Abstract
A novel coupled-mode model is developed for the wave–current–seabed interaction problem, with application in wave scattering by non-homogeneous, sheared currents over general bottom topography. The formulation is based on a velocity representation defined by a series of local vertical modes containing the propagating and evanescent modes, able to accurately treat the continuity condition and the bottom boundary condition on sloping parts of the seabed. Using the above representation in Euler equations, a coupled system of differential equations on the horizontal plane is derived, with respect to the unknown horizontal velocity modal amplitudes. In the case of small-amplitude waves, a linearized version of the above coupled-mode system is obtained, and the dispersion characteristics are studied for various interesting cases of wave–seabed–current interaction. Keeping only the propagating mode in the vertical expansion of the wave potential, the present system is reduced to a one-equation, non-linear model, generalizing Boussinesq models. The analytical structure of the present coupled-mode system facilitates extensions to treat non-linear effects and further applications concerning wave scattering by inhomogeneous currents in coastal regions with general 3D bottom topography.
Highlights
In coastal areas, steep bathymetries and strong currents are often observed
First numerical results concerning the simulation by the present CMS for waves propagating in variable bathymetry regions will be presented
We will first investigate in Sections and 5.2 the behavior of the linearized CMS in cases of waves propagating in variable bathymetry
Summary
Steep bathymetries and strong currents are often observed. Among several causes, the presence of cliffs, rocky beds, or human structures may cause strong variations of the sea bed, while oceanic circulation, tides, wind action, or wave breaking can be responsible for the generation of strong currents. Massel [4] suggested a multi-modal expansion to improve the representation of the vertical distribution of the wave field deriving a set of coupled equations, involving both propagating and evanescent modes The latter approach has been further improved by Athanassoulis and Belibassakis [6] by including extra terms to consistently treat the boundary conditions. If the vorticity conservation equation involves a source term, due to the interaction of the flow with the sheared current, water waves cannot be treated as irrotational anymore For this purpose, the present novel coupled-mode system is obtained in the framework of Euler equations by using a velocity-based formulation, allowing one to describe wave–vorticity interactions in more generic conditions. The main conclusions are presented, including directions for further research
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