Stochastic evolution equations with localized nonlinear shoaling coefficients

Nonlinear interactions between sea waves and the bottom are a main mechanism for energy transfer between the different wave frequencies in the near-shore region. Nevertheless, it is difficult to account for this phenomenon in stochastic wave models due to its mathematical complexity, which consists of computing either the bi-spectral evolution or non-local shoaling coefficients. Recent advances allowed the localization of the nonlinear shoaling coefficients, setting a simpler way to apply this mechanism in these models for one-dimensional interactions. This was done by taking into account only mean energy transfers between the modes while neglecting oscillatory transfers. The present work aims to improve these localized coefficients in order to make them more consistent with the dominating resonance mechanism—the class III Bragg resonance. The approximated stochastic models are tested with respect to a deterministic nonlinear mild-slope equation model, where a significant advantage of the improved coefficients is observed.