Explaining extreme waves by a theory of stochastic wave groups

Abstract

It is well known that in a Gaussian sea an extreme wave event is a particular realization of the space–time evolution of a well defined linear wave group, in agreement with the theory of quasi-determinism of Boccotti [Boccotti P. On mechanics of irregular gravity waves. Atti Acc Naz Lincei, Memorie 1989;19:11–170] and the Slepian model of Lindgren [Kac M, Slepian D. Large excursions of Gaussian processes. Ann Math Statist 1959;30:1215–28; Lindgren G. Some properties of a normal process near a local maximum. Ann Math Statist 1970;4(6):1870–83]. In this paper, the concept of stochastic wave groups is proposed to explain the occurrence of extreme waves in nonlinear random seas, according to the dynamics imposed by the Zakharov equation [Zakharov VE. Statistical theory of gravity and capillary waves on the surface of a finite-depth fluid. J Eur Mech B—Fluids 1999;18(3):327–44]. As a corollary, a new analytical solution for the probability of exceedance of the crest-to-trough height is derived for the prediction of extreme wave events in nonlinearly modulated long-crested narrow-band seas. Furthermore, a generalization of the Tayfun distribution [Tayfun MA. On narrow-band representation of ocean waves. Part I: Theory. J Geophys Res 1986;91(C6):7743–52] for the wave crest height is also provided. The new analytical distributions explain qualitatively well recent experimental results of Onorato et al. [Onorato M, Osborne AR, Cavaleri L, Brandini C, Stansberg CT. Observation of strongly non-Gaussian statistics for random sea surface gravity waves in wave flume experiments. Phys Rev E 2004;70:067302] and the numerical simulations of Socquet-Juglard et al. [Socquet-Juglard H, Dysthe K, Trulsen K, Krogstad HE, Liu J. Probability distributions of surface gravity waves during spectral changes. J Fluid Mech 2005;542:195–216].