The influence of the Mach number on the stability of radiative shocks

Abstract

We study the stability properties of hydrodynamic shocks with finite Mach numbers. The linear analysis supplements previous analyses which took the strong shock limit. We derive the linearised equations for a general specific heat ratio as well as temperature and density power-law cooling functions, corresponding to a range of conditions relevant to interstellar atomic and molecular cooling processes. Boundary conditions corresponding to a return to the upstream temperature ($R$ = 1) and to a cold wall ($R$ = 0) are investigated. We find that for Mach number $M > 5$, the strong shock overstability limits are not significantly modified. For $M < 3$, however, shocks are considerably more stable for most cases. In general, as the shock weakens, the critical values of the temperature power-law index (below which shocks are overstable) are reduced for the overtones more than for the fundamental, which signifies a change in basic behaviour. In the $R$ = 0 scenario, however, we find that the overstability regime and growth rate of the fundamental mode are increased when cooling is under local thermodynamic equilibrium. We provide a possible explanation for the results in terms of a stabilising influence provided downstream but a destabilising effect associated with the shock front. We conclude that the regime of overstability for interstellar atomic shocks is well represented by the strong shock limit unless the upstream gas is hot. Although molecular shocks can be overstable to overtones, the magnetic field provides a significant stabilising influence.