Bragg scattering of random surface gravity waves by irregular seabed topography

The Bragg scattering of random, non-stationary surface gravity waves by random topography on a gently sloping bottom is investigated. A correction is given of previously published expressions for the triad wave-wave-bottom interaction source term in the spectral energy balance equation, and the result is reconciled with deterministic theories for the reflection of waves from sinusoidal seabed undulations. For both normal and oblique incidence, the stochastic and deterministic theories are equivalent in the limit of long propagation distances. Even for relatively short distances (for example two bottom undulations), the reflected energy predicted by the stochastic source term formulation is generally within 15% of values predicted by deterministic theories. The detuning of Bragg resonance by refraction and shoaling is discussed, suggesting practical validity conditions for the stochastic theory. The effect of bottom scattering on swell propagation is illustrated with numerical model computations for the North Carolina continental shelf using high-resolution bathymetry and an efficient semi-implicit scheme to evaluate the bottom scattering source term and integrate the energy balance equation. Model results demonstrate the importance of forward scattering of waves that propagate at large oblique angles over bottom features with typical scales of one to several surface wavelengths. This process contributes significantly to the directional spread of swell on the continental shelf by diffusing energy, in the spectrum, around the mean wave direction. Back-scattering, caused by bottom features with crests parallel to those of the surface waves and wavelengths close to half the surface wavelength, is weak, owing to the sharp roll-off of the bottom elevation spectrum at high wavenumbers. Model predictions are consistent with field measurements.