NONLINEAR EVOLUTION OF A UNIDIRECTIONAL SHOALING WAVE-FIELD

Nonlinear energy transfer in the wave spectrum is very important in the shoaling region. Existing theories are limited to weakly dispersive situations (i.e. shallow water or narrow spectrum). A nonlinear evolution equation for shoaling gravity waves is derived, describing the process all the way from deep to shallow water. The slope of the bottom is taken to be smaller, or of the order of the wave steepness (epsilon). The waves are assumed unidirectional for simplicity. The shoaling domain extends up to, and excluding, the first line of breaking of the waves. Reflection by the shore is neglected. Dispersion is fully accounted for. The model equation includes terms due to quadratic interactions, which are effective over characteristic time and spatial scales of order (T/epsilon) and (lambda/epsilon), respectively, where lambda and T are wavelength and period at the spectral peak. In the limit of shallow water, the quadratic interaction model tends to the Boussinesq model. By discretizing the wave spectrum, mixed initial and boundary value problems may be computed. The assumption of the existence of a steady state, transforms the problem into a boundary value one. For this case, solutions for a single triad of waves describing the subharmonic generation and for a full discretized spectrum were computed. The results are compared and found to be in good agreement with laboratory and field measurements. The model can be extended to directionally spread spectra and two dimensional bathymetry.