Stability of stratified shear flows

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 J₀ the minimum of the local Richardson number over the flow domain, it is further established that kc_i → 0+ as $\text{J}{}_0\text{→}\text{1}\text{4}\text{?}$. The case of free upper surface and conditional reduction dependent on the curvature of the basic velocity of the unstable region is also studied.