Compressible Analysis of Combined Kelvin-Helmholtz Rayleigh-Taylor Instabilities in the Supersonic Regime

The Kelvin-Helmholtz (KH) and Rayleigh-Taylor (RT) instabilities are well studied in classical fluid dynamics. The stabilizing effect of magnetic fields is also well known. However, the analysis does in general assume incompressibility. It is known from fluid dynamics that compressibility stabilizes the KH instability at high velocities. In this work we begin developing a model for the combined hydrodynamic KH and RT instabilities for a compressible fluid in the presence of an equilibrium magnetic field. We report on both (1) the case with constant magnetic field aligned with the velocity using ideal MHD and (2) the case without magnetic field for benchmark and comparison. For the first case, no analytic expression for the dispersion relation is present in the literature to the best of our knowledge. We present our results as numerical solutions of the obtained dispersion relations, from which we derive approximate analytic expressions. We investigate the behavior of the instability with increasing magnetic field starting from the unmagnetized case, and the field strength required to completely suppress the instability given some initial parameters. We compare our results with the unmagnetized case, in particular with analytic expressions for the asymptotic behavior of the growth rate at supersonic velocities, beyond the point where the pure KH is stabilized by compressibility.