Shoaling transformation of wave frequency‐directional spectra

Abstract

A Boussinesq model for the nonlinear transformation of the frequency‐directional spectrum and bispectrum of surface gravity waves propagating over a gently sloping, alongshore uniform beach is compared with field and laboratory observations. Outside the surf zone the model predicts the observed spectral evolution, including energy transfers to harmonic components traveling in the direction of the dominant waves, and the cross‐interactions of waves traveling in different directions that transfer energy to components with the vector sum wavenumber. The sea surface elevation skewness and asymmetry, third‐order moments believed to be important for sediment transport, also are predicted well. Effects of surf zone wave breaking are incorporated with a heuristic frequency‐dependent dissipation term in the spectral energy balance equation and an empirical relaxation of the bispectrum to Gaussian statistics. The associated coefficients are calibrated with observations that span a wide range of surf zone conditions. With calibrated coefficients, the model predicts observed surf zone frequency spectra well and surf zone skewness and asymmetry fairly well. The observed directional spectra inside the surf zone are broader than the predicted spectra, suggesting that neglected scattering effects associated with the random onset of wave breaking or with higher‐order nonlinearity may be important.