Boussinesq evolution equations: Numerical efficiency, breaking and amplitude dispersion

This paper deals with the possibility of using methods and ideas from time domain Boussinesq formulations in the corresponding frequency domain formulations. We term such frequency domain models "evolution equations". First, we demonstrate that the numerical efficiency of the deterministic Boussinesq evolution equations of Madsen and Sorensen [Madsen, P.A., Sorensen, O.R., 1993. Bound waves and triad interactions in shallow water. Ocean Eng. 20 359-388] can be improved by using Fast Fourier Transforms to evaluate the nonlinear terms. For a practical example of irregular waves propagating over a submerged bar, it is demonstrated that evolution equations utilising FFT can be solved around 100 times faster than the corresponding time domain model. Use of FFT provides an efficient bridge between the frequency domain and the time domain. We utilise this by adapting the surface roller model for wave breaking to frequency domain evolution equations. An equation for the variation of the mean water level is derived. Results for regular and irregular waves are presented and compared to results of conventional breaking formulations for evolution equations as well as for results of the corresponding time domain model. Emphasis is given to the shape of the breaking waves. The amplitude dispersion of evolution equations is analysed using a third-order perturbation approach. It is found to exceed the amplitude dispersion of the corresponding time domain model, and the approximation causing this deviation is pinpointed. (C) 2004 Elsevier B.V. All rights reserved.