Applicability of a Stochastic Model to Nonlinear Shoaling of Surface Waves

Abstract

This paper examines the applicability of a stochastic model for the propagation of random waves in shallow water including triad near-resonance interactions. The model is derived from evolution equations for complex Fourier amplitudes of fully dispersive multidirectional waves. The closure used in the stochastic equations is adopted from the theory of weak turbulence based on the assumption of quasi-normal sea state. In the model formulation, the second- and third-order wave statistics are formulated by evolution equations for the wave energy spectrum and bispectrum. These equations are implemented numerically, for the case of unidirectional waves, to compute the evolution of the wave energy spectrum and the bispectrum. Experimental measurements for random wave transformation over a shoal (submerged bar) are compared against the numerical-model results. The stochastic model accurately predicted the wave spectra and third-order statistics, in regions where the Ursell number is smaller than 1.5. For larger Ursell-number values, the stochastic model tends to over-estimate the content of high-frequency energy, and the skewness and asymmetry, compared to its deterministic counterpart.