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Stability of stratified shear flows. (English) Zbl 0575.76106
For a steady plane parallel flow of an inviscid, incompressible fluid of variable density under gravity, it is shown that the complex wave velocity for any unstable mode lies in a semiellipse-type region whose major axis coincides with the diameter of Howard’s semicircle, while its minor axis depends on the stratification. If \(kc_ i\) denotes the complex part of wave frequency and \(J_ 0\) the minimum of the local Richardson number over the flow domain, it is further established that \(kc_ i\to 0+\) as \(J_ 0\to 1/4-\). The case of free upper surface and conditional reduction dependent on the curvature of the basic velocity of the unstable region is also studied.

MSC:
76V05 Reaction effects in flows
76E05 Parallel shear flows in hydrodynamic stability
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76F10 Shear flows and turbulence
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