Quasi-steady dissipative nonlinear critical layer in a stratified shear flow

Abstract

When a wave with small but finite amplitude ε propagates towards the CL, where the effects of nonlinearity and dissipation are essential, the jump of mean vorticity over the CL appears. For the dynamically stable stratified shear flow with the gradient Richardson number Ri≳1/4 the jump of vorticity has the same order as the undisturbed one [J. Fluid Mech. 233, 25 (1991)]. The process of formation of the flow with this substantial jump of vorticity (or ‘‘break’’ of the velocity profile) in the CL is studied at large time after beginning of the process. The transition region between the CL and the undisturbed flow, the dissipation boundary layer (DBL), is shown to be formed. Its thickness grows in time proportional to √t (t being time), and the CL moves towards the incident wave. When the jump of the wave momentum flux over the CL is constant in time, the flow characteristics can be found in the most simple way. The velocity profile in the DBL appears to be self-similar, the displacement of the CL is proportional to √t and the values of vorticity at the both sides of the CL do not depend on time and they are determined only by the constant wave momentum flux. It is shown that, to provide the constant jump of the wave momentum flux the amplitude of the wave radiated by the source in the undisturbed flow region should vary in a certain complicated manner, because it reflects from the time-dependent (broadening) velocity profile in the DBL. On the other hand, the wave momentum flux from the steady source (for example, the corrugated wall) depends on time. When the coefficients of reflection from the CL (R) and from the DBL (r) are small, this dependence is weak and the wave and flow parameters depending on time are found as series in R and r. The wave–flow interaction for this case is studied.