Diffuse-Boundary Rayleigh-Taylor Instability

For density profiles, $N(y)$, making a smooth transition from N(-∞)=0 to N(+∞)=const with dlnN/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, kc. For k>kc the growth of the response to an initial perturbation is slower than exponential, ~t^-1/2exp(γbt). For k<kc an unstable eigenmode (analogous to that in the sharp boundary case) exists, and purely exponential growth occurs.