Linear stability analysis of compressible Kelvin–Helmholtz and Rayleigh–Taylor instability and the effect of a flow-aligned magnetic field

The Kelvin–Helmholtz (KH) and Rayleigh–Taylor (RT) instabilities are well-known phenomena in classical fluid dynamics and plasma physics. We investigate the stability of an ideal MHD fluid system subject to both KH and RT instabilities, considering the effects of compressibility and a magnetic field. While it is well established in the literature that compressibility can stabilize the KH instability at high Mach numbers, we find a new instability in the presence of a finite gravity. This instability persists regardless of whether the system is RT stable or unstable. Even though this residual instability is small, the eigenfunctions are very broad, affecting large portions of the fluid. We also demonstrate that under large gravitational accelerations, or small stratification lengths, the maximum growth rate of the instability with respect to the velocity difference deviates significantly from the case without gravity. In the presence of a flow-aligned B-field, we show an analytic expression that accurately approximates the field required for stabilization. Our results indicate that a stronger magnetic field is required for stabilization compared to the incompressible case.