2-DIMENSIONAL MODELS FOR NONLINEAR VORTICITY WAVES IN SHEAR FLOWS

The evolution of weakly nonlinear wave perturbations in shear flows of stratified fluid is investigated for large Reynolds numbers, The stud y is focused on the vorticity waves, i,e., the wave-like motions cause d by the mean flow vorticity gradient, A situation typical of the uppe r ocean is considered. The shear flow is supposed to be localized near the surface and to have no inflection points, The vertical scale of s tratification is much larger than that of the shear current. Descripti on 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 sm all 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 evoluti on it describes occurs owing to small but finite viscosity, The viscos ity inhibits generation of strongly nonlinear vortices in the critical layer. Possible existence of localized two-dimensional stationary sol utions of the equation is investigated, Axially symmetric soliton solu tions are found for a fluid of arbitrary depth in tile limit of vanish ing stratification, In stratified flows a linear resonant interaction between shear flow perturbations and internal waves is found to play t he major role, The radiation damping of vorticity waves due to these r esonances makes the existence of similar lump solitary structures in s tratified fluid impossible.