Higher order numerical schemes for propagation of wind wave spectra

Abstract

Spectral wind wave models seek to solve a four-dimensional energy (or action) balance equation for values of the spectrum discretized in frequency and direction of propagation at fixed positions in space. When modeling an ocean area of any appreciable size, computational time and storage capacity limit resolution to relatively coarse grids in all four dimensions. Propagation schemes used in these models, typically the 1st order upwind scheme, encounter difficulty arising from the poor directional resolution (typically 30 degrees) in regions of varying depth and current where wave energy is refracted and concentrated into a small number of directional bins. Since the widely used 1st order upwind scheme is not appropriate for such a rapid bin to bin variation, higher order numerical schemes are investigated to identify one which will produce better results for this wind wave propagation problem. After evaluating the characteristics and performance of the 2nd-order upwind scheme, Lax-Wendroff scheme, and modified Lax-Wendroff scheme suggested by Gadd, for both steady and transient cases, a new propagation scheme is proposed using a time-splitting method and a limiter which combines the modified Lax-Wendroff scheme with the 1st order upwind scheme. For varying depth and current fields, it is shown that the new scheme gives results superior to the ordinary 1st order upwind scheme without any increase of storage capacity at an increased cost in computing time which is minor to the overall wind wave model.