Coupled Ostrovsky Equations for Internal Waves, with a Background Shear Flow

We study the behaviour of weakly nonlinear oceanic internal waves in the presence of background rotation and shear flow, when two distinct linear long wave modes have nearly coincident phase speeds. The waves are described by a system of coupled Ostrovsky equations, derived from the full set of Euler equations for incompressible density stratified fluid with a free surface and rigid bottom boundary conditions. We report here some preliminary results obtained when there is a background shear flow.