An analytical approach for calculating energy spectra of relativistic runaway electron avalanches in air

Simplified equations describing the transport and energy spectrum of runaway electrons are derived from the basic kinematics of the continuity equations. These equations are useful in modeling the energy distribution of energetic electrons in strong electric fields, such as those found inside thunderstorms. Dwyer and Babich (2011) investigated the generation of low-energy electrons in relativistic runaway electron avalanches. The paper also developed simple analytical expressions to describe the detailed physics of Monte Carlo simulations of relativistic runaway electrons in air. In the current work, the energy spectra of the runaway electron population are studied in detail. Dependence of electron avalanche development on properties such as the avalanche length, radiation length, and the effective Møller scattering efficiency factor are discussed in detail. To describe the shapes of the electron energy spectra for a wide range of electric field strengths, the diffusion term responsible for random deviation of electron energy ionization loss from the mean value is added to the kinetic equation. We find that the diffusion in energy space helps maintain an exponential energy spectrum for electric fields that approach the runaway electron threshold field.