Structure of a Plasma Shock Wave

Abstract

The one‐dimensional, steady‐state structure of a shock wave in a fully ionized plasma is investigated in the absence of external applied magnetic or electric fields. The structure is assumed to be described by the Navier‐Stokes equations written for the electron and ion fluids, together with Poisson's equation for the self‐induced electric field. When the Debye length downstream of the shock (λD2 ) is small compared to the ion‐ion mean free path there (l 2), the plasma remains essentially neutral and the equations governing the charge separation and electric field can be uncoupled from the system. The remaining equations are reduced by a boundary layer type analysis, which uses the fact that different dissipative mechanisms are important over different length scales, and an integration in phase space is carried out. It is found that when the free stream Mach number M 1 is less than 1.12, the plasma behaves like a single fluid and there is only one shock layer whose thickness is proportional to l 2/ε, where ε is the square root of the electron to ion mass ratio. When M 1 > 1.12, an ion shock appears imbedded in a wider electron thermal layer and the respective thicknesses are of order l 2 and l 2/ε. Within the ion shock the electric field and charge separation each show a single damped oscillation with an amplitude proportional, respectively, to λD2 /l 2 and (λD2 /l 2)2. When the shock is strong, a precursor electric shock layer in which the electric field and the charge separation again show a single damped oscillation appears upstream at the beginning of the thermal layer. With λD2 and l 2 of the same order, the electrical effects interact directly with the flow in the two electric shock layers. The equations are simplified by the fact that the electrical gradients are reduced by the increased tendency towards charge neutrality. Calculations for M 1 = 10 show that in the imbedded electric shock layer there is a relative excess of electrons of approximately 50% locally and that there are damped vestiges of ion oscillations present.