Scattering of surface gravity waves by bottom topography with a current

A theory is presented that describes the scattering of random surface gravity waves by small-amplitude topography, with horizontal scales of the order of the wavelength, in the presence of an irrotational and almost uniform current. A perturbation expansion of the wave action to order η2 yields an evolution equation for the wave action spectrum, where η = max(h)/H is the small-scale bottom amplitude normalized by the mean water depth. Spectral wave evolution is proportional to the bottom elevation variance at the resonant wavenumbers, representing a Bragg scattering approximation. With a current, scattering results from a direct effect of the bottom topography, and an indirect effect of the bottom through the modulations of the surface current and mean surface elevation. For Froude numbers of the order of 0.6 or less, the bottom topography effects dominate. For all Froude numbers, the reflection coefficients for the wave amplitudes that are inferred from the wave action source term are asymptotically identical, as η goes to zero, to previous theoretical results for monochromatic waves propagating in one dimension over sinusoidal bars. In particular, the frequency of the most reflected wave components is shifted by the current, and wave action conservation results in amplified reflected wave energies for following currents. Application of the theory to waves over current-generated sandwaves suggests that forward scattering can be significant, resulting in a broadening of the directional wave spectrum, while back-scattering should be generally weaker.