Rayleigh-Taylor and Richtmyer-Meshkov instabilities and mixing in stratified spherical shells

We study the linear stability of an arbitrary number of spherical concentric shells undergoing a radial implosion or explosion. The system consists of N incompressible fluids with small amplitude perturbations at each of the N-1 interfaces. We derive the evolution equation for the perturbation ${\mathrm{\eta }}_{\mathit{i}}$ at interface i; it is coupled to the two adjacent interfaces via ${\mathrm{\eta }}_{\mathit{i}\pm 1}$. We show that the N-1 evolution equations are symmetric under n⇆-n-1, where n is the mode number of the spherical perturbation, provided that the first and last fluids have zero density (${\mathrm{\rho }}_1$=${\mathrm{\rho }}_{\mathit{N}}$=0). In plane geometry this translates to symmetry under k⇆-k. We obtain several analytic solutions for the N=2 and 3 cases that we consider in some detail. As an application we derive the shock timing that is required to freeze out an amplitude. We also identify ‘‘critical modes’’ that are stable for any implosion or explosion history. Several numerical examples are presented illustrating perturbation feedthrough from one interface to another. Finally, we develop a model for the evolution of turbulent mix in spherical geometry, and introduce a geometrical factor G relating the mixing width h in spherical and planar geometries via ${\mathit{h}}_{\mathrm{spherical}}$=${\mathit{h}}_{\mathrm{planar}}$G. We find that G is a decreasing function of R/${\mathit{R}}_0$, implying that in our model ${\mathit{h}}_{\mathrm{spherical}}$ evolves faster (slower) than ${\mathit{h}}_{\mathrm{planar}}$ during an implosion (explosion).