Spectral modeling of wave breaking: Application to boussinesq equations

The nonlinear transformation of wave spectra in shallow water is considered, in particular, the role of wave breaking and the energy transfer among spectral components due to triad interactions. Energy dissipation due to wave breaking is formulated in a spectral form, both for energy-density models and complex-amplitude models. The spectral breaking function distributes the total rate of random-wave energy dissipation in proportion to the local spectral level, based on experimental results obtained for single-peaked spectra that breaking does not appear to alter the spectral shape significantly. The spectral breaking term is incorporated in a set of coupled evolution equations for complex Fourier amplitudes, based on ideal-fluid Boussinesq equations for wave motion. The model is used to predict the surface elevations from given complex Fourier amplitudes obtained from measured time records in laboratory experiments at the upwave boundary. The model is also used, together with the assumption of random, independent initial phases, to calculate the evolution of the energy spectrum of random waves. The results show encouraging agreement with observed surface elevations as well as spectra.