Interactions of currents and weakly nonlinear water waves in shallow water

Two-dimensional Boussinesq-type depth-averaged equations are derived for describing the interactions of weakly nonlinear shallow-water waves with slowly varying topography and currents. The current velocity varies appreciably within a characteristic wavelength. The effects of vorticity in the current field are considered. The wave field is decomposed into Fourier time harmonics.A set of evolution equations for the wave amplitude functions of different harmonics is derived by adopting the parabolic approximation. Numerical solutions are obtained for shallow-water waves propagating over rip currents on a plane beach and an isolated vortex ring. Numerical results show that the wave diffraction and nonlinearity are important in the examples considered.