Exponentially self-similar impact ionization waves

The existence of generalized self-similar solutions to the system of continuity and Poisson equations is analyzed for the problem of evolution of impact ionization waves (IIWs). It is shown that, for any physically reasonable electric-field dependence of the impact ionization coefficients, there exist only exponentially self-similar (“limiting”) asymptotic solutions. These solutions describe IIWs whose spatial scales and propagation velocities increase exponentially with time. Conditions are found for the existence of plane, cylindrical, and spherical waves of this type; their structure is described; analytical relations between the key parameters are derived; and effects of recombination (or attachment) and tunnel ionization are analyzed. It is shown that these IIWs are intermediate asymptotics of numerical solutions to the corresponding Cauchy problems. The most important and interesting type of exponentially self-similar IIWs are streamers in a uniform electric field. The simplest comprehensive and explicit model describing their evolution is a spherical IIW.