The WRT method for the computation of non-linear four-wave interactions in discrete spectral wave models

An overview is given of the WRT method for the computation of weakly resonant non-linear four-wave interactions in a gravity wave spectrum and its application in discrete spectral wave models. The WRT method is based on Webb's [Webb, D.J., 1978. Nonlinear transfer between sea waves. Deep-Sea Res., 25, 279–298.] transformation of the Boltzmann integral and the numerical method introduced by Tracy and Resio [Tracy, B.A., Resio, D.T., 1982. Theory and calculation of the nonlinear energy transfer between sea waves in deep water. WIS technical report 11. US Army Engineer Waterways Experiment Station, Vicksburg, Mississippi, USA, 47 pp.]. It is shown that Webb's method produces an attractive set of integrable equations. Moreover, the Jacobian term arising from the integration over the frequency delta-function in the Boltzmann integral has a singularity well outside the energy containing part of the wave spectrum. A description is given of methods for computing the integration space for a given discrete spectral grid, both for deep and finite depth water. Thereafter, the application of Webb's method to discrete spectral wave models is described, followed by a summary of techniques reducing the computational workload while retaining sufficient accuracy. Finally, some methods are presented for the optimal inclusion of the WRT method in operational discrete spectral wave prediction models.