Past exam of the astrophysical fluid dynamics course of the University of Cambridge / 2016 / Paper 1 / 2 / a

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 isA 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 $s=k|U_1-U_2|/2$. 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.