Stability Analysis of Compressible Kelvin-Helmholtz Raleigh-Taylor Instability

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 both cases, 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 an approximate analytic expression. We show the behavior of the instability with increasing magnetic field starting from the unmagnetized case, obtained from numerically solving the dispersion relation, where it displays nonlinear behavior distinct from the incompressible case.