Compressibility Effect on the Rayleigh–Taylor Instability with Sheared Magnetic Fields

Abstract

We study the effect of plasma compressibility on the Rayleigh–Taylor instability of a magnetic interface with a sheared magnetic field. We assume that the plasma is ideal and the equilibrium quantities are constant above and below the interface. We derive the dispersion equation. Written in dimensionless variables, it contains seven dimensionless parameters: the ratio of plasma densities above and below the interface $\zeta$ , the ratio of magnetic field magnitude squared $\chi$ , the shear angle $\alpha$ , the plasma beta above and below the interface, $\beta_{2}$ and $\beta_{1}$ , the angle between the perturbation wave number and the magnetic field direction above the interface $\phi$ , and the dimensionless wave number $\kappa$ . Only six of these parameters are independent because $\chi$ , $\beta_{1}$ , and $\beta_{2}$ are related by the condition of total pressure continuity at the interface. Only perturbations with the wave number smaller than the critical wave number are unstable. The critical wave number depends on $\phi$ , but it is independent of $\beta_{1}$ and $\beta_{2}$ , and is the same as that in the incompressible plasma approximation. The dispersion equation is solved numerically with $\zeta= 100$ , $\chi= 1$ , and $\beta_{1} = \beta_{2} = \beta$ . We obtain the following results. When $\beta$ decreases, so does the maximum instability increment. However, the effect is very moderate. It is more pronounced for high values of $\alpha$ . We also calculate the dependence on $\phi$ of the maximum instability increment with respect to $\kappa$ . The instability increment takes its maximum at $\phi= \phi_{\mathrm{m}}$ . Again, the decrease of $\beta$ results in the reduction of the instability increment. This reduction is more pronounced for high values of $|\phi- \phi_{\mathrm{m}}|$ . When both $\alpha$ and $|\phi- \phi_{\mathrm{m}}|$ are small, the reduction effect is practically negligible. The theoretical results are applied to the magnetic Rayleigh–Taylor instability of prominence threads in the solar atmosphere.