Transient critical-level effect for internal gravity waves in a stably stratified shear flow with thermal forcing

We have examined the transient critical-level effect for internal gravity waves produced by thermal forcing in a stably stratified shear flow. For this, we have solved analytically the equations governing small-amplitude perturbations in a two-dimensional, hydrostatic, nonrotating, inviscid, and Boussinesq flow system. In the case of the point pulse forcing, there is only one transient critical-level line at any time and every internal gravity wave passing through the critical level is attenuated by a factor of e ^-πμ[μ = ( Ri - 1/4)^1/2, Ri: Richardson number (=N²/α², where N is the buoyancy frequency and α the vertical shear of the basic-state horizontal velocity)]. In the case of the line-type pulse forcing (bell-shaped in the horizontal), there are an infinite number of transient critical-level lines at any time. The attenuation factor for internal gravity waves is a function of space and time. This is because internal gravity waves passing through any point at any time coasist of internal gravity waves which have already experienced the transient critical-level effect and those which have not experienced the effect yet.