Density gradients to reduce fluid instabilities in multishell inertial-confinement-fusion targets

The growth of Rayleigh-Taylor instabilities in intertial-confinement-fusion targets can be reduced if smooth density gradients are introduced at the interfaces. Keeping the total mass fixed, we consider spreading a heavy shell either continuously or in a discrete number of subshells. We calculate the rate for the fastest growing mode as a function of the fraction of mass spread and find the minima for the cases of 4, 6, and 8 subshells. The rates are reduced by 1.4-1.6. If all the mass is spread continuously into an exponential profile, we find that the rates are reduced by approximately $\sqrt{2\pi }\approx 2.5$.