A transition from Kelvin–Helmholtz instabilities to propagating wave instabilities

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Abstract
The object of this paper is to study the linear stability of an unbounded stably stratified shear layer in an inviscid, Boussinesq fluid. The flow is modeled by the velocity and buoyancy frequency profiles: U=U0 tanh(z/d) and N2=N12+N22 ‖tanh(z/d)‖α, where α≳0. It represents a shear layer which has already been mixed, to a certain extent, since the stratification is smaller inside the shear layer than outside. This flow can generate propagating wave instabilities if the layer of less static stability is sufficiently broad (i.e., α sufficiently large) as compared to the layer of large velocity shear: when α<2, the flow only generates Kelvin–Helmholtz instabilities; when α≳2, the flow generates both Kelvin–Helmholtz and propagating wave instabilities. In three specific cases (α=0,2,4), the neutral modes are derived systematically using an analytical transform of the Taylor–Goldstein equation into the hypergeometric equation. Furthermore, the neutral modes, associated to propagating wave instabilities, correspond to gravity waves with infinite critical level reflection and transmission (i.e., resonant overreflection). It is to be noted that resonant overreflection is possible in the present model as long as the minimum Richardson number of the flow is smaller than 0.25. In the conclusion, the importance of the results obtained is discussed, in relation with the spontaneous generation of gravity waves in a stratified shear layer.