A Novel Coupled-Mode System for Nonlinear Wave Propagation over Variable Bathymetry Based on a Velocity Formulation

Fast, convergent local-mode expansions of nonlinear water waves are discussed for the representation of the velocity and stream function. Subsequently, the representations are used to derive and study a novel nonlinear coupled-mode system of differential equations on the horizontal plane, with respect to unknown horizontal velocity modal amplitudes and free-surface elevation. The coupled-mode system, in conjunction with the convergence properties of the local-mode series, facilitates the numerical solution of the water wave propagation problem over general bottom topography. The efficiency of the present method is demonstrated through various examples, including the simulation of periodic waves in a constant depth and over trapezoidal bar test cases. The results show the robustness and accuracy of the coupled-mode system in capturing the complexities of wave transformations over non-uniform bathymetric features. Moreover, truncating the modal expansions of the wave velocity field by keeping only the first mode leads to a low-cost, single-mode nonlinear wave model with enhanced dispersion characteristics that is useful for engineering applications.