Steady-state nonlinear internal gravity-wave critical layers satisfying an upper radiation condition

Abstract

We consider the behaviour of an internal gravity wave encountering a critical level in a stratified fluid, assuming the critical-level flow to be dominated by nonlinear effects. The background flow is a shear layer, and the stratification is sufficiently strong to support wave propagation everywhere. Incident and reflected waves are permitted below the critical level, and a radiation condition is imposed far above it. For this geometry we construct, by a combination of asymptotic and numerical means, steady, nonlinear solutions, and discuss the associated transmission coefficients, reflection coefficients, phase shifts, and resonance positions when the system is forced from below.The inviscid solutions we exhibit have continuous density and velocity everywhere, and so do not require the introduction of internal viscous boundary layers. Further, the streamlines bounding the recirculating cat's-eye regions have corners, just as in the unstratified case. For weak stratification, the transmitted wave is nearly as strong as the incident wave, and there is accompanying strong over-reflection. As the stratification increases, the critical level becomes a nearly perfect reflector. The amount of transmission depends on wave amplitude, and the sensitivity increases with increasing stratification.There are regions of parameter space for which steady solutions could not be found. The critical-layer structure appears to break down by unbounded thickening when the stratification becomes too strong, suggesting that in these cases some neglected physical process must intervene to limit growth of the recirculating region.