University of Toronto
Toronto, Ontario, Canada
The work is concerned with long nonlinear internal waves interacting with a shear flow localized near the sea surface. The most intense resonant interaction occurs when the phase velocity of internal waves matches the flow velocity at the surface. Within the weakly-nonlinear long-wave approximation a system of evolution equations governing nonlinear dynamics of the waves in resonance is derived and an asymptotic solution to the basic equations is constructed. The derived equations are found to possess solitary wave solutions of different polarities propagating slightly faster or slower than the surface velocity of the shear flow. The amplitudes of the `fast' solitary waves are limited from above, the crest of the limiting wave forms a sharp corner. The solitary waves of the amplitudes smaller than a certain threshold are shown to be stable and localized pulses tend to such solutions while localized pulses of amplitudes exceeding this threshold form vertical slopes in finite time, which indicates wave breaking.