Two‐dimensional Models for Nonlinear Vorticity Waves in Shear Flows

Abstract

The evolution of weakly nonlinear wave perturbations in shear flows of stratified fluid is investigated for large Reynolds numbers. The study is focused on the vorticity waves, i.e., the wave‐like motions caused by the mean flow vorticity gradient. A situation typical of the upper ocean is considered.The shear flow is supposed to be localized near the surface and to have no inflection points. The vertical scale of stratification is much larger than that of the shear current. Description of the dynamics of essentially three‐dimensional wave perturbations is reduced by a systematic asymptotic procedure to a single nonlinear evolution integrodifferential equation for (2+1)‐dimensions. The small parameters are the ratio of the vertical scale of the shear to the typical wavelength of the perturbations and the amplitude parameter. The equation does not contain viscous terms, but the regime of evolution it describes occurs owing to small but finite viscosity. The viscosity inhibits generation of strongly nonlinear vortices in the critical layer. Possible existence of localized two‐dimensional stationary solutions of the equation is investigated. Axially symmetric soliton solutions are found for a fluid of arbitrary depth in the limit of vanishing stratification. In stratified flows a linear resonant interaction between shear flow perturbations and internal waves is found to play the major role. The radiation damping of vorticity waves due to these resonances makes the existence of similar lump solitary structures in stratified fluid impossible.