📝 Abstract

The instability and nonlinear evolution of directional ocean waves is investigated numerically by means of simulations of the governing kinetic equation for narrow-band surface waves. Our simulation results reveal the onset of the modulational instability for long-crested wave-trains, which agrees well with recent large-scale experiments in wave-basins, where it was found that narrower directional spectra leads to self-focusing of ocean waves and an enhanced probability of extreme events. We find that the modulational instability is nonlinearly saturated by a broadening of the wave-spectrum, which leads to the stabilization of the water-wave system. Applications of our results to other fields of physics, such as nonlinear optics and plasma physics are discussed.

💡 Analysis

The instability and nonlinear evolution of directional ocean waves is investigated numerically by means of simulations of the governing kinetic equation for narrow-band surface waves. Our simulation results reveal the onset of the modulational instability for long-crested wave-trains, which agrees well with recent large-scale experiments in wave-basins, where it was found that narrower directional spectra leads to self-focusing of ocean waves and an enhanced probability of extreme events. We find that the modulational instability is nonlinearly saturated by a broadening of the wave-spectrum, which leads to the stabilization of the water-wave system. Applications of our results to other fields of physics, such as nonlinear optics and plasma physics are discussed.

📄 Content

Instability and Nonlinear Evolution of Narrow-Band Directional Ocean Waves Bengt Eliasson and P. K. Shukla1 1Fakult¨at f¨ur Physik und Astronomie, Ruhr–Universit¨at Bochum, D-44780 Bochum, Germany, and Department of Physics, Ume˚a University, SE-90187 Ume˚a, Sweden (Received 2 December 2009; Revised 3 June 2010) The instability and nonlinear evolution of directional ocean waves is investigated numerically by means of simulations of the governing kinetic equation for narrow-band surface waves. Our simulation results reveal the onset of the modulational instability for long-crested wave-trains, which agrees well with recent large-scale experiments in wave-basins, where it was found that narrower directional spectra leads to self-focusing of ocean waves and an enhanced probability of extreme events. We find that the modulational instability is nonlinearly saturated by a broadening of the wave-spectrum, which leads to the stabilization of the water-wave system. Applications of our results to other fields of physics, such as nonlinear optics and plasma physics are discussed. PACS numbers: 47.35.Bb; 47.20.-k; 47.35.-i; 92.10.Hm Giant freak waves, or rogue waves, have been observed in mid-ocean and coastal waters [1], in optical systems [2], and in parametrically driven capillary waves [3]. The freak/rogue waves are short-lived phenomena appearing suddenly out of normal waves, and with a small probabil- ity [4]. The study of extreme gravity waves on the open ocean has important applications for the sea-faring and offshore oil industries, where they may lead to structural damage and injuries to personnel [1]. It is, therefore, very important to understand the physical mechanisms that lead to the formation of freak waves. Since the lin- ear theory cannot explain the number of extreme events that occur in the ocean and in optical systems, one has to account for nonlinear effects (e.g. wave-wave inter- actions) in combination with the wave dispersion. This can lead to the modulational instability (for water waves called the Benjamin-Feir instability [5, 6]), followed by focusing and amplification of the wave energy. Wind-driven waves on the ocean often have wide fre- quency spectra that are peaked in the direction of the wind [11–13]. The statistics of directional spectra for narrow-band gravity waves have also recently been stud- ied experimentally in water basins [14–16], where it was found that sea states with narrow directional spectra (long–crested waves) were more likely to produce extreme waves. Examples of statistical models that govern col- lective interactions of groups of water waves are Hassel- mann’s model [7] for random, homogeneously distributed waves and Alber’s model [8] for narrow-banded wave trains. Wave-kinetic simulations in one spatial dimension have shown Landau damping and coherent structures [9], and recurrence phenomena [10] for random water wave fields. In this Letter, we derive a nonlinear wave-kinetic (NLWK) equation for gravity waves in 2 + 2 dimensions (two spatial dimensions and two velocity dimensions) and carry out simulations to study the stability and nonlinear spatio-temporal evolution of narrow-band spectra waves that were observed in the recent experiments by Onorato and coworkers [14]. The present NLWK model, which is similar to Alber’s model [8], is particularly suitable for studying the nonlinear dynamics of narrow-band water waves due to its relative simplicity. Similar nonlinear wave-kinetic equations also appear in the description of optical systems, photonic lattices, and plasmas [17]. Deep water gravity waves are governed by the disper- sion relation ω = √gk, where g is the gravitational con- stant, k = q k2x + k2y is the modulus of the wave vector k = kxbx + kyby, and bx and by are the unit vectors in the x−and y−directions. Assuming surface displacements of the form η = (1/2)A(r, t) exp(−iω0t+ik0x)+ complex conjugate, where A is the slowly varying (|∂/∂t| ≪ω0, |∇| ≪k0) envelope, r = xbx + yby is the spatial coordi- nate, and ω0 = √gk0, the nonlinear interaction of water waves is governed by the nonlinear Schr¨odinger equation (NLSE) i ∂A ∂t + vgr ∂A ∂x  +Dx ∂2A ∂x2 +Dy ∂2A ∂y2 −ξ|A|2A = 0, (1) where vgr

∂ω/∂kx

ω0/2k0 is the group ve- locity, Dx = (1/2)∂2ω/∂k2 x = −ω0/8k2 0 and Dy = (1/2)∂2ω/∂k2 y = ω0/4k2 0 are the group dispersion coef- ficients, and the nonlinear coupling coefficient is ξ = ωk2 0/2. Introducing the two-dimensional Wigner trans- form [18] f(r, v, t) = 1 2(2π)2 Z A∗(R+, t)A(R−, t)eiλ·(v−vgrbx) d2λ, (2) where we have denoted R± = r ± ¯¯D · λ and ¯¯D · λ = Dxλxbx + Dyλyby, we obtain the evolution equation for the pseudo-distribution function f as ∂f ∂t + v · ∇f − 2iξ (2π)2 Z Z [I(R+, t) −I(R−, t)] × f(r, v′, t)eiλ·(v−v′)d2v′ d2λ = 0, (3) where I(r, t) = R f(r, v, t) d2v is the variance of the sur- face displacement (the wave intensity). The transfor- arXiv:0912.0474v3 [physics.flu-dyn] 10 Jun 2010 2 mation (2) between (1) and (3) is valid in both direc- ti

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