Diffraction in a spectral wave model

Abstract

Waves that approach the coast and encounter obstacles such as small islands, rocks or breakwaters may be reflected backwards and in lateral directions, but the wave crest may also bend around the obstacle. This phenomenon can be described with refraction-diffraction models based on the Boussinesq equation or the mild-slope equation of Berkhoff. However, these models are computationally very demanding since they require a high spatial resolution. Moreover, physical phenomena such as wind generation or depth induced breaking are not readily accounted for. Inclusion of diffraction in a spectral wave model would eliminate these drawbacks. Similar to refraction the effect of diffraction can be represented as a transport of wave energy through spectral space (in the directional domain). Two ad hoc proposals are made to include diffraction in the model SWAN, which is a fully spectral model based on the action balance equation. The first proposal to describe this diffraction-induced turning rate is derived from the mild-slope equation for monochromatic, long-crested waves. It depends on the second order spatial derivative of the wave amplitude. Adding the diffraction term made the model unstable. The second proposal to describe the diffraction-induced turning rate is based on the first-order spatial gradient of the wave field. The transport of wave energy along the wave crests is proportional to the first-order derivative of the energy along the crest. The model is tested for three different cases: the academic case of monochromatic, unidirectional waves near a semi-infinite breakwater, a realistic harbour and the Bay of Viano do Castelo (Portugal). In areas with considerable wave motion the influence of diffraction is relatively unimportant. In other regions the gradientapproach for diffraction seems to give a realistic estimate for the wave field.