Critical layer formation in a stratified medium with mean shear flows

The problem of the excitation of internal waves with a given wave number k and frequency in a stratified medium with shear flows is considered. The internal wave field of the form v(z)exp(–it+ikx) established as t in a medium without dissipation has a singular point at the level z=z₀ (critical level), at which the flow velocity U(z) coincides with the phase velocity /k. Dissipative effects (viscosity and heat conduction) smooth out this singularity. An exact solution of the model equation describing as t and zz₀ the field excited by oscillating sources activated at t=0 is constructed with allowance for dissipation. This makes it possible to describe the limiting steady-state field, determine the critical layer as the neighborhood of the critical level in which dissipation effects are important, and to estimate its width and the rate of convergence to the limiting steady-state regime. The asymptotic behavior of the fields is examined for Ri1, where Ri is the Richardson number. It is shown that when the well-known Miles stability condition Ri>1/4 is satisfied there are no natural oscillations with a critical level.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 82–93, May–June, 1990.