Shock Wave and Solitary Wave Structure in a Plasma

Abstract

Plane steady shock waves and solitary waves in a hydrogen plasma without external magnetic fields are studied using a simple kinetic theory model to describe the plasma. This model uses a Mott-Smith distribution for the protons and a local Maxwellian distribution for the electrons. Charge separations occur inside the shock front because of the proton-electron mass difference. The proton and electron densities, and the electric field, have an oscillatory fine structure with characteristic length ∼10 MλD, where M is the Mach number and λD the Debye length, going through the shock. In these oscillations, the densities overshoot their final Rankine-Hugoniot condition values. The distance in which the oscillations decay to 1/e, which we take as the shock width, is ∼4λ, where λ is the mean free path in the unshocked gas, in the range of Mach numbers considered. There are no continuous solutions for this model above Mach 2.19. The peak electric fields inside the shock are large; in a plasma of λ/λD = 2 × 104, the electric field reaches 2.2 × 106 v/cm in a Mach 2.1 shock. The large-amplitude solitary waves, which are the basic ingredient of the fine structure of the shock, are studied separately, and a heuristic picture is presented of the growth of these solitary waves, which is due to the coupling between the bulk flow of the plasma and the internal electric fields generated by charge separation. The rate of decay of these solitary waves is estimated.