Nonlinear effects on shoaling surface gravity waves

Abstract

Two nonlinear models that describe the shoaling of unidirectional surface gravity waves are developed. Based on variants of Boussinesq’s equations, the models are cast as a set of coupled evolution equations for the amplitudes and phases of the temporal Fourier modes of the wave field. Triad interactions across the entire wind wave frequency band (0.05-0.25 Hz) provide the mechanism for cross spectral energy transfers and modal phase modifications as the waves propagate shoreward through the shoaling region (10-3 m depth). A field experiment, designed to test the operational validity of the nonlinear shoaling models, provided data on wave parameters over a wide range of conditions. Three representative data sets illustrating different initial spectral shapes and subsequent evolutions are compared with predictions of the nonlinear shoaling models and linear, finite-depth theory. Power spectral comparisons, as well as spectra of coherence and relative phase between model predictions and data, indicate that the nonlinear models accurately predict Fourier coefficients of the wave field through the shoaling region sets. Differences between the predictions of the various models are related to differences in the models' dispersion relations. Although generally inferior to the nonlinear models, linear, finite-depth theory accurately predicts Fourier coefficients in regions of physical and frequency space where nonlinear evolution of the power spectrum is not observed, thus verifying the validity of the linear, finite-depth dispersion relation in limited portions of physical and frequency space in the shoaling region.