Diffuse-Boundary Rayleigh-Taylor Instability

Abstract

For density profiles, $N(y)$, making a smooth transition from $N(\ensuremath{-}\ensuremath{\infty})=0$ to $N(+\ensuremath{\infty})=\mathrm{const}$ with $\frac{d\mathrm{ln}N}{\mathrm{dy}}$ decreasing monotonically with $y$, it is shown that the Rayl\'eigh-Taylor instability exhibits essentially different behavior above and below a certain critical wave number, ${k}_{c}$. For $k>{k}_{c}$ the growth of the response to an initial perturbation is slower than exponential, $\ensuremath{\sim}{t}^{\ensuremath{-}\frac{1}{2}}\mathrm{exp}({\ensuremath{\gamma}}_{b}t)$. For $k<{k}_{c}$ an unstable eigenmode (analogous to that in the sharp boundary case) exists, and purely exponential growth occurs.