Freak Waves in Random Oceanic Sea States

Abstract

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...

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