Diffuse-Boundary Rayleigh-Taylor Instability

For density profiles, $N\left(y\right)\mathrm{}$, making a smooth transition from $N(-\infty )=0\mathrm{}$ to $N(+\infty )=\mathrm{const}$ with $\frac{d\ln N}{\mathrm{dy}}$ decreasing monotonically with $y$, it is shown that the Rayléigh-Taylor instability exhibits essentially different behavior above and below a certain critical wave number, $k_c$. For $k>\mathrm{}{k}_c$ the growth of the response to an initial perturbation is slower than exponential, $\sim t^{-\frac{1}{2}}\exp \left({\gamma }_{b}t\right)$. For $k<\mathrm{}{k}_c$ an unstable eigenmode (analogous to that in the sharp boundary case) exists, and purely exponential growth occurs.