Electron acceleration by a stochastic electric field in the atmospheric layer

Electron acceleration by a stochastic (in space) electric field is considered in the atmosphere taking into account limited scales of the acceleration layer. The general kinetic equation for the isotropic part of the electron distribution function is thereby derived, which includes runaway effects and influence of a stochastic electric field on spatial diffusion. A tail of accelerated electrons in the distribution function appears when the electric field intensity overtakes the threshold value, which is close to the threshold in a constant electric field. Stationary solutions of the kinetic equation in a finite layer are analyzed numerically in the presence of source of energetic electrons. These solutions are discussed in connection with γ- and x-ray emissions observed inside thunderclouds.