NONLINEAR TRANSFORMATION OF DIRECTIONAL WAVE SPECTRA IN SHALLOW-WATER

A shallow water, nonlinear spectral wave transformation model is developed for conditions of a mild sloping bottom (mu = DELTAh/kh much less than 1) and small amplitude effects (epsilon = zeta/h much less than 1). Nonlinearities and combined shoaling and refraction effects act on the same time and length scales. The evolution equation of the wave action is prescribed by the wave Boltzmann equation, whereby resonant collinear triad interactions transfer energy among Fourier components. Combined shoaling and refraction effects are taken into account through the geometrical optics approximation. A numerical solution of the three-wave collision integral is presented, and the steady state wave Boltzmann equation is integrated using a piecewise ray method. The model is tested using the high-resolution frequency-directional wave spectra of Freilich, Guza, and Elgar (1990) that show nonlinear transfers of energy between both harmonic and nonharmonic frequencies. A digitized version of the measured frequency-directional spectrum at 10-m depth is evolved 246 m shoreward over a bathymetry of straight and parallel bottom contours to 4-m depth. The model predicts the prominent spectral features in the measured wave field. The model results are in general superior to estimates using linear, finite depth wave theory, and they compare well with the observations in the region of the spectrum dominated by nonlinear effects.