Counterpropagating wave instability

Waves travelling in opposite directions relative to their local background currents can share a laboratory phase velocity. Their interaction can then lock their phases and extract energy from the mean shear, producing growing disturbances. In a layered flow, the coupling is often exponentially weak when the interface separation exceeds the waves' decay length. A dispersion relation for two density interfaces in uniform shear gives a simple explicit example.

For stable density jumps in the three-layer stratified piecewise-linear shear flow, the upper interface's intrinsically left-going gravity-vorticity interface wave and the lower interface's intrinsically right-going wave have opposite laboratory phase velocities. They coincide at zero when $1+J=2\alpha$. At large wavenumber, interaction is of order $e^{-2\alpha }$ and the unstable band of a three-layer stratified shear flow has width $4\alpha e^{-2\alpha }+O(\alpha e^{-6\alpha })$ around this resonance.