The topographical scattering of gravity waves is investigated using a numerical model based on a spectral energy balance equation that accounts for first order wave-bottom Bragg scattering (Ardhuin and Herbers, 2002). This model represents the bottom topography with a spectrum that is used to compute a Bragg scattering source term, it is theoretically valid for small bottom slopes and uniform bottom spectral properties. The robustness of the model is tested for 1D topography: sinusoidal, nearly sinusoidal and linear ramp profiles. Results are compared with a known accurate method that uses integral matching along vertical boundaries, using Rey's (1992) algorithm.
Wave propagation over complex 2D topography can now be predicted with boundary element methods or other accurate numerical techniques. However, wave forecasting still relies exclusively on phaseaveraged spectral wave models based for examples on the energy balance equation (Gelci et al; 1957). For large bottom slopes waves can be reflected and this reflection is currently not represented in these operational models, and the significance of this process is poorly known. For waves propagating over a sinusoidal seabed topography, a maximum reflection or resonance is observed when the seabed wavenumber is twice as large as the surface wave wavenumber (Heathershaw, 1982). A theory of this phenomenon called Bragg resonance can be extended to any seabed topography that is statistically uniform, using a Fourier transformation of the bottom (Hasselmann, 1966). Ardhuin and Herbers (2002) further included slowly varying bottom spectra and mean water depths in Hasselmann's (1966) perturbation expansion. The resulting spectral energy balance equation contains a bottom scattering source term S Bragg, that is formally valid for small bottom slopes and weak evolutions of the wave field on scales larger than the wavelength. S Bragg is easily introduced in existing energy-balance based spectral wave models, allowing to compute this reflection.
