The last decade freak waves have become an important topic in engineering and science and are sometimes featured by a single and steep crest causing severe damage to offshore structures and ships. Freak wave studies started in the late 80's (, ) and the high-order nonlinear effects on the freak waves were discussed in the early 90's (e.g. , ,). Due to the many research efforts, the occurrence of freak waves, their mechanism and detailed dynamic properties are now becoming clear (e.g. , ,,,,,). The state of the art on freak waves was summarized at the Rogue Wave Conference, held in 2000 and 2004 (, ,). It was concluded that the third order nonlinear interactions enhance freak wave appearance and are the primary cause of freak wave generation in a general wave field except for the case of strong wave-current interaction or wave diffraction behind the islands.
Numerical and experimental studies have demonstrated that freak-like waves can be generated frequently in a two-dimensional wave flume without current, refraction or diffraction (, ,,,). Moreover, the numerical studies clearly indicate that a freak wave having a single, steep crest can be generated by the third order nonlinear interactions in deep-water (, ). Also, the theoretical background of freak wave generation has become more clear (, ), but the quantitative occurrence probabilities in the ocean remain uncertain. In addition, it is still questionable how to characterize the dominant statistical properties of the freak wave occurrence in terms of nonlinear parameters, spectral shape, water depth and so on.
Nevertheless, although there is no doubt that the third order nonlinear interactions relate with the steep wave generation in the random wave train, the theoretical background of the relationship between the freak wave generation and the third order nonlinear interactions is not well-established. Freak wave generation is sometimes discussed in the context of the Benjamin-Feir instability in deep-water waves because of the similarity of the steep wave profile itself (, ,). The last two decades, the Benjamin-Feir type instability of the deep-water gravity waves has been studied by many researchers using the nonlinear Schrödinger type of equations (, ,,), mode-coupling equations (, ), pseudo-spectral methods (, ) and experiments (, ). However, there is disparity between the periodic wave instabilities and random wave behavior, because the broad banded spectra and random phase approximation are essential describing the ocean waves in nature (i.e. , ,). Thus, the energy transfer of random waves due to four-wave interactions has been studied for describing spectral evolutions (, ,). By means of a series of numerical investigations () stated that the instability is confined within an initially unstable range and becomes weak if the spectral bandwidth broadens. () mathematically demonstrated that for a random sea the Benjamin-Feir instability vanishes if the wave spectrum is sufficiently broad. Therefore, there is discrepancy between the nonlinear behavior of periodic waves and random waves.
Recently, () investigated the freak wave occurrence as a consequence of four-wave interactions including the effects of non-resonant four-wave interactions. He found that the homogeneous nonlinear interactions give rise to deviations from the Gaussian distribution for the surface elevation on the basis of the Monte Carlo simulations of the Zakharov equation. Surprisingly, inhomogeneities only play a minor role in the evolution of the wave spectrum. He also formulated the analytical relationship between spectral shape and the kurtosis of the surface elevation. These results have the potential to unify previous freak wave studies covering nonlinear interactions, spectral profiles to nonlinear statistics and etc.
The purpose of this study is to investigate the relationship between kurtosis and the occurrence probability of freak waves through the nonlinear four-wave interactions. First, for a nonlinear stochastic wave field the relationship between high-order moments including kurtosis of surface elevation and nonlinear transfer function is derived. Second, the wave height and maximum wave height distributions are formulated as a simple function of kurtosis by the non-Gaussian theory. Third, the wave height distribution is compared with laboratory experiments and the occurrence probabilities of freak waves are compared with field observations. Finally, the dependence of the occurrence of freak waves on the number of waves and kurtosis will be analyzed and discussed.