Nonlinear Instabilities in Shock-Bounded Slabs

Abstract

(substantial changes to section 3.2, otherwise minor) We present an analysis of the hydrodynamic stability of a cold slab bounded by two accretion shocks. Previous numerical work has shown that when the Mach number of the shock is large the slab is unstable. Here we show that to linear order both the bending and breathing modes of such a slab are stable. However, nonlinear effects will tend to soften the restoring forces for bending modes, and when the slab displacement is comparable to its thickness this gives rise to a nonlinear instability. The growth rate of the instability, above this threshold but for small bending angles, is $\sim c_sk (kη)^{1/2}$, where $η$ is the slab displacement. When the bending angle is large the slab will contain a local vorticity comparable to $c_s/L$, where $L$ is the slab thickness. We discuss the implications of this work for gravitational instabilities of slabs. Finally, we examine the cases of a decelerating slab bounded by a single shock and a stationary slab bounded on one side by thermal pressure. The latter case is stable, but appears to be a special case. The former case is subject to a nonlinear overstability driven by deceleration effects. We conclude that shock bounded slabs with a high density compression ratio generically produce substructure with a strong local shear, a bulk velocity dispersion like the sound speed in the cold layer and a characteristic scale comparable to the slab thickness.