NONLINEAR DISSIPATIVE CRITICAL-LEVEL INTERACTION IN A STRATIFIED SHEAR-FLOW - INSTABILITIES AND GRAVITY-WAVES

A two-dimensional numerical model is used to investigate the nonlinear dissipative interaction between a disturbance and an unbounded stratified shear flow. The disturbances considered are Kelvin-Helmholtz instabilities and forced gravity waves. The nonlinear stabilization (destabilization) of Kelvin-Helmholtz instabilities at Prandtl number, Pr < 1 (Pr > 1), found by Brown et al. (1981) is recovered. The model confirms that it is mostly due to a stabilization (destabilization) of the mean flow by the wave. The nonlinear evolution of instabilities existing when thermal dissipation is large and when the Richardson number is everywhere larger than 0.25 is also investigated. lt is shown that such a mode stops growing when the nonlinear distortion of the mean flow becomes significant. For forced gravity waves, and when the initial minimum Richardson number, J = 0.25, it is found that mean flow stabilization (destabilization) also occurs at the critical level for Pr < 1 (Pr > 1). More generally, the value of Pr above (below) which critical level destabilization (stabilization) occurs increases (decreases) when J increases (decreases). The nonlinear reflection and transmission of a wave are partly related to those stability changes. They are also related to the mean flow distortions, located below the critical level, and where the incident wave can be strongly reflected. In the present study, the critical level interaction is weakly nonlinear, quasi-steady and dissipative. The amplitude of the fundamental mode is such that the nonlinear effects are significant while secondary modes remain small and convective overturning does not occur.