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An amplitude-evolution equation for linearly unstable modes in stratified shear flows

A nonlinear amplitude equation of second order in time, which governs the temporal evolution of linearly unstable modes in stratified shear flows, is derived. It applies to a class of flows with continuous velocity and density profiles, and two examples of such flows are studied. One of the flows that is studied is the stratified Couette flow with the buoyancy frequency equal to Qy2, where Q is a constant and y the vertical co-ordinate. The nonlinear amplitude equation is studied for various values of Q. For the Garcia flow the nonlinear amplitude equation for the long-wave modes is evaluated, and it is compared with the corresponding equation in the Kelvin–Helmholtz flow, which has been found previously.