Freak waves are very large, rare events in a random ocean wave train. Here we study the numerical generation of freak waves in a random sea state characterized by the JONSWAP power spectrum. We assume, to cubic order in nonlinearity, that the wave dynamics are governed by the nonlinear Schroedinger (NLS) equation. We identify two parameters in the power spectrum that control the nonlinear dynamics: the Phillips parameter α and the enhancement coefficient γ. We discuss how freak waves in a random sea state are more likely to occur for large values of α and γ. Our results are supported by extensive numerical simulations of the NLS equation with random initial conditions. Comparison with linear simulations are also reported.Freak waves are extraordinarily large water waves whose heights exceed by a factor of 2.2 the significant wave height of a measured wave train [1]. The mechanism of freak wave generation has become an issue of principal interest due to their potentially devastating effects on offshore structures and ships. In addition to the formation of such waves in the presence of strong currents [2] or as a result of a simple chance superposition of Fourier modes with coherent phases, it has recently been established that the nonlinear Schroedinger (NLS) equation can describe many of the features of the dynamics of freak waves which are found to arise as a result of the nonlinear self-focusing phenomena [3][4][5]. The self-focusing effect arises from the Benjamin-Feir instability [7]: a monochromatic wave of amplitude, a 0 , and wave number, k 0 , modulationally perturbed on a wavelength L = 2π/∆k, is unstable whenever ∆k/(k 0 ε) < 2 √ 2, where ε is the steepness of the carrier wave defined as ε=k 0 a 0 . The instability causes a local exponential growth in the amplitude of the wave train. This result is established from a linear stability analysis of the NLS equation [8] and has been confirmed, for small values of the steepness, by numerical simulations of the fully nonlinear water wave equations [5,6] (for high values of steepness wave breaking, which is clearly not included in the NLS model, can occur). Moreover, it is known that small-amplitude instabilities are but a particular case of the much more complicated and general analytical solutions of the NLS equation obtained by exploiting its integrability properties via Inverse Scattering theory in the θ-function representation [11,12].Even though the above results are well understood and robust from a physical and mathematical point of view, it is still unclear how freak waves are generated via the Benjamin-Feir instability in more realistic oceanic conditions, i.e. in those characterized not by a simple monochromatic wave perturbed by two small side-bands, but instead by a complex spectrum whose perturbation of the carrier wave cannot be viewed as being small. Furthermore, the focus herein is not to attempt to model ocean waves but instead to study leading order effects using the nonlinear Schroedinger equation, as suggested by [3][4][5]. Research at hi...
Here we consider a simple weakly nonlinear model that describes the interaction of two-wave systems in deep water with two different directions of propagation. Under the hypothesis that both sea systems are narrow banded, we derive from the Zakharov equation two coupled nonlinear Schrödinger equations. Given a single unstable plane wave, here we show that the introduction of a second plane wave, propagating in a different direction, can result in an increase of the instability growth rates and enlargement of the instability region. We discuss these results in the context of the formation of rogue waves.
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.
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