Analytic solutions for Rayleigh-Taylor growth rates in smooth density gradients.

Abstract

The growth rate of perturbations on the shell of a laser fusion target can be estimated as \ensuremath{\surd}gk , where g is the shell acceleration and k is the transverse wave number of the perturbation. This formula overestimates the growth rate, and should be modified for the effects of density gradients and/or ablation of the unstable interface. The density-gradient effect is explored here analytically. With the use of variational calculus to explore all possible density profiles, the growth rate is shown to exceed \ensuremath{\surd}gk/(1+kL) , where L is a typical density-gradient scale length. Density profiles actually exhibiting this minimum growth rate are found.