Evolution of ocean wave statistics in shallow water: Refraction and diffraction over seafloor topography

Abstract

We present a stochastic model for the evolution of random ocean surface waves in coastal waters with complex seafloor topography. First, we derive a deterministic coupled‐mode model based on a forward scattering approximation of the nonlinear mild slope equation; this model describes the evolution of random, directionally spread waves over fully two‐dimensional topography, while accounting for wide angle refraction/diffraction, and quadratic nonlinear coupling. On the basis of the deterministic evolution equations, we derive transport equations for the wave statistical moments. This stochastic model evolves the complete wave cross‐correlation matrix and thus resolves spatially coherent wave interference patterns induced by topographic scattering as well as nonlinear energy transfers to higher and lower frequencies. In this paper we focus on the linear aspects of the interaction with seafloor topography. Comparison to analytic solutions and laboratory observations confirms that (1) the forward scattering approximation is suitable for realistic two‐dimensional topography, and (2) the combined effects of wide angle refraction and diffraction are accurately captured by the stochastic model.