Rayleigh–Taylor instability

Remarks

The normal modes satisfy $$s^2 = \frac{g k (\rho_2 - \rho_1) - \sigma k^3}{\rho_1 + \rho_2}.$$ So the wavenumber $k$ is unstable exactly when $\rho_2 > \rho_1$ and $k^2 < g (\rho_2 - \rho_1) / \sigma$, with every $k > 0$ unstable when $\sigma = 0$.

Example: with $\sigma = 0$ and heavy fluid on top, the growth rate is $s = \sqrt{A g k}$, where $A = (\rho_2 - \rho_1) / (\rho_2 + \rho_1)$ is the Atwood number. Non-example: if $\rho_2 < \rho_1$, the light fluid on top, then $s^2 < 0$ for every $k$, and the modes are oscillating interfacial gravity waves, not growing ones.

By the equivalence principle, the same instability occurs at an interface accelerated with acceleration $a$ directed from the light fluid towards the heavy fluid, with $g$ replaced by $a$. The configuration is the special case $U_1 = U_2 = 0$ of the Kelvin–Helmholtz problem.