plasma-expert/reference/sources/bazelyan-noaa-preprint.txt

=== PAGE 1 ===
XV International Conference on Atmospheric Electricity, 15-20 June 2014, Norman, Oklahoma, U.S.A. 
 
 
1
Non-Stationary Corona around Multi-Point System in 
Atmospheric Electric Field:  
Discharge Current and Vertical Electric Field Profile  
 
Eduard M. Bazelyan1, Yuri P. Raizer2, Nickolay L. Aleksandrov1,* 
 
1. Krzhizhanovsky Power Engineering Institute, Moscow, Russia  
2. Institute for Problems in Mechanics, Moscow, Russia 
3. Moscow Institute of Physics and Technology, Dolgoprudny, Moscow region, Russia  
 
 
ABSTRACT: The properties of a non-stationary glow corona maintained near the tips of a multi-point 
ground system in a time-varying thundercloud electric field have been studied numerically. The discharge 
was simulated from a system of identical vertical conductive electrodes that is a model of the earth’s 
surface extremities coronating under a thundercloud. The effect of system geometry and dimensions on 
the discharge properties and on vertical electric field profile above the coronating system was investigated. 
Conditions were determined under which the corona properties of a multi-point system are similar to the 
properties of a plane surface that emits ions into the atmosphere. The obtained results were used to 
estimate the temporal evolution of corona current density and corona space charge emitted during 
thunderstorms from the earth’s surface covered with dense vegetation. 
 
INTRODUCTION 
Corona discharges developed from the earth’s surface extremities (the tips of trees, bushes, leaves, 
grass and other sharp objects) under a thundercloud leads to the space charge injection into the atmosphere 
and make a contribution to the global electric circle. In addition, the corona space charge layer affects the 
local electric field at ground level and is practically important for lightning protection.  
Laboratory studies of a corona discharge cannot be directly extended to thunderstorm conditions 
because a discharge occurring near grounded objects in a time-varying atmospheric electric field is 
non-stationary and the corona current depends on the manner in which the ambient field evolves in time, 
rather than on its instantaneous values. The reason is that, in this case, the space charge front has no time to 
bridge the gap and to reach the thundercloud, whereas the corona space charge reaches usually the opposite 
electrode on a laboratory scale.  
The properties of a corona discharge developed from a solitary grounded hemispherically-tipped rod in 
a thundercloud electric field was considered analytically and numerically [Bazelyan and Raizer 2000; 
Aleksandrov et al. 2001; Bazelyan et al. 2008] on the basis of a simple 1D approximation. It was shown that 
the corona current varies in time as icor (t) ~ t(3k-1)/2μ1/2, when the cloud electric field varies as E0(t) ~ tk, k > -1. 
                                                        
 Contact information: Nickolay L. Aleksandrov, Moscow Institute of Physics and Technology, Dolgoprudny, Moscow region, 
Russia, Email: nick_aleksandrov@mail.ru; nick_aleksandrov@hotmail.com 

=== PAGE 2 ===
XV International Conference on Atmospheric Electricity, 15-20 June 2014, Norman, Oklahoma, U.S.A. 
 
 
2
Here, μ is the ion mobility. In this case, the discharge current is constant only when the electric field rises in 
time. In a steady electric field (k = 0), the current decreases with time. The effect of ion mobility on the 
current is smaller than this effect for laboratory gaps when icor (t) ~ μ. 
Recent time-consuming numerical 2D simulations for a solitary grounded rod [Becerra 2013] and for a 
long horizontal grounded wire [Mokrov et al. 2013] supported the use of a much simpler 1D calculations for 
a qualitative analysis when the focus is on the processes in the vicinity of the coronating surface or when 
computational time is limited. This simplification is especially important when considering the properties of 
a corona developed from a grounded multi-point system with a complicated geometry. 
In a thundercloud electric field, the corona current even from an extremely high solitary electrode does 
not exceed 1 mA that is not important from the standpoint of the global electric circuit. Multi-point ground 
coronating systems (forest, bushes, grass and urban areas) make much larger contribution to the total current 
from the earth’s surface. In this case, the local electric field near a given coronating point is affected not only 
by the space charges developed from this point, but by the space charges emitted by others corona sources as 
well. Numerical simulation of a corona discharge from a multipoint system is much more complicated than 
that from a solitary electrode since it is necessary to consider interaction between coronating points and 
individual corona space charge layers.  
In this work, we extended the 1D approach developed in [Bazelyan and Raizer 2000; Aleksandrov 
et al. 2001; Bazelyan et al. 2008] for a solitary grounded electrode to a multi-electrode system. The 
properties of a non-stationary (transient) corona initiated and developed from a model multi-point ground 
system in a thundercloud electric field were numerically studied for different geometrical parameters of 
the system (see also [Bazelyan et al. 2014a]). A simplified method to determine the corona current density 
and injected corona space charge under real conditions was suggested. The evolution in time of vertical 
electric field profiles in the space charge layer above a multi-point system was also considered (see also 
[Bazelyan et al. 2014b]). 
 
CORONA INITIATION FROM MULTI-POINT SYSTEM IN EXTERNAL ELECTRIC FIELD 
In this work, the model of a corona discharge around a solitary electrode (see [Bazelyan and Raizer 
2000; Aleksandrov et al. 2001; Bazelyan et al. 2008]) was generalized to study the discharge from a 
multi-point system. We considered a system of vertical grounded hemispherically-tipped electrodes under 
practically important conditions when the electrode height h is much higher than the curvature radius of 
the electrode top, r0, and the distance between adjacent electrodes, D, is comparable with h. Electrodes 
were uniformly distributed over concentric circles with the radii rk = kD (k = 1,2…) around a given 
electrode (see figure 1). It was assumed that 6k electrodes are located on the k-th circle and that the total 
number of electrodes is such large that almost every coronating point is surrounded by numerous similar 
points. This allowed calculation of discharge properties only for the central electrode under the 
assumption that the discharge properties for other electrodes are similar. (Here, the peculiarities of the 
corona discharge near the electrodes at the outer boundary of the system were neglected.) 
The same approximation was used to calculate the corona onset atmospheric electric field, E0cor, at 
which the local electric field near the electrode tips reaches the corona onset field, Ecor, and corona is 
ignited. The value of Ecor was determined from the empirical formula suggested by Bazelyan et al. (2007).  

=== PAGE 3 ===
XV International Conference on Atmospheric Electricity, 15-20 June 2014, Norman, Oklahoma, U.S.A. 
 
 
3
 
Fig. 1. The distribution of electrodes over the ground surface. 
 
A quantitative relation between E0cor and Ecor for a given multi-electrode system can be calculated 
using available electrostatic numerical methods. Figure 2 shows the threshold atmospheric electric field 
E0cor calculated with the charge simulation method [Malik 1989] for a system of grounded spherical 
electrodes as a function of the number of circles with surrounding electrodes. The value of E0cor increases 
with the number of surrounding electrodes and is affected even by electrodes located at a distance of 100 
m. This is explained by the fact that the number of surrounding electrodes distributed over a given circle 
increases with the circle radius; that is, the distant circles contain much more surrounding electrodes and 
each of these electrodes makes a contribution into the potential of the central electrode. The value of E0cor 
even for the multipoint system with closely packed electrodes (D/h =0.1) is only double that E0cor for a 
solitary electrode (N = 0).  
 
Fig. 2. The threshold ambient electric field required for corona initiation in a multi-point system as a 
function of the number of circles with surrounding electrodes. The calculation was made for h = 10 m, D = 
1m and r0 = 1 cm. 

=== PAGE 4 ===
XV International Conference on Atmospheric Electricity, 15-20 June 2014, Norman, Oklahoma, U.S.A. 
 
 
4
 
CALCULATED MODEL OF CORONA DISCHARGE 
A physical approach to simulating a non-stationary, streamer-free, glow corona of positive polarity 
initiated from grounded electrodes in an atmospheric electric field and algorithms applicable to the 
simplest electrode geometries has been given in detail elsewhere [Aleksandrov et al. 2002]. In this model, 
the ionization layer in the immediate vicinity of the electrode tip was not considered because its thickness 
is much smaller than the radius of curvature of the tip. Here, the corona-producing surface was assumed to 
be an emitter of ions and the boundary condition for electric field was reduced to a condition widely used 
to determine the current-voltage characteristic of a stationary glow corona in long gas gaps [Raizer 1991], 
namely, that electric field at a coronating surface is equal to the onset corona field, Ecor. For a 
hemispherically-tipped rod with radius r0, the boundary condition was reduced to  
E(r0) = Ecor.          (1) 
For the sake of definiteness, we assumed that an external electric field was produced by a 
time-varying thundercloud negative charge. The expansion of the corona space charge layers was 
described by the electrostatic equation for electric field  
div E(r) = /0      (2) 
and continuity equations for ions 


j
j
j
j
S
E
n
div
t
n





,      (3) 
where  = enj is the space charge density, e is the charge of a singly charged ion, nj and j are the number 
density and mobility of ions of species j, respectively, and Sj is a source term describing ion-molecule 
reactions that affect the ion composition and, hence, the ion transport. The potential  introduced as E = 
-  was assumed to tend to zero at the grounded plane and at grounded electrodes, whereas, away from 
them and from the ion “cloud”, the electric field tended to the undisturbed external electric field, E0(t). 
Electric field above every coronating electrode was calculated taking into account not only the corona 
space charge emitted by this electrode, but the charges emitted by other electrodes as well. The effect of 
these charges was considered approximately, assuming that they are point charges.  
 
NUMERICAL SIMULATION OF CORONA CURRENT AND INJECTED SPACE CHARGE 
Our numerical simulation showed the following peculiarities of a corona discharge from a multi-point 
system. 
Corona current decreases with increasing the number of coronating sources (see figure 3), whereas the 
rate of decrease of the corona current at E0 = const increases in this case. The temporal evolution of the 
corona current, icor(t), is easy to analyze in figure 4 where the values are normalized to the peak corona 
currents, imax.  
In a multi-point system with a few thousand of electrodes, where the corona current is stabilized in a 
linearly rising thundercloud electric field, the value of the stabilized current, icor max is almost independent 
of the electrode height (see figure 5) and depends strongly on the distance between electrodes, D (see 
figure 6). It follows from the data that icor max ~ D2.  

=== PAGE 5 ===
XV International Conference on Atmospheric Electricity, 15-20 June 2014, Norman, Oklahoma, U.S.A. 
 
 
5
 
                                                              
Fig. 3. The evolution in time of the corona current from the top of the central electrode in a multi-rod 
system with rods for h = D = 1 m and r0 = 10-1 cm. The external electric field rises linearly from zero to 
E0m at t < tm and is equal to E0m at t > tm, where E0m = 40 kV m-1 and tm = 1 s. 
 
Fig. 4. The evolution in time of the corona current from the top of the central electrode in a multi-rod 
system with rods for h = D = 10 m and r0 = 1 cm. The external electric field rises linearly from zero to E0m 
at t < tm and is equal to E0m at t > tm, where E0m = 20 kV m-1 and tm = 10 s. 

=== PAGE 6 ===
XV International Conference on Atmospheric Electricity, 15-20 June 2014, Norman, Oklahoma, U.S.A. 
 
 
6
 
 
Fig. 5. The evolution in time of the corona current from the top of the central rod in a multi-point 
system with rods of height h = 10 and 50 m. The number of circles with surrounding rods is N = 50. Other 
conditions are similar to those in figure 4. 
 
 
Fig. 6. The value of the stabilized corona current from the top of the central rod in a multi-point system 
with rods of height h = 10 m as a function of the distance between electrodes. The number of circles with 
surrounding rods is N = 50. The external electric field rises linearly from zero to 40 kV m-1 for 30 s.  

=== PAGE 7 ===
XV International Conference on Atmospheric Electricity, 15-20 June 2014, Norman, Oklahoma, U.S.A. 
 
 
7
 
The time it takes to saturate the corona current for a multi-point system in a linearly rising external 
electric field also depends on the distance between electrodes; this dependence is close to a linear one (see 
figure 7).  
 
Fig. 7. The time it takes to saturate the corona current for a multi-point system in a linearly rising 
external electric field as a function of the distance between electrodes. Conditions are similar to those in 
figure 6. 
  
Analysis of our calculations shows that the properties of a multi-point coronating system 
asymptotically tend to those of a prefect emitting plane with the surface electric field that is equal to the 
corona onset atmospheric electric field Е0cor [Bazelyan et al. 2008]. Stabilization of the surface electric 
field is due to ion emission. Indeed, the plane space charge layer and its image in the conducting ground 
form a double electrostatic layer; that is, the electric field is equal to E0(t) at the upper boundary of the 
layer and to E0cor at the ground surface. In this case, it follows from the Poisson equation (the Gauss 
theorem) that, to stabilize the surface electric field at the level E0cor, the corona space charge injected into 
the atmosphere per unit area must be [Bazelyan et al. 2008]  
]
)
(
[
)
(
cor
E
t
E
t
q
0
0
0



.        (4) 
Then, the corona current density is expressed as 
dt
t
dE
dt
dq
t
jcor
)
(
)
(
0
0
0




.      (5) 
It follows from (5) that in the asymptotic limit the corona current density depends only on the rate of rise 
of the external electric field, E0(t). In particular, the current must be constant for a linearly rising electric 
field and must tend to zero for a constant electric field. It is precisely this manner of the temporal 

=== PAGE 8 ===
XV International Conference on Atmospheric Electricity, 15-20 June 2014, Norman, Oklahoma, U.S.A. 
 
 
8
evolution of the corona current is obtained from our calculations for multi-point systems when the number 
of coronating electrodes is sufficiently large. The current through one electrode in multi-point systems 
studied is obtained by taking the product of jcor and the area per one electrode in the system, S = D2N2/nel, 
where N is the number of circles covered with electrodes and nel is the total number of electrodes in the 
system. Then, we have  
dt
t
dE
n
N
D
t
j
n
N
D
t
i
el
cor
el
cor
)
(
)
(
)
(
0
2
2
0
2
2




.   (6) 
From (6), icor max ~ D2, in agreement with our calculations (see figure 6). Moreover, there is good 
quantitative agreement between equation (6) and our calculated results. For instance, it follows from the 
results shown in figure 6 that icor max = 5.04 μA for the system with D = 20 m, whereas the current obtained 
from (6) under the same conditions is 4.85 μA. Here, the difference is less than 5%. 
The calculated corona current actively increases in time due to the development of individual 
corona space charges from their sources until a united corona space charge layer is formed. In the end, 
individual space charges unite into one plane corona space charge layer (see figure 8) and then the model 
of emitting plane (equations (4) and (5)) becomes adequate.  
 
Fig. 8. A schematic diagram of the space charge layer formed above a ground multi-points system in an 
atmospheric electric field E0.  
 
According to our calculations, the duration of the phase of active current growth in a multi-point 
system corresponds to the time it takes for the fronts of the individual space charge “clouds” to develop 
from the coronating sources until the formation of a united space charge layer. This time can be estimated 
as the time when the radius of the front of an individual space charge “cloud”, Rf, reaches D/2 (see figure 
9).   

=== PAGE 9 ===
XV International Conference on Atmospheric Electricity, 15-20 June 2014, Norman, Oklahoma, U.S.A. 
 
 
9
 
Fig. 9. The evolution in time of the radius of the front of an individual space charge “cloud” 
developed from a central electrode in a multi-point system with D = 20 m. Conditions are similar to those 
in figure 6. 
 
It may be concluded that, to calculate the corona current emitted from a unit area of the ground 
surface during thunderstorms, there is no need to consider geometry of coronating extremities on the 
ground surface. With a good accuracy, current density could be estimated from the rate of rise of an 
undisturbed thundercloud electric field using equation (5). The corona space charge emitted from a unit 
area of the ground surface can be estimated in a similar way. From (4), this charge depends on the 
geometry properties of a coronation system only indirectly, via the corona onset atmospheric electric field, 
E0cor. Under most practically important thunderstorm conditions, we have E0 >> E0cor. In this case, the 
value of q turns out to be independent of the system parameters and is equal to  
qmax  ε0E0max ,           (7) 
where E0max is the peak thunderstorm electric field. For instance, we have qmax  0.53 μC m-2 for E0max = 60 kV 
m-1 [Soula and Chauzy 1991].  
 
ELECTRIC FIELD PROFILES ABOVE MULTI-POINT CORONATING SYSTEM  
Our calculations showed that corona properties for a multi-point system are controlled by an 
undisturbed thundercloud electric field, E0(t). Its direct measurement is not easy to make because of the 
effect of corona space charge layer. The local electric field near coronating sources is stabilized at the 
level of the corona onset electric field. Electric field in the corona space charge layer is lower than E0 due 
to this charge and, only outside of the layer (outside of the double electrostatic plane layer), a 
thundercloud electric field is not disturbed. 

=== PAGE 10 ===
XV International Conference on Atmospheric Electricity, 15-20 June 2014, Norman, Oklahoma, U.S.A. 
 
 
10 
 In an 1D approximation, electric field profiles above an emitting plane can be exactly found from 
equations (2) and (3) in an analytical way [Bazelyan et al. 2014b]. Figure 10 shows the temporal evolution 
of the electric field at different altitudes in this case when the thundercloud electric field rises linearly up 
to 60 kV m-1 for 30 s and then is kept constant. Electric field at any altitude is equal to the thundercloud 
electric field until the front of the space charge layer reaches this altitude. Then, the local electric field, 
E(t), is stabilized. Stabilization is obtained only for a linearly rising thundercloud field, E0(t) ~ t. In the 
general case the local electric field inside the corona space charge layer increases in time for d2E0/dt2 > 0 
and decreases in time at d2E0/dt2 < 0. This means that a sensor, being placed inside the corona space 
charge layer, registers a local electric filed that not only can differ quantitatively from the undisturbed 
thundercloud electric field, but can have even opposite temporal tendency as well. This is demonstrated in 
figure 11 that shows the temporal evolution of the electric field at different altitudes above an emitting 
plane when the thundercloud electric field E0(t) rises in time in a relaxation manner, 






/
max
)
(
t
e
E
t
E
1
0
0
.     (8) 
Here, we have d2E0/dt2 < 0 and the local electric field inside the space charge layer decreases in time 
although dE0/dt > 0. 
 
 
Fig. 10. The evolution in time of the electric field at different altitudes above an emitting plane at Е0cor 
=1.65 kV m-1. The dashed curve corresponds to the thundercloud electric field that rises linearly in time up 
to E0 max = 60 kV m-1 for tm = 30 s and then is kept constant. 

=== PAGE 11 ===
XV International Conference on Atmospheric Electricity, 15-20 June 2014, Norman, Oklahoma, U.S.A. 
 
 
11 
 
Fig. 11. The evolution in time of the electric field at different altitudes above an emitting plane at Е0cor 
=1.65 kV m-1. The dashed curve corresponds to the thundercloud electric field that varies as (8) at E0 max = 
60 kV m-1 and τ = 10 s. The arrows indicate the instants at which the top boundary of the space charge 
layer reaches given altitudes.  
 
Fig. 12. The evolution in time of the electric field at different altitudes above the central rod in a 
multi-point system with rods of height h = 10 m and radius r0 = 2 cm. The distance between rods is D = 10 
m. The number of circles with surrounding rods is N = 100. The altitude is reckoned from the ground. The 
dashed curve corresponds to the thundercloud electric field that rises linearly in time up to E0 max = 60 kV 
m-1 for tm = 30 s and then is kept constant. The arrows indicate the instants at which the top boundary of 
the space charge layer reaches given altitudes. 

=== PAGE 12 ===
XV International Conference on Atmospheric Electricity, 15-20 June 2014, Norman, Oklahoma, U.S.A. 
 
 
12 
 
Stabilization of the thundercloud electric field at t > tm leads to a collapse of the corona current. In this 
case, the corona space charge layer ascends and expands because the top front of the layer moves with a 
velocity vf  = E0max, whereas the velocity of the bottom boundary of the layer is lower, vb = E0cor.    
The total electric field behind the top front of the layer decreases in time and tends to E0cor, the electric 
field at the bottom boundary of the layer.  
 Figure 12 shows the temporal evolution of the electric field inside the space charge layer above a 
model multi-point coronating system. The distance between the rods in the system was equal to the rod 
height. Similarity between the data in figures 12 and 10 is close. In both cases, the total electric field E(t) 
(i) is close to the undisturbed thundercloud electric field, E0(t), at altitudes above the space charge front, 
(ii) is stabilized (although with some delay) inside the space charge layer at E0 = At and (iii) sharply 
decreases at E0 = const. Our calculations show that the vertical electric field profile above a multi-point 
coronating system tends to the electric field profile above a plane surface emitting ions as the number of 
electrodes in the system increases. 
 
CONCLUSIONS 
The developed computer model allows quantitative estimation of the properties of a non-stationary 
glow corona in the system of grounded hemispherically-tipped electrodes in a thundercloud electric field 
E0. The properties of the multi-point coronating system asymptotically tend to those of a prefect emitting 
plane with the surface electric field that is equal to the corona onset atmospheric electric field Е0cor. The 
field Е0cor is controlled by the dimensions of the individual electrodes and by the distance between them. It 
is shown that the model of an emitting plane is valid when the individual space charge layers from 
different coronating points reach each other and form a unite plane layer. The time it takes for the 
formation of the united layer depends on the distance between coronating electrodes. 
In the asymptotic approximation, the corona current density is equal to ε0dE0/dt. In this case, the 
current through each coronating point is independent of the dimensions of the electrodes and depends only 
on the distance between them. The total corona space charge injected into the atmosphere per unit area of 
a multi-point system tends asymptotically to the expression q = 0(E0 - E0cor) and depends on the 
geometrical parameters of the electrodes only indirectly, through the corona onset atmospheric electric 
field E0cor. Under practically important thunderstorm conditions, it is generally follows from field 
observations that E0 >> E0cor. In this case, the value of q turns out to be independent of the system 
parameters.  
The vertical electric field profile above a multi-point coronating system tends to the electric field 
profile above a plane emitting surface as the number of electrodes in the system increases. As a result, the 
electric field distribution tends to be independent of the height of coronating points, whereas the spacing 
between the electrodes affects only the time it takes to stabilize the electric field profile. 
Electric field at a given altitude above the ground coronating surface in a thundercloud electric field is 
equal to this field until the space charge layer reaches this altitude. The evolution in time of the electric 
field E measured in the space charge layer depends on the rate of change of the thundercloud electric field 
Е0. The field E (i) undergoes a stabilization when the value of Е0 rises linearly in time, (ii) increases in 
time at d2E0/dt2 > 0 and decreases in time at d2E0/dt2 < 0. Consequently, simultaneous measurements of 

=== PAGE 13 ===
XV International Conference on Atmospheric Electricity, 15-20 June 2014, Norman, Oklahoma, U.S.A. 
 
 
13 
electric field at various levels could produce not only various results, but radically different evolutions in 
time as well. 
ACKNOWLEDGMENTS 
This work was partially supported by the Russian Ministry of Education and Science under the program 
“5Top100”. 
REFERENCES 
Aleksandrov, N. L., Bazelyan, E. M., Carpenter Jr., R. B., Drabkin, M. M., Raizer, Yu. P., 2001: The effect of coronae 
on leader initiation and development under thunderstorm conditions and in long air gaps. J. Phys. D: Appl. Phys., 
34, 3256-3266. 
Aleksandrov, N. L., Bazelyan, E. M., Drabkin, M. M., Carpenter Jr., R. B., Raizer, Yu. P., 2002: Corona discharge at 
the tip of a high object in the electric field of a thundercloud. Plasma Phys. Rep.., 28, 953-964. 
Bazelyan, E. M., Raizer, Yu. P., 2000: Lightning attraction mechanism and the problem of lightning initiation by 
lasers. Physics-Uspekhi, 43, 753 – 769. 
Bazelyan, E. M., Aleksandrov, N. L., Raizer, Yu. P., Konchakov, A. M., 2007: The effect of air density on 
atmospheric electric fields required for lightning initiation from a long airborne object. Atmos. Res., 86, 
126-138. 
Bazelyan, E. M., Raizer, Yu. P., Aleksandrov, N. L., 2008: Corona initiated from grounded objects under 
thunderstorm conditions and its influence on lightning attachment. Plasma Sources: Sci. Technol., 17, 024015 
(17pp). 
Bazelyan, E. M., Raizer, Yu. P., Aleksandrov, N. L., 2014a: Non-stationary corona around multi-point system in 
atmospheric electric field: I. Onset electric field and discharge current. J. Atm. Solar-Terr. Phys., 109, 80-90. 
Bazelyan, E. M., Raizer, Yu. P., Aleksandrov, N. L., 2014b: Non-stationary corona around multi-point system in 
atmospheric electric field: I. Altitude and time variation of electric field. J. Atm. Solar-Terr. Phys., 109, 91-101. 
Becerra, M., 2013: Glow corona generation and streamer inception at the tip of grounded objects during 
thunderstorms: revisited. J. Phys. D: Appl. Phys., 46, 135205. 
Malik, N. H., 1989: A review of the charge simulation method and its applications. IEEE Trans. Electr. Insul., 24, 
3-20. 
Mokrov, M. S., Raizer, Yu. P., Bazelyan, E. M., 2013: Development of a positive corona from a long grounded wire 
in a growing thunderstorm field. J. Phys. D: Appl. Phys., 46, 455202. 
Raizer, Yu. P., 1991: Gas Discharge Physics, Springer, Berlin, Germany. 
Soula, S., Chauzy, S., 1991: Multilevel measurement of the electric field underneath a thundercloud. 2. Dynamical 
evolution of a ground space charge layer. J. Geophys. Res., 96, No D12, 22327-22336.