The modified Boussinesq equations given by Nwogu (1993a) are rederived in terms of a velocity potential on an arbitrary elevation and the free surface displacement. The optimal elevation where the velocity potential should be evaluated is determined by comparing the dispersion and shoaling properties of the linearized modified Boussinesq equations with those given by the linear Stokes theory over a range of depths from zero to one half of the equivalent deep-water wavelength. For regular waves consisting of a finite number of harmonics and propagating over a slowly varying topography, the governing equations for velocity potentials of each harmonic are a set of weakly nonlinear coupled fourth-order elliptic equations with variable coefficients. The parabolic approximation is applied to these coupled fourth-order elliptic equations for the first time. A small-angle parabolic model is developed for waves propagating primarily in a dominant direction. The pseudospectral Fourier method is employed to derive an angular-spectrum parabolic model for multi-directional wave propagation. The small-angle model is examined by comparing numerical results with Whalin's (1971) experimental data. The angular-spectrum model is tested by comparing numerical results with the refraction theory of cnoidal waves (Skovgaard & Petersen 1977) and is used to study the effect of the directed wave angle on the oblique interaction of two identical cnoidal wavetrains in shallow water.

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Abstract.Predictions from Boussinesq-equation-based models for the evolution of breaking surface gravity waves in shallow water are compared with field and laboratory observations. In the majority of the 10 cases investigated, the observed spectral evolution across the surf zone is modeled more accurately by a dissipation that increases at high frequency than by a frequency-independent dissipation. However, in each case the predicted spectra are qualitatively accurate for a wide range of frequency-dependent dissipations, apparently because preferential reduction of high-frequency energy (by dissipation that increases with increasing frequency) is largely compensated by increased nonlinear energy transfers to high frequencies. In contrast to the insensitivity of predicted spectral levels, model predictions of skewness and asymmetry (statistical measures of the wave shapes) are sensitive to the frequency dependence of the dissipation. The observed spatial evolution of skewness and asymmetry is predicted qualitatively well by the model with frequency-dependent dissipation, but is predicted poorly with frequencyindependent dissipation. Although the extension of the Boussinesq equations to breaking waves is ad hoc, a dissipation depending on the frequency squared (as previously suggested) reproduces well the observed evolution of wave frequency spectra, skewness, and asymmetry.