Nonlinear transformation of directional wave spectra in shallow water
Abreu, Manuel A.
Thornton, Edward B.
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A shallow water, nonlinear spectral wave transformation model is developed for conditions of a mild sloping bottom (u = vh/kh < 1) and small amplitude effects ( c = n/h < 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 developed, 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 spectrum of freilich, Guza and Elgar (1990) that shows nonlinear transfers of energy between both harmonic and non-harmonic frequencies. A digitized version of the measured frequency-directional spectrum at meter depth is evolved 246 meter shoreward over a bathymetry of straight and parallel bottom contours to 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.
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