430 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 29, NO. 3, JUNE 2001 Comparison of Relativistic Runaway Electron Avalanche Rates Obtained from Monte Carlo Simulations and Kinetic Equation Solution Leonid P. Babich, Evgenii N. Donskoy, Igor M. Kutsyk, Andrei Yu. Kudryavtsev, Robert A. Roussel-Dupré, Boris N. Shamraev, and Eugene M. D. Symbalisty Abstract—New computer simulations of the relativistic runaway electron avalanche mechanism were carried out to remove prior approximations and reassess the space and temporal scales pre- dicted by previous Boltzmann calculations. Two Monte Carlo tech- niques, a finite-difference solution, and a finite-volume solution of the kinetic equation for high-energy electrons were employed. Re- sults obtained for the length and time scales at sufficiently high electric fields by the different methods are consistent with each other and with analytical estimates. The physical reason for the remaining discrepancy at small fields is discussed. Index Terms—Avalanche, kinetic equation, Monte Carlo, run- away electrons. I. INTRODUCTION I N AN EFFORT to understand the source of -ray emissions observed in thunderstorm clouds, Gurevich et al. proposed a new breakdown mechanism based on a relativistic runaway electron avalanche (RREA) that is capable of developing in an electric field that is weak compared to the conventional break- down threshold [1]. This mechanism has also been invoked to model recent observations of high-altitude transient luminous events (e.g., sprites and blue jets) that occur above thunder- storms. The rates of RREA development have been character- ized both in terms of -fold spatial scales and temporal scales . The scales were calculated previously from solu- tions of the kinetic equation (KE) [2]–[4], [6], from Monte Carlo (MC) simulations [5]–[7], and using the macroparticle approach [8]. Significant improvements were obtained since the first re- sults on the RREA scales were published, however, there still re- mained a 3–4-fold discrepancy between the KE and MC scales compared to the recent calculations of Symbalisty et al. [4]. This discrepancy is sufficiently large to have important impli- cations for the modeling of high-altitude and intracloud dis- charges using the idea of RREA. Therefore, it is important to identify a cause or causes for the discrepancy between the KE and MC scales. Potential sources for the discrepancy include differences in the numerical algorithms (statistical MC versus Manuscript received April 4, 2000; revised March 14, 2001. L. P. Babich, E. N. Donskoy, I. M. Kutsyk, A. Yu. Kudryavtsev, and B. N. Shamraev are with the Russian Federal Nuclear Center-VNIIEF, Sarov 607190, Russia (e-mail: babich@expd.vniief.ru). R. A. Roussel-Dupré and E. M. D. Symbalisty are with the Earth and Environmental Sciences Division, Atmospheric and Climate Sciences, Los Alamos National Laboratory, Los Alamos, NM 87545 USA (e-mail: rroussel-dupre@lanl.gov). Publisher Item Identifier S 0093-3813(01)04956-6. finite-volume solutions of the KE), the treatment of the colli- sion processes (detailed cross sections with discrete interactions versus a Fokker–Planck diffusion approximation in the KE), the different accuracy of collision cross sections in different treat- ments, and the particular formulation and numerical implemen- tation of the ionization source term in KE for runaway electrons. In this paper, new results are presented for both MC simu- lations and for a KE solution that incorporates a more precise relation between directions of motion of two electrons partici- pating in an ionizing event, relative to the direction of the elec- tric force. The results are compared with analytical estimates. For sufficiently strong electric fields, the KE yields and scales consistent with those predicted by direct MC simulations of RREA. The analytical estimates are also in basic agreement with the detailed numerical results. In our KE approach, calcula- tions were performed for air at atmospheric pressure assuming an average molecular number of (or mean atomic number of ). II. ANALYTIC ESTIMATES FOR THE SPACE AND TEMPORAL SCALES OF RREA The first estimates of RREA rates were obtained by Gurevich et al. [1] without taking into account elastic scattering of elec- trons. With this approximation it is possible to obtain simple estimates for the rates and it is sufficient to know the differen- tial ionization cross section, the drag force due to inelastic interactions of electrons with atomic particles, the electric force which is convenient to characterize by overvoltage relative to the relativistic minimum of the drag force, and the neutral number density . To better appreciate the basic RREA process, it is helpful to review the assump- tions inherent to the analytical estimates obtained in [1] and to compare them with more detailed MC calculations performed without electron scattering. A relativistic electron moving through air experiences drag as a result of collisions with air molecules. The dominant in- teractions that lead to energy loss in the energy range of in- terest (from 1 keV to tens of megaelectronvolts) are Coulomb in nature and result in the transfer of sufficient energy to the “bound” electrons to ionize them. We note that the binding en- ergy even for the inner shell electrons in the air ( 500 eV) is so small compared to the energy of the incident relativistic elec- tron that the “bound” electron can be considered to be “free” and the resulting interaction is, therefore, Coulomb in nature. 0093–3813/01$10.00 ©2001 IEEE