point (that of contact with the earth) restrict the possibilities of 
both a theoretical physicist and a practical engineer. Data on the current 
wave damping along the leader are important for the former because then 
he may try to reconstruct the plasma conductivity variation. The latter 
needs them to be able to calculate the lightning electric field at the earth 
and in the troposphere, because it is hazardous to both ground objects and 
aircraft. 
Lightning current statistics deserves special attention. Normally, they 
are used in calculations of the occurrence probability of lightning with 
hazardous parameters, e.g., a critically fast rise of the impulse front and/or 
amplitude. The practical requirements on the calculation reliability are 
extremely high. Indeed, it is impossible to provide the necessary accuracy, 
using lognormal parameter distributions. Any approximation of an actual 
distribution lognormally would be approximate, especially in the range of 
large values important for lightning protection. The error may be as high 
as 100%. One should keep this in mind when comparing calculations of 
hazardous lightning effects and the available experience in object protection. 
This is the reality not to be ignored either by a theorist attempting to 
create a lightning model or by an engineer working on lightning protection. 
No matter how ingenious a theorist may be, he will not be able to check his 
model, filling the gaps by laboratory spark data or by general physical 
considerations. As for practical lightning protection, one usually gained 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 144 ===
136 
Available lightning data 
the unfortunate experience from analyses of emergencies that resulted from 
the lack of knowledge of atmospheric electricity. 
References 
[l] Uman M 1987 The Lightning Discharge (New York: Academic Press) p 377 
[2] Golde R H (ed) 1977 Lightning (London, New York: Academic Press) vols 1, 2 
[3] Imyanitov I M 1970 Aircraft Electrization in Clouds and Precipitation (Lenin- 
grad: Gigrometeoizdat) p 210 
[4] Gunn R 1948 J. Appl. Phys. 19 481 
[5] Gunn R 1965 J. Atmos. Sci 22 498 
[6] Evans W H 1969 J. Geophys. Res. 74 939 
[7] Winn W P, Schwede G W and Moore C B 1974 J. Geophys. Res. 79 1761 
[8] Winn W P, Moore C B and Holmes C R 1981 J. Geophys. Res. 86 1187 
[9] Kazemir H W and Perkins F 1978 Final Report, Kennedy Space Center Contract 
CC 69694A 
[lo] Newman M M, Stahmann J R, Robb J D et all967 J. Geophys. Res. 72 4761 
[ll] Kito Y, Horii K, Higashiyama Y and Nakamura K 1985 J. Geophys. Res. 90 
[12] Hubert P and Mouget G 1981 J. Geophys. Res. 86 5253 
[13] Hubert P, Laroche P, Eybert-Berard A and Barret L 1984 J. Geophys. Res. 89 
[14] Idone V P and Orville R E 1984 J. Geophys. Res. 89 7311 
[15] Fisher R G, Schnetzer G H, Thottappillil R et a1 1993 J. Geophys. Res. 98 22887 
[16] Wang D, Rakov V A, Uman M A et a1 1999 J. Geophys. Res. 104 4213 
[17] Malan D J and Schonland F G 1951 Proc. R. Soc. London Ser. A 209 158 
[18] Malan D J 1963 Physics of Lightning (London: English Univ. Press) p 176 
[19] Malan D J 1963 J. Franklin Inst. 283 526 
[20] Bazelyan E M and Raizer Yu P 1997 Spark Discharge (Boca Raton: CRC Press) 
[21] Chalmers J A 1967 Atmospheric Electricity (2nd edn) (Oxford: Pergamon) p 418 
[22] Berger K and Fogelsanger E 1966 Bull. SEV 57 13 1 
[23] Schonland B 1956 The Lightning Discharge. Handbuch der Physik 22 (Berlin: 
Springer) p 576 
[24] Schonland B, Malan D and Collens H 1938 Proc. Roy. Soc. London Ser. A 168 
455 
[25] Schonland B, Malan D and Collens H 1935 Proc. Roy. Soc. London Ser. A 152 
595 
[26] Jordan D M, Rakov V A, Beasley W H and Uman M A 1997 J. Geophys. Res. 
102 22.025 
[27] Fisher R G, Schnetzer G H, Thottappillil R et a1 1992 Proc. 9th Intern. Conf. on 
Atmosph. Electricity 3 (St Peterburg: A I Voeikov Main Geophys. Observ.) p 873 
[28] McCann G 1944 Trans. AIEE 63 11 57 
[29] Berger K and Vogrlsanger E 1965 Bull SEV 56 No 1 2 
[30] Bazelyan E M, Gorin B N and Levitov V I 1978 Physical and Engineering Funda- 
mentals of Lightning Protection (Leningrad: Gidrometeoizdat) p 223 (in Russian) 
6147 
251 1 
p 294 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 145 ===
References 
137 
[31] Saint Privat d’Allier Research Group 1982 Extrait de la Revue Generale de 
[32] Gorin B N and Shkilev A V 1974 Elektrichestvo 2 29 
[33] Idone V P, Orville R E 1985 J. Geophys. Res. 90 6159 
[34] Rubinstein M, Uman M A and Thomson P 1992 Proc. 9th Intern. Conf. on 
Atmosph. Electricity 1 (St Peterburg: A I Voeikov Main Geophys. Observ.) p 276 
[35] Rubinstein M, Rachidi F, Uman M A et a1 1995 J. Geophys. Res. 100 8863 
[36] Thomson E M 1985 J. Geophys. Res. 90 8125 
[37] Kolechizky E C 1983 Electric Field Calculation for High-Voltage Equipment 
[38] Jordan D M and Uman M A 1983 J. Geophys. Res. 88 6555 
[39] Schonland B and Collens H 1934 Proc. Roy. Soc. London Ser. A 143 654 
[40] Idone V P and Orville R E 1982 J. Geophys. Res. 87 9703 
[41] Idone V.P, Orville R E, Hubert P et a1 1984 J. Geophys. Res. 89 1385 
[42] Berger K, Anderson R B and Kroninger H 1975 Electra 41 23 
[43] Berger K 1972 Bull. Schweiz. Elekrtotech. Ver. 63 1403 
[44] Gorin B N and Shkilev A V 1979 in Lightning Physics andLightning Protection 
[45] Gorin B N and Shkilev A V 1974 Elektrichestvo 2 29 
[46] Eriksson A J 1978 Trans. South Afr. ZEE 69 (Pt 8) 238 
[47] Anderson R B and Eriksson A J 1980 Electra 69 65 
[48] Alizade A A, Muslimov M M et a1 1974 in Lightning Physics and Lightning 
[49] Master M J, Uman M A, Beasley W H and Darveniza M 1984 ZEEE Trans. PAS 
[50] Krider E P and Guo C 1983 J. Geophys. Res. 88 8471 
[51] Cooray V and Lundquist S 1982 J. Geophys. Res. 87 11203 
[52] Cooray V and Lundquist S 1985 J. Geophys. Res. 90 6099 
[53] McDonald T B, Uman M A, Tiller J A and Beasley W H 1979 J. Geophys. Res. 
[54] Lin Y T, Uman M A et a1 1979 J. Geophys. Res. 84 6307 
[55] Krehbiel P R, Brook M and McCrogy R A 1979 J. Geophys. Res. 84 2432 
[56] Thottappillil R, Goldberg J D, Rakov V A, Uman M A et a1 1995 J. Geophys. 
[57] Rakov V A, Thottappillil R, Uman M A and Barker P P 1995 J. Geophys. Res. 
[58] Malan D J and Collens H 1937 Proc. R. Soc. London A 162 175 
[59] Uman M A 1969 Lightning (New York: McGraw-Hill) 
[60] Orvill R E 1968 J. Atmos. Sci. 25 827 
[61] Orvill R E 1968 J. Atmos. Sci. 25 839 
[62] Orvill R E 1968 J. Atmos. Sci. 25 852 
[63] Kuznetsov N M 1965 Thermodynamic Functions and Shock Adiabata for High 
[64] Orvill R E 1968 J. Geophys. Res. 73 6999 
[65] Orvill R E 1977 in Lightning, vol 1, R Golde (ed) (New York: Academic Press) 
[66] Positive Discharges in Air Gaps at Les Renardieres - 197s 1977 Electra 53 31 
I’Electricite, Paris, September 
(Moscow: Energoatomizdat) p 167 (in Russian) 
(Moscow: Krzhizhanovsky Power Engineering Inst.) p 9 
Protection (Moscow: Krzhizhanovsky Power Engineering Inst.) p 10 
Pas-103 2519 
84 1727 
Res. 100 25711 
100 25701 
Temperature Air (Moscow: Mashinostroenie) (in Russian) 
p 281 
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=== PAGE 146 ===
Chapter 4 
Physical processes in a 
I ig h t n i ng d isc ha rge 
Here we shall discuss the basic phenomena occurring in a lightning discharge: 
a descending negative leader, an ascending positive leader, the return strokes 
of the first and subsequent components, a dart leader, and some others. 
Lightning may travel not only from a cloud towards the earth, or from a 
grounded object towards a cloud, but it may also start from a body isolated 
from the earth - a plane, a rocket, etc. About 90% of all descending 
discharges are negative and about as many ascending discharges are positive. 
For this reason, an ascending leader is said to be positive. Available experi- 
mental data on lightning as such are of little use in our attempts to explain the 
mechanisms underlying the above processes. There are very few observations 
that might shed light on their physical nature. So, one has to resort to spec- 
ulations, invoking both theory and experimental data on a long laboratory 
spark, which relate primarily to a positive leader. Since this process is most 
simple (to the extent a lightning process may be considered simple), we 
shall begin with the discussion of an ascending positive leader. 
4.1 
An ascending positive leader 
4.1.1 The origin 
The lightnings people observe most frequently are descending discharges, 
which originate among storm clouds and strike the earth or objects located 
on its surface. However, constructions over 200m high and those built in 
mountainous regions suffer mostly from ascending lightnings. These are of 
nearly as much interest to the physicist as the seemingly common, descending 
discharges. An ascending leader is initiated by a charge induced by the elec- 
tric field of a storm cloud in a conducting vertically extending grounded 
object. If a metal conductor of height h with a characteristic radius of the 
rounded top r << h is fixed on the earth and then affected by a vertical 
138 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 147 ===
An ascending positive leader 
139 
external field Eo, a field El x Eoh/r >> Eo is created by the induced charge at 
the conductor top (see section 2.2.7). This field rapidly decreases in air (for a 
distance of several r values), creating a potential difference between the 
conductor end and the adjacent space, AU x Eoh. When the cloud bottom 
is charged negatively and the vector Eo is directed from the earth up to 
the cloud, the grounded conductor becomes positively charged, since the 
field makes some of the negative charges leave the metal to go down to 
the earth. 
No stringent conditions are necessary for the field E, to initiate the air 
ionization (at sea level El x E, x 30 kV/cm) or for a corona discharge to 
arise at the pointed parts of a high structure (it is necessary to have 
El x40-31kV/cm for r = 1-10cm). The conditions for a leader to be 
initiated in the streamer corona stem are much more rigorous. The energy 
estimations made in section 2.6 show that there is no chance for a leader to 
arise if the leader tip potential U,, or, more exactly, its excess over the external 
potential at the tip, A U  = U, - U,, is less than AVrm,, x 300-400 kV. This 
estimate is supported by experiments with leaders, whose streamer zones 
have no contact with the electrode of opposite sign at the initial moment of 
time. Therefore, for the desired potential difference A Ut,, to be produced, 
the structure must have, at least, h x AUrm,,/EO 
x 20-30m if the average 
field of the storm cloud at the site of the grounded object is -150 V/cm. 
On the other hand, even if a leader is produced at such a low potential, 
AVfm,,, it can hardly travel for a large distance. The leader current will be too 
low to heat the channel to a sufficiently high temperature. As a result, the 
channel resistance will be too high so that a very strong field will be required 
to support the current in the channel. The channel field E, is, however, 
limited by the external field Eo. Indeed, a grounded body of height h, from 
which a positive leader has started, possesses zero potential. Having covered 
the distance L, the leader tip acquires the potential U, = -E,L. Here, the 
potential of the unperturbed external fields is U. = -Eo(L + h), and we have 
AUt = Ut - U0 = AU, + (Eo - E,)L. 
AU, = Eoh. 
(4.1) 
For a leader to develop from the initial threshold conditions, the potential 
difference AU, should not decrease relative to the initial value of AU,. For 
this, the average channel field E, must be lower than the external field Eo. 
However, a mature channel possesses a falling current-voltage characteristic 
E, (i). A decrease in E, to N 100 V/cm requires a channel current higher than 
1 A. We discussed this issue in sections 2.5.2 and 2.6. With the approximation 
accepted there (E, x b/i and b =300 VAjcm), the leader current is to exceed 
i,, 
= b/Eo x 2 A at Eo x 150 V/cm. 
Let us see how large the potential difference Aut should be to make the 
current exceed i,. 
In chapter 2, we derived formula (2.35) relating the 
channel current behind the leader tip to the tip potential U, and the leader 
velocity vL. That formula was applicable to the laboratory conditions 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 148 ===
140 
Physical processes in a lightning discharge 
considered in that chapter, when a leader travelled through the rapidly 
decreasing field of a high-voltage electrode. Having covered a distance of 
only a few radii of the electrode curvature, usually very small, the leader 
tip found itself in a space with a nearly zero potential, U, << U,. The neglect 
of the external field was justifiable in that case. An ascending leader is quite 
another matter: the potential difference AU, = U, - U, continuously 
increases with the leader velocity, as its tip approaches a charged cloud, 
since the tip enters a region of an ever increasing external field. Hence, we 
have I Uti << I Uoi and the value of AU, largely determined by U,. Therefore, 
the approximate formula of (2.35) must be rewritten in its general form, with 
U, replaced by AU,: 
. 
~TEOAU~VL aut = U, - U, 
’ = ln(L/R) ’ 
where R is the effective radius of the leader charge cover. 
The available leader theory fails to provide a clear and convincing 
physical expression to describe the relationship between vL and AU,. So 
we shall further use the empirical relation suggested in section 2.6: 
wL = a(AU,)”2, 
a = 1 5 m / ~ V ’ / ~ .  
(4.3) 
This relation was derived from experimental data on rather short gaps, in 
which the tip potential could be taken to be identical to that of a high voltage 
e1ectrode.t In accordance with (4.2), expression (4.3) corresponds to the 
relation vL = ill3 also supported by some laboratory experiments. Now, 
using the value of imin, we shall find AUi which provides the leader viability: 
Assuming L = 10m and R = 1 m for a still-short initial leader at 
Eo = 150V/cm, we obtain AVjmln = 3.1 MV and hmin = 210m. The result 
of this simple estimation agrees with that of lightning observations. In a 
flat country, ascending lightnings make up an appreciable fraction of the 
total number of strikes affecting grounded objects of about that height. 
The continuous ascending leader of a triggered lightning (initiated from a 
grounded wire raised by a rocket above the earth) is also excited at about 
200m. Note that the value of Eo used in the calculations is somewhat 
larger than those measured at the earth surface. The storm cloud field 
near the earth is always attenuated by the space charge introduced in the 
air by corona discharges from thin conductors of small height, such as 
tree branches, shrubs, grass, constructions, etc. Some measurements show 
that the field at the earth is half that at a height of 10-20m. 
t The streamer theory has been advanced further. Note, for comparison, that the streamer 
velocity is V, RZ AU, from formulae (2.6) and (2.8) with AU, instead of U, and E, = const. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 149 ===
An ascending positive leader 
141 
4.1.2 Leader development and current 
Two main leader parameters are accessible to measurement: its velocity and 
the current through the channel base contacting a grounded object. The 
current is due to the charge pumped by an electric field into the growing 
leader. If the field in the leader channel is lower than the external field 
(otherwise the leader is non-viable), the difference between the tip potential 
and the unperturbed potential at its site becomes larger as the leader becomes 
longer (see expression (4.1)). According to (4.2) and (4.3), the leader current, 
proportional to i N AU:”, also rises. The current rise becomes more rapid as 
the leader becomes longer, especially when the leader reaches the region of a 
very high cloud field. The rising current heats the channel more, so that its 
linear resistance and field drop. With time, the channel becomes a nearly 
perfect conductor. Grounded at its base, the channel possesses the same 
potential U ( t ,  x) everywhere along its length, including the tip, which is 
low relative to the absolute external potential IUo(x)I. 
In this case, the 
value of AU = U ( t , x )  - Vo(x) 
z -Uo(x) 
varies only slightly with time at 
every point x along the channel. 
The linear leader capacitance C1, given by formula (2.8) with length L 
and cover radius R instead of 1 and Y, also varies very little. Indeed, the 
cover radius behind the tip is about the same as the streamer zone radius 
which, according to (2.39), is R = AUt/2E,, where E, M 5 kV/cm is the 
streamer zone field under normal conditions. The height of the charged 
region centre in the cloud, H PZ 3 km, is much greater than that of the 
leader starting point, h PZ 200m. Suppose the leader length is greater than 
h but smaller than H by such a value that the cloud field non-uniformity 
along the channel can be neglected. We then have AU, M IUo(L)I M IEoLI, 
and the value of LIR M 2E,/Eo under the logarithm in C1 of (4.2) and 
(4.4) is independent of time, If this relation does change, which happens 
when a leader rises so high that it enters the region of a rapidly increasing 
external field, the logarithm changes much more slowly. Thus, the linear 
charge ~ ( x )  
M C,AU remains nearly constant in time at every leader point. 
But if there is no charge redistribution along the channel, the current in it, 
i(t, x), does not change along its length but changes only in time. Entering 
the channel through its grounded base, the current supplies charge only to 
the front leader portion. The current in the base is the same as in the channel 
right behind the tip. It is defined by formula (4.2) with Aut M I Uo(L)I, close 
to the unperturbed potential of the cloud charge at the tip site. Similarly, the 
leader velocity can be found from (4.3). 
Therefore, the velocity and current of a fairly long leader (long relative 
to the start height), which develops in the average field Eo, are described as 
WL = a(ALJ,)1/2, 
1 ‘ = 2TEoa(AUt)3’2 
AU, = EoL, e = 2.72.. . 
(4.5) 
ln(2&E,/eEo) ’ 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 150 ===
142 
Physical processes in a lightning discharge 
The numerical factor fi/e in the second expression of (4.5) has resulted 
from a more rigorous calculation. This result is obtained if the linear charge 
T(X) is calculated directly from external field Eo with r(x) = const x, as in 
section 3.6.2, rather than from average linear capacitance C1 and AU(x), 
as was done in the derivation of (4.2). The current is found from 
i = dQ/dt, where Q is the net charge of the conductor (the integral of ~(x) 
in x). 
It follows from (4.5) that the current rises rapidly with time as the 
leader develops, whereas the velocity increases much more slowly: 
U, = dL/dt x L1I2 M t and i M L3I2 M t3 at Eo = const. In stronger external 
fields, the leader current also rises with EO much faster than the velocity. 
Numerically, a leader with L = 500 m has i = 4.5 A and 
= 4 x lo4 cmjs 
in an average field Eo = 150V/cm, i.e., about the same values as for an 
extremely long laboratory spark. An increase of L to 2000m and Eo to 
300V/cm 
gives the 
typical 
lightning 
parameters: 
i = 120A and 
U, = 12 x 105cm/s. A rapid rise of the leader current and a much slower 
increase of its velocity were inevitably registered in observations of both 
natural and triggered lightnings [ 1,2]. These estimations reasonably agree 
with measurements. 
In contrast to (4.2) and (4.3), expression (4.5) ignores the voltage drop 
across the leader channel because U, << lUo(L)l. It is easy to see the validity 
of this assumption in the next approximation using the derived formulae. 
With the voltage-current characteristic E M [ lip', the voltage drop across 
the channel decreases as U, E EcL M L'I2 M t-' with the leader develop- 
ment. At the tip site, on the contrary, IUo(L) = lEoiL grows even faster 
than L 
t2 if one takes into account the increase of the average external 
field along the channel during its travel up to the cloud. Note that E, and 
U, do not drop to zero in reality but only decrease to a certain limit, because 
the field Ec(i) 
in a very heated channel with high current is stabilized due to 
the greater plasma energy loss for radiation (the current-voltage character- 
istic should be expressed as E, = c + b/i rather than as E, = b/i). This issue, 
however, is of no importance to an ascending leader, since its current 
becomes very high only when the tip reaches the region with 1 Uo(L)/ >> U,. 
If one desires to refine these simple results by taking account of the 
voltage drop, charge redistribution, and current variation along the channel, 
one should regard it as a long line, as was done with the streamer in section 
2.2.3. The distributions of potential U ( t . x ) ,  charge per unit length r(x, t ) ,  
and current i(x, t )  along the line can be described by equations similar to 
(2.13) and (2.14): 
d r  
di 
dU 
at 
dx 
dX 
- + - 
= 0, 
-- = E,(i), 
i(L) = r(L)vL 
(4.6) 
where E, is the longitudinal channel field expressed through current i(x, t) 
from the current-voltage characteristic (the field in (2.13) was expressed 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 151 ===
An ascending positive leader 
143 
through current and linear resistance, E, = iR1). The leader velocity uL is 
given, for example, by formula (4.3): dL/dt = uL. 
Equations (4.6) must involve an electrostatic relation between charges 
and potentials. In a simple approximation, expressions (2.13) and (4.2) 
were allowed to contain a local relation, ~(x) 
= C1 [ U ( x )  - Uo(x)], through 
linear capacitance C1. This approximation was shown by many calculations 
to be quite acceptable to the case of a uniform or weakly non-uniform 
external field, but it appears insufficiently rigorous for a strongly non- 
uniform field, which a leader crosses on entering a storm cloud. In fact, 
the potential at every point along the channel length is also created by 
charges located at adjacent channel sites. To simplify the non-local relation, 
the leader charge can be assumed to be concentrated on a cylindrical surface 
with an effective cover radius R; then the desired relation takes the form 
r(z, t) dz 
AU(x, t) = U ( X ,  t) - U~(X) 
= 
* 
(4.7) 
[(z - x ) ~  
+ R2I1l2 
The boundary conditions for the set of integral differential equations (4.6) 
and (4.7) are described by the third equality in (4.6) and U(0, t) = 0, since 
the leader base is grounded. Practically, it is convenient to subdivide the 
channel into N fragments and consider the charge density in each fragment 
to be dependent only on time, thus replacing the integral equation of (4.7) by 
a set of linear algebraic equations. Each of them will relate the potential 
U(xk) at the middle point xk of the kth fragment to the intrinsic and all 
other linear charges. After integrating (4.7), one can easily see that radius 
R enters logarithmically the factors of the set of equations (compare with 
(4.2)), thereby justifying the use of linear leader charge r instead of its 
cover space charge. The set of algebraic equations for U(xk) and '(Xk) is 
solved in time at each step, and the progress is made by using equations 
(4.6). We are presenting the result of this solution. 
As the leader tip approaches the cloud, the external field at the tip site 
becomes stronger and the ever increasing portion of the channel finds itself 
in a strongly non-uniform field. Since the velocity and current are largely 
defined by the potential Uo(L) at the tip site, formulae (4.5), in which Eo is 
an average field, remain valid. In a simple model of a cloud with a spherical 
unipolar charged region, the potential distribution in the space free from 
charges is the same as for a point charge. If H is the height of the spherical 
charge centre, Q,, the potential at height x at the point displaced from the 
vertical charge axis for distance r (with the account of the mirror reflection 
by the earth's plane) is 
- 
}. 
(4.8) 
"{ 
1 
= 4T&o [(H - x ) 2  + r2I1l2 [(H + 
+ y2I1l2 
Figure 4.1 presents the parameter calculations for an ascending leader 
propagating in such a non-uniform field. The calculations were made from 
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=== PAGE 152 ===
144 
Physical processes in a lightning discharge 
3 00 
~ 
200 
5 
s 
8 
U 
100 
- 1.4 
1.2 e 
3 
- 
E
.
 
E 
- 1.0 *z 
6 
- -0.8 'g 
8 3 .  
d 
3 
? 
2 - 0.6 
L 
- 0.4 2 
- 0.2 
Y 
1 
3 
Figure 4.1. The propagation of an ascending leader from a grounded object in a 
negative cloud field. 
Indicate current iL calculated from (4.5); Q, = 5C, 
H = 3km, r = 0.5km. 
the set of equations (4.6) and (4.7), as described above. The current at the 
channel base is defined by the total charge Q and the velocity by expression 
(4.5): 
For comparison, the current was also calculated from (4.5). The results show 
the good accuracy of this simple formula, so the use of average linear 
capacitance C1 can be considered justifiable in the calculation of 
T(L) = CIAUo(L) and in the case of a sharply non-uniform field. 
4.1.3 Penetration into the cloud and halt 
There are two questions to be answered here: how high the maximum leader 
current is and where the leader halts. To answer the first question, one should 
keep in mind that the cloud charge is concentrated in a certain volume but 
not at a point. Suppose it is a sphere of radius R, with the centre at height 
H in (4.8). Measurements made during flights through storm clouds indicate 
that R, is most likely to be by an order of magnitude smaller than H .  The 
maximum potential at the centre of a uniformly charged sphere is by a 
factor of 1.5 higher than on its surface and equals U,,,, = 3Qc/8mORc. 
Penetrating into the charged region, an ascending leader acquires a 
considerable velocity and a very high current. To illustrate, calculated 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 153 ===
An ascending positive leader 
145 
from (4.8) for H = 3 km, the field near the earth under the charge centre, 
EO = -Qc/27qH2 = 150V/cm, is created by charge Q, = -7.4C and 
~Uo,,,I 
= 340MV for R, = 300m. From (43, a leader that has reached 
the charged region centre acquires the velocity wL = 2.8 x 105m/s and the 
current i,, 
= 5 kA (the field at the sphere boundary is EOmax 
= 7.5 kV/cm, 
decreasing to zero towards the centre; in the current estimation, the 
logarithm was taken to be 1). The lifetime of this high current is short, 
about Rc/w~,,, N lO-'s, with the total duration of the leader ascent of 
about 3 x 
s (these estimations ignore the effect of air density, which is 
1.5 times lower than normal at a height of 3 km). Maximum currents of 
the kiloampere scale were registered during observations of ascending 
lightnings. 
On its way up through the charged region, the leader enters an area of 
reciprocal external field at height x > H. The potential difference AU, is, 
at first, positive but decreases as the leader elongates. Its velocity and current 
now decrease with time, but this process has its limits. There is a region of 
positive charge of nearly the same value high above the negative charge 
region. Representing it as a sphere with the centre at height H + D and 
taking the mirror reflection effect into account, as in (4.8), we can find the 
potential of the dipole thus formed: 
- 
1 
+ 
1 
}. 
(4.10) 
0) the 
leader tip will reach the point x,, where the absolute potential U. drops to 
AU, E 400 kV, remaining negative as before. Since AUlmn is small relative 
to huge potentials of charged regions (I Uolm,, N 100 MV), a positive ascend- 
ing leader halts at a slightly lower height than the zero equipotential surface 
of the external field. Because of the effect of charges reflected by the earth, the 
zero potential line lies somewhat lower than the dipole centre. For example, 
at D = H, which corresponds, more or less, to reality, we have x, = 1.4868 
exactly on the vertical axis (Y = 0) instead of 1.5H, as would be the case with 
a solitary dipole. With greater radial displacement Y, the zero equipotential 
line comes closer to the earth, slowly at first but then more rapidly at 
r > H (figure 4.2). This is the reason why ascending leaders taking different 
vertical paths halt at different heights. 
It has just been mentioned that the equipotential line Uo(x, 
Y) = 0 
corresponds to the maximum height attainable by a single ascending 
leader. With allowance for the voltage drop across the channel, which 
may appear appreciable in some situations, AU, drops to the threshold 
value AUc, below the maximum height. This is supported by numerical 
[(H + D - x ) ~  
+ r2I1l2 [(H + D + x ) ~  
+ r2I1l2 
Without allowance for the voltage drop across the channel (U, 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 154 ===
146 
Physical processes in a lightning discharge 
0.00 
0.0 
0.5 
1 .o 
1.5 
2.0 
r/H 
Figure 4.2. The zero potential line of a cloud dipole with the allowance for charges 
reflected by the earth for D = H. 
calculations made from the set of equations (4.6) and (4.7) and illustrated in 
figure 4.3. They also indicate the leader retardation rate. As the leader 
velocity decreases, the channel current becomes lower, causing the field E, 
to rise. The tip potential decreases respectively, together with the potential 
difference AU,, which limits the current still more, and so on. 
2.0 
Q 1.5 
E 
0 
d 
.s 1.0 
8 
U 
8 0.5 
el 
-, 
0 
0 - 
0.0 
Leader length, km 
200 
150 
$ 
loo z 
50 
0 
1000 
800 s 
+ 
600 
a 
400 $ 
el 
200 
0 
Figure 4.3. Numerical simulation of an ascending leader propagating in a cloud 
dipole field (Qc = 12 C, H = D = 3 km, r = 0.5 km), with allowance for the voltage 
drop across the channel. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 155 ===
An ascending positive leader 
147 
When a leader goes beyond the lower cloud charge region, the external 
field changes its direction along the channel: below the negative charge 
centre, x = H ,  its vertical component is directed upwards but above the 
centre it is directed downwards. Correspondingly, the external potential U, 
is non-monotonic and has an extremum at height H (absolute maximum). 
The leader continues to develop beyond the maximum point, as long as 
the relation 
Aut 
ut(L) - Uo(L) = IUo(L)I - Iut(L)/ > AUt,,,,, 
is valid. But now, the leader velocity decreases continuously, because I Uo(L)I 
drops with the leader elongation and I U,(L)I rises due to the rising channel 
field. It is clear that a leader can develop successfully in any other direction, 
since it is capable of propagating in the direction strictly opposite to the 
external field. The calculations show the leader path along the equipotential 
line in a zero external field. Here, AU,, i and wL decrease slowly, only due to 
the greater voltage drop across the channel; otherwise, the leader would 
travel for an infinitely long time. 
We have focused on this circumstance because it is here that the princi- 
pal features of a leader process manifest themselves clearly. The external field 
at the tip site is usually low and cannot affect the instantaneous leader velo- 
city, current and direction of motion. The direction may vary randomly, a 
fact well known to those making lightning observations. What is important 
is the voltage U, created by this field along the leader path, rather than the 
field strength. The propagation of a positive leader is provided by the trans- 
port of a fairly high positive charge to its streamer zone. The current of many 
streamers taking the charge out accumulates in the channel, heating it and 
providing its viability. But for many streamers to be excited off from the 
leader tip, the latter must possess a high potential relative to the unperturbed 
potential AU, = U,(L) - Uo(L) x IUo(L)I. This is indicated by the absence 
of appreciable discrepancies between the current calculations made straight- 
forwardly from the linear density of induced charge T(X) in a strongly non- 
uniform external field and from formula (4.5) containing only AU,. 
Therefore, it is not surprising that the lightning paths exhibit the diversity 
illustrated in figure 4.4. No random change of the leader path can disturb its 
viability. A leader can follow any direction: it can move along the external 
field or in the opposite direction, along the equipotential line, etc. - all ways 
are open as long as the condition AU, > AUtmn is valid. But the leader 
acceleration does depend, of course, on its direction of motion. Moving 
along the field, the leader is accelerated, because the voltage drop is compen- 
sated excessively by the increase in I U, 1, When the leader moves in the opposite 
direction, it is decelerated. The maximum acceleration is achieved in the 
direction of the maximum gradient AU,, and this seems to be the reason for 
the fact that the main leader branch darts in the direction of the rising field, 
i.e., towards a charged cloud, a high object, etc. (for details see section 5.6). 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 156 ===
148 
Physical processes in a lightning discharge 
Figure 4.4. A photograph of a well-branched lightning with the path bendings. 
4.1.4 Leader branching and sign reversal 
Leader branches are nearly always visible in photographs of ascending 
lightnings. Branching may start almost from the channel base or after the 
leader has covered many hundreds of metres (figure 4.4). The currents of 
branches are summed up at the branching points, so it is higher at the channel 
base than in any branch. It is very unlikely that branches would start 
simultaneously and that the potential differences AU, at their tips would 
be the same at any moment of time. Rather, the values of AU, are distributed 
randomly. An abrupt decrease or even an entire cut-off of current in one of 
the branches does not at all mean that a similar thing has happened in 
another branch or in the base. Therefore, at least one of the ‘main’ 
branches will have a relatively high current and, hence, a greater probability 
to go up very high and even to reach the maximum leader height x, than a 
single leader does. This event is stimulated by the decreasing voltage drop 
along the branching leader ‘stem’, where the total branch current has accu- 
mulated and where the field is low (in accordance with the current-voltage 
characteristic), especially if the stem is long and branching occurs at different 
heights. 
Branching can send the leader up above the zero equipotential surface, 
where its sign reversal occurs. Imagine the situation, in which a branch, 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 157 ===
An ascending positive leader 
149 
running up far from the charged cloud region, has reached the maximum 
height x,(r) and stopped. The channel plasma cannot decay immediately 
but persists for some time. During this time, another, luckier branch has 
approached the negative cloud bottom and even partly penetrated it. In 
this region, the cloud has potential U,,,,, so that the branch portion that 
has entered it acquires the positive charge 
which may be as large as S x 10% of the negative cloud charge. Due to the 
partial compensation of the lower cloud charge, with the upper charge 
being constant, the zero equipotential surface will become lower by the 
length Ax, which is about the same percentage of x, - H ,  so that 
Ax x S(x, - H). As a result, the upper portion of the first halted branch 
(from the tip down to the new zero equipotential surface) will be in a field 
directed downwards. The new external potential at the tip site, 
U; x IdUo/dxlxsAx, will become positive and the potential difference 
AU, = U,(L) - U;(L) will be negative. For the example given in the previous 
section with x, - H x 1.5 km at S x 0.1, we have Ax x 150 m; from formula 
(4.10) with r << H ,  we have IdUo/dx~xs 
x 600V/cm, so that eventually 
U;(L) x 9MV. Even if the branch penetrating the cloud charge misses its 
centre to enter a region with a potential several times lower than U,, 
(as 
a result, UA(L) will be reduced as much), this will still be sufficient to 
revive the first leader branch. 
Therefore, the halted leader has a chance to revive and move on up to 
the upper positive cloud charge but as a negative leader this time. The 
leader position at the point of the first stop is unstable. Even a slight 
perturbation, such as a decrease in the lower cloud charge (in the example 
presented, due to the penetration of another branch) may stimulate its 
further growth with the opposite sign. As the leader develops, it will 
penetrate into an increasingly higher field of the upper charge and 
become accelerated. Having passed the upper charge centre, H + D, it 
will be retarded and stop, for good this time, at a height H,,, 
> H + D, 
where the potential of (4.10) will drop to a relatively low value of AUtmin. 
The height H,,, 
may be 10-20km or higher if one accounts for the air 
density decrease. The currents flow in different directions in different 
portions of this leader. Above the equipotential surface, the current flows 
downwards, as in a negative leader. In the lower leader portion which 
serves as a stem for many positive branches, the current remains directed 
upwards. The observer, who registers the current at the earth, may not 
suspect the sign reversal occurring up in the clouds. The channel field is 
established in accordance with the current. It reverses in the upper channel 
portion, thereby reducing the total voltage drop across the branch that 
went far up and stimulating its further development. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 158 ===
150 
Physical processes in a lightning discharge 
4.2 
Lightning excited by an isolated object 
Like a high grounded body, a large object isolated from the earth can become 
a source of lightning in a high electric field of a storm cloud. Discharges can 
be induced by fields extending not only between the earth and a charged 
cloud but also between oppositely charged clouds. A lightning discharge 
can be excited by a large aircraft, rocket or spacecraft when it travels through 
the troposphere, and this is a serious hazard to its flight. Therefore, this 
phenomenon is of primary practical importance. 
4.2.1 A binary leader 
In contrast to an ascending leader starting from a high grounded body 
(section 4.1), an isolated body produces two leaders, one going along the 
external field vector and the other in the opposite direction. The physical 
reason for the excitation of two leaders is the same. The external field induces 
charge in the conductor, so that a large difference between its potential and 
the external potential arises at the conductor end. If the body is extended 
along the field, the electrical strength at its end increases abruptly. In contrast 
to the situation with a grounded conductor, the opposite charge does not 
flow down to the earth but accumulates at the other end, polarizing the 
isolated body. A grounded conductor in an external field possesses the earth’s 
potential, while an isolated conductor acquires a potential corresponding to 
an average external potential along its length. Large differences between the 
body’s and external potentials (of opposite signs) now arise at the ends of the 
body, and both ends are capable of exciting leaders of the respective signs. A 
long conductor absolutely symmetrical relative to its average cross section 
transversal to the uniform field acquires potential U equal exactly to the 
external field at the body’s centre. The distribution of unlike charges in 
each of its halves is identical to the charge distribution in a grounded 
conductor of the same size and shape as the isolated conductor half. 
The process described here can be easily reproduced in laboratory 
conditions. Figure 4.5(a) shows streak pictures of leaders which have started 
from a rod of 50 cm in length, suspended by thin plastic threads in a 3-m gap 
in a uniform field. One can see all characteristic features of a positive leader 
propagating continuously to the upper negative plane and those of a stepwise 
negative leader travelling down towards a plane anode. Generally, leaders 
arise at different moments of time because of the threshold field difference 
for the excitation of positive and negative initial streamer flashes or due to 
the difference in the curvature radii of the rod ends. The leaders may have 
different velocities because the same voltage drop A U, creates streamer 
zones of different sizes at the positive and the negative ends. The instanta- 
neous currents at the growing channel ends may also differ. But on average, 
every leader transports the same charge, since the net charge remains to be 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 159 ===
Lightning excited by an isolated object 
151 
Figure 4.5. Streak photographs of leaders from the ends of a metallic rod placed in a 
uniform electric field: (a) general view; (b) fast streak photograph demonstrating the 
relationship between the positive and the negative leaders; (1) rod, (2), (3) tip and 
streamer zone of positive leader, (4), (5) tip and streamer zone of negative leader, 
(6), (7) negative and positive leader flashes. 
zero in a system isolated from the voltage source. The discharges appear to be 
interrelated. Any fluctuation - say, a flash - of one leader appreciably acti- 
vates the other: the space charge (e.g. positive) incorporated in front of the 
rod stimulates the accumulation of negative charge across the conductor, 
thereby enhancing the field at its negative end. The high-speed streak pictures 
in figure 4.5(b), 
resolving individual streamer flashes, show an activation of 
the positive leader channel following a negative leader flash. 
The conditions for the start of leaders from a long isolated conducting 
body are the same as from a grounded conductor, and they are also defined 
by expression (4.4). But now, when estimating the threshold field Eo from the 
value of AVimin = Eod, one should keep in mind that d is a half length of an 
isolated leader. For a field capable of exciting a discharge from a conductor 
of length 2d, we find 
(4.11) 
Eo=-[ 1 
bln(L/R) ] 2/5 . 
d3/5 
27qa 
As in the illustration in section 4.1, we take the ratio of the channel length of 
a young leader to the equivalent charge cover radius to be L/R M 10. Then 
we have Eo = 440 V/cm for an aircraft of length 2d = 70 m. This estimate 
describes the external field component along the aircraft axis. But an aircraft 
often flies at an angle to the field vector, so that the threshold external field 
may be several times higher. Fortunately, the lightning excitation threshold is 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 160 ===
152 
Physical processes in a lightning discharge 
not very low, otherwise airline companies would suffer tremendous losses 
from lightning damage. On the other hand, fields of this scale are not very 
rare: much higher fields were registered during airborne cloud surveys. For 
this reason, the problem of lightning protection in aviation is regarded as 
being very serious. 
Having started from an isolated body, each leader develops as long as 
the external field permits. This process is basically the same as that discussed 
in section 4.1 for an ascending leader. Below, we shall consider the specific 
behaviour of two differently charged leaders developing simultaneously. 
This specificity becomes especially clear in a non-uniform field typical of a 
storm cloud. 
4.2.2 Binary leader development 
The principal features and quantitative characteristics of a binary leader can 
be understood from a simple model. The x-coordinate will be taken along the 
leaders. The leader paths should not necessarily be straight lines but they may 
have various bends, as is the case in reality. Denote the external field poten- 
tial along the leader lines as Uo(x) and their tip coordinates as x1 and x2. In 
figure 4.6 Uo(x) corresponds to the field of a negatively charged cloud. The 
leaders were excited by a conducting body somewhere half way between 
the cloud and the earth. The x-axis is directed upwards, the leader with the 
subscript 1 travels downwards and the one with the subscript 2 upwards. 
Let us neglect the voltage drop across the leader channels, ascribing the 
same potential U to the channels and the initiating body. The whole system 
now represents a single conductor. In the satisfactory approximation above, 
in which the capacitance per unit length C1 at every moment of time was 
Figure 4.6. A schematic diagram of a binary leader channel in a cloud dipole field. 
x1 : descending leader tip coordinate; x2: ascending leader tip coordinate; xo: position 
of zero charge point. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 161 ===
Lightning excited by an isolated object 
153 
assumed to be the same along the conductor length, the general condition for 
an uncharged conductor is 
U~(X) 
dx. 
(4.12) 
C1[U- Uo(x)]dx=O; 
U=-/ 
1 
x* 
x2 - X I  
XI 
The condition of (4.12) defines the conductor potential U ,  which generally 
varies in time during the leader development (it is constant only if the electric 
field and the binary leader are symmetrical relative to the centre of the 
initiating body, which is also symmetrical). The conductor potential is 
equal to the average external potential along its length. The leader velocity 
can be calculated from (4.2). It was pointed out above that the available 
theory cannot provide a clear physical expression for the velocity of even a 
relatively simple, continuous positive leader, let alone a negative stepwise 
one. It is this circumstance which makes one resort to the empirical formula 
(4.2) derived from results of laboratory experiments with currents up to 
lOOA and justifiable, to some extent, for positive leaders. No similar 
measurements for negative leaders are available, and this is especially true 
of natural lightning observations. So, one has to rely on close experimental 
data on breakdown voltages in superlong gaps at the sign reversal of the 
high voltage electrode, as well as on the moderate velocity differences 
between positive and negative lightning leaders. The deviations of their 
measured values usually overlap. These facts provide good grounds for 
extending expression (4.2), as a first approximation, to negative leaders. In 
the latter case, we mean the average velocity neglecting the instantaneous 
effects of stepwise development. This approximation is the more so justifiable 
that the direct dependence of the leader velocity on the potential difference at 
the tip, AU,, raise no doubt and that the variation of the factor a or of the 
power index in (4.2) cannot change the picture qualitatively. 
Thus, with the account of the x-axis directions and velocities, as well as 
the signs of AU at the leader tips, the equations for the leader development 
can be written as 
1 12 
(4.13) 
dx2 
lI2 
-=a[Uo(x2) 
- U ]  . 
= -a[U - Uo(x1)] ’ 
dt 
dt 
Together with (4.12), expressions (4.13) describe the evolution of the two 
leaders starting from the body ends, whose coordinates xl0 and x20 are 
given as the initial conditions for equations (4.12) and (4.13). The sign 
reversal point of the conductor, xo(t), defined by the equation 
U(t) = UO(x0) is displaced, during the leader propagation, in accordance 
with the nature and degree of field non-uniformity along the channels. 
Having solved the equations, one can find the currents at the leader tips 
from (4.3) with L = x2 - xl. Generally, they differ quantitatively from one 
another and from the current in other channel cross sections, including the 
sign reversal at point xo, through which the total charge flows during the 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 162 ===
154 
Physical processes in a lightning discharge 
polarization. The current i(xo) = io is defined as 
l o = - = - -  
' 
Ql = C1 r [ U -  Uo(x)]dx. 
(4.14) 
dt 
dt ' 
XI 
This current is used for changing the charge of the old leader portions, 
increasing or decreasing them as AU(x, t), and for supplying charge to its 
new portions. This leads to the current variation along the channels, which 
can be found by solving the problem. 
For some simple distributions of Uo(x), the division of equations (4.13) 
by one another 
(4.15) 
allows the functional relationship between x2 and x1 to be found from 
squaring, after which finding the final result x1 ( t ) ,  x2(t) reduces to squaring, 
too. This becomes possible if the cloud field is approximated by the point 
charge field Uo(x) - IxI-', and if a new variable z = x2/x1 is introduced. 
The resultant formulas allow an analytical treatment of some characteristic 
relationships. To avoid cumbersome derivations, we invite the reader to do 
this independently, while we, instead, shall present some numerical calcula- 
tions for several variants. 
The calculations prove to be quite simple in integrating the set of 
equations (4.13) and (4.14) as well as in the case of a more rigorous approach 
to the problem, when the charge distribution along the conductor length is 
found from an equation similar to (4.7). Figure 4.7 demonstrates the propa- 
gation of vertical leaders in the field of a cloud dipole (with the allowance 
for the earth's effect). The calculation was made using an equation similar 
to (4.7). The initiating vertical body is located between the lower negative 
charge of the dipole and the earth, being displaced horizontally by 
Y = 500m from the charge line. As the ascending leader moves up, its tip 
approaches the bottom charge centre and enters a region of an ever increas- 
ing field. The descending leader moves more slowly towards a weaker field. 
The external field potential approaches zero at the earth but increases rapidly 
near the charged cloud. As a result, the negative potential of the conductor 
made up of the leader channels, U ,  rises with time, with the sign reversal 
point xo going up closer to the cloud. At the initial moment of time, the 
potential is U = -27 MV and the point is at an altitude xo = 1603m. 
When the ascending leader reaches the charged centre 17 ms later, we have 
the altitude xo = 2040 m and U = -64 MV. The absolute potential rise 
stimulates the descending leader, increasing its velocity by a factor of three 
during this time in spite of its propagation through an ever decreasing 
external field. The calculations made with (4.13) and (4.14) have yielded 
similar results. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 163 ===
Lightning excited by an isolated object 
155 
" . " I  
. 
I 
. 
, 
. 
I 
. 
I 
, 
0 
5 
10 
15 
20 
Time, mc 
Figure 4.7. The propagation of leaders from a metallic body located between the 
cloud and the earth (Q, = -1OC, H = D = 3 km, Y = 0.5 km). 
Figure 4.8 illustrates the propagation of one of the leaders in a zero 
external field and refers to the situation when the descending leader has 
suddenly changed its direction for some reason at a certain height to 
follow the equipotential surface, i.e. along the zero field. The calculation 
was made with (4.13) and (4.14). Similar to the first variant, this situation 
exhibits a remarkable property of a binary leader. The leader developing 
along a rising field sustains the other leader, which has travelled in less 
favourable conditions, allowing it to move with a certain acceleration even 
in a zero field. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 164 ===
156 
1.4- 
1.2- 
1.0- 
0
.
 
6 
0.8- 
B
.
 
7 0.6- 
8
.
 
0.4- 
0.2 - 
2
.
 
.e 
S
.
 
Physical processes in a lightning discharge 
leader moving along \ 
equipotential path 
0
.
0
-
I
 
10 
15 
20 
25 
Time, ms 
4.0 
3.5 
l 3 . 0  
J 
.s 2.5 
U 
7 
2.0 
1.5 
downward 
leader 
upward leader 7 
/rd 
I 
1'5 
' 
20 
25 
Time, ms 
1.6 
1.4 
1.2 .E 
5 
1.0 2 
0 - 
0.8 3 
0.6 
0.4 $ 
0.2 
0.0 
* 
.I 
Figure 4.8. The development of a leader pair from a metallic body at 1.5 km above the 
earth in a cloud dipole field (H = 3 km, D = 3 km, Q, = - 10 C). At the moment of time 
N 10 ms and 1 km altitude, the descending leader turned to follow an equipotential path: 
(top) leader velocities; (bottom) position of the zero charge point (T = 0), the altitude of 
the ascending leader tip and the length of the portion along the equipotential path. 
Where an isolated conducting body may initiate a lightning discharge 
depends, to some extent, on a mere chance. A leader may start under a 
storm cloud, as in the illustrations just described, or inside a cloud at the 
height of the lower or upper charges or somewhere between them. These 
variants differ considerably in the polarization charge distribution along 
the conductor and, hence, in the leader propagation conditions. A situation 
may arise when the positive leader penetrates into the field of the negative 
lower cloud, thereby transporting a positive charge to the earth, which 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 165 ===
Lightning excited by an isolated object 
157 
must apparently be ‘attracted’ by the negative cloud. (In the two situations 
above, it was the negative leader that travelled to the earth, ‘naturally’ 
extracting a negative charge from the cloud). This exotic situation arises 
when two leaders are excited from a body located somewhat above the 
negative charge centre. The negative leader then goes up to a positive 
cloud. The strong field created by the cloud dipole induces a large negative 
charge in the ascending leader, displacing the positive charge down. The 
average external potential between the two leader tips, U < 0, appears to 
be ‘more positive’ than the external potential of the lower tip, Uo(xl) < 0, 
so that AU,, = U - Uo(xl) > 0. This is the reason why the descending 
leader is positive. With time, when the ascending tip comes closer to the 
positive charge, the potential U of the binary system does become slightly 
positive (3-3.5 MV), making up several percent of 1 Uol,,,. 
Then it persists 
as such, sustaining the descending leader travel to the earth. By the 
moment of contact with the earth, the positive charge is distributed along 
the channel in about the same way as in a grounded conductor in a negative 
cloud field, being mainly concentrated at the height of this charge. For this 
reason, the return stroke current, which only slightly contributes to the 
charge after the contact, is weak. The return stroke can be said to make no 
contribution to the charge redistribution, since the channel potential 
should be corrected only slightly (as compared with I Uolm,, x 100 MV), by 
reducing it from 3 MV to zero. This reduction enhances, though only slightly, 
the ascending leader, which travels on until it stops high above the positive 
charge of the storm cloud. 
When an isolated body initiates two oppositely directed leaders, it does 
not always happen that the descending positive leader reaches the earth. For 
the contact with the earth to take place, the potential U of the conductor 
made up of the two leaders must become positive at a certain moment. Other- 
wise, the descending leader will stop at the point xls, where the negative 
leader tip potential U,, will be by a small value of AVtm,, higher than the 
negative potential of the external field (assuming AVtm,, = 0, when U is 
equal to Uo(xls)). The condition for the average conductor potential to be 
positive at the moment of contact with the earth is described by the inequality 
JZ u0(x) 
dx > o 
(4.16) 
which follows from (4.12). Here, Uo(x) is the cloud dipole potential given by 
(4.10) with the allowance for its reflection from the earth, and x2 is the 
altitude the ascending leader tip has reached by that moment. In principle, 
there are no reasons for this inequality to be violated, since the integral of 
(4.16) in the limit x2 = CO is necessarily positive (and equal to 
-21n(l + D/H)Q,/4mo at r << H )  for the negative lower cloud charge 
(Q, < 0). This means that the descending positive leader has a chance to 
reach the earth - this only requires that the ascending leader should reach 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 166 ===
158 
Physical processes in a lightning discharge 
a sufficient altitude. If the horizontal channel displacement from the vertical 
line crossing the centres of the cloud charges, Y << H ,  we obtain x2 > 2.478 
from (4.16) and (4.10) at D = H. The ascending leader must cover a distance 
of about 0.5H above the upper positive charge centre. 
4.3 
The descending leader of the first lightning component 
4.3.1 The origin in the clouds 
Although lightning observers are familiar with the propagation of a descend- 
ing (negative) stepwise leader, the conditions and the mechanisms of its 
origin are literally foggy. No one has ever observed the lightning start or 
its development in the clouds. Its origin cannot be totally reproduced in 
laboratory conditions, although negative stepwise leaders have been pro- 
duced experimentally (section 2.7). But the conditions for their initiation 
by a high-voltage metallic electrode connected to the condenser of a impulse 
generator have little in common with what actually occurs in the clouds - a 
cloud is not a condenser winding and, of course, not a conductor. The 
negative cloud charge is scattered throughout the dielectric gas on small 
hydrometeors. It is very hard to perceive how the charges, fixed to particles 
with low mobility and dispersed in a huge volume, can come together to form 
a plasma channel in a matter of a few milliseconds. 
In our terrestrial practice, we encounter events somewhat similar to the 
spark initiation in the clouds. Investigation of what has caused an explosion 
or a fire in industrial premises containing an abundance of electrostatic dust 
particles or droplets can provide evidence for a spark discharge arising in a 
medium with a dispersed charge. Lately, there have been reports of studies 
with gas jet generators ejecting into the atmosphere miniature electrically 
charged clouds [3,4]. Sometimes, extended bright structures of about 10 cm 
in size were observed along a charged spray boundary; on some occasions, 
they were observed to form spark channels of about 1 m in length. Unfortu- 
nately, no measurements could be made of the field at the discharge start, so 
the fact of discharge excitation was only stated. Therefore, one can do 
nothing more than just make conjectures about the excitation mechanisms 
of lightning in the clouds and of sparks in laboratory sprays. 
Speculations concerning these mechanisms (the only type of conclusion 
we can draw today) have to be arrived at via the process of elimination. A 
cloud medium cannot be considered as being conductive when we speak of 
current supply to the leader channel. Common charges are not transported 
directly to the leader, nor do they leave the cloud by themselves during the 
fast leader process. Therefore, the cloud charges play a different role - 
they are the source of electric field which ionizes the air molecules, producing 
the initial plasma, and then sustains the leader process. To fulfil the first task, 
the field somewhere in the charged region is to exceed the ionization 
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=== PAGE 167 ===
The descending leader of theJirst lightning component 
159 
threshold (Ei M 20-25 kV/cm at the height of the cloud charge) or the cloud 
is to contain inclusions enhancing the field locally via the polarization charge. 
It seems that neither mechanism should be discarded entirely, although cloud 
probing rarely registered fields exceeding several kilovolts per centimetre. 
These results do not testify to the absence of higher fields, because most of 
the measurements concerned fields averaged over lengths of several dozens 
of metres. No measurements were made at the moment of lightning initia- 
tion, because the probability of a detector registering a field at the right 
place at the right moment is extremely low. On the other hand, the conditions 
necessary for the excitation of a leader process in a cloud are quite rare; 
otherwise, the number of lightning strikes per square kilometre of the earth’s 
surface would greatly exceed 2-5 per storm season. 
Let us estimate the volume to be occupied by a cloud charge capable of 
creating an ionization field. It was mentioned above that the field Eo at the 
earth was often found to be lOOV/cm during thunderstorms. This value 
should not be considered to be the cloud dipole field, since the near-earth 
charge provided by microcoronas from various pointed objects attenuates 
the cloud field at the earth. A similar value is obtained from a small positive 
charge supposed to lie under the principal negative charge [5]. Taking, for 
estimations, the intrinsic dipole fields Eo to be 200V/cm and the heights of 
the lower (negative) and the upper (positive) charges to be x = H = 3 km 
and H + D = 6 km, respectively, we find, from (4.10), the dipole charges 
Q, = 13.3 C. These values will serve as guidelines in further numerical calcu- 
lations. The charge Q, can create field Ei M 25 kV/cm at its boundary if it is 
distributed throughout a sphere of radius R, = 220 m. Measurements show 
that the charged region is, in reality, 2-3 times larger, but one should not 
discard the possibility of a short accidental charge concentration in a smaller 
volume due to the action of some flows in the clouds. 
More probable is the situation when a macroscopically averaged maxi- 
mum field of cloud charge is several times lower than Ei and local fields, 
enhanced to Ei x 25 kV/cm, arise near polarized macroparticles. Note that 
the maximum field near a metallic ball polarized in an external field E is 
E,,, 
= 3E. Similarly enhanced is the external field of a spherical water 
droplet, since water possesses a very high dielectric permittivity E = 80 and 
E,, 
= 3 E ~ / ( 2  + E ) .  Therefore, if charge Q, is concentrated in a sphere of 
& times larger radius, R, = 380 m, the field three-fold enhanced by polari- 
zation can achieve the ionization threshold. Following the ionization onset, 
streamers may be produced around large droplets, giving rise to a possible 
leader, because streamers may be branched and extended in an average 
field of -10 kV/cm. 
Leaving aside the mechanisms of ionizing fields and leader origin, 
because they are still poorly understood, we shall take for granted only the 
mere fact that a leader does occur. At its start, a descending leader is 
devoid of the possibility of taking the charge it needs away from the cloud. 
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=== PAGE 168 ===
160 
Physical processes in a lightning discharge 
Observations show that this charge is quite large: an average negative leader 
transports to the earth a charge QL RZ -5 C and, sometimes, it is as large as 
-20 C [l], a value close to the evaluations of Q, for the storm cloud. But if the 
cloud charge remains ‘intact’, the only thing that can provide the charge 
balance is the ascending leader of opposite sign, which is to develop simulta- 
neously with the descending leader. This idea was suggested in [6], which 
presented a qualitative distribution of the charge induced along a vertical 
conductor made up of two leaders prior to and following its contact with 
the earth. What happens is principally the same as in the excitation of two 
leaders by a conducting body isolated from the earth and is affected by an 
external field (section 4.2). This process is independent of the descending 
leader sign; therefore, one should not think that a negative cloud can produce 
only a negative leader while a positive cloud always produces a positive one. 
In any case, two oppositely charged leaders are produced simultaneously, 
and which of them will travel to the earth depends on the charge position 
in the cloud and on the leader starting point. 
A binary leader is most likely to be initiated near the external boundary 
of the charged region, because the field there is highest. The field at the 
centre of an isolated charged sphere is zero. In the case of a uniform 
charge distribution throughout its volume, the field rises along the radius 
as E N r but decreases from the outside as e N rP2 with the maximum 
E,, 
= Qc/47r~& at the boundary. For a dipole configuration of real 
charges, the field does not practically vary across the boundary surface of 
the charged region. For the above values of D, 
H and R,, the field at the 
upper point of the lower sphere is about 5% higher than at the lower 
point. The probability of a binary leader being initiated at either point is 
nearly the same. However, the final result of the binary leader development 
will differ radically, and this circumstance was essentially demonstrated in 
section 4.2. If both leaders are initiated at the bottom edge of the lower 
negative charge, the negative leader will go down and the positive one will 
go up. The negative leader has a real chance to reach the earth with a high 
negative potential equal to that of cloud charges averaged over the whole 
conductor length. The conductor is mostly in the region of high negative 
potential, nowhere entering the positive potential domain. The closing of 
this highly charged channel to the earth leads to the wave processes of 
charging and charge exchange (the return stroke) involving high current. 
The latter represents a real hazard. This is what happens in the case of a 
negative lightning. If a binary leader is initiated at the upper boundary of 
the lower charge, the positive leader goes down to the earth and the negative 
one goes up. A positive descending leader can never reach the earth unless it 
acquires a positive potential. For this, its ascending partner must necessarily 
go beyond the zero potential point, closer to the upper positive charge of the 
dipole. Owing to the compensation of positive and negative charges at 
various sites along the path, the average potential transported down to the 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 169 ===
The descending leader of the first lightning component 
161 
earth is quite low. This actually cancels the return stroke current. Positive 
lightnings with very low currents of about 1 kA present no danger and are 
quite frequent. The number of their registrations by the observer increases 
with increasing sensitivity of the detectors used. 
4.3.2 Negative leader development and potential transport 
The stepwise propagation pattern has been believed by many to be the 
principal problem for a theoretical description of a negative leader [7]. 
However, it is of little importance to the leader evolution whether it develops 
continuously or by relatively short steps. The leader current and velocity are 
averaged over many steps. Averaged also is the channel energy balance, 
although the energy release at a distance of several step lengths from the 
tip has a well defined periodic pulse character. We shall discuss the stepwise 
effect in section 4.6, following the consideration of the return stroke, since 
this process is involved in every step as the main component. 
The evolution of the descending channel of a binary leader is intimately 
related to that of its ‘twin brother’ - the ascending leader. (In ths sense, the 
term ‘Siamese twins’ would be more appropriate.) A characteristic feature of 
the twins is the break-off of their potential, which varies but little along 
their highly conductive channels, from the external potential at the start. In 
this respect, a lightning leader differs considerably from a laboratory leader 
starting from an electrode connected to a high-voltage source. Being ‘tied 
up’ to the electrode, a laboratory leader with a well conducting channel carries 
the electrode potential, which may be close to the source emf. Generally, it is 
lower than the emf by the value of the voltage drop across the external circuit 
impedance when a discharge current is flowing through it. The underestima- 
tion of the principal difference between a laboratory spark initiated from a 
high-voltage electrode and a natural electrodeless lightning leads to erroneous 
attempts to derive from observations the voltage drop value across the leader 
channel. The reasoning is usually as follows. The potential U ,  transported by a 
lightning leader to the earth can be estimated from the return stroke current 
and the characteristic channel impedance (section 4.4). The cloud potential 
U,, can also be estimated (see formula (4.17) below). The leader channel 
base has the same potential - as if the cloud were an electrode. Therefore, 
the voltage drop across the leader length, from the cloud to the earth, is 
AU, = lUoRl - lull, and the average field in the channel is expressed as 
E, = AUJL, where L is the leader length ( L  M H ,  or 30-50% greater with 
the allowance for the path bendings). Such estimations lead to incredibly 
large values of A U, 
100 MV and E, M 1 kV/cm. A mature leader channel 
with i ~ 1 0 0 A  
current cannot have such high fields. Its state is very much 
like that of the quasi-equilibrium hot plasma in an arc, which has a field 1-2 
orders of magnitude lower. This follows from theory and from evaluations of 
fields in superlong laboratory sparks. 
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=== PAGE 170 ===
162 
Physical processes in a lightning discharge 
The attempts to solve the ‘inverse’ problem using the expression 
1 U,,l = AU, + 1 U1 1 to calculate the storm cloud potential have also failed. 
When one includes in this expression the generally correct values of arc 
field E,, one gets unjustifiably low cloud potentials U,, inconsistent with 
atmospheric probing measurements and other calculations. 
The methodological error of both approaches is due to the rigid relation 
of the base potential of a descending leader to the external field potential at 
the leader start. In actual reality, the channel potential undergoes a consider- 
able time evolution, being determined by the polarization charge distribution 
along the binary leader length. When the descending leader approaches the 
earth, its base potential may differ significantly from the potential created 
by the cloud charge at the start site at the moment of start. 
A simple calculation of the leader development can be made from 
equations (4.13) and (4.12), but a more rigorous solution can be obtained 
from equations (4.13) and (4.7), which were used for that purpose in section 
4.2. One should also bear in mind that if both leaders start from the 
boundary of a charged cloud region, at least one of them will enter the 
charged volume and may even pass through its centre. Then, we have to 
discard the point model of a cloud dipole and make the next approximation 
by assuming that charge Q, is distributed uniformly with the density 
3QC/47rRf in a sphere of radius R,. Inside the sphere, the potential of its 
intrinsic charge is radially symmetrical and is equal at point r to 
r d R,. 
(4.17) 
3 Qc 
The potentials from the upper dipole charge and from charges reflected by 
the earth can be found as from point charges. They do not contribute 
much to U,$. For example, for the centre of a negative sphere with 
Q, = -13.3C and R, = 500m, we have U,, = -360MV 
and at the 
boundary U,, = 5 U,, = -240 MV. With all other charges taken into 
account, we get U,, = - 196 MV for the bottom edge of the lower sphere 
at H = D = 3km. 
Figure 4.9 presents the results of this calculation including those for the 
charge distribution along the conductor length from an equation similar to 
(4.7). We have evaluated the development of both leaders along the dipole 
axis, following the start from the bottom edge of the lower negative 
Figure 4.9. (Opposite) The model of a descending leader from the lower boundary 
of the negative dipole charge (Qc = -12.5C, H = 3 km, D = 3 km, R, = 0.5 km). 
Vertical channels have no branches: (top) tip positions of the negative descending 
leader, xl, and its positive ascending partner, x2, with the points of zero potential 
differences, xo; (centre) charge distribution along the leader channel; (bottom) 
potential and velocity of the descending leader. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 171 ===
The descending leader of the first lightning component 
163 
E 
U 
E --. 
I-" 
2 -  
1- 
0 
4 
-1 - 
-2 - 
-3 
Time, ms 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 172 ===
164 
Physical processes in a lightning discharge 
sphere. The dipole potential from (4.10) was used as Uo(x), 
except for the 
length in the charged region, where expression (4.17) was employed with 
r = jx - HI. The descending negative leader of a binary system is accelerated 
quickly after the start. Having covered about 500 m, it travels farther to the 
earth with a slightly decreasing velocity wL x (1.6-1.7) x 105m,’s, a value 
close to observations. The leader strikes the earth in 16ms. By that time, 
the ascending leader has reached the height x2 = 3.6 km. This is far even 
from the zero potential point located at x x 4.5 km, let alone from the 
upper positive charge located at an altitude of 6km. The descending 
leader, which started from a site with the local potential - 185 MV, trans- 
ports to the earth nearly half of this value, U1 % - 105 MV, in spite of the 
initial assumption of the zero voltage drop across the channel assumed to 
be a perfect conductor. 
The reader should not feel discouraged by the large calculated value of 
U1, which is more appropriate to record strong lightnings rather than to a 
common lightning discharge, especially considering that the cloud param- 
eters taken for the calculation were quite moderate. It will be demonstrated 
in section 4.3.3 that leader branching, which is a rule rather than an excep- 
tion, reduces considerably the potential transported down to the earth. It 
is quite likely, however, that lightnings of record intensities are produced 
in ordinary clouds rather than in those having a record high charge, but 
only if the descending leader does not branch (or does so slightly). 
The above calculation for an ideal situation with unbranched leaders is 
interesting and useful for two reasons. First, one should understand the 
physics of a simple observable phenomenon before one turns to its complex 
modifications. The other reason is, probably, more important. Practical 
lightning protection requires the knowledge of both typical average lightning 
parameters and their record high values. It is the latter that become more 
important in designing prospective measures for especially valuable con- 
structions and objects. As was pointed out above, the case of an unbranched 
leader just discussed is likely to be one of the rare but most hazardous 
phenomena. 
The potential U1 transported by a lightning leader to the earth is an 
important parameter for practical lightning protection. The return stroke 
current (section 4.4), the most destructive force of lightning, is proportional 
to U1. The nature of U1 becomes clear from the above conception of 
descending leader development in a binary leader process. Ideally, potential 
U1 is that of a perfect conductor, made up of two leader channels, at the 
moment of its contact with the earth. But the ascending and descending 
leaders develop differently, because their paths cross regions possessing 
different distributions of cloud potentials Uo(x). 
The descending leader 
travels nearly without retardation because the potential difference at its 
tip, AU,, = U - Uo(xl), 
remains almost constant (a decrease in 1 U1 is largely 
compensated by 
1 Uo(x)I 
decreasing towards the earth). The ascending 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 173 ===
The descending leader of the first lightning component 
165 
I” 
Figure 4.10. Estimation of the potential transported by a negative leader to the earth. 
positive leader moves ‘against’ the field and soon enters a region of a rapidly 
rising external potential; as a result, AU,, = U - U0(x2) 
becomes relatively 
low soon after the start. This leads to a lower velocity of the ascending 
leader, which now goes up ‘unwillingly’, being affected by its more active 
twin which moves faster, pumping its charge into it. For this reason, just 
before the descending leader contacts the earth, the total potential of the 
system nearly coincides with the external field potential U0(x2) 
at the site 
of the ascending leader tip (AU,, = U - Uo(x2) 
<< 1 Ul). 
This circumstance makes it possible to determine the transported poten- 
tial U1 = U just from the condition U = U0(x2) 
at x1 = 0. The condition has 
a clear geometrical interpretation (figure 4.10). The shaded regions between 
the external potential curve Uo(x) 
and the horizontal line intercepting it must 
be identical on both sides of the left-hand interception point (the point of the 
conductor sign reversal, xo). This results from the net polarization charges of 
both signs being identical; they are proportional to the shaded regions (see 
formula (4.12)). This approach can be used to find U1 in different charge 
distribution models and for different horizontal deviations of the vertical 
leader path from the dipole axis. In a simple case when both leaders propa- 
gate along the dipole axis, formulae (4.12) and (4.10) with D = H and r = 0, 
together with expression (4.17), yield a dimensionless equality for finding 
point x2 and then U1: 
(4.18) 
For the variant shown in figure 4.9 with n = i, expressions (4.18) give 
e2 = 1.27 and U/UOR = 0.63, in a fairly good agreement with the 
calculations of the leader evolution (note that U,, 
RZ U,, is the external 
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=== PAGE 174 ===
166 
Physical processes in a lightning discharge 
field potential at the bottom edge of the lower cloud charge that has 
triggered both leaders). 
A negative descending leader can start from any point on the lower 
hemisphere of the bottom negative charge of the cloud. The location of 
this point is quite likely to be a matter of chance, since the field across the 
surface is nearly uniform. Depending on the location of the starting point, 
the ascending twin crosses the charged region along chords of different 
lengths, and this, along with the other factors, affects the potential trans- 
ported to the earth. The maximum potential U1 at the moment of contact 
with the earth is characteristic of a leader that has started from the lowermost 
point of the charged sphere, when the paths of both twins cross the regions of 
maximum potentials and the ascending leader path in it is longest. The value 
of U1 max is about 60% of the external potential U,, at the start and 40% of 
the maximum U,, value at the centre of the negative cloud charge. But even 
if the descending leader is initiated near a lateral point of the hemisphere 
located at a maximum distance from the dipole axis, it transports a 
considerable potential found from calculations to be 0.65U1 max x O.4UoR. 
Therefore, an unbranched negative leader transports to the earth a high 
potential, (0.6-0.4)UOR, no matter where it has started from the lower 
hemisphere. 
4.3.3 The branching effect 
Measurements of return stroke current show that a descending negative 
lightning rarely transports to the earth a potential as high as 100MB 
(Z, = U 1 / Z ,  where 2 is the channel impedance; see section 4.4.2). The 
reason for this is not the supposedly lower potentials of most clouds. The 
value of -100MV is characteristic of cloud charges moderate in size and 
density. The reason is most likely to be the leader branching, since an 
unbranched leader is an exception rather than the rule. Numerous downward 
branches of a descending leader can be well seen in photographs. Although 
ascending leaders are screened by the clouds, their branching can be 
registered by radio-engineering instruments [8- lo]. However, the potential 
U1 is affected by the branches of a descending negative leader rather than 
of its positive ascending twin brother. 
Let us make sure first that the branches of an ascending leader do not 
change the situation much. In the limit of a very intensive branching, the 
negative cloud bottom, pierced by numerous conductive channels, is electro- 
statically identical to a continuous conductive sphere of capacitance 
C, = 4mORc, 
whose charge has been pushed out on to the surface. The net 
charge of a system made up of a sphere and a negative leader attached to 
it remains equal to the initial charge -Qc. 
With the neglect of the voltage 
drop across the descending channel, the binary system possesses the same 
potential U along its length. At the moment of contact with the earth, 
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=== PAGE 175 ===
The descending leader of the first lightning component 
167 
when the leader capacitance CL corresponds to the length L x H ,  this 
potential is 
(4.19) 
U=-=-- Qc 
cc UOR 
cc + CL 
Cc + C, - 1 + L/2R, ln(L/R) 
where UoR would be the boundary potential of a charged sphere, were it 
isolated (the small capacitance gain due to charges induced in the earth are 
ignored). For L/R x 100 and H/Rc = 6, as in the previous numerical 
illustration, the potential U1 x O.6UoR is nearly the same as that transported 
to the earth in the absence of ascending leader branching. The potential of 
the cloud-leader system drops because of the outflow of some of the cloud 
charge to the new capacitance of the descending leader just produced. A 
similar effect has been observed in long laboratory sparks. The capacitance 
of an extremely long spark is often only one order of magnitude smaller 
than the output capacitance of a impulse voltage generator, connected 
directly to a gap without a large damping resistor. The charge inflow into 
the leader is quite appreciable and reveals itself as a voltage drop across 
the gap. 
A branched descending leader possesses a larger capacitance than an 
unbranched one; it takes away a higher charge from the cloud and decreases 
the potential more. To estimate this effect, let us represent a branched leader 
as a bunch of n identical conductors of radius R and length L, spaced at 
distance d (L > d >> R). Supplied by the same power source, they possess 
the same potential U and linear charge T .  The potential at the centre of 
any of these conductors is found by summing the potentials of all charges 
of all conductors, including the intrinsic potential. Integration with the 
neglect of the small effect of the earth yields 
The total capacitance of the n conductors, Ctn = nrL/U, is larger than that 
of a single isolated conductor, but this gain is less than n-fold: 
Ct n 
nln(L/R) 
ct1 
ln[(L/R)(L/d)n-'] 
- 
The reduction in the potential transported to the earth roughly follows 
the distribution of the cloud charge Q, between the capacitances of the 
charged cloud cell, C,, and of the leader, Ctn, described by the first equality 
of (4.19). For n = 10 branches separated at distances d = L/3 and 
L/R x 100 derived from photographs, the capacitance is Ctl0 x 3.2Ct1. This 
well-branched leader will transport to the earth potential U1 x 0.3Uo~. In 
view of the real length of a leader (especially, a well-branched one) which is 
about 1.5 times longer than the charge height H ,  i.e. L M 1.5H, the 
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=== PAGE 176 ===
168 
Phjsical processes in a lightning discharge 
potential decreases to O.2UoR, in good agreement with the data on negative 
lightning currents. The number of branches and their lengths vary randomly 
with the lightning. The potential U1 determining the return stroke current 
vary together with them. This variation is likely to produce a wide range 
of current amplitudes. The variation of storm cloud charges seems to be 
less significant. 
There is another source of reduction in the potential transported by a 
negative leader to the earth. In more complex models than the vertical 
dipole variant, the reduction is due to a low positive charge assumed to be 
present at the very bottom of a cloud [5]. Calculations show that if a positive 
4C charge of 0.25 km radius with the centre at 2 km above the earth is added 
to the above dipole with Q, = 113.3C, R, = 0.5 km and H = D = 3 km, the 
negative leader initiated from the bottom edge of the negative charge will 
transport half of the potential to the earth. 
4.3.4 Specificity of a descending positive leader 
Positive leaders do not occur very frequently. Statistics indicate that in 
Europe their number is 10 times smaller than that of negative ones. But it 
is quite likely that their actual number is larger than the number of their 
registrations. It was pointed out in section 4.3.1 that a descending positive 
leader does not carry high potential to the earth and that its return stroke 
current is low. For this reason, the electromagnetic field of a positive 
lightning discharge can be detected at a much shorter distance than that of 
a negative discharge and, probably, not all of them are registered. 
If the bottom charge of a cloud dipole is negative, a positive descending 
leader may start either from the upper negative hemisphere or from the 
bottom hemisphere of the upper positive charge. The leader will reach the 
earth, transporting to it a positive potential, provided the condition of 
(4.16) is met. With a small deflection of the leader vertical axis from the 
dipole axis (Y << H), the transported potential found from (4.12) and (4.10) 
with x = 0 will be 
where H1 and H2 are the heights of the bottom and top charge centres and x2 
is the ascending leader height at the moment the descending leader contacts 
the earth. We mentioned at the end of section 4.2.2 that an ascending leader 
must go up at least to x2 = 2.47H1 at H2 = 2H1; then we have U = 0. 
At x2 M 4H1, the function U ( x 2 )  crosses the smooth maximum, 
U,,, 
KZ -Qc/207r~OH1 
M 8MV, if Q, = -13.3C and H1 = 3 km, as in the 
previous examples. Even the maximum potential transported to the earth 
is small. This means that the return stroke current of a descending positive 
leader travelling along the dipole axis will be low. The potential and the 
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=== PAGE 177 ===
The descending leader of the$rst lightning component 
169 
current will be still lower, with the real voltage drop U, across a channel of 
total length 4H x lOkm taken into account. Even for the channel field 
E x 10 V/cm, the value of U, 
10 MV is comparable with U,,. 
This light- 
ning is so weak that it has little chance of being registered and included in the 
statistics. 
Vertical channels demonstrate maximum positive potentials transported 
to the earth. They go through more or less identical regions of negative (at 
the bottom) and positive (at the top) external potentials, and the respective 
contributions to the integral of (4.12) are mutually compensated. Positive 
lightnings, however, can possess very high currents. With the foregoing 
taken into account, one can suggest at least two reasons for this. One is a 
favourable random deviation of the channel path from the vertical line. 
Suppose the ascending leader of a binary system, starting from the upper 
positive charge point closest to the earth, xo = H2 - R,, moves up vertically, 
while the other leader, having descended to the zero potential point between 
the charges, turns aside and goes along the zero equipotential line. After it 
has deviated for a large distance r from the dipole axis, it turns down 
vertically to contact the earth this time. In this case, the descending leader 
misses the region of high negative potential, and positive contribution to 
the integral of (4.12) remains uncompensated. Calculations with formulae 
(4.12), (4.10) and (4.17) made at H I  = 3 km, H2 = 6km, R, = 0.5 km, and 
r = 1 km show that the descending leader will transport to the earth a poten- 
tial 4.3 times greater than that to be transported along the dipole axis. 
Another principal possibility is the deviation of the dipole axis itself 
from the vertical line, with the vertical leader path preserved. The centres 
of the top and bottom charges can be shifted from the same vertical line 
because of the difference in the wind forces at different heights. Then the 
leader that has started up vertically from the top charged region passes 
through the region of high positive potential, while its twin, descending ver- 
tically, will appear to be shifted aside relative to the bottom charge and go 
through the region of low negative potential. The effect will be the same as 
in the first case. Quantitatively, it may even appear to be stronger, since 
the length and capacitance of the descending leader are smaller due to the 
lack of an extended path along the zero potential line. 
4.3.5 
A counterleader 
The descending lightning leader does not reach the earth or a grounded body, 
because it is captured by the ascending leader developing in the electric field 
of cloud and earth-reflected charges. This field is enhanced by the charge of 
the descending leader approaching the earth. This can also happen in labora- 
tory conditions, especially if the descending leader is negative. Then the 
counterleader is positive and requires a lower field for its development. 
Streak pictures of laboratory sparks clearly show the counterleader start 
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=== PAGE 178 ===
170 
Physical processes in a lightning discharge 
Figure 4.11. A streak photograph of a long spark with a counterleader coming from a 
grounded electrode. 
and motion towards the descending leader (figure 4.11). The altitude at which 
their encounter occurs depends on the descending leader sign and charge. 
The length of the counterleader at the moment of their contact is important 
for lightning protection practice, because it defines the number of strikes at 
bodies of different heights and, to some extent, the current rise parameters 
of the return stroke from the affected body. 
Let us estimate the altitude z, which the descending leader tip is to 
reach to be able to create a field at the earth high enough to produce a 
viable counterleader. The latter does not differ from any other ascending 
leader, and its development from a body of height d requires that the 
near-terrestrial field should exceed the value of Eo from formula (4.1 1). 
For the height d = 30m characteristic of industrial premises, the field 
must be Eo x 480V/cm. If the cloud field is -lOOV/cm, 
the field 
AE = 380V/cm must be created by the descending leader with charge. 
The main contribution to the near-terrestrial field is made by the charge 
concentrated at the leader channel bottom. Therefore, the calculation of 
the field AE under a very long vertical conductor should utilize the constant 
value of r averaged over this bottom of length -z, rather than the linear 
density of the non-uniform charge r(x). With the charge reflected by the 
earth, we have 
(4.21) 
r 
"dx 
r 
u-uo - 
U 
N 
- 
*E(z) =GIZ 
T - G - z l n ( L / R )  
-zln(H/R) 
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=== PAGE 179 ===
Return stroke 
171 
where U is the channel potential and L x H is its length, which is about the 
cloud height at the tip height z << H. Here, we have used the conventional 
expression T = C1 ( U  - Uo) with average linear capacitance and accounted 
for the near-terrestrial potential of the cloud, 1 Uoi << 1 UI. For an unbranched 
descending leader carrying high potential U x 50 MV, we obtain z x 260 m 
at ln(H/R) x 5. 
The counterleader arises at the last stage of the descending leader 
development, i.e., near the earth. Its velocity is not high and is equal to 
wL, x 2 x 104m/s from the first formula of (4.5), because the potential 
difference on the leader tip is quite low, A U x Eod x 1.5 MV. The descend- 
ing leader has an order of magnitude higher velocity. For this reason, the 
counterleader acquires the length L1 x (wL, 
/vL)z x 25 m by the moment of 
encounter. This is a large value, since the length L1 is summed with the 
body’s height d, so that the total height of the grounded conductor becomes 
nearly doubled. This affects the frequency of the body’s damage by lightning 
strikes. 
It follows from formulae (4.21) and (4.11) that the height z, 
to which the 
leader descends before it can initiate a counterleader, is greater for higher 
premises, from which the counterleader starts, z 
N d3I5, although this depen- 
dence is not very stringent. It is important that as the altitude of a body and z 
become greater, the counterleader has more time for its acceleration and can 
acquire a longer length. It is important for applications that it is not only the 
length L1 which increases but also the L1 / d  ratio. 
The simple estimation obtained from (4.21) and (4.11) can be refined by 
accounting for the T ( X )  non-uniformity in the integral of (4.21) arising from 
the proportionality T N U - U0(x) and by rejecting the approximation of 
constant linear capacitance C1. In the latter case, T ( X )  should be found 
from equation (4.7). Calculations show that the two corrections are rather 
small, so the estimations above can be considered to be satisfactory. 
4.4 
Return stroke 
4.4.1 The basic mechanism 
A return stroke, or the process of lightning channel discharging, begins at the 
moment the cloud-earth gap is closed by a descending leader. After the 
contact with the earth or a grounded body, the leader channel (it will be 
taken to be negative for definiteness) must acquire zero potential, since the 
earth’s capacitance is ‘infinite’. Zero potential is also acquired by the ascend- 
ing leader, which is a continuation of its descending twin brother. The 
grounding of the leader channel carrying a high potential leads to a dramatic 
charge redistribution along its length. The initial channel distribution prior 
to the return stroke was T~ = C1 [Ui - U0(x)]. Here and below, the potential 
transported to the earth, which acts as the initial potential for the return 
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=== PAGE 180 ===
172 
Physical processes in a lightning discharge 
Figure 4.12. Schematic recharging of a lightning channel after the contact of the 
descending leader with the earth. Shaded regions, charge; (a) moment of the leader 
contact with the earth; (b) the return stroke reaching the upper channel end; (c) 
charge change. 
stroke, will be denoted as Vi. As before, it will be taken to be constant along 
both leader lengths, and the voltage drop across the channel will be ignored 
as an insignificant parameter. We shall assume that the channel is character- 
ized by linear capacitance C1, which does not vary along its length or in time 
during the return stroke process. After the whole channel has acquired zero 
potential, U = 0, the linear charge becomes equal to T~ = -C1 Uo(x). 
The 
channel portion belonging to the negative descending leader does not just 
lose its negative charge but it acquires a positive charge (Uo < 0, r0 < 0, 
T~ > 0). Not only does it become discharged but it is also recharged. The 
twin positive channel high in the cloud acquires a larger positive charge 
(figure 4.12).The linear charge variation for the return stroke lifetime is 
AT = T~ - T~ = -CoUi. At Ui(x) = const, the charge variation is constant 
along the channel length and has such a value as if a long conductor (a 
long line) pre-charged to the voltage Ui becomes completely discharged (as 
if it were r0 = C1 Vi to become T~ = 0). 
It has been emphasized that the leader charge is concentrated in its 
cover. The charge in a non-conducting cover changes due to the charge 
incorporation from the conductive channel, owing to the streamer corona 
excitation at the channel surface. This is an exceptionally complicated pro- 
cess, whose rate can be found only from an adequate theoretical treatment. 
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=== PAGE 181 ===
Return stroke 
173 
For this reason, the assumption of capacitance C1 being constant, which 
implies a zero-inertia charge variation in the cover with varying channel 
potential, is quite problematic. But if we discard this seemingly essential 
assumption, nothing will change qualitatively or even quantitatively. 
Indeed, suppose the cover charge does not change at all during the time 
the whole channel acquires zero potential. This is equivalent to the assump- 
tion that the channel capacitance is determined, during the return stroke pro- 
cess, by the conductor radius r, rather than by the cover radius R. Because of 
the logarithmic dependence of linear capacitance (2.8) on the radius, it 
decreases to the value Ci equal to about a half of C1; for example, we 
obtain C1 % 10pF/m and Ci RZ 4.4pF/m at I = 4000m, R = 16m and 
Y, = 1.5 cm. Then the charge variation during the stroke 
A T = T ~  
- ~ o = [ ( C l  -Ci)(Ui-Uo)+C’1(0-Uo)]-[C1(Ui-Uo)] 
= -c:ui 
(4.22) 
remains the same in order of magnitude. Consequently, when considering 
fundamental stroke mechanisms, one can take C: % C1 and assume the 
equivalent line to be charged uniformly. 
Measurements made at the earth show that a descending leader is 
discharged with a very high current. For negative lightnings, the current 
impulse of a return stroke with an amplitude ZM -10-100 kA lasts for 50- 
100 ps on the 0.5 level. A short bright tip of the return channel well seen in 
streak photographs runs up for approximately the same time. Its velocity 
v, M (0.1-0.5)~ is only a few times less than light velocity c. It would be 
natural to interpret this fact as the propagation of a discharge wave along 
the channel; this wave is characterized by a decreasing potential and rising 
current. Due to an intensive energy release, the channel portion close to 
the wave front, where the potential drops from U, and a high current is 
produced, is heated to a high temperature (from 30000 to 35000K, as 
shown by measurements). This is why the wave front is so bright. The 
channel behind it is cooled due to expansion and radiation losses, becoming 
less bright. A return stroke has much in common with the discharge of a 
common metallic conductor in the form of a long line. The line discharge 
also has a wave nature, and this process was taken to be a model discharge 
in shaping the ideas concerning the return lightning stroke. 
A lightning channel is discharged much faster than it was charged 
during its development with the leader velocity vL % (10-3-10-2)~,. But 
the variations in potential and linear charge during the charging and the 
discharge are expressed as values of the same order of magnitude: ro - AT. 
In agreement with the velocity, the channel is discharged with current 
1, 
M ATV, by a factor of v,/uL %102-103 higher than the leader current 
iL M r0uL ~ ~ 1 0 0 A .  
The linear channel resistance RI 
decreases approximately 
as much during the leader-stroke transition. This decrease is due to the 
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=== PAGE 182 ===
174 
Physical processes in a lightning discharge 
channel heating by high current. As a result, the plasma conductivity 
increases and the channel expands, making the conductor cross section 
larger. In this respect, a lightning discharge certainly differs from a discharge 
of a common conductive line, whose resistance remains constant (if the skin- 
effect is ignored). Since resistance is a plasma characteristic, its decrease can 
be found straightforwardly only if the physical processes occurring in the 
channel are taken into account. (This situation will be analysed in section 
4.4.3.) But this conclusion can be arrived at indirectly from general energy 
considerations. Over the return stroke lifetime, t, N H/wr, where H is the 
channel length, the energy dissipated in the channel must be approximately 
equal to the initial electrical energy C1 U:/2 
per unit length: 
C1 U:/2 
N IhRlt, N IhR1 H/v, N ATHI-WR~. 
(4.23) 
About as much energy was dissipated in the leader when the capacitance 
C1 was charged. If the leader develops in the optimal mode (see section 2.6), 
to which a natural lightning process is, probably, very close, because Nature 
usually takes optimal decisions, the voltage drop across the channel is com- 
parable with the excess of the leader tip potential over the external potential. 
Therefore, the resistances of the channel and the streamer zone are compar- 
able, because the same current flows through them. Therefore, the unit length 
of the leader dissipates the same energy C1U,'/2 (in order of magnitude), 
expressed by the leader parameters il, vL and R I L  similar to (4.23). This 
yields RIIM 
N RILiL, i.e., R1/RIL 
N 10-2-10-3. It is also found that the 
average electric field in the leader channel and behind the discharge wave 
in the return stroke, E, M R1 IM x RILiL, have the same order of magnitude. 
This is consistent with the conclusion to be made from a straightforward 
analysis of the established states in both channels. The situation there is 
similar to that in a steady state arc. But the channel field E, in a high current 
arc does vary but slightly with the current [ 111. 
It follows from the foregoing that if a leader has iL M 100 A, E, z 10 Vi 
cm and ROL M 0.1 R/cm, the return stroke must have Ro x 10-3-10-4 R/cm 
in the steady state behind the wave front; the total resistance of a channel of 
several kilometres in length appears to be lo2 R. This value is comparable 
with the wave resistance of a long perfectly conducting line in air, 2, whereas 
the total ohmic resistance of a leader of the same length is two orders of 
magnitude larger than Z. The ratio of the ohmic resistance of the line portion 
behind the wave to the wave resistance indicates the degree of the wave 
attenuation during its travel along the line (section 4.4.2). If the channel 
resistance were constant and remained on the leader level, the lightning chan- 
nel discharge wave would attenuate, being unable to cover a considerable 
channel length. The current through the point of the channel closing on 
the earth would also attenuate too quickly. Experiments, however, point to 
the contrary: the visible bright tip has a well-defined front, and a high current 
is registered at the earth during the whole period of the tip elevation. The 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 183 ===
Return stroke 
175 
transformation of the leader channel during the wave travel decreases its 
linear resistance considerably, determining the whole return stroke process. 
4.4.2 
Long line equations with the allowance for the main factor - variation in 
linear resistance - can be solved only numerically (section 4.4.4). However, 
the nature of the process and its essential physical characteristics can be 
understood from the analysis of well-known analytical solutions for simple 
situations. Their comparison with lightning observations indicate the impor- 
tant points for the formulation and solution of the real problem. 
In the absence of transverse charge leakage due to imperfect insulation, 
a long line is described by the equations 
Conclusions from explicit solutions to long line equations 
dU 
=cl--. 
au 
di 
di 
-L1-+Rli, 
-- 
d X  
dt 
d X  
at 
(4.24) 
They generalize equations (2.12) by accounting for inductance. The induc- 
tance per unit conductor length, L1, as well as its capacitance C1, can be 
assumed to be approximately constant. For an isolated conductor of 
radius Y, and length H >> rc, it is 
Po 
H 
H 
27r 
rc 
YC 
L1 x -In - = 0.2 In - 
pH/m 
Here, the channel length H is about equal to the height 
leader tip. For a perfectly conducting line with R1 = 0, 
are re-differentiated to produce a simple wave equation: 
- 0, 
21 = (L1C1)y2. 
a2u 
1a2u 
8x2 
212 at2 
(4.25) 
of an ascending 
equations (4.24) 
(4.26) 
If the line is charged to voltage Ui and short-circuited on the earth by its base 
x = 0 at the moment of time t = 0, a rectangular wave of complete voltage 
elimination (from Ui to 0) and an unattenuated current wave of the same 
shape will propagate with velocity U from the grounding point (figure 4.13): 
While the wave propagates along the line, a detector mounted at its 
beginning will register direct current. If voltage Ui is low and there is no 
charge cover around the conductor, the capacitance and inductance are 
characterized by the same radius Y, in the logarithms of (2.8) and (4.26). In 
this case, we have 
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176 
Physical processes in a lightning discharge 
t i  
Figure 4.13. Distributions of potential and current during the discharge of a perfectly 
conducting line. 
where c is light velocity. If one describes the capacitance, in contrast to the 
inductance, with the leader cover radius R = 16m, then Y, = 1.5cm, 
H = 4km (as above), C1 = 10pF/m, L1 = 2.5pH/m, v = 0.67C, and 
Z = 500R. The wave velocity is now lower than light velocity, but not 
much. Current of amplitude Z, 
= 30 kA, typical of the return stroke of the 
first negative lightning component, arises at I Uil = ZZM = 15 MV. The 
values of Vi and w are correct in order of magnitude, but the wave velocity 
w exceeds several times the observable velocity, and it is impossible to 
reduce this discrepancy by varying the reactive line parameters. In a line of 
preset length, C1 and L1 vary only slightly (logarithmically) with the conduc- 
tor radius. What remains to be done is to focus on the only parameter that 
has not been accounted for - resistance R1 which is very high in a leader 
but reduces by 2-3 orders during a return stroke. 
Let us discuss the exact solution of equations (4.24) describing the line 
discharge at R1 = const: 
i(x, t) = - 
Z 
(4.29) 
where Zo(z) is the Bessel function of a purely imaginary argumentjz: 
ZO(Z) 
1 + (z/2)2 
at z << 1 
z0(z) M eZ(2~z)-1/2[1 
+ ~ ( z - ' ) ]  at z >> 1. 
(4.30) 
The wave has the same velocity as an ideal line, without losses, but the 
current at the wave front falls exponentially, as it propagates: 
(4.31) 
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=== PAGE 185 ===
Return stroke 
111 
The attenuation if is described by the ratio of the ohmic resistance Rlxf of 
the line behind the wave to the wave resistance. The line base current, 
which is just the current registered in lightning observations, arises instanta- 
neously (with instantaneous short-circuiting of the line on the earth) and, at 
the first moment, is determined exclusively by the wave resistance, indepen- 
dent of the value of RI: i(0,O) = -Ui/Z. As the wave moves on towards the 
cloud, the ohmic resistance the current has to overcome becomes increasingly 
higher, so the base current decreases. At at >> 1, or at Rlxf/2Z >> 1, the 
current through the base is 
(4.32) 
This current decreases much more slowly than at the wave front, because in 
spite of the negligible front current, the line far behind the wave front is 
discharged all the same, and all the charge that flows down from it goes 
through the base. 
The wave front propagates at a rate of electromagnetic perturbation. It 
is independent of the line ohmic resistance but is determined exclusively by 
its reactive parameters and is close to light velocity. This is a 'precursor' 
which exists under any conditions, no matter whether the line has a 
resistance or whether it changes behind the wave front. The precursor 
carries information about the changes in the line, in our case about the 
line grounding. If the resistance is zero or, more exactly, has no effect yet 
because it is much less than the wave resistance (Rlxf << Z), the line is 
discharged in a resistance-free way, and its initial potential and charge 
practically vanish right behind the front of the primary electromagnetic 
signal, the precursor. When the resistance becomes much higher (practically 
several times higher) than the wave resistance, the charge and potential dis- 
appear gradually, and the rate of their reduction decreases as the linear 
resistance RI increases. At R1 = 10 n/m corresponding to the leader chan- 
nel resistance, the time constant is a = 2ps-l and the precursor current 
decreases, in accordance with (4.3 l), by an order of magnitude as compared 
with the initial value of i(0, 0) over the period of time t N 1 ps, for which the 
precursor covers only 200m (U = 0.67C). Half way up to the cloud 
(x = 1500m), the front current decreases by a factor of 3 x lo6. According 
to (4.32), the line base current at that moment will be 3 x lo5 times higher 
than at the front. Therefore, the current somewhere behind the precursor 
will inevitably rise to a much larger value. Let us see where this happens 
and what will be the velocity of the high current region carrying the 
charge away from the line. 
For the analysis of the relatively late stage in the discharge process with 
at >> 1, we shall employ the second, asymptotic formula of (4.30) for the 
Bessel function. For the region x << xf = ut located fairly far from the 
weak precursor, the root in the argument of Io can be expanded. Using 
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=== PAGE 186 ===
178 
Physical processes in a lightning discharge 
formulae (4.29) for a and w, we get 
This expression is the explicit solution to equations (4.22) without the 
inductance term but with the same boundary and initial conditions. The 
potential is 
s/(4xr)’:2 
U ( x ,  t )  vi- 
exp( -t2) d[ = U, erf 
f
i
0
 
? [  
Expressions (4.33) and (4.34) have a clear physical sense, demonstrating the 
nature of a non-ideal line discharge. 
When the perturbation front (the precursor due to the action of induc- 
tance) goes far away, the current decreases slowly from the line base to the 
front. It also varies slowly in time at every point, except for the region 
close to the front. This is the reason why the inductance effects in the main 
discharge region are very weak. With the neglect of the inductance term, 
equations (4.24) transform to equations similar to those for heat conduction 
or diffusion: 
(4.35) 
To use an analogy, the potential acts as temperature, current as heat flow, 
and x as thermal conductivity (heat diffusion). We did not take x = const 
out of the derivative deliberately to be able to come back to this equation, 
also valid at RI # const. 
The process of line discharge is similar to the cooling of a uniformly 
heated medium, when a low (zero) temperature is maintained at its bound- 
ary, beginning with the moment of time t = 0. Formulas (4.34) and (4.33) 
describe the diffusion of the earth potential along the channel (figure 
4.14(a)). The current-potential wave, smeared in contrast to the precursor, 
Figure 4.14. The potential wave (a) in linear diffusion with x = const and (b) in non- 
linear diffusion with rising x. 
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=== PAGE 187 ===
Return stroke 
179 
propagates in such a way that its characteristic point x l ,  say, where the 
potential is reduced by half relative to the initial value of Vi, x 1 / ( 4 ~ t ) ~ / *  
= 
0.477, obeys the diffusion law x1 x (xt)ll2 with a decreasing velocity 
v1 x ( ~ / t ) ” ~  
M x / 2 x 1 .  From expressions (4.33), the current at point x1 is 
20% lower than that at the channel base x = 0. Substituting the leader 
resistance R1 x 10R/cm and C1 = 10pF/m into the formulae, we get 
x = 10 m / S .  Over the time t = 10 ps (at at = 20), during which a weak 
precursor will cover a distance of 2000 m, the half-potential point character- 
izing the propagation of the line discharge wave will diffuse for 3 15 m only 
and will be moving at velocity v1 x 0.05~. By that time, the base current 
i(0, t )  will have dropped by a factor of 11 relative to the initial current 
i(0,O) (formula (4.32)). 
The calculated values of x l ,  wl, and i(0, t )  can be brought closer to 
measurements at a certain stage of the lightning discharge. Instead of the 
leader resistivity, one should then deal with a lower resistivity averaged 
over the perturbed region. This makes sense in some evaluations. But the 
illusion of a satisfactory numerical agreement with measurements in a 
short stage of the process is destroyed, as soon as we recall one of the 
important qualitative observations. At the return stroke stage, a bright 
and well-defined wave front - the channel tip, which becomes smeared 
only slightly with time - is moving up to the cloud. This indicates that the 
energy release and, hence, the current rise occur faster than in the solution 
to (4.33). Clearly we deal with a wave possessing a steep front, at least for 
powerful lightnings, rather than with diffuse current profiles. This contra- 
diction can be resolved by rejecting the approximation R1 = const and by 
including, in the theoretical treatment, the time evolution of the leader 
channel and its transformation to a return stroke channel. 
Note that the simple and attractive model of an immediate transforma- 
tion of the leader channel at the wave front to an ideal conductor cannot 
rectify the situation. This model would take us back to equalities (4.26) 
and (4.27) describing the wave of immediate voltage removal and sustained 
current, which propagates with the velocity of an electromagnetic signal 
close to light velocity. But this possibility was already refused above. It 
was mentioned in section 4.4.1 that the key to the phenomenon of return 
stroke should be the analysis of the channel transformation dynamics. 
The effect of the gradual resistance reduction during the Joule heat 
release can be understood from equations (4.35) and (4.24) without the 
inductance term. It would be justifiable to replace U by the potential varia- 
tion AU = U - Ui, since Ui = const in our approximation, so we get 
10 
2 
(4.36) 
d A U  
1 
i = - C  1
X
X
’
 
x = -  
dAU 
a 
dAU 
- X -  
-- 
RlCl . 
- 
at 
ax 
ax 
The resistance decreases while x increases, as the amount of charge flowing 
through the particular channel site becomes larger, or with the increase in 
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=== PAGE 188 ===
180 
Physical processes in a lightning discharge 
AU. Consequently, the rear sites of the diffusion wave, where AU and 
diffusion coefficient x are already higher, propagate faster than the front 
sites, where A U  and x are still low. To supply current to the region close 
to the discharge wave front (a weak precursor is out of the question now), 
the potential gradient there must be large because of the small diffusion 
coefficient. Both circumstances indicate that the wave acquires a sharp 
front, its profile becomes steeper and convex. In contrast to the gradual 
asymptotic approximation at x = const, the curves U ( x )  and i(x) for a 
given moment of time look as if they stick into the abscissa (figure 4.14; 
the same will be seen from the numerical simulation illustrated in figure 
4.17). The effect described here is well known [12]; this is a non-linear heat 
wave driven, for example, by radiative heat conduction, whose coefficient 
drops with decreasing temperature T approximately as x = T3.t 
The variant with RI = const, for which the solution to (4.33) and (4.34) 
is valid, probably corresponds to low current lightnings, when the energy 
release is too small to provide an essential reduction in the former channel 
resistance. In any case, there are streak pictures of return strokes with unclear 
wave fronts or those becoming smeared after the propagation for a few 
hundreds of metres [13,14]. To obtain conclusive evidence, stroke streak 
pictures should be analysed at different currents. Regretfully, no simulta- 
neous recordings of currents and stroke waves are available. 
One can draw another conclusion from the solution to the set of 
equations (4.24) at RI = const # 0, which is important for the analysis of 
observations and for the formulation of boundary conditions necessary for 
finding a numerical solution. According to (4.29), when the line closes on 
the earth instantaneously the discharge current through the closed end also 
reaches its maximum instantaneously. As mentioned above, the maximum 
is independent of R I ,  being determined exclusively by the wave resistance. 
Clearly, the same will also be true for any time-variable resistivity, and the 
only question is how fast the current will decrease after the maximum. 
However, the current in a real return stroke rises for several microseconds, 
sometimes for several dozens of microseconds, and this time may become 
even comparable with the total impulse time. Such a slow current rise may 
t Equations (4.36) with AU(x.0) = 0, AU(0. t )  = -L'i and no inductance terms allow self- 
similar solutions. The simplest of them are (4.33) and (4.34) for x = const. The process is self- 
similar in a more complex approximation for x = b( 1 U, l)ntv, which corresponds qualitatively 
to the RI - x-' evolution during the channel transformation. Constants 6, n, and U can be 
chosen from the analysis of RI behaviour (section 4.4.3): n x 1-2; U 
0.5-1. The wave front 
follows the relations 
x/ = E[b(lC:l)"]'!?t'"-'~:? 
Vf = i ( V +  l)E[b(lC:l)"]':2t-('-V'". 
where E of about 1 is to be found by solving an ordinary differential equation [12]. 
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=== PAGE 189 ===
Return stroke 
181 
be only due to the properties of the commutator, whose role is played by the 
streamer zones of the descending leader and the counterleader. Their contact 
actually gives rise to the return stroke. The streamer zone field rises, as the 
streamer zones are reduced and the leader tips come close to each other or 
as the descending tip approaches the earth with no counterleader formed. 
The streamers are accelerated to a velocity lo7 m/s, transporting kiloampere 
currents even in laboratory conditions [ 15,161. Thus, the rate of current rise 
and the impulse front duration at the earth, 9, are determined by processes 
occurring in the vanishing streamer zone rather than in the former leader 
channel. Measurements provide indirect evidence for this, showing that the 
impulse rise time tf in positive lightnings possessing a longer streamer zone 
than negative ones, at the same voltage, is several times longer. 
4.4.3 Channel transformation in the return stroke 
It has been shown above that electromagnetic perturbation propagates along 
a line with a velocity equal to or somewhat lower than light velocity, indepen- 
dent of the initial resistivity. When the resistance is high, as in the leader 
channel, the current and the potential variation induced by the perturbation 
attenuate rapidly. But the precursor is followed by a stronger perturbation 
propagating at a lower velocity, which reduces the potential considerably, 
to zero with time. The potential of a negative lightning drops to zero at 
this channel site due to positive charge pumping; this compensates the initial 
negative potential there. This process is accompanied by Joule heat release 
with a linear power i2R1, 
which is high at first since the impulse front of 
the ‘genuine’ (not the precursor) current is quite short and the initial 
(leader) resistance RI is relatively high. The processes that follow - the 
channel heating, its radial gas-dynamic expansion, the shock wave propaga- 
tion, and the resistivity reduction - have much in common with those in 
powerful spark discharges in short laboratory gaps. The latter have been 
extensively studied experimentally, theoretically, and numerically [ 17-24]. 
Also, calculations have been made with the initial parameters characteristic 
of a lightning return stroke, accounting for radiative heat exchange which is 
especially important in this large-scale phenomenon [22-241. The stroke 
channel gas is heated up to 35000K. Most of the Joule heat is radiated by 
the highly heated gas in the ultraviolet spectrum. The emission from this 
spectral region is absorbed by the adjacent colder air, adding the newly 
heated gas to the conductive channel. 
Such a treatment of the process would take us far from the point of 
interest, so we shall restrict ourselves to a description of two numerical results 
for atmospheric air, obtained with a rigorous allowance for radiative heat 
exchange [23,24]. In both calculations, the shape and parameters of the 
current impulse were preset, as is usually done in lightning calculations. Of 
course, the current behaviour here depends on what happens in the whole 
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=== PAGE 190 ===
182 
Physical processes in a lightning discharge 
10 
15 
20 
25 
30 
35 
T, kK 
Figure 4.15. The conductivity of thermodynamically equilibrium air at atmospheric 
pressure. 
of the perturbed region. But the formulation of a self-consistent problem 
requires a combined solution of a set of equations for a long line discharge 
and extremely cumbersome equations describing the physical evolution of 
each section along the line. A problem of this complexity has not been 
approached yet.? 
Both calculations for one-dimensional cylindrical geometry were made 
with the current impulse 
i(t) = Z M t / t f  
at t < tf, 
i(t) = ZMexp[-(t - tf)/tp] at t > 9 
possessing a linearly rising front and exponentially decreasing tail. The 
calculation in [23] was made with moderate parameters tf = 5ps, 
ZM = 20 kA, and tp = 50 ps corresponding to a moderate power lightning. 
The other calculation [24] was for tf = 5 ps, ZM = 100 kA, and tp = 100 ps 
of a very powerful lightning. It is generally believed that the air conductivity 
0 corresponds to its thermodynamic equilibrium and is determined by 
temperature (figure 4.15). 
Figure 4.16 shows the evolution of pressure, gas density, temperature, 
and radial velocity distributions behind the shock front for a powerful 
current impulse [24]. The curves for moderate current impulses are qualita- 
tively similar. 
tThe problem of a short laboratory spark is much simpler. The set should include a simple 
discharge equation for a capacitor bank as a high-voltage source for a spark gap with the desired 
resistance and allowance for the circuit inductance. Note that this kind of LRC circuit usually 
registers damped oscillations unobservable in lightning. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 191 ===
Return stroke 
183 
2 5 -  t = 5 p  
20 - 
$15 
a 
20 
10 - 
50 
5 -  
OO 
5 
IO 
15 
20 
0 
5 
10 
15 
20 
" 
' 
" " 
r, cm 
r. cm 
6 
4 s 
a 
2 
0 
r, cm 
0 
r, cm 
Figure 4.16. The radial distributions of pressure p ,  density p, temperature T ,  and 
the velocity v behind the cylindrical shock wave of a return stroke: Z, 
= 100 kA, 
tf = 5 ps, tp = 100 ps; po = 1 atm, pa and co are the initial presure, air density and 
sound velocity; To = 300 K. 
The point of primary interest in a return stroke treatment is the beha- 
viour of the integral channel parameter - its linear resistance: 
RI = [ 1; 27rra(r) dr] -'. 
(4.37) 
Table 4.1 presents, among other parameters, the linear resistance values 
obtained from T(r) data borrowed from [23,24]. One can see that the 
resistivity drops at first for 1 ps but then falls rather slowly. This decrease 
ceases closer to the pulse tail, and the resistance begins to rise gradually. 
The dramatic initial drop in RI is due to the primary heating of a very thin 
initial channel by high density current.? As T increases to about 20000K, 
the conductivity a rises but remains nearly constant with further temperature 
t The gas is assumed to be in thermodynamic equilibrium at every moment of time. This assump- 
tion is justified by a fast energy exchange (for lO-*-lO-'s) 
between electrons and ions, resulting 
in a small difference between the gas and electron temperatures. The ionization is of thermal 
nature: a Maxwellian distribution is established in the electron gas, and the amount of ionizing 
electrons is defined directly by the electron temperature, rather than by the field. The electron 
temperature, in turn, is determined by the Joule heat release and energy balance of the gas. 
Equilibrium ionization is also established rapidly (for details, see [ll]). 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 192 ===
184 
Physical processes in a lightning discharge 
0.074 8 
5 
2 
>24 
0.1 
- 
1600 
3.7 
28 
18 
120 
5.5 
0.3 
8 
3.1 
11 
24 
15 
170 
4 
0.7 
26 
0.38 
39 
17 
12 
110 
2 
1.2 
38 
0.20 
91 
14 
10 
60 
1 
1.4 
46 
0.27 
Current impulse 100 kA with tf = 5 p, 
tp = 100 ps, Q = 1OC 
5 
35 
25 
180 
16 
0.8 
- 
0.28 
20 
22 
20 
140 
5.7 
2 
- 
0.057 
50 
18 
15 
90 
2.6 
3 
- 
0.039 
100 
14 
12 
70 
1.8 
4 
- 
0.028 
200 
12 
11 
40 
1.0 
5 
-150 
0.032 
300 
10 
10 
20 
1.0 
5 
- 
0.064 
Note. T,,, 
is the temperature along the channel axis, Teff is the average temperature in the 
conductive channel, oeff is an average channel conductivity, p is channel pressure, reff is the effec- 
tive radius of the conductive channel, W is the total energy released (no data for the second 
variant; the given values was estimated as W x i~axR1tp)r 
and Q is the charge transported 
during the current impulse. 
rise. In a strongly ionized plasma, with ions of constant charge o - T3I2, but 
doubly charged ions appear with increasing T .  Since U - ZC2, where Zi is the 
ion charge multiplicity, the two effects compensate each other. The resistance 
of a highly heated channel decreases with time due to its expansion only. 
Some time later, however, the pressure at the channel centre drops to 
atmospheric pressure, and the expansion ceases. The conductive channel 
cross section is reduced gradually because of the gas cooling caused by 
thermal radiation. The channel resistance begins to rise slowly because of 
decreasing reff and Teff. 
The expansion time of the channel becomes longer and its minimal 
resistance decreases for the stronger current impulses. Physically, the linear 
resistance is affected by the energy released per unit channel length, W ,  
rather than by the current. This value is not described unambiguously by 
the current amplitude; what is more important is the amount of the 
transported charge Q: W1 - i2Rlt - QiRl - QE and the field does not 
vary much. The calculations, however, deal with the current impulse but 
not with W1. Semi-quantitatively, the time dependence of resistance can be 
understood using the relations for the shock wave of a powerful cylindrical 
explosion. The explosion can be considered to be strong as long as energy 
is released in a thin channel and the pressure of the explosion wave does 
not fall close to the atmospheric pressure. In this case, the flow is self-similar. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 193 ===
Return stroke 
185 
The shock front radius rs and pressure p in the affected region depend, within 
the accuracy of numerical factors, on W1 and t as r, N ( W1/po)1/4t1/2 
and 
p N Wl/rz N ( Wlpo)'/2t-', where po is cold air density. The channel 
expansion is completed when the pressure drops to a certain value close to 
atmospheric pressure. This sets the limit to the validity of formulae for self- 
similar motion. This means that they are still applicable, and the duration of 
the resistance reduction then is t N ( Wlpo)1/2. 
It can be shown [12] that for 
a self-similar cylindrical explosion in the central region with the pressure 
equalized along the radius (figure 4.16), the internal specific energy depends 
on r, t and W1 as E N ( Wy/2t2-Tr-2)1/(T- 
'I, where y is the adiabatic exponent. 
A point with fixed temperature, e.g., T M 10 000 K, can be regarded as the 
conductive channel boundary, since the plasma conductivity below this 
point is relatively low. The radius of a point with fixed T and E (  T )  varies 
with time as r N W:'4t1-y/2, reaching a value proportional to r,,, 
N W;I2 
by the moment the channel stops expanding, t N W:12. Therefore, the linear 
channel resistance drops to a value proportional to Rl,, 
N rmax N W-' , and 
this occurs for the time t N W'/'. These relationships are qualitatively 
consistent with the calculations for the two variants described in table 4.1. 
-2 
4.4.4 Return stroke as a channel transformation wave 
The first substantiated attempt to make a numerical simulation of the light- 
ning return stroke with allowance for the resistance variation was undertaken 
as far back as the 1970s [25,26]. The most important features of the process, 
which are due to an abrupt conductivity rise at the site of intensive Joule heat 
release, became evident at once. The simulation showed that a weak initial 
perturbation (precursor) propagating up along the channel at an electro- 
magnetic signal velocity close to light velocity does not change the plasma 
state and cannot be treated as the return stroke wave front visible in streak 
photographs. The main wave of current and decreasing potential travels 
several times slower; its velocity is defined by the transformation of the 
low conductivity leader to the low resistivity stroke channel. This conclusion 
was formulated explicitly in [25-281; it reflects the nature of the lightning 
return stroke. 
Turning to numerical simulation today, we should like to formulate this 
problem in a simple and clear physical language and to try to outline 
problems to be solved within this model. An obviously essential aspect of 
the theory still is the resistivity dynamics of the lightning channel. An 
exhaustive formulation of this problem would involve a simultaneous 
solution of equations describing the propagation of a current-voltage 
wave and the channel dynamics at every point along its length, affected by 
the ever varying energy release. So we shall restrict the discussion to a 
simple model, having accepted a probable law for the linear conductivity 
rise, G = R:', and focusing on the qualitative results of the solution. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 194 ===
186 
Physical processes in a lightning discharge 
Let us describe G in the simplest way reflecting the main qualitative 
features of the channel evolution. It will be assumed that the linear conduc- 
tivity increases with current. This partly reflects the fact that resistance 
decreases with increasing charge through a particular channel site. But the 
resistance is stabilized with time, even though the current continues to 
flow. In principle, the stable state of a lightning channel hardly differs 
from that of an arc. The field E in a hgh current arc only slightly varies 
with current; in other words, the linear conductivity of the channel is 
G,, = i/E N i. Assume E to be equal to the field EL in the lightning leader, 
whose current is not low on the arc scale; then the conductivity is 
G,, = i/EL. In a mature gas-dynamic process when the shock wave is still 
strong, the resistance will drop with time. As the shock wave becomes 
weaker, the decrease in R1 and the increase in G become slower. These 
tendencies are described by the relaxation-type formula 
(4.38) 
where Tg is the characteristic time of linear conductivity variation (relaxation 
time). In a simple case with i = const, Tg = const, and G(0) = 0, we have 
G = GSt[1 - exp(-t/T,)]. 
Equations (4.24) are solved with the initial conditions U ( x ,  0) = U, and 
i(x, 0) = 0, RI (x3 0) = RI,-; the reactive parameters are taken to be constant: 
C1 = 10 pF/m and L1 = 2 pH,”. 
The channel does not close on the earth 
in an instant but does so through the time-decreasing resistance of the 
commutator (similarly to the real lightning length decreasing through the 
streamer zone). The accepted values of R,,, = R(0) exp(-cut), R(0) = 10 R 
and cu = 1 ps-’ provide a typical duration of the negative current impulse 
front tf FZ 5 p .  The boundary condition at the grounded end of the line 
raises no doubt: U(0. t )  = i(0, t)R,,,. 
The problem of the far end up in the 
clouds, x = H ,  is much more complex. Conventionally, it is considered as 
being open, assuming i(H, t) = 0. In reality, the situation is far from being 
self-evident. When the line gets discharged and its end in the clouds takes 
zero potential, a high electric field must arise near it due to the voltage 
difference A U  = -Uo(H). This gives impetus to very intensive ionization 
processes, probably involving high current. This situation will be partly 
discussed below. Now, we shall assume the upper end to have no current. 
The results to be presented were obtained for a vertical unbranched channel 
with the total length H = 4 km. This is the height the ascending leader tip 
reaches when the descending leader, which has started from the point closest 
to the earth in the bottom negative sphere with the centre 3 km high, contacts 
the earth. It is normal practice to use the following averaged parameters 
of the leader prior to the contact: EL = 10V/cm, iL = lOOA, and 
RIL = EL/iL = 10 O/m. For a realistic description of the resistivity dynamics 
(section 4.4.3), the relaxation time should be taken to be Tg = 40 ps, when the 
dG 
i/EL - G(t) - Gst(i) - G(t) 
- 
- 
- 
- 
dt 
Tg 
Tg 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 195 ===
Return stroke 
187 
current at this channel site rises, and Tg = 200p, when the current 
decreases. The model calculation reproduces the distributions of current 
i(x, t )  and potential U ( x ,  t )  along the channel; the linear charge is ~ ( x ,  
t )  = 
C1 [ U ( x ,  t )  - Uo(x)]. Generally, the external field potential Uo(x) 
can also 
vary in time, because we should not discard a possible partial 
neutralization of the charge in one of the regions of the cloud dipole. The 
latter point will not be discussed for the time being. 
The calculations are presented in figures 4.17-4.2 1. The precursor tra- 
velling with velocity 0 . 6 4 ~  is damped so fast that this is not shown in the 
plots after a noticeable break from the principal wave re-charging the chan- 
nel (we shall term it a discharge wave for simplicity). The wave in figure 4.17 
travels along the channel with velocity U, x 0.4c, i.e., 1.6 times slower than 
the precursor. This velocity somewhat decreases as the wave moves up. Its 
variation can be conveniently followed from the change in the well-defined 
maximum linear power of the Joule losses i2Rl (figure 4.17 (centre)). The 
wave front power rises abruptly along a 100-200 m length, then it decreases 
towards the earth, making the channel tip with intensive energy release stand 
out clearly. It seems that it is this region which is clearly discernible in streak 
photographs. The linear power proportional to the squared current drops 
remarkably on the way up the cloud, and the maximum becomes smeared. 
This is also consistent with observations of radiation intensity [14,29]. A 
photometric study has shown that the radiation from the wave front is 
attenuated and the front loses its clear boundary. The current wave is not 
attenuated so rapidly (figure 4.17 (top)). For the time of its earth-cloud 
travel lasting for 3 4 p ,  the current at the channel base drops from the 
maximum of 35 kA to 24 kA. This agrees with observations indicating that 
an average current impulse duration in a negative lightning is close to 
75ps on the 0.5 level. The wave front deformation depends on the initial 
potential Ui delivered by the leader to the earth. The higher the value I Uil, 
the higher the discharge current. The rate of resistivity decrease at the 
wave front grows respectively, so the front steepness increases. This is evident 
from a comparison of figures 4.18 (top) and 4.18 (bottom). At 1 Uii = 50 MV, 
the current wave goes along the channel practically without elongating the 
front? while at lUil = 10MV it has a lower velocity and a smooth front. 
Unfortunately, there have been no registrations of current and streak 
photographs of the return stroke taken simultaneously. A comparison of 
the relationships between current and wave velocity could provide a good 
test for the return stroke theory. 
As the current impulse amplitude rises, the linear resistance falls more 
quickly and to a lower level, so the wave is damped more slowly during its 
t The motion of a high current wave with attenuation but without noticeable distortions 
facilitates the electromagnetic field calculation necessary in many applied problems of lightning 
protection and in substantiation of remote current registration methods. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 196 ===
188 
Physical processes in a lightning discharge 
0 
1 
3 
4 
X,2 km 
t = 8 p  
0 
1 
2 
3 
4 
x, km 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 197 ===
Return stroke 
189 
3 
U 
\ 
.3 
I,,,= = 67 kA 
0.4 - 
0.2 - 
0.0 I 
. 
, 
. 
, 
. 
, 
. , 
0 
1 
2 
3 
4 
x, lan 
1- 
= 8,15 kA 
1 
.I 
0.8 
0.6 
0.4 
0.2 
0.0 
0 
1 
2 
3 
4 
Figure 4.18. Deformation of the current wave front at leader potential (top) 
Ui = -50MV and (bottom) -10 MV; for the other parameters, see figure 4.17. 
propagation along the channel. There is no damping at a very high current and 
the impulse front becomes steeper, as was discussed in section 4.4.2 (figure 
4.18). Non-linearity is also observed in the current amplitude dependence 
on the initial potential U, at the earth. If the commutator were perfect 
(R,,, = 0), the current at the earth at the moment of contact would instantly 
Figure 4.17. (Opposite) Numerical simulation of the return stroke excited by a 
descending leader with potential -30 MV: (top) current and (centre) voltage dis- 
tributions; (bottom) the power of Joule losses. The initial leader resistance, 10 n/m. 
Steady state field in the channel behind the wave, 10 V/cm. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 198 ===
190 
Physical processes in a lightning discharge 
200 1 
r 0.6 
Voltage, MV 
Figure 4.19. Calculated dependencies of the current amplitude and average wave 
velocity in the return stroke on the leader potential Vi. 
take the maximum value Z, 
= I Ui l/Z, independent of the actual channel resis- 
tance, and would be ZM N Ui. With the finite time of R,,, decrease to zero, the 
current wave is able to cover some distance along the channel and to include in 
the circuit the ohmic resistance of this channel portion. For this reason, the 
current amplitude appears to be lower than Ui/Z and rises somewhat faster 
than potential Ui (figure 4.19). It is important that the lightning current 
amplitude ZM is found to be appreciably smaller than its theoretical limit 
Ui/Z: e.g., Z, 
M 0: 6Ui/Z at Vi = 30 MV. This is another source of errors 
in evaluations using the equality Vi = ZZ,, 
in particular, in the calculation 
of cloud potential from lightning current data. 
4.4.5 Arising problems and approaches to their solution 
The current at the earth is independent of the boundary condition at the 
upper channel end, until the reflected wave comes back to the earth with 
the information about the processes occurring there. Before that moment, 
the positive charge is pumped into the line from the earth. In virtue of the 
boundary condition - zero current at the upper end - the current wave is 
reflected there with the sign reversal. As a result, the current behind the 
reflected wave, i.e., between its front and the channel end (figure 4.20), 
decreases (it would drop to zero in the absence of damping). The incoming 
positive charge now re-charges the line making it positive (an ideal line 
would be re-charged to -Ui). The reflected wave moves faster and is 
damped more slowly, because the linear resistance in most of the channel 
has dropped by an order of magnitude or more due to the action of the 
forward current wave. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 199 ===
Return stroke 
191 
0 
1 
2 
3 
4 
x, km 
& 15- 
- 
a
-
 
8 10- 
5 -  
0- 
-5 - 
Figure 4.20. Current and potential distributions during the propagation of waves 
reflected by the cloud end of the channel. 
When the reflected wave reaches the earth, delivering a positive potential 
to it, a new discharge cycle begins. The newly acquired positive charge flow- 
ing into the earth is equivalent to the extracted negative charge. The current 
sign at the grounded end is reversed (figure 4.21). In the absence of 
dissipation in a distributed system such as a long line, undamped oscillations 
with a period T = 4H/w, would arise similar to those in an LC circuit. 
Nothing of the kind is observed in lightning registrations, nor is there a 
single change in the current direction. This means that the discharge wave 
is either not reflected by the upper end of the line or the reflected wave 
becomes so damped on the way back to the earth that it is unable to manifest 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 200 ===
192 
Physical processes in a lightning discharge 
.- 
U 
-0.4 j 
Figure 4.21. Calculated current impulse through the grounded channel end. 
itself against the background of other variations in the current. By changing 
the parameter Tg or the quasi-stationary channel field EL, one can reduce 
or even cancel part of the current impulse of opposite sign, but it is 
impossible to attain a portrait likelihood between the calculated and 
observable currents. The suppression of the reflected wave by raising the 
instantaneous values of the channel resistance RI (x, t )  inevitably results in 
an excessive reduction of the impulse duration at the grounded end of 
the line. There seems to be no way of avoiding this even by changing the 
resistivity reduction law. 
The first thing that seems to be suitable for rectifying this situation is to 
question the boundary condition at the upper end. This idea appears reason- 
able because it generally agrees with lightning current registrations at the 
earth for the double path time t FZ 2H/w, while the model solution for 
i(0, t )  remains independent of the boundary condition. It is obvious that 
the open circuit condition is an excessively rough idealization. It was 
mentioned in section 4.3.3 that if the negative cloud bottom is filled with a 
large number of branches which stem from the ascending leader, this 
region becomes similar to a metallic sphere. Assuming such a 'metallization' 
of the cloud, it would be more reasonable to consider the upper end to be 
connected to a lumped capacitance C, = 4 7 r ~ ~ R ~ ,  
defined by the cloud 
charge radius R,, instead of being open. The boundary condition at x = H 
would have the form i(H, t )  = C, dU/dt. When the current wave reaches 
the line end, the delivered current also discharges the negative 'metallized' 
cloud region. This, however, does not prevent the appearance of the reflected 
wave. At the first moment of time, the capacitance still preserves its charge and 
is similar, in accordance with the reflection condition, to a short-circuited 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 201 ===
Return stroke 
193 
end of the line, which generates a reflected current wave of the same sign and 
amplitude as the incident one. As the capacitance becomes discharged, the 
reflected wave amplitude decreases and then the sign is reversed. The com- 
pletely discharged capacitance, incapable of supporting current, eventually 
becomes equivalent to an open line end. It is clear even without a numerical 
calculation how much the current changes at the earth after the arrival of a 
reflected wave of such complexity. It should be emphasized again that 
nothing of the kind has ever been observed in real lightning. 
One can also try to rectify the situation by complicating the boundary 
condition with the allowance for the final resistivity of the ‘metallized’ 
cloud region. The streamer and leader branches filling the cloud possess a 
resistance at the moment of their generation. A leader branch can hardly 
be heated as much as a single descending leader. The resistance of the 
‘metallized’ cloud, R,,, is quite likely to be high during the whole return 
stroke stage. If this is so, the boundary condition should be formally repre- 
sented as i(H, t) = C, dU/dt - R,, dildt. Strictly, it is not only the boundary 
condition that changes in this case, like in the case of ideal metallization, but 
also the set of equations. The cloud potential U. can no more be considered 
as being constant in time. The second equation of (4.24) should be re-written 
as 
having taken into account the change in U, due to the change in the cloud 
charge Q,. Then the function Uo(Q,) must allow for the delay because of 
the finite rate of the electromagnetic field propagation. The problem becomes 
extremely complicated. Although radar registrations do indicate the develop- 
ment of a wide network of branches in clouds, there has been no investigation 
of cloud ‘metallization’. The reason for this, no doubt, is the lack of initial 
data. 
One should not discard two other factors unaccounted for by the numer- 
ical model. First, a leader channel can practically never be single. Owing to 
the numerous branches of different lengths developing at different heights, 
numerous reflected waves will arrive at the earth at different moments of 
time, creating a sort of ‘white noise’ with a nearly zero total signal. This 
will deprive the current of its characteristic bending which is usually created 
by a single reflected wave at the earth. Second, constant linear capacitance 
only approximately describes the real re-charging of a lightning leader. We 
have mentioned above that the cover charge around a leader channel is 
changed by numerous streamers starting from it. Their velocity decreases 
rapidly when the streamer tips go away from the channel surface with its 
high radial field. So, when the voltage at the wave front changes, the 
charge near the channel changes almost immediately, while its change at 
the external cover boundary occurs with a delay. In other words, a lightning 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 202 ===
194 
Physical processes in a lightning discharge 
discharge can proceed for a fairly long time. The quasi-stationary current 
from the discharge of the cover periphery, having the same direction as the 
current in the forward wave, can compensate for the reverse current induced 
by the wave reflected by the earth. 
To conclude, the model of a single long line with varying linear resis- 
tance allows elucidation of many aspects of the return stroke but cannot 
claim to give reliable quantitative description of all characteristics of this 
phenomenon. The much more simplified models of return stroke are usually 
used calculating electromagnetic field for technical application. A review of 
this model is given [30]. 
4.4.6 The return stroke of a positive lightning 
Two kinds of current impulse can be distinguished in oscillograms taken at 
the earth after the arrival of a positive leader. Common impulses are similar 
to those registered in negative lightnings, although they have a slightly longer 
duration t p  and less steep fronts. Such impulses can be naturally interpreted 
as return stroke currents corresponding to the wave discharge of the leader 
channel, as described above. Sometimes, however, quite different impulses 
are registered with an order longer duration and an amplitude as large as 
200 kA. A closer examination shows that impulses with an ‘anomalous’ 
duration cannot be interpreted as resulting from a grounded leader dis- 
charge. They appear to result from another process, and we shall offer 
some suggestions concerning their nature in section 4.5. Here, only 
common impulses will be discussed. 
It was shown in section 4.4.2 that the stroke current front is unrelated to 
the wave discharge process in the channel but, rather, is associated with an 
imperfect commutator closing the channel on the earth. The leader streamer 
zone acting as a commutator possesses a finite resistance and is reduced 
during a finite period of time. The front steepness is determined by the rate 
of resistance reduction in this transient link between the channel and the 
earth. But the streamer zone length of a positive leader at the same voltage 
is about twice as large as that of a negative leader and takes more time to 
be reduced. It is quite likely that this is the main reason why, with the 
50% probability, the current front duration in positive lightnings, 
tr = 22 ps, is four times longer than in negative leaders [l]. Approximately 
the same proportion is characteristic of the maximum pulse steepness. 
The duration of the pulse itself, tp, is primarily determined by the stroke 
channel length. It was shown in section 4.3 that it is only positive leaders 
starting from the top positive region of a storm cloud which have a real 
chance to reach the earth. This region is twice as high as the negative 
cloud bottom. Hence, the channel length of a positive descending leader is, 
at least, twice as long. But the vertical positive channel transverses the 
negative cloud region, delivering a very low potential U, to the earth, so it 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 203 ===
Anomalously large current impulses of positive lightnings 
195 
is incapable of producing a return stroke with an appreciably high current 
(section 4.3.6). Only those lightnings, whose positive descending leaders 
bypass the negative cloud region along a very curved path, can actually be 
identified in the registrations. The statistics shows that the total length of 
such a leader, including the path bendings, is 1.3-1.7 times greater than 
that of a straight leader. Therefore, a positive channel length and its stroke 
pulse duration appear on average to be three times greater than in a negative 
leader. As for other characteristics, common positive pulses are the same as 
the negative ones described above. 
4.5 
Anomalously large current impulses of positive 
lightnings 
Anomalous impulses of a positive lightning have the duration t, M 1000 ps of 
the 0.5 amplitude level and the rise time q x 100 ps. The current in some of 
them is as high as 100 kA or more [l]. Although such lightnings are rare, their 
effects on industrial objects are so hazardous that they should not be 
underestimated. A current impulse delivers to the earth a charge 
Q M 1OOC; therefore, as large a charge must be located in the cloud cell 
where the lightning originated. The potential at the boundary of a charged 
cloud region of radius, say, R, M 1 km is U,, M 1000 MV, with 1500 MV 
at its centre. Any attempt to treat a long current impulse as return stroke 
current inevitably leads to contradictions. Indeed, in order to reduce the 
near-earth current by half of its maximum value for lOOOps, it would be 
necessary to assume in (4.29) at - 0.7 and a = R1/2L1 = 700s-’; hence, 
the average linear resistance behind the wave front of the return stroke 
would be Rl M 3 . 5 ~  a/m. The total resistance of a channel of length 
H = 4000m would be R1H M 14R, i.e., 40 times less than the wave 
resistance. The line would seem to be discharged as an ideal line, i.e., for 
20 ps instead of 1000 ps, with the velocity of an electromagnetic signal. 
Excessively smooth impulses are sometimes observed in ascending 
leaders. A positive impulse IM M 28 kA with t, = 800 ps was registered 
during the propagation of a negative ascending leader from a 70-m tower 
on the San Salvatore Mount in Switzerland [31]. This fact in itself is of 
interest, but its analysis may offer an explanation of ‘anomalous’ currents 
of descending positive lightnings. Note, at first, the unusual situation at 
the start. Since the negative charge of the dipole is located at the cloud 
bottom, the ascending leader is to be positive rather than negative. Therefore, 
the dipole axis has either deviated from the normal or the bottom negative 
charge was neutralized earlier by, say, an intercloud discharge. This situation 
occurs rarely but it is possible. 
We mentioned in section 4.1 that ascending lightnings have no return 
strokes because their channels are grounded from the very beginning. 
p. - 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 204 ===
196 
Physical processes in a lightning discharge 
However, when the ascending leader penetrates the charged cloud region 
(positive, in this case), a large potential difference arises between the front 
end of its grounded channel and the space around it, so the leader current 
has been found from many registrations to rise to several kiloamperes. 
This event seems to be triggered by the same mechanism, but its effect is 
greatly enhanced by the leader hitting the very centre of a large cloud 
charge of, say, Q, x 30C and radius R, M 500m, where the potential is as 
high as U, M 500-800MV. At such voltages, the streamer zone and cover 
appear much extended. Negative streamers develop until the average field 
in their streamers drops below E, M 10 kV/cm under normal conditions (or 
1.5 times less at a 5-6km height [16]). Streamers elongate very quickly 
when the field is higher. Therefore, a very powerful streamer corona con- 
sisting of numerous branched streamers (they are likely to originate not 
only from the stem but from its numerous branches, too) will fill up a 
space of size R M Uo/Es M R,. The negative charge of the streamer zone 
will partly neutralize the positive charge of the cloud cell. If the streamers 
have velocity U, x 106m/s, they will fill the charged cloud region for 
t x Rc/vU, M 
s. Since the capacitance of the leader portion inside the 
cloud, CL, is comparable with that of the charged cloud region, Ccl, 
a charge of opposite sign, comparable with the cloud intrinsic charge, 
penetrates the cloud. The resultant effect is such that most of the cloud 
charge would seem to run down to the earth with current i x Qo/t M 30 kA 
for t M lop3 s. Microscopically, the cloud medium remains non-conductive, 
as before. Charges do not recombine but neutralize one another on average. 
The process of current organization reduces to the neutralization of the 
cloud rather than leader charge. 
Returning to long current impulses after the positive leader arrival at the 
earth, let us imagine that the leader has been developing along a vertical line 
somewhat away from the axis of a powerful cloud dipole with Q, x lOOC or 
more, R, x 1 km, and U,, M 1000 MV. Suppose the leader cover has no 
contact with the cloud charge boundary but is close to it. All the same, a 
huge, actually induced charge comparable with Q, arises in the vicinity of 
the cloud charges. Note that the arrival of a vertical positive leader does 
not reveal itself in any way, since its potential is close to zero because of a 
nearly complete symmetry of charges induced in the lightning channel. 
Suppose now that while this leader still preserves conductivity (this 
period of time is measured in dozens of milliseconds because of the current 
supply of -100 A), an intercloud discharge occurs, connecting the lower 
negative charge of the dipole to another positive charge. Intercloud dis- 
charges have been observed to be a much more frequent phenomenon than 
cloud-earth discharges. So our suggestion is not at all improbable. The 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 205 ===
Stepwise behaviour of a negative leader 
197 
charges of opposite signs connected by intercloud leaders will gradually 
neutralize each other via the same mechanism as the one underlying an 
ascending 1eader.t As the neutralization goes on, the earlier induced but 
now liberated positive charge of the grounded leader will flow down to the 
ground. This will occur at a lower velocity than the return stroke velocity, 
in accordance with the neutralization rate of the negative cloud charge. 
This is a likely explanation for long powerful current impulses in positive 
lightnings. 
4.6 
Stepwise behaviour of a negative leader 
When discussing the negative leader in section 4.3.2, we put off the considera- 
tion of its stepwise behaviour until the reader has become familiar with the 
return stroke, since a similar phenomenon is the principal event occurring 
in each step. It would be reasonable, at this point, to turn to the nature 
and effects of the stepwise leader behaviour. But we should like to warn 
the reader that there is no clear answer to the question why a negative 
leader has a stepwise structure while a positive one has not. Nonetheless, 
some observations of stepwise positive lightning leaders were presented in 
[32]. This phenomenon has never been observed in laboratory conditions. 
4.6.1 The step formation and parameters 
The only thing one can rely on today in discussing the nature of leader steps is 
the results of laboratory experiments with long negative sparks (section 2.7). 
Natural lightning observations are not informative, except for the step 
lengths Ax, x 5-100m [13,32-371 and the registrations of leader channel 
flashes occurring at the step frequency. Streak photographs indicate that 
only the front channel end of 1-2 steps in length shows bright flashes. But 
weak flashes may appear even along a kilometre length (the vision field of 
a photocamera does not always cover the whole channel). 
Laboratory streak pictures show that a step originates from two second- 
ary twin leaders at the front end of the streamer zone in the main negative 
leader (in the Russian literature, these are termed bulk leaders). The positive 
leader moves towards the main leader tip while the negative one develops 
along the latter (figure 4.22). During the pause between two steps, the 
secondary negative and the main leaders do not have a high velocity, but 
the positive leader moves faster for two reasons. First, as the distance to 
the main leader tip becomes shorter, the difference between the positive tip 
t The fact that intercloud discharges do neutralize charged regions is supported by electric field 
measurements, and high currents that flow through lightning channels are indicated by peals of 
thunder. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 206 ===
198 
Physical processes in a lightning discharge 
L 
4 
Figure 4.22. The potential distribution for various stages of a step formation. The main 
leader potential U(x) is counted from the external potential U,: 
(top) secondary leaders 
1 and 2 are formed at point 3 at the streamer zone end; (centre) the tip of a positive sec- 
ondary leader has reached tip 4 of the main leader, and a discharge wave has started its 
travel along the secondary leader channel (dashed line); (bottom) the main leader tip 
after the step formation has taken a new position, and the process is repeated. 
potential U1 and the external (for the tip) potential U ( x l )  at the tip site x1 
increases (figure 4.22 (top)). Second, the streamer zone field of the main 
leader, E, % 10 kV/cm, which must support the generated negative streamers, 
is higher than the field E, x 5 kV/cm required for the development of positive 
leaders. For this reason, the streamers generated by the secondary positive 
leader tip develop in a fairly strong field, become accelerated and all reach 
the main leader tip. Since the long channel of the main leader has a 
capacitance greatly exceeding that of the short secondary leader, it absorbs 
completely all charges carried by the positive streamers. In other words, 
the secondary positive leader develops in the final jump mode. We know 
from section 2.4.3 that this leads to its acceleration. The secondary negative 
leader, on the contrary, develops in a decreasing field beyond the streamer 
zone of the main leader, whose streamers stop in space, so it moves much 
more slowly, similarly to the main leader. 
When the tip of the secondary positive leader comes in contact with the 
main channel, the positive leader experiences the transition to the return 
stroke. Charge variation waves run along both channels, as described in 
section 4.4, and their potentials tend to become equalized (figure 4.22 
(centre)). But the capacitance of the kilometre length channel is much 
higher than that of the shorter secondary channel, so their fusion results in 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 207 ===
Stepwise behaviour of a negative leader 
199 
establishing a potential only slightly differing from the initial potential of the 
main leader tip, U,. 
The moment at which the potential U1 is taken off the secondary leader 
channel and the latter joins the main leader, manifests the end of the step. 
The main leader tip ‘jumps’ over to a new place, the one occupied previously 
by the tip of the secondary negative leader, delivering to it its potential U, 
(figure 4.22 (bottom)). The tremendous potential difference that arises in the 
vicinity of the newly formed tip at this moment produces a flash of a powerful 
negative streamer corona, which transforms to the novel streamer zone of the 
main leader. Then the sequence of events is repeated. The combination of the 
charge utilized for the short recharging of the secondary positive leader and 
for charging the secondary negative one, plus the charge incorporated into 
the new streamer zone, create the step current impulse. (Recall that there is 
always a local current peak at the streamer tip or in the leader streamer 
zone, related to the displacement of the charge in this region; see sections 
2.2.3 and 2.3.2.) Part of the step impulse creates a current impulse in the 
main channel and the other part is spent for the formation of a new cover 
portion. The charge Q, pumped into the main channel can be evaluated in 
terms of the mean velocity of the stepwise leader, wL M 3 x 10m/s, the 
length of a step Ax, M 30m, and the current iL x lOOA averaged over the 
whole duration of the process. Since the time between two steps is 
At, x Ax,/wL M lop4 s, the charge is Q, M iLAt, x lop2 C. 
4.6.2 Energy effects in the leader channel 
The energy pumped by the charge pulse Q, into the channel can be evaluated 
if the effect is assumed to be similar to that observed when the small capaci- 
tance (of the secondary leader) is added parallel to the large capacitance (of 
the main leader). While a common potential is being established, there is a 
dissipation of energy nearly equal to the electrical energy stored by the 
small capacitance at a voltage equal to the difference between the initial 
capacitance voltages U, - U1, where U1 is the potential of secondary leaders. 
It was pointed out in section 2.4.1 that the leader tip potential is shared 
nearly equally between the streamer zone and the space in front of it. 
Secondary leaders are produced at the streamer zone edge, so we have 
U1 - U, M 4 (U, - U,); hence, U, - U1 = f (U, - U,,). With the accepted 
average values of current i x C1 (U, - Uo)vL and of velocity wL and assuming 
C1 x 10pF/m, we find U, - U, = 30MV and U, - U1 = 15MV. Thus, the 
step energy is 
W M Q,( U, - U1)/2 x 7.5 x lo4 J. 
(4.39) 
Of course, not all of this energy is released in the main channel. During the dis- 
sipation of charge Q,, the channel potential rises appreciably. This leads to the 
radial field enhancement and to the excitation of a streamer corona which 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 208 ===
200 
Physical processes in a lightning discharge 
pumps some of the charge into the leader cover. This process, the cover 
ionization in particular, requires much energy. But even if the energy released 
in the channel is assumed to be W M lo4 J, ths is still a very large energy. 
The power required to support an average current of lOOA in remote 
channel portions, where the effects of current impulses are averaged and 
smeared, should be P1 M lo5 Wjm. It is removed from the channel by heat 
conduction and, partly, by radiation. These parameters correspond to the 
maximum channel temperature T M 10 000 K, field E x 10 Vjcm, and resis- 
tance R1 M 10R/m taken for the above estimations (section 2.5.2). At this 
power, the energy released between two flashes per unit channel length will 
be Wl,, M 10J/m for the time At, M lOP4s. Therefore, the single pulse 
energy W would be sufficient to support a channel 1 km long. In reality, a 
step pulse is damped at a much shorter length. At a distance of about 
1 km, the step effects are smeared and the energy released in the channel 
becomes totally averaged. But at a short distance from the tip, the energy 
effect of the step is very strong, as is indicated by the intensive flash. The 
temperature registered in some measurements was as high as 30 000 K [35], 
i.e., the same as at the wave front in a return stroke. 
Let us evaluate the distance at which the energy effect of an individual 
step is still essential. When a short step joins a long channel, charge Q, is 
pumped into the channel for a short time. Since we are interested in distance 
and time much larger than the real length and duration of a charge source, let 
us assume the source to be instantaneous and point-like, as is usually done in 
physics: a point charge Q, is introduced at the initial point of the line, x = 0, 
at the initial moment of time t = 0. The resistance of not very short channel 
fragments, Rlx, is higher than the wave resistance, so the inductance effect 
will be neglected. At an average resistance of 10R/m, this distance is just 
about the step length Ax,. At shorter distances, the instantaneous point 
source model is invalid, since it implies an infinite initial voltage and 
energy. They drop to realistic values only if the charge affects a length 
exceeding Ax,, at which the source was actually placed. Therefore, with 
the neglect of inductance (and the precursor), the line charging to potential 
U,(x, 
t )  above the background potential is described by equations (4.36). 
On the assumption of R1 = const ,t they have an exact solution correspond- 
ing to heat flow from an instantaneous lumped source: 
I. The value of resistance RI to be taken for evaluations may be smaller than that in the leader, 
having in mind a transformation of the channel due to the step current. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 209 ===
Stepwise behaviour of a negative leader 
20 1 
The power released by the current step per unit channel length is 
described as 
(4.42) 
At the point x, the power reaches a maximum at moment t, = x2/6x, and 
(4.43) 
For the time of the pulse action, the energy released per unit length at point x 
is 
Q2 
W 
Ax, 
W, M 1; iiR, dt = 2 
x - - , 
x >  Ax, 
(4.44) 
d 1 x 2  Ax, ( x 
where W is the total energy injected into the channel by the pulse, with its 
upper limit given by formula (4.39). The effective duration of energy release 
from a single step at point x is expressed as 
X2 
Wl 
N 2.2t - -. 
- 2.7% 
At, - 
PImax 
(4:45) 
Consequently, the contribution of charge injection to the energy release at 
a given channel site decreases in the direction of perturbation propagation as 
Wl x x - ~  
and is independent of R1. The latter fact justifies the use of 
R1 = const without reservations concerning the resistance variation during 
the current impulse passage. The energy pulses released at point x owing to 
the two subsequent steps superimpose at x > (2.7xAts)ll2; ths critical 
distance follows from the condition At, 
M At,. For example, at the average 
resistance RI = 10 R/m, with x = 10 m /s and step frequency At, = lop4 s, 
this happens at a distance x x 1.6 km in t, M AtJ2.2 M 45 ps, after the pulse 
arrival here. Thus, the effects of energy release from individual steps are 
detectable even along an extended lightning path, and this is the cause of 
observable flashes of almost the whole channel. For a flash to arise, there is 
no need for a strong energy effect. A short temperature rise of, say, above 
l000K over l0000K would be sufficient for a flash to be detected by 
modern photographic equipment. 
The channel energy is affected by the temperature rise above the average 
background, rather than by the time separation of the energy pulses between 
two subsequent steps. In this respect, the impulse effect on the channel during 
the wave propagation is damped at distances close to the tip. The plasma 
temperature modulation determining the flash intensity at large distances is 
due to the imbalance between the energy release and heat removal from 
the channel during the pauses between the steps. There is no imbalance 
at a large distance from the tip after the channel development has been 
10 
2 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 210 ===
202 
Physical processes in a lightning discharge 
established. At T x 10 000 K, the losses for air plasma radiation are not par- 
ticularly great but become appreciable at T M 12- 14 000 K. The Joule heat of 
current is eliminated from the channel primarily by heat conduction. This 
process occurs at constant (atmospheric) pressure when the energy release 
is moderate, as is the case for distances of hundreds of metres from the tip. 
At T M l0000K, the air heat conductivity is X x 1 . 5 ~  WjcmK and 
the thermal conductivity at pressure p = 1 atm is XT = X/pcp x 180cm2/s, 
where p is air density and cp is heat capacity. The average conductivity in 
the channel 
corresponds to a temperature lower than the maximum tem- 
perature. To illustrate, at c x 10 (Cl cm)-’ corresponding to T = 8000 K, 
the effective radius of a conductive channel is r M (mrR1)-”2 M 0.6cm 
for R1 = 10R/m. The heat is removed from the channel for time 
t N r /2xT N 
s, an order of magnitude longer than the pause between 
the steps. The repeated energy pulses dissipate rather slowly, and the 
temperature modulation relative to its average value T M 10000K is not 
large at long distances x. Indeed, the energy released in the remote channel 
portions during a pause is Wla, x PlavAt, x 10 Jim at an average power 
Plav M lo5 Wjm. Even if we assume that all energy of a step is released in 
the channel and W/Ax, x 2500 Jim in (4.45), the excess of the pulse release 
over the average heat removal, which is equal to the average energy release 
Wla,, will be small at x > Ax,( Wl/Wlav)1’2 x lOAx, zz 500m. With allow- 
ance for other energy expenditures, this reduction in the pulse effect will be 
evident even at shorter distances. This circumstance makes the use of average 
parameters reasonable in the consideration of the evolution of long stepwise 
lightning leaders, ignoring the stepwise behaviour effects. In any case, labora- 
tory experiments show that there is no appreciable difference between a 
positive continuous and a negative stepwise spark discharge as for the 
velocity, average leader current or breakdown voltage in superlong gaps. 
However, even a small excess of the average temperature over its aver- 
age value may be sufficient for a flash to be registered optically. As for chan- 
nel portions located at a distance of one or two steps from the tip, the energy 
pulses and outbursts of temperature and brightness are found to be very 
strong there. A gas-dynamic expansion of the channel is also possible, as 
happens in the return stroke (section 4.4), although it occurs on a smaller 
scale. No doubt, a flash is also produced by a powerful impulse corona 
giving rise to a new streamer zone of the elongated leader. Photographs 
show that the transverse dimension of a step flash is about 10m [38]. 
2 
4.7 
The subsequent components. The M-component 
The processes in the lightning channel following the first component are 
known as subsequent components. Of interest among these are so-called 
M-components and dart leaders. In the first case, the current impulse 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 211 ===
The subsequent components. The M-component 
203 
registered at the earth has a very smooth front (0.1-1 ms), a similar duration 
and an amplitude of several hundreds of amperes, sometimes of 1-2 kA. The 
channel radiation intensity increases abruptly during the impulse, but one 
can hardly identify in the photographs a structure similar to the impulse 
front. The current impulse of an M-component is always registered against 
the background of about 100 A continuous current of the interpulse 
pause. For a dart leader to arise, this current must necessarily be cut off 
[39,40]. A few microseconds after the cut-off, a short high-intensity region 
- a dart leader tip - runs down to the earth along the previous channel 
with a velocity of -107m/s. The contact of the dart leader with the earth 
produces a return stroke with its typical characteristics but having a much 
shorter impulse front than in the first component (less than 1 ps or even 
0.1 ps in some impulses). It is hard to say anything definite about the lower 
limit of the front duration: it is likely to lie beyond the time resolution of 
the measuring equipment. 
The papers published almost simultaneously [41, 421 interpret the sub- 
sequent component qualitatively as representing the discharge, into the 
earth, of an intercloud leader after its contact with the upper end of the preced- 
ing grounded but still conductive channel. Here we describe the evolution of an 
M-component in terms of a numerical simulation. 
The model underlying the simulation is as follows (figure 4.23). Initially, 
there is a grounded plasma channel of length HI with zero potential, which 
was left behind by the preceding lightning component. At time t = 0, a leader 
channel of length H2 and potential Ui joins it in the clouds (the voltage drop 
from the leader current and from the intercomponent current is neglected). 
The short process of the channel commutation through the streamer zone 
X 
Figure 4.23. The formation of a subsequent component: (a) the grounded channel of 
the previous component and an intercloud leader; (b) channel charging-discharging 
waves. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 212 ===
204 
Physical processes in a lightning discharge 
d 
g 
1 500 - 
C 
E 6 
1000- 
500- 
0- 
of the intercloud leader is ignored. At the moment of closure, the leader 
channel possesses a typical resistance RIL % 10O/m. The resistance RI, of 
the previous channel depends on the duration of the intercomponent 
pause. After the return stroke current impulse of the previous component 
2000 
25001 
f 
Figure 4.24. Simulation of the M-component on closing an intercloud leader 2 km in 
length and lOMV potential on a 4-km grounded channel. The initial linear resistances 
of the channels RI = 10 n/m and the steady field EL = 10 Vjcm. The waves of potential 
(this page, top), current (this page, bottom), the power of Joule losses (opposite, top) 
and the current impulse at the grounded end of the channel (opposite, bottom); for 
comparison, the latter is also given for RI = 20 G/m and EL = 20 V/cm (curve B). 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 213 ===
The subsequent components. The M-component 
205 
is damped, the channel resistance increases gradually due to the gas cooling. 
But if the intercomponent current is comparable with the leader current, as 
is usually the case by the moment the M-component arises, the increased 
resistance of the grounded channel may be suggested to be limited by the 
value of RI, M rlL. The reactive parameters of both lines, which are not 
very sensitive to the channel plasma state, can be taken to be identical to 
those of the leader: C1 M 10pF/m and L1 M 2.7 pH/m. 
During the intercloud leader discharge into the earth via the preceding 
channel path, the channel resistances change, as in the return stroke 
14 
12 . 
10 
2 8  
b7- 
E 
3 
g 6  
&
4
 
2 
0 
0 
800 - 
< 
600- 
m 
a, 
a, 
c, 
E 
400- 
4-3 
t, 
200- 
i 
2 
3 
4 
x, km 
0 
100 200 300 400 
500 600 700 
Time, ps 
Figure 4.24. Continued. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 214 ===
206 
Physical processes in a lightning discharge 
(section 4.4.4). Suppose that these changes follow the relaxation law expressed 
by formula (4.38). This formula describes adequately the qualitative tenden- 
cies; there are no quantitative results to compare them with. 
This process is described by the long line equations of (4.24) with the 
following initial and boundary conditions: 
U(x,O) = 0 at 0 < x < H I ,  
at H I  < x < H I  + H2, 
U(x,O) = U, 
(4.46) 
i(x, 0) = 0; 
U(0, t) = 0, 
i(H1 + Hz, r )  = 0. 
At the site of contact of the two lines, their potentials are identical at r > 0. 
After the contact of the intercloud leader with the grounded channel, current 
and voltage waves start running along both lines away from the point of 
contact. As a result, the grounded channel becomes charged while the 
leader channel is discharged. If the initial parameters of the lines are 
identical, the initial current at the point of contact is i = Ui/2Z, where Z 
is the wave resistance of the lines. Rapidly attenuated precursors run in 
both directions at velocities of electromagnetic signal, while the main current 
and voltage waves propagate via the diffusion mechanism (figure 4.24). These 
waves spread much stronger than in the return stroke, since the current and 
voltage are lower here (the more so that the initial voltage amplitude is 
half the value of Ui). The channel resistance decreases more slowly and the 
wave fronts become smooth instead of becoming steeper. The initial voltage 
Ui in the subsequent components seems to be lower on average than in 
the stepwise leader of the first component because this process involves the 
increasingly less mature cloud cells with lower charges. If we ignore the 
weak displacement current induced by the changing charges of the recharged 
channels, the current impulse at the earth can be registered only after the 
diffusion wave front has reached the earth. By that moment, the wave has 
become very diffuse, so the current impulse front appears to be very 
smooth (figure 4.24 (bottom, page 205)).t The more or less uniform power 
distribution along the channel is to look as a uniform enhancement of its 
radiation intensity. The calculations of this distribution and such current 
impulse characteristics as the front steepness, duration, and amplitude are 
similar to their observations in M-components. A still better agreement 
with the measurements can be attained by varying the parameters in the cal- 
culations, primarily RLL and quasistationary field EL in fully transformed 
channels. These arguments favour the above suggestions concerning the 
nature of lightning M-components. 
t The current impulse of a return stroke is of a different form. The current amplitude is registered 
right after the short-term commutation process when the leader contacts the earth via its 
reducing streamer zone, which takes a few microseconds. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 215 ===
Subsequent components. The problem of a dart leader 
207 
4.8 
Subsequent components. The problem of a dart leader 
There is still no clear understanding of the nature of a dart leader, so we shall 
discuss the scarce experimental data available and suggest a hypothesis based 
on them. Then we shall consider some possible consequences of this hypoth- 
esis and the difficulties that may arise. The dart leader problem remains 
unsolved but it cannot be put aside because of the importance of the dart 
leader process. 
4.8.1 
There are no grounds to believe that the mechanism of dart leader initiation 
in the clouds is essentially different from that of a M-component. Rather, 
both processes result from the closure of an intercloud leader on a grounded 
channel remaining after the passage of the return stroke in the previous light- 
ning component. But the potential wave running along this track to the 
earth, known as a dart leader, differs radically from that of an M-component. 
It has a well-defined front identifiable by the intense radiation of dart leader 
tip travelling to the earth with velocity ?&L N lo7 m/s, which is at least by an 
order of magnitude hgher than the typical velocities of the first stepwise 
leader. The ability of a dart leader to travel so fast is especially remarkable 
because its potential is most likely to be lower than that of a stepwise 
leader. This is indicated by the return stroke currents, which are on average 
2-2.5 times lower (I, M U,/Z). The potential drop from the dart leader tip 
in the previous, still untransformed channel towards the earth must occur 
very quickly. This is indicated by a very fast front rise of the return current 
impulse, tf. To gain the full current Z, 
M U / Z ,  where U is the potential 
carried by the dart leader, the return wave must run along a leader section 
with a rising potential Sx and reach the totally charged portion of the 
channel. Therefore, we have Ax N ‘U,?, and if the return wave velocity is 
w, M 10’ m/s and 
M 0.1 ps, the length of the region with an abrupt poten- 
tial drop in the dart leader front is Sx M 10m. It is quite possible that this 
value is actually smaller because the return wave cannot at first gain the 
full return stroke velocity U, M lo’,/,. 
On the other hand, the potential 
drop region should not be shorter than Ax M vdL?f x 1 m, since the cross 
section of a channel with total potential U approaches the earth at velocity 
‘udL. Such a steep front of 1-10m is unattainable not only by a diffusion 
wave with its potential varying along many hundreds of metres (figure 
4.24) but even by an ‘ordinary’ leader of the first lightning component, in 
which Ax is determined by the streamer zone length. At the moment of 
contact with the earth, the latter is measured in dozens of metres at the tip 
potential of 20-30MV. This is the reason why the time necessary for the 
return wave current to grow to its maximum value is dozens of times 
longer than that for the dart leader. 
A streamer in a ‘waveguide’? 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 216 ===
208 
Physical processes in a lightning discharge 
It follows from the foregoing and the fact that a dart leader travels as 
fast as a very fast streamer that the former has no streamer zone which 
would serve as the primary prerequisite for a leader mechanism. It appears 
that the dart leader, contrary to its name, is essentially not a leader, although 
it has a charge cover, which it has to acquire under somewhat different 
circumstances (see below). Nor does it look like a diffusion wave of the M- 
type. The latter would have an order of magnitude higher velocity and a 
very diffuse front. 
A dart leader looks more like the oldest of the known types of propa- 
gating plasma channel - a streamer, whose head represents an ionization 
wave. The velocity of a dart leader is close to that of a high voltage streamer. 
The principal reason for a streamer channel being non-viable in air - a rapid 
loss of conductivity by the cold plasma - is very weak in this case. A dart 
leader follows the track heated by the preceding component, so that the 
still-hot track serves as a kind of waveguide to the leader. The high gas 
temperature greatly retards electron losses. Therefore, the possibility for the 
region behind the tip to be heated to arc temperatures increases considerably. 
This provides a stable highly conductive state inherent in a ‘classical’ leader. 
The preheated air pipe serves another, probably more important, func- 
tion. Its hot and rarefied air is surrounded laterally by cold dense air, Since 
the rate of ionization due to the field is described by the E / N  ratio, the 
radial expansion of the channel region behind the streamer tip is abruptly 
retarded as compared with the forward motion of the ionization wave. So 
the air mass to be heated by the current is reduced, permitting the channel 
gas to be heated to a higher temperature. The cold air restricts the channel 
expansion because it acts as a charge cover produced by the streamer zone 
of the leader. 
One should not think that the channel does not expand through the 
ionization mechanism at all. This process is just much slower than the 
forward motion of an ionization wave towards the earth, so most of the 
Joule heat is released into the yet unexpanded channel having a smaller 
radius. The radial field leads to the channel expansion only at the beginning, 
as is the case with common streamers (section 2.2.2). When the radial field is 
somewhat reduced, the channel becomes the source of a radial streamer 
corona which does not require a high field. Radial streamers rapidly lose 
their conductivity in cold air, and their immobile charges form a cover of 
the type that surrounds a common leader channel. Now, though with some 
delay, the mechanism of radial field attenuation and hot channel stabilization 
comes into action. Thus, a dart leader, being essentially a streamer (i.e., an 
ionization wave having no streamer zone in front of the tip), must possess 
a charge cover, as a leader. Unlike the case with a common leader, the 
cover is not inherited from a streamer zone but is formed entirely behind 
the tip, which is the seat of the principal processes driving the dart leader. 
(In a classical leader, the cover formation partly continues behind the tip, 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 217 ===
Subsequent components. The problem of a dart leader 
209 
as described in section 2.2.4.) 
The streamer mechanism of the dart leader development due to impact 
ionization of the gas in the strong field of the tip can manifest itself only if the 
conductivity in the channel of the previous component has dropped below a 
critical value by the time the next lightning component is to arise. This is 
unambiguously supported by the following observations. The M-component 
is produced against the background of a continuous current of the inter- 
pause, whereas the dart leader arises some time after this current is cut off. 
As long as the medium preserves a high conductivity, the diffuse penetration 
of the field and current prevents the ionization wave propagation. The diffu- 
sion wave has practically no ionization due to a direct action of the low field. 
The medium pre-ionization does not stimulate but rather hampers the propa- 
gation of the ionization wave. The latter requires a strong field, but the high 
conductivity of the medium in front of the wave leads to the field dissipation. 
In order to focus the potential drop to a narrow region, one must stop the 
charge flux (electric current) by concentrating charge in a narrow region to 
produce a strong field. The charge flux can be ‘locked in’ only by creating 
resistance to it if one places a poor conductor in front of the well-conducting 
portion of the channel. 
4.8.2 The non-linear diffusion wave front 
At this point, we have to make a short digression to discuss the structure of 
the near-front region of a diffusion potential wave. One will see later that this 
is directly related to the ionization wave problem. The diffusion wave velocity 
w is determined by the propagation process along the whole wave length. 
Its variation along the path from the cloud to the earth is illustrated in 
figure 4.24. By order of magnitude, the velocity of a non-linear wave is 
U FZ x,,/-xf, where -xf is its total length from the source to the initial front 
point and xav is an averaged diffusion coefficient in the transformed channel 
behind the front, which better fits the final linear resistance of the channel 
than to its initial resistance. If constant potential Ui is applied to the initial 
channel end, the value of xav does not change much. The velocity changes 
appreciably over the time, during which the wave covers a distance compar- 
able with that between the cloud and the earth. But its change is relatively 
small over the time the wave covers a distance of the order of its front 
width where the potential U(x) rises steeply. This means that we have 
U(x, t )  M U(x - wt) in the wave front, and the distributions of all parameters 
along the x-axis are quasistationary in the coordinate system related to the 
moving front (as in a non-linear heat wave [12]). With this circumstance in 
mind, we can rewrite the potential diffusion equation (4.35) as 
(4.47) 
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210 
Physical processes in a lightning discharge 
Taking into account E = -dU/dx = 0 and U = 0 in front of the wave at 
x -+ m, the integral of this equation is 
(4.48) 
The familiar relation i = rv is valid at every point of the quasistationary 
wave portion but not only at the site of the current cut-off in front of the 
streamer or leader tip. 
The energy W1 per unit length of the quasistationary channel is 
described as 
(4.49) 
and is expressed directly through the local potential. Indeed, reducing the 
rank of the set of equations (4.48) and (4.49) by dividing them by one 
another, we find 
w, - w,o = c1u2/2 
(4.50) 
where Wlo is the initial energy in the channel far out the wave front. Thus the 
statement repeatedly used in evaluations that the energy dissipated in the 
channel is of the same order as that stored in its capacitance is valid exactly 
in the stationary case.t 
We shall consider moderate waves, when the gas is heated at constant 
pressure, and the Joule heat is released at constant mass m = r r 2 p  = mopo 
per unit channel length (yo and p are the initial radius and gas density in 
front of the wave). Then we have W1 = mh, where h is the specific gas 
enthalpy. Assume for simplicity that thermodynamically equilibrium ioniza- 
tion is established at every point of the wave, so that conductivity 0 and 
x = ma(pC1)-’ are the functions of temperature T or h(T). Then x is 
unambiguously related to U through formula (4.50).$ With (4.48)-(4.50), 
finding the distributions along the wave reduces to the quadrature 
2 
wlo 
(4.51) 
1- 
= -2vx. 
U = [  
cl ] 
. 
ho=-. m 
2m(h - h,) 
1’2 
Let us calculate the integral by approximating the relationship x E o / p  
by the power function x = Ahn in the temperature range typical of the wave. 
The coordinate origin x = 0 is taken at an arbitrary point of the wave front 
t This is quite natural because under the problem conditions the channel is not created anew but 
exists from the very beginning with its linear capacitance C,. Then every channel portion is 
charged as lumped capacitance (cf. the comment on formula (2.17) in section 2.2.4). 
$In sections 4.7 and 4.4, the quantity x was related to electrical parameters through relation 
(4.38) which refers to strong waves with a high energy release. If desired, one can use this relation 
after substituting aG/ar by -U dC/dx and doing the above operations. 
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=== PAGE 219 ===
Subsequent components. The problem of a dart leader 
21 1 
start, in front of which (x > 0) the channel transforms very slightly, so that x 
increases slightly, followed by (x < 0) where it changes noticeably. The 
parameters of the initial front point will be marked by the subindex 1, assum- 
ing for definiteness x1 = 2x0, where xo corresponds to the initial channel 
conductivity. Then we have hl - ho = Sho, where S = 2l/" - 1. An exponen- 
tially damping tail of the electric field and current extends forward along the 
wave where the diffusion is 'linear': 
Within the front, where h exceeds ho considerably or, asymptotically at 
x + -CO, we have 
By matching the approximate solutions asymptotically valid at x + +x and 
x -+ -CO, the parameters of the front start and the matching point coordi- 
nate x1 can be found as 
Ax 
2nxl ' 
2n 
x1 = -. 
(4.54) 
U1 [($)r5(9)1'n]1'2, 
El =- U1 
This point is closer to the a priori position of the front start x = 0 than Ax, 
which justifies the approximations. 
Let us illustrate this situation numerically with reference to the conditions 
typical of the M-component (figure 4.25). Suppose the diffusion wave has 
velocity v = 10' mjs running along a channel with the initial radius ro = 1 cm, 
temperature To = 5900 K (h, = 14.8 kJ/g) and po = 5 x lop5 g/cm3 which 
is by a factor of 25 less than the normal; m = 1.54 x 10-4g/cm; 
ne 
1 . 8 ~  
10'4cm-3, the initial linear resistance R1 = 10R/m, xo = 10 m /s 
and C1 = 10pF/m. For the temperature range T x 6-10000K in air at 
1 atm, we have a l p  = 17h3 (where .[(a 
a cm)-'], p[g/cm3], and h[kJ/g]). 
Hence, A = 2.7 x lo6 (m*/~)(kJ/g)-~ 
and S = 0.25. From formulas (4.51)- 
(4.53), we find for the initial front point U1 = 3.5 MV, El = 2.2 kV/cm, and 
10 
2 
x, 0 A x  
Figure 4.25. Schematic diagram of the nonlinear diffusion wave front. 
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212 
Physical processes in a lightning discharge 
the effective field length before the wave front Ax = 100m. The point behind 
the wave with U = lOMV (h M 50 kJ/g, T M 100OOK) lies at a distance 
x = 500m from the front. There, x M 3x10" m2/s, the resistance is by a 
factor of 30 lower than before the front, and the field drops to 33 V/cm. The 
field maximum lies near the initial front point. The qualitative picture presented 
in figure 4.25 agrees with the numerical results of figure 4.24(a). 
4.8.3 
The possibility of diffusion-to-ionization wave transformation 
Let us define the conditions, under which a diffusion wave can transform to 
an ionization wave which is supposed to be a dart leader. Consider a simple 
situation. It is suggested that potential U, is applied to the upper end of a 
grounded conductive channel of the previous lightning component. It 
begins to diffuse into the channel. It is assumed that there is no transfor- 
mation and the initial conductivity corresponding to the diffusion coefficient 
xo is preserved. The diffusion is 'linear' in this case. The potential and field 
vary as 
E = iR1 = 
exp (- A). 
(..xot) lI2 
4x0 t 
(4.55) 
At every point x, the field first rises with time but then falls after the 
maximum E,, 
= 2(7re)-li2U,/x at moment t = x2/2x0. The point E,, 
moves at velocity wg = xo/x, and the potential at this point is U, = 0.33U1. 
An ionization wave can be formed if the maximum field is sufficiently 
high and exceeds a certain critical value E,. The ionization wave is assumed 
to propagate at velocity w, supposed to be equal to that of a dart leader. Since 
E,, 
N x-l N tC1I2, the ionization wave could principally arise at earlier 
times when E,, 
> E,, but if its velocity is U, < wg, is immediately overcome 
by a diffusion wave. This will not happen if wg drops below U, while E,, 
is 
still higher than E,, i.e., if the conditions vs 3 wg, E,, 
2 E, are fulfilled 
together. For this to happen, the diffusion coefficient must be smaller and 
the linear channel resistance larger: 
(4.56) 
For this, the gas temperature in the initial channel should not be high. On 
the other hand, for the 'waveguide' properties to manifest themselves, the 
temperature must be as high as possible to make the air rarefied. Because 
of the very sharp temperature dependence of conductivity (when it is 
low), these conditions are met only in a very short temperature range, 
T M 3000-4000K, where the air density is by a factor of 10-15 lower than 
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=== PAGE 221 ===
Subsequent components. The problem of a dart leader 
213 
normal. For the estimations, we take Ei = 3 kV/cm corresponding to a field 
characteristic of initial air ionization, 30-40 kV/cm under normal conditions. 
Suppose w, = 107m/s, Ui = 5MV (such potential usually provides current 
IM M lOkA for the next component at 2 = 500R during the return stroke), 
and C1 = 10pFjm. We find xOcr x l.lx10sm2/s and Rlcr = 880R/m. The 
resistance is two orders of magnitude higher than that supposed to 
precede the M-component. For this reason, a dart leader can appear only 
after the current cut-off during the interpause and a partial cooling of the 
channel. 
In reality, the channel undergoes transformation due to the diffusion 
wave, its conductivity rises, and the field dissipates faster than what is 
expected from the second formula of (4.55). To provide for the critical con- 
ditions, the initial conductivity may seem to be lower than the estimated 
value. So we should consider the other extremal case when the diffusion 
wave heats a limited amount of air and an equilibrium ionization is estab- 
lished. The diffusion wave is now non-linear. Its maximum field is near the 
initial point of the wave front, and one should use, instead of (4.56), the last 
relation of (4.52) similar to it with El = Ei. One should keep in mind that of 
interest are the temperatures 3000-6000K, at which 0 varies with T much 
more strongly: a l p  M 1.8 x 10-6h9 (the dimensionalities are the same as in 
the illustration of section 4.8.2). Now we have n = 9; at U, = 107m/s and 
channel radius yo = 1 cm, we have U1 = 1.2MV, xo = 4x lo7 m2/s, and 
RI, M 2500 a/m. The field extends before the wave front only for Ax = 4m. 
The electron density in the initial channel under critical conditions is 
ne x 6x 10l2 ~ m - ~ ,  
corresponding to its temperature 4000 K. 
4.8.4 
The ionization wave in a conductive medium 
The values obtained in section 4.8.3 on two extremal assumptions do not 
differ much and seem to be reasonable. The problem of the conditions 
necessary for a dart leader to arise may seem to have been solved. This 
optimism will, however, disappear as soon as one evaluates the parameters 
of an ionization wave when it propagates through a medium with critical 
conductivity. 
Consider a wave in the front-related coordinate system, as was done in 
section 4.8.2. The equation for field (4.48) will be supplemented by an ioniza- 
tion kinetics equation written directly for x because x N CT N ne: 
vj = N f ( E / N ) .  
dX 
-U- 
= uix, 
dx 
(4.57) 
This equation describes a new law for the channel transformation. Owing to 
(4.48), the ionization frequency 4 turns to the potential function. Then, by 
dividing (4.48) and (4.57), the problem is reduced to the equation for x( U )  
and the quadrature, as in section 4.8.2. 
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214 
Physical processes in a lightning discharge 
To advance further, one should choose the functionf(E/N) in a way 
suitable for integration.1 But difficulties and doubts arise immediately 
here. The approximation of vi by the power function vi = bEk, as in the 
streamer theory when this approximation with k = 2.5 provided fairly 
good results, does not work in this case. The ionization wave propagating 
through a conductive medium appears absolutely diffuse, as in any other 
channel transformation law: (4.38) or on the assumption of equilibrium 
ionization (section 4.8.2). Let us impart a threshold nature to the function 
vi(E) in a simple way - 4 = 0 at E < E* and 4 = const at E > E*. This is 
what was done by the authors of [43] when solving a similar problem for 
the laboratory ionization wave in a tube. The integration of (4.48) and 
(4.57) by the above method yields the following result. The change in the 
electron density and x in the ionization wave is defined by the ratio of 
potentials U2 and U1 at the points of ionization outset and onset, where 
E = E*. A potential 'tongue' of effective length Ax = xo/u extends in front 
of the ionization wave, as in other diffusion modes. The potential at the 
front is U1 = E*Ax. The parameter ratios at the wave boundaries are 
(4.58) 
This relation can be regarded as the dependence of the wave velocity on an 
'external potential' U2 applied to its back. On the other hand, the velocity 
is expressed by a formula similar to (2.2) for the streamer: 
where Axi is the extension of the ionization region from the initial to the final 
point of the wave. For a wave to survive, its parameters must meet the last 
inequality of (4.59). Otherwise, the field within the wave will be unable to 
exceed E*, so no ionization will occur. 
The capabilities of an ionization wave are limited, and this limit increases 
with increasing initial conductivity of the medium. For example, if the initial 
parameters ne0 M lOl2cmP3 and x0 x 107m2/s were even lower than the 
critical values found in section 4.8.3 and if the threshold field was 3 kV/cm, 
it would be necessary to have the ionization frequency 4 = 2.1 x lo6 s-' 
and potential U2 = 300MV in order to increase ne and x by three orders of 
magnitude (Ax = 1 m, U1 = 0.3 MV) and U2 = 30 MV by two orders. The 
wave width in t h s  case is Axi E 22m, i.e., it is very extended. Only when 
t Sometimes, it seems better to describe vi the function of E and, on the contrary, to remove U 
from (4.48) 
and (4.57). Instead of (4.48), we then get 
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=== PAGE 223 ===
Subsequent components. The problem of a dart leader 
215 
the initial conductivity is still an order of magnitude lower (nCo = 10" cmP3, 
xo = lo6 m2/s, and R = lo5 fl/m), the wave width begins to approach what 
would be desired for a dart leader. In the same medium at the same U 
and E*, the parameters necessary for the ratio x2/x0 = lo3 would be 
vi = 1 . 4 ~  
lo7 sP1, U2 = 30 MV, Ax = 10 cm, Axi = 5 m, and U ,  = 30 kV. 
A still narrower region of the potential rise would be obtained at a still 
lower initial conductivity. But then we approach the applicability limits of 
the basic concepts of the theory of perturbation propagation in a conductive 
medium and of the long line theory, and we are probably coming closer to the 
understanding of criteria for the dart leader production. 
4.8.5 The dart leader as a streamer in a 'nonconductive waveguide' 
The diffusion mechanism of field evolution in a channel, or in a long line, is 
incompatible with abrupt potential changes and, hence, with strong fields. If 
abrupt changes do arise, they are rapidly smeared by diffusion. We believe 
for this reason that neither a narrow ionization wave nor a dart leader can 
be formed in a well-conducting channel. To find the conditions, in which a 
very strong field can be induced, we should remind ourselves of the prerequi- 
sites for the long line equations. 
The electrostatics equation for cylindrical geometry has the form: 
-+--rE 
dEx 
1 d 
-- 
P 
r -  
dx 
r dr 
EO 
(4.60) 
where p is space charge density. By integrating, in the cross section, a conduc- 
tor of radius ro and neglecting the dependence of the longitudinal field E, on 
r, we obtain 
7- , 
r = 1; 27rrpdr 
m-0 - 
+ 27rroEr, = - 
2 8Ex 
dX 
EO 
(4.61) 
where Er0 is the radial field on the surface of a conductor of length I >> yo: 
(4.62) 
If the longitudinal field varies along the channel so slowly that the axial diver- 
gence can be neglected (the characteristic length for the variation of E, is 
Ax >> ro), we arrive at one of the basic conceptions of the long line theory, 
~ ( x )  
= C1 U(x), whose implication is the potential diffusion mechanism. It 
is suggested implicitly that the resistance varies very slowly along the 
channel, so this variation cannot be an obstacle to a charge flux, making 
the flux velocity decrease abruptly and create a space charge due to its 
local accumulation (a long line has no 'jams'). 
However, space charge does accumulate at a sharp boundary between a 
poorly- and a well-conducting channel portion. A charged tip is formed at 
the end of an ideal (or non-ideal) conductor, the potential in front of it 
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=== PAGE 224 ===
216 
Physical processes in a lightning discharge 
drops abruptly, at distances about equal to ro, inducing there a strong field 
capable of sustaining an ionization wave. This is what happens in a 
common streamer in a non-conductive medium. It is then clear what is 
necessary to support a sharp potential drop at the ionization wave front for 
a long time. The conductivity along the perspective trajectory must drop to 
a value low enough for the diffusion field tongue to be unable to smear the 
sharp potential drop. Therefore, the tongue length must become comparable 
with the channel radius Ax x xo/v N yo. Because of the strong temperature 
dependence of the degree of equilibrium ionization in air at low temperatures, 
a drop to T x 3000K would be sufficient. The equilibrium electron density 
established for the long zero-current pause will be neo - 10'o-lO1l cmP3; 
hence, Rlo - 105-1060jm, and xo - 106-105m2/s. But the air density in 
the cooled channel of the previous component at T x 3000K is by an order 
of magnitude lower than that of cold air, so that the conductivity drop will 
not interfere with the 'waveguide' properties of the track. 
The velocity of a dart leader as an ionization wave is defined, in order 
of magnitude, by the same formula (2.2) as the streamer velocity. But the 
'pre-ionization' in this case (nee - 10'o-lO'l cmP3) is considerable, and a 
much smaller number of electron generations (1n(ne2/neo) x 5 )  is to be 
produced in the wave. With the account of the similarity law for vi at 
an order of magnitude lower gas density, the ionization frequency is 
vi - 10" sC1 and ro - 1 cm; then we obtain a correct order of the velocity 
U = qro/ In(ne2/ne0) - io7 mjs. 
One cannot say that all the details of the dart leader behaviour have been 
clarified by the above considerations. For the dart leader channel to be well 
conductive, the electron density in it must be at least 5-6 orders of magnitude 
higher than the initial value for the track. But the capabilities of the ioniza- 
tion wave to produce more electrons are limited. The maximum conductivity 
of an ionization wave propagating through a non-conductive medium is 
defined, in order of magnitude, by the relation ~
F
~
~
~
/
E
~
 
- vi (section 2.2.2), 
because the space charge of the streamer tip, providing a strong ionization 
field is dissipated with the Maxwellian time TM = Eo/cF.t After the wave 
t It also determines the rate at which the linear charge T = C1 U is established, if it is, in the 
channel. Let us integrate the relation for charge conservation in the conductor cross section. 
Neglecting, for simplicity, the variation in 0 along the channel length, we obtain 
Using (4.61) and (4.62), we arrive at a refined equation for the relation between T and U: 
The postulate of the long line theory, T = C, U, is valid if the changes in the system, which also 
define ~ ( t ) ,  
occur slower than with T , ~  
= co/u. When applied to the wave front moving at 
velocity U in a line with conductivity uo, this happens at 00 >> uio/ro and xo >> vr0. 
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=== PAGE 225 ===
Experimental checkup of subsequent component theory 
217 
has passed, the channel still needs to be heated and ionized, but both pro- 
cesses are to occur in a moderate electric field, as in a classical leader channel. 
Besides, this must take place before a strong radial field makes the channel 
expand beyond the hot gas tube, or if it has already become enveloped by 
a stabilizing charge cover (section 4.8.1). 
There are still many questions about the processes in a dart leader that 
remain to be answered; the development of its quantitative theory is also a 
task of further research. 
To conclude, it is worth noting some specific features of a current 
impulse in the return stroke of subsequent components. Generally, the 
impulse duration is related to the time it takes the return stroke to run 
along the whole channel. For the subsequent components, this time must 
be longer than for the first component due to the attached intercloud 
leader. But the impulse duration in the subsequent components is about 
twice as short, although the return wave velocities are generally the same. 
The reason for this difference is likely to be the absence of branches in a 
dart leader. It is quite possible that the relatively slow process of their re- 
charging elongates the current impulse tail of the first component. The 
impulses of the subsequent components do not reverse the sign, similarly 
to those of the first one. In the absence of branches, the action of the reflected 
wave can no longer be screened by the randomly reflected waves of the 
numerous branches (section 4.4.5). The hypothesis of ‘white noise’ should, 
probably, be discarded as being inadequate. This problem, like the others 
above, awaits its solution. 
4.9 
Experimental checkup of subsequent component theory 
The theoretical treatment of processes occurring in the channel of the 
previous component has been reduced to the various wave propagation 
modes - the diffusion mode in the M-component and the ionization wave 
mode in the dart leader. The former has a strongly elongated front with a 
slowly varying potential, and the latter must possess a tip with a concentrated 
charge, producing an abrupt potential change. Indirect evidence for the 
significant difference in the field distribution is the registrations of current 
impulses at the earth. The impulse front durations are found to differ by 
2-4 orders of magnitude between an M-component and a dart leader. 
There is a possibility for a direct experimental evaluation of the potential 
distribution in a wave approaching the earth. This can be done by measuring 
the electric field gain at the earth during the wave motion. If the potential 
slowly rises along the whole wave length, as in an M-component (figure 
4.24(a)), the distributions of the potential and linear charge from the initial 
front point, located at height h, to the cloud can be considered to be 
linear, ~ ( x )  
= a,(x - h) (x is counted from the earth and x 2 h). For the 
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=== PAGE 226 ===
218 
Physical processes in a lightning discharge 
field at distance r from a vertical channel. we find 
- 
H - h  ] 
(4.63) 
aq [ lnH + (H2 + r 2 y 2  
AE(r) = - 
27rEO 
h + (h2 + r2)1'2 
(H2 + r2)1/2 
where H is the height of the grounded channel. If H is, at least, several times 
larger than r, the dependence of field AE on the distance between the 
registration point and the channel line will be only logarithmic. The same 
is true of the front duration of a field pulse. 
The situation must be quite different for a dart leader with the abrupt 
potential drop at the wave front, since the first approximation in the field 
calculation may assume a uniform potential along the channel and 
~(x) 
= const at x > h. This gives formulae (3.6) and (3.7), which yield the 
maximum value AE,,,(r) 
N r-'. Such a large difference in the field variation 
is easily detectable experimentally, especially if we remember that it concerns 
not only the field pulse amplitude but also its front rise time. To see that this 
is so, it is sufficient to introduce into (4.63) and (3.6) the h-coordinate for the 
wave front, expressed through the respective velocities: h = H - ut. 
Triggered lightning is a perfect source for such measurements. A triggered 
lightning is initiated by launching a small rocket raising a very thin wire which 
evaporates during the development of the first component. The point of 
contact of the lightning with the earth is strictly defined, so it is easy to 
position current detectors at the necessary distances. Besides, the channel at 
the earth follows the wire track and is strictly vertical, as is implied in 
the numerical formulae. Such measurements have been partly made [44-451. 
In section 3.5, we discussed the measurements of field AE at distances 
r1 = 30m and r2 = 500m from the channel during the dart leader develop- 
ment. These measurements were not synchronized. However, the ratio 
AE(30)/AE(500) = 17.4 for approximately equal currents is nearly the 
same as r2/r1 = 16.7. 
The field measurements for M-components have been reported only for 
r = 30m [42]. The oscillogram of AE(t) is accompanied by a simultaneous 
registration of a current impulse with the amplitude of 800A and the front 
rise time -100 ps. The duration of the impulse front A E  is approximately 
the same, but the field reaches its maximum of 1350 V/m earlier, when the 
current has reached half of its maximum amplitude (until the potential 
wave arrives, the current at the earth is zero, whereas the field begins to 
rise since its start). Figure 4.26 shows the calculated functions i(t) and 
AE(t) at the observation points with r = 30m and 500m. The long line 
model described in section 4.7.1 was used with the same C1 = 10pF/m, 
L1 = 2.7 pH/m, and RI (0) = 10 njm. The length of the grounded channel 
was 4000m and that of the intercloud leader contacting it was 2000m. The 
experimentally observed current of 800 A was reproduced in the calculation 
at the leader potential U, = 9.7MV. Under these conditions, the field 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 227 ===
References 
219 
Figure 4.26. Calculated variations of the electric field at the earth’s surface due to the 
M-component under the conditions of figure 4.24. The dashed curve shows the 
current impulse I .  
amplitude of 1500 V/m at the point r = 30 m is close to the measured value. It 
follows from figure 4.26 that the temporal parameters of the current impulse 
are also consistent with the measurements. At the point r = 500m, the 
calculated field amplitude is a factor of three smaller and the time for the 
maximum amplitude is nearly the same as for r = 30m. Both parameters 
would differ by an order of magnitude in a dart leader with this increase in 
r. Therefore, the diffusion model of the M-component reproduces fairly 
well the available observations. It would, certainly, be most desirable to 
make simultaneous field registrations at different distances from a grounded 
lightning channel. 
References 
[l] Berger K, Anderson R B and Kroninger H 1975 Electra 41 23 
[2] Idone V P and Orville R E 1985 J. Geophys. Res. 90 6159 
[3] Antsurov K V, Vereschagin I P, Makalsky L M et a1 1992 Proc. 9th Intern. Con$ 
on Atmosph. Electricity 1 (St Peterburg: A I Voeikov Main Geophys. Observ.) 
360 
[4] Vereschagin I P, Koshelev M A, Makalsky L M and Sysoev V S 1989 Izvestiya. 
Akad. Nauk SSSR, Energetika i transport 4 100 
[5] Simpson G C and Robinson G D 1941 Proc. R. Soc. London A 117 281 
[6] Kasemir H W 1960 J. Geophys. Res. 65 1873 
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=== PAGE 228 ===
220 
Physical processes in a lightning discharge 
[7] Gorin B N and Shkilev A V 1976 Elektrichestvo 6 31 
[8] Proctor D A 1971 J. Geophys. Res. 76 1078 
[9] Mazur V, Gerlach J C and Rust W D 1984 Geophys. Res. Lett. 11 61 
[lo] Mazur V, Rust W D and Gerlach J C 1986 J. Geophys. Res. 91 8690 
[ll] Raizer Yu P 1991 Gas Discharge Physics (Berlin: Springer) p 449 
[12] Zel’dovich Ya B and Raizer Yu P 1968 Physics of Shock Waves and High- 
Temperature Hydrodynamic Phenomena (New York: Academic Press) p 916 
[13] Schonland B 1956 The Lightning Discharge. Handbuch der Physik 22 (Berlin: 
Springer) 576 
[14] Orvill R E 1999 J. Geophys. Res. 104 
[15] Gorin B N and Shkilev A V 1974 Elektrichestvo 2 29 
[16] Bazelyan E M and Raizer Yu P 1997 Spark Discharge (Boca Raton: CRC Press) 
[17] Abramson I S, Gegechkori N M, Drabkina S I and Mandel’shtam S L 1947 Zh. 
[18] Drabkina S I 1951 Zh. Eksper. i Teor. Fiz. 21 473 
[19] Dolgov G G and Mandel’shtam S L 1953 Zh. Eksper. i Teor. Fiz. 24 691 
[20] Braginsky S N 1958 Soviet Phys. JETP 7 (Eng. Trans.) 1068 
[21] Zhivyuk Yu N and Mandel’shtam S L 1961 Soviet Phys. JETP 13 (Eng. Trans.) 
[22] Plooster M N 1971 Phys. Fluids 14 2111 and 2124 
[23] Paxton A N, Gardner R L and Baker L 1986 Phys. Fluids 29 2736 
[24] Sneider M N 1997 Unpublished report 
[25] Gorin B N and Markin V I 1975 in Research of Lightning and High-Voltage 
Discharge (Moscow: Krzhizhanovsky Power Engineering Inst.) p 114 (in 
Russian) 
[26] Bazelyan E M, Gorin B N and Levitov V I 1978 Physical and Engineering 
Fundamentals of Lightning Protection (Leningrad: Gidrometeoizdat) p 223 (in 
Russian) 
p 294 
Eksper. i Teor. Fiz. 17 862 
338 
[27] Gorin B N 1985 Elektrichestvo 4 10 
[28] Gorin B N 1992 Proc. 9th Intern. Conf. on Atmosph. Electricity 1 (St Peterburg: 
[29] Jordan D M and Uman M A 1983 J. Geophys. Res. 88 6555 
[30] Rakov V A and Uman M A 1998 IEEE Trans. on E M  Compatibility 40 403 
[31] Berger K 1977 in Lightning, vol. 1, Physics Lightning (R Golde (ed) New York: 
[32] Berger K and Vogrlsanger E 1966 Bull. SEV 57(13) 1 
[33] Schonland B, Malan D and Collens H 1935 Proc. Roy. Soc. London Ser A 152 
[34] Schonland B, Malan D and Collens H 1938 Proc. Roy. Soc. London Ser A 168 
[35] Orvill R E 1968 J. Geophys. Res. 73 6999 
[36] Orvill R E and Idone V P 1982 J. Geophys. Res. 87 11 177 
[37] Krider E P 1974 J. Geophys. Res. 79 4542 
[38] Uman M A 1969 Lightning (New York: McGraw Book Company) p 300 
[39] Fisher R G, Rakov V A, Uman M A et a1 1993 J. Geophys. Res. 98 22887 
[40] Fisher R G, Rakov V A, Uman M A et a1 1992 Proc. 9th Intern. Con$ on Atmosph. 
Electricity 3 (St Petersburg: A I Voeikov Main Geophys. Observ.) p 873 
A I Voeikov Main Geophys. Observ.) 206 
Academic Press) p 119 
595 
455 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 229 ===
References 
22 1 
[41] Bazelyan E M 1995 Fiz. Plazmy 21 497 (Engl. transl.: 1995 Plasma Phys. Rep. 21 
[42] Rakov V A, Thottappillil R and Uman M A 1995 J. Geophys. Res. 100 25701 
[43] Sinkevich 0 A and Gerasimov D N 1999 Fiz. Plazmy 25 376 (Engl. transl.: 1999 
[44] Rubinstein M, Rachidi F, Uman M A et a1 1995 J. Geophys. Res. 100 8863 
[45] Rakov V A, Uman M A, Rambo K J et a1 1998 J. Geophys. Res. 103 14117 
470) 
Plasma Phys. Rep. 25 339) 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 230 ===
Chapter 5 
Lightning attraction by objects 
In this chapter, we shall describe the way a lightning channel chooses a point 
to strike (a terrestrial or a flying body). This is the principal issue for lightning 
protection technology. In any case, a direct stroke is more hazardous than a 
remote lightning effect via the electromagnetic field or shock wave in the air. 
Historically, direct lightning strokes were observed earlier than indirect ones, 
and the first research into lightning protection problems was associated with 
direct strokes. 
Everyday experience and scientific observations, including those made 
as far back as the 18th century, indicate that lightning most often strikes 
individual structures elevated above the earth. These may be towers, 
churches, houses on high open hills, and just high trees. Today, this list is 
much longer and includes power transmission lines, transmitting and 
receiving antennas, and the like. The experience in maintaining such 
structures indicates that the frequency of strokes increases with the 
object’s height. This observation was used as a basis for the most common 
lightning protection techniques. A grounded rod higher than the object to 
be protected - a lightning rod - put up in the vicinity of the object is 
supposed to attract most strokes, thus protecting the object. The underlying 
principle of this approach has not changed since the first lightning rod was 
constructed two and a half centuries ago. What has changed is the require- 
ment for the protection reliability, which have become extremely stringent. 
For this reason, the specialists have to deal with exceptions rather than the 
rules, focusing on the rare cases of lightning breakthroughs to the object 
being protected, because they lead to emergencies and sometimes to 
catastrophes. 
The study of lightning attraction mechanisms is extremely time- 
consuming and expensive. Even a simple measurement of the number of 
lightning strokes at objects of various heights is very hard to arrange. 
Most apartment houses and industrial premises in Europe are less than 
222 
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=== PAGE 231 ===
The equidistance principle 
223 
50m high. On the average, a lightning strokes a 50m building once in five 
years. Every kilometre of a power transmission line 30m high attracts 
approximately one lightning discharge per year. Long-term observations of 
a large number of buildings and multi-kilometre transmission lines are 
necessary to accumulate a representative statistics. The difficulties increase 
many-fold when one needs to extract information on the protection reliabil- 
ity from the observational statistics. To illustrate, 10-20 years of continuous 
observations of a 50 m building would be required to obtain information on 
the lightning discharge frequency, and at least 1000 years would be necessary 
to check whether its lightning rod can really provide a ‘99% protection’ 
promised by the rod producers. 
In a situation like this, one has to resort to theoretical evaluations, 
and this is one reason why lightning attraction theory has been the focal 
point of research for many lightning specialists. Here, as in many other 
lightning problems, there is an acute lack of factual data. The available 
evidence obtained from laboratory investigations on long sparks does not 
always provide an unambiguous interpretation, and this makes one treat 
with caution many, even generally accepted, concepts. We shall focus on 
the most advanced approaches, discussing, where necessary, alternative 
hypotheses. 
5.1 The equidistance principle 
This approach is oldest and clearly correct in its theoretical formulation. 
Suppose that an object of small area and height h is located on a flat earth’s 
surface (it is a rod electrode in laboratory simulations). Let us assume further 
that a lightning channel is shifted from it horizontally at a distance r, and the 
channel tip is at an altitude Ho (figure 5.1). In order to predict whether 
the lightning will strike the object or the earth, we shall take into account 
the breakdown voltage measurements of long air gaps with a sharply non- 
uniform electric field. They show that the longer the gap, the higher the 
average voltage required for its breakdown and the longer the time necessary 
for the discharge formation. This means that the shortest gap has the best 
chance to experience a breakdown, provided that the same voltage is applied 
simultaneously to several gaps. Let us keep in mind that the distance from the 
lightning tip to the object, [(Ho - h)’ + r2I1/’, is shorter than that to the 
earth’s surface Ho at 
The distance Re, is known as the equivalent attraction radius for an object of 
height h. It indicates the surface area, from which lightning discharges that 
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=== PAGE 232 ===
224 
Lightning attraction by objects 
Figure 5.1. Estimating the equivalent attraction radius. 
have descended to the altitude Ho are attracted by the object. For a compact 
object of small cross section, this is a circle of area Seq M T R ; ~ ;  
for an 
extended object of length L >> h and width b << h (e.g., a power transmission 
line), this is a stripe of area Seq x 2ReqL. The average number of strokes per 
storm season is evaluated from Seq as 
NI = nlSeq 
(5.4 
where nl is the year density of lightning discharges into the earth at the 
object’s site. Global and regional maps of storm intensity are made from 
meteorological survey data [l, 21. The n1 data are usually given per 1 km2 
per year. Quite often, the maps indicate the number of storm days or 
hours, together with empirical formulae to relate this parameter to nl. 
The equidistance principle, simple and clear as it may seem, is of little 
use, because one can employ formulas (5.1) and (5.2) to advantage only if 
one knows the altitude Ho (the attraction altitude), at which a descending 
lightning leader begins to show its preference and selects the point to 
strike. The condition of the earth’s surface and the objects located on it 
cannot influence the lightning behaviour high up in the clouds. A lightning 
develops by changing its path randomly. As it approaches the earth, the 
field perturbation by charges induced by terrestrial objects become 
increasingly comparable with random field fluctuations. Eventually, the 
perturbation begins to play the dominant role, determining the channel 
path more or less rigorously. The average altitude Ho at which this happens 
is known as the attraction altitude. 
It is unlikely that the altitude Ho should be determined only by the 
terrestrial object’s height h. It must also depend on the leader field varying 
statistically with the lightning due to the variations in the storm cloud 
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=== PAGE 233 ===
The equidistance principle 
225 
charge, the starting point of the descending leader, its path, number of 
branches, etc. This diversity of lightning conditions is uncontrollable. The 
only parameter that can, to some extent, depend on observations is the 
attraction altitude averaged over all descending discharges. It deserves 
attention because the averaging will require only the statistics of descending 
lightnings which have struck objects of various heights. These statistics 
cannot be said to be reliable but it provides some factual information 
important for lightning protection practice. 
Before we use the stroke statistics in a theoretical treatment, we think it 
worthwhile defining the range of object heights. Unfortunately, one has to 
discard the stroke data concerning high constructions. Ascending discharges 
become dominant at heights h > 150m. Data on such strokes cannot be 
included in the statistics without reservations, even though they were 
obtained from well-arranged observations, in which every discharge was 
identified unambiguously. The point is that ascending lightnings partly 
discharge the clouds, reducing the number of descending discharges. This 
interference into the storm cloud activity is so appreciable that a further 
increase of h above 200 m does not practically change the stroke frequency 
of an object by descending lightnings. Of little use are the data on low 
structures (10-15m). The number of strokes in this case is greatly affected 
by the nearest neighbours and the local topography. Account should be 
taken of the statistics for low buildings, but such observations are scarce. 
The overall data have a too large spread. 
The authors of [3] selected the most reliable data and, by averaging 
many observations, derived the relationship between the number of descend- 
ing strokes and the terrestrial object height. Figure 5.2 shows individual 
representative values to demonstrate the data spread. All of the results are 
normalized to the intensity of the storm cloud activity, which is 25 storm 
days per year. In the range h 9 150 m considered, we can admit, with some 
reservations, the existence of a quadratic height dependence of the number 
of lightning strokes for concentrated objects and a linear dependence for 
extended ones. Both dependencies mean Re,/h 
Figure 5.2 shows that the expression Re, = 3h, sometimes used for 
rough estimations of the expected number of strokes, agrees fairly well 
with the averaged values of Re, derived from observations. The substitution 
of Re, = 3h into (5.1) yields for the average attraction altitude for descending 
leaders 
const. 
Ho = 5h. 
(5.3) 
This does not seem to be a large height. A lightning is insensitive to the 
earth’s surface along most of its path and it is only its last 50-500m which 
are predetermined. Below, we shall discuss the mechanism of the more or 
less rigid determination of the leader behaviour by a particular site on the 
earth (section 5.6). 
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=== PAGE 234 ===
226 
Lightning attraction by objects 
0. 
Figure 5.2. The average number of strokes per year for compact (top) and extended 
(bottom) objects of height h. The dashed curves bound spread zones in observation 
data. Solid curves are plotted using the equivalent attraction radius. 
5.2 
The electrogeometric method 
Popular among some lightning specialists, this method of calculating the 
number of lightning discharges into a grounded structure [4-81 should be 
considered as a modification of the equidistance principle. The main 
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=== PAGE 235 ===
The electrogeometric method 
227 
Figure 5.3. Lightning capture regions. 
calculation parameter in this method is the striking distance r,. Surfaces 
located at a distance r, from the upper points of a structure (roof), the 
adjacent buildings, and from the earth’s surface define by means of their 
interception lines the lightning capture regions (figure 5.3). 
The further path of a lightning channel which has reached the capture 
region is considered unambiguously predetermined. The leader will move 
to the object (or to the earth), whose capture surface it has intercepted. 
With these initial assumptions, the calculation of the number of strokes NI 
reduces to geometrical constructions, since the lightning density nl at an 
altitude z > h + r, is considered to be uniform, and the value of NI can be 
calculated if one knows the area S, of the capture surface projection on to 
the earth’s plane, NI = ytlSs. 
The long history of the electrogeometric method has witnessed only one 
improvement - that of the selection principles concerning the striking 
distance r, [l, 31. Discarding inessential details, the quantity Y, is found from 
an average electric field E, between the object’s top (or the earth’s surface) 
and the lightning leader tip which has reached the capture region. Usually, 
the values of E, were taken to be equal to the average breakdown strengths 
of the longest laboratory gaps. As the laboratory study of increasingly longer 
sparks progressed, the values of E, introduced into the calculation method 
decreased from 6 to 2 kV/cm, entailing larger striking distances. In this 
approach, the parameter r, is independent of the grounded object’s height 
but is sensitive to the leader tip potential U, (Y, zz U,/&). However for applica- 
tions, attempts are made to find the relation to the current amplitude Z,w of the 
return stroke, rather than U,, using simulation models of the kind discussed in 
section 4.4. If the function Y, =f(ZM) and, hence, S,(ZAw) 
are known, the 
number of lightning strokes at an object with current 1, > Zlwo is found as 
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=== PAGE 236 ===
228 
Lightning attraction by objects 
Figure 5.4. The dependence of striking distance on lightning currents. The lower 
curve is plotted using [4] data, the upper using [5] data. The spread region is hatched. 
where p(Z) is the probability density of current of amplitude Z found from 
natural measurements. To find the total number of strokes, the lower limit 
of the integral in (5.4) should be taken to be zero. 
The generally correct idea of differentiating distances r, in current 
amplitude actually fails to refine the calculation of NI. There is no factual 
information to determine the function rs = f ( Z M )  experimentally, while 
theoretical evaluations suffer from an unacceptably large spread, so that 
the values obtained by different authors differ several times (figure 5.4). 
The stroke statistics for objects of various heights could, to some extent, 
be used for fitting the calculated total number of strokes NI, but it proves 
unsuitable for finding the function r, =f(Z,w). 
The calculation procedures in the electrogeometric approach do not 
involve a strong dependence of stroke frequency on an object's height. 
Indeed, for a single construction, like a tower, the capture region projection 
on to the earth's plane is a circle of radius 
R = (2r,h - h2)'I2 at rs 2 h 
R = r, 
at r, < h 
and for an extended object, like a transmission line, it is a stripe of width 2R. 
Therefore, the number of strokes of low power lightnings with a small stroke 
distance (rs < h) will be entirely independent of the object's height, while the 
frequency of powerful discharges with r, > h must increase with height 
slower than h for compact objects and as 
for extended ones. Actually 
both of these dependencies are steeper. 
( 5 . 5 )  
5.3 
The probability approach to finding the stroke point 
A predetermined choice of the discharge path through an air gap contradicts 
the experience gained from long spark investigations. Neither a spark nor a 
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=== PAGE 237 ===
The probability approach to Jinding the stroke point 
229 
lightning travel along the shortest path. When voltage is simultaneously 
applied in parallel to several air gaps of various lengths, it is the longest 
gap that is sometimes closed by a spark. This is supported by the large 
spread of breakdown voltages: the standard deviation u for multi-metre 
gaps with a sharply non-uniform field is 5-10% of the average breakdown 
voltage. 
If two gaps, tested individually, possess the distributions of breakdown 
voltage of the probability densities cpl ( U )  and p2( U ) ,  then, provided that the 
voltage of a common source is applied simultaneously, the breakdown prob- 
ability for one of the gaps, say, the first one, is described as 
where a2 is the integral distribution defining the probability of the gap break- 
down at a voltage less than U .  If the distributions cpl and p2 are described by 
the normal law with the standard deviations u1 and u2 and by the average 
values of Uavl and Uav2, the breakdown voltage difference AU = U1 - U, 
also obeys this law, with AU,, = Uavl - Uav2 and a = (a: + 
This 
allows us to rewrite (5.6) using the tabulated probability integral: 
(5.7) 
P1 = A  [I - g/:exp(-x2/2)dx 
, 
A = Uavl - uav2 
2 
1 
(0: + a;, li2 . 
The expressions of (5.7) are valid for Uavl 2 Uav2. Otherwise, one should find 
the breakdown probability P2 for the second gap, writing for the first one 
The formal relations of the probability theory (5.6) and (5.7) are valid if 
the discharge processes in the gaps do not affect one another and if every 
individual breakdown can be considered as an independent event. Multi- 
electrode systems of this kind can be termed uncoupled. A classical example 
of an uncoupled multi-electrode system is an insulator string of a power 
transmission line. The distance between the adjacent towers is so large that 
there is no electrical or electromagnetic effect of discharges occurring in 
one string on those of its neighbours. The earth’s surface and an object 
located on it can also be regarded as an uncoupled system, with a descending 
lightning leader acting as a common high voltage electrode. Such systems 
have been studied in laboratory conditions [9], in which the distribution of 
breakdown voltage was used as the indicator of an uncoupled nature of 
the system. If the individual gaps comprising a system are tested individually 
and have the integral distributions Q1( U )  and a2( 
U )  with the probability 
densities cpl(U) and cp2(U), the system will have the following distribution 
of the breakdown voltages: 
P1 = 1 - P2. 
(5.8) 
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=== PAGE 238 ===
230 
Lightning attraction by objects 
Figure 5.5. The breakdown voltage probability for the uncoupled multielectrode 
system involving the high-voltage and two grounded electrodes. x : measured 
QsYs(U), 
0: measured @(U) for the single gap. The dashed curve is evaluated for 
the system using @ ( U ) .  
In the particular case of equal gap lengths with @ I  ( U )  = (a,( U )  = @( U ) ,  we 
have 
@&q = 1 - [l - qU)]”*. 
(5.9) 
Therefore, the uncoupled character of a system can be tested experimentally 
by comparing the measured distribution of its breakdown voltages with those 
calculated from formulas (5.8) and (5.9) and the distributions in the 
individual gaps. Experiments show that if the distance between grounded 
electrodes is comparable with their height, the leader processes in each gap 
develop independently, so that the system they comprise can, indeed, be 
regarded as an uncoupled one (figure 5.5). 
Suppose now that the attraction of a descending leader begins when its 
tip reaches the altitude Ho. The problem reduces to finding the breakdown 
path in an uncoupled system with a common high voltage electrode - the light- 
ning leader - and two grounded electrodes, namely, the earth’s surface and an 
object of height h located on it. The probability of lightning attraction towards 
the object from the point with the x- and y-coordinates in the attraction plane 
is equal to that of the gap bridging between the leader tip and the object’s top, 
P,(x,y). 
This probability is defined by the integral of (5.7). When the relative 
standard deviations are identical, al/UaVl = a2/Uav2 = a,, at identical 
average breakdown voltages, the upper probability limit is expressed through 
the shortest distance from the leader tip to the earth, d,(x,y), and to the 
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=== PAGE 239 ===
The probability approach to jnding the stroke point 
23 1 
object, do(x,y): 
(5.10) 
The expected total number of lightning strokes at the object, NI, is found by 
integrating Pa over the attraction plane. If the earth's surface is flat and the 
lightning discharge density nl is constant, then we have for a compact object 
of height h and for an extended object of average height h and length L, 
respectively: 
NI = 2nnl 1: 
P,(r)rdr, 
NI = 27rnlL[r P,(y) dy, 
[z2 + (Ho - h)2]'12 - Ho 
ua [z2 + (Ho - h)2 + Hi] 
A, = 
z = r,y. 
! 
(5.11) 
The relations obtained from the equidistance principle are identica, to (5.11) 
at cr, = 0. 
In virtue of the approximate symmetry of the function Pa(r) 
relative to 
the point with Pa = 0.5 (r/h E 3; see figure 5.6), the calculations of NI slightly 
depend on the standard attraction deviation 0,. When cra varies from zero 
to 10% (there are practically no greater deviations in pure air), the value 
of NI increases only by 15% for a compact object and by less than 5% for 
an extended one. It would be unreasonable to discard the simple and clear 
equidistance principle for the sake of this small correction, but for the greatly 
inclined (almost horizontal) paths of lightnings attracted by objects. 
r h  
Figure 5.6. Evaluated attraction probability for the attraction altitude Ho = 5h (h is 
the object height, Y is the object-lightning stroke distance). 
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=== PAGE 240 ===
232 
Lightning attraction by objects 
hr 
I / , ,  
lightning 
rod 
h0 
I 
a ,,,//,/,,/; 
f 
I 
r 
-I 
Figure 5.7. Why a lightning rod is less effective when an inclined lightning approaches 
from a side of the protected object. 
It is clear from the foregoing that the larger the distance between the 
lightning and the object (compared with that from the lightning to the rod) 
the greater the protective effectiveness of a lightning rod. For a lightning 
travelling in the attraction plane strictly above the lightning rod, the 
difference between the two paths (figure 5.7) is largest: 
A d  = [(Ho - ho)’ + 
at a << [2(h, - ho)(Ho - h0)]1/2. 
- Ho + h, 
A d  z h, - ho 
With increasing side shift of the lightning in the attraction plane r, the value 
of A d  decreases, and this decrease is especially noticeable when the lightning 
approaches from the side the object being protected, as is shown in figure 5.7: 
A d  = [(Ho - h0)’ + ( r  - a)2]1/2 - [(Ho - A,)’ + ~’1”’. 
In the limit r + x, 
the distance to the object is smaller than to the lightning 
rod ( A d  x -a), and it is quite ineffective for lateral strongly inclined lightnings 
coming from the object side. In the equidistance approach, there should be no 
events like ths, since lightnings are not to strike an object at a distance r > Req. 
In reality, the proportion of lateral strokes is found to be fairly large. The fact 
that the probability method considers this circumstance correctly (figure 5.6) is 
very important for the evaluation of the lightning rod effectiveness. 
5.4 
Laboratory study of lightning attraction 
Laboratory investigations of lightning attraction were initiated in the 1940s 
by simulating a descending lightning by a long spark and placing small model 
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=== PAGE 241 ===
Laboratory study of lightning attraction 
233 
rods and objects to be protected on the grounded floor [lo, 111. At that 
time, experimental researchers expected to derive information necessary for 
a numerical evaluation of lightning rod effectiveness. The naive optimism 
has long vanished. The measurements showed that the attraction process 
did not obey similarity laws. Essentially different results were obtained 
from gaps of different lengths and different time characteristics of the voltage 
pulses applied [12-141. But the interest in laboratory investigations of 
lightning has survived, and they are currently performed in an attempt to 
understand the attraction mechanism of long leaders. 
The primary question is when the attraction begins. Clearly, the 
condition of the earth’s surface does not affect the leader propagation 
while its tip is far from the earth. Here, the spark paths become distributed 
randomly. If one projects a multiplicity of paths on a sheet of paper and 
finds the mean deviation Ax from the normal passing through a high-voltage 
electrode with the account of the sign (e.g., plus on the right and minus on the 
left), one obtains Ax = 0 for altitudes z > Ho. The mean path in a gap 
perfectly symmetrical relative to the normal proves strictly vertical down 
to the altitude Ho. The attraction onset is indicated by the mean path 
deviation towards the electrode simulating a terrestrial object (figure 5.8). 
Data treatment for determining the attraction altitude was made in [14] for 
spark discharges of up to 12 m in length. The path statistics involved different 
time characteristics of the voltage pulse. In the case of a steep pulse front 
(6p), a leader was attracted from the moment of its origin; for a smooth 
front (250p), it had enough time to cover an appreciable gap length 
before the deviation towards a grounded electrode became noticeable 
(figure 5.9). 
I 
I 
I- r-I 
Figure 5.8. Determination of the attraction altitude Ho by the bend point of a mean 
path deviation onset. 
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=== PAGE 242 ===
234 
Lightning attraction by objects 
0.0 
0.2 
0.4 
0.6 
0.8 
1.0 
L l d  
Figure 5.9. The average leader deviation from the high-voltage electrode axis toward 
the grounded electrode, Ar, depending on the leader length L. The results are given 
for configuration of figure 5.8 and two voltage fronts, tf; d = 3m. In fact each 
curve presents a leader trajectory in cylindrical z - r coordinates, averaged over 
many experiments. 
The reason for this is as follows. When the voltage rises rapidly, the 
streamers of the initial corona flash reach the grounded cathode, and the leader 
develops in the jump mode from the very beginning. But when the voltage rises 
slowly, the leader channel covers about one third of a 3 m gap before the transi- 
tion to the jump mode. This suggests that the streamer zone imposes a definite 
direction on the leader as soon as it touches the grounded electrode. If ths is 
the case, the attraction of a laboratory leader must begin later in a long gap 
than in a short one. The experimental data presented in figure 5.10 show that 
the attraction delay time does increase with the length of the discharge gap d: 
Ho = d at d = 0.5 m but Ho x (0.4-0.5)d at d = 10 m. The ratio of the streamer 
zone length to d decreases nearly as much by the moment of the final jump. 
Another independent method for the study of spark attraction is to use a 
blocking electrode. Suppose we are able to set up instantaneously a metallic 
electrode on a grounded plane in the right place at the right moment of time. 
Let us do this many times for different lengths of a developing leader channel 
L and plot the probability of the electrode striking, Bo, as a function of L. 
The possible curves are presented in figure 5.1 1. The first version corresponds 
to the ‘instantaneous’ choice of the striking point at an altitude Ho < d (at 
the critical leader length L,, = d - Ho). A probability of the electrode striking 
falls sharply if the electrode is set up with delay (at L > Lcr) since the leader has 
already chosen some other point for stroke at the moment of the leader start 
(Ho = d, L,, = 0) while in the third version the leader chooses a striking 
point gradually also but beginning from the altitude Ho < d when L,, > 0. 
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=== PAGE 243 ===
Laboratory study of lightning attraction 
235 
1.0 
0.8 
0.6 
0.4 
0.2 
Figure 5.10. The deviation Ar at various gap length d for tf = 250ps under the 
conditions of figure 5.9. 
D 
Figure 5.11. ‘Blocking electrode’ experiment. A supposed qualitative probability @ of 
a stroke to the electrode when: ( A )  a leader is instantly chooses the striking point when 
its length reaches the critical length L,,, (B) a leader is gradually attracted from the 
very beginning, (C) a leader is gradually attracted reaching Lc,. (D) 
measurements 
for d = 3 m, tf = 6 ps (curve 1) and 250 ps (curve 2). 
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=== PAGE 244 ===
236 
Lightning attraction by objects 
This can be done experimentally if the electrode displacement is replaced 
by its screening by an electric field. The electrode should be insulated from 
the earth and a high voltage of the same sign as that of the leader should 
be applied to it. This will create a counter-propagating field which will 
block the electrode from the leader. The electrode will become accessible 
only after the blocking voltage is cut off. The electric circuit provides a 
precise control of the voltage cut-off [ 131. The experimental relationships 
in figure 5.11 show again that the attraction begins since the leader origin, 
if it develops in the final jump mode from the very beginning. In long gaps 
with a smooth voltage pulse, when the initial leader phase is well defined, 
the attraction is delayed as much as the transition to the final jump (curve 
2 in figure 5.1 1). 
Experiments on negative leaders have yielded similar qualitative results 
[15], but the attracting effect of a grounded electrode on a negative leader 
proves to be stronger. 
5.5 
Extrapolation to lightning 
The scale of laboratory experiments, 1 : 100 or 1 : 1000, is too small to resolve 
the details or to make long-term predictions. Laboratory studies have so far 
failed to clarify an important point: Does the attraction onset really coincide 
with the moment of the leader transition to the final jump, or is this process 
controlled by a counterleader starting from the grounded electrode? The 
interest in the counterleader and its relation to lightning attraction arose 
long ago [4]. The counterleader seems to increase the altitude of the grounded 
electrode. The difference between the tip potential and the external potential, 
AU, increases, so the counterleader goes up with acceleration (section 4.1.2) 
to meet the descending leader. 
One of the difficulties is that the moments of the descending leader tran- 
sition to the final jump and of the counterleader origin in a laboratory are 
hardly discernible. The experiment accuracy is insufficient to separate them 
reliably in conventional laboratory gaps of about 10 m long. For lightning, 
these moments may be considerably separated, but a direct measurement is 
practically unfeasible. So, one has, as usual, to rely on numerical evaluations. 
Let us first evaluate the field perturbation in the atmosphere by the 
charge of the grounded electrode of height h and radius ro before a counter- 
leader starts from it. The external threshold field necessary for a counterlea- 
der to arise and develop is defined by formula (4.1 l), in which d must be 
equalized to h. The field Eo for an industrial building of height h = 50m 
was found to be 350 V/cm at the parameters used in section 4.1.1. Note 
that this field results not so much from cloud charges as from the charge 
of the descending leader approaching the earth. The field induces a charge 
on the grounded rod, whose density per unit length can be considered to 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 245 ===
Extrapolation to lightning 
237 
depend linearly on height: ~ ( z )  
= aqz (section 3.6.2). The value of a, is 
defined by (3.11) where d = h and r = r0. For Eo = 350V/cm, h = 50m 
and ro = 0.1 m, we have aq M 3 x lop7 C/m2. 
The field gain AEo at the altitude zo above the rod, associated with the 
rod charge and its reflection by the earth, is 
AE, = 4 
= & (p 
2zoh - l n e ) .  
zo - h 
(5.12) 
At the attraction altitude zo = Ho M 5h = 250m and the found value of aq, 
we get AEo M 30V/m. This is about IOp3 of the unperturbed atmospheric 
field at the altitude Ho and 3 x lo4 times lower than the field in the streamer 
zone of a negative leader. It is hard to imagine a lightning leader which would 
respond to such weak perturbations. In any case, laboratory experiments 
have failed to reveal changes in the breakdown probability of a gap for 
such a small relative increase in the voltage. Consequently, the attraction 
process cannot begin before the counterleader is excited. 
Let us follow the excitation of a counterleader by the field of a descend- 
ing leader, relating the tip altitude of the latter to the grounded rod height. 
For this, expression (4.1 1) should be supplemented by the dependence of 
the average near-earth field on the descending leader charge. Consider a 
simple situation. Suppose a descending leader starts at altitude H1 and 
moves together with its partner, a positive ascending leader, vertically 
without branching right above the grounded rod of height h. At the 
moment the descending channel acquires the length L, with its tip having 
descended to the altitude Ho = H1 - L, the potential of the leader charge 
at the rod top, together with the charges reflected by the earth, is 
2H1 - Ho -t '1. (5.13) 
2H1 - Ho - h 
Ho + h 
- (Hl + h) In 
(p, = - 
a, [(Hl - h) In 
4T&o 
Ho - h 
The field average in the rod height is E,, = pq/h + Eo (with the account of 
the cloud field Eo). By equating Eay to the threshold field necessary for the 
excitation of a viable counterleader (formula (4.1 l)), we find the attraction 
altitude Ho from (5.13), assuming the attraction to begin at the moment of 
the counterleader start. 
We shall not be interested in the quantity Ho linearly dependent on the 
poorly known parameter aq to be averaged over all descending lightnings. 
Rather, we shall focus on the tendency in the variation of the Ho/h ratio 
with varying h in the range 10-150m. Buildings lower than 10m are rarely 
affected by lightning, while the picture for high structures is greatly distorted 
by ascending lightnings, as pointed out above. Suppose that the attraction 
altitude for an object of average height, say, h = 50 m, is found from (4.11) 
and (5.13) to be really close to the experimental value Ho = 5h. This yields 
the estimate for a, (which is aq/4neo E 1.5 kV/m at Eo = lOOV/cm and 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 246 ===
238 
Lightning attraction by objects 
H I  = 3 x 103m), permitting the calculation of Ho/h for constructions of 
other heights. The calculations are 
h.m 
10 
20 
30 
50 
100 150 
Holh 9.3 7.0 6.0 5.0 4.0 
3.6 
Of course, this is not the linear dependence Ho FZ h obtained from a prelimin- 
ary treatment of observational data. The attraction altitude definitely rises 
with the object’s height, as Ho x ho65 according to the calculation. A 
better agreement could hardly be expected since the observational data are 
limited and have a very large spread. 
Another result would be obtained if one related the attraction onset to 
the moment of the leader transition to the final jump. The attraction altitude 
would then be determined by the streamer zone length L,, Ho = h + L,. The 
length L, only slightly depends on the grounded rod height. At its zero 
height, the streamer zone is totally created by anode-directed streamers of 
the negative descending leader, which require an average field of 10 kV/cm 
(there is no counter-discharge). If the rod has a large height, the active vol- 
tage is shared equally between the streamer zones of the descending leader 
and the positive counterleader. Cathode-directed streamers of the latter 
can develop in a 5 kV/cm field, thereby decreasing the average field in the 
common streamer zone, at most, by a factor of 1.5-2. This would set a 
limit to the possible variation in the attraction altitude. It may seem that 
the result obtained is quite promising. The attraction altitude at the leader 
potential U x 100MV is also found to be close to lOOm for low objects 
and about 300m for structures 100-150m high. But one should keep in 
mind that only unique unbranched leaders are capable of delivering to the 
earth such a large cloud potential (section 4.3.2). Such leaders occur rarely 
in nature; the potential of a normally branched lightning is several times 
lower. As smaller will be the streamer zone length proportional to U .  The 
attraction altitude would then become equal to the object’s height, provided 
it is not too low. 
In other words, the attempt to relate the attraction process to the final 
jump unambiguously relates the quantity Ho to the potential of a descending 
leader, making it strongly dependent on the factors discussed in section 4.3.2, 
which change this potential (e.g., branching). If one relates the attraction to 
the excitation and development of a counterleader, the dominant factor will 
be the total charge delivered by all the components of a descending leader to 
the earth. 
The idea that the attraction onset is associated with the excitation of 
a counterleader leaves little hope for an unambiguous relationship between 
Ho and the return stroke current IM, as was implied, for example, by 
the electrogeometric method. Indeed, the current IM is determined by the 
potential U, delivered by the leader to the earth (section 4.4. l), whereas the 
field at the grounded rod is due to the total charge of the descending 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 247 ===
On the attraction mechanism of externaljeld 
239 
leader. The branching and path bending typical of a descending lightning 
greatly affect the value of Ui and, hence, 1, (section 4.3.3), but they do 
not much change the total leader charge. 
5.6 
On the attraction mechanism of external field 
There is no doubt that lightning attraction is due to the electric field which is 
related to the object. It is difficult to imagine another remote way to affect a 
leader. As for the field source, the evaluations made in section 5.5 show that 
the field created by the charge induced in the object itself proves very weak. 
When the distance between the descending leader tip and the object is 
sufficient for an attraction effect to reveal itself, the object charge field at 
the leader tip is by a factor of 102-103 lower than the cloud charge field. 
There are no reasons why such a slight perturbation should make the 
leader change its path, which is subject to various random bendings even 
without the influence of any terrestrial objects. No doubt, a counterleader 
excited by the object serves as a mediator between the object and the 
descending lightning. It looks as if it elongates the object, thereby increasing 
the charge acting on the descending leader. The counterleader travelling 
towards its tip attracts it to itself, and this eventually results in the lightning 
stroke at the object. The mutual attraction of the two leaders becomes 
especially pronounced when the fields they excite at the tip are comparable 
with or, better, exceed the differently directed cloud field. It is only then 
that the descending leader changes its path to go to the object, and the 
counterleader is attracted by the descending leader rather than by the 
cloud charge centre, as is usually the case. It is the excess of the perturbation 
field over the cloud charge field which imparts a quasi-threshold character to 
the attraction process. 
This unquestionable and fairly trivial reasoning is certainly useful for 
lightning protection practice. Physically, however, it remains quite meaning- 
less until the mechanism of the external field effect on the leader is known. 
This is equally true of the cloud field which is also involved in the attraction 
of the leader, generally directing it to the earth. It is not clear at first sight 
what exactly is affected by the external field, which may be very weak. The 
fact is that the leader moves along the field even at Eo M lOOV/cm. Fields 
of this scale cannot affect directly the leader development - we have 
emphasized this several times above. The leader propagation, which occurs 
via turning the air into the streamer and leader channel plasmas, requires 
much stronger fields. These are present in the leader tip, in the tips of numer- 
ous streamers, as well as in the streamer zone where the strength (the lowest 
of the three) exceeds 10 kV,km in a negative leader and 5 kV/cm in a positive 
one. High driving fields are created by the charges of the tips, streamer zones 
and, partly, by the nearest portions of the channel and leader cover. They 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 248 ===
240 
Lightning attraction by objects 
cannot be created by the cloud or any other remote objects. The 
instantaneous leader velocity is entirely independent of the low external 
strength Eo but is determined by the potential difference between the 
leader tip U, and the external field U. at the tip site. This great difference, 
A U  = I U, - Uoi x 10-100 MV, along the relatively short length of the 
streamer zone creates in it the field E, x 5-10 kV/cm >> Eo necessary for 
the streamer and, eventually, leader development. What is then the instanta- 
neous effect of the negligible field Eo and its weak perturbations produced by 
the remote counterleader on the motion of the descending leader? 
Apparently, this effect is that the external field accelerates the leader. We 
mentioned this at the end of section 4.1.3 and shall now discuss it at length. 
The underlying mechanism is as follows. Voltage determines the leader 
velocity, while the voltage gradient determines its acceleration. Velocity is 
a function of the absolute potential change at the tip, VL = f ( A U ) ,  with 
U. in the expression for AU being a function of the space coordinates or 
of the tip vector radius r. A particular form of the functionf(AU) in this 
case does not matter; what is important is that VL grows with AU. So, retain- 
ing the generality, we can use the empirical approximation of (4.3), 
VL N jAUl’ (y = 4). The algebraic value of the leader acceleration is 
=&- 
-- 
dUt f-- 
duo dr) =*y- :( 
dUt E0VL) (5.14) 
dVL 
dt i: 
( 
dt 
dr dt 
dt 
where plus refers to a negative leader and minus to a positive one. The first 
term in the sum of (5.14) does not depend on the direction of the external 
field. One of the reasons for the variation of U, with time was discussed in 
section 4.3.2. Another reason is the increasing voltage drop across the 
channel with its elongation. Normally, a variation in U, has a retarding 
effect on descending leaders of both signs. 
The second term in (5.14) leads to acceleration if the negative leader 
moves in the direction opposite to the field vector, with the positive leader 
moving along the field. The accelerating effect of the external field increases 
as the field becomes higher and the angle between the field and velocity 
vectors becomes smaller. Both terms have been estimated to have the same 
order of magnitude (10’ m/s2); the second term may sometimes be even 
larger. For this reason, the attractive action of the external field proves 
essential. 
We can now make clear the attraction mechanism. The actual mechan- 
ism, by which a leader chooses its propagation direction, has a statistical 
nature. This is indicated by numerous random path bendings and branching. 
Clearly, there is a high probability that the leader moves towards a site where 
it can acquire the greatest acceleration or the least retardation. It will be able 
to develop a maximum velocity in this direction, bypassing other competitors 
on its way. Large-scale leader photographs taken with a very short exposition 
nearly always show several leader tips on short, variously oriented branches 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 249 ===
How lightning chooses the point of stroke 
24 1 
Figure 5.12. A still photograph of the leader channel front with exposure of 0.3 ps. 
(figure 5.12). Among these, only one tip has a real chance of survival - for a 
positive leader, it is the one which belongs to the branch oriented along the 
external field; for a negative leader, the respective branch must be oriented 
against the field vector. The other tips usually die. 
The mutual attraction of the descending leader and the counterleader, 
mediated by the electric fields created by their charges, is a self-accelerating 
process. This is due to a positive feedback arising between them. An 
enhanced field of one leader accelerates the other leader towards the first 
one. Because the distance between the leaders becomes shorter, the field of 
each leader rises at the site of the other leader tip, and the mutual acceleration 
proceeds at an increasing rate. This goes on until the streamer zones of the 
leaders come in contact and their channels unite. As a result, the common 
channel appears to be tied up to the object, from which the counterleader 
started. 
5.7 How lightning chooses the point of stroke 
Suppose the descending lightning leader has deviated from the vertical line to 
go to some high terrestrial structures. The highest structure is a lightning rod, 
or several lightning rods. If the objects to be protected are much lower, the 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 250 ===
242 
Lightning attraction by objects 
Figure 5.13. Increasing the grounded electrode effective height by a counterleader. 
lightning usually bypasses them to strike one of the rods. This can be predicted 
from the equidistance principle. But when designing lightning protection 
devices, one usually focuses on exceptions rather than the rules. So the question 
arises of how large is the probability that the leader will miss the rod and strike 
the object, having taken a longer path. It seems justifiable to apply the concepts 
of a multi-electrode system to this problem. The lightning which has become 
oriented towards a group of grounded ‘electrodes’ has to choose among 
them. Let us make an estimation from formulas (5.10) and (5.1 l), substituting 
the distance from the leader tip to the earth, de, by the distance to the rod top, d, 
(do is, as before, the distance to the object’s top). Lightning protection 
experience shows that there is no need to make a lightning rod much higher 
than typical terrestrial constructions (ha < 50m). Arranging them close to 
each other, one can provide a reliable protection of the 0.99 level (of 100 light- 
nings, 99 are attracted by a protection rod) if the rod height h, is only 15-20% 
larger than ho. For an ‘average’ lightning, displaced at a distance equal to the 
attraction radius Re, x 3ho relative to the grounded system, we have 
Ad = do - d, RZ (0.12-0.15)hr at Ho = 5h, (figure 5.13(a)). The substitution 
of these values into (5.10) with oa RZ 10% gives A, x 0.2. After taking the 
integral of (5.7,) one gets the probability of the lightning stroke at the object 
Po x 0.4 instead of the experimental value 0.01. 
The complete failure of the theory was predictable. A system with a close 
arrangement of grounded electrodes cannot be considered to be discon- 
nected, Its counterleaders affect one another. The first leader that has started 
from one of the electrodes decreases the electric field behind it, via its cover 
space charge, preventing the upward development of counterleaders from the 
other electrodes. Appearing with a delay, if they do, these counterleaders 
cannot retard their faster competitor, because the field is enhanced in the 
direction of the first leader propagation (figure 5.13(b)). This makes all of 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 251 ===
How lighttiitig choo.scs 11w poiti t r!/'.strokc 
'43 
Figure 5.14. The oscillogram shows how the counterleader started from the 'active' 
grounded electrode of 1.1 m height screens an electric field on similar 'passive' 
electrode located at a distance of 10cm. The gap length is 3 m. E 
the field at the 
passive electrode tip. Q - the counterleader charge. I .  
the counterleaders interconnected: therefore. one now deals with a connected 
multielectrode system. 
Turn to laboratory experiments [9]. The oscillograms in figure 5.14 
illustrate the field variation on the grounded. 'passive' electrode when a 
counterleader develops from the nearby 'active' electrode. To simulate this pro- 
cess for a sufficiently long time, a planeeplane gap 3 m long was used with two 
rod electrodes on the grounded plane. A high negative voltage pulse was applied 
to the other plane. A possible discharge from the passive electrode was excluded 
by placing a thin dielectric screen totally covering the rod top. Before discharge 
processes came into action. the passive electrode field rose in a way similar to the 
voltage pulse. After a leader had started from the active electrode, the field rise 
on the passive electrode became slower. and the shorter the distance between 
the electrodes. the greater the rate of slowdown. At a very short inter-electrode 
distance, the passive electrode field stopped rising with voltage and even 
decreased somewhat. This obvious result indicates that the degree of mutual 
effects of discharge processes and grounded electrodes becomes greater with 
decreasing distance between them. Eventually. the role of passive electrodes 
becomes negligible the grounded electrode system behaves as if it is replaced 
by one active electrode which attracts nearly all descending leaders. 
Owing to the feedback mechanism considered. the choice of the 
stroke point made by a lightning become more definite. Even the slightest 
the gap voltage. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 252 ===
244 
Lightning attraction by objects 
x 
c, 
.& 
3 
0.6- 
s 
a 0.4- 
E! 
0.8 ’g 
o.21 
0.0 .r/ 
, 
, 
, 
. 
, 
, 
0.90 
0.95 
1.00 
1.05 
1.10 
U”% 
Figure 5.15. The breakdown voltage distribution for the system of figure 5.5, but with a 
small distance 10 cm between the grounded electrodes, which makes the system coupled. 
advantages in the conditions in which a counterleader arises acquire an 
additional significance, being enhanced by the weakening electric field in the 
vicinity of the passive electrode, below the leader channel. It seems as if the 
passive electrode entirely disappears from the system. The breakdown voltage 
distribution in it nearly exactly coincides with that characteristic of a solitary 
active electrode (cf. figures 5.15 and 5.5). Formally, this can be accounted for 
by introducing a smaller relative standard deviation for the distribution of the 
breakdown voltage difference a, in expressions (5.10) and (5.11). We shall term 
it a choice standard. The upper limit of the probability integral 
do - dr 
A, = a,(di + &)‘I2 
(5.15) 
defines, as in (5.7), the probability of choosing the stroke point on grounded 
electrodes: 
(5.16) 
Formula (5.16) describes the probability of a lightning striking a body more 
remote from its leader, and expression (5.15) is valid as long as do > dr. 
Otherwise, instead of finding the probability of a lightning stroke at an 
object (P,,), one should find this probability for a lightning rod (Per), then 
defining P,, as 1 - PCr.t If the height of a lightning rod is h,, that of an 
?At A, >> 1 and, hence, P, << 1, one can use the approximate expression P, Y 
(27r-’/*A;’ exp(-Af/2), which is valid and more convenient for estimations. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 253 ===
How lightning chooses the point of stroke 
245 
object is ho and the distance between their top projections on to the earth’s 
surface is A r ,  we have 
(5.17) 
[(Ho - h
~
)
~
 
+ ( r  - Ar)2]1’2 - [(H, - 
o,[(Ho - h0)2 + ( r  - Ar)’ + (Ho - h,)’ + r2I1l2 ’ 
+ r 2 ] 1 / 2  
A, = 
Here, as before, o, is given in relative units, r is the horizontal distance from 
the descending leader tip to the lightning rod axis, and Ho is the attraction 
altitude. With increasing r, A, and the probability integral values become 
smaller. As a consequence, the probability of a lightning stroke at the 
object increases. Therefore, remote lightnings make protection measures 
complicated, especially when their paths are greater deflected from a vertical 
line (section 5.3). 
It would be useless today to try to define the choice standard from 
theoretical considerations. One should also bear in mind that the final 
result of the integration of (5.16) in area for finding the number of lightning 
strokes at an object strongly depends on o,, in contrast to (5.11). The quan- 
tity U, can no longer be taken to be constant, since it must decrease as the 
distance between the rod and object tops is made shorter. It is only the prac- 
tical experience gained with various lightning protection systems which can 
give some hope. The choice of objects to be observed is strictly limited. 
Bulk registrations of stroke locations are made only for power transmission 
lines of high and ultrahigh voltages. Sometimes, registration equipment is 
mounted on unique constructions such as skyscrapers or very high television 
towers [3,16]. In order to derive the values of choice standard from observa- 
tions, it is necessary to calculate the expected number of lightning strokes at 
the object of interest at various oc values, trying to get the best possible 
agreement with the observations. As a first approximation, one may consider 
the lightning attraction by a system of grounded electrodes and the choice of 
the stroke point within the system to be independent events described by the 
probabilities Pa and P,. Then, by analogy with (5.1 l), the expected number 
of breakthroughs to a compact and an extended object (of length L) will be 
described as 
Nb = 27rnl 
P a ( r ) P c ( r ) r d r ,  
Nb = 27rnl 1 Pa(y)PC(y) 
dy. 
(5.18) 
The observational data processing made in [3,17] revealed the dependence 
of the choice standard U, on the distance between the object and the 
lightning-rod tops D. For a relative choice standard, the following formula 
is recommended: 
a, = 7 x 
+ 8 x ~ o - ~ D .  D [m]. 
(5.19) 
Its use in the calculations of (5.16)-(5.18) provides reasonable agreement 
with observations of 0.9-0.999% reliability rods. There are no data on 
rods with a higher reliability. 
x 
sol 
0 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 254 ===
246 
Lightning attraction by objects 
Note the following important circumstance concerning the protective 
action of a lightning rod. Even common sense indicates that the rod height 
must be increased with increasing distance between the object and the rod. 
Let us discuss the opposite situation when a rod is mounted directly on an 
object of small area. How large must be the excess Ah = h, - ho to provide 
a given protection reliability? Essentially, we deal here with the frequency 
of lightning strokes below the lightning rod top. This question is justified 
by observations of such a high construction as the Ostankino Television 
Tower in Moscow (540m). During the 18 years of observations, descending 
lightnings have struck it at various distances below the top, down to 200 m 
(figure 1.10). The rod has been unable to protect itself. This sounds ridicu- 
lous, but this is the reality. 
The results of a numerical integration of the first expression in (5.18), 
using (5.10), (5.11), (5.16), and (5.17) at Ar = 0, are presented in figure 
5.16. The stroke probability @b = Nb/Nl shows the fraction of lightnings, 
attracted by the whole system of grounded electrodes, NI, which have 
missed the lightning rod to strike the object. The calculations were made 
with the attraction standard ra = 0.1 for objects of height ho = 30- 150 m. 
The actual protective effect is achieved only if the height of the lightning 
rod considerably exceeds that of the object. For short constructions with, 
say, ho = 30m, a 99% protection reliability ( a b  = lo-*) requires the light- 
ning rod height excess of Ah x 0.2ho, which is quite feasible technically 
because it is equal only to 6 m  above the object. An object 150m high will 
require a lightning rod 50m higher than the object (Ah x 0.3ho), which 
Figure 5.16. The evaluated probability of a lightning breakthrough to an object of 
height ho, protected by an adjacent lightning rod of height h, > ho. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 255 ===
Why are several lightning rods more effective than one? 
247 
will be more expensive and complicated. In technical applications, the ten- 
dency of the Qb(Ah) curves to saturation is very important. This tendency 
becomes greater with increasing construction height, which means that a 
single rod will be ineffective for a high protection reliability. It is hard to 
protect an object with ho > lOOm with a reliability above 0.999% 
(ab = lop3), an object with ho > 150m above 0.99%, etc. The higher the 
construction, the more complicated is the problem, and this is the reason 
why the Ostankino Tower is unable to protect itself. Nine lightning strokes 
were registered photographically along its length of 200 m from the top [18]. 
The protection efficiency decreases as the distance between the top of a 
high lightning rod and that of an object of similar height increases, reducing 
the mutual effect of counterleaders. Formally, this manifests itself as a larger 
choice standard U,, in accordance with (5.19). Sooner or later, its effect begins 
to dominate over that of lengths in formula (5.17), so the upper limit of the 
probability integral A, stops rising. 
5.8 
Why are several lightning rods more effective than one? 
The answer to this question can be found geometrically. Let us consider two 
lightnings which travel in the same vertical plane going through an object 
and its lightning rod in opposite directions. Suppose both leader tips are at 
an attraction altitude Ho at the same distance from the rod. They have, there- 
fore, an equal chance to be attracted by the object-rod system. The only 
difference is that one leader will approach it on the lightning rod side 
(version 1) and the other on the side of the object to be protected (version 2). 
Assume, for definiteness, that the displacement of the lightnings relative to the 
rod axis is equal to the attraction radius Re, = 3h, (i.e., an average displace- 
ment), Ho = 5h,, and the horizontal distance between the rod and the object 
is AY = h, - ho << h,. From (5.17), the upper limit of the probability integral 
for version 2 is nearly seven times less than for version 1: 
Ar 
7Ar 
u,25fih, 
u,25&hr 
’ 
> 
4
2
 = 
4
1
 = 
Consequently, the Probability integral from (5.16) for version 2 also 
decreases, increasing sharply the probability of striking the object. To 
illustrate, for A r  = 0.2hr and uc = 0.01, the parameter A, takes the values 
of 4 and 0.57, respectively. When the lightning approaches on the lightning 
rod side, the probability of striking the object is, according to (5.16), 
nearly zero, but on the object side it is 0.28. Therefore, a single lightning 
rod can protect an object reliably only from the ‘back’, while its protection 
efficiency from the ‘front’ is much lower. This situation can be rectified if 
the object to be protected is placed half-way between two rods; it is still 
better if there are three rods and so on - this becomes only a matter of 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 256 ===
248 
Lightning attraction by objects 
cost. No rod palisades are known from the protection practice; nevertheless, 
it is tempting to surround the object of interest with a protecting wire, 
especially if it is not very high but occupies a large area. 
As an illustration, let us consider a case simple for the calculations. This 
will allow us to get numerical results and demonstrate the calculation 
procedures. Suppose a circle of radius Ro = lOOm is densely filled by 
constructions of height ho = 10m. All of them must be protected with a 
0.99% reliability, i.e., the probability of a lightning stroke should not exceed 
@bmu = 
Let us now place a circular grounded wire at a distance of 
10m from the external perimeter of the premises. This distance is necessary 
for technical considerations. For example, we must prevent a sparkover 
between the grounded wire and the communications systems and other struc- 
tures, whether it occurs across the earth’s surface or through the air due to high 
current pulses of the lightning discharge. Therefore, the circular grounded wire 
will have a radius R, = 1 10 m. Let us find the wire height h,, whch will provide 
the necessary value of Bb,,,. For the radial symmetry, the probability of the 
lightning breakthrough is found from formulae (5.11) and (5.18) as 
(5.20) 
The probabilities of attraction P,(r) and point choice P,(r) for a lightning, 
whose tip (in the horizontal plane at the attraction altitude Ho) is at the 
instantaneous distance r from the area being protected, are defined by similar 
expressions (5.7) and (5.16). These differ only in the values of the upper limit 
of the probability integral. For the attraction probability, the limit A,, 
according to (5.10), is described by the difference between the minimal 
distances from the leader tip at the attraction altitude Ho to the system of 
grounded electrodes and to the earth, Ad, = d, - de. In the case being 
considered, A ,  is defined by the smaller of the values (at Y < Ro): 
Ad,, = ho. 
Ad,, = [(R, - r)2 + (Ho - hr)2]1’2 - Ho, 
At r > Ro, we have Ad, = Ad,, . In the calculation of the choice probability, 
the upper integral limit is given by formula (5.15). When calculating the 
difference between the minimal distances to the object and the protector, 
A d  = do - d,, one has to keep in mind that we have domi, = H - ho at 
r < R and do,,, 
= [(r - R0)2 + (Ho - hO)2]1’2 at r > Ro. 
The calculation procedure reduces to finding, for every value of r, the 
upper limits A, and Act in the integrals of (5.7) and (5.16) to calculate 
(extract from tables) these integrals, which give P,(Y) and P,(r), and to 
calculate the integrals of (5.20). Practically, it is sufficient to make the 
f The value of the choice standard oc necessary for the calculation of A, is found from formula 
(5.19). taking into account the distance D between the protector top and the point on the object’s 
surface nearest to the lightning with the instantaneous coordinate r; ua 
0.1. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 257 ===
Some technical parameters of lightning protection 
249 
I 
.1 
10” 
1 O‘* 
10’‘ 
Lightning breakthrough probability 0 
Figure 5.17. The object of 10 m height and of 100 m radius is protected by a bounding 
circular wire. In the graph is presented the evaluated wire height h necessary to 
decrease the probability of a lightning breakthrough to the object up to the value 
of CJ shown on the abscissa axis. 
calculations with the step Ar M (0.1 - 0.2)hr and finish them when P,(r) 
drops to 10-6-10-7 with growing r .  If the probability integral is given reason- 
ably (by an empirical formula or by borrowing it from a table, e.g., using a 
spline), the volume of calculations proves so small that they can be made with 
a programmed calculator. With a modern computer, the time necessary for 
numerical computations is only that for the data input. 
The calculations made for the above example are shown in figure 5.17. 
The probability of a lightning breakthrough to the object decreases to the 
given value of lo-* when the protective wire is suspended at a reasonable 
height h, 
34m. Note, for comparison, that a single lightning rod placed 
at the centre of a similar area provides the same protection reliability only 
if its height is h, > 150m. Even if one builds such a rod, the result may 
prove disappointing. Quite often, it is impossible to provide a safe delivery 
to the earth of a high lightning current impulse, when conductors with 
current pass close to structures being protected. Electromagnetic induction, 
sparking capable of setting a fire, etc. may also be dangerous. 
5.9 Some technical parameters of lightning protection 
5.9.1 The protection zone 
It follows from the foregoing that a lightning-rod has a better chance of 
intercepting descending lightnings if it has a greater height above the 
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Lightning attraction by objects 
object and is closer to it. Practically, it is important to identify a certain area 
around a protector, which would be reliably protected. This is the protection 
zone. Any object located within this zone must be considered to be protected 
with a reliability equal to or higher than that used for the calculation of the 
zone boundary. There is no doubt that this idea is technically constructive. 
When the configuration of the protection zone is known, the determination 
of the grounded rod or wire height reduces to a simple calculation or geo- 
metrical construction - this was an important factor in the recent age of 
‘manual’ protection designing. At that time, the general tendency was to 
simplify the zone configuration as much as possible. In Russia, for instance, 
a single lightning rod zone was usually a circular cone, whose vertex 
coincided with the rod top [lo]. When lightning protection engineers realized 
that the height of the rod was to exceed that of the object to be protected 
(section 5.7), the cone vertex was placed on the rod axis under its top [19]. 
The greater the protection reliability required, the more pointed and lower 
was the zone cone. For a grounded wire, the protection zone had a double 
pitch symmetry; when intersected transversally by a plane, it produced an 
isosceles triangle with nearly the same dimensions as those of a vertical 
cross section made through the rod cone half. Lightning protection manuals 
give a set of empirical formulas to design protection zones for simple types of 
lightning protector [2,4]. 
The long-term practice has somewhat screened the principal ambiguity 
of the notion of protection zone. Indeed, having only one parameter - the 
admissible probability of a lightning stroke abmaX 
- one is unable to determine 
exactly the zone boundary. So one has to resort to some additional consid- 
erations of one’s own choice. In particular, there is nothing behind the 
concept of a conic zone except for the consideration of an axial symmetry 
and the desire to make the geometry simple. The value of BbmaX 
corresponds 
to a wide range of zone configurations, so the chosen configuration may 
appear to be far short of optimum. A protection zone is rarely filled up. 
When an object occupies a small fraction of this area, which is frequently 
the case in practice, the lightning rod height proves excessive. For high 
objects and still higher lightning rods, this results in unjustifiably large 
costs, which increase when high reliability is required. When the engineer 
places an object within a protection zone, he has no idea about its actual 
protection. But by decreasing the distance from the zone boundary inward, 
the probability of a lightning stroke may decrease by several orders of 
magnitude. To specify its value, one has to make numerical calculations 
similar to those illustrated in section 5.8. 
Finally, the most important thing is that protection zones can be built 
with sufficient validity only for two types of lightning-rods - rods and 
wires. Even an attempt to combine them causes much difficulty. The same 
is true of multirod protectors, non-parallel two-wire protectors, and sets of 
rods of different height. All of them find application, especially when natural 
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Some technical parameters of lightning protection 
25 1 
‘protectors’ are used, such as neighbouring well-grounded metallic structures 
or high trees. The analysis of protection practice shows that preference is 
often given to easily-calculated designs rather than to effective designs. How- 
ever, the statistical techniques used for the calculation have no limitations on 
the protector type, their number, or the geometry of the objects to be pro- 
tected. Some problems may arise only in finding the shortest distance from 
the lightning leader tip to the lightning-rod and to the object. But they are 
surmountable with the use of modern computers. One should also bear in 
mind that the calculation provides the engineer not only with the break- 
through probability but with the number of expected breakthroughs over 
the time a particular object is in use. The latter parameter is more definite 
and cost-significant. 
5.9.2 The protection angle of a grounded wire 
The concept of protection angle cy is used in designing wire protectors for 
power transmission lines (figure 5.18). The protection angle is considered 
positive when the power wires are suspended farther from the axis than the 
grounded wires, so they are open, to some extent, to descending lightnings. 
The value of Icy1 decreases with the grounded wire suspension height and 
with decreasing horizontal displacement of the power wire relative to the 
grounded one. The protection reliability is lower when the positive angle is 
larger. The angle was introduced as a parameter necessary for the generaliza- 
tion of observations of lightning strokes at transmission lines of various 
designs. It turned out that the angle cy could not serve as an unambiguous 
characteristic of the protective quality of a grounded wire. A transmission 
line must also be described in terms of the grounded wire height above 
power line wires, Ah, and of the grounded wire height above the earth, h,. 
Figure 5.18. Positive and negative protection angles. A: grounding wire; B power 
wire; C: insulator string. 
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Lightning attraction by objects 
This determines the distance between the grounded wire and the power wire 
at fixed a, which defines, through the choice standard a,, the degree of the 
system connectivity. Of lines with an identical protection angle, the best pro- 
tected line is the one with the largest value of Ah and the lowest value of h,. 
Empirical formulas, which relate the lightning breakthrough probability 
to wires with a and h,, have found wide practical application. Their accuracy, 
however, is not very high because they do not include Ah. For example, there 
are expressions identical in composition [21,22]: 
4, 
lg@b =-- 
3.95, h, [m], a [degree]. 
(5.21) 
ah;'= 
90 
75 
lg@b =--- 
They give the probability of a lightning stroke with a 300% error related to a 
value supported by practical observations. These formulae should be treated 
with caution when the line supports are higher than 50m at small positive 
and, especially, at negative protection angles. This is because most main- 
tenance data refer to lines of up to 40m high with positive protection 
angles of 20-30". Besides, very few of the data used for deriving empirical 
formulas represent direct measurements. Usually, the data are derived 
from registrations of storm cut-offs minus the calculated return sparkovers 
(section 1.6.1). The latter calculations often give a large error. Still, expres- 
sions (5.21) demonstrate that negative protection angles are quite attractive. 
The action of protection wires placed farther from the tower axis than line 
wires (cy < 0) is similar to that of a closed grounded wire surrounding a 
region being protected (section 5.8). This type of protector could provide 
an exceptionally low probability of a lightning stoke at line wires, but the 
implementation of negative angle protection requires larger towers and, 
hence, a higher cost. This approach is, for this reason, unpopular. 
5.1 0 Protection efficiency versus the object function 
No doubt, there is a close relationship between the protector efficiency and a 
particular function (purpose) of a protected construction, especially when it 
is under high potential relative to the earth (e.g., ultrahigh voltage wires) or 
ejects a highly heated gas into the atmosphere. By raising the object potential 
to values comparable with the absolute potential of a descending leader, one 
can either increase the field at its tip, making the leader move towards the 
object, or lower it, suppressing the lightning attraction. It is a matter of 
the quantitative effect produced rather than its principal feasibility. High 
object potential U,, may affect both the process of lightning attraction 
and the choice of the stroke point. The latter is more sensitive to external 
effects owing to the positive feedback in the connected system. Expressions 
(5.10) and (5.15) are suitable for estimations. They include the standards 
of attraction, ca, and of choice, ac. It is easier to control the process when 
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Protection efjciency versus the object function 
253 
their values are lower. For objects of regular height (-30m), 
we have 
ua 
lo-' and uc x 
This enables us to focus on the choice only. The 
effects of the high potential action of the object, U&, will noticeable at 
U& 
u,U,, where U, is the tip potential at the moment the leader has 
descended to the attraction altitude. An 'average' 
lightning has 
U, = 50MV. Therefore, in order to get an effect on the process of choice, 
one must apply Uob M 500 kV to the object (or to the protecting wire). In 
order to affect the attraction process, the applied voltage must be -5 MV. 
The latter value is, certainly, not feasible for the present power industry, 
but available operation voltage of an ultrahigh voltage (UHV) line is high 
enough to affect the lightning preference to a protecting wire or to a line 
wire [22,23]. 
Most UHV lines operate at alternative voltage of frequency f = 50 Hz 
(60Hz in the USA). Over the time Ho/VL x 
s along the flight path 
Ho, during which the lightning chooses a point to strike, the wire potential 
changes but little, and its values U&([) = Ufmax 
sinwt (U = 2759 can be 
taken to be equally probable. By the initial moment of attraction, uob(t), 
may have the same or opposite sign relative to the lightning. If the sign is 
the same, the development of a counterleader from the wire will be delayed, 
so the probability of the lightning striking the wire will be reduced. In the 
other situation, the effect will be opposite. To get a total result over a 
long-term observation of the line operation (or a short-term observation of 
a very long line), one should average the operating voltage effects over an 
oscillation period. For this, expression (5.15) for the parameter A, must be 
extended to the case in question. Expression (5.15) was based on the differ- 
ence in the average fields along the lengths from the leader tip at the attrac- 
tion altitude to the protector and to the object. Now, this difference can be 
calculated with the potential U& to get, instead of (5.15), 
where U, is the descending leader tip potential at altitude Ho. 
The qualitative result of the calculations to be given below is predictable. 
We are interested in the effect of alternative voltage on the preferential choice 
of the stroke point between a protection wire and a power wire, since the 
operating line voltage is too low to affect appreciably the lightning attraction. 
In the half period when Uob(t) and U, have the same sign (suppose it is a 
negative descending leader and negative voltage), the lightning is 'repelled' 
by the power wire; in the positive half period, it is attracted by it. Owing 
to the protecting wire, the probability of a lightning stroke at the wire in 
the off-voltage mode is low, 10-2-10-3. Therefore, the favourable effect of 
all negative half periods is small. Even if no lightning strikes the power wire 
during this time, the number of strokes at it will, for a long time, be reduced 
only by half relative to the no-load mode, because negative half periods take 
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254 
Lightning attraction by objects 
Figure 5.19. Effect of the AC transmission line operation voltage on the lightning 
breakthrough probability. The lower and upper curves correspond to probabilities 
of 
and lo-', respectively, without voltage. 
only half of the on-voltage time. The unfavourable effect of positive half 
periods may be much stronger. In principle, the potential difference U,, > 0 
and U, < 0 at a high alternative voltage amplitude may produce such a 
strong 'attracting' field that all lightnings going to the power line will strike 
its wires. The probability of a strike at the line wire during the positive half 
periods may rise by 2-3 orders of magnitude (even as much as unity) against 
its two-fold reduction during the negative half periods. As a result, the 
stroke probability averaged over a long time for the line wire grows. The 
operating voltage effect on power lines reduces the reliability of lightning 
protection. 
The numerical calculations of this effect are illustrated in figure 5.19. 
The probability of lightning breakthrough to AC lines increases by an 
order relative to the probability 
x lop2 for the off-voltage mode at 
y = Uobmax/ccUt x 3.75; at ab x lop3, this effect is produced at 1.5 times 
lower voltage. For the typical size of modern power line towers with 
oc x 0.008, the stroke probability for the power wire at U, x 30MV rises 
from lop3 to lop2 at phase voltage amplitude Uobmax x 625kV. Such are 
the line voltages (750 kV) in some countries. Only the next generation of 
power lines with 11 50 kV can be expected to produce as strong effect on light- 
ning at U, x 50MV. An experimental line of this kind has been in use in 
Russia for a short time. 
Direct current line has a more pronounced effect on lightning. Lightning 
separation is possible in DC lines: a positive line wire more strongly attracts 
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Lightning attraction by aircraft 
255 
negative lightnings and a negative wire more strongly attracts positive ones. 
Since the frequency of positive descending lightnings is an order of magni- 
tude smaller, a positive UHV DC line wire will attract a larger fraction of 
strokes. This effect may become well pronounced at the wire potential of 
f500 kV and higher. 
The treatment of a hot air flow from an object to be protected is 
generally similar to the above analysis. The density and electrical strength 
of hot air are lower, and the strength is proportional to the density in the 
first approximation [25-261. Formally, this is equivalent to the reduction 
of distance do from the lightning to the object in expression (5.15), as if the 
object height were increased. As a result, the lightning protector has a 
lower efficiency. Consider, as an illustration, a chimney lOOm high with a 
10m lightning rod fixed on its top. With the practically zero horizontal 
distance between the rod and the object and in the absence of hot smoke 
gases, the rod will intercept about 90% of all lightnings attracted by the 
chimney (figure 5.16). But if the chimney ejects a hot gas flow with the 
temperature of 100°C along the length of 30m, the probability that 
the lightning will miss the rod to strike the chimney will rise from 10 to 
50%. Actually, the lightning rod becomes ineffective. The question is whether 
it is worth constructing this purely decorative device on the chimney top. 
5.1 1 Lightning attraction by aircraft 
Protection of aircraft and spacecraft has always been a complex and demand- 
ing problem - poor protection may have serious repercussions. It has been 
mentioned that an aircraft can be damaged by an ascending lightning starting 
from its surface or by an attracted descending discharge in the atmosphere, as 
happens with a terrestrial construction. Naturally, the concept of attraction 
refers only to descending lightnings. There are no observational data on the 
interaction between aircraft and descending lightnings, and one has to resort 
again to laboratory experiments. Figure 5.20 gives a set of static photographs 
taken from the screen of an electron optical converter. Of many pictures, we 
have selected the most typical ones. The electronic shutter was shut at differ- 
ent moments of time, so the result is not exactly a movie film but something 
close to it. One can see that a vertical rod insulated from the earth has 
attracted one of the leader branches together with its streamer zone, 
having first excited a streamer flash and then a counterleader. Its contact 
with the descending leader has produced a short luminosity enhancement 
of their, now common, channel, like a step of the negative leader with its 
miniature return stroke (sections 2.7 and 4.6). As a result, the channel and 
the rod have become the extension of a high voltage electrode. The leader 
has started off towards the earth from the lower end of the rod which now 
seems to be part of the leader channel. 
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256 
Lightning attraction by objects 
Figure 5.20. Attraction of the spark leader by the isolated metal rod suspended in the 
gap middle. 
It appears that the attraction of a descending lightning by an insulated 
conductor, as well as by a grounded one, is stimulated by the excitation of a 
counterleader. The similarity in their mechanisms accounts for the similarity 
in the basic parameters of attraction. Below, we present some laboratory 
measurements of equivalent radii Re, for spark attraction by a vertical metal- 
lic rod of length 1 = 0.5m, suspended at height H above a grounded plane. 
The spark was produced by a positive voltage pulse with a loops front in 
a rod-plane gap of 3 m. The front provides a more or less reliable field rise 
time for a real object during the development of a descending lightning 
leader. The measured values of Req(H) are normalized to the value of 
Re,(0) for a rod that has descended to a plane to become grounded: 
HI1 
0 
1 
2.8 3.4 
R,,(H)/R,,(O) 
1.0 0.9 0.9 0.8 
The response to the conductor rise above the earth is fairly weak. A 10-20% 
decrease in Re, seems to be regular, although it lies within the experimental 
error range. To extrapolate this result to lightning, one should assume that 
the number of descending lightning strokes for aircraft with the maximum 
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Lightning attraction by aircraft 
257 
size I is not larger than that for a grounded object of the height h = 1. This 
limit for the number of strokes does not follow only from the experimental 
fact of a certain decrease in Re, with H .  Of greater importance are the 
possible variations in the aircraft position relative to the external field 
vector, Eo, during the flight. The field enhancement at the ends of its fuselage 
of length I is defined by field projection on to the aircraft axis, rather than by 
the value of EO. High terrestrial constructions are always aligned with the 
field since it is vertical at the earth. 
Let us now estimate the possible number of descending lightning strokes 
at an aircraft of length I = 70m, using the concept of attraction radius 
Re, M 31. We shall have Nd x n17rR&kh, where n1 is an average annual 
frequency of lightning strokes at the earth and kh is the ratio of the total 
flight hours per year to the total number of hours in a year. For kh = f 
and nl x 3 kmP2 per year, we get Nd M 0.1 per year. This is at least an 
order of magnitude less than what follows from official statistics. One 
should not think that the discrepancy is due to the neglect of intercloud 
discharges, whose number is 2-3 times larger than that of lightnings striking 
the earth. In order to be attacked by intercloud lightnings, aircraft must 
penetrate through the storm front, but this is absolutely forbidden and 
may happen only as an accident. Rather, the result was overestimated 
because any pilot tries to stay as far away from a storm as possible. 
Therefore, descending lightnings are responsible for fewer than 10% of 
strokes at aircraft. The other 90% or more are due to ascending lightnings 
excited by aircraft and spacecraft themselves (section 4.2). However, the 
interest in descending lightnings remains active because of the poor predict- 
ability of the stroke points on the aircraft surface. A similar situation but for 
high terrestrial constructions was discussed in section 5.7. The probability of 
a lightning striking much below the top is rather high. This situation can be 
readily simulated in the laboratory for a long positive spark excited by a 
voltage pulse with a smooth front, tf M loops and higher. The photograph 
in figure 5.21 illustrates a spark stroke almost at the rod centre, together 
with the integral distribution of the stroke points along its length. The 
wide, if not random, spread of stroke points over the aircraft surface creates 
additional problems. The aircraft has many vulnerable areas. In addition to 
the cockpit and fuel tanks, these are hundreds of antennas and external 
detectors providing a safe flight. It would be desirable to hide them from 
descending lightnings but the chances for this are quite limited. One con- 
solation is that most lightnings affecting aircraft are of the ascending type 
starting mostly from the ends of the fuselage and wings, where the external 
electric field is greatly enhanced. 
The excitation of ascending lightnings by aircraft was considered in 
section 4.2. Formula (4.11) allows estimation of the hazardous field Eo for 
an aircraft of length I = 2d. The field Eo decreases with growing d, some 
slower than d-315. Note that the parameter 2d is not necessarily the fuselage 
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258 
Lightning attraction by objects 
Figure 5.21. The stroke probability at various points of an isolated rod for two 
voltage front durations. The photograph shows how the spark has struck at the 
rod centre. 
length; this may be the wing length if it is larger. In general, the experience 
indicates a direct relationship between the aircraft size and the frequency 
of lightning strokes. There are exceptions, of course. The statistics of flight 
accidents shows that aircraft of identical size may differ considerably in the 
capacity to excite lightnings. In one design, the engines are mounted on 
the wing pylons, and the ejected hot gas jet passes near the metallic fuselage, 
where the low fields cannot excite a leader. In another design characteristic of 
rockets also, the engine nozzle is placed in the tail, so that the hot jet serves as 
the fuselage extension. This is a perfect site for a counterleader to be excited 
since the leader development needs a lower field in a low density gas. 
In the estimation, we shall assume the jet length to be half the fuselage 
length, lj = d, and its average temperature to be twice as high as the ambient 
air temperature. Suppose that the jet radius is large enough for the streamer 
zone to be entirely within it and that the leader develops in a gas of relative 
density S = 0.5. When the gas density becomes lower due to the heating, the 
field providing the streamer propagation decreases at a rate 6 [25,26]. The 
rate of the electric strength decrease in long gaps is approximately the same 
for mountainous regions, although the density variation range in these 
experiments was narrower, 6 M 0.7 [25]. We shall assume from these data 
that a leader developing within a hot jet requires a potential drop 5-' times 
smaller than that given by formula (2.49), i.e., AU = 36A3/5(3hd/2)2/5 
(here, the leader length L has been replaced by the jet length d). The total 
length of a conductor consisting of a fuselage of 2d long and a leader of 
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Are attraction processes controllable? 
259 
length d is equal to 3d. Hence, we have AU = 3Eod/2, and the estimated 
external field providing the leader formation in the jet is 
The field is found to be Eo(6) x 165V/cm at the values of A = ( 2 7 ~ ~ o a ) - ~ ' ~ ,  
a = 1.5 x 103V-'/* cm/s, b = 300VA/cm, used in chapters 2 and 4, and 
d = 35m. At this field, there will be a breakdown of the jet, increasing the 
aircraft size by the jet length value. This will create favourable conditions 
for an ascending lightning to develop from the fuselage in a low field. To 
estimate the field, d should be replaced by Le, = (2d + lj)/2 in formula 
(4.11); for the present example, it should be 1.5d. The 25% decrease of the 
threshold field which will follow may greatly change the total number of 
lightning strokes at the aircraft. 
5.12 
Are attraction processes controllable? 
We gave an affirmative answer to this question, when discussing the effects of 
operating voltage in ultrahigh voltage lines and hot gas flows. The further 
consideration of this problem should be concerned with quantitative aspects 
and particular methods of lightning control. Lightning control has two aims: 
to raise the reliability of lightning protection of nearby objects and to expand 
the area being protected by using conventional techniques. These may only 
seem to be two sides of the same effect. For example, increasing the lightning 
rod height increases both the protection reliability for a particular object and 
the maximum radius of the protected area. This, in principle, is the case, but 
quantitatively the two results differ considerably. 
Turn to the estimations above. It follows from the calculations in figure 
5.15 that the increase in the lightning rod height h, by only 4 m (from 36 to 
40 m) reduces the probability of a stroke at an adjacent 30 m object from lo-* 
to 
or by an order of magnitude. The effect is significant. As for the 
expansion of the protected area on the earth, its radius Ar does not grow 
faster than h,. This can be demonstrated by putting ho = 0, r = 0, and 
HO = 5h, in formula (5.17). Then we shall have A r  N h, for a given proba- 
bility of choosing a stroke point, i.e., at fixed A, and D~ = const. In actual 
reality, the standard oC grows with h,, due to which A r  rises still slower. In 
our example, A r  increases by less than lo%, and this insignificant effect is 
of no interest to us. 
Lightning control eventually reduces to a change either in the electrical 
strength of the discharge gaps between the descending leader tip and the 
protector and the earth or in the gap voltage. For this reason, the particular 
conclusion that follows from the above example can be extended to any 
control measures - their effectiveness falls with distance between the object 
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Lightning attraction by objects 
top and the lightning-rod, since the mutual effects of the components in a 
multi-electrode system become weaker. Formally, this weaker effect mani- 
fests itself in increasing standard cc. It appears that the lightning control is 
easier for objects of low height and area, when conventional protectors are 
sufficiently effective. It is much more difficult to deviate a lightning from 
an object without mounting a metallic rod on top of it. The application of 
destructive technologies to storm clouds and their charge neutralization 
are not discussed in this book, because this is a special problem having no 
direct relation to lightning processes. 
The physics of the effect of a voltage pulse rise on a descending lightning 
leader is clearer than that of other effects. The effect can be expected to be 
favourable when the potential applied to the lightning rod is of opposite 
sign to that of the lightning, or the potential applied to the object is of the 
same sign. In the former case, the conditions for a counterleader to start 
from the rod are quite favourable. To initiate a preventive start of a counter- 
leader from a lightning rod is to deviate the stroke point from the object. But 
in order to produce a noticeable effect, the counterleader must have a channel 
length comparable with the length difference between the object and the rod, 
or between their tops (the latter quantities are comparable). Only then does 
the effective rod height really grow and the charge space of the counterleader 
considerably limits the field at the object top. Therefore, one deals with 
channels of metre lengths, sometimes of tens or even hundreds of metres, 
especially if one takes into account the multi-fold increase in the radius of 
the area to be protected. This is a fairly complicated task. 
A short-term ‘elongation’ of the rod by exciting a plasma channel from its 
top is very similar to the counterleader behaviour. A laser spark or a short- 
term long plasma jet would be sufficient for ths. Laboratory studies have 
shown that a man-made plasma conductor affects a long spark path as a 
metallic conductor. The problem is the technological complexity and consider- 
able cost of the project rather than the principal possibility of control. 
Imagine an ideal pulse generator, whose effectiveness is so high that it 
blocks a lightning breakthrough to the object with 100% probability. The 
protection reliability will then be determined by the reliability of the genera- 
tor itself, primarily by its synchronizing unit. It is a difficult task to design a 
reliable synchronizing unit capable of responding to a nearby descending 
lightning leader. A leader always chooses a complicated, poorly predictable 
path and has many branches. It is necessary either to distinguish a branch 
from the main channel or to trigger the control unit repeatedly. The latter 
is undesirable not only because this is resource-consuming. A control pulse 
can stimulate a branch to become the main channel, which is the first to 
reach the grounded electrode, producing a powerful return stroke pulse. 
The close vicinity of a strong current may be as hazardous to the object 
being protected as a direct stroke. Finally, we should not discard multi- 
component lightnings - 50% of subsequent components do not follow the 
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Are attraction processes controllable? 
26 1 
channel of the first component [27]. So it is necessary to design a control unit 
capable of generating a series of pulses with millisecond intervals. Such a pro- 
ject would be very costly. 
High costs have been the main reason for the decreasing interest in 
lightning control among specialists. They think of using nonmetallic rods 
or other unconventional measures only in exceptional situations when the 
common approaches are incompatible with the technological functions of 
the object being protected. Designers have suggested some exotic ways of 
lightning protection. Specialized firms advertise lightning rods with radio- 
active, piezoelectric and other wonder tops. The performance of radioactive 
sources has been tested in a laboratory, and no noticeable effect has been 
registered even on the leader start, let alone its propagation along the dis- 
charge gap. This should have been expected, because a leader arises from a 
pulse corona flash resulting from a long application of an electric field (as 
happens during a storm). Every pulse flash represents a streamer branch 
with a channel electron density of 10'2-10'4cm-3 [28]. A radioactive top 
can hardly add anything to this density, unless its power is so high that it 
kills everything alive around it. 
It appears that leader suppression may be more promising than its 
excitation. Laboratory experiments have long been known [29], in which 
an ultracorona was successfully used to suppress the leader start. The 
corona arises as a thin uniform cover on the anode or the cathode made of 
a thin wire (-0.1-1 mm). A slow voltage rise does not change the corona 
structure or the ionization region thickness. The electric field strength on 
the electrode is stabilized by the space charge of ions drifting slowly on the 
corona periphery. The field stabilization prevents the formation of an ioni- 
zation wave, or a streamer flash, which would otherwise produce a leader. 
With no consequences, the average field in a gap of several dozen centimetres 
long could be raised to 20-22 kV/cm, whereas 5 kV/cm was commonly 
sufficient to produce a breakdown in the absence of an ultracorona. 
It would be tempting to extend the laboratory effect to lightning protection 
practice to suppress the counterleader start from the object being protected. An 
obstacle here is the rate of external field variation at the top of a grounded 
electrode of height h, rather than the much greater gap length. The electrode 
possesses zero potential, U = 0. The potential of the external field EO at the 
top is U, = Eoh, 
so that the air at the electrode top is affected by the potential 
difference equal to U - U, = - U,. The linear charge of a leader descending 
directly on to the object creates field AEo at the earth, given by formula 
(3.5). As the leader approaches the earth, potential U, rises at the rate 
(5.23) 
where z is the altitude of the descending leader tip and VL = -dz/dt is its 
velocity. 
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262 
Lightning attraction by objects 
Let us find the maximum rate of the field rise at which the ultracorona 
can still survive. Assume, for simplicity, that a corona (positive, for definite- 
ness) arises at a sphere of radius yo, attached to the electrode top. TO 
prevent the corona transformation to an ionization wave capable of 
initiating a streamer flash and then a leader, the field on the sphere 
should not rise in time with AEo. The surface with maximum field should 
not detach from the sphere to move into the gap interior. In an ultracorona, 
the field on the sphere is stabilized by space charge on the level of E, 
depending on radius yo. The sphere concentrates a constant charge 
Q, = 4neor&. A short time At after the corona ignition, the voltage 
increases by the value A U  = A,At, which is supposed to increase the posi- 
tive sphere charge by AQ1 = CAU = 4mOrOAuAt. 
To avoid this, the sphere 
charge AQ1 must be compensated. The compensation occurs owing to the 
gas ionization in the thin surface layer. Positive ions transport the charge 
AQ for the distance Ar = plE,At (where p, is the ion mobility), so that 
the negative charge induced in the sphere AQ, = -AQro/(ro + Ar) is able 
to neutralize AQ1. The charge actually induced in the sphere is transported 
into it by electrons produced in the near-surface layer, whose number is 
excessively large since lAQll = AQ, < AQ. ‘Excessive’ electrons leave for 
the external circuit and then to the ‘opposite’ electrode - the earth. The 
field on the radius r = yo + Ar now becomes equal to E(r) = (Q, + AQ)/ 
[4mO(r0 + AY)^] and should not exceed E,. To the small value of about 
Ar/ro, this requirement is met at A, d 2p,E; M 3.6 kV/p (E, M 30 kV/cm, 
p1 x 2 cm2/V SI. 
We have analysed the other extrema1 situation when the corona exists so 
long that charge Q >> Q, is incorporated into space and the ion cloud radius 
becomes r1 >> YO. A well-developed corona can exist at the sphere for a long 
time if the voltage U. does not grow in time faster than U, = Aut. The 
maximum admissible growth rate A, coincides, in order of magnitude, 
with the above estimate but is slightly lower. At a fast voltage growth, 
say, U M t” with n > 1, there necessarily comes the moment when the 
ion cloud field becomes higher than E,, stimulating the transition to a 
streamer flash. For the typical values of h = 50m, rL x 5 x 
Cjm, and 
VL x 3 x lo5 m/s, the voltage growth rate reaches the estimated critical 
value when the leader descends to the altitude z x 200m, at which the 
attraction process begins. A little later, A, N z-* becomes even more critical, 
and the ultracorona dies giving way to a counterleader. 
To conclude, lightning can be controlled but this task is costly and very 
complicated technologically. So it would be unreasonable to discard 
conventional protection technologies where they can solve the problem suc- 
cessfully. One should not expect miracles in lightning protection. If particular 
circumstances make one turn to unconventional measures, one must be ready 
to create complex devices, whose protection reliability will be determined by 
their operation, rather than by the interaction with a lightning. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 271 ===
If the lightning misses the object 
263 
5.13 
This is likely to happen more often than direct strokes at an object. Some- 
times, the object attracts a lightning branch which could hit the object if it 
had enough time before the return stroke develops from the main channel. 
Such a situation is illustrated in figure 5.22. The counterleader, which has 
started from the television tower top, has no time to transform to an ascend- 
ing lightning or intercept the descending leader, because the latter has struck 
a metallic tower below the tower top. As a result, the counterleader remains 
uncompleted. The counterleader channel has, however, become several 
dozens of metres longer. This is now a mature channel, whose temperature 
is at least 5000-6000K. If it had touched a hot gas jet, it would inevitably 
ignite the gas. Practically a leader of any length is suitable for ignition of 
inflammable exhausts into the atmosphere. To excite and develop a leader 
in air under normal conditions, a voltage of 300-400 kV would be sufficient. 
Such a potential difference AU = Eoh can be produced in objects of height 
h > 30m even in the absence of lightning because this would require a 
storm cloud field of Eo M lOOV/cm. If the object is lower, uncompleted 
counter-leaders can be excited even by remote lightnings. From formula 
(3.7), a descending leader that has started at an altitude of H = 3 km and 
has touched the earth creates a field Eo = lOOV/cm at a distance R = 1 km 
from the stroke point if it carries the linear charge T~ M 8 x lop4 Cjm. This 
charge is characteristic of a descending leader with average parameters. 
This is one of the long-range mechanisms of lightning, which should be 
If the lightning misses the object 
Figure 5.22. The long incomplete counterleader (2) started from the top of the 
Ostankino Tower while the descending lightning struck lower than the top (1). 
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=== PAGE 272 ===
264 
Lightning attraction by objects 
taken into account when treating possible emergencies for objects containing 
large amounts of inflammable fuels. 
References 
[l] Uman M A 1987 The Lightning Discharge (New York: Academic Press) p 377 
[2] Operating Instruction for Lightning Protection of Buildings and Works RD 
31.21.122-87 1989 (Moscow: Energoatomizdat) p 56 (in Russian) 
[3] Bazelyan E M, Gorin B N and Levitov V I 1978 Physical and Engineering Funda- 
mentals of Lightning Protection (Leningrad: Gidrometeoizdat) p 223 (in Russian) 
141 Golde R H 1967 J. Franklin Inst. 286 6 451 
[5] Linck H and Sargent M 1974 CIGRE, Sec. N 33/09 (Paris) 11 
[6] Wagner C F 1963 AZZZ Trans. 83 (Pt 3) 606 
[7] Wagner C F 1967 J. Franklin Inst. 283 (Pt 3) 558 
[8] Darveniza M. Popolansky F and Whitehead E R 1975 Electra 41 39 
[9] Bazelyan E M, Levitov V I and Pulavskya I G 1974 Elektrichestvo 5 44 
[lo] Stekolnikov I S 1943 Lightning Physics and Lightning Protection (Moscow, 
[ll] Akopyan A A 1940 Res. All-Union. Electr. Inst (Moscow) 36 94 
[12] Bazelyan E M, Sadychova E A and Filippova E B 1968 Elektrichesrvo 1 30 
[13] Bazelyan E M and Sadichova E A 1970 Elektrichesrvo 10 63 
[14] Aleksandrov G N, Bazelyan E M, Ivanov V L et a1 1973 Elektrichesrvo 3 63 
[I51 Bazelyan E M. Burmistrov M V, Volkova 0 V and Levitov V I 1973 Elektri- 
[16] Cann G 1944 Trans. AIEE 63 1157 
[17] Gorin B N and Berlina N S 1972 Elektrichesrvo 6 36 
[18] Gorin B N, Levitob V I and Shkilev A V 1977 Elektrichesrvo 8 19 
[19] Bazelyan E M 1967 Elektrichesrvo 7 64 
[20] International Standard Protection Structures against Lightning 1990 IEC 1021 
[21] Burgsdorf V V 1969 Elektrichesrvo 8 31 
[22] Kostenko M V, Polovoy I F and Rosenfeld A N 1961 Elektrichesrvo 4 20 
[23] Bazelyan E M 1981 Elektrichesrvo 5 24 
[24] Larionov V P, Kolechitsky E S and Shulgin V N 1981 Elektrichesrvo 5 19 
[25] Bazelyan E N, Valamat-Zade T G and Shkilev A V 1975 Zzvestiya. Akad. Nauk 
[26] Aleksandrov N L and Bazelyan E M 1996 J. Phys. D: Appl. Phys. 29 2873 
[27] Rakov V A, Uman M A and Thottappillil R 1994 J. Franklin Inst. 99 10745 
[28] Bazelyan E M and Raizer Yu P 1997 Spark Discharge (Boca Raton: CRC Press) 
[29] Uhlig C A 1956 Proc. High Voltage Symp. Nut. Res. Council of Canada 
Leningrad: Izdatelstvo Akademii Nauk SSSR) p 229 (in Russian). 
chesrvo 7 72 
P 48 
SSSR, Energetika i transport 6 149 
p 294 
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=== PAGE 273 ===
Chapter 6 
Dangerous lightning effects on 
modern structures 
This chapter is concerned with the mechanisms of hazardous lightning effects 
on various objects in the atmosphere, having no contact with the earth, on 
terrestrial constructions and underground communications lines. The dis- 
cussion will be restricted to those effects which are, in this way or other, 
produced by the electrical and magnetic fields of lightning. No doubt, a 
hot lightning channel can ignite inflammable material but their direct contact 
is a rare phenomenon, whereas a remote excitation of sparks in such material 
due to electrostatic or magnetic induction is a regular thing. Lightning can 
destroy constructions by a purely mechanical action but this does not 
happen often. The burn-offs and holes at the site of contact of a hot lightning 
channel with metal are hazardous only to thin (one millimetre thick) metallic 
coatings. On the other hand, the range of electromagnetic effects is very wide. 
They can damage both microelectronic devices and ultrahigh voltage lines. 
The test maintenance of a 1150kV transmission line in Russia has shown 
that it is not resistant to powerful lightning discharges. Most of the material 
presented in this chapter concerns the physical mechanisms of electrical, 
magnetic and current effects of lightning. We shall discuss simple and clear 
qualitative models illustrating the physics of these processes. We believe 
that this is the key issue to lightning protection theory. The process of 
equation solution, so important two decades ago, is not so essential today. 
If a physical model describes the reality adequately and the respective 
equations are available, modern computers are able to overcome almost 
any mathematical complexity. 
When a lightning strikes a grounded metallic construction, a high return 
stroke current I, passes through it. Because of an imperfect grounding 
having a resistance R,, the construction potential rises by the value 
U = IMR,, for example, by 1 MV at I, = 50 kA and R - 20R. This is 
one of the reasons for the overvoltage due to a direct lightning stroke. 
Another reason is the emf of magnetic induction (the intrinsic induction 
g.- 
265 
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266 
Dangerous lightning effects on modern structures 
due to an abrupt current change in the construction and the mutual induction 
produced by the current wave running through the lightning channel). But 
lightning overvoltages may result not only from a direct stroke but from 
remote lightning discharges as well. Their effect is associated with electro- 
static and electromagnetic inductions. In the former case, an overvoltage 
results from the time variation of the electric field strength at the object, 
created by the lightning channel charges during the leader and return 
stroke stages (sometimes, by the slowly changing charge of the storm 
cloud). Another reason for a remote excitation of overvoltage is the varying 
magnetic field of the rapidly changing lightning current. Overvoltages 
became a very serious hazard at the beginning of the twentieth century 
when the first power transmission lines were built, and the engineer still 
associates an overvoltage with a powerful effect of tens and hundreds of kilo- 
volts. This is true of transmission lines of high and ultrahigh voltages (UHV 
lines). However, an overvoltage as small as several hundreds or dozens of 
volts may become hazardous to electric circuits with a low operating voltage. 
Especially vulnerable in this respect are the circuits of microelectronic 
devices. 
Historically, the theory of overvoltages has developed with reference to 
power transmission lines. Naturally, the mechanisms of ultrahigh voltage 
excitation were the first to attract the researchers’ attention. So this theory 
is now very detailed [l-41 and the numerical procedures suggested are 
capable of solving engineering problems with a desired accuracy. We shall 
not describe these approaches here but rather focus on the physical aspects 
of the overvoltage problem, because in many practical applications they 
are not as self-evident as in a lightning stroke at a power line. 
The calculation of overvoltage includes the solution of two equally 
important problems. One is to find the electromagnetic field of a lightning 
discharge at the site where the object to be protected is located. These calcu- 
lations may prove very cumbersome and time-consuming, especially when 
one tries to take into consideration such parameters as the real path and 
length of a leader channel, the non-uniform charge distribution along the 
channel length, and the lightning current spread over the metallic parts of 
a particular object and underground service lines. The physical aspects of 
this problem, however, are quite clear and the numerical methods are well 
known. The other problem is to determine the response of an object and 
its electrical circuits to the electromagnetic field of lightning. The physical 
aspects of this problem are much more diverse, and the basic mechanisms 
of overvoltage excitation are not always obvious. So the latter are the subject 
of special interest in this chapter. 
An induced overvoltage is normally smaller than an overvoltage pro- 
duced by a direct stroke, especially by remote strokes, but it affects the 
object more frequently. When one calculates the frequency of emergencies 
for a high-voltage circuit with an insulation designed for hundreds of 
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=== PAGE 275 ===
Induced overvoltage 
261 
kilovolts, one usually deals with direct strokes, because induced overvoltages 
cannot damage the insulation. Objects with metallic shells which can screen 
well the internal electric circuits (including low-voltage ones) are designed in 
a similar way. However, unscreened low-voltage circuits suffer equally from 
overvoltages due to direct strokes and from induced overvoltages. Since the 
latter are more numerous, they should not be discarded when choosing the 
protective measures. 
6.1 Induced overvoltage 
6.1.1 
The atmospheric electric field varies in time during a storm. The slowest 
changes, lasting for several seconds or tens of seconds, are due to the 
accumulated charges of the storm cloud cells and their transport by the 
wind. Field variations associated with the leader propagation last for several 
milliseconds. Changes of microsecond duration arise from the charge re- 
distribution during the return stroke. In any field variation, the electrostatic 
potential of a perfectly grounded object would remain equal to zero. In 
reality, however, the grounding resistance R, is always finite. If the change 
in the charge induced on the object surface creates current i, = dqi/dt 
through the grounding rod, the object acquires potential U = -i,R, 
relative 
to the earth. 
A grounded body of capacitance C possesses a potential difference 
AU = U - U, relative to the adjacent space (here, U, is the average potential 
of the external field Eo at the object’s site). The charge induced on the body is 
q, = CAU; hence, current i, is defined by the equations 
‘Electrostatic’ effects of cloud and lightning charges 
-+A--- 
d i, 
A, . 
d U, 
A = -  
dt 
R,C- 
R, 
’ dt ‘ 
2, = - exp(-t’RgC) 1: A,(t‘) exp(t‘/R,C) dt’ 
R, 
where we assume ig(0) = 0. In a simple case with A, = const, we have 
i, = -A,C[l - exp(-t/R,C)], 
U = A,R,C[l - exp(-t/R,C)]. 
(6.2) 
For the estimation, we put C = 100 pF, corresponding to a sphere of 1 m 
radius, and set the overvoltage amplitude below 1 kV. During a storm 
without lightning discharges (the field variation A, N lo4 Vjs), the desired 
grounding resistance should be R, < 1OOOMR. But in the presence of a 
close descending lightning leader with the field variation A ,  N lo9 Vjs, the 
grounding resistance must be reduced to 10 kR. With the account of the 
return stroke at A, N 10” Vjs, this value must be decreased further to 
1000. Therefore, a good grounding of an object seems to be an effective 
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268 
Dangerous lightniTzg effects on modern structures 
tool for its protection against overvoltages excited by electrostatic induction. 
No doubt, faster variations in the external field impose more stringent 
requirements on the grounding rod. Resistances exceeding 1000 MO are 
hardly realistic because of the leakage across the unclean surface of even a 
perfect insulation. For this reason, overvoltages due to a slow variation in 
the storm cloud charge present a problem only in exceptional situations 
(for example, in providing protection to the explosives industries or to 
storages of explosives). It is not difficult to provide a 100Q resistance but 
special designs are necessary. 
We should like to mention an exotic but fairly realistic situation when 
the object capacitance is subject to a change. This happens, for example, in 
apparatus with remote wire control. When the apparatus goes away horizon- 
tally from the operator and the cable elongates with a constant velocity 
v = const, the capacitance grows linearly in time, C(t) = Clvt. At a constant 
external field, the grounding electrode current and the object voltage relative 
to the earth do not change in time and are 
During the object motion up to a cloud, the overvoltage will be larger 
because of the higher average potential of the conductor, U, x Eowt/2. 
Let us calculate the overvoltage due to the return stroke current. Its 
specificity results from a high velocity of the recharging wave through the 
channel, wr, which is comparable with light velocity c. Strictly, this requires 
account to be taken of the delay time of an electromagnetic signal in the 
calculation of charges induced on the object. When faced with this complex 
task, engineers sometimes feel a mystic horror. In actual fact, the delay 
changes little in many situations, especially in the case of a compact object. 
To illustrate this, consider the limiting case when a terrestrial compact 
object is located right under a vertical, descending leader, more exactly, 
when the horizontal distance to the stroke point is r << z, where z is the 
height of the return wave front. At the moment of time t, the leader charge 
is neutralized, and the channel is recharged along its portion from the 
earth to the altitude z = vrt. But the object 'is aware' of the charge change 
along a shorter portion only, z, = cv,t/(c + vr). The effect of the delay is 
equivalent to a decrease in the return wave velocity by a factor of 
(1 + vr/c). The equivalent velocity is v,, = ze/t > vr/2, because U, < c. For 
a lightning of medium power with v, x 0.25c, the velocity is wre RZ 0 . 8 ~ ~ .  
A 
20% correction is of little importance, particularly as the neglect of the 
delay leads to an overestimated overvoltage, thus providing a certain reserve 
for the engineering solution. The effect of the delay will be smaller at 
comparable values of r and z. Indeed, the distance between the charge 
neutralization front and the object, (r2 + z2)lI2, increases more slowly than 
z. It remains nearly unchanged at r >> z. Therefore, the time evolution of 
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=== PAGE 277 ===
Induced overvoltage 
269 
the field, Eo(t), at the object's site will not differ from that calculated neglect- 
ing the delay. The phase delay which acts for the time At = r/c does not 
affect the overvoltage. 
Let us make a direct evaluation of the 'electrostatic' component of 
overvoltage during the return stroke, assuming that a rectangular charge 
neutralization wave (section 4.4) is moving along a vertical, perfectly 
conducting channel towards a cloud. At any point of the channel behind 
the wave front z = wrt, the charge changes by the same value r. The electric 
field follows the charge variation. Without the account of the delay, its 
change AE, at the distance r from the channel is described by an expression 
similar to (3.5) (with h = 0, H = z and R = z): 
The time constant for real electric circuits, R,C < 0.1 ps, is several orders of 
magnitude smaller than the time of the return stroke flight from the earth to 
the cloud. Then, according to (6.2), the electric component of the overvoltage 
(relative to the earth) for a compact object is defined as 
2 
dAE, 
rR Ch 
vr t 
U, 
R,Ch- 
- 
- 
dt 
2 m  (vft2 + r2)3'2 
where h is an average object height. The short-term action of this overvoltage 
load must be endured by all the insulation gaps separating the object from 
the adjacent constructions and service lines, whose potentials were not 
changed by the lightning or, if they were, to a different extent. 
At the moment of time tmax, the pulse Ue(t) reaches its maximum 
In the second formula of (6.6), we have substituted I, = TU,. A lightning of 
medium current IM = 30 kA, which has contacted the earth at the distance 
r = lOOm from the object of medium height h = 10m and capacitance 
C = lOOOpF (a wire l00m long), is capable of exciting an overvoltage 
pulse with an amplitude Uem,, = 2 kV at R, = 10 R because of the channel 
recharging during the return stroke. 
Most of the parameters in (6.6), are beyond the engineer's capacity when 
he requires a high protection reliability. It is hardly possible to change the 
capacitance or average height of the object being protected. It seems more 
feasible to reduce the overvoltage to a safe level by decreasing the grounding 
rod resistance R,. This is an effective way of overvoltage protection against 
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210 
Dangerous lightning effects on modern structures 
electrostatic induction. However, this measure, like any other technological 
tool, has its limitations. It is difficult to provide R, < 1 R in a impulse 
mode. The obstacles are the relatively low conductivity of the earth and 
the inductance of the grounding conductors, which are fairly long when 
the grounding mat occupies a large area. After the potentialities of R, reduc- 
tion have been exhausted, there remains only one way - increasing the dis- 
tance r to the nearest lightning discharge. To do this, one has to protect 
from direct strokes not only the object itself but the area around it together 
with the other constructions located on it, some of which are higher than the 
object to be protected. In that case, all lightning rods must necessarily be 
mounted outside this area; otherwise, the protectors will be able to attract 
lightnings, bringing their charges close to the object. 
In contrast to the amplitude, the duration of the overvoltage pulse front 
is practically independent of the object's parameters, being primarily deter- 
mined by the distance to the stroke point, r. From the first formula of 
(6.6), we have t,, 
0 . 7 ~ ~  
at r = lOOm and TI, 
x 0 . 3 ~ .  Overvoltages of 
microsecond duration are typical of the lightning return stroke. Pulses 
with the front duration of 1 - 1 . 2 ~ ~  
are still used as standards in insulation 
tests for resistance to lightning overvoltages, although they do not always 
reflect the reality. 
6.1.2 Overvoltage due to lightning magnetic field 
The problem of overvoltage induced by the magnetic field of a lightning 
discharge is the most common one among overvoltage problems. The 
lightning current varying in time and space induces the emf in any circuit. 
If a circuit is formed by conductors, the emf excites electric current. If the 
circuit is disconnected, the voltage equal to the induced emf is applied to 
the break. Let us estimate the maximum effect produced by an infinitely 
long straight conductor with current i. At the distance r from the conductor, 
the magnetic field is H = pOi/27rr. Consider a rectangular frame in a plane 
intercepting the conductor (figure 6.1). Suppose the side parallel to the 
conductor has a length h and the side normal to it has r2 - rl = d; the short- 
est distance between the frame and the conductor is rl. The magnetic flux 
through the frame is defined as 
--In-. 
r1 
The emf induced in the circuit, U, = -dQ/dt, is 
At the maximum rate of the current change, Ai z 10" A/s, characteristic of the 
return stroke of subsequent lightning components, the emf'induced in a circuit 
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=== PAGE 279 ===
Induced overvoltage 
I Pr 
271 
Figure 6.1. Estimating the overvoltage magnetic component. 
with the sides h = d = 10 m at the distance rl = 100 m from the conductor with 
current is U, = 19 kV. The emf for a smoother current impulse of the first 
component of a moderate lightning with Ai = 5 x lo9 A/s is U, = 1 kV. 
Overvoltages excited electrostatically and electromagnetically are gener- 
ally comparable. The former can be coped with using an effective grounding 
of the object, but overvoltages due to the electromagnetic mechanism do not 
respond to the grounding efficiency. Imagine metallic columns buried deep in 
the ground, which support rails for a mobile overhead-track crane mounted 
high up at the ceiling of industrial premises. The whole construction has a 
perfect grounding owing to the metallic columns which provide a complete 
absence of electrostatic overvoltages from close lightning strokes. However, 
a pair of columns with a rail and the conducting earth forms a closed circuit 
with an area of several hundreds of square metres, in which the time-variable 
lightning current excites an emf. The same thing occurs in a circuit formed by 
columns, fixed at the opposite sides of the premises, and an overhead crane. 
A possible disconnection at any site of the metallic construction cannot be 
ignored either. A disconnection may arise due to metal erosion, poor welding 
or inadequate contact between the crane wheel and the rail. In that case, 
practically all emf of the circuit will appear to be applied to the site of 
defect, provoking a spark discharge through the air or a creeping discharge 
across the surface to bypass the defective site. A spark-induced emergency is 
inevitable if there is an explosive gas mixture in the premise. 
The fact that any construction may serve as a circuit capable of inducing 
an emf increases the hazard - this may be a metallic ladder on a conductive 
floor, a metallic pipe leaning against a wall, etc. Such casual circuits present 
an even more serious hazard, because their parts may have only a slight 
contact between them, so that the probability of a spark gap is extremely 
high. An explosion would, no doubt, destroy the casual circuit, creating a 
mystery to the fire brigade in the spirit of Agatha Christie’s stories. 
The sequence of procedures for the calculation of overvoltages due to 
lightning current is similar to that for lightning charge calculation. One 
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=== PAGE 280 ===
212 
Dangerous lightning effects on modern structures 
should first find the magnetic flux through the circuit in question and 
calculate the induced emf. The magnetic flux is often replaced by the 
vector potential A(t) to simplify the calculations. For current i in a thin 
conductor such as a lightning channel, the vector-potential is 
where the integral is taken in the conductor length and r is the distance from 
the current element id1 to the point, at which A is determined. The emf 
induced in the circuit of interest is defined as 
(6.10) 
where EM is the strength of a vortex electric field excited by the time-variable 
magnetic field of the lightning. For a straight conductor with current, the 
vector EM is parallel to the current. If the lightning channel is vertical, the 
vector EM is also vertical. 
Let us represent a lightning return stroke as a rectangular wave of 
current ZM propagating at velocity vr along a vertical channel from the 
earth up to the cloud. Without accounting for the delay, leading to a certain 
overestimation of the result, we have 
Factor 2, instead of 4, in the denominator results from the allowance 
for the current spread in the earth. The field EM is vertical, so the horizontal 
sections of the circuit do not contribute to U,. 
In the vertical sections, the 
values of EM are summed algebraically. For a metallic frame with an air 
gap, like the one shown in figure 6.1, the magnetic component of overvoltage 
in a small gap of A << h is 
u.44 = h P d r 1 )  - E M ( Y 2 ) I  
(6.12) 
where EM are taken to be values averaged over the conductor heights h. 
All the results obtained within the model of a rectangular current wave 
of the lightning return stroke overestimate the overvoltage; the smoother the 
front of the real current wave, the greater is the overestimation. 
6.2 Lightning stroke at a screened object 
6.2.1 
Overvoltages due to a lightning stroke at the metallic shell of a body, such as 
a plane or other objects, occur very frequently. To get an idea of what 
happens in this case, let us look at the schematic diagram in figure 6.2. 
A stroke at the metallic shell of a body 
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=== PAGE 281 ===
Lightning stroke at a screened object 
273 
Figure 6.2. Lightning current flows along a pipe with a conductor inside. 
Suppose lightning current runs along a closed metallic shell of an object, inside 
which there is a conductor connected to the shell at one of its ends, say, at the 
lightning current input. The potential at this contact will be taken to be zero. If 
RI is the linear resistance of a shell of length 1 and L1 is its linear inductance, 
the voltage applied to the shell will be Uf = -(L1 di/dt + Rli)l. The lightning 
current does not branch into the inner conductor disconnected at the other end 
(the capacitance is neglected). The conductor potential changes only due to 
the mutual inductance, U, = Mlldi/dt. Since the magnetic flux of the shell 
current is entirely attributed to the inner conductor, the linear mutual induc- 
tance M I  is equal to the linear inductance of the frame, L1 . Then the potential 
difference between the shell and the inner conductor at the far end of the latter 
is described as 
U, = U, - Uf = iR1l. 
(6.13) 
The remarkable property of a cylindrical system with an inner wire to 
compensate completely the induction emf is well known to impulse measure- 
ment technology. This property is the basis for making shunts for measuring 
current impulses with very short fronts (to a few fractions of a nanosecond). 
The respective theory, useful for the understanding of the overvoltage 
mechanism, is discussed in detail in [5]. We shall turn to it when evaluating 
the skin-effect in a shell. Here, it should be noted that the shape of an 
overvoltage pulse, U,(t), in the absence of a skin-effect is similar to that of 
a current impulse, i(t). This is valid as long as the time of the electromagnetic 
wave propagation along the frame is much shorter than the impulse 
duration. 
No principal changes will occur when the conductor ends are connected 
to the shell via resistances Rkl and Rk2. The voltage U, will then appear to be 
operative in the inner closed circuit consisting of the shell, conductor, and 
resistors. When the resistances of the conductor and the shell are small, the 
current i = Ue/(Rkl + Rk2) arising in the circuit will distribute the over- 
voltage U, between the turned-on resistors in reverse proportion to their 
values. The same will happen when the conductor is connected to the shell 
via spurious capacitances. Of course, if a massive aluminium shell has a 
cross section of 100 cm2 and the linear resistance is R1 E 3 x lop6 n/m, the 
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=== PAGE 282 ===
214 
Dangerous lightning effects on modern structures 
overvoltages may be small, U, RZ 30V, even at the maximum lightning 
current Z,w = 200 kA and a long 1 = 50 m. But one should bear in mind 
that modern microelectronic units stuffed into an aircraft or spacecraft can 
be damaged easily even by lower voltages. 
6.2.2 How lightning finds its way to an underground cable 
The problem of lightning access to an underground cable deserves special 
attention, because the spark propagation under these conditions has a 
peculiar physical mechanism. A direct stroke of a descending lightning at an 
underground cable is a rare event, since the leader cannot ‘sense’ its presence. 
The current flows to the cable along a spark channel creeping along the earth’s 
surface. Its path can sometimes be identified easily because of the bulging 
loosened soil. Normally, the spark path is as long as several dozens of 
metres, or even hundreds of metres in low conductivity soils. It seems unlikely 
that a creeping spark should move towards a cable purposefully; rather, this is a 
matter of chance. But the local topography can stimulate the spark access to the 
cable. Suppose a cable is laid along a forest path, and the current of the light- 
ning that has struck a nearby tree flows down to its roots, giving rise to a spark 
channel, which propagates across the path until it hits the cable. 
A high current spreading through a poor conductor such as soil induces a 
fairly high electric field which initiates ionization. This fact has long been 
known, so the calculations of grounding resistance take into account the 
increasing radius of the metallic conductor owing to the larger ionization 
zone around the metal. It has also been suggested that a strong electric field 
induced by high current may cause a breakdown of some gaps by a spark fila- 
ment in the soil [6]. The soil ionization creates a natural ’grounding electrode’, 
when a lightning channel contacts the earth’s surface. The mechanism of 
current spread through the soil can become clear from analysis of the simple 
case of a spherically symmetrical distribution of current I,. 
At the distance 
Y from the point of lightning stroke, the current is sustained by the electric 
field E = p1M/(27rr2), where p is the soil resistivity. The soil represents a 
porous medium with the pores filled by air. Experiments show that the soil 
air is ionized more readily than the atmospheric air at E > EIS E 10 kV/cm 
[6,7]. This is due to the local field enhancement around sand grains, etc. 
(cf. section 4.3.1). Therefore, the medium in a hemisphere of radius 
Y, = (I~p/27rE1g)’i2 
is ionized to become a natural well-conducting grounding 
electrode. The grounding electrode resistance, i.e., the resistance to the current 
spread through a non-ionized soil, is defined as 
R --I 
1
”
 
Edr=---. 
P 
- IM 
r, 
~ T Y ,  
(6.14) 
For example, the grounding resistance is found to be R, = 7 2 0  at 
ZM = 30 kA, p = lo3 0 .  m (sandy soil), and ri = 2.2m. 
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Lightning stroke at a screened object 
275 
This situation is very unlikely because the process is unstable. Even a 
slight asymmetry, which is always present in nature, say, the asymmetry 
created by tree roots at the site of the lightning strike, may produce a creeping 
discharge. A plasma channel similar to a leader channel originates at the 
strike site. It acts as a long grounding electrode, from which the lightning 
current spreads through the soil. The leakage current per unit channel 
length, Zl, is proportional to the channel potential U at this site, Zl = G, U. 
The linear conductivity GI of the leakage through the channel surface 
contacting the soil is defined by an expression similar to (6.14) but with 
allowance for the cylindrical (or, rather, semi-cylindrical) geometry. The 
radial field at radial distances r smaller than the conductor length I is 
E M Z1p(7rr)-', where I, = Z M / I  is the leakage current per unit channel 
length. When integrating the field over the radius to find the channel poten- 
tial U ,  one should take the upper limit I ,  x I ,  because at r > I the field 
decreases as l/r2 and the integral converges quickly. Hence, we have 
(6.15) 
Here, ri is the radius of a well-conducting channel. Because of the logarithmic 
dependence of G1 on ri and 11, these values do not affect G1 much. 
Laboratory experiments [8] have shown that the principal difference 
between a classical leader in air and a spark running along a conducting 
surface is the mechanism of current production providing the energy for 
the channel heating. In the former case, the current is produced by the 
streamer zone in front of the leader tip (section 2.4.3) and in the latter, 
owing to the transverse current leakage from the surface of the channel 
contact with a conducting medium. A streamer-free leader process was 
clearly observed under these conditions in laboratory experiments [5,8]. 
The streak picture in figure 6.3 does not show even a trace of the streamer 
zone, whereas the air gap of the same length is filled by streamers nearly 
from the very beginning of the leader process, in the absence of a conducting 
surface. The spark process occurring along a conducting surface is very 
effective. A creeping leader requires an order of magnitude lower voltage 
for its development than an ordinary leader - 135 kV instead of 1300 kV - 
for bridging a gap of 5m long. Of primary importance here is the medium 
conductivity and the current supplied to the channel. To make a streamer- 
free leader move on, the field at its tip must be E > Elg to be able to initiate 
the ionization, to supply the initial channel with a current as high as the 
ordinary leader current, it > if,,, N 1 A, to heat the gas rapidly, and to 
maintain the channel conductivity (section 2.4.3). 
A small portion of the lightning current, it << ZAw, delivered to the leader 
tip is sufficient for a creeping leader to develop successfully. The tip current 
is, at first, very high. But as the channel grows, more and more lightning 
current leaks down to the earth because of the increasing contact area of 
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216 
Dangerous lightning effects on modern structures 
Figure 6.3. Streak photographs of a leader creeping along the soil (top) and an air 
leader (bottom). 1: channel, 2: tip, 3: streamer zone. 
the plasma column with the conducting soil. So, the leader eventually stops. 
Let us evaluate the maximum length I of the leader channel. Suppose current 
Z, 
is delivered to the channel through its base at the stroke point. The current 
value is determined by the recharging of the lightning leader channel at the 
return stroke stage and is independent of the creeping spark length. For 
simplicity, we take the delivered current I.w and the longitudinal field E,, 
supporting the creeping leader current, to be constant. At high currents 
(i > 1 A), the dependence E,(i) is, indeed, not particularly strong. By the 
moment the leader has stopped, the tip potential U, and current it are low 
relative to U(x) and i(x) at distances .x from the tip, comparable with the 
channel length. We then have 
(6.16) 
With i(Z) = 1, at the channel base, the maximum channel length is defined, 
with the account of (6.15), as 
di 
GI Ecx2 
dx 
_ -  
- Zl = G,U(.x), 
i(x) = 
~ 
2
.
 
U(x) M E,x, 
(6.17) 
( 2zM )'/2 - [ 21,p In ( I / ~ J  1 'I2 
1 %  - 
N 
GI E, 
..E, 
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Lightning stroke at a screened object 
277 
If the longitudinal field E, is lOOV/cm, as in the case with a common 
leader channel which is usually close to the arc state, the channel length I 
will be 40m (ri x 1 cm) for an average lightning with IM x 30 kA and a 
well conducting soil with p x 100 R - m. The channel length will grow with 
rising lightning current and decreasing soil conductivity: its value is 
I M 220 m at the maximum current 1, x 200 kA and p x 1000 R/m. These 
estimates are consistent with observations. If a creeping leader encounters 
a cable, the still available current in it will penetrate to the cable sheath. 
6.2.3 Overvoltage on underground cable insulation 
If one digs the soil to expose the site of the lightning current input into a 
cable, one can observe the cable cores with damaged insulation, which are 
in contact with the metallic sheath. The damage may be stimulated by the 
presence of a gas-generating dielectric in the cable. The dielectric is decom- 
posed, because of the heating by high current, to produce an electrical 
hydraulic effect, so that the cable appears literally compressed by the 
shock wave. A similar effect can be produced by an explosive evaporation 
of soil water. The elimination of the damage at the current input may not 
remove the emergency, because there may be several others along a distance 
of several hundred metres, on both sides of the strike point. These damages 
result from overvoltages arising between the core and the sheath during the 
lightning current flow along the cable. The overvoltage mechanism is similar 
to that described in section 6.2.1, except that the conductor with a sheath has 
a longer length, sometimes of many kilometres. When the lightning has 
incorporated its current into the sheath, the cable in a soil of infinite 
volume should be regarded as a long line with distributed parameters, or, 
more exactly, as two lines. One is the sheath in a conducting soil. The light- 
ning current flowing along the sheath gradually leaks into the soil and goes to 
‘infinity’, thus raising the sheath potential U,(., 
t )  relative to an infinitely far 
point on the earth. The other line is the core with the sheath. It is affected by 
the magnetic field of the sheath current, giving rise to an induction emf and 
voltage drop in the conductive sheath due to its finite linear resistance RI,. 
As 
a result, the cable core acquires potential U,(x, 
t )  relative to infinity, which is 
generally different from U,(x. t). The difference U, = U, - U, represents the 
overvoltage on the cable insulation capable of damaging it. 
A rigorous solution to the problem of Ue(x. t) follows from a combined 
solution of the set of equations describing the lightning current flow along a 
cable sheath and the voltage wave propagation (between the core and the 
sheath) along the cable core. This would be a correct approach, provided 
that the waves in the sheath and inside the cable had approximately the 
same velocities. But we shall show that these velocities differ by several 
orders of magnitude, which necessitates the subdivision of this problem 
into two problems. One will describe the lightning current flow along the 
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278 
Dangerous lightning effects on modern structures 
sheath and the other the propagation of waves, excited by this current, inside 
the cable. 
Let us first follow the fate of lightning current i(x, t )  in the cable sheath. 
Its variation along the length due to the displacement current associated with 
the charging of the sheath linear capacitance C1, to the voltage U,(x, t )  can be 
assumed to be negligible, as compared with the large current leakage into the 
soil through the linear conduction G1 of the sheath grounding. One can also 
neglect the mutual induction emf in the sheath, produced by the core current 
i,, because it is small compared to the self-induction emf. Since the total 
magnetic flux of the sheath current i involves both the sheath and the core, 
the mutual inductance M1 per unit length of the sheath-core system is 
equal to the linear sheath inductance L1. However, the current in the core 
is i, << i. Indeed, the current in the core screened from the earth by the 
sheath is only due to the charging of the cable capacitance C1, to the voltage 
U, acting between the core and the sheath. The value of U, does not exceed 
the electrical strength of the cable insulation, U, M 2 kV. Even if the current 
wave velocity in the core were close to light velocity, the core current would 
be of the order i, M C1,Uec M 10-30A (for a cable of a small cross section, 
C1, M 20-50 pF/m), which is much lower than the lightning current 
i M 10kA. So, one can ignore the current deviation into the core even 
when its insulation is damaged at the lightning current input into the cable 
so that the core appears to be connected to the sheath. 
Therefore, on the above assumptions, the current flow along the sheath 
is defined by the equations 
aU, 
ai 
ai 
ax 
at 
-L1-+Rli, 
-- 
ax = G1 us 
(6.18) 
where L1 is given by formula (4.25) and R1 is its linear resistance. If the cable 
were on the earth's surface, with the lower half of the sheath touching the 
earth, formula (6.15) would be valid for G1. When a cable is buried at a 
large depth, the current spreads radially from it in all directions uniformly, 
so the value of GI is doubled. In intermediate situations, one can use the 
empirical formula 
2n 
p In ( 12/2hr) ' 
G1 = 
Y Q h K 114 
(6.19) 
where h is the cable depth. The boundary condition for (6.18) is expressed by 
the equality i(0, t )  = Io, where Z,, is one half of the current delivered by the 
lightning to the cable at the input x = 0 (the current flows in both directions 
from this point). 
The cable sheath possesses a low active resistance and a fairly high 
inductance because it is a solitary conductor. The self-induction emf has 
a greater effect on the distribution of the rapidly varying lightning current 
in the sheath than the active voltage drop. If Rli is neglected in the first 
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Lightning stroke at a screened object 
279 
0.0 
0.5 
1 .o 
1.5 
2.0 
Z 
Figure 6.4. The function 1 - erf(z). 
approximation, the set of equations (6.18) changes to the familiar diffusion 
equation, with the only difference that the diffusion coefficient xo is now 
defined by LIGl rather than by RIC1. The current varies along the sheath 
length in both directions from the input as 
1 
2P 
O - LlGl 
Po 
x -. 
(6.20) 
x -- 
X 
The latter expression for xo corresponds to a surface cable; at a large depth, this 
would be xo = p/po: xo x 160-600m2/ps at p x 102-103 Rjm (p x 500 s2jm 
for a common sandy soil). The point with a fixed value of illo is shfted with 
a decreasing velocity ‘U x xo/x x ( ~ , / t ) ” ~ ,  
in agreement with the diffusion 
law x 
(4xor)’/*. The current covers a 1 km cable length for t x 2000- 
200 ps, decreasing rapidly at the wave front (figure 6.4). The sheath potential 
from (6.18) and (6.20) is 
The potential at the current input drops with time, from an ‘infinite’ value at 
t = 0, which results from the neglect of Cls.t The equivalent sheath resistance 
t If C1, is taken into account, there is a weak precursor which propagates with the velocity of an 
electromagnetic signal (L, 
overtaking the diffusion wave described by (6.20) and (6.21) 
(cf. section 4.4.2). 
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280 
Dangerous lightning effects on modern structures 
also decreases with time. It is defined by the resistance of the soil around the 
elongating cylindrical surface, through which the current leaks. 
Now turn to the wave process inside the cable. The type of overvoltage 
under consideration is dangerous only to communications lines, whose linear 
inductance L1, is small because of the narrow gap between the core and the 
sheath. The self-induction term is usually small relative to the voltage drop 
on the active core resistance RI,. The current leakage through a high quality 
insulation can also be neglected, since it is small relative to the displacement 
current charging the linear core capacitance C1, (relative to the sheath). With 
these assumptions, the core potential U, relative to infinity and the core 
current i, are described by the equations 
(6.23) 
di 
ai, 
a(uc - Us) 
at 
- Rlcic + L1 at! 
- - 
= Clc 
au, 
d X  
a x  
which account for M 1  = L1. The electrical signal induced in the core by the 
lightning stroke has a much higher propagation rate than the process of filling 
the sheath with current. Indeed, we have U, = 0 and ailat = 0 far ahead of 
the filled part of the sheath. Equations (6.23) transform to the diffusion equation 
for U, and i, with the coefficient xc = (RlcClc)-l x 2.5 x 106-2 x lo5 m2/ps 
(RI, x 0.01-0.1 fl em) exceeding xo by several orders of magnitude. This 
means that the charging of the cable capacitance occurs very quickly, and a 
quasi-stationary mode is established in the cable, in which U, and i, follow a 
relatively slow variation of the sheath current. 
By subtracting the first equalities of (6.18) and (6.23) from one another 
and keeping in mind RI, N RI and i, << i, we obtain the equations for over- 
voltages in the cable: 
= Rli. 
U,(x. t )  FZ 
i ( x ,  t ) R 1  dx + UJO, t). 
(6.24) 
au, 
d X  
If the cable insulation at the lightning current input is intact, V,(CQ, t )  = 0 
and U2(m. t )  = 0, the overvoltage value is maximal at the input point and 
is defined as 
U2(0. t )  = - 
i ( x .  t)R1 dx KZ -ZoRlxl, 
= ( 4 ~ ~ t ) ” ~  
(6.25) 
where x1 is the equivalent sheath length with the lightning current at the 
moment of time t. Overvoltages rise in time as long as the lightning 
current is high; more exactly, as long as its decrease is compensated by the 
elongation x1 (the situation for a realistic current impulse will be discussed 
below). 
If the cable insulation is damaged at the current input ‘instantaneously’ 
and the core contacts the sheath, then we have U,(O, t) = 0 and the over- 
voltage grows with distance from the stroke point up to the maximum 
value of (6.25) at x > x1 = ( 4 ~ ~ t ) ’ / ~ ,  
provided of course that the lightning 
sox 
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Lightning stroke at a screened object 
28 1 
current is still high at the moment t.t For example, at Io = ZM/2 = 10 kA and 
R1 = 3.5 x lop4 sljm (the aluminium sheath is 1 mm thick and 30“ 
in 
diameter), we have U, M ZoRlxl x 2 kV at a distance x1 M 600m from the 
current input. This happens at the moment of time t M x:/4x0 x 6 0 p  (at 
xo x 1000 m2/ps, if the lightning current is still high relative to its amplitude. 
This is the duration of comparatively short current impulses of negative 
lightnings. For anomalously long impulses (- 1000 ps) of positive lightnings, 
the length of the ‘active’ cable portion where the overvoltage arises can 
increase to 1-10 km, with the overvoltage amplitude becoming appreciably 
larger. It is clear now why the repair of the damaged insulation at the 
lightning input is insufficient and other damaged sites must be found and 
removed along several kilometres of the cable length. In regions with 
poorly conducting soils (rocks, permafrost), a damaged line may extend to 
dozens of kilometres. 
So far, we have evaluated the overvoltage for a rectangular current 
impulse. To calculate it for a real lightning pulse, we should first find a 
more rigorous solution for the current input into the cable with an intact 
insulation. This will provide the maximum value of U,. We shall apply the 
operator approach to equation (6.18), omitting the term R1i, as before. As 
a result, we get the expression 
A = (P/xo)1/2 = (PPo/2P)1/2 
in which the last term corresponds to a cable on the earth’s surface. If unit 
current i(0, t )  = Io = 1 flows into the sheath, the integration constant is 
A = 1. The operator form of the overvoltage is 
The inverse transform of (6.27) is the function 
(6.28) 
which coincides, within the accuracy of the numerical coefficient of the order 
of unity, with (6.25) at Io = 1. Expression (6.28) for unit current Zo(t) = 1 
represents the unit step function of y(t) providing the solution for arbitrary 
lightning current i( t )  by taking the Duhamel-Carson integral. In particular, 
we get the following expression for a bi-exponential current impulse 
tThe equivalent core resistance Rlcx2, with x2 x ( 4 ~ ~ t ) ’ / ~ ,  
grows in time, in contrast to the 
decreasing input resistance of the sheath. This is another argument in favour of the current enter- 
ing primarily the sheath rather than the core, even if they come in contact at the input. 
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282 
Dangerous lightning effects on modern structures 
0.0 1
.
1
 
0.01 
0.1 
1 
10 
100 
Z 
Figure 6.5. The function h(z). 
i = 210 = I,w[ exp(-at) - exp(-Pt)], allowing for its spread in both direc- 
tions from the stroke point: 
U,(O. t )  = 
IMR1 
[ h ( 4  h(9r)] 
(7rp0/2p)”2 
a1J2 fill2 
(6.29) 
h(z) = exp(-z) 
exp(y2) dy. 
The function h(z) has a maximum h,, 
M 0.54 at z M 1 and goes down to 
0.5hm,, at z E 5 (figure 6.5). Therefore, the overvoltage pulse front Ue(O, t )  
is close to the duration of a lightning current impulse and U, decrease several 
times more slowly than the current. The duration of the current pulse front 
affects the overvoltage value only slightly. The estimation from (6.29) for a 
current impulse of duration t - 100 ps (a = 0.007 ps-’, P = 0.6 ps-’) gives 
Uem,,/ZMR1 M 145m at resistivity p = l000R .m. For IM = 30kA and 
R1 = 3.5 x 
R/m (an aluminium sheath of 1 mm thickness and 30” 
diameter), the maximum overvoltage is 1.5 kV. 
The diffusion equation for sheath current can be solved numerically for 
any shape of the current impulse in a cable of finite length, when the core 
contacts the sheath at the lightning stroke point and when the insulation 
is intact. Figure 6.6 illustrates the results for the former situation. At 
xo = 2p/p0 = 1.6 x lo9 m2/s (p = 1000 R m), the bi-exponential current 
impulse i(t) = ZM[ exp(-of) - exp(-Pt)] with a 5 ps front and duration of 
tp = loops (on the 0.5 level) excites an overvoltage pulse with the reduced 
amplitude Uem,,/RIIO = llOm along the cable length of 500m. The over- 
voltage maximum occurs at the moment of time t, = 60 ps; the overvoltage 
p .-. 
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Lightning stroke at a screened object 
283 
E 
" I  
I
.
 
0 
5b 
100 
140 . 260 ' 240 
300 
Time, pi 
Figure 6.6. Evaluated overvoltage pulses in the cable of a length x with a core con- 
tacting a sheath at the stroke point. The bi-exponential lightning impulse current 
with cy = 0.007 ps, p = 0.6 ps; p = 1000 njm. 
drops by half in 230 ps. At a distance of 1000 m from the lightning current 
input, the overvoltage pulse is somewhat higher, smoother and longer. For 
current IM = 30 kA, its amplitude rises to 1.1 kV in an aluminium sheath 
with R1 = 3.5 x lop4 O/m and to 7.5 kV for a cable with a lead sheath of 
the same cross section. All of the calculated parameters are quite comparable 
with those estimated from (6.29). 
6.2.4 The action of the skin-effect 
One of the manifestations of the skin-effect is that the current turned on at a 
certain moment takes some time to penetrate into the conductor bulk. The 
characteristic time for a conductor of thickness d and conductivity 0 to be 
filled by current is T, = p o d 2 ;  for example, T, M 6 ps at d M 1 mm and 
0 M lo7 (a. 
m)-'. One can neglect the skin-effect when treating overvoltages 
in underground cables with about the same sheath thickness but with an 
order of magnitude longer time of lightning current flow along the cable. 
One can assume the current to flow through the whole sheath thickness, as 
was suggested above in the treatment of linear sheath resistance RI. 
But 
when one considers short sheaths, especially those of terrestrial objects, in 
which current runs very rapidly (at light velocity), it is often impossible to 
ignore the finite time of current penetration in the transverse direction, i.e., 
through the sheath thickness. 
The electric field and current diffuse from the conductor surface into its 
bulk with the diffusion coefficient xs = (pOo)-' (hence, d2 N xsTs). Due to 
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284 
Dangerous lightning effects on modern structures 
this fact, the effective resistance of the conductor is higher than in the case of 
direct current. The formal use of this fact in (6.13) would result in an increase 
in the overvoltage which is proportional to RI. But an opposite effect is 
observed in reality. Owing to the skin effect, the overvoltage pulse front 
becomes smoother than the current pulse front, reducing the overvoltage 
at a finite pulse duration. 
The reason for this paradox is that the last equality of (6.13), which is 
strictly valid only for an infinitely thin sheath or for direct current, should 
not be used in any situation. If the current varies in time and the sheath 
has a finite thickness, its voltage can also be represented as a sum of the 
resistance UR(t) = f ' j l  ( j  is the current density) and the induction 
Ui(t) M d@/dt components (@ is the magnetic flux). But with the same sum 
U, = UR + Vi, the value of each component varies with the point r of the 
sheath cross section, for which the calculation is being made, since the 
proportion between the current density j ( r )  and the magnetic flux @ ( r )  
varies when the total current over the cross section changes. Calculations 
of overvoltages between the conductor and the sheath, U, = U, - U,, are 
generally indifferent to which r the value of U, is being found, because the 
potential does not vary with the thickness. For simplicity, however, it is 
reasonable to make calculations for the inner sheath surface: this surface 
and the internal conductor are the only elements of the system affected by 
equal magnetic fluxes, mutually excluding the induction components of 
overvoltage on the conductor and the sheath. Consequently, formula 
(6.13) can be replaced, without any restrictions, by the expression 
U,(?) =j&)cr-ll 
= Ein(?)l 
(6.30) 
where j,, and Ein are the current density and longitudinal electric field, respec- 
tively, on the inner surface of the object's sheath. 
The current penetration into a thin sheath is described by the equation 
for one-dimensional plane diffusion. It has been mentioned that the diffusion 
coefficient is expressed by the quantity xs = (pea)-'. For a rectangular 
current impulse of infinite duration, the longitudinal field strength on the 
inner surface of a sheath of thickness d can be written as 
The exponential series at t > y-' converges very rapidly, so one can restrict 
oneself to the first term only. Therefore, the field Ei, rises with the time 
constant T :  = y-l = p0ad2/7r2; its value is 6 ps at cr FZ 5 x lo7 (Cl. m)-' 
and d M 1 mm. This permits the neglect of the skin-effect action on over- 
voltages in long underground cables, in which the current diffusion along 
the sheath and, hence, the time of the overvoltage rise to the maximum 
take 1 or 2 orders of magnitude longer than T:. However, the skin-effect 
in objects located on the earth's surface and having relatively short sheaths, 
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Lightning stroke at a screened object 
285 
0.0 ' . 
I 
I 
50 
100 
150 
200 
Time, p 
Figure 6.7. Overvoltage pulse deformation in a cable sheath due to skin-effect with 
the time constant T, = 10 p. An exponential current impulse is duration of 100 ps 
(dashed curve). 
in which lightning current propagates over the time t << Ti, decreases the 
overvoltage with a greater efficiency in the case of a shorter current impulse. 
For an exponential current impulse i(t) = ZMexp(-at), we have from for- 
mula (6.31) with the first series term only and the Duhamel integral 
1 
27 
exp(-at) - - 
exp(-yt) 
t > y-'. 
(6.32) 
7 - a  
The results of the calculations presented in figure 6.7 show that the skin effect 
elongates the overvoltage pulse front to T i  and the amplitude decreases by 
several dozens of percent. 
6.2.5 
We have assumed so far that the sheath has the shape of a circular cylinder. 
In that case, the current is distributed uniformly along the cross section 
perimeter and there is no magnetic field inside. But this model is inapplicable 
to many real objects, for example, the fuselage or wing of an aircraft, having 
very complex cross section profiles with different curvatures. The current 
flowing through a non-circular sheath is distributed non-uniformly along 
its perimeter and the magnetic field is present inside. These factors affect 
the mechanism of overvoltage excitation by lightning current. 
Let us consider a two-dimensional sheath shaped as a cylinder of a non- 
circular cross section and a considerable length I ,  when the end effects are 
weak and the current and field distributions are plane-parallel. The sheath 
is considered to have a uniform resistivity and thickness. Let us subdivide 
The effect of cross section geometry 
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286 
Dangerous lightning tlffects on modern structures 
the sheath into a set of N parallel conductors of a short length Ark along the 
cross section perimeter, such that the current J k  per perimeter unit length in 
the kth conductor could be regarded as varying only with time (the total 
current in the kth conductor is ik = JkArk). In a steady-state mode when 
the current becomes direct, all J k  values are the same, since they are 
determined by equal ohmic voltage drops in all of the conductors. A mere 
summing of the magnetic fields of the conductors will indicate that a 
magnetic field may be present inside a sheath of an arbitrary cross section 
geometry. 
When lightning current is introduced into the sheath very quickly, the 
magnetic induction emf in the conductors greatly exceeds the ohmic voltage 
drop. But in this approximation, all of the conductors will indeed form an 
integral ‘perfectly conducting’ sheath, namely, they will be connected in 
parallel. This means that all of them will have equal potentials. Hence, the 
magnetic Aux coupling @ for each conductor is the same. This provides a 
set of equations for finding the currents ik at the initial stage of the process: 
N 
N 
Lkik(0) + 
M k m i m ( 0 )  
m # k, 
im(0) = IM 
(6.33) 
m =  1 
k =  1 
where Lk is inductance, Mkm is the mutual inductance of the conductors k 
and m, and IM is the lightning input current. Now, in contrast to the 
steady-state mode, the currents will be different even in identical conductors 
if they are located at different sites of the sheath. We shall illustrate this with 
reference to three parallel conductors of length I and radius r located in the 
same plane so that the distances between the adjacent conductors are 
identical and equal to D. If il is the current in the central conductor and iz 
is that in the end conductors, with M12 as the mutual inductance of the 
adjacent conductors and M23 of remote ones, we shall have 
Lil + 2M12iz = Liz + M12il + M23i2. 
il t 2i2 = IAM 
The current in the central conductor is lower than in the end conductors 
because of M12 > M23. 
It is easy to solve a set of equations of the type (6.33) even for a large 
number of conductors simulating a sheath. Only the calculation of inter- 
conductor distances is somewhat cumbersome, requiring knowledge of the 
cross section profile coordinates. We shall leave this problem to the reader 
and illustrate, instead, the analytical solution for the current distribution 
in a long cylindrical sheath with an elliptical cross section [9]. This solution 
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Lightning stroke at a screened object 
287 
is useful for the evaluation of many real profiles and for testing computation 
programmes: 
J ( x )  = 
24a2 - x2( 1 - b2/a2)]’I2 
‘ 
(6.34) 
Here, a is the large and b the small semiaxis of the ellipse and x is the distance 
between the ellipse centre and the calculation point projection on the large 
axis. The ratio of the minimum linear current density (on the plane part of 
the ellipse) to the maximum one (on its tip) is Jmax/Jmin 
= a/b. The current 
non-uniformity may be great in real structures, such as the aircraft wing, 
a/b > 100. 
There is no magnetic field in the sheath at the moment of time t = 0. This 
is the result of the initial current distribution among the conductors owing to 
the magnetic induction emf. With the redistribution of the currents under the 
action of ohmic resistance, a magnetic field will gradually arise in a non- 
circular sheath. The field becomes the source of overvoltages in the inner 
circuits of the object. By integrating numerically the set of equations 
where U( t) is also the unknown voltage drop along the length of the sheath 
‘made up’ of conductors, one can find the variation in the current distribu- 
tion along the sheath perimeter. The initial condition for the integration is 
the solution to (6.33). The calculation accuracy increases with the number 
N of simulating conductors. But the limiting case of N = 1 is also suitable 
for the evaluation of the time constant of a transient process: TI, = L1/R1, 
where L1 and RI are the linear sheath inductance and resistance. The current 
is redistributed slowly, T,, N 0.1 s, in the sheaths of large objects with radius 
Y pv 1 m and thickness d N 1 mm. During the action of a common lightning 
current impulse with t, N 100 ps, the current distribution along the sheath 
perimeter differs but little from the initial distribution profile. The results 
of a computer simulation support this conclusion. The computation for 
a sheath of complex geometry (figure 6.8) with L1 = 0.57 pH/m and 
RI = 1.05 x lop5 Rjm (Ttz = 54ms) has shown that the linear current 
density at all characteristic points of the sheath takes the steady-state value 
for a time about 20 ms. During the first 200 ps typical of lightning current, 
the density cannot change appreciably. 
Let us consider overvoltages across the insulation between an inner 
conductor and the sheath. Suppose the conductor is placed very close to 
the inner sheath surface. The contour area between the conductor and the 
wall will be very small, and the internal magnetic flux will be unable to 
create an appreciable induction emf. The voltage between the conductor 
and the sheath will be equal to the integral of the ohmic component of the 
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288 
Dangerous lightning effects on modern structures 
O ’  
fo 
20 
30 
4-0 
t,ms 
Figure 6.8. Evaluated evolution of a linear current density at indicated points of the 
wing-like sheath shown. J, = J ( t  + XI). 
longitudinal electric field &(x) at the site of the conductor location x. But 
now, the evaluation of Ein should not be based on the average current density 
in the sheath, using the total current and linear resistance R I .  For a sheath of 
thickness d, we have 
= J ( x ) ~ / d .  
(6.36) 
The nearer the current line with the maximum linear density, the higher the 
overvoltage across the conductor insulation relative to the object’s shell. One 
practical conclusion is quite evident. To reduce manifold the overvoltage in 
the electrical line inside an aircraft wing and along its thinnest back end, 
where the current density is maximal, it is sufficient to shift the wire closer 
to the upper wing plane or, better, to the lower one, where the current density 
is minimal due to the wing curvature (figure 6.8). Laboratory measurements 
have confirmed this suggestion [lo]. 
Note the seemingly ambiguous character of the evaluations. The sheath 
cross section is practically equipotential, so the inner conductor must be 
under the same voltage with respect to any point of the sheath in a particular 
cross section. However, the ohmic overvoltage component for a conductor 
inside an elliptical cylinder (figure 6.9) with respect to points 1 and 2 of the 
large and small semiaxes differ by a factor of Jmax/Jmin 
= a/b, in agreement 
with (6.34). This contradiction is superficial. In the presence of a magnetic 
field, there is the magnetic component, in addition to the electrical one, 
U = U, + U,. 
The distance between the conductor and current line 1 is 
practically zero, and the magnetic flux induces nothing in such a narrow 
circuit, U ,  = 0. On the contrary, a wide circuit, made up of a conductor 
and remote current line 2, is affected by most of the internal magnetic flux. 
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Lightning stroke at a screened object 
289 
Figure 6.9. The conductors inside an elliptic cylinder. 
The emf induced by the flux adds the ohmic voltage to the necessary value U. 
The evaluation of the magnetic flux direction will show that the signs of U, 
and U, coincide if the circuit, in which U, is induced, is composed by the 
current line with a linear density less than the average value; otherwise, U, 
and U, have opposite directions. Therefore, the values of U, and U, vary 
with the design circuit chosen, but the sum remains the same. 
We shall make use of this circumstance to find the time variation of the 
magnetic field inside the sheath. It has been pointed out above that lightning 
current i(t) acts for such a short time that it cannot be redistributed radically 
along the sheath perimeter; therefore, we have J (  t )  w i(t) at any point. Hence, 
we get Ein(t) 
N i(t) for a thn sheath where the slun effect is inessential. Choos- 
ing a design circuit with U, = 0, we find U( t) = U, ( t )  N E,, ( t )  N i( t) . But in 
the general case, this is U(t) = Ue(t) + UM(t), with the values of U, and U, 
being comparable; hence, U, ( t )  N i( t )  . 
Thus, the induction component of overvoltages in a sheath is pro- 
portional to lightning current rather than to its derivative! Therefore, the 
magnetic flux penetrating into the sheath varies in time as the integral of 
current i( t). This remarkable result has been confirmed by experiments. 
The oscillograms in figure 6.10 illustrate a test current impulse, similar in 
shape to a lightning current impulse, and a magnetic pulse H(t) inside a 
i- 
0.2 
0.'4 
0.'6 
' t, ms 
OV 
Figure 6.10. Oscillograms of the test current and magnetic field inside the wing-like 
sheath. 
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290 
Dangerous lightning effects on modern structures 
sheath simulating an aircraft wing [l 11. The response time of the magnetic 
field detector did not exceed 0.5ps, so that the H(t) pulse front close to 
300ps and an order of magnitude higher than the current impulse front 
causes no doubt. 
6.2.6 Overvoltage in a double wire circuit 
Although the use of a metallic sheath as a reverse wire saves on metal, most 
internal circuits of objects to be protected consist of two wires, because they 
are better screened from noises. When the magnetic field inside an object is 
zero, as is the case with a perfectly circular sheath, the lightning current 
raises the potential of each wire relative to the shell but no overvoltage 
arises between the wires. This is important because the electromagnetic 
induction can damage the insulation and produce noises in information 
transmission systems. The consequences of an information line disorder 
are often as hazardous as a failure in an electronic unit. 
It follows from the previous section that the magnetic induction emf 
inside the sheath strongly depends on the wire location. The emf value is 
maximal when one of the wires goes near the inner sheath surface along 
the current line of maximum linear density and the other wire is immediately 
adjacent to the line with J,,,. 
The overvoltages U1 and U, of the two wires 
relative to the shell are determined only by the ohmic components, since 
the wires immediately adjacent to the inner sheath produce with it zero 
area circuits: U1 = J,,,pl/d 
and U2 = J,,,pl/d. 
The voltage between the 
wires AU,,, = U1 - U2 = (J,,, - J,,,)pl/d 
is due to the internal magnetic 
field, so the variation rate of the magnetic flux penetrating through the circuit 
composed of the wires is dQ,,/dt = (J,,, - Jmln)pl/d. We can again con- 
clude that the magnetic field pulse in the sheath is not similar to the lightning 
current but to its time integral. During a current impulse of a negative light- 
ning tp = 100 ys (on the 0.5 level) with J,,, 
= const, the magnetic field within 
a non-circular sheath rises as H (  t )  N t (for a circular sheath, Jmax 
= J,,, 
and 
H = 0). At this lightning current, the higher the conductive sheath resistivity 
and the greater the non-uniformity of the initial current distribution along 
the sheath perimeter, the higher is the internal magnetic field. 
To illustrate, an estimation will be made for an elliptical cylinder of 
length I = 100 m. The following parameters will be used: a = 1 m, b = 1 cm, 
the aluminium sheath thickness d = 1 mm, and p = 3 x 
fl- m. The 
lightning current amplitude will be taken to be 1, = 200kA, a value used 
in aircraft tests for lightning resistance. Using formula (6.34), we obtain 
J,, 
x 3200 kA/m, J,,, 
NN 32 kA/m, and AU,, 
9.5 kV. In a real construc- 
tion, such a great overvoltage could have resulted from a poor design of the 
internal electrical network. Wires running to the same electronic unit must 
not be separated so much from each other, nor should they be placed on the 
inner side of a metallic shell at places differing much in the surface curvature 
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Lightning stroke at a screened object 
29 1 
and, hence, in the linear current density. A compact paclung of cable assem- 
blies at sites of minimum surface curvature is a good and nearly free means 
of limiting overvoltages in internal circuits of objects with metallic shells. 
Overvoltages rise considerably if the shell is made from a plastic and if 
its electric circuits are located in a special outer metallic jacket extending 
from the head to the tail. The linear resistance of the jacket may be 1-2 
orders higher than that of the totally metallic fuselage. The ohmic component 
of overvoltage will increase respectively. To eliminate the magnetic compo- 
nent, associated with the penetration of the magnetic field into the jacket, 
it is very desirable to make it as a pipe with a circular cross section. 
6.2.7 Laboratory tests of objects with metallic sheaths 
The lightning protection practice involves a great many technical problems 
associated with the formation of test current and the measurement of all 
parameters of interest. Here, we shall be concerned with the physical aspects 
of testing, which could allow prediction of the object's response to lightning 
current in a real situation, generally different from laboratory conditions. 
Let us begin with a laboratory current simulating lightning current. The 
best thing to do would be to make a laboratory generator reproduce light- 
ning current exactly. The high requirements on the protection reliability 
make one apply maximum currents with an amplitude up to 200kA, 
especially for testing aircraft. Tests on the 1 : 1 scale are attractive because 
they do not require overvoltage measurements. It is sufficient to examine 
the object's equipment after the tests to see that there is no damage. However, 
the generation of a high current creates problems when the object has a long 
length or when the current impulse front to be reproduced must be short. For 
example, the maximum steepness for the impulse i( t )  = Zo [ 1 - exp( -pt)] 
with an exponential front is (dildt),,, 
= pZo. To generate such impulses, 
the source must develop the voltage U,,, 
=DIOL, where L is the circuit 
inductance close to that of the test object; L z L1l for an object of length 
1. The maximum voltage is U,, 
z 12MV for L1 x 1 pH/m, I x loom, 
Io = 200 kA, and p x 0.6 ps-', corresponding to the front tf % 5 ps average 
for the current of the first negative lightning component. A generator with 
such parameters would have an enormous size and great cost. 
The intuitive desire to elongate the current impulse front rather than to 
reduce its amplitude in the testing of objects with a solid metallic sheath has a 
reasonable physical basis. Due to the longer front duration tf, the ohmic 
overvoltage in the internal circuits of the object to be designed could 
change only when there is an appreciable current redistribution along its 
cross section perimeter during the time t x tf. This would require the time 
tf > 100 ,us. Therefore, the application of impulse fronts with a duration of 
dozens of microseconds, instead of typical lightning impulses, cannot affect 
the test results. The same is true of overvoltages in a double wire circuit, 
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292 
Dangerous lightning effects on modern structures 
induced by an internal magnetic flux. Consequently, the increase of the 
impulse front duration by one order of magnitude will be unable to affect 
appreciably the overvoltage in internal electrical circuits. This considerably 
reduces the requirements on the laboratory source of impulse current, 
because its operating voltage decreases in proportion with the increase in 
tf. The decrease in U by an order of magnitude reduces the costs, because 
the costs of high-voltage technologies rise faster than the actual voltage. 
Much attention should be given to the simulation of the lightning 
impulse duration in laboratory conditions. Anyway, the test impulse 
should not be shorter than the real one, for the overvoltage amplitude may 
thus be underestimated because of the skin-effect. It would be unreasonable 
to reproduce on the test bed the actual amplitude of the lightning current 
impulse if there are no non-linear elements in the test object’s circuit and 
the overvoltages can be registered by detectors. Since the electrical and mag- 
netic components of overvoltage are similar in shape and equally depend on 
the applied current amplitude, one can recalculate the measurements in pro- 
portion with higher currents and select the test impulse amplitude in terms of 
the highest possible accuracy and registration convenience. 
Quite another matter is the situation when the test object’s sheath is not 
solid but has slits or technological windows. The ‘external’ magnetic field of 
the lightning partly penetrates through the sheath; the field is proportional to 
the current and the induced overvoltages are proportional to the current 
impulse steepness. The total overvoltage now depends on both the current 
rise time and amplitude, so the engineer has no chance to select a convenient 
test impulse shape. In principle, the recalculation of measured pulses to real 
ones is also possible, but this requires a detailed analysis of the overvoltage 
mechanism and the responses of the object’s circuits, which does not raise the 
testing reliability. 
Another problem is to connect the test object to the laboratory 
generator. It is obvious that a conductor with the generated current should 
be connected to the site of a possible lightning stroke. In the case of terrestrial 
objects in natural conditions and on a test bed, the problem of current output 
is solved in a simple way - by using a grounding bus. The situation for 
aircraft and spacecraft is more complicated. In real conditions, the lightning 
current first flows through a metallic sheath (say, the fuselage) and then 
enters the ascending leader channel, whose length is much greater than 
that of the object. It is difficult to reproduce the real current path in 
laboratory conditions - this would require a very high voltage to make the 
impulse current run through the long conductor simulating a lightning 
channel. Besides, the test object and the numerous detectors would be 
under a very high potential relative to the earth. 
The return current wire is normally located close to the test object. Its 
magnetic field interacts with the object’s metallic sheath, through which 
forward current flows. As a result, the current distribution along the sheath 
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Lightning stroke at a screened object 
293 
Point number 
01 
1 
2 
3 
4 
1 
Figure 6.11. Measured angular distributions of the linear current density along the 
circular pipe perimeter at various locations of the reverse current conductor. 
Marked points (on the pipe scheme) are presented on the abscissa axis. The curve 
A I  corresponds to a single reverse wire A for a = 2r, curve A2 is that for a = 4r, 
curve B depicts three reverse wires B placed as shown in the scheme. Uniform 
distribution C corresponds to the coaxial reverse current cylinder of radius 2r. 
perimeter changes, the redistribution being considerable if the return current 
wire is close to the object. Inducing the emf of the opposite sign, the reverse 
current increases the current load in the nearby parts of the metallic sheath 
but decreases it in the remote parts. 
For a particular geometry, the current distribution should be found 
numerically from the set of equations (6.33) by adding, to each equation, a 
term for the magnetic flux from the reverse current wire, -ZMMko, where 
Mko is the mutual inductance between the return current wire and the kth 
conductor simulating the sheath. Under conditions typical of test beds, the 
distortions due to the return current path may be appreciable. The results 
presented in figure 6.11 have been obtained from the tests of a sheath 
shaped as a circular pipe. In order to avoid the effects of currents induced 
in the conductive soil, the sheath was raised above the earth at a height 
H = 7r, where r is the pipe radius. The role of the return current wire was 
performed by a thin conductor running parallel to the pipe at the distances 
a = 2r and a = 4r from it, three conductors located at 120" at the same 
distance, and a coaxial cylinder of radius 2r. The latter design provides a 
perfectly uniform current distribution along the perimeter of the sheath 
cross section. The return current of a single wire distorts the current distribu- 
tion to the greatest extent: its minimum linear density drops to 0.5ja, and the 
maximum density rises to 2.3ja, (jav 
= Z0/27rr). The current distribution 
becomes more uniform when the number of reverse conductors is increased. 
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294 
Dangerous lightning effects on modern structures 
When a sheath has a complex geometry, it is hard to predict the reverse 
current effect on the test results. The current redistribution in the sheath may 
lead to both the overestimation and underestimation of overvoltages in the 
internal circuits. Much depends on the arrangement of the internal conduc- 
tors and return current wire. 
6.2.8 Overvoltage in a screened multilayer cable 
Overvoltages in screened multilayer cables are due to the skin-effect. As a 
result the cable wire screens in the layers are loaded differently by the light- 
ning current. Every layer is formed by wires arranged in a circle and having 
their own screens (figure 6.12(a)). Depending on the reliability requirements, 
a cable may have an outer metallic sheath or a dielectric coating protecting it 
from mechanical damage. However, a direct lightning stroke produces a 
breakdown of dielectric material, and the lightning current is distributed 
among the screens. The adjacent screens in a layer contact each other 
along the whole cable length. It can be assumed in a first approximation 
that they form a solid sheath of circular cross section with resistance 
Rk = R/nk and inductance Lk, where R is the resistance of an individual 
wire screen and nk is the number of screened wires in the kth layer 
(figure 6.12(b)). For simplicity, we shall consider a double layer cable, mark- 
ing the inner layer with k = 1 and the outer layer with k = 2. The adjacent 
screens of wires from the adjacent layers are also in contact with one another. 
Figure 6.12. Multilayer cable (a) and solid sheath model (b). 
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Lightning stroke at a screened object 
295 
For this reason, a set of circular sheaths can be regarded as a solid conductor, 
and the current penetration along its radius (from the second layer to the first 
one) can be considered as a skin effect. Such a system can also be treated as a 
set of discrete circular layers. 
In the latter case, the current distribution among the layers at the initial 
moment of time t = 0 can be found from the condition of magnetic flux 
coupling equality (6.33). Equations (6.35) are valid at t > 0 and have the 
following solution for two layers at constant current IM = il + i2 = const: 
[Rl + R2 exp(-Xt)l 
I M  
[l - exp(-Xt)], 
il(t) = ____ 
i2(t) = ~ 
R1+ R2 
RI + R2 
R2rM 
(6.37) 
where X = (R, + R2)/(L1 - L2). Equations (6.35) allowed for the mutual 
inductance of the layers, M12 = L2, as in the treatment of the screen-wire 
system in section 6.2. In accordance with the skin-effect law, the lightning 
current first loads the outer sheath and then gradually penetrates into the 
inner sheath. The current is distributed uniformly between the individual 
screens in each circular layer, iSl = il/nl and is2 = i2/n2. The overvoltage 
across the insulation between a wire and its own screen (providing that the 
skin-effect in an individual screen is neglected) is similar to the current in the 
layer, U,(t) = Rlil(t) and U2(t) = R2i2(t), but not to the lightning current. 
If a double wire circuit uses the cores of one layer, there is no over- 
voltage in the instruments connected to it, because the potentials of the 
layer cores are identical. If the instruments are connected to the cores of 
different layers, the voltage between them is 
U12 = U2 - U1 = I M R ~  
eXp(-Xt). 
(6.38) 
At I M  = 1, expression (6.38) is a unit step function for the set of equations 
providing the solution for the lightning current impulse of an arbitrary 
shape. In particular, at i(t) = IM[exp(-at) - exp(-Pt)], we have 
U12 = IMR2[Bexp(-Pt) - A exp(-at) - ( B  - A )  exp(-At)] 
(6.39) 
Owing to the relatively small value of L1 - L2 x (p0/27r) In (r2/r1) at close 
layer radii r2 and r l ,  the layer current ratio is redistributed rapidly, for 
T = A-' 
FZ l o p .  This is the reason for a fast damping of the overvoltage 
pulse U12 (figure 6.13), which may be remarkably shorter than the current 
impulse. It follows from (6.39) that the pulse U12 reverses the sign; its 
opposite tail is damped approximately at the rate of lightning current reduc- 
tion. The overvoltage amplitude in a double wire cable is close to that in a 
wire-shell system, exactly as in a sheath with a sharply non-uniform current 
distribution. If the screens are thin and have a high resistance, the hazard of 
damaging the connected measuring instruments is fairly great. 
A = ./(A 
- a), 
B = p/(X - p). 
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Dangerous lightning effects on modern structures 
-0.2 J 
Figure 6.13. Overvoltage pulse on a two-layer cable for the bi-exponential current 
impulse with cy = 0.007 ps, ,3 = 0.6 ps and the redistribution time constant T = 50 ps. 
The problem for a multilayer cable can be solved in a similar way. The 
overvoltages between the cable cores grow with distance between the respec- 
tive layers. Other conditions being equal, the overvoltages drop with the layer 
depth in the cable. The use of cores of one cable layer reduces considerably 
the overvoltage in a double wire system but does not eliminate it entirely. 
There are no perfectly circular cables - the cable is pressed under its own 
weight and becomes deformed during its winding on a drum. The result is 
that the current distribution along the sheath cross section perimeter 
becomes non-uniform, producing additional overvoltages between the 
cores of the same layer. To minimize these overvoltages, it is desirable to 
connect the equipment to the adjacent cores of the same layer. High precision 
equipment should be connected to the cores of deeper layers. Overvoltages 
arising in a multilayer cable can be evaluated from the same set of 
equations (6.3 5). 
6.3 
Metallic pipes as a high potential pathway 
Modern constructions have an abundance of underground metallic pipes, 
and the lightning protection engineer must take them into account as a 
possible pathway for currents from remote lightning strokes. This actually 
happens when a pipe lies close to a high lightning rod or another object 
preferable to lightnings. Spreading through the earth away from the 
grounding electrode in a way described in section 6.2.2, some of the current 
enters a metallic pipe and runs along its length. A pipe is sometimes 
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Metallic pipes as a high potential pathway 
291 
Figure 6.14. Underground pipe as the pathway for a lightning current and the design 
circuit for a simple evaluation of the object potential. 
connected directly to the object grounding electrode. Figure 6.14 illustrates 
the typical situation when a metallic pipe line connects the grounding 
electrode (with grounding resistance Rgl) of an object, struck by lightning, 
to the grounding (with resistance Rg2) of a well-protected object. Although 
the lightning is unable to reach the latter directly, some of the current finds 
its way to its grounding electrode - the pipe. For applications, it is 
important to know the dependence of this current on the line length 1 and 
on the soil conductivity. 
Section 6.2.3 considered the problem of current distribution for an under- 
ground pipe of infinite length. The limited line length in the present case is an 
important parameter, especially because it has the grounding resistances at its 
ends. Generally, ths problem can be solved analytically using the Laplace trans- 
formation. But the final result is represented as a functional series too complex 
for a treatment, so numerical computations are necessary. It is, therefore, more 
expedient to solve this problem numerically from the very beginning. Before 
presenting the results of a computer simulation, we shall make a simple evalua- 
tion. Let us replace an underground pipe by the lumped inductance L = L1 I and 
its intrinsic grounding resistance R, = (G1 l)-'. The latter will be represented as 
two identical resistors R = 2R, by connecting them to the ends of the line in 
parallel to the grounding resistances Rgl and Rg2 of the objects it connects 
(figure 6.14). This rough approximation makes sense, since we are interested 
in the value of current i2 at the far end connected to the grounding mat, 
rather than in its distribution along the line. In this approximation, we have 
RR . 
(6.40) 
d i2 
dt 
R+Rd 
L- + Re2i2 = (i - i2)Rel, 
R . - 2 
, j = 1,2. 
Putting the lightning current to be i = ZM exp(-at) and i2(0) = 0, we find 
Re1 + Re2 
L 
i2(t) = 
Re 1 AIM 
[exp(-at) - exp(-At)], 
X = 
(Re1 + Re2)(A - a) 
(6.41) 
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298 
Dangerous lightning effects on modern structures 
At the beginning, while the effect of self-induction emf is still noticeable, the 
current largely flows through the equivalent resistance Rel at the front end of 
the line. After time T = A-', the current gradually penetrates to the far end 
of pipe. Some of it, i82 = i2R/(R + Re]!, finds its way to the grounding elec- 
trode of the object of interest, raising its potential to the value U2 = ig2Rg1 
relative to a remote point on the earth. For a longer line, the values of ig2 
and U2 decrease for two reasons. An increase in L = L1l and G = G1l 
raises the time constant T ,  and by the time the current has reached the far 
end of the pipe, the initial lightning current is considerably damped. Besides, 
a smaller portion of the current i2 that has reached the far end enters the 
object's grounding electrode because of the greater pipe leakage. The depen- 
dence of ig2 and U2 on 1 proves to be rather strong, especially when the 
effective duration of the lightning current, t, x a-', is comparable with 
T = A-'. Suppose we take t, = 100 ps on the 0.5 level (a = 0.007 ps-'), the 
grounding resistances Rgl = Rg2 = 10 R, and L1 = 2.5 pH/m. A metallic 
pipe with a lOcm diameter and lOOm in length, lying at the surface of the 
soil with p = 200 R/m (G1 = 2.1 x 
(a/m)-', R = 9.7 R), will deliver 
the current igz 0.171ZAv to the ground of the object located at its far end. 
The object's potential will be raised to U, x 50kV at ZM = 30kA. At 
I = 200m, we have ig2 x 0.0861Zjw and, at the same lightning current, 
U2 z 25 kV. But even this voltage is quite sufficient for a spark to be ignited 
between closely located elements of two metallic structures, provided that 
one of them is connected to the grounding electrode and the other is not. 
Such a spark can induce an explosion or fire in explosible premises. 
In low conductivity soils, current can be transported through metallic 
pipes for many kilometres. This refers, to a still greater extent, to external 
pipes and rails mounted on a trestle which are grounded only locally, through 
the supports separated by dozens of metres. Here, evaluations can also be 
made with expression (6.42), putting R = 2Ri/n, where RL is an average 
resistance of the support grounding and n is the number of supports. 
A comparison of the estimates and computations is shown in figure 6.15 
for the above example with I = 200m. The estimates for the current 
amplitude at the far end of the pipe and for the moment of maximum 
current show a satisfactory agreement with the numerical computations. 
The computations will be unnecessary if one finds it possible to ignore the 
initial portion of the pulse front and can put up with a 20-25% error. 
Let us calculate the potential at the far end of the pipe unconnected to 
the grounding electrode at either end. This may happen due to careless design 
or poor maintenance of communications lines. The soil will be considered to 
have a low conductivity, p = lOOOQ/m; L1 = 2.5 pH,". 
The curves in 
figure 6.16 show the variation in the voltage and current amplitude ratio 
UmaX/ZAv 
for impulses of negative lightnings with tp = 100 ps and for those 
of 'anomalous' positive lightnings, which are an order of magnitude 
longer. The pipe is capable of delivering a potential of dozens of kilovolts 
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Metallic pipes as a high potential pathway 
299 
Time, ~s 
Figure 6.15. Portion of a lightning current passed to the object through the communi- 
cation pipe of 200 m length. Curve 1: numerical computation, 2: simple evaluation. 
for a distance of 1 km to the object even at a moderate lightning current of 
30 kA. Damage of the contact between the pipe and the object's grounding 
electrode may be fatal if a spark arising in the air gap encounters an 
inflammable substance. 
20 - 
E 
0 15- 
+- . 
f 
J 
10- 
5 -  
0 
200 
400 
600 
800 
1000 
1, m 
Figure 6.16. Computed maximum overvoltages transferred to an object at the far end 
of the underground pipe of 10 cm diameter and of length 1. The pipe is not connected 
with the grounding of both an object and a lightning rod. Computations were made for 
the usual lightning current impulse of 100 ps duration, and for an 'anomalous' impulse 
of 1000 ps, Lightning stroke to the other end of the pipe. 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 308 ===
300 
Dangerous lightning effects on modern stsuctuses 
The delivery of high potential can be controlled in a simple way - all 
communications lines must be connected to the same grounding mat. In 
that case, the voltage of all mat components will be raised equally by the 
brought current of a remote lightning stroke. It should be noted that this 
is a reliable means to cope with the overvoltage of kilovolt values. A 
simple connection of metallic sheaths to the grounding mat cannot remove 
pulse noises of tens or hundreds of volts having a short rise time. Steep cur- 
rent impulses spreading across the buses and components of the grounding 
mat always create an induction emf, producing abrupt voltage changes 
even in conductors of about l m  in length. Electrical circuits must be 
mounted in such a way as to avoid the appearance of closed contours or 
joints of the conductor screens to points remote from each other in the 
grounding mat. This sometimes becomes such a delicate matter that the 
result depends on the engineer’s intuition rather than on exact knowledge, 
6.4 
Direct stroke overvoltage 
We described the manifestations of overvoltage due to a direct lightning 
stroke when discussing the lightning current propagation across a metallic 
sheath. The highest current enters the sheath when a lightning discharge 
strikes an object directly (section 6.2.1). This happens, for example, when 
an aircraft is affected by the return stroke current recharging the descending 
leader which has connected the aircraft to the earth. Below, we discuss a 
direct lightning stroke at a grounded terrestrial object. Specifically, we 
shall be interested in the voltage applied to the insulation of the object 
relative to the earth or another construction located nearby. The classical 
situation is that a voltage arises between the lightning rod that has 
intercepted the lightning and the nearby object being protected. A rough 
treatment of this situation was made in section 1.5.1. The fast variation of 
a high lightning current i along the metallic parts of a construction raises 
its potential by U = R,i + Ldi/dt relative to a remote point on the earth. 
Much depends on what is understood by the grounding resistance R, and 
inductance L. These issues are discussed in much detail in the books on 
direct stroke overvoltages (e.g., [6]). Here, we outline the most important 
physical aspects of the problem. 
6.4.1 The behaviour of a grounding electrode at high current impulses 
An important parameter of a grounding electrode is the stationary grounding 
resistance usually measured during the spread of direct or low frequency 
alternative current of several amperes. The value of Rgo found from the 
measurements may be several times larger or smaller than R, = Ue/ZM 
corresponding to a rapidly varying kiloampere lightning current (here, U, is 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 309 ===
Direct stroke overvoltage 
301 
the potential at the current input into the protector). We have discussed, at sev- 
eral points in the book, the two physical mechanisms affecting differently the 
ability of a metallic conductor to tap off the lightning current to the earth: the 
self-inductance and ionization expansion of the surface contacting the soil. 
The voltage drop across the inductance prevents current flow into the 
conductor. A long conductor has to be treated as a line with distributed 
parameters. The input resistance of the line, Ri, = U(0, t)/i(O: t )  varies in 
time, since the current diffuses along the line, and it takes some time for 
the whole conductor to be loaded by current more or less uniformly. As 
the limiting case, consider an infinite conductor in a soil with resistivity p .  
From formulae (6.21) and (6.22), the voltage at the conductor input is 
Ue(t) z U(0, t )  N t-1/2 for the current i(0, t )  = const = lo and t > 0. At 
Io = 1, formula (6.21) can be treated as a unit step function of the system, 
y(t). This allows us to follow the input voltage of a horizontal grounding 
conductor at the lightning current i(t) with a real impulse front by using 
the Duhamel-Carson integral: 
U(0, t )  = y(t)i(O) + y(T)i’(t - 7) dr. 
s:, 
(6.42) 
For a impulse with an exponential front i( r )  = Io [ 1 - exp( -pt)] we have 
U(0, t )  = 210 ( g y 2 1 1 ( 3 t )  
(6.43) 
where h(@) is a function given by the last integral in (6.29) and figure 6.5. Its 
maximum h,, 
at pt, M 0.9 permits the calculation of the maximum voltage 
drop across the grounding electrode: 
(6.44) 
The effective input resistance of an extended horizontal grounding electrode, 
corresponding to U,,,, 
is expressed as 
(6.45) 
In contrast to a lumped grounding electrode with R, M p ,  the input resistance 
of an extended one varies much less with the soil resistivity, R,,, 
N pli2. 
Extended grounding electrodes are ineffective, because only a short initial 
portion of their length , leff M (RgerG1)-’, is actually operative during the 
impulse front time. For example, the effective resistance is R,,, M 13 R and 
the effective length of a long grounding pipe with L1 = 2.5 pH/m at the 
earth’s surface is leff x 22m in the case of the first component current of a 
negative lightning with the rise time 
tf M 5ps (p M 0 . 6 ~ ~ - I )  
and 
p = 100 R - m. In a soil with an order of magnitude lower conductivity, the 
respective values are R,, 
M 42 R and leff M 75 m. 
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=== PAGE 310 ===
302 
Dangerous lightning effects on modern structures 
Extending the grounding bus beyond the limit leff, we are still unable to 
reduce appreciably the maximum voltage drop across the bus. For this 
reason, it is better to introduce current at the centre of a long bus rather 
than at its end, such that two current waves would run in opposite directions 
along the half-length conductors. Still more effective are three conductors 
arranged at an angle of 120”, and so on. When a grounding mat with the 
lowest possible value of R,, 
is desired, it is preferable to load, more or less 
uniformly, the whole of the adjacent soil volume. For this aim, a set of 
horizontal conductors or a conductor network is combined with vertical 
rod electrodes. To avoid the interaction effect of the grounding elements 
and to achieve the maximum loading of them by current, the distance 
between the elements should be made comparable with their length (or 
with the height, for vertical rods). But even in that case, only part of the 
grounding mat, within the radius of leE from the current input, will operate 
effectively at the impulse front. 
Thus, the resistance of a grounding electrode for rapidly varying cur- 
rents is much higher than for direct current. A grounding mat network 
with numerous horizontal buses and vertical rods is able to reduce the effec- 
tive resistance to the value of R,,, x 1 0. But when a large number of objects 
is being constructed, for example, the towers of a power transmission line, 
one has to deal with resistances as high as R,, 
x 10 R and more. 
Laboratory experiments show that the grounding resistance of an 
electrode delivering to the earth very high currents is lower than for low 
currents. The grounding resistance decreases with the current rise. The 
grounding resistance ratio of a high impulsed current and low direct current, 
cui = R,/Rpo, is often called the impulse coefficient of a grounding. The 
coefficients ai used in the literature are sometimes as small as ai x 0.1. To 
illustrate, we shall cite the generalized function cui =f(plM) which has 
been suggested for a vertical rod of 2.5m in length from the results of 
small-scale laboratory experiments [7] (figure 6.17). The grounding resistance 
is reduced by a factor of four at p = 1000 R - m and IM = 30 kA. 
In principle, this reduction in resistance might be due to a larger effective 
radius of the grounding electrode because of the soil air ionization. In section 
6.2.2, we gave formula (6.15) for the linear conductivity of a long rod lying on 
the earth with one half of its surface contacting the soil. If the rod is fixed in 
the vertical position, the whole of its surface contacts the soil but its leakage 
conductivity is lower by a little less than a factor of 2 at the same length (due 
to the poorer operation of the upper end of the rod located at the earth’s 
surface, because current cannot flow upward into the air). The linear conduc- 
tivity GI and the grounding resistance R, of a rod of radius ro, fixed vertically 
into the earth for a length 1, are 
(6.46) 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 311 ===
Direct stroke overvoltage 
1.0’ 
0.8- 
0.6- 
0.4. 
0.2 - 
0 ,  
303 
ai 
PI, MVm 
1 
I 
I 
2.5 m 
Figure 6.17. Impulse coefficient for the grounding rod of 2.5 m length. 
To reduce R, by a factor of 4 at the initial rod radius ro = 1 cm and 1 = 2.5 m, 
the radius must be increased to r1 = 105 cm. The field at the ionized volume 
boundary must exceed the ionization threshold in the soil, Eig x 10 kV/cm, 
and the current density at p = IOOOR/m must be j = Eig p = 1 kA/m2. 
For the surface area of the ionized volume S M 27rrll + ,,f = 24m2, the 
total leakage current would be Z = j S  = 24 kA, corresponding to the current 
of a moderate lightning power. 
However, the uniform radial ionization expansion of the initial ground- 
ing volume at a rate r l / t f  = 2 x 105m/s (this process must be completed 
within the rise time of the current impulse, tf = 5ps) can hardly occur in 
reality. Anyway, there is no experimental indication for this. More probable 
would be the rod ‘elongation’ owing to the leader development into the soil, 
because the current density and the field at the rod end are higher than at its 
lateral surface. The elongation of a grounding electrode is a more effective 
means of reducing the grounding resistance R,, because of R, - 1/1, since 
the resistance decreases only logarithmically with increasing radius (but 
only at r << 1). However, even this process seems unlikely. There is no 
experimental evidence for the existence of long leaders in the soil bulk. 
The leakage area of a grounding electrode is likely to increase due to the 
elongation of the leader creeping along the soil surface from an element of the 
grounding mat. The grounding resistance will then decrease, as 1/L at a long 
leader length L. This mechanism, observed in model laboratory experiments, 
seems optimal for a natural reduction in R, due to the lightning current. To 
change appreciably the grounding resistance of a typical lightning protector, 
the leader must grow to L M 10 m in length (the total length of the protector 
electrodes) for the time tf M 5 ps of the lightning current rise. For this to 
happen, the leader must elongate at a rate of 2 x lo6 mjs. A creeping leader 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 312 ===
304 
Dangerous lightning effects on modern structures 
develops in the air adjacent to the surface of a conductive soil and is shown by 
laboratory experiments to be devoid of a streamer zone and a charge sheath 
(section 6.2.2). In t h s  respect, it is similar to a dart leader whch develops a 
velocity of about lo7 mls at current i - 1 kA. Assuming that the development 
of a fast leader along the soil surface requires this current in the leader tip, it, let 
us estimate the tip radius, at which this appears possible. 
If the grounding resistance is R, and the lightning current is I, 
the leader 
is supported by the voltage U zz R,IM applied to its base. The tip possesses 
approximately the same potential, because the leader channel is a good 
conductor and not a large part of the voltage drops across it. The resistance 
of the leakage current from a hemispherical tip is equal, from formula 
(6.14), to R, % p/27rrt. The tip current is it N U/R,. Only part of the lightning 
current enters the channel, Io. This current mostly leaks into the soil through 
the lateral channel surface possessing, according to (6.15), a leakage resistance 
RI, = (G,L)-' = pln(L/ro)/rL. Keeping in mind U N Rl,Zo. we obtain the 
formula to be used for the estimation of the tip radius: 
(6.47) 
The 6cm radius obtained at Io = IAw/2 = 15 kA, it = 1 kA, L =10m, and 
yo zz rt, appears to be quite reasonable. The field at the channel lateral surface 
behind the tip, E - pZo/*irr,L 80 kV:cm, is high enough for the ionization 
expansion of the leader channel to occur there. Radii much larger than 
those of a conventional leader in air under similar conditions have been 
registered for laboratory leaders creeping along the soil. The photographs 
in figure 6.18 illustrate this quite clearly. 
To conclude, the spread of high lightning currents reduces the ground- 
ing resistance, probably due to the excitation of one or several leaders 
creeping along the soil surface, thereby increasing the length of the ground- 
ing electrodes. But for a fast leader growth (otherwise, the leakage surface 
has no time to become larger for the short lightning rise time), a high 
current of about 1 kA must be delivered to the leader tip. This restricts 
the process of grounding resistance reduction by the condition under 
which the electrodes are arranged in compact groups. A fast reduction is 
hardly possible for a modern substation having an extended grounding 
network. The reduction is, however, quite possible in the case of a con- 
centrated protector consisting of 2-3 horizontal conductors or several 
vertical rods. 
It is worth saying a few words about the testing of lightning grounding. 
The great complexity of a large-scale simulation of lightning current makes 
one turn to model tests. in which the surface current density of small 
electrodes is preserved while the total current is reduced manifold. The 
laboratory studies indicates that the similarity laws are invalid for the 
leader process. The questions of how to interpret the small-scale simulation 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 313 ===
Direct stroke overvoltage 
305 
Figure 6.18. Still photographs of leader creeping along the soil during its develop- 
ment (a) and at the moment of gap bridging (h). 
results and how well they reproduce the real process of lightning current 
spread are open to speculation. 
6.4.2 Induction emf in an affected object 
Let us consider a descending lightning stroke at an object of height h. The 
induction emf for the object is proportional to its inductance, L = Llh. 
The linear inductance can be estimated in a simple way, assuming that the 
current fills up a conductor composed of the object and the lightning channel 
of length 1 >> h. If we assign to the conductor a constant radius yo << I and 
assume a perfectly conducting soil, we shall obtain L1 M (pO/27r) In (21/r0). 
This value will be L1 M 2.3 pH/m at 1 M H = 3 km ( H  is the altitude of the 
negative cloud charge centre) and ro = 5cm. Actually, the return stroke 
wave covers a much smaller distance during the time of the current impulse 
rise, when di/dt and the induction emf have maximum values. But owing to 
the logarithmic dependence of L, on the geometrical size of a long conductor, 
the change in the length of the lightning channel filled by current will affect 
but little the value of L1. For example, we obtain 1 = w,tf = 500m and 
L1 M 2pH/m for the return stroke velocity w, = IO8 m/s and tf =5 ps, 
corresponding to the first component of a negative lightning. At tf = 1 ps 
(the rise time of the subsequent component), L1 will be only 20% lower. 
The same result is obtained when one uses the vector potential A([) and 
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=== PAGE 314 ===
306 
Dangerous lightning effects on modern structures 
vortex electric field EM = -aA/dt for the calculations. Suppose the current 
rises linearly with a distance from the current wave front, i(x) = b(xf - x), 
where xf = v,t and b = const. The current at the point x of the channel 
rises linearly with time, Ai = ai/& = bur. Neglecting the delay time, as was 
done in (6.9) and (6.1 l), we have 
where ro is the average radius of the object affected by lightning. One can see 
that the formula for L1 in (6.48) coincides with the one above, provided 1 is 
understood as the length of the channel loaded by current by the moment of 
time t. 
If the finite velocity of an electromagnetic signal is taken into account 
and the object is located directly under the lightning channel, which happens 
in the case of a direct stroke, the evaluations made in section 6.1.1 give 
Once again, we should like to emphasize the small contribution of the delay: 
the logarithms in formulae (6.48) and (6.49) differ less than by 3% at 3 = 0.3, 
tf = 5 ps, and yo M 1 m. 
Expressions (6.48) and (6.49) define rigorously the vortex electric field 
EM at the earth's surface. When the object's height is h << urtf, which is 
valid for many practical situations, the variation in EM along the object 
can be ignored and the induction emf is U, = EMh. The emf rises linearly 
with increasing h. In particular, if we have h = 30m, w, = 0 . 3 ~  
and the 
lightning current rises to the amplitude IM = 100 kA for the time tf = 5 ps, 
the maximum value of the induction component of the voltage at yo = 1 m 
is U.Mm,, = 780 kV, a value comparable with the electrical component 
Uem,, = R,ZM M 1000 kV at R, M 10 R. Of course, the effects of the electrical 
component may be more serious because of its longer action. Indeed, in the 
first approximation, the pulse Ue(t) is similar in shape to the lightning current 
impulse and U,w(t) to its time derivative. 
Formula (6.49) can also be used to evaluate the magnetic component 
after the current impulse maximum. For this, the real current entering the 
lightning channel should be represented as a sum of the two components: 
i, = Ait, i2 = -A,(t - tf), and i2 = 0 at t < tf. It is not surprising that U, 
is non-zero behind the impulse front, since the magnetic field continues to 
grow, as the lightning channel is filled by current. The total overvoltage 
pulse of a direct stroke, Ud(t) = U,(t) + U M ( t ) ,  is very different from the 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 315 ===
Direct stroke overvoltage 
2.0 1 
2.0 - 
1.5- 
$1.0- 
8
.
 
0.5 - 
0 . 0 t .  
1 
. 
, 
. 
, 
. 
, 
, 
I 
0 
2 
4 
6 
8 
10 
0.0 4
q
1
 
0 
2 
4 
6 
8 
10 
307 
Time, ks 
Figure 6.19. Computed overvoltage of the direct stroke at the transmission line tower 
with the grounding resistance R, = 10 s2 for the lightning current with tf = 5 ps and 
amplitude of 100 kA; U, = 0 . 3 ~ .  
lightning current impulse because of an abrupt rise and an equally abrupt fall 
of U, with time (figure 6.19). 
6.4.3 Voltage between the affected and neighbouring objects 
It is important for many applications to know the voltage affecting the 
insulation gap between an object of height h, affected by a direct lightning 
stroke, and another object of height hl < h, located nearby. For this, one 
should find the difference between the evaluated overvoltage of the direct 
stroke, Ud,,,, and the maximum overvoltage Uinmax, induced on the 
neighbouring object. The latter value strongly depends on the object’s 
construction. So, let us analyse two extrema1 situations. 
Suppose a lightning strikes a lightning rod located near the mast it 
protects (figure 6.20). The magnetic components of the overvoltages are 
determined by the vortex field strengths EM from formula (6.49). For the 
rod, yo can be taken to be equal to its average radius rl. For the mast, yo 
can be assumed to be equal to the distance d between the rod and the 
mast. The maximum time-dependent difference between the magnetic 
components of the voltage at the height h l ,  where the distance between the 
constructions is minimal, is equal to 
(6.50) 
Expression (6.50) allows for the value v,t >> d at the moment this maximum 
occurs. The magnetic component of the overvoltage across the insulation 
Copyright © 2000 IOP Publishing Ltd.

=== PAGE 316 ===
308 
Dangerous lightning effects on modern structures 
Figure 6.20. Estimation the voltage between a lightning rod struck and an object. 
gap AUMm,, = UMrOd 
- U,wob, increases with distance d, because UMrOd is 
independent of d and 
drops as the distance between the object and 
the current increases. The return stroke velocity v, has practically no effect 
on AUMm,,. Its upper limit is the value of 
for the affected lightning 
rod (AUMm,, M UMm,,/2 at d/rl RZ 100). 
The situation with the electrical component of overvoltage is less definite. 
The overvoltage is also determined by the difference between the two values, 
AU, = Uerd - Ueobl, but Uerd = -Rg,IM is a definite quantity and Ueobl 
varies with the design of the object’s grounding mat. The latter may be 
common with the lightning rod grounding grid and quite compact; in that 
case, we have AU, = 0 because the bases of the rod and the object are 
interconnected. There may be another extreme situation: the grounding mat 
of the object may be so far from that of the lightning rod that it may be 
unaffected by the electric field of the lightning current spreading through the 
soil. In that case, we shall not have UeOb, = 0 and AU,” 
= Rg,IM, because 
this would be possible only in the absence of current through the object’s 
grounding mat. In reality, there is an electric charge induced on the object’s 
surface due to the electrostatic induction (section 6.1. l), so a current flows 
across the object, creating the electrical component of the overvoltage. Its 
value can be found from formula (6.5) and the maximum value from (6.6), 
provided that the return stroke is simulated by a rectangular current wave 
in a vertical lightning channel. Let us evaluate the possible voltage from 
formula (6.6). 
To go beyond the zone of the current spread away from the lighting rod 
grounding, it is necessary to move away at a distance -20m from it. The 
radius of the grounding grid, within which the electrodes are located, is 
hardly larger than 5m, so that the distance between the rod and the 
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=== PAGE 317 ===
Direct stroke overvoltage 
309 
object in formula (6.6) can be taken to be r = 25m. Assuming that the height 
of a typical object is h = 30m, its linear capacitance C1 z 10pF/m, 
C = Clh = 300pF, and R, = loa2, we shall have UeDb, 30kV at the light- 
ning current Z,v = 30 kA. Although Uerod and Ueobi have different signs and 
iAU,I > IUerodI, the additional value is not essential because it is an order 
of magnitude less than R,ZM in the above example. The situation when a 
lightning rod is put up at a distance sufficient for the separation of its own 
grounding grid and that of the object is quite realistic. This is done for the 
protection of especially important constructions to avoid pulse noises or 
sparking due to the induction emf, when some of the current finds its way 
to the object’s grounding through the soil. 
Another extreme case, in which the electrical component of the object 
overvoltage is dominant, is a lightning stroke at a metallic grounded tower 
of a power transmission line. The direct stroke overvoltage affects an 
insulator string, to which a power wire is suspended. Consider first a 
simple and frequent variant (in lines with an operation voltage below 
1lOkV) when the line has no protecting wire. In that case, we do not have 
to solve the difficult problem of lightning current distribution between the 
affected tower and the wire, repeatedly grounded by the adjacent towers. 
Nor should we bother about the electromagnetic effect of the protecting 
wire on the power wire (section 6.4.4). As in the previous situation, the 
insulator string is affected by the overvoltage A U  equal to the potential 
difference of the tower at the point of the string suspension and the power 
wire. The calculation of the tower overvoltage is similar to that for a light- 
ning rod, just described. A specific feature of this problem is the existence 
of the wire. Being suspended horizontally, it does not respond to the mag- 
netic field of the current in the lightning vertical channel. The power wire 
is well insulated from the tower grounding by the insulator string. Owing 
to its far end being grounded, it would be able to maintain zero potential, 
but for the current created by the redistribution of the charge induced on 
the wire. The induced charge is very high because some of the wire length 
is located close to the lightning channel, and the total capacitance of a 
long wire is very large. Naturally, the small distance between the wire and 
the lightning channel does not mean the existence of a direct contact between 
them, so we can speak only of the effect of electrical induction on the wire. 
Even though the wire is connected to the earth at zero resistance, the 
induced charge cannot respond immediately to the lightning charge variation 
and the wire potential cannot remain at zero. The grounding point is located 
far away, at the end of the wire, so the charge liberated by the induction 
cannot be delivered to it faster than with light velocity c. For the induced 
charge q,n to appear at the point x, a current wave must be excited at this 
point, which will eventually transport the charge -q,n out of the wire to 
the earth. This wave will propagate at light velocity. During its motion, 
the potential at the wave front will rise due the voltage drop on the wave 
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=== PAGE 318 ===
310 
Dangerous lightning effects on modern structures 
resistance of the line with distributed parameters, i.e., along a long wire. 
Elementary current and potential waves arise at any point on the wire, 
where the induced charge is changed by the lightning field. Propagating 
with light velocity to the left and to the right of the origin, the currents of 
elementary waves are summed, raising the voltage between the wire and 
the earth. After the waves are damped, this voltage, naturally, drops to 
zero, because the wire is grounded. The response of a long line to the external 
field Eo,(x, t )  acting along a horizontally suspended wire is described by the 
equations 
aue 
(6.51) 
-- 
aue =Rli+Ll--Eo,(x,t), 
--= 
C 1 T  
di 
di 
dx 
at 
dx 
where the potential Ue(x, t )  is due exclusively to the line response to the 
field Eo,(x, t ) .  The total potential of the wire relative to the earth, Upe(x, t )  = 
Uo(x, t) + Ue(x, t ) ,  contains another component, Uo(x, t), defined by the 
charges of the lightning return stroke. Neglecting the ohmic voltage drop 
relative to the induction term and taking b’Uo/b’x = -Eox into account, we 
arrive at the wave equation with a distributed driving force and containing 
no damping term: 
The solution to this equation represents a general solution to a homogeneous 
equation and a particular solution to an inhomogeneous one, corresponding 
to the two identical waves propagating in opposite directions along the line: 
Uge(x,t) 
= zIo 
1 ‘ 8  
dOUo(Xl,O)dO+ 
(6.53) 
XI = x - C ( t  - 0). x, = x + c(t - 0). 
The integrals give the sum of the above elementary waves moving at light 
velocity. The waves are excited by the time variation of the external field 
potential U,. For the elementary wave to arrive at the point x at the 
moment of time t, the causative variation in U0 must occur at the points 
x f Ax earlier, by the time 0 = Ax/c. If the time is counted from the 
moment of the lightning contact with the line tower, the lower, zero limit 
of the integrals of (6.53) should be replaced by the time-of-flight of light 
for a minimum distance between the lightning channel and the wire, 
In the general case, the difficulties that arise in the calculation of the 
integrals depend on how one approximates the lightning current related to 
the linear charge in the return stroke wave inducing the field Eo, as well as 
on the lightning channel position relative to the wire. Of significance are 
the following factors: what object the lightning strikes (the earth or an 
element of a power transmission line raised above the ground), the channel 
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=== PAGE 319 ===
Direct stroke overvoltage 
311 
deviation from the normal, as well as its bendings and branching. It is impos- 
sible to solve this problem without numerical computations. The question 
then arises as to the stage in the study, at which a computer simulation is 
most helpful. One should not ignore a numerical integration of initial 
equations (6.51), allowing the control of the effect of active resistance RI 
which sometimes has a large value. The effective value of RI may be much 
higher than the resistance of the line wire to direct current because of the 
skin-effect, the soil resistance, used by the wave as a ‘return wire’, and due 
to the energy consumed by the impulse corona. The corona is excited in 
the wire by overvoltages and absorbs some of the propagating wave 
energy, contributing to its damping. The impulse corona also increases the 
effective linear capacitance of the wire, since the electrical charge is localized 
not only on the wire surface but in the adjacent air. The charge is delivered 
there by streamers of metre lengths. The capacitance CleR depending on the 
local wire overvoltage varies together with the velocity of perturbations in 
the wire, U = (C1,,L1)-1/2. This greatly distorts the wave front, since different 
sections of the wave front have different velocities. The problem becomes 
greatly non-linear and definitely requires a numerical solution. 
The calculation formulas given below describe simple situations 
neglecting the wave damping in the wire. They have been derived by direct 
integration of (6.53) and borrowed from [3]. The lightning channel is 
considered to be vertical; the return stroke wave moves towards the cloud 
at constant velocity U,. For a rectangular charge wave in the channel of 
lightning that has struck the earth (but not a line tower) at a horizontal 
distance r from the wire, we obtain for the wire point nearest to the lightning 
channel 
where p = ur/c, U, is the return stroke velocity, and h is the wire height above 
the earth. The time in (6.54) is counted from the moment of the lightning 
channel contact with the earth. This formula can be used at t > r/c, i.e., 
after the electromagnetic signal has covered the distance between the channel 
and the wire. The overvoltage is still active at a large distance from the stroke 
point (x + CQ), where the lightning field effect is negligible, Eox x 0. The 
wave reaches that point through the wire, as in the case of a communications 
line (this occurs without damping at RI = 0). Such overvoltage waves are 
known as wandering waves. For these waves, we have 
(6.55) 
where the time is counted from the moment of the wave front arrival at the 
‘infinitely’ remote point of interest. At t ,  = r/wr, the function -Uge(m, t )  has 
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312 
Dangerous lightning effects on modern structures 
Figure 6.21. Evolution of an overvoltage at the wire point nearest to the lightning 
stroke point (solid curves) and a wandering wave voltage (dashed curve). For a 
return stroke, a rectangular current wave model is used. 
a maximum: 
(6.56) 
which is independent of q.. 
At t >> t,, the overvoltage is damped as f-’. 
Typically, the amplitude of a wandering wave is somewhat higher than 
the voltage amplitude relative to the earth at the site where the wire is closest 
to the stroke point (figure 6.21). The reason for this is the opposite signs of U. 
and U,, causing a reduction in the value of Uge = U, + U, in the close 
vicinity of the wire, where U. # 0. Far from the stroke point, we have 
U, M 0, and the overvoltage is totally defined by the wire response. Although 
the overvoltage maximum at the closest point does vary with wr, this 
variation is not appreciable. This is good because there are few measurements 
of the return stroke velocity and practically no synchronized measurements 
of the lightning current. 
If the lightning current is supposed to rise at the impulse front as 
i(t) = A,t with A, = const, the overvoltage U,,(O. t), for the same conditions 
and designations as in (6.54), is 
(2 + 1 - 32)”2 - 3 K  
(6.57) 
1 - 32 
Uge(0. t )  = -- 
In 
27r&oCWu, 
Formula (6.57) has a sense at r / c  6 t 6 t f  + r / c  (3 < K < (t+tf/r) + 3). After 
the current impulse maximum, t > tf, the calculation can be made using this 
formula and a superpositior. by representing a real current wave as two 
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Direct stroke overvoltage 
313 
waves of different signs shifted in time, as was done in the comments on 
formula (6.49). As the time increases within the lightning current rise time, 
the value of Ug,(O, t )  rises monotonically. The calculated pulses V,,(O, t) 
for a rectangular current wave and for a wave with a linearly rising current 
have something in common at r/v, x tf, which is valid for remote lightning 
strokes with r 2 100 m. Both approximations lead to an increase in the over- 
voltage during the current front. But for close strokes, especially for a direct 
stroke at a line tower, the discrepancy between the calculated pulses becomes 
remarkable. This is the reason for the sceptical attitude to analytical solu- 
tions, which we showed at the beginning of the discussion. No doubt, a 
linearly rising current is closer to reality than a rectangular impulse, but it 
cannot simulate the actual current rise accurately. The same is true of the 
impulse amplitude. The discrepancy in the calculations made within the 
models considered grows with decreasing distance r. 
Nevertheless, another analytical solution [3] may be useful for the estima- 
tions. It concerns the case of a direct lightning stroke at a transmission line 
tower, when the shortest distance between the lightning channel and the 
wire is determined only by the height difference between the tower, h,, and 
the wire, h. If a charge wave corresponding to the lightning current front 
i(t) = Air moves up along the vertical lightning channel from the tower top 
to the cloud with constant velocity w,, we shall have at the point of the wire 
suspension 
Let us calculate the overvoltage due to the first component of a negative 
lightning with the average parameters IM = 30 kA, tf = 5 ps, Ai = 6 kA/ps, 
and w, = 0 . 3 ~  
= 90m/ps. We shall have U,,(O, tf) = 320kV for ho = 30m 
and h = 20m at t = tf. Similar, but of the opposite sign, is the potential 
rise on the tower grounding resistance R, x 10 R due to the lightning current, 
RgIM =300 kV. This doubles the electrical component of the overvoltage 
across the insulator string. 
It is worth noting the specific effect of overvoltage on the line insulation. 
Overvoltage is not strictly related to any point on the line, as is the case with 
the voltage drop across the tower grounding. It has been mentioned that the 
charge liberated by electrical induction moves along the wire, creating a 
wandering overvoltage wave. With a negligible damping, it can cover a 
distance of several kilometres, affecting, on its way, all the insulator strings it 
encounters. An insulation breakdown may occur even far from the lightning 
stroke, where the line insulation is poor for some reason. Really hazardous is 
the encounter of the wandering wave with a hgh-voltage substation, because 
the overvoltage wave penetrates to the internal insulation of its transformers 
and generators, which is always poorer than the external insulation. 
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Dangerous lightning effects on modern structures 
A wandering wave also arises when a lightning strikes a power wire. The 
lightning current spreads along the wire from the stroke point in both 
directions, producing very strong overvoltage waves, U ( x ,  t )  = Zi(x, t)/2. 
Since the wave resistance is 2 x 250-400R (the smaller value is typical of 
ultrahigh voltage lines with split wires of a large equivalent radius), the 
current ZM = 30 kA would produce an overvoltage with an amplitude of 
3.5-6MV. In reality, the overvoltage is limited to the value of breakdown 
voltage for the tower insulator string closest to the lightning stroke, where 
the flashover does occur, A wave with an amplitude equal to the string break- 
down voltage is a wandering wave in this case. Of course, the overvoltage 
may rise again, after the string flashover, due to the self-induction emf of 
the tower and to the voltage drop across its grounding, to which the lightning 
current runs after the string flashover. Wandering waves are damped by the 
same processes that determine the resistance RI in (6.51). 
It is important for lightning protection practice to compare the over- 
voltages due to direct strokes at a line tower and a wire. In the former 
case, the voltage drop across the insulator string is the sum of three 
components. The voltage drop across the tower grounding and the induced 
voltage of the wire are approximately the same quantitatively but have 
opposite signs. This totally gives about -2R,ZM over the string. The mag- 
netic component L,Ai has a real effect only on the current impulse front, 
and its average value is equal to L,Z,w/tf (L, is the tower self-inductance). 
The magnetic component for the first leader of a moderate negative lightning 
(ZM = 30 kA, tf = 5 ps) and for a tower of standard size (h, M 30 m) does not 
exceed 200 kV but 2R,Iy > 600 kV because of R, 2 10 Cl. It appears that 
overvoltages due to a direct moderate stroke at a tower can flashover the 
insulation only in lines with voltages less than llOkV, which have strings 
less than 1 m in length. For a 220 kV transmission line, a hazard may arise 
when the currents are twice as high as the average value, but such lightnings 
occur only with a 10% frequency. The hazard of a lightning stroke at a tower 
is not high for 500-750 kV transmission lines, since they have long strings. A 
reverse flashover may arise from a lightning with 100 kA currents and more, 
but their number is less than 1 YO of the total. If the lightning current strikes 
the wire, the current spreads in both directions along it. With the wave resis- 
tance Z > 200 52, we get ZZM/2 > 3 MV even for a moderate lightning. This 
is sufficient to flashover the insulation of any of the currently operating lines. 
A lightning stroke at a wire should always be considered to be hazardous. 
6.4.4 Lines with overhead ground-wires 
When a lightning strikes a tower of a power transmission line protected by a 
grounded wire, the current is split between the tower and the grounded wire, 
due to which the current load on the tower is reduced. However, the engineer 
is then faced with a complex problem of calculating the current distribution. 
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Direct stroke overvoltage 
315 
Another aspect of this problem is the account of the screening effect of a 
protecting wire. Since the wire is connected to the tower, it acquires, in a 
first approximation, the tower potential, thus creating a voltage wave of 
the same sign. Owing to the electromagnetic induction, a similar wave of a 
lower amplitude is excited in the power wire. As a result, the voltage in the 
insulator string, equal to the potential difference between the tower and 
the power wire, drops. These additional problems complicate the calculation 
of direct stroke overvoltage for a line with an overhead ground-wire. The 
problems that arise here relate to electric circuit theory rather than to 
physics, so we shall discuss them only briefly. 
Many engineers try to calculate the current distribution between a tower 
and an overhead wire within the model of an equivalent circuit with concen- 
trated parameters. The lightning channel is regarded as a source of current 
i(t). The tower is replaced by its inductance L, and grounding resistance R,, 
the two grounding wire branches (on the left and on the right of the stroke 
point) are represented by the branch inductances L,/2 and their grounding 
resistances in the adjacent towers, R,/2, connected in parallel. One also intro- 
duces the mutual-induction emf M, dildt, induced by the lightning current in 
the wire-towers-earth circuit (figure 6.22). This circuit can be simplified 
further by putting R, = 0, because the principal interest is focused on the 
current front of tf = 1-10 ps and because the cable inductance along the 
many hundreds of metres of its length is as high as hundreds of microhenries 
and the time constant is usually taken to be LJR, > 100 ps. This model circuit 
then presents no calculation problems, provided that the mutual inductance of 
the vertical lightning channel and the circuit including the ground-wire is 
known. As the return stroke wave moves up, the channel is filled by current 
so that the value of M, rises in time. The calculations similar to those for 
the derivation of formula (6.49) and allowing for the time delay yield [3] 
(6.59) 
Figure 6.22. The design circuit for a current a tower of the line with the protective 
cable. 
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316 
Dangerous lightning effects on modern structures 
The considerable simplification of the real process can be justified only at low 
grounding resistances of the towers, when the current in the wire circuit is 
limited mostly by its inductance, and one can neglect the current branching 
off to the grounding resistances of all other towers except the one nearest to 
the affected tower. 
The value of R, in a real transmission line in areas with low conductivity 
soils may be several times higher than the normal value, reaching 100 0. Then 
the current distribution problem must also take into account the removal of 
some of the lightning current to 2-5 towers away from the stroke point. The 
equivalent circuit becomes more complicated (chain-like), representing a 
series of link circuits identical to the first one. For a more rigorous solution, 
the ground-wire is to be considered as a long line with a wave resistance Z, 
and many local non-uniformities produced where the ground-wire contacts a 
tower. Each tower is then represented as a chain of L, and R, connected in 
series. Figure 6.23 illustrates the variation of the current impulse in the 
tower with the design circuit. For a circuit with lumped parameters, the neglect 
of the tower grounding resistance R, x 10 R does not affect the result much, 
while at R, x 1000, the tower current impulse shape changes radically. A 
circuit with distributed parameters permits one to follow the effect of con- 
secutive wave reflections at the contacts between the ground-wire and the 
towers. The current impulse distortion by the reflected waves is especially 
Time, ps 
Figure 6.23. Current impulse on the struck tower with (solid curves) and without 
(dashed curves) allowance for a grounding resistance of the nearest tower. The 
model with a linearly raising current front is used for return stroke. 
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=== PAGE 325 ===
Direct stroke overvoltage 
317 
appreciable for a short impulse front tf characteristic of the subsequent light- 
ning components. As the linear resistance of the line, R I ,  rises, the effect of 
the reflections becomes less pronounced because the reflected waves are 
damped more strongly. In any case, the overhead wire takes some of the 
lightning current away, thereby unloading the affected tower; this current 
fraction cannot be less than 2R,/Z,. 
Let us evaluate the screening effect of the protecting wire, which also 
reduces the direct stroke overvoltage. Engineers had become aware of this 
effect long before overhead wires were used as lightning protectors. Some 
even supposed that a wire could reduce the voltage of an insulator string 
to a value lower than the flashover voltage. The wire acquires the electrical 
potential of the tower, which has increased by the value of the voltage 
drop across the grounding resistance. As a result, a high-voltage wave runs 
along the wire. The nearby power wire finds itself in its electromagnetic 
field inducing a similar wave. If U, is the voltage wave amplitude in the 
ground-wire, the voltage produced in the power wire is Ucoup = kcoupUc, 
where kcoup = ZCw/Z, 
is a coupling coefficient and Z,, is the wave resistance 
of the grounded wire-power wire system whch can also be regarded as a long 
line. We have Z,, = (L~cw/Cl~)1’2 
by definition. The linear inductance L1, 
and the capacitance Clcw between this two wires are calculated in a 
conventional way, with the allowance for the earth’s effect. With Llcw N 
ln[(ho + h)/(ho - h)] and Clcw - {ln[(ho + h)/(ho - h)]}-’, the coupling 
coefficient is 
(6.60) 
where Y, is the ground-wire radius. For a rigorous calculation, the geometri- 
cal radius in (6.60) should be replaced by an equivalent radius of the space 
charge region at the wire, (the space charge is incorporated by a impulse 
corona under the action of high voltage). This somewhat increases the 
value of kcoup. Measurements give approximately kcoup = 0.25, instead of 
the calculated ‘geometrical’ value of kcoup z 0.2. Therefore, the electrical 
component of power wire overvoltage is reduced once more, this time by 
the value Ucoup = kcoupUc. The total overvoltage reduction owing to the 
ground-wire makes up several dozens percent, decreasing the tower-stroke 
effect on the transmission line insulation. 
We should like to mention a certain relationship between the type of 
lightning action and the transmission line cut-off. Even induced overvoltages 
are hazardous for low voltage lines (primarily those of 0.4- 10 kV). Induced 
overvoltages are much more frequent than direct strokes and are the main 
reason for the line cut-offs. A protecting wire is useless in this case, so low 
voltage transmission lines do not have it at all. For a line of 35kV or 
more, induced overvoltages are practically harmless and direct lightning 
strokes are dominant. The favourable effect of an overhead grounded wire 
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=== PAGE 326 ===
318 
Dangerous lightning effects on modern structures 
becomes apparent at an operating voltage U 2 110 kV, when the lightning 
current leading to the insulation flashover after a stroke at a line tower 
exceeds an average value ~ 3 0  
kA. About 50% of cut-offs for 110-220 kV 
lines equipped with a grounded wire are due to strokes at towers and 50% 
of cut-offs occur when lightning breaks through to get to the power wire. 
Beginning with 500 kV, an increasing number of cut-offs are due to lightning 
breakthroughs to the power wire. 
6.5 
Concluding remarks 
We finish this chapter and the book by describing the lightning effect on 
power transmission lines. Scientists are still unable to offer a clear 
mathematical description of its complicated mechanism. Modern computer 
simulations can infinitely specify and refine a mathematical model of the 
lightning effect, with respect to both the electromagnetic field and the object’s 
response to a stroke. This is, to some extent, interesting, useful and makes 
sense. The process of refining computations has no limit. To illustrate, a 
detailed analytical treatment of long line parameters with the account of 
the earth’s effect has taken several hundreds of pages in the work by 
Sunde [6]. Suppose a superprogramme has been created for the solution of 
the lightning protection problem; its application will immediately show 
that the great efforts it has required can change but little the existing low 
predictability of lightning-induced cut-offs. The key problem today is not a 
rigorous mathematical solution of the available equations but an adequate 
physical description of the principal physical processes producing a lightning 
discharge, its electromagnetic field, and the object’s response to it. For this 
reason, we have tried to present simple qualitative models rather than 
stringent solutions to the equations. On the other hand, many aspects of 
this problem have been omitted, partly because they are not directly related 
to lightning as a physical phenomenon and partly due to the lack of space or 
to the limited knowledge about the key physical phenomena. 
Let us look back at the material presented in this book in order to 
emphasize the points of primary importance. After the numerical value of 
an overvoltage has been calculated, it is necessary to compare the result 
obtained with the flashover voltage of the insulation in order to identify its 
possible flashover. Most of the voltage-time characteristics of insulation 
strings have been found from tests by standard 1.2/5Ops impulses (here, 
the first value is the front duration and the second is the impulse duration 
on the 0.5 level). Such a refined impulse has little to do with lightning over- 
voltages, and this is clear from figure 6.24. A lightning-induced overvoltage 
has necessarily a short-term overshoot arising not only from the current wave 
reflection by the grounding of the neighbouring towers but also from the 
magnetic induction emf. It is not quite clear how this rapidly damping 
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=== PAGE 327 ===
Concluding remarks 
319 
04- 
0 
5 
10 
15 
20 
Time, ps 
Figure 6.24. Current impulses in a struck tower (is) and in the few neighbouring 
towers (il-i3). The wave problem was solved allowing for wave reflection from the 
places where grounded wire is connected with the towers. 
overshoot affects the electrical strength of an insulation string. Under certain 
conditions, a powerful corona flash saturates the gap with a large space 
charge and can ‘lock up’ the leader process, increasing the strength [12]. As 
for ‘anomalously’ long overvoltages induced by positive lightnings, the 
electrical strength of air may, on the contrary, be several dozens percent 
smaller than what standard tests give [5,13]. The question of the real 
electrical strength of the UHV transmission line insulation is still to be 
answered. 
The return stroke models discussed above ignore the lightning channel 
branching and bending, whereas an actual discharge channel is far from 
being a straight vertical conductor. The channel can deviate from the 
normal by dozens of degrees, especially when it approaches high con- 
structions. Another complicating point in a return stroke model is the 
counterleader. The assumption that the return wave starts directly from 
the top of an affected construction, say, from a line tower, is far from the 
reality. The length of a counterleader is comparable with the height of the 
construction it starts from. Together with the total length of the streamer 
zones of the descending and ascending leaders, this will give a value 1.5-3 
times greater than the object’s height. Such a high altitude of the return 
stroke origin and its propagation in both directions from the point of 
contact, not only towards the cloud, may have a considerable effect on 
the electromagnetic field of the lightning. The available theoretical models 
do not take these facts into account. There are no data on counterleaders 
related to the subsequent lightning components. Today, it is even impossible 
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=== PAGE 328 ===
320 
Dangerous lightning effects on modern structures 
to confirm, or to disprove, the mere existence of a leader travelling to meet a 
dart leader. 
Another weak point of the models is the set of statistical data on the 
amplitude and time characteristics of the lightning current impulse. There 
is some information on medium current lightnings, because these are numer- 
ous, whereas lightnings of extremal parameters are poorly understood. The 
consequences of this are quite serious. The choice of protection means and 
measures depends, to a large extent, on what has actually caused the storm 
cut-off of a particular transmission line - a reverse flashover of the insulation 
string, when the lightning strikes a tower, or the lightning breakthrough to 
the power wire bypassing the overhead protecting wire. Underestimating 
or, on the contrary, overestimating the high current probability by ignor- 
ance, one may arrive at the wrong conclusion concerning the contribution 
of reverse flashover in UHV transmission lines, which may cause great 
losses. The determination of extremal lightning parameters is one of the 
key problems in natural investigations. We should like to emphasize again 
that the exceptions are more important than the rules to lightning protection 
practice. 
Ref e re nces 
[l] Wagner C F 1956 Trans. AIII 75 (Pt 3) 1233 
121 Lundholm R, Finn R B and Price W S 1958 Power Apparatus and Systems 34 
[3] Razevig D V 1959 Thunderstorm Overvoltage on Transmission Lines (Moscow: 
[4] Golde R H (ed) Lightning 1977 vol. 2 (London, New York: Academic Press) 
[5] Bazelyan E M and Raizer Yu P 1997 Spark Discharge (Boca Raton: CRC Press) 
[6] Sunde E D 1949 Earth Conduction Effects in Transmission Systems (Toronto: 
[7] Ryabkova E Ya 1978 Grounding in High-Voltage Installations (Moscow: 
[8] Bazelyan E M, Chlapov A V and Shkilev A V 1992 Elektrichesrvo 9 19 
191 Kaden H 1934 Archivfur Electrotechnik 12 818 
1271 
Gosrenrgoizdat) p 216 (in Russian) 
p 294 
Van Nostrand) p 373 
Energiya) (in Russian) 
[lo] Babinov M B and Bazelyan E M 1983 Elektrichesrvo 6 44 
[ll] Babinov M B, Bazelyan E M and Goryunov A Yu 1991 Elektrichesrvo 1 29 
[12] Bazelyan E M and Stekolnikov I S 1964 Dokl. Akad. Nauk SSSR 155 784 
[13] Burmistrov M V 1982 Elektrot. promyshlennost’; Ser. Appar. wysokogo napryaz- 
heniya 1 123 
Copyright © 2000 IOP Publishing Ltd.