Phase-averaged equation for water waves

• Published in: Journal of fluid mechanics Vol. 718; pp. 280 - 303
• Main Authors: Gramstad, Odin, Stiassnie, Michael
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 10.03.2013
• Subjects: Dynamics of the ocean (upper and deep oceans); Earth, ocean, space; Exact sciences and technology; External geophysics; Fluid mechanics; Kinetics; Monte Carlo simulation; Numerical analysis; Physics of the oceans; Spectrum analysis; Surface water; Surface waves; Water waves; surface gravity waves; waves/free-surface flows; Free surface flow; Wave interaction; digital simulation; Modeling; ocean waves; Surface gravity wave
• ISSN: 0022-1120; 1469-7645
• DOI: 10.1017/jfm.2012.609

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’ $O({\epsilon }^{- 2} )$ time scale, where $\epsilon $ 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 $O({\epsilon }^{- 2} )$ 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.