A Generalized Multiple Discrete Interaction Approximation for resonant four-wave interactions in wind wave models

Abstract

For several decades, the Discrete Interaction Approximation (DIA) for nonlinear resonant four-wave interactions has been the engine of third-generation wind-wave models. The present study presents a Generalized Multiple DIA (GMD) which expands upon the DIA by (i) expanding the definition of the representative quadruplet, (ii) formulating the DIA for arbitrary water depths, (iii) providing complimentary deep and shallow water scaling terms and (iv) allowing for multiple representative quadruplets. The GMD is rigorously derived to be an extension of the DIA, and is backward compatible with it. The free parameters of the GMD are optimized holistically, by optimizing full model behavior in the WAVEWATCH III® wave model as reported in a companion paper. Here, a cascade of GMD configurations with increasing complexity, accuracy and cost is presented. First, the performance of these configurations is discussed using idealized test cases used to optimize the GMD. It is shown that in deep water, GMD configurations can be found which remove most of the errors of the DIA. The GMD is also capable of representing four-wave interactions in extremely shallow water, although some remaining spurious behavior makes applications of this part of the GMD less suitable for operational wave models. Finally, several GMD configurations are applied to an idealized hurricane case, showing that results from idealized test cases indeed are representative for real-world applications, and confirming that such GMD configurations are economically feasible in operational wind wave models. Finally, the DIA results in surprisingly large model errors in hurricane conditions.