Numerical simulations of local thundercloud field enhancementscaused by runaway avalanches seeded by cosmic raysand their role in lightning initiation

Leonid P. Babich,1 Evgeniı^ I. Bochkov,1 Joseph R. Dwyer,2 and Igor M. Kutsyk1

Received 4 April 2012; revised 22 June 2012; accepted 27 July 2012; published 22 September 2012.

[1] We investigate the possibility of the production of an ionized domain by theionization of the atmosphere by cosmic rays amplified by relativistic runaway electronavalanches (RREAs), the polarization of which could lead to the localized boosting ofthundercloud fields, triggering the sequence of processes leading to lightning development.For this goal, a 2D numerical model was developed of atmospheric discharges in aself-consistent electric field allowing for the kinetics of high-energy REs, low-energyelectrons, and positive and negative ions. The effects are tested for cosmic ray air showersand for the continuous background of low-energy cosmic radiation. It is shown thatlightning cannot be triggered by the joint action of the cosmic ray shower and RREAs.On the contrary, with realistic rates of the cloud charging, the RREAs seeded bylow-energy cosmic rays, produce a plasma domain, at the edges of which the electricfield of the cloud is enhanced above the magnitude required for the selfsustained growthof the free electron population, potentially resulting in lightning initiation.

Citation: Babich, L. P., E. I. Bochkov, J. R. Dwyer, and I. M. Kutsyk (2012), Numerical simulations of local thundercloud fieldenhancements caused by runaway avalanches seeded by cosmic rays and their role in lightning initiation, J. Geophys. Res., 117,A09316, doi:10.1029/2012JA017799.

1. Introduction

[2] The gas-discharge phenomena in thundercloud electricfields have been studied since the 18th century, since theprocesses underlying these high-power phenomena haveboth scientific and applied importance. Despite considerablesuccesses, complete understanding of the mechanism ofatmospheric discharges has not yet been reached. In partic-ular, the intriguing problem of the lightning discharge initi-ation still remains unresolved. Obviously, for the lightning tobe initiated, the electric field, at least locally, should providethe growth of the free electron density required for electricalbreakdown, caused by the collisional ionization of the airmolecules by the electrons themselves. For the breakdown indry air, the required strength of the homogeneous fieldshould be of Ebr(kV/cm) ≈ 30 � P(atm.) [Raizer, 1991, 1992],where P is the pressure, i.e., Ebr decreases with the altitude.Precipitation particles also reduce the breakdown thresholddown to Ebr, pr(kV/cm) ≈ (10–14) � P(atm.) depending onthe size and form of the particles [Solomon et al., 2001]. Thedifficulty is that, when reduced to the STP conditions, themeasured magnitudes of the electric field in thunderclouds

are less than Ebr, pr by a factor of 3–4 and do not appear toexceed 3–4 kV/cm [Marshall et al., 1995a, 1995b, 2005].[3] Among the ideas of the lightning initiation, some of

which were reviewed by Petersen et al. [2008], one of themost promising seems to be the very weakly elaboratedhypotheses connecting the start of the lightning leader withrelativistic runaway electron avalanches (RREAs) seeded byhigh-energy electrons of secondary space radiation [Gurevichet al., 1997]. According to this hypothesis, the RREAs, gen-erating a considerable number of electron-ion pairs, producean ionized domain, polarization of which in the thundercloudelectric field leads to a local rise of the field at the edges of thedomain up to the value required to “…initiate the spark typelocal electric breakdown…which services for the triggering oflightning” [Gurevich et al., 1997]. Obviously a developmentof a discharge driven by low-energy electrons is meant. Thisprocess is very similar to selfsustained laboratory discharges inair at atmospheric pressure initiated by sub-nanosecond pulsesof runaway electrons [Babich et al., 1990; Babich, 2005;Babich and Loı^ko, 2009] or to discharges with external sourceof seed ionization in high-pressure gas lasers [Mesyats andKorolev, 1986]. The title of the paper [Gurevich et al., 1999]implies that local boosting of the thundercloud field leads to“lightning initiation,” however, more precise interpretation isthat this mechanism could be a “trigger” step in the larger“lightning initiation” sequence of processes.[4] One of the sources of seed REs in the atmosphere are

high-energy cosmic ray particles. It is known that a particlewith an energy above 1016 eV, as a result of a cascade pro-cess, produces 106–107 secondary particles of lower energies,

1Russian Federal Nuclear Center –VNIIEF, Sarov, Russia.2Department of Physics and Space Sciences, Florida Institute of

Technology, Melbourne, Florida, USA.

Corresponding author: J. R. Dwyer, Department of Physics and SpaceSciences, Florida Institute of Technology, Melbourne, FL 32901, USA.(jdwyer@fit.edu)

©2012. American Geophysical Union. All Rights Reserved.0148-0227/12/2012JA017799

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 117, A09316, doi:10.1029/2012JA017799, 2012

A09316 1 of 10

mainly, electrons, positrons and gamma photons, constitutingan cosmic ray air shower (CRS) covering a large area[Murzin, 1988]. A flux of particles with energies above1016 eV per unit time of 0.1/(km2�s) is close to the frequencyof lightning discharges in a thunderstorm per 1 km2. Resultsof analytical estimations carried out byGurevich et al. [1999]led them to conclude that lightning initiation is possible dueto the ionization of the atmosphere by CRS amplified byRREAs. However, Dwyer [2010, p. 1] has shown that “…with the avalanche multiplication limit from X-ray and pos-itron feedback … even 1017 eV extensive cosmic-ray airshowers do not produce high enough conductivities to sig-nificantly alter the electric field inside a thundercloud. As aresult, … no compelling theoretical argument exists to sug-gest that extensive cosmic-ray air showers initiate lightning.”To test a validity of the idea ofGurevich et al. [1999] accuratenumerical simulations are required, providing conditions atwhich the mechanism of the lightning initiation by CRS isvalid.[5] Besides CRS the other main source of seed REs at the

thundercloud altitudes is the continuous low-energy spaceradiation. Dwyer [2005] has executed numerical simulationsof the ionized domain formation as a result of the atmosphereionization by RREAs seeded by this radiation. He obtained amagnitude of the field intensity 12 kV/cm reduced to the STPconditions that is sufficient for the breakdown in the atmo-sphere with precipitation particles. Dwyer carried out simu-lations with and without feedback and found that such fieldenhancements occur in both cases. However, when the rela-tivistic feedback by X-rays and positrons [Dwyer, 2003;Babich et al., 2005] is allowed, this may greatly increase therate of RREA generation and, as a consequence, lead to thefast growth of the atmosphere conductivity. The feedbackalso leads to very sharp, during a time on the order of milli-seconds, rise in the number of high-energy REs and, conse-quently, of the flux of hard photons. However, airborneobservations in thunderclouds have demonstrated onlysmooth growth of flux of hard X-rays with characteristic timein the second range [Parks et al., 1981;McCarthy and Parks,1985] suggesting that relativistic feedback may not beimportant in the generation of lightning.[6] Therefore it is expedient, to further analyze whether

2D numerical model omitting the effects of the relativisticfeedback, is capable of predicting a development in thun-derstorm atmosphere of conducting channel resulting fromthe joint action of cosmic rays and RREA. Furthermore, aswill be discussed in section 3, even if relativistic feedback isoccurring during the cases being studied, it is not expected toalter the basic conclusions about the field enhancements,only the timescales over which the enhancements occur. Ourstudy is limited to numerical simulations of production of theseed ionized domain, possibly leading to the development ofa conducting channel of lightning owing to the thundercloudfield enhancement at the edges of the domain stemming fromthe plasma polarization. Development of such a domain withlocally enhanced field may be the first “trigger” link in thechain of lightning initiation processes. For example, thismechanism may enhance the field to a value where streamersmay be initiated from hydrometeors, leading to the formationof a hot leader channel. We simulated effects of both CRSand continuous low-energy space radiation, as RREA

sources. On the basis of the obtained results, conclusions arededuced on the adequacy of the hypothesis.

2. Mathematical Formulation of the Problem

[7] The processes in thunderstorm fields with RE partici-pation are analyzed in terms of the electric field overvoltaged = eE/(FminP) relative to the minimum of the drag forceacting on electrons Fmin = 2.18 � P kV/(cm � atm.) as a resultof inelastic interactions with atmosphere molecules. Magni-tudes of the overvoltage dbr ≈ 13.5 and dbr,pr ≈ 4.5–6.5 cor-respond to the breakdown field strength at STP conditions,respectively, of Ebr ≈ 30 kV/cm in dry air and Ebr,pr ≈10–14 kV/cm in atmosphere with precipitation particles.Allowing for the scattering by molecules, as a result of whichelectrons move under some angle relative to the direction ofthe electric force�e~E, the threshold value of the overvoltage,above which the RREA development is possible, is of dth ≈1.3 [Dwyer, 2003; Babich et al., 2004], corresponding to acritical field intensity Eth ≈ 2.83 kV/cm at STP conditions.[8] The multigroup approach proposed by Babich and

Kutsyk [1995] and developed further using a rigorous set ofmultigroup equations [Babich and Kudryavtseva, 2007],allowing to accurately describe the RE kinetics, is extremelycomputer time-consuming while simulating discharges indense gas accompanying by high-energy emissions, espe-cially atmospheric discharges [Babich et al., 2008a, 2008b,2011]. Therefore, Babich and Bochkov [2011] have analyzedthe use of the conventional diffusion-drift equation anddemonstrated that transport of REs in dense gas with electricfields is described with rather high accuracy by this equation:

∂nre∂t

þ div

�nre~vre � DT � ~rnre �~vre � ~vre � ~rnre

� �=v2re

� �

� DL �~vre � ~vre � ~rnre� �

=v2re

�¼ nre � nre þ Sre; ð1Þ

where nre is the RE density; nre is the rate of the RE pro-duction by REs themselves;~vre ¼ �u �~E=E is the vector andu absolute value of the RE directed velocity; DT and DL arecoefficients of the transverse and longitudinal diffusion rel-ative to the vector�e~E; Sre(z, d) is the volumetric rate of seedREs (above the runaway threshold ɛth(d)) production by CRSor steady flux of low-energy cosmic rays.[9] However, unlike the multigroup equations, equation (1)

does not allow calculating the RE energy spectrum, however,it is very efficient for calculating the space-time evolution ofREs provided that magnitudes of u, DT, DL and te are known.The dependencies of u, DT, DL and te on d were calculated byMonte Carlo (MC) technique [Babich et al., 2004; Babichand Bochkov, 2011; Coleman and Dwyer, 2006; Dwyer,2010]. The speed of REs, u, is practically independent of din the range 1.5–100, and with good accuracy u = 0.9 � cwhere c is the speed of light in free space. Here we use theformulae for the diffusion coefficients from [Babich andBochkov, 2011]:

DL m2=s½ � ¼ 2:687 � 108 � d�1:715=P;DT m2=s½ � ¼ 1:246 � 109 � d�1:973=P;

ð2Þ

approximating the results of MC simulations in the range ofthe overvoltage magnitudes d = 1.5–14 with a margin error,

BABICH ET AL.: LIGHTNING INITIATION BY COSMIC RAYS A09316A09316

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not exceeding 13%. These diffusion coefficients agree wellwith the values found by Dwyer [2010].[10] For the dependence of nre on d the following formula

nre s�1� � ¼¼ 7:24 � 106 � d � 1:27ð Þ1:08 � P; d ≥ 1:27;

1:03 � 107 � d � 1:27ð Þ � P; d < 1:27

�ð3Þ

was used, obtained as follows. For high d the dependence on dof the RE e - fold production time te was used as calculated in[Babich et al., 2004, 2007b] byMC code ELIZA, which givesnre = 1/te with inaccuracy less than 3.5% in the range d = 1.4–14.For lower d we carried out calculations using a simplifiedMC code similar to that in [Lehtinen et al., 1999; Babichet al., 2001]. The negative nre(d) in the range d < 1.27means that in domains of atmosphere with d in this range thenumber of REs decreases as they lose energy and are reab-sorbed by the air. The inaccuracy of nre in the range d < 1.27is less than 7%. Note, in stochastic MC simulations the run-away threshold ɛth(d), artificial magnitude appearing in deter-ministic approach, is absent; therefore under REs we considerelectron population with energies above 1 keV [Babich et al.,2004, 2007b; Babich and Bochkov, 2011]. Exactly kinetics ofthese electrons is described by equation (1).[11] The rates of RE generation by continuous low-energy

space radiation Sreb and its dependence on d, calculated byMC

technique, are available in the paper of Babich et al. [2007a].[12] To calculate the source of seed REs initiating RREAs,

Sreeas caused by an CRS, using a MC technique and the same

approach as in the paper [Babich et al., 2007a], we simulatedthe passage through the Earth’s atmosphere of an CRS pro-duced by a proton with energy of 1016 eV, letting the protonmove along a vertical axis and its first nuclear interaction withof the air species occurring at the altitude of 40 km [Murzin,1988]. The linear concentration Lre(z, ɛth(d)) ≡ Lre(z, d) ofREs with energies above the local runaway threshold ɛth(d) inthe thundercloud field was calculated. Obviously, the shapeof the curve Lre(z, d) at the fixed value of d depends both onthe angle, at which the proton enters the atmosphere, and onthe altitude of the first interaction with air nuclei. But themaximum magnitudes of Lre(z, d), being functions of thethickness of the air layer traversed by the CRS, are indepen-dent of these quantities; therefore the maximum value ofLre(z, d) at a given d is determined only by the energy of theprimary proton. The maximum values of Lre(z, d), given inthe Table 1 for the d range 1.3–14, were used for furthercalculations of the RE source Sre

eas. We consider that all par-ticles in the CRS are in one plane parallel to the surface of theEarth and move downward with the speed of light. Then thesource of seed REs, created by the CRS initiated by the pri-mary proton with energy ɛeas is calculated as follows:

Seasre z; r; tð Þ ¼ gNKG rð Þ � dD t � zt � zð Þ=cð Þ � Lmaxre d zð Þð Þ

� ɛeas eV½ �1016

ð5Þ

where dD is the Dirac’s delta-function, gNKG rð Þ ¼ 0:37R2eas� s2 �

2:07� sð Þ5=4 � rReas

� �s�2� 1þ r

Reas

� �s�4:5is the formula of

Nishimura-Kamata-Greisen [Murzin, 1988], approximating adistribution of charged particles in the CRS in a planeorthogonal to the CRS symmetry axis, which is reasonable tolet coinciding with the direction of motion of primary protonproducing CRS (in our case this a vertical axis). In this for-mula Reas is a characteristic transverse size of the CRS, s isthe dimensionless parameter depending on the atmospheredepth. As in the paper of Gurevich et al. [1999], we letReas = 100 m, s = 1; the coefficient 0.37 provides normalizinggNKG to unity with the above values of Reas and s.[13] The kinetics of low energy electrons (e), positive (p)

and negative (n) ions obeys the following set of the equations:

∂ne∂t

þ div ne~ve ¼ nion � nattð Þ � ne � bep � ne � np þ nep � nre þ Sep

∂np∂t

þ div np~vp ¼ nion � ne � bep � ne � np � bpn � np � nnþ nep � nre þ Sep

∂nn∂t

þ div nn~vn ¼ natt � ne � bpn � np � nn ð6Þ

Here ne, np, nn are densities; ~ve ¼ �me~E , ~vp ¼ mp

~E , ~vn ¼�mn

~E are drift velocities, me and mp,n are electron and ionnobilities; nion is the ionization rate of air molecules by low-energy electrons; bep and bpn are respectively coefficients ofrecombination of electrons with positive ions and positive andnegative ions; natt = [Kdiss + Kthr � NO2

(z)] � NO2(z) is the rate of

the electron attachment to the oxygen molecules,Kdiss andKthr

are the rates of dissociative and threefold attachment,NO2

(z) = 0.2 � N(z) and N(z) = 2.688 � 1025m�3 � P(z) are localconcentrations of oxygen molecules and air at the altitude z;nep is the rate of generation of low-energy electrons owing tothe RE collisions with molecules; Sep is the external source ofelectron - positive ion pairs via the low-energy radiation.[14] The rate of production of low-energy electrons due to

ionizing impacts of REs is calculated dividing the power ofthe drag force by the energy price of producing one electron-ion pair in air Dɛion ≈ 32 eV:

nep s�1� � ¼ Fmin � P � u=Dɛion ≈ 2 � 1012 � P: ð7Þ

[15] We used for the ionization rate by low-energy elec-trons, electron attachment rates, and ion mobility the for-mulae from [Aleksandrov et al., 1981a, 1981b] and[Bazelyan and Raizer, 2000] and the recombination coeffi-cients from [Raizer, 1991, 1992]:

nion s�1� � ¼ 10�14:8�28:1=z � N zð Þ;

Kthr m6=s

� � ¼ 4:7� 0:25 � zð Þ � 10�43;

Kdiss m3=s� � ¼ 10�9:42�12:7=z ; z ≤ 9;

10�10:21�5:7=z ; z ≥ 9;

(

mp;n m2= V � sð Þ� � ¼ 2 � 10�4=P;

bep m3=s� � ¼ 2 � 10�13 � P;

bpn m3=s� � ¼ 2 � 10�12 � P: ð8Þ

Table 1. Maximal Values of Linear Concentration of Seed REsGenerated by Cosmic Ray With Energy 1016 eV in the Field WithOvervoltage d

D 1.3 1.5 2 3 4 5 6 7 8 10 12 14

Lremax, 103 m�1 10 28 93 233 384 528 680 850 1010 1350 1750 2000

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where z = 1020(V �m3)�1E/N(z). For the electron drift velocitythe approximation of Golubev et al. [1985] was used:

ue m=s½ � ¼

1:47 � 104 � x; 0 ≤ x < 0:251;

0:737 � 104 � x1=2; 0:251 ≤ x < 6:65;

0:459 � 104 � x3=4; 6:65 ≤ x < 421;

2:08 � 104 � x1=2; 421 ≤ x ≤ 6860;

8>>><>>>:

ð9Þ

where x ¼ E V=m½ �3 � 104 � P atm:½ �. Note, the absorption of the ions

by cloud particles reduces, to some extent, the conductivity ofthe ionized channel, possibly affecting the channel dynamics[Dwyer, 2005]. However, since the absorption rates dependupon the water content of the cloud as well as the cloud par-ticle density, these rates are not known with good accuracyunless very specific assumptions are made about the cloudenvironment (e.g., the absorption length is found to be in thewide range 3–20 m [Dwyer, 2005]). As a result, we choose toomit this process.[16] The source Sep in the (6) was calculated from a sta-

tionary set of equations for the background densities of thecharged particles ne

b, npb, nn

b

�natt � nbe � bep � nbe � nbp þ Sep ¼ 0;

�bep � nbe � nbp � bpn � nbp � nbn þ Sep ¼ 0;

natt � nbe � bpn � nbp � nbn ¼ 0;

ð10Þ

as follows:

Sep ¼ nbe � natt þ bep � nbp� �

;

nbe ¼bpn � nbp � nbpnatt þ bpn � nbp

:ð11Þ

The background density of positive ions npb was calculated

from formula for the background conductivity of atmospheresb = e � (me � neb + mp � npb + mn � nnb), for which the approx-imation by Roussel-Dupré and Gurevich [1996] is used:

sb W �mð Þ�1h i

¼ ɛ0 � 10� 28�z km½ �ð Þ=30 ð12Þ

Combining (11) and (12) and solving the obtained equation

e � me þ mp

� �� bpn � nbp

� �2þ e � mp þ mn

� �� natt � sb � bpn

� �� nbp � sb � natt ¼ 0 ð13Þ

one can calculate npb and then using (10) and (11) calculate

neb, nn

b, and Sep.[17] The set of equations (1) and (6) is closed by the

electric field equations:

~E ¼ �~rj int þ~Eext;

Dj int ¼ � e

ɛ0np � nn � ne � nre

;ð14Þ

where jint is the potential of the field of space charge causedby charge separation due to the motion of charged particles(plasma polarization), ~Eext is the vector of the external fieldintensity (the field of thundercloud).[18] In the case with the low-energy space radiation as the

source of seed REs, we let the external field to be generatedby the top positive Qp and a bottom negative Qn charges of

equal magnitude. The charges are distributed in space underthe Gauss’s law in the cylindrical frame of coordinates:

rp z; r; tð Þ ¼ Qp tð Þp � ffiffiffi

pp � lz;p � l2r;p

� exp � z� zp 2

=l2z;p

� �

� exp �r2=l2r;p

� �;

rn z; r; tð Þ ¼ Qn tð Þp � ffiffiffi

pp � lz;n � l2r;n

� exp � z� znð Þ2=l2z;n� �

� exp �r2=l2r;n

� �: ð15Þ

Here z and r are vertical and radial coordinates, zp and zn arecoordinates of the centers of the charges lz,n, lr,n, lz,p, lr,p arewidths of the distributions.[19] We let the cloud to be charged following the linear

law with a rate kch:

Qp tð Þ ¼ �Qn tð Þ ¼ kch � t ð16Þ

For the CRS case, two profiles of the stationary externalelectric field along the altitude were analyzed, namely, afield created by two flat parallel layers of charges of differentsign with thickness Δz (a linear profile L) and a parabolicprofile (P), as analyzed by Gurevich et al. [1999]:

ELext zð Þ ¼

Em � zb � zð ÞDz

; zb < z < zb þDz

�Em; zb þDz < z < zt �Dz

Em � z� ztð ÞDz

; zt �Dz < z < zt

8>>>><>>>>:

EPext zð Þ ¼ Em � 2z� zb þ ztð Þð Þ2

zt � zbð Þ2 � 1

! ð17Þ

Here z is the altitude, zb and zt are the bottom and top bordersof the area occupied by the field, Em = dext � Fmin(0.5 � (zb + zt))is the maximum value of the absolute value of the thunder-cloud field intensity. The vector of the external field intensityis directed downward in agreement with a typical configu-ration of charges in thundercloud with the bottom negativeand top positive charges. The layer thickness Δz was putequal to 100 m.[20] Physical quantities controlling the initial conductivity

produced by RREAs, namely, the electric field overvoltage dand corresponding number of RREA e-folding lengths aremore important than the specific model geometry (i.e., spe-cific profiles of the external electric field). The measuredmagnitudes of the field strength in thunderclouds, forexample [Marshall et al., 1995a, 1995b, 2005], correspond tothe d of �3; therefore our simulations are carried out aroundthis magnitude. Note, however, the chosen geometries ofcharge distribution both for the case of low-energy spaceradiation and CRS with profile L are dipole-like geometrieswidely used to model thundercloud field.[21] Obviously, the initial conditions for the set (1), (6)

and (14) are as follows:

ne ~r; t ¼ 0ð Þ ¼ nbe ;

nn ~r; t ¼ 0ð Þ ¼ nbn zð Þ;np ~r; t ¼ 0ð Þ ¼ nbp zð Þ;nre ~r; t ¼ 0ð Þ ¼ 0;~E ~r; t ¼ 0ð Þ ¼ ~Eext:

ð18Þ

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We placed the boundary of the calculation area far awayfrom the domain where the REs are produced and wherethey generate secondary low-energy charged particles;therefore, as at the boundary surface ~rbound the REs areabsent and the following boundary condition is true [Babichet al., 2008b]:

∂F∂~n

����~rbound

¼ 0; ð19Þ

where F = {np, nn, ne, nre} and~n is a vector of a normal to theboundary surface~r bound.[22] The boundary electric potential, as a rule, is calcu-

lated as follows:

j int ~rboundð Þ ¼ 1

4pɛ0

ZV

e � np � nn � ne � nre � dV

~rbound �~rj j ; ð20Þ

where V is a volume occupied by the space charge. Such anapproach, however, is inefficient and in our case demandshuge and unjustified expenditures of time for calculating theintegral in each boundary point, exceeding the expenses forthe numerical solving of the Poisson’s equation for thepotential jint. As the space charge is mainly concentrated faraway from the border of the calculation area, near to itscenter, we calculated j int ~rboundð Þ using the multipoledecomposition up to the quadrupole moment:

j int ~rboundð Þ ¼ 1

4pɛ0

"Qþ

~rbound �~rþj j þQ�

~rbound �~r�j j

þ~rbound �~rþ;~dþ� �

~rbound �~rþj j3 þ~rbound �~r�;~d�� �

~rbound �~r�j j3 þ

þ 1

2

X3i¼1

X3j¼1

Dijþ

xibound � xiþ � x j

bound � x jþ

~rbound �~rþj j5

þ 1

2

X3i¼1

X3j¼1

Dij�

xibound � xi� � x j

bound � x j�

~rbound �~r�j j5

#; ð21Þ

where[23] Q� ¼

ZV

r� ~rð Þ � dV are positive and negative

charges,

[24] ~r� ¼ZV

~r � r� ~rð Þ � dV=Q� are radii-vectors of the

centers of the charges,

[25] ~d� ¼ZV

~r�~r�ð Þ �r� ~rð Þ �dV~d� ¼ZV

~r�~r�ð Þ �r� ~rð Þ �

dV are vectors of the dipole moments,

[26] D�ij

ZV

(3 � (xi� x�i ) � (x j� x�

j )� dij � j~r�~rj2) � r� (~r ) �dV are tensors of the quadrupole moment and dij is theKronecker’s symbol[27] r� = 0, 5�(r�|r|).[28] As the positive charge, mainly, is accumulated in the

head of the growing gas-discharge channel, and negativecharge is accumulated in the channel body, it is expedient toallocate separately subsystems with positive and negativecharges and by that to increase the accuracy of calculationsof the border potential.[29] In view of the cylindrical symmetry of the problem

specified by the charges distribution (15), solving of theproblem was carried out in the framework of cylindricalcoordinates ~r; ~zð Þ.

3. Results of Calculations and Their Analysis

[30] First we analyzed if the lightning can be triggered byCRS. As mentioned above, secondary electrons in CRS withenergies above the runaway threshold initiate RREAs inthundercloud fields. The REs generate large numbers ofelectron-ion pairs. The overwhelming majority of electrons,born as a result of ionization, are distributed in the range onthe order of tens of electronvolts. During a period of 10 nselectrons lose this energy in inelastic interactions with airmolecules and attach to molecules O2 and H2O, producingthus negative ions. The life-time of free electron at thun-dercloud altitudes is limited to the attachment time of tatt = 1/natt � 100 ns [Raizer, 1991, 1992]. The field amplificationcaused by the polarization due to the motion of electronsprior their attachment is insignificant. Most of the fieldenhancement is due to the ion motion. Therefore it is pos-sible to believe, that as a result of joint action of an CRS andRREAs in the atmosphere a quasi-neutral domain is devel-oped of plasma of positive and negative ions, polarization ofwhich in the external electric field leads to the fieldstrengthening at the domain edges.[31] The simulations were carried out for three maximum

values of the external (cloud) field overvoltage dext = 2, 3, 4 inthe center of the domain. The energy ɛeas of the space particleinitiating CRS, and the number of the RREA e-fold amplifi-

cation lengths in external field xext ¼Zztzb

dz

u dext zð Þð Þ � te dext zð Þð Þwere varied. Here dext(z) is above 1.27 according to equation (3).The integration is carried along the CRS symmetry axis. Themaximum values of the overvoltage dmax of the fieldenhanced due to the plasma polarization and the time tmax

when dmax is achieved are presented in Table 2 for the fieldprofiles L and P. Figures 1 and 2 illustrate the temporalevolution of the overvoltage at the symmetry axes of thedischarge for dext = 2.

Table 2. Maximal Magnitudes of the Overvoltage dmax at Momenttmax in the Channel Initiated by Cosmic Ray With Energy ɛeas

a

dext x ext ɛeas, eV dmax tmax, s

2 (L) 11 1020 3.4 4.5�10�1

3 (L) 11 1020 5.6 2.4�10�1

4 (L) 11 1020 7.5 8.1�10�2

4 (L) 11 1018 5.5 1.3�10�1

2 (P) 11 1020 3.9 3.4�10�1

3 (P) 11 1020 5.3 9.0�10�2

4 (P) 11 1020 6.8 4.5�10�2

2 (L) 22 1016 3.9 5.5�10�1

3 (L) 22 1016 5.9 1.4�10�1

4 (L) 22 1016 7.8 7.5�10�2

aNotation: xext is the number of the RREA multiplication lengths in thecloud field with linear (L) and parabolic profiles (P) as described byequation (17) with zb = 5 km.

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[32] No more than double the strength of the thundercloudfield is reached, even for the greatest magnitudes of dext, ɛeasand xext. That is natural, as the level of the field strengtheningis defined by the shape of the conducting domain. Increasingxext and ɛeas leads to the rise of the plasma channel conduc-tivity that affects only the time of polarization of the channel.To reach higher level of the field strengthening, a compactnarrow conducting channel is required to be developed alongthe vector of the external electric force �e~E . In our case,however, for both field profiles (17) a diffuse ionized domaindevelops with characteristic transverse and longitudinal sizesof �1 km as seen in Figures 3 and 4 illustrating spatial dis-tributions of conductivity in atmosphere for linear (L) andparabolic (P) field profiles given in (17) at the moment oftime when the last RREA, initiated by CRS, has crossed thedomain occupied by the thundercloud field (t ≈ 2�(zt � zb)/c)and practically all electrons have attached.[33] According to Dwyer [2003] a number x of the RREA

e-fold amplification lengths in the discharge in homoge-neous electric field is limited to about 11 (ex � 104–105), asat higher x the relativistic feedback by X-rays and positronsis switched on exterminating the field due to intensivemultiplication of REs and, as a consequence, increase of theconductivity. With x = 11 the initiating cosmic particle withthe energy not less than 1020 eV is capable of producing theion density of 1014–1015 m�3 required for noticeableenhancement of the electric field at the conducting domainedge. However, the flux of such particles per unit time in theEarth’s atmosphere is extremely small; it is significantly lessthan the frequency of the lightning discharges. On the otherside, the same ionization is produced by the particle with theenergy 1016 eV, however the field is required to be strongenough for the number of the RREA e-fold lengths to be ofx ≈ 22. Thus, for the lightning discharge to be triggered byjoint action of CRS and RREA either strong large-scaleelectric field, not observed in nature, or initiating CRS cos-mic ray with unreal energy or both are required.[34] To analyze whether the lightning can be triggered by

low-energy cosmic rays, we carried out simulations with thefollowing dimensions of the charge configuration defined by(15): lz,n = lz,p = 300 m, lr,n = lr,p = 600m, zn = 5 km, zp = 7 km.

The rate of the cloud charging kch was varied. The results arepresented in the Table 3, where dmax is the maximal magnitude

of the field overvoltage, xmax ¼Zznzp

dz

u d zð Þð Þ � te d zð Þð Þ is the

maximum number of the RREA multiplication lengths duringthe discharge development, and Qcl

f is the cloud charge atthe moment of time when dmax is achieved. It is seen that thebreakdown magnitude dbr ≈ 13.5 is achieved with kch in therange from 1.3 С/s to 13 C/s. With kch = 0.65 C/s the maximalovervoltage, dmax = 7.2, is sufficient for the lightning trigger-ing in thunderstorm atmosphere with precipitation particles(dbr,pr ≈ 4.5–6.5). The number of the RREA multiplicationlengths x is varied from about 15 to 24, above the positivefeedback limit x = 11 by Dwyer [2003]. However, unlike theCRS case, where the feedback threshold and the number ofparticles in the shower limit the total amount of ionization thatoccurs, given enough time, the low-energy cosmic rays willeventually supply sufficient ionization to discharge the field.As a result, the time structure of the discharge may be differentthan when feedback effects are included, but the key result thatthe field is enhanced to the point where lightning may betriggered remains unchanged by including feedback.[35] Figure 5 displays the temporal evolution of the field

overvoltage at the symmetry axis of the discharge forkch = 1.6 C/s. This field amplification is qualitatively similarto that calculated by Dwyer [2005]. At the moment of thefield strength maximum when the difference of the potentialis of 350 MV, the cloud is charged up to the magnitude 15 C,within the range of the observed cloud charges [Uman,1972, 2001; Rakov and Uman, 2003]. Figure 6 illustrates adistribution of conductivity and field overvoltage at themoment when the latter achieves a maximum magnitude.

4. Conclusions

[36] To verify the idea of Gurevich, Zybin and Roussel-Dupré [1999], connecting the lightning discharge trigger-ing with the development of relativistic runaway electronavalanches (RREAs) triggered by secondaries of cosmic

Figure 1. Distribution of the field overvoltage along thesymmetry axis of the discharge for a few moments of time.Extensive cosmic ray air shower, ɛeas = 1020 eV, dext = 2(L),xext = 11.

Figure 2. Distribution of the field overvoltage along thesymmetry axis of the discharge for a few moments of time.Extensive cosmic ray air shower, ɛeas = 1020 eV, dext = 2(P),xext = 11.

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rays, and thus allowing for the circumvention of the funda-mental difficulty of thundercloud fields being below thebreakdown limit, using the fluid approach, we developed atwo-dimensional numerical model of atmospheric dis-charges in self-consistent electric fields allowing for thekinetics of high-energy REs, low-energy electrons, positiveand negative ions, both background and secondary. Wesimulated the formation of conducting plasma channelsproduced by RREAs seeded either by cosmic cosmic ray air

showers (CRSs) as assumed by Gurevich et al. [1999] or bythe continuous low-energy cosmic radiation.[37] As follows from results of our simulations, carried out

with realistic thundercloud configurations, rates of the cloudcharging, and thunderstorm field strengths, the magnitude ofthe self-consistent field overvoltage, of dbr ≈ 13.5, at whichtriggering of the lightning is possible, was reached at theedges of the polarized plasma channel produced by the jointaction of RREAs and steady flux of low-energy cosmic rays.

Figure 3. Plots of 2D spatial distribution of conductivity of ionized channel created by joint action ofcosmic cosmic ray air shower and RREAs. ɛeas = 1020 eV, dext = 2(L), xext = 11.

Figure 4. Plots of 2D spatial distribution of conductivity of ionized channel created by joint action ofcosmic cosmic ray air shower and RREAs. ɛeas = 1020 eV, dext = 2(P), xext = 11.

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[38] On the contrary, the simulations of the dischargetriggered by RREAs seeded by CRS, produced by the pri-mary proton with the energy ɛeas in the range 1016–1020 eVin the intracloud field with a variety of profiles and initial(external) maximal values of the overvoltage dext = 2, 3, 4,demonstrate that the achieved magnitudes of the overvoltagedmax = 3.4–7.8 of the self-consistent field, caused by thepolarization of the secondary plasma, are only sufficient forthe breakdown and development of selfsustained dischargein atmosphere with precipitation particles (dbr, pr ≈ 4.5–6.5).However, either strong large-scale electric field, notobserved in nature, or initiating CRS cosmic ray with unrealenergy of �1020 eV are required. As the flux of such parti-cles per unit time in the Earth’s atmosphere is many ordersof magnitude less than the frequency of the lightning dis-charges, it is very unlikely that the lightning discharge canbe triggered by joint action of CRS and RREA even inatmosphere with precipitation particles.[39] There are serious experimental evidences in favor of

the simulated scenario of lightning triggering and the rela-tionship between lightning triggering and generation ofhigh-energy REs in thunderstorm environments. During a

lightning stroke, fields with strength close to the runawaythreshold were recorded inside thunderclouds [Marshallet al., 1995a, 1995b, 2005]. The enhancements of penetrat-ing radiation [Parks et al., 1981; McCarthy and Parks,1985; Eack et al., 1996a, 1996b, 2000; Torii et al., 2009;Tsuchiya et al., 2007, 2009] whose source was in thunder-clouds [Babich et al., 2010], were recorded during thun-derstorm activity. The enhancements observed by Parkset al. [1981] and McCarthy and Parks [1985], precededthe lightning discharge and were “switched off” by the dis-charge itself. The bremsstrahlung of electrons multiplyingand acquiring a high energy in the cloud electric field can bethe only source of the enhancements.[40] Though lightning triggering relied particularly upon

short-living CRSs, as was assumed by Gurevich et al.[1999], seems very unlikely in view of the results of our

Table 3. Maximal Magnitudes of the Overvoltage dmax in the Dis-charge Initiated by the Continuous Low-Energy Space Radiationa

kch, C/s dmax xmax Qclf , C

13 13.5 24.6 15.54.3 13.2 21.7 15.12.2 13.1 19.5 15.01.6 12.4 18.4 15.01.3 11.3 17.5 14.90.65 7.2 14.7 15.20.32 2.8 10.1 16.6

aNotation: xmax is the maximal number of the RREA multiplicationlengths,Qcl

f is the cloud charge at the moment of time when dmax is achieved.

Figure 5. Distribution of the field overvoltage along thesymmetry axis of the discharge for a few moments of time.Low-energy cosmic rays, kch = 1.6 C/s.

Figure 6. The 2D distribution of the conductivity (left panel) and overvoltage (right panel) at the momentt = 9.2 s. Low-energy cosmic rays, kch = 1.6 C/s.

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simulations, the idea itself about local thunderstorm fieldenhancement to the breakdown threshold due to the polari-zation of a plasma channel developing through atmosphericionization, even if not by CRSs but by the continuous low-energy cosmic radiation followed by the development ofRREAs, seems fruitful and requires a further study. Ourresult that the continuous weaker cosmic ray flux may pro-vide a slower but still strong local electric field enhancementin a thundercloud seems intriguing. It is distinct from themechanism of Gurevich et al. [1999] in that it does notrequire CRSs but only a steady supply of seed electrons fromthe more common lower energy cosmic rays.

[41] Acknowledgments. The VNIIEF co-authors are grateful toC. Haldoupis and T. Neubert, the international collaborators in the ISTCProject 3993-2010 funded by European Union, within the framework ofwhich this work was carried out, to N. Crosby, A. van Deursen, S. Cummer,R. Roussel-Dupré, D. Smith, T. Torii, and E. Williams for their support ofthe Project proposal. L. P. Babich and I.M. Kutsyk thank A.V. Gurevich,R.A. Roussel-Dupré and K.P. Zybin for long-term collaboration on thephysics of atmospheric electricity whose continuation is this paper. Thework done by Joseph R. Dwyer has been supported in part by DARPA grantHR0011-10-1-0061. The authors are grateful to two reviewers whose com-ments allowed significantly improving the paper.[42] Robert Lysak thanks the reviewers for their assistance in evaluat-

ing this paper.

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