Kelvin–Helmholtz shear instability

Take two semi-infinite, inviscid, incompressible fluids separated by a horizontal interface at $y=0$. Label the upper fluid 1 and the lower fluid 2, with densities $\rho_1,\rho_2$ and uniform horizontal speeds $U_1,U_2$. Gravity points downward. The interface is material and has surface tension T. In each uniform region, a small irrotational disturbance admits a velocity potential that decays away from the interface. This does not require the combined base flow to have zero sheet vorticity.

The kinematic condition says that the displaced interface moves with the fluid on either side. Linearized Bernoulli relations provide pressure disturbances. Their jump must balance the linearized capillary pressure, while hydrostatic pressure supplies the gravity contribution. Use a real positive wavenumber k and dependence $e^{i(kx-\omega t)}$. Positive imaginary frequency then gives temporal exponential growth. These matching conditions, rather than a drawing of rolled-up vortices, determine the initial linear behavior.

The density-weighted mean velocity moves the pattern. It contributes a real frequency and therefore no envelope growth by itself. Relative velocity appears squared with a destabilizing sign in the radicand. For a stably stratified interface, the denser fluid is below and the gravitational term is positive. Positive surface tension also restores a flat interface, becoming especially important at large wavenumber. Comparing these terms predicts which wavelengths can amplify within this model.

If gravity and surface tension are both neglected, any nonzero velocity jump destabilizes this ideal sheet. Its growth rate is proportional to k and to the magnitude of the velocity difference. Equal densities maximize the density factor at fixed total density. Unequal densities reduce the growth factor and move the pattern toward the denser stream’s speed. A Galilean boost changes both base velocities by the same constant; it changes propagation while leaving the velocity difference and growth unchanged.

The unbounded short-wave growth as wavelength approaches zero is a warning about the idealization. A real shear layer has finite thickness, viscosity, and sometimes compressibility, stratification, or surface tension. A disturbance whose wavelength is comparable with the shear thickness samples the internal profile rather than a sharp jump. Its growth must be calculated from that profile and the appropriate fluid equations. There is no license to extrapolate the sheet formula to molecular scales or to an arbitrary atmospheric interface.

Linear theory predicts initial amplification, not the eventual vortex roll-up, entrainment rate, or mixing efficiency. Once slopes or velocities cease being small, the interface shape and surrounding flow alter one another nonlinearly. An experiment can also contain several wavelengths and three-dimensional secondary instabilities. To compare data with the linear prediction, measure a small-amplitude Fourier component during a well-defined early interval and report the densities, layer thickness, velocity difference, and omitted restoring effects.