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A wave basin experiment has been performed in the MARINTEK laboratories, in one of the largest existing three-dimensional wave tanks in the world. The aim of the experiment is to investigate the effects of directional energy distribution on the statistical properties of surface gravity waves. Different degrees of directionality have been considered, starting from long-crested waves up to directional distributions with a spread of ±30• at the spectral peak. Particular attention is given to the tails of the distribution function of the surface elevation, wave heights and wave crests.Comparison with a simplified model based on second-order theory is reported. The results show that for long-crested, steep and narrow-banded waves, the second-order theory underestimates the probability of occurrence of large waves. As directional effects are included, the departure from second-order theory becomes less accentuated and the surface elevation is characterized by weak deviations from Gaussian statistics.

We propose a new approach for modeling weakly nonlinear waves, based on enhancing truncated amplitude equations with exact linear dispersion. Our example is based on the nonlinear Schrödinger ͑NLS͒ equation for deep-water waves. The enhanced NLS equation reproduces exactly the conditions for nonlinear four-wave resonance ͑the ''figure 8'' of Phillips͒ even for bandwidths greater than unity. Sideband instability for uniform Stokes waves is limited to finite bandwidths only, and agrees well with exact results of McLean; therefore, sideband instability cannot produce energy leakage to high-wave-number modes for the enhanced equation, as reported previously for the NLS equation. The new equation is extractable from the Zakharov integral equation, and can be regarded as an intermediate between the latter and the NLS equation. Being solvable numerically at no additional cost in comparison with the NLS equation, the new model is physically and numerically attractive for investigation of wave evolution.

Simulations have been performed with a fairly narrow band numerical gravity wave model (higher-order NLS type) and a computational domain of dimensions 128 × 128 typical wavelengths. The simulations are initiated with s 6 × 10 4 Fourier modes corresponding to truncated JONSWAP spectra and different angular distributions giving both short-and long-crested waves. A development of the spectra on the so-called Benjamin-Feir timescale is seen, similar to the one reported by Dysthe et al. (J. Fluid Mech. vol. 478, 2003, P. 1). The probability distributions of surface elevation and crest height are found to fit theoretical distributions found by Tayfun (J. Geophys. Res. vol. 85, 1980Res. vol. 85, , p. 1548) very well for elevations up to four standard deviations (for realistic angular spectral distributions). Moreover, in this range of the distributions, the influence of the spectral evolution seems insignificant. For the extreme parts of the distributions a significant correlation with the spectral change can be seen for very long-crested waves. For this case we find that the density of large waves increases during spectral change, in agreement with a recent experimental study by Onorato et al. (J. Fluid Mech. 2004 submitted).

Nonlinear modulational instability of wavepackets is one of the mechanisms responsible for the formation of large-amplitude water waves. Here, mechanically generated waves in a three-dimensional basin and numerical simulations of nonlinear waves have been compared in order to assess the ability of numerical models to describe the evolution of weakly nonlinear waves and predict the probability of occurrence of extreme waves within a variety of random directional wave fields. Numerical simulations have been performed following two different approaches: numerical integration of a modified nonlinear Schrödinger equation and numerical integration of the potential Euler equations based on a higher-order spectral method. Whereas the first makes a narrow-banded approximation (both in frequency and direction), the latter is free from bandwidth constraints. Both models assume weakly nonlinear waves. On the whole, it has been found that the statistical properties of numerically simulated wave fields are in good quantitative agreement with laboratory observations. Moreover, this study shows that the modified nonlinear Schrödinger equation can also provide consistent results outside its narrow-banded domain of validity.

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We discuss two independent, large scale experiments performed in two wave basins of different dimensions in which the statistics of the surface wave elevation are addressed. Both facilities are equipped with a wave maker capable of generating waves with prescribed frequency and directional properties. The experimental results show that the probability of the formation of large amplitude waves strongly depends on the directional properties of the waves. Sea states characterized by long-crested and steep waves are more likely to be populated by freak waves with respect to those characterized by a large directional spreading. DOI: 10.1103/PhysRevLett.102.114502 PACS numbers: 47.35.Bb, 47.55.NÀ An important task in the study of surface gravity waves is the determination of the probability density function of the surface wave elevation. The knowledge of the probability of the occurrence of large amplitude waves is essential for different engineering purposes such as the prediction of wave forces and structural responses or the design of offshore structures. A deep comprehension of the physical mechanisms of the generation of such waves is also a first step towards the development of an operational methodology for the probabilistic forecast of freak waves. It is well known that surface gravity waves obey nonlinear equations and, to date, a universal tool suitable for deriving the probability distribution function of a nonlinear system has not yet been developed. Fortunately, water waves are on average weakly nonlinear [1,2] and solutions can be generally written as power series, where the small parameter, in the case of deep water waves, is the wave steepness ". Strong departure from Gaussian statistics of the surface elevation can be observed if third order nonlinearities are considered. At such order it has been shown numerically [3] and theoretically [4] that, for long-crested waves, a generalization of the Benjamin-Feir instability [5] (or modulational instability [2]) for random spectra can take place [6]. This instability, that corresponds to a quasiresonant four-wave interaction in Fourier space, results in the formation of large amplitude waves (or rogue waves) [7] which affect the statistical properties of the surface elevation (see, for example, [8]). This is particularly true if the ratio between the wave steepness and the spectral bandwidth, known as the Benjamin-Feir Index (BFI), is large [4]. We mention that rogue waves have also been recently observed in optical systems [9] and in acoustic turbulence in He II [10] where giant waves are observed during an inverse cascade process.We emphasize that in many different fields of physics (plasmas [11,12], nonlinear optics [13,14], chargedparticle beam dynamics [15,16]) the modulational instability plays an important role; under suitable physical conditions a nonlinear Schrödinger equation can be derived and the modulational instability can be analyzed directly with this equation [2]. A major question which has to be addressed (and is the subject of the pre...