Abstract
A new semi-Lagrangian advection scheme called multistep ray advection is proposed for solving the spectral energy balance equation of ocean surface gravity waves. Existing so-called piecewise ray methods advect wave energy over a single time step using “pieces” of ray trajectories, after which the spectrum is updated with source terms representing various physical processes. The generalized scheme presented here allows for an arbitrary number N of advection time steps along the same rays, thus reducing numerical diffusion, and still including source-term variations every time step. Tests are performed for alongshore uniform bottom topography, and the effects of two types of discretizations of the wave spectrum are investigated, a finite-bandwidth representation and a single frequency and direction per spectral band. In the limit of large N, both the accuracy and computation cost of the method increase, approaching a nondiffusive fully Lagrangian scheme. Even for N = 1 the semi-Lagrangian scheme test results show less numerical diffusion than predictions of the commonly used first-order upwind finite-difference scheme. Application to the refraction and shoaling of narrow swell spectra across a continental shelf illustrates the importance of controlling numerical diffusion. Numerical errors in a single-step (Δt = 600 s) scheme implemented on the North Carolina continental shelf (typical swell propagation time across the shelf is about 3 h) are shown to be comparable to the angular diffusion predicted by the wave–bottom Bragg scattering theory, in particular for narrow directional spectra, suggesting that the true directional spread of swell may not always be resolved in existing wave prediction models, because of excessive numerical diffusion. This diffusion is effectively suppressed in cases presented here with a four-step semi-Lagrangian scheme, using the same value of Δt.
Numerical and Physical Diffusion: Can Wave Prediction Models Resolve Directional Spread?
