Consistent nonlinear deterministic and stochastic wave evolution equations from deep water to the breaking region

Abstract

Modelling the evolution of the wave field in coastal waters is a complicated task, partly due to triad nonlinear wave interactions, which are one of the dominant mechanisms in this area. Stochastic formulations already implemented into large-scale operational wave models, whilst very efficient, are one-dimensional in nature and fail to account for the majority of the physical properties of the wave field evolution. This paper presents new two-dimensional (2-D) formulations for the triad interactions source term. A quasi-two-dimensional deterministic mild slope equation is improved by including dissipation and first-order spatial derivatives in the nonlinear part of equation, significantly enhancing the accuracy in the breaking zone. The newly defined deterministic model is used to derive an updated stochastic model consistent from deep waters to the breaking region. It is localized following the approach derived in Vrecica & Toledo (J. Fluid Mech., vol. 794, 2016, pp. 310–342), to which several improvements are also presented. The model is compared to measurements of breaking and non-breaking spectral evolution, showing good agreement in both cases. Finally, the model is used to analyse several interesting 2-D properties of the shoaling wave field including the evolution of directionally spread seas.