Edinburgh Napier University

  To get an insight into dynamics of the oceanic surface boundary layer we develop an asymptotic model of nonlinear dynamics of linearly decaying three-dimensional long10 wave perturbations in weakly stratified boundary layer flows. Although in nature the free surface boundary-layers in the ocean are often weakly stratified due to solar radiation and air entrainment caused by wave breaking, weak stratification has been invariably ignored. Here, we consider an idealized hydrodynamic model, where finite amplitude three-dimensional perturbations propagate in a horizontally uniform unidirectional weakly stratified shear flow confined to a boundary layer adjacent to the water surface. Perturbations satisfy the no16 stress boundary condition at the surface. They are assumed to be long compared to the boundary layer thickness. Such perturbations have not been studied even in linear setting. By exploiting the assumed smallness of nonlinearity, wavenumber, viscosity and the Richardson number, on applying triple-deck asymptotic scheme and multiple-scale expansion, we derive in the distinguished limit a novel essentially two-dimensional nonlinear evolution equation, which is the main result of the work. The equation represents a generalisation of the two22 dimensional Benjamin-Ono equation modified by the explicit account of viscous effects and new dispersion due to weak stratification. It describes perturbation dependence on horizontal coordinates and time, while its vertical structure, to leading order, is given by an explicit analytical solution of the linear boundary value problem. It shows the principal importance of weak stratification for three-dimensional perturbations.