On analytical approximations for the structure of a shock wave in a fully ionized plasma

Abstract

Two approximate solutions for the shock wave structure in a fully ionized plasma are given for weak and moderately strong shocks. Both solutions are algebraically very simple in the phase space of the electron and ion temperatures as functions of the plasma velocity, being algebraically more involved in the physical spatial coordinate, except when constant electron conductivity is assumed. One solution is based on the observation that for weak, relaxation shocks, the electron and ion temperatures are very close to each other. However, for sufficiently large ionization (atomic) number Z, this solution is valid even for any difference between both temperatures, capturing quite accurately the ion temperature overshoot appearing in moderately strong relaxation shocks for large Z. For stronger shocks with an internal ion shock, this first approximate solution remains quite accurate in the preheating region upstream of the inner shock but not in the relaxation downstream region. For the latter region, we find another good algebraic approximation based on the almost constancy of the electron entropy. The combination of these two approximations upstream and downstream of the inner shock, connected through the algebraic Rankine-Hugoniot relations for the inner shock, provides a good approximation for the entire shock structure even for moderately strong shocks. These alge- braic approximate solutions are compared with exact numerical solutions for several values of the Mach and ionization numbers. Some rele- vant features such as the shock thickness and the ion temperature overshoot are analyzed.