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

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 ζζ, the ratio of magnetic field magnitude squared χχ, the shear angle αα, the plasma beta above and below the interface, β2β2 and β1β1, the angle between the perturbation wave number and the magnetic field direction above the interface ϕϕ, and the dimensionless wave number κκ. Only six of these parameters are independent because χχ, β1β1, and β2β2 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 ϕϕ, but it is independent of β1β1 and β2β2, and is the same as that in the incompressible plasma approximation. The dispersion equation is solved numerically with ζ=100ζ=100, χ=1χ=1, and β1=β2=ββ1=β2=β. We obtain the following results. When ββ decreases, so does the maximum instability increment. However, the effect is very moderate. It is more pronounced for high values of αα. We also calculate the dependence on ϕϕ of the maximum instability increment with respect to κκ. The instability increment takes its maximum at ϕ=ϕmϕ=ϕm. Again, the decrease of ββ results in the reduction of the instability increment. This reduction is more pronounced for high values of |ϕ−ϕm||ϕ−ϕm|. When both αα and |ϕ−ϕm||ϕ−ϕm| 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.