Lightning initiation by simultaneous effect of runaway breakdown and cosmic ray showers

5 April 1999

PHYSICS

ELSEVIER

Physics Letters A 254

LETTERS

A

( 1999) 79-87

Lightning initiation by simultaneous effect of runaway breakdown and cosmic ray showers ’ A.V. Gurevich ‘, K.l? Zybin ‘, R.A. Roussel-Dupre b a i?N. Lebedev Institute of Physics, Russian Academy of Sciences, I 17924 Moscow. Russia b Space and Science Technology Division, Los Alamos National Laboratory, Los Alamos, NM 87545. USA Received 8 January 1999; accepted for publication 21 Communicated by V.M. Agranovich

January 1999

Abstract The combined action of cosmic ray showers (CRS) and runaway breakdown (RB) on thunderstorm atmosphere is studied. The ionization of atmosphere produced by CRS core being’strongly amplified in RB conditions is shown to be sufficient to generate a highly ionized plasma region. Electric polarization of this plasma region is manifested in a significant local amplification of a thunderstorm electric field, what could result in a local electric breakdown and lead to initiation of lightning. Experiments which could check the predictions of the theory are proposed. @ 1999 Published by Elsevier Science B.V.

1. Introduction Following Wilson’s [ 1] suggestion that thunderstorm electric field could accelerate cosmic ray secondary electrons to high energies, researches tried to observe the X-rays produced in air by high energy electrons. However, most of the early observations could not obtain a definite conformation of correlations between lightning flashes and X-rays. This situation changed decisively through the last few years. First of all, McCarthy and Parks [ 21 flew an X-ray detector on an aircraft into a thundercloud and found a significant (two to three orders of magnitude) increase in X-ray intensity. The increase lasts for about ten seconds, what means that the region of intensive Xrays has several kilometers scale. A detailed analysis

’ Note that in the cosmic ray physics “cosmic ray showers” are usually called “extensive atmospheric showers”. 0375-9601/99/$ - see front matter @ 1999 Published PNS0375-9601(99)00091-2

done by McCarthy and Parks [2] showed that the observed effect could not be explained by acceleration of cosmic ray electrons, as supposed in Ref. [ 11. Second, a new physical concept of an avalanche type increase of a number of energetic electrons in air under the action of the thunderstorm electric field was proposed by Gurevich, Milikh, and Roussel-Dupre [ 31. The avalanche can grow in the thundercloud at the heights 3-10 km in electric fields E 2 EC x l-2 kV/cm, which is almost an order of magnitude less than the threshold electric field of conventional air breakdown Ee x lo-20 kV/cm. The growth of the number of electrons with energies E > E, M 0. I- 1 MeV is determined by the fact that under the action of electric field E > EC fast electrons in thundercloud could become runaways, what means that they are accelerated by the electric field E. Due to collisions with air molecules they can generate new fast electrons having energies E > q.. Directly

by Elsevier Science B.V. All rights reserved.

80

A. E! Cureviclz et al. /Physics

&his process-acceleration and collisions lead to the avalanche type growth of the number of runaway electrons, which was called in Ref. [ 31 “runaway air breakdown”. The detailed kinetic theory of runaway air breakdown and its application to the high-altitude lightning phenomena was developed recently by a number of authors (see for example Refs. [ 4-61) . When the electric field in a thundercloud reaches the critical value E > E,, every cosmic ray secondary electron initiates a runaway breakdown. This process called multiple runaway breakdown serves as a source of intensive X-ray emission, observed in Refs. [2,7]. On the other hand, accelerated electrons produce an intensive ionization of the air and as a result a significant growth of electric conductivity, which results in a fast charge transfer in a thundercloud predicted by Gurevich et al. [ 31. The simultaneous measurements during 90 set of X-ray emission and electric field E changes in a thundercloud were done on balloons by Eack et al. [ 71. The detailed comparison of the observational results with multiple runaway breakdown theory [ 81 allowed them to establish the existence of a fast charge transfer process. Runaway breakdown in thunderstorm atmosphere is stimulated by cosmic ray secondaries. In all previous papers it was considered only the main flux of cosmic ray secondaries. This flux is initiated by the cosmic ray particles with the energies EO 3 109-10’o eV and at these energies it is distributed rather homogeneously. The high energy cosmic ray particles with E 2 1014lOi eV fell down rarely. But, on the other hand, they induce cosmic ray showers (CRS) which are accompanied by much more intensive localized flux of secondary electrons. The goal of the present paper is to study the combined effect of runaway breakdown (RB) and cosmic ray showers in thunderstorm conditions. We will show that in the core of the shower the ionization of atmosphere in RB conditions is growing strong enough to produce a local (- 0. l-l m) highly conductive plasma. The thunderstorm electric field is reconstructing in the highly conducting region generating in the 100 ns time scales a Iocalized zone where the electric field is signi~cantly ampli~ed. As a result the conventional electric spark breakdown sets in, what can serve to the triggering of lightning.

Letters A 2% (1999)

79-87

Table I Flux F of CRS with E > ~1) at 10’3 4x I@

&II (eV) F

I km2 per I xc

10’4 102

10’5 2

10’6 2 x 10-Z

2. Secondary electrons in CRS The total number of secondary electrons with energies E > 1 MeV precipitating in air, no, depends on the energy of cosmic ray particle, ~0, generating CRS. According to Belenky [ 91, 0.3Ec (1)

’

no = PJlh?ZDi

where p is constant, for air p z 72 MeV (see Ref. [ 91, Table 1) . The density distribution of secondary electrons in the plane orthogonal to the axis is given by the well-known NKG (Nishimura, Ksimata, Greizen) empirical formulae, pr = 0.366s

~~(2.07 - s)5/4

0

;

2-s

(2) Here r is the distance from the axis (in other words, from the CR particle trajectory), R is a characteristic scale of CRS (for air R M 115m) and s is a nondimensional parameter, depending on the height in atmosphere. For a thunderstorm heights h x 3-10 km one can take 0.8 < s < 1.2. In estimates we will use s= 1,

R= IOOm.

(3)

The flux of CRS in atmosphere depends significantly on the CR particle energy ~0. The results of observations are presented in Ref. [ lo]. The total number F of CRS at 1 km2 per 1 second caused by cosmic ray particles having the energy I > EOis given in Table 1.

3. Avalanche of runaway electrons stimulated CRS

by

The electric field E in a thundercloud is height inhomogeneous, E = E(z) [7,8]. In the vicinity of the maximum value E,, it can be presented in a form

A.V Gwevich

et al./Physics

(4) Here E,,, is a maximal value of the electric field (we choose z = 0 at the maximum point and axis z is directed downwards) and L, is a characteristic altitude size of electric field. As is well known, the runaway avalanche is growing when the electric field is higher than the critical field [3],

Letters A 254 (1999)

both by a newborn electrons CRS electrons D,, [ 111,

E,, =

4rN,,, Ze3

a = 2P( atm) kV/cm

1713

,

(5)

where N,,, is the molecular density, P is the atmospheric pressure, 2 = 14.5 is the mean molecular charge of the air and a = 11 is the Coulomb logarithm. We note also that only electrons having kinetic energy greater than E, M mc2 E,/2E can trigger the avalanche. Taking into account that the cosmic ray secondaries have high enough energy, E 2 1 MeV, we can suppose that they all take part in the avalanche process if condition (5) is fulfilled. The growth of the avalanche takes place in the heights region -L,

< z < L,,

where the value of L, is determined

by the conditions

(4)> (5),

$ _

L,.=L, /--

1.

‘

of

Here D,,b according to analytical [ 121 and numerical (Monte Carlo) methods [ 131 has the value

D nb = 0.03c2th, m2,3

x 2.1 x 10” cme3 S.

4re4 ZN,,a

(8)

N,,,

The dtffuston coefficient D,,, which usual scattering, has a form [ 111

is defined

by

Here Y is the collision frequency of a runaway electron with a nucleus of air molecules,

Here Zerr M ~yZ/2, and LYM 0.5-0.7 is a shielding factor. We took into account in (9)) ( IO) that the cosmic secondaries have relativistic energies. Along the z-axis the beam particles are moving with the high velocity c’, and diffusion in the z-direction is not important. So, finally the equation which describes the avalanche growth and spreading of a beam of runaway electrons N,, has the form

aN, z

Propagating in the region z > -L,, the beam of CRS electrons is not only growing with z with the exponential factor hj,

D,,b and by scattering

D = &b + D,, .

t/1 = E > EC,

81

79-87

a

+

--(‘:‘Nr)

a=N N + -L art r; ’

= D'

Here N, is the density their drift velocity,

of beam electrons

(11) and u: is

z

(12)

N,f rx exp

-L hj=Aj

(1

=(CTi)-',

f

c

but also spreading in the direction perpendicular to the z-axis rl. The spreading process has a diffusive character in the r~. plane. The diffusion coefficient D is determined

The full derivation of Eq. ( 11) was obtained using the kinetic approach to the theory of runaway breakdown in inhomogeneous plasma in Ref. [ 111. To determine the growth of the beam we have to take into account the source of fast electrons also. The source in our case is caused by a flux of secondary CRS electrons generated by a high energy cc cosmic ray particles. Supposing that this particle moves along

82

AI! Cmvich

et a&/Physics

the vertical z we can write the secondary electron flux (SE) q in a form q(rl,

2, t> = cp,(r~)&z

- cr) .

N

Green function C, of Eq. ( 11). After some simple calculations one can find

xs(t-t,

-

p(atm) .

f++(‘))

j&) z1

So SE loose their initial energy on the lengths L x e/l%. Taking this fact into account we can add the source term in Eq. ( 11) in a form

S=$

79-87

(13)

Here pr (rl > is the cross section density of SE determined according to (2). Every SE looses its energy in collisions with air molecules. The friction force F is approximately constant for E 3 I MeV (see Ref. 131)> F e F. = 2 g

Letters A 254 fI999)

b

-

(4

Fo ’

(14)

Here(4 M 30 MeV is an average energy of SE (see Ref. [ 91). We took into account also that the typical velocity of the main bulk of runaway electrons along the electric field v, is Iess than the speed of light c. Identical estimation of the length Le could be obtained if we take into account the diffusion along the z-axis. in fact, using equation

x exp

’ ;Aj

(s

dz2 -

Zi

(Ti

-011*

JZ; (D/v,)

dz2

x B(j$dz2).

(16)

7.1 Integrating (16) with initial conditions ( 15) we find the distribution of fast electrons N,, N, _ 0’8no co la(2nixR2/~) exp [-R2.r:/~] (1 +X,)3.5 K(77)Lo s

dx I

0

R2x2

x exp

--+Aj(77*t)

(17)

KC% t>

one can find that

Here la is a modified Bessel function and

This estimation of the length Lc coincides with a previous one ( 14). The source S ( 14) being included in Eq. ( 11) could be considered as the density N( r, 0) of runaway electrons simultaneously created at the point z in initial moment t = 0 by CRS particle ea,

ql= =J-drli J- Nr),t) x=f. (18) JD(rll)dw, v-1 ‘I

’

=

0%) 2 kV p(atm)

Uz(Zl)

'

0

Ti(rfl)

'

7-f

??

K(q,t) = 2

Taking into account that the diffusion time t&f NN R*/D is big enough (t&f > L, /u, ), we can rewrite the formulae ( 17) in a simple form,

Pe(rl) N(r,O)= -, Lo &

dzt

cm’

(15)

It is natural that for a given atmospheric pressure p the source N does not depend on z. To find the growth of the avalanche of runaway electrons we have to solve Eq. ( 11) with initial conditions ( 15). It is convenient first to determine the

N, = 0.6

exp A; _.fk K&J X( 1 +X)3.5 ’

J;;
(19)

Here no is determined according to formulae ( 1) . The factor Ai characterizes the exponential growth of the number of runaway electrons (7), ( 18).

A.V

4. The distribution runaway ekctrons

Gurevich

et al./Physics

of slow electrons generated by

Letters A 254 (1999)

dNse = ANtln dt

Previously we discussed the runaway particles only. But according to the theory [ 31 under runaway conditions high energy electrons produce either fast or lower energy electrons. Futhermore, the number density of lower energy electrons is essentially higher than the density of fast electrons. To take this fact into account let us consider the production of slow electrons generated by a single high energy electron. According to the theory [ 14], an ionization rate is given by a formulae

79-87

83

(23)

%in

Taking into account the structure of dis~ibution function (22)) we can estimate the number Nt , NI

Thus we see that the ionization rate of slow electrons essentially increase in runaway conditions, (24) or in ionization length terms,

dN,e = A,‘N,,

-

(20) To determine an ionization rate for a monoenergetic beam of electrons with density Nr, one has to choose the distribution function in a form f(g)

= N&E

dt

Q

=G

Nr

(21)

1

or in terms of ionization length /\I, Eq. (21) can be rewritten in a form

dN[, -=: dS

ALIN,.,

= :,

Irim2 -!??-

At = 6 x 10m4cm.

( %lin )

(25)

In numerical estimations we take EO x 1 MeV and eminM 1 keV. As a result the source of slow electrons qe is given by the relation Nr (

Here N, is the number of fast particle which is defined by formulae ( 17). To estimate the ionization produced by runaway electrons it is necessary to determine the ionization parameter .4i. According to analytical [ 3,121 and numerical calculations [5,6] the ionization time ri depends on electric field as

At x 3 x lo-* cm.

Here S is the length in the direction of motion of an energetic electron, and AI is a constant determined for air by Bethe’s theory [ 15 1 and multiple experiments. We have to take into account now that in our case the distribution function f(e) is not monoenergetic but increases significantly at low energies e. According to the theory [ II] and numerical calculations [ 51 the distribution function f(e) cc l/e at ekn < E < EO where EOis the lower boundary of runaway particles. So choosing the distribution function in a form f(e)

& =

qe = CA,’

- q,) .

After a simple calculations, we obtain

dNie

dS

Emin< E <

EO ,

%in -K co ,

(22)

In the following numerical estimations we will use CY= 1. According to ( 12) the drift velocity of fast electrons is proportional to electric field E. Rewriting formulae ( 12) in a form E

u,(z) =uo---

EC

and substituting into ( 18)) one can find the ionization parameter A; =

A0 UO {

one can calculate the avalanche of slow electrons,

z

-

I;,

1 f z/L, - (1 - z/Lc)e’iT 1 + z/L, + f 1 - z/Lc)ef/7 1 ’ (27)

84

A. V Gurevich et d/Physics

Here the scale L, is defined by relation (6)) and time r = EcL,/2voEo. F rom (27) one can see that the point where ionization takes a maximum value moves downwards to the point z = L,. At t > r, the maximum of ionization is located near this point and takes the value 2L, n; Z Ai0 ’

(28)

nouoLc DA,L+)AiO

where the parameters determined according

exp(ZJ~~0) (r/R)

(

1 + r/R)3.5

’

(29)

no, D, L,, A;0 LO and A, are to (I), (9), (6), (26), (25),

(15).

5. Conductivity and plasma polarization runaway breakdown region

in the

Because of runaway breakdown the number density of free electrons in the core of CRS increase exponentially (29). This lead to the strong growth of the conductivity and because of this the local thunderstorm electric field could significantly change too. Let us determine first the conductivity in the core region. To describe this process we can use the hydrodynamic approach,

=

-vaN, f qr ,

dN_ = v,,N, , at

j=g,OE,

creo=-.

e2N,

mvpn

(30)

Here N, is an electron density, N- is the density of negative ions, E is the electric field, v, is a frequency of electron attachment to oxygen atoms and (T,O is an electronic conductivity of plasma determined by collision frequency of electrons with neutrals v,,. These equations are valid in the following conditions, V,,? B Vtr3

of positive N+ and negative ions N- is very small, and their hydrodynamic motion could be neglected. To complete the hydrodynamic system (30) it is necessary to consider the equation for the electric field E, divE=-4rre(N,+N_-N,),

z = L, .

Thus, according to (27), ( 19)) the maximum electron density N,,, of slow electrons at t > r is located near bottom point of runaway region (28) and is equal to N max = 0.3

Letters A 254 (1999) 79-87

(31)

which is always fulfilled at thunderstorm heights [ 83. In (30) we also took into account that the mobility

(32)

and initial conditions N,(r,O)

= N+(r,O),

E(r, 0) = Eo >

N_(r,O)

=0, (33)

where N,(r, 0) is given by the formulae (29). Note that here we took into account that the cosmic ray particle generating by CRS moves through the runaway breakdown zone with the speed of light and because of this the initial conditions (30)) (33) are establishing simultaneously. The electric field EO = E( z = L,). Eqs. (30)-( 33) describe the relaxation of conducting region which is generated in the CRS core. The relaxation process has two characteristic times. One is the electron attachment time v<;’ which characterizes the change of electron density, it is of the order of 10P7 s [ 81. The other one is polarization time t,’ N (4rcr) -‘, characterizing the change of electric field E due to electric polarization of ionized region. In the central region of the core of CRS generated by energetic particle co, rmin < l-10 cm,

(34)

under the RB conditions the maximal growth of electron density for EO > lOI eV near the bottom of runaway region z z L, can be strong enough. According to ( 29)) N,,, can reach here the values N,,, N 3 x 108-10’ cm-j. The following conditions could then be fulfilled in this region, V‘{< t;’

=4%-a.

(35)

In these conditions the polarization process could significantly change the distribution of the thunderstorm electric field. Let us analyze the problem at times 7, which satisfied to inequalities -‘>r>t,,. Vn

(36)

Under conditions (36)) due to the polarization process the electric tield inside the highly conducting re-

A.V Gurevich et (11./Physics

gion will diminish to a very small value (as in usual conducting body). At times

85

Letters A 254 (1999) 79-87

t =4rzEo

E,,

e-‘+‘N, s 0

7 > v,i

,

O,rl,z -zEot

x electrons attache to oxygen molecules fixing the polarization. To determine the boundary of the “conducting region” let us first consider the core, where conditions (36) are fulfilled. Taking into account a small parameter (~~~7)-’ we can neglect attachment in the first approximation. Eqs. (30), (32) take then the “classical” form $

+ div(a,&)

d&E=

(37)

Their solution in initial conditions

> 1.

(41)

we will find from (40), E,: = -4~7~Nov, (N,(O, ~1)) 7 ‘

.

+...

(38)

,

one can find after linearization

(NetOvr~))=

& /- N,(O,rl,z)dz.

of the system (30))

So, we see that the disturbances of electric field El are small and proportional to the disturbances of N,(r) . Thus the boundary of “conducting region” could be defined as 4nu,0Np(r,

N,(r,

0) =;.

0) z vaNo mvrnVcl

For thunderstorm

or (43)

conditions.

ve z 8 x 10” s-’ ,

v,, x lo7 s-’ ,

and as follows from (43)) N,(r,O)

= 3 x lo8 cmm3 .

(44)

It should be emphasized that N, (r, 0) (44) is a high enough value. It could be reached only at the core region of strong CRS (EO > lOI eV) under the RB conditions.

6. Electric field amplification initiation

C?N_ = v,,Nr .

(42)

-L;

(33) is well known,

Here we took into account that the change of electron density is very small, N, M N, (r, O), because of quasineutrality conditions rmin > RD, where RD = ( T/4?re2 N,) ‘j2 is Debye radius, and T is the temperature of slow electrons (practically equal to the temperature of the air). At N, M IO8 cmd3, RD z low2 cm. So due to polarization the value of the electric field inside the conducting region is falling down with time scale t,, = (4ra)-i. Now let us on the contrary analyze the solution of (30) in the region when the density N, is small enough and condition Y, > 477-a is fulfilled. In this case taking E=Eo+E,

4lrs

L:

- N,).

E = EO exp( -47rg,at)

(40)

In a limit t --+ co, bearing in mind that

= 0,

-4rre(N,

dt.

and lightning

at

div El = -4re(

N, + N- ) .

(39)

We remain with the fact that parameter a,a/Na does not depend on NO. The solution of the system (39) could be introduced in a form

As is well known the electric field near the surface of conducting bodies changes significantly. The strongest enhancement of electric field takes place near singular points of conducting surface. The structure of the surface of “conducting region” in the CRS core has

86

A.1! Gurevich et al. /Physics

naturally the cony form. It is well known that the electric field near a thin conically pointed body (190< 1) is amplified according to the law [ 161 Eixr”-‘,

n=

1 2 lnW80)

<

1.

Here 00 is the angle of a cony. This amplification of electric field in the high density plasma generated by combined effects of CRS and RB could lead to streamer generation. According to the multiple observations and the theory [ 17,181 to generate a streamer in air the following conditions have to be fulfilled. ( 1) The region should be created where the plasma density is big enough to establish strong plasma polarization what leads to a significant increase of the electric field at the leading front and can generate the ionization wave. (2) To generate the streamer the full number of electrons in the polarized region n at atmospheric pressure should be higher than the minimal number n&,,

/ ns. %2X

n >

ml”

109.

(3) The electric field at the streamer front should exceed the ionization threshold Ett, M (3060) kV/cm. All these requirements could be fulfilled for the case of joint action of CRS and RB effects. In fact, the fulfillment of conditions 1 and 2 follows directly from (34)) (40) and (44). The last condition 3 could be fulfilled because of a strong amplification of electric field near the front of the streamer: it should be taken into account that at the atmospheric heights l-10 km the streamer has a very small radius T,~M (0.1-0.2) cm and as r,,,in N 10 cm the electric field at the front according to formula (45) could be strong enough. So, at the region near the bottom point z = L, the streamer could be generated. In other words, the combination of atmospheric cosmic ray shower and runaway electron air breakdown could serve to trigger the lightning. Note that recently [ 201 it was supposed that the CRS itself can trigger lightning in thunderclouds. As is evident from the here presented theory this supposition is incorrect. Without runaway breakdown the ionization of atmosphere by CRS is four-five orders of magnitude less than needed to generate streamer.

Letters A 254 (1999) 79-87

7. Conclusions In conclusion we will formulate briefly the main results of the theory developed in this paper and propose some experiments which can check the theoretical predictions. It was demonstrated that a strong amplification of the thunderstorm electric field can take place in the vicinity of special “singular” point under the combined action of CRS-RB effects. The amplification is due to electric polarization which comes from the high level of local ionization in the core of CRS. We suppose that the rapidly growing electric field in the “singularities” initiate the spark type local electric breakdown [ 181, which serves for the triggering of lightning. The amplification of electric field takes place near both ends of a polarized region, what means that the lightning triggered by this mechanism can go downwards and upwards as well. So the considered process could relate to the high-altitude lightning also. The amplification effect depends on the number of secondary electrons which is proportional to the energy of cosmic ray particle cc ( 1) . According to formulae (45) strong enough E-field amplification takes place if EO is of the order of lOI eV or higher. We see from Table 1 that during one second several particles with such energy cross 1 km2 of atmospheric surface. This value is in general agreement with the ambient frequency of lightnings during thunderstorm. So, if the runaway breakdown conditions are fulfilled in thundercloud the CRS could serve to initiate the lightning. The correlations of lightning with RB conditions indicated by strongly amplified X-ray emission [ 31 is seen in observations [2,7]. In this sense one can suppose that the qualitative agreement of our theory with experiments exists. Of course, to prove the lightning initiation effect by CRS-RB combined action, new much more definite experiments should be fulfilled. We can propose the following experiments: ( 1) The mostly informative would be direct simultaneous observations of cosmic ray showers (CRS), runaway breakdown (RB), and lightning (L). This experiment could be done at the special cosmic ray stations. For example, the Lebedev Physical Institute (LPI) group proposes to use the LPI mountain cosmic ray station (MCRS) which is situated in Tien-Shan at the height 3300 m above sea level. This installation can measure simultaneously CRS electrons with ener-

A.V. Gurevich et aLlPhysics Letters A 254 (1999) 79-87

gies 0.1-10 MeV, muons with energies above 5 GeV and neutrons, hadrons with energies above 10 MeV. The thunderstorm clouds are practically at the same height as MCRS. Because of this the runaway breakdown effect which lead to a strong ampli~~ation of the X-ray and y-ray emission could be observed by y-ray and X-ray counters installed around the station, for example of the type used in balloon or satellite experiments [ 7,191. The lightning could be fixed by its strong electromagnetic radiation using special antennas. Direct correlation of the whole complex of RBCRS and lightning phenomena would give invaluable information. (2) Another method is to study and locate the electromagnetic emission which is generated by the polarization currents which arise in the CRS-RB process, The currents are of two types. The first is produced by the llux of energetic secondary electrons. The second is produced by thermal plasma electrons moving under the action of thunderstorm electric field during polarization time. The first current is responsible for the electromagnetic emission in the diapason of several 100 kHz, the second for the order of 10 MHz. The experimental study of this emission, its statistics and correlation with lightnings looks very promising for the understanding of lightning initiation effects. The development of the detailed theory of radio emission and radar cross sections generated in CRS-RB process will be presented in our future paper.

Acknowledgement The authors are grateful to Drs. A.P. Chubenko and G.M. Milikh for helpful discussion. This work was supported by Grant ISTC 490 and partially by Grant RFBR 98-02-16715.

87

References C.T.R. Wilson, Proc. Cambridge Philos. Sot. 22 ( 1924) 34. M.P. McCarthy, G.K. Parks, Geophys. Res. Lett. 12 ( 1985) 393. A.V. Gurevich, G.M. Milikh, R.A. Roussef-Dupre. Phys. Lett. A 165 (1992) 463. R. Roussel-Dupre. A.V. Gurevich. J. Geophys. Res. 101 ( 1996) 2297. E. Symbahsty, R. Roussel-Dupre et al.. Trans. AGU 78 (1997) 4760. L.P. Babich. I.M. Kutsyk et al., Phys. Lett. A 245 ( 1998) 460. IX. Eack. W.H. Beasley et al.. J. Geophys. Res. 101 (1996) 29637 A.V. Gurevich, GM. Milikh, J.A. Valdivin. Phys. Lett. A 231 (1997) 402. S.Z. Belenkij, Cosmic Ray Showers (Atomi~at. Moscow, 1948). A.P Chubenko et al., Yademaya Fizica 57 ( 1994) 22. A.V. Gurevich. R. Roussef-Dupre, A.V. Lukyanov. K.P. Zybin, Trans. NATO Meting St. Petersburg (1996) p. 34. A.V. Gurevich, G.M. Milikh, R.A. Roussel-Dupre, Phys. Lett. A 187 (1994) 197. [ 131 A.V. Gurevich, J.A. Valdivia, G.M. Milikh, K. Papadopoulas, Radio Sci. 31 (1996) 1541. [ 141 A.V. Gurevich R.A. Roussel-Dupre. K.P. Zybin, Phys. Lett. A 237 (1998) 240. [ 15 1 H.A. Bethe, J. Ashkin, Passge of radiation through matter, in: Ex~~mentai and Nuclear Physics, E. Segre, ed. (Wiley, New York, 1953). [ 161 L.D. Landau, EM. Lifshits, Elec~~ynamics of Continuous Media (Pegamon, New York, 1958). [ 171 Yu.P Raizer. G.M. Milikh, M.N. Shneider, S.V. Novakovski, J. Phys. D 31 (1998) I. [ 181 Yu.P. Rair..r. Gas Discharge Physics (Springer. Berlin, 1991). [ 19 I R.J. Nemiroff, J.T. Bonnell, J.P. Norris, J. Geophys. Res. 102 (1997) 9659. 1201 V.I. Ermakov, Yu.1. Stozhkov. Fisika Atmosfery i Okeana (1999), in press.