GEOPHYSICAL RESEARCH LETTERS, VOL. 32, L09809, doi:10.1029/2004GL021744, 2005
The feedback mechanism of runaway air breakdown L. P. Babich, E. N. Donskoy, and I. M. Kutsyk Russian Federal Nuclear Center – All-Russian Scientific Research Institute of Experimental Physics (VNIIEF), Nizhegorodskaya Oblast, Russia
R. A. Roussel-Dupre´ Los Alamos National Laboratory, Los Alamos, New Mexico, USA Received 13 October 2004; revised 16 March 2005; accepted 25 March 2005; published 12 May 2005.
[1] The feedback mechanism presented in the paper published by Dwyer (2003) is examined in the context of runaway breakdown and thunderstorm electrical conditions. A detailed comparison between previous kinetic and Monte Carlo simulations performed by the authors and others and those of Dwyer are presented. The role of positrons and bremsstrahlung photons in contributing to the runaway avalanche rate is discussed. An alternative and simplified treatment of the feedback mechanism is presented. Our preliminary results are in basic agreement with Dwyer and indicate that positron feedback is more important. In addition, we find that there is a temporal ‘stepping’ effect associated with the feedback process and that this effect could also lead to a spatial stepping reminiscent of stepped leaders. Citation: Babich, L. P., E. N. Donskoy, I. M. Kutsyk,
[4] - For photons: non-coherent (Compton) scattering with allowance for bound electrons, coherent (Rayleigh) scattering, photoabsorption with emission of fluorescent photons and Auger electrons, and production of electronpositron pairs and triplets. [5] - For electrons: elastic scattering by atomic nuclei, ionization and excitation of atomic electron shells, and bremsstrahlung. [6] - For positrons: elastic scattering by the nucleus, scattering by free electrons, bremsstrahlung, and twophoton annihilation.
and R. A. Roussel-Dupre´ (2005), The feedback mechanism of runaway air breakdown, Geophys. Res. Lett., 32, L09809, doi:10.1029/2004GL021744.
[7] Table 1 shows the RREA rates obtained by ELIZA in comparison with the rates obtained by two simplified MC simulations [Lehtinen et al., 1999] and SMC [Babich et al., 2001] and two kinetic equation approaches (KE): Symbalisty and MigDesk [Babich et al., 2001]. Processes with photon participation were not included in the simplified MC simulations and in the KE solution. It is seen that with electron angular scattering switched off, the rates given by SMC and the KE are very close to the ELIZA rates, thus indicating that photon and positron effects are negligible. Even when angular scattering is included, the rates obtained by various techniques agree rather well. The differences are possibly connected with the different cross sections for angular scattering and the inherent shortcomings of a KE approach relative to the comprehensive ELIZA treatment. [8 ] Dwyer [2003] performed a test simulation with Bremsstrahlung ‘‘temporarily suppressed, the resulting avalanche rates. . .agree within 10% with published rates by Lehtinen et al. [1999]’’. As Table 2 shows the rates given by the ELIZA code allowing for all manner of electron, positron, and photon interactions, agree to better than 10% with the results of Lehtinen et al. and SMC. This result indicates that photon and positron contributions to the RREA rates are very small in agreement with Gurevich et al. [2000]. The direct comparison given in Table 3 demonstrates that the difference between ELIZA and Dwyer’s rates is significant for weaker fields. [9] We repeated self-consistent RREA simulations for d = 2 and d = 8 in air at P = 1 atm. Using the ELIZA code, the same simulations were carried out by Babich et al. [2004] in the range d = 1.2– 14. However this time we followed up temporal dependencies of positron and photon numbers for post-processing and display. N0 (=10,000) seed electrons with initial energy 2 MeV were injected
1. Introduction [2] Runaway breakdown was first developed by Gurevich et al. [1992]. A number of subsequent papers were dedicated to an analysis of the kinetic theory [e.g., Roussel-Dupre et al., 1994; Gurevich et al., 2000] and to numerical calculations of the avalanche rates [e.g., Symbalisty et al., 1998; Lehtinen et al., 1999; Babich et al., 2001]. Dwyer recently published results of Monte Carlo (MC) simulations that showed a significant contribution originating from secondary photons and positrons to the runaway relativistic electron avalanche (RREA). A feedback mechanism associated with the initiation of secondary RREAs by these secondary emissions was also demonstrated [Dwyer, 2003]. The goal of this paper is to elaborate on the results of Dwyer.
2. Monte Carlo Simulations [3] Our remarks will be based on a comparison between Dwyer’s calculations and the results of recent and past simulations including those conducted using the full VNIIEF MC code ELIZA [Babich et al., 2001, 2004]. ELIZA incorporates new libraries of interaction crosssections including new data on the relaxation of atomic electron shells. The following elementary processes are included.
Copyright 2005 by the American Geophysical Union. 0094-8276/05/2004GL021744
3. Bremsstrahlung (Photons and Positrons) Contribution to the RREA Rate
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Table 1. Characteristic RREA Enhancement Time te (ns)a
values of d the deposition of photon-generated electrons varies proportionally to the photon number. In particular at d = 8 the photon number is 10 fold less than at d = 2. At d < 2 the photon and positron depositions should be significantly larger.
Electron Angular Scattering Not Taken Into Account Overvoltage d = E/(2.18 kV/cm)
2
5
Taken Into Account
8
2
5
8
16.3
161 197
34.4 39.9
18.9 21.2
11.2 10.7
174.4 200 189.7
33.2 35.6 34.3
17.3 18.6 17.8
KE Symbalisty MigDesk
98
31 MC
Lehtinen et al. [1999] SMC ELIZA [Babich et al., 2004]
77.6 81
20.8 20.1
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a
Air, P = 1 atm.
along the electric field. Results for the relative densities of electrons, photons, and positrons are presented in Figure 1. The temporal evolution of the electron and photon populations achieved an exponential form for both overvoltages while the positron population only achieved that form for d = 2 but not quite for d = 8. The results in Figure 1 are rather illuminating. For d = 2 at t = 750 ns (4 te) the positron population is 103 times the electron number (1s = 7.5%). Because the lifetime of the RREA photons (energies above 1 MeV) in air at P = 1 atm. is of order tfree 500 – 1700 ns, the portion of positrons in the exponential mode, will increase, but most likely, insignificantly. For d = 8 the simulation time t = 100 ns 6 te was insufficient for positrons to achieve an exponential growth. Therefore the positron portion, namely, 1.66 105 with 1s = 27%, is less than the steady state value of 104 expected from the fact that at d = 8 the number of the bremsstrahlung photons is 10 times less than at d = 2. The lower number of the photon-generated electrons coincides with the positron number (pair production). Because at d = 2 the Compton interaction rate exceeds by almost 500 times the rate of electron – positron pair production the number of photongenerated electrons is correspondingly larger. Figure 1 shows that at d = 2 each avalanche electron emits one photon. About half of the REs are capable of producing secondary REs. During the time tfree 500– 1700 ns (3 – 9) te the number of electrons due to direct avalanche multiplication increases by 20 –8000 times indicating that at d = 2 the photon-generated electrons constitute no more than 1 – 3% of the total electron population. For other
4. Feedback or Irradiation by Cosmic Rays? [10] Feedback due to the initiation of secondary RREAs by photons and positrons away from the primary avalanche ensuring a series of new RREAs is similar to that of the conventional streamer breakdown: a few long – range photons emitted by the primary avalanche produce seed electrons that initiate new avalanches, thus speeding up the streamer propagation. This promising idea needs more thorough analyses concerning the external (cosmic-ray) source of seed electrons and the polarization effect. [11] To address this issue we adopt the equation set used by Gurevich et al. [2000] to represent the 1D spatial growth of energetic electrons, positrons, and X-rays in the avalanche. We assume that an applied electric field exceeds the threshold for runaway by a factor to be specified below in a region extending from x = 0 to x = xmax. The relevant equations are: @nrun nrun ¼ @x la
ð1Þ
@nX nrun nX ¼ @x lbr lc
ð2Þ
@nþ ¼ sþ Nm f nX @x
ð3Þ
where nrun, nX, and n+ are the number densities of REs, X-ray photons, and positrons, respectively, la cte is the RREA length, lbr is the bremsstrahlung production scale length, lc is the Compton attenuation scale, s+ is the pairproduction cross-section, f is the fraction of photons with energy greater than twice the rest energy of the positron, and Nm is the air density. For constant la, lbr, lc, and s+ the solutions to equations (1) – (3) are straightforward and given in terms of the local density of REs n0 produced by cosmic rays. The solution for nX represents the total density of photons over all energies (less than the RE energy) and
Table 2. Difference Between ELIZA Rates and Rates Given by Simplified MC Techniques Overvoltage d = E/(2.18 kV/cm)
te ½ELIZA te ½Lehtinen % Lehtinen et al. ðte ½ELIZA þ te ½ðLehtinenÞ Þ=2 te ½ELIZA te ½SMC % SMC ðte ½ELIZA þ te ½SMC Þ=2 te ½ELIZA te ½SMC % SMC ðte ½ELIZA þ te ½SMC Þ=2 Switched off angular scattering
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2
5
8
8.4
3.3
2.9
5.3
3.7
4.4
4.3
1.7
2.3
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Table 3. Difference Between ELIZA and Dwyer’s Rates (%) Field
Dwyer
d
E = dFmin(1) kV/m
l, m [Dwyer, 2003]
1.4 2.5 5 8 12
305 545 1090 1744 2616
240 27 8.9 4.9 3.1
ELIZA te = l/c, ns
te, ns [Babich et al., 2004]
l = c te, m
l½Dwyer l½ELIZA ðl½Dwyer þ l½ELIZA Þ=2; %
800 89 29 16 10
1236 110 34.3 17.8 10.45
371 33 10.3 5.3 3.14
43 20 15 7.8 2.2
emitted in all directions by the ‘braking’ of REs. For this analysis we are interested in those photons that can propagate back to x = 0 and produce a population of seed energetic/REs by Compton scattering. Let hb equal the fraction of total photons that are emitted/scattered backward into a solid angle that encompasses the source region at x = 0 and let he equal the fraction of total photons that have sufficient energy to produce REs. Then, after integration (from x ! 0) of the photon production term folded against Compton scattering losses, it is possible to write the total density of photons that reach x = 0 and produce a seed population of energetic electrons as: nX ;run ¼ hb he n0
1
la
lbr 1
la lc
e
0 1 la x la x 1 lc la B lc la C @1 e A: ð4Þ
A similar result applies for positrons except that no attenuation occurs given that the applied electric field accelerates the positrons in the same way as for the REs but in the opposite direction. Thus, we have: þ nþ;run ¼ hþ b he nþ ;
ð5Þ
where h+b and h+e have the same meaning as hb and he. In order for feedback to compete with the cosmic ray source of seed energetic electrons we must have nX,run and/or n+,run equal to n0, assuming that each photon or positron produces one RE. With this condition it is possible to derive an expression for the critical length scale (xcr) over which the RREA must proceed in order to produce sufficient radiation for feedback to compete. We find in each case,
xcr;X la
3 2 l 1 a 6 lbr lc 7 7 ¼ ln6 4 la hb he 5
1 1
la lc
xcr;þ lbr 1 ¼ ln ; þ la la h þ b he f Nm sþ la
;
radius of 300 m (comparable to the dimensions of charge regions in thunderstorms and of extensive air showers of cosmic rays), that the minimum energy of photons needed to produce energetic seed electrons at x = 0 is 500 keV, and that the corresponding minimum energy required of photons to pair-produce 500 keV positrons is 1.5 MeV. We note here that Dwyer assumed a transverse dimension of infinity in his calculations and that this choice will maximize the effect of photon scattering. The choice of 500 keV as a representative value of the photon energy is appropriate for d = 2 because the mean energy of the corresponding Compton electrons is 170 keV which is the minimum energy for the electron to runaway at this overvoltage [e.g., Babich et al., 2001]. However, this choice is somewhat arbitrary for other values of d and for the simple reason that higher energy photons have a larger mean free path (so that more of them reach x = 0) and will also create REs. In order to accommodate these considerations and others we present results for representative photon energies eg = 500 keV and 4 MeV. At 4 MeV the dominant factor outside the ln term in equation (6) begins to saturate because la lc for most d and xcr,X is insensitive to the various parameters in the ln term. Thus we expect the results with eg = 0.5 and 4 MeV to be upper and lower bounds for feedback to proceed. The positron critical length is insensitive to our choice of parameters (i.e., it is proportional to a ln term). With these considerations we find that at sea level air density, lbr = 72 m, lc = 90 m, 252 m (taken from the Nuclear Data Tables for eg = 0.5 and 4 MeV respectively), and la = 57 m for d = 2. With these parameter values and the results of our detailed Monte Carlo calculations for d = 2 that n+/nrun = 103 as noted previously we can derive a value for f = 3.4 102. From the detailed results of kinetic calculations we also find
ð6Þ
ð7Þ
where we have dropped the exponential term inside the parentheses of equation (4) to obtain equation (6) because la < lc and (1 la/lc)(xcr,X/la) is generally 1. We now obtain estimates for the critical lengths needed for feedback. As a first approximation we assume that the applied field has a magnitude such that d = 2, that the source region for the RREA (at x = 0) has a transverse
Figure 1. Particle and photon production. Time dependence of the number of electrons, bremsstrahlung photons, and positrons for different overvoltages.
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of electrons remains constant and then exponentiates for a time texp = (xmax xcr)/c at the RREA rate. We can depict this behavior mathematically as: x ðn 1Þðxmax xcr Þ la la nrun ¼ n0 e e tn1 t tn1 þ 2xcr =c x ðn 1Þðxmax xcr Þ ðt tn1 Þ la la te n ¼n e e e
Figure 2. Critical scale lengths for development of feedback as a function of the overvoltage. The solid and dotted curves represent feedback driven by X-rays (with eg = 0.5 and 4 MeV, respectively). The dashed curve is for positron driven feedback. The dot-dash curve represents the scale length above which the electric field is eliminated. hbhe 105 for all d, 2 107 for d 2 (for eg = 0.5 and 4 MeV respectively) and h+b h+e 0.5 in the electrified region for all d. Substituting these results into equation (6) and (7) we find xcr,X/la = 31.6 and 19.5, respectively, and xcr,+/la = 9.5. At an altitude of 8 km (typical midpoint for charge layers in clouds) the critical lengths for feedback to occur are 5.6, 3.5, and 1.7 km for X-rays (0.5 and 4 MeV) and positrons respectively. The upper critical length for X-rays is too large to be feasible both because thunderstorm electric fields with these amplitudes are not observed over this scale length and because a RREA over that scale length would eliminate the electric field before feedback could occur (see discussion below on polarization and shielding). The lower results for X-rays and positrons on the other hand are marginal. We note that the positron feedback result is in basic agreement with Dwyer [2003]. From Dwyer [2003, Figure 3] we would deduce a critical scale length of approximately 490 m for d = 2 at sea level compared to our values of 1.8 and 1.1 km for X-rays and 542 m for positrons. A plot of the dependence of xcr,X/la and xcr,+/la on d including the effect of photon and positron backscattering is shown in Figure 2. We see that the feedback mechanism becomes marginally possible due to X-rays beyond d 2.7 independent of the representative photon energy and remains plausible due to positrons over the entire range of d. The results at higher d are sensitive to the transverse dimension chosen for the charge region. We have not addressed values of d 5 because for larger d, xcr la la (the photon attenuation length) and as a result photon feedback may be irrelevant for our assumed field configuration. In addition the higher d are not measured in thunderstorms. [12] An important point not addressed by Dwyer concerns the temporal growth of the feedback mechanism. In general we would expect feedback to undergo a temporal stepping. Suppose that the length over which the avalanche occurs, xmax, is larger than xcr then the feedback mechanism will work to enhance the number of seed electrons at x = 0 and therefore the number of REs with time even though xmax is a constant in time. However, the enhancement process can only occur after the photons created by the REs propagate back to x = 0 from xcr and the now enhanced RE seed population avalanches and reaches xcr. Thus, during a period of time tint = 2 xcr/c the seed population
run
0
tn1 þ 2xcr =c t tn
ð8Þ
with tn = [2xcr/c + (xmax xcr)/c]n + x/c and where n represents the number of cycles over which feedback occurs. Observations of stepping in the leader activity preceding the return stroke and in other lightning discharge processes could well be an indication of the feedback mechanism at work. However in the leader process the discharge also steps forward in space. This characteristic could result from the enhanced charge at the end of the avalanche (due to feedback) that in turn would enhance the electric field and permit the avalanche to move forward. [13] As noted above shielding of the cloud field by the polarizing plasma produced by the avalanche is capable of stopping the process before feedback can switch on. The field relaxation time tp = e0/s is determined by the air conductivity s = ene (me/P) due to low-energy secondary electrons. Here me 300 cm2 atm./(Vs) [Huxley and Crompton, 1974] is the electron mobility in air, ne is the density of secondary electrons, and P is the air pressure. The density of secondary electrons is approximated as ne = x (erun/De)nrun = (erun/De)n0e l , where erun is the mean energy of the REs (7 MeV), De is the ‘‘price’’ for the production of one electronion pair, and n0 has the same meaning as before. The seed density n0 = FcosmicDW/c, where DW is the solid angle over which the seed population is produced. The cosmic ray showers produce a mean electron flux Fcosmic 2000 1/(m2ssr) above 10 MeV with a spectrum @Fcosmic/@e e1.29 [Daniel and Stephens, 1974]. The flux above the runaway threshold (eth 0.1 MeV) is likely more powerful at least by two or three orders of magnitude: Fcosmic 2 105 – 2 106 1/(m2ssr) assuming @Fcosmic/@e e1.29 for lower energies. Taking DW = 2p (electrons directed against the electric field only) we find n0 4.2 103 – 4.2 102 m3. With these results it is possible to write the field relaxation time in terms of the RE density and to derive a critical length xcr,p beyond which the electric field is eliminated and the RREA ceases. For the latter to apply we must have tp/te 1. We find, a
Xcr;p De e0 ; ¼ ln la erun h0 me ete
ð9Þ
where tp/te = 1 and e is the charge of the electron. A plot of the critical length for field relaxation is provided in Figure 2. Note that xcr,p/la is less than xcr,X/la for d < 2 assuming eg = 0.5 MeV so that the field in this case is eliminated before feedback can occur. For d > 2 xcr,p/la lies above the other
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critical lengths so that feedback can proceed. Initiation of feedback driven by positrons is unaffected by the field relaxation.
5. Conclusions [14] (1) The RREA rates obtained using ELIZA differ from those reported by Dwyer [2003] particularly at near threshold values of the overvoltage. [15] (2) Our preliminary results suggest that feedback could occur under certain thunderstorm conditions and over scale lengths similar to those obtained by Dwyer. We find that positron feedback is more important than photon feedback. [16] (3) We find that there is a temporal ‘stepping’ associated with the feedback process and that this effect could also lead to a spatial stepping reminiscent of stepped leaders. [17] (4) Significantly more work is required to establish the existence and role of feedback in the runaway process under thunderstorm electrical conditions.
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Monte Carlo simulations and from kinetic equation solution, IEEE Trans. Plasma Sci., 29, 430 – 438. Babich, L. P., E. N. Donskoy, I. M. Kutsyk, and R. A. Roussel-Dupre (2004), Characteristics of relativistic electron avalanche in air (in Russian), Dokl. Phys., 49, 35 – 38. Daniel, R. R., and S. A. Stephens (1974), Cosmic-ray produced electrons and gamma rays in the atmosphere, Rev. Geophys., 12, 233 – 258. Dwyer, J. R. (2003), A fundamental limit on electric fields in air, Geophys. Res. Lett., 30(20), 2055, doi:10.1029/2003GL017781. Gurevich, A. V., G. M. Milikh, and R. A. Roussel-Dupre (1992), Runaway electron mechanism of air breakdown and preconditioning during a thunderstorm, Phys. Lett. A, 165, 463 – 468. Gurevich, A. V., H. C. Carlson, Y. V. Medvedev, and K. P. Zybin (2000), Generation of electron-positron pairs in runaway breakdown, Phys. Lett. A, 275, 101 – 108. Huxley, L. G., and R. W. Crompton (1974), The diffusion and drift of electrons in gases, 669 pp., Wiley – Interscience, Hoboken, N. J. Lehtinen, N. G., T. F. Bell, and U. S. Inan (1999), Monte Carlo simulation of runaway MeV air breakdown with application to red sprites and terrestrial gamma ray flashes, J. Geophys. Res., 104, 24,699 – 24,712. Roussel-Dupre, R. A., A. V. Gurevich, T. Tunnell, and G. M. Milikh (1994), Kinetic theory of runaway air breakdown, Phys. Rev. E, 49, 2257 – 2271. Symbalisty, E. M. D., R. A. Roussel-Dupre, and V. Yukhimuk (1998), Finite volume solution of the relativistic Boltzmann equation for electron avalanche studies, IEEE Trans. Plasma Sci., 26, 1575 – 1582.
References Babich, L. P., E. N. Donskoy, I. M. Kutsyk, A. Y. Kudryavtsev, R. A. Roussel-Dupre, B. N. Shamraev, and E. M. D. Symbalisty (2001), Comparison of relativistic runaway electron avalanche rates obtained from
L. P. Babich, E. N. Donskoy, and I. M. Kutsyk, Russian Federal Nuclear Center VNIIEF, pr. Mira 37, Sarov, Nizhegorodskaya Oblast, 607180 Russia. (
[email protected]) R. A. Roussel-Dupre´, EES-2, Mail Stop F665, Los Alamos National Laboratory, Los Alamos, NM 87545, USA. (
[email protected])
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