Phase-averaged equation for water waves - Gramstad, Odin; Stiassnie, Michael (Journal of fluid mechanics, 2013)
We investigate phase-averaged equations describing the spectral evolution of dispersive water waves subject to weakly nonlinear quartet interactions. In contrast to Hasselmann’s kinetic equation, we include the effects of near-resonant quartet interaction, leading to spectral evolution on the ‘fast’ $O({ε }^{- 2} )$ time scale, where $ε $ is the wave steepness. Such a phase-averaged equation was proposed by Annenkov & Shrira (J. Fluid Mech., vol. 561, 2006b, pp. 181–207). In this paper we rederive their equation taking some additional higher-order effects related to the Stokes correction of the frequencies into account. We also derive invariants of motion for the phase-averaged equation. A numerical solver for the phase-averaged equation is developed and successfully tested with respect to convergence and conservation of invariants. Numerical simulations of one- and two-dimensional spectral evolution are performed. It is shown that the phase-averaged equation describes the ‘fast’ evolution of a spectrum on the $O({ε }^{- 2} )$ time scale well, in good agreement with Monte-Carlo simulations using the Zakharov equation and in qualitative agreement with known features of one- and two-dimensional spectral evolution. We suggest that the phase-averaged equation may be a suitable replacement for the kinetic equation during the initial part of the evolution of a wave field, and in situations where ‘fast’ field evolution takes place.