Modelling of ocean waves with the Alber equation: application to non-parametric spectra and generalization to crossing seas

Abstract

We study the Alber equation, a phase-averaged second-moment model for the statistics of a wavefield, derived under a narrowbandedness assumption. More specifically, we use the Alber equation and its associated instability condition to quantify how close a given non-parametric spectrum is to being modulationally unstable, and apply this to a dataset of 100 non-parametric spectra provided by the Norwegian Meteorological Institute. The vast majority of realistic spectra turn out to be stable, but the "proximity to instability" metric is seen to correlate strongly with their steepness and Benjamin-Feir Index (BFI). Moreover, upon comparing with phase-resolved broadband Monte Carlo simulations, the kurtosis and probability of extreme waves for each sea state wavefield are also seen to correlate well with its "proximity to instability". Finally, in order to extend this approach to crossing seas, we derive a generalized Alber equation starting from a system of nonlinear Schrödinger equations modelling crossing seas. We also derive the associated 2-dimensional instability condition.

Freak waves caused by reflection

“…These model equations can only produce a different outcome when correlations between wave modes are introduced into the system, the effects of which have gained increasing attention in recent years (see e.g. Ribal et al (2013); Gramstad (2017); Athanassoulis and Gramstad (2021)). The source of such initial correlations is not yet fully understood, despite some preliminary results of Regev et al (2008) and Andrade and Stiassnie (2020a).…”