= Solution
The <Rayleigh-Taylor instability> extracts <gravitational potential energy> when a heavy fluid is supported above a lighter one. A small downward excursion of heavy material and upward excursion of light material can grow instead of experiencing restoring <buoyancy>. The <Kelvin-Helmholtz instability> instead extracts <kinetic energy> from a <velocity> difference across an interface or <shear layer>. The interface displacement and its induced flow reinforce one another by tapping the background shear; a reversed density ordering is not required.
A useful common model has two inviscid, incompressible, semi-infinite fluids separated at $z=0$, with lower density/<velocity> $\rho_1,U_1$, upper density/<velocity> $\rho_2,U_2$, downward <gravity> $g$, and <surface tension> $\sigma$. For a disturbance proportional to $e^{i(kx-\omega t)}$, with $k>0$, the <two-fluid interfacial gravity-shear dispersion relation> is
$$
\left(\omega-k\frac{\rho_1U_1+\rho_2U_2}{\rho_1+\rho_2}\right)^2=\frac{gk(\rho_1-\rho_2)+\sigma k^3}{\rho_1+\rho_2}-\frac{\rho_1\rho_2}{(\rho_1+\rho_2)^2}k^2(U_1-U_2)^2.
$$
A negative right side gives exponential growth. With no shear or <surface tension> and $\rho_2>\rho_1$, the <growth rate> of the <Rayleigh-Taylor instability> is $s=\sqrt{\mathcal A gk}$, where $\mathcal A=(\rho_2-\rho_1)/(\rho_2+\rho_1)$ is the <Atwood number>. With equal densities and no restoring force, the <growth rate> of the <Kelvin-Helmholtz instability> is $s=k|U_1-U_2|/2$. \b[Heavy-over-light stratification drives Rayleigh-Taylor growth; <velocity> shear drives Kelvin-Helmholtz growth.] A heavy lower fluid supplies gravitational stabilization, and <surface tension> stabilizes sufficiently short waves in this ideal interface model.
Finite shear-layer thickness, <viscosity>, compressibility and <magnetic tension> can change the <growth rates> and unstable <wavelength> range. In particular, unlimited growth with increasing $k$ is a limitation of the zero-thickness, zero-surface-tension idealization. Density-interface overturning and jet-boundary mixing are appropriate physical applications of the two mechanisms, but the nonlinear mixed state is not determined by the linear <growth rate> alone.
\Image[/past-exam-of-the-astrophysical-fluid-dynamics-course-of-the-university-of-cambridge/2016/paper-1-interface-instabilities.png]
{title=Original schematics of <Rayleigh-Taylor instability> and <Kelvin-Helmholtz instability>}
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