--

y

Indian Journal of Radio & Space Physics Vol. 3 1 , February 2002, pp. 2 1 -27

Electric fields due to lateral corona current

M P Singh

Irrigation Research Institute, Roorkee 247 667

and

Jagdish Rai

Department of Physics, University of Roorkee, Roorkee 247 667

and

J S Tomar

Department of Physics, DA V College, Muzaffarnagar, U P

Received 19 February 2001;revised received 6 July 2001; accepted 7 September 2001

In this paper calculations have been made for the electric fields due to lightning lateral corona currents. The electrostatic, induction and radiation fields have been found to be highly dependent on ground conductivity. It has been found that the cut-off distances for electrostatic and induction fields increase with the increasing conductivity of the soil. The ratio of horizontal to vertical field components becomes independent of time after a distance of above 5 km and is only a function of ground conductivity.

1 Introduction

Electromagnetic radiation from the lightning has been studied by many workers l .13, who have considered the return stroke as a vertical dipole with positive and negative charge centres over a closed surface. Also, they have dealt with the case above a perfectly conducting ground. Levine and Meneghini 14 considered the lightning with arbitrary orientation, and have used an approximation where the current wave propagation velocity equals the velocity of light. It is well known, however, that the conductivity of the ground is finite and current wave propagation velocity is less than the velocity of light. This apart, most of the lightning return strokes become horizontal after entering into the cloud and extend many kilometres in length8,15,16.

The horizontal and vertical electric field compo­nents due to an arbitrarily oriented lightning channel above a finitely conducting ground have been studied by some workers6,17 , I9, The theory has been applied to the vertical return stroke and horizontal lightning, Rao l2 calculated the vertical component of ELF field from corona current over a perfectly conducting ground. Pathak8, in his investigations on radio spectrum of atmospherics, included the corona current in his studies. However, no rigorous calculations of

vertical and horizontal components of electric field have yet been considered. The purpose of the present paper is to calculate parallel and perpendicular components of electric field above an imperfectly conducting ground due to the corona current.

2 Theoretical background

In a thundercloud there is continuous generation and separation of charges due to certain charge generation mechanism. Due to this, the lower base of the cloud contains net negative charge, while the top of the cloud contains net positive charge. The accumulated charge in the lower base of the cloud funnels down to a distance of about tens of centimetre, thereby increasing the potential till the breakdown occurs. This initial breakdown progresses earthward in steps and is known as stepped leader. When stepped leader reaches close to earth, the air between this positive tip and ground breaks down. A highly luminous discharge moves continuously towards cloud, This is known as return stroke.

The return stroke tip essentially moves in the partially ionized channel left by the stepped leader. The propagation of stepped leaders and return strokes can be described 11,20 by considering the potential gradient waves.

22 INDIAN J RADIO & SPACE PHYS, FEBRUARY 2002

In general, the lightning channel is assumed to be straight whose length increases with time, and time­dependent current flows through it.

The vector potential of such a current, in general, at the observation point, has been described successfully by Divya and Rai l 7 and Rai2 1 • Applying appropriate Maxwell equations, the vertical and horizontal electric field components are calculated from the vector potential A and image vector potential Am at a point of observation P (Fig. 1 ), as follows: .

dE)t) = dz' 1 47rEo [/1 (8 , ¢) 1 r3 f� I (z ' , ( ' ) dt'

+ 12

(8, ¢) 1 r2c I ( z' , t ' ) + 1

2(8 , ¢ ) 1 rc2d I ( z' , t ' ) / dt] . . . ( 1 )

where, dEy (t) i s the vertical component of the electric field due to dz' and r the radial distance from the point of observation.

z z

� � / I r , I , , / , Y I I I "-, $l '\ , , I X '\ , X �

H P ( r, 8, � )

z

o

� '

H

Z l Fig. I -Geometry o f an arbitrarily oriented lightning channel ( P is the point of observation in space)

dz'=V.dt' . . . (2)

Similarly, the parallel component dEh(t) for the same can be written as:

dE , ( t ) � dz' 1 4""0 [{f3 (9 ,�) 1 r3 1 I (z' , t ' ) dt'

+ f3 (8, ¢) / r 2c l (z' , t ' )

+ 14 (8 ,¢ ) 1 rc 2 dl(z' , t ' ) 1 dt Y + {ls (e, ¢) l r3 l1 ( Z' , t ' ) dt '

+ fs (8, ¢) / r 2c l (z ' , t ' )

+ fs (e, ¢) / rc 2 dl I z' , t ' ) / dt }2 J

I I 2 . . . (3)

The functions It(8, ¢), 12( 8, ¢), h(8, ¢) , 14(8, f/J) and

Is( e, ¢) depend upon the orientation of the lightning source and the point of observation and also upon the ground conductivity. Functions II (e, ¢), h(e, ¢) and

Is( e, ¢) correspond to electrostatic field, h( e, ¢), 13( e, ¢) and Is( e, ¢) to induction field and 1

2( e, ¢),

14( e, ¢) and h( e, ¢) to radiation field components of atmospherics and are given as:

II (8, ¢)=( l +Rv)[cos 81 (2-3sin2 8) +3 sin el sin 8cos 8cos(¢I-¢)]

Iz (e, ¢)=( 1 +Rv)[-cos el sin2 e + sin 81 sin ecos eCos(¢I-¢)]

13 (8, ¢)=( 1 +Rv)[ 3 cos 81 sin 8 cos e + sin 81 (3 sin2 8- 1 ) COS(¢I-¢)]

14 (8, ¢)=( 1-Rv)[ cos el sin e cos e -sin el coi ecos(¢I-¢)]

Is (e, ¢)=( 1 +Rh) [sin el sin (¢I-¢)] . . . (4)

Here, (r, e, ¢) are the polar co-ordinates of the point of observation with respect to source and image, respectively. Angle el is the orientation of the lightning channel from the vertical and ¢I is the angle between x-axis and the channel ' s projection in the x-y plane (Fig. 1 ) . The parameters Rv and Rh are the reflection coefficient of the vertical and horizontally polarized components of atmospherics and are given by:

Rv=[ I (E+cr/iw)/Eo ) cos \JI-I (E+cr/iw)/Eo - sinz \JI ) '12]1 [ { (E+cr/iw)/Eo } cos\JI + { (E+cr/iw)/Eo - sin2 \JI } \/']

. . . (5)

-

-r

t-

T

,.

SINGH et at.: ELECTRIC FIELDS DUE TO LATERAL CORONA CURRENT 23

and

Rh=[cos \If- { (e+cr/iw)/Eo - sin2 \If } 'h]/ [cos \If + { (e+cr/iW)/Eo - sin2 \If } II,] . . . (6)

where, \If = 1t-8, E is the permittivity of the medium, Eo the permittivity of the free space, cr the ground conductivity and W the angular frequency of the wave.

The lateral corona current is generally taken as vertical flowing through the return stroke channel and extending from ground to the cloud base. Therefore, we put 8, = 0 i n the above equations. Srivastava22

obtained a double exponential velocity expression from the photographs of the lightning channeI3. 1 3.23

and showed that the double exponential velocity expression accounts well for the electric field observations on the ground surface. Rai I I obtained the double exponential expression theoretically as:

V 11 ( --<II -hI) t= YO e -e

where, Vo=9.0x I 07 ms-I ; a=6.0x l 04 S- I and b=7.0x l 05 S-I

(7)

The expression for lateral corona current is given by lS

ic(z,t)=KV2VoIab [(b-a) e-211CR _be-(a+21CR)1 +ae-(b+2/CR)I] . . . (8)

where, K is the lateral corona current constant, V the potential difference between the return stroke and the leader sheath in volts, C the distributed capacitance of the leader sheath-return stroke core in Farad/m, R the distributed resistance of the above configuration and Vo is the velocity constant of the return stroke. The values of these parameters are a=6.0x I 04 S-I , b=7.0x I 05 S-I , V= 1 .0x I08 volts, Vo=9.0x I 07 mis, CR=4.6855 S, K= 1 .0x l O-16 A y-2 m-I S-I . From the above current and velocity expressions we obtain EyCt'), Eh(t) by using Eqs ( l ) and (3). The parallel and perpendicular components of the electric field are then calculated. The total electric field in the parallel and perpendicular components separately is the sum of electrostatic, induction and radiation fields. The ratio of the parallel and perpendicular components of electric field R is given by

R=I E,,(t) / Ey(t) I . . . (9)

The ratio R is time dependent and is a function of ground conductivity cr.

3 Results A computer programme developed in Fortran was

run on PC 486 to solve Eqs ( 1 ), (3) and (9) by using the current Eqs (8) for the solution of parallel and perpendicular components of atmospherics. The calculations have been made in time domain for different parameters, viz conductivity, maximum current and velocity of current waveform.

Close to a l ightning discharge the electrostatic field is dominant and at a certain distance the induction field becomes dominant. Further apart the radiation field dominantes and continues so till very large distances. Figures 2 and 3 show the variation of cut -off distance for horizontal and vertical field components (curve 1 is a plot where the dominance of electrostatic field finishes and curve 2 the distance where induction field vanishes and radiation term starts) with conductivity of the soil . The cut-off distances for electrostatic and radiation fields increase with increasing conductivity of the soi l . In Fig. 2, for dry sandy soil (cr= I O-3 mho/m), the electrostatic cut­off distance Dee is 20 km, while for wet soil i t is 35 km. For mountainous terrain it i s as low as 5 km. The corresponding values for induction field cut-off distance Dic are 30, 50 and 1 0 km, respectively. For still higher values of conductivity, Die increases very fast. For the horizontal component, Dee is 5 km for mountainous region, 1 0 km for sandy soil and 20 km for wet soil . The corresponding values of Die are 1 0, 25 and 35 km, respectively. For conductivity of 1 0- 1 mho/m (water surface) the two curves merge together, indicating that there is no presence of the i nduction field.

5 0r---------------------------------,

E 40 ..: ui U Z � '0 (/) Ci u. u. 2 2 0 ::J U

1 0

RADIATION FIE l D

E l E C TI10STATIC F i E l D

CONDUCTIVITY ( cr ). mho/m

Fig . 2-Cut-off distances for electrostatic and radiation fields for horizontal field component

24

1 0 0

.0

e o

7 0

� 6 0 ui U Z ;S 5 0 (/) 0 LL LL 0 ' 0 ...:. ::> u

30

20

1 0

10�t..

' .

INDIAN J RADIO & SPACE PHYS. FEBRUARY 2002

RADIATION FIEL 0

E L E CTROSTATIC F IE L D

CONDUCTIVITY ( a ). mho/m

The calculations of the time variation of electrostatic field for different distances have been shown in Figs 4-6. At 2 km the maximum field decreases with increasing conductivity. So is the case for 20 km. However, for a conductivity of 10-4 mho/m the electrostatic field has a sharp peak at 10 km. This peak disappears as the conductivity increases. Similarly, the induction field at a given time decreases with increasing conductivity, and the behaviour at 25 km for 10-4 mho/m conductivity is quite different. This peak disappears for higher conductivities. The radiation fields start falling from a peak value at 5 �s and no peculiar behaviour at any particular time has been observed for the whole conductivity range in the present calculations .

Fig. 3-Same as Fig. 2, but for vertical field components

Calculations for the ratio Eh/Ev of electric fields with time have been performed for distances ranging from 100 m (from the foot of the channel) to 100 km. As an example, these calculations have been shown for distances 100 m, I km, 5 km and 20 km [Figs 7-9] distances. For 100 m the ratio peaks at 2 �s becoming very small after about 5 !!s. The ratio at 2 �s increases with decreasing conductivity. The non-linearity remains up to about 1 km. However, at this distance two peaks have been found. A negative peak at 10 �s and a positive peak at 20 �s is visible in Fig. 8 . The

zoo�--------------------------------------------·-------.

__ ---�------------------------------------rokm 2 0 0 �-----------------------IOkm

. 0 0 __ -----,km E F I ELD, X 1 0-5 Vim :> 600 T I M E , � s III 0 X 6 00

0 Skm ...J LLJ u::: 1000

12 00

.. 00

10 20 JO '0 50 60 70 eo 90 100 T I M E, � s

Fig. 4-Time variation of electric field due to corona current at different distances for ground conductivity 1 0-4 mho/m

SINGH et al.: ELECTRIC FIELDS DUE TO LATERAL CORONA CURRENT 25

20

0 20 km

10 km

20

§ .. '0

... X c::i ...J UJ iI: 5 km

I�·L--T-----T-----T-----T-----r-----r-----r-----r-----r------� , 10 1 0 40 '0 60 70 80 '0 l Oa

TIME, 11 s

Fig. 5-Time variation of electrostatic field due to corona current at different distances for ground conductivity 1 0-2 mho/m

80r-----------------------------------------------------�

60

.E > 1 40 o ...

ci -' w u: 20

5 10 2 0 30

2 k m

50 50 70 80 90

TIME, !!s

Fig. 6-Time variation of radiation field due to corona current at different distances for ground conductivity 10-2 mho/m

ratio becomes almost constant after about 5 km (Fig. 9) and is a function of only ground conductivity.

4 Discussion Divya and Rai 17 have obtained the ratio of parallel

to the perpendicular component of atmospherics in frequency domain. They found that the ratio is independent of the source parameters and is only a function of frequency and soil conductivity. Hazarika24 verified experimentally the theoretical

findings of Di vya and Rai l 7 at a frequency of 5 kHz. Divya and Rai 17 found that the radiation field becomes independent of distance of propagation after about a distance of 40 km. The upper l imit of this distance is beyond 1 00 km til l the ionospherically reflected components become important. However, they completely excluded the electrostatic and induction fields from their studies. The present study takes into account the electrostatic, induction and radiation fields and it has been found that for

26

· 30

INDIAN J RADIO & SPACE PHYS. FEBRUARY 2002

distances greater than 5 km, the ratio is constant with time_ Further, the present study corresponds to lateral

cr = 1 0-4 corona current, while the works of Rai 2 1 and Divya ...

b 200 � x 0 � � 100

cr = 1 0-2

° 1 4 TIME, J.ls

5 , 7 8 9 '0'

and Rai 17 correspond to the return stroke. The lightning electric fields from lateral corona

currents are highly dependent on soil conductivity . The cut-off distances for electrostatic, induction and radiation fields increase with increasing conductivity. The ratio of parallel to the perpendicular components becomes constant after a distance of 5 km. This information can be used in geophysical exploration as predicted by Divya and Rai l7 for return strokes.

Fig. 7-Time variation of electrostatic field amplitude ratio R at a distance of 100 m for different ground conductivities ( in mho/m)

In the present calculations we have used the soi l conductivities in the range 10-4_ 1 0-1 mho/m. Leite and Barker25 have used the ground conductivity in the range 1 0-3_ 1 0-2 mho/m in their calculations. Mallick

x

5 6 7 I , 101 TIME, J.ls

2

- 4 1(010

Fig. 8-Time variation of electrostatic field amplitude ratio R at a distance of I ()()() m for different ground conducti vities (in mho/m)

x o � <! cr:

0

40

8C

12(

16(

z J 4 5 6 7 I , lot z , 4 TIME, J.l S

L a = 1 0- 1

5

L a = 1 0-2

L a = 10-3

a = 1 0'"

, 7 8 9 16

Fig. 9-Time variation of electrostatic field amplitude ratio R at a distance of 5 km for different ground conductivities ( in mho/m)

. -<t- -

SINGH et 01.: ELECTRIC FIELDS DUE TO LATERAL CORONA CURRENT 27

and Verma26 considered the homogeneous, two layer, three layer and four layer cases of earth in their studies on electromagnetic prospecting. Excepting the homogeneous case, the half-space in two, three and four layer cases has the ground conductivity of about 5 .0x I0-4 mho/m. They have given that the soil conductivity varies from 1 .5x1O-2 mho/m to 5.0x 1 O-4

mho/m for the homogeneous case. Verma and Bhuin27

have observed the peak conductivity values for different coal seams to vary from 2 .4x 1O-3 to 10-3

mho/m. The ground soil conductivity of coal seams in B ihar (India) area was determined and reported28 to be in the range of 10-4_ 1 0-3 mho/m. Divya29 has shown that the muddy soil has conductivity of 10-1 mho/m, while the dry mountainous terrain has the conductivity of 1 0-5 mho/m. Gokarn et al.3D and Gupta3 1 showed that the top layer ground electrical conductivity varies from O.5x 1 O-3 to O.33x 1 O-3

mho/m. Xu et al.32 showed that the ground electrical conductivity of earth's mantle is influenced by many factors like temperature, pressure and coexistence of mUltiple mineral phases in the ground soil. These phases can affect the ground soil conductivity. They found that the top layer electrical conductivity varies from O.Olx1O-3 to 6.3x 1 O-3 mho/m. The above discussion shows that the soil electrical conductivity generally varies from 1 0-4 to 1 0-1 mho/m. Therefore, these values have taken into account for the present calculations.

References 1 Bruce C E R & Golde R H, J Instn Elect Engrs ( UK), 88

( 194 1 ) 487. 2 Evans W F J, Rainhart C & Puckrin E, J Geophys Res lett

(USA), 22 ( 1 995) 2 1 35 . 3 Iwata A, Proc Res Inst Atmos Nagoya Univ (Japan), 1 7

( 1 970) 1 1 5. 4 Khastgir S R & Saha S K, J Atmos & Terr Phys (UK), 34

( 1 972) 1 1 5 . 5 Kumar K, Studies on the Vertically and Horizontally

Polarized Components of Atmospherics, Ph D Thesis, University of Roorkee, Roorkee, 1 988.

6 Kumar R, Lightning and Its Interaction with the

Troposphere, Ph D Thesis, University of Roorkee, Roorkee, 1994.

7 McLain D K & Uman M A, J Geophys Res (USA ), 76 ( 1 97 1 ) 2 10 1 .

8 Pathak P P, Rai J & Varshneya N C, J Geophys J R Astr Soc ( UK), 69 ( 1 982) 1 97.

9 Rai J, Rao M & Tantry B A P, Nature (UK), 238 (1 972) 59. 10 Rai J, Gupta S P & Bhattacharya P K, Int J Electron (UK),

36 ( 1974) 649. 1 I Rai J, J Atmos & Terr Phys (UK), 40 ( 1 978) 1 275. 1 2 Rao Manoranjan, J Radio Sci (USA), 2 ( 1 967) 24 1 . 1 3 Schonland B F J, Hodges D B & Collens H , Proc R Soc

Edinb A (UK), 1 66 ( 1 938) 56. 14 Levine D M & Meneghini R, J Geophys Res (USA), 83

( 1 978) 2377. 1 5 Brantley R D , Tiller J A & Uman M A, J Geop/zys Res

( USA), 80 ( 1 975) 3402. 1 6 Teer T L & Few A A, J Geophys Res(USA), 79 ( 1 974) 3436. 1 7 Divya & Rai J , Indian J Radio & Space Phys, 1 5 ( 1986) 96. 1 8 Master M J , Uman M A , Lin Y T & Standler R B , J Geophys

Res (USA), 86 ( 1 98 1 ) 1 27. 1 9 Nikoshinen K I & Eloranta E H . Radio Sci (USA ). 3 1 ( 1 996)

03447. 20 Loeb L B, J Geophys Res ( USA), 73 ( 1 968) 5 8 1 3 . 2 1 Rai J , Ann Geophys (France), 3 6 ( 1980) 263. 22 Srivastava K M L, J Geophys Res ( USA), 7 1 ( 1 966) 1 283. 23 Rai J & Bhattacharya P K, J Geophys (Germany), D 4 ( 1 97 1 )

1 252. 24 Hazarika S, Some Experimental Studies on Lightning and

VLF Atmospherics, Ph D Thesis, University of Roorkee, Roorkee, 1 987.

25 Leite J L & Barker R D, Geoexploration (Netherlands), 16 ( 1 978) 25 1 .

26 Mallick K & Verma R K, Geoexploration (Netherlands), 1 6 ( 1 978) 29 1 .

27 Verma R K & Bhuin N C, Geoexploration (Netherlands), 17 ( 1 979) 1 63 .

28 Koyal J R, Geoexploration (Netherlands), 1 7 ( 1 979) 243. 29 Divya, Studies on the VLF Emissions from Lightning, Ph D

Thesis, University of Roorkee, Roorkee, 1986. 30 Gokarn S G, Rao C K, Singh B P & Nayak P N, Phys Earth

& Planet Inter (Netherlands), 72 ( 1 992) 58. 3 1 Gupta G, Gokarn S G & Singh B P, Phys Earth & Planet

Inter (Netherlands), 83 ( 1 994) 2 1 7. 32 Xu Y, Shankland T J & Poe B T, J Geophys Res (USA), 105

(2000) 27865.