Rayleigh-Taylor instability in spherical geometry

An analysis of the Rayleigh-Taylor instability in a spherical geometry is presented. Expanding any initial perturbation at a spherical surface between two fluids in spherical harmonics $Y_{\mathrm{lm}}$ and further assuming an exponential time growth of the expansion coefficients, an eigenvalue equation for the growth rate is obtained. The free-surface and jump boundary conditions are obtained from the eigenvalue equation. The eigenvalue equation is solved for solid spherical targets and analytical formulas for the growth rate of the instability are presented in the cases where the initial plasma density profile has a step function or exponential variation in space. An analytical expression for the growth rate is also presented for the shell targets and its variation with the aspect ratio of the shell is discussed.