Statistical properties of higher order moments for random and monochromatic waves

Various phase-resolving (deterministic) and phase-averaged (stochastic) models are commonly employed to characterize the nonlinear wave evolution. Deterministic models can resolve the exact behavior of wave systems in time and space, while stochastic models evaluate their long-term statistical evolution. We investigate several properties of higher-order moments which are used in stochastic modeling, with an emphasis on the disparities between deterministic and stochastic models in describing wave self-interactions. For regular waves (and for truly narrow spectra), it is necessary to adjust the typical Gaussian closure. Properly adjusting the definition of higher order moments and taking into account the initial phasings yields a perfect agreement between these two types of models. Evolution of a monochromatic surface water wave over bathymetry, and evolution of a degenerate wave quartet are used to illustrate these properties.