Nonlinear Evolution of Surface Wave Spectra on a Beach

Abstract

The shoaling evolution of wave spectra on a beach with straight and parallel depth contours is investigated with a stochastic Boussinesq model. Existing deterministic Boussinesq models cast in the form of coupled evolution equations for the amplitudes and phases of discrete Fourier modes describe accurately the shoaling process for arbitrary incident wave conditions, but are numerically cumbersome for predicting the evolution of continuous spectra of natural wind-generated waves. The stochastic formulation used here, based on the closure hypothesis that phase coupling between quartets of wave components is weak, predicts the shoaling evolution of the continuous frequency spectrum and bispectrum of the wave field. The general characteristics of the stochastic model and the dependence of wave shoaling on nonlinearity, initial spectral shape, and bottom profile are illustrated with numerical simulations. Predictions of stochastic and deterministic Boussinesq models are compared with data from a natural barred ocean beach. Both models accurately reproduce the observed nonlinear wave transformation for a range of conditions.