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A coupled-mode model for the scattering of water waves by shearing currents in variable bathymetry

A coupled-mode model is presented for wave–current–seabed interaction, with application to the problem of wave scattering by ambient shearing currents in variable bathymetry regions. We consider obliquely incident waves on a horizontally non-homogeneous current in a variable-depth strip, which is characterized by straight and parallel bottom contours. The flow associated with the current is assumed to be directed along the bottom contours and it is considered to be steady and known. In a finite subregion containing the bottom irregularity, we assume that the horizontal current profile is general and smoothly varying. Outside this region, the current is assumed to be uniform (or zero). Based on a variational principle, in conjunction with a rapidly convergent local-mode series expansion of the wave pressure field in the finite subregion containing the current variation and the bottom irregularity, a new coupled-mode system of equations is obtained, governing the scattering of waves in the presence of variable bathymetry and longshore shearing currents. By keeping only the propagating mode in the local-mode series, a new one-equation model is derived, having the property to reduce to the modified mild-slope equation when the current is zero, and to the enhanced mild-shear equation when the bottom is flat. An important aspect of the present model is that it can be further elaborated to treat shearing currents with general, depth-dependent vertical structure, and to include the effects of weak nonlinearity.