Freak wave events and the wave phase coherence

Abstract. In this paper we demonstrate the strong correlation in the spectrum area close to the spectral peak in cases when the Benjamin-Feir instability causes intense wave groups of unidirectional deep-water surface waves referred to freak events. A simple phase coherence estimator in the form of an autobicorrelation function is suggested and tested on the basis of the results of numerical simulations within different frameworks, including the primitive Euler equations. The correlation reaches the value of a unity, and, thus, the random phase approximation is definitely violated for these waves.

[1]  Miguel Onorato,et al.  Extreme wave events in directional, random oceanic sea states , 2001, nlin/0106004.

[2]  Vladimir E. Zakharov,et al.  Statistical theory of gravity and capillary waves on the surface of a finite-depth fluid , 1999 .

[3]  E. Kartashova.  Nonlinear resonances of water waves , 2009, 0905.0050.

[4]  Harald E. Krogstad,et al.  Oceanic Rogue Waves , 2008 .

[5]  C. Schober,et al.  Predicting rogue waves in random oceanic sea states , 2004, nlin/0411025.

[6]  E. Pelinovsky,et al.  Dynamics of modulationally unstable ion-acoustic wavepackets in plasmas with negative ions , 2008, Journal of Plasma Physics.

[7]  V. Zakharov,et al.  On the formation of freak waves on the surface of deep water , 2008 .

[8]  A. Osborne,et al.  On the extreme statistics of long‐crested deep water waves: Theory and experiments , 2007 .

[9]  Alexey Slunyaev.  Nonlinear analysis and simulations of measured freak wave time series , 2006 .

[10]  Congming Li,et al.  Stabilizing the Benjamin–Feir instability , 2005, Journal of Fluid Mechanics.

[11]  Karsten Trulsen,et al.  Evolution of a narrow-band spectrum of random surface gravity waves , 2003, Journal of Fluid Mechanics.

[12]  C T Stansberg,et al.  Statistical properties of directional ocean waves: the role of the modulational instability in the formation of extreme events. , 2009, Physical review letters.

[13]  Karsten Trulsen,et al.  On weakly nonlinear modulation of waves on deep water , 2000 .

[14]  Kharif Christian,et al.  Rogue Waves in the Ocean , 2009 .

[15]  Vladimir E. Zakharov,et al.  Modulation instability of Stokes wave → freak wave , 2005 .

[16]  Stéphane Leblanc,et al.  Amplification of nonlinear surface waves by wind , 2007 .

[17]  Lev Shemer,et al.  Applicability of envelope model equations for simulation of narrow-spectrum unidirectional random wave field evolution: Experimental validation , 2010 .

[18]  Efim Pelinovsky,et al.  Physical Mechanisms of the Rogue Wave Phenomenon , 2003 .

[19]  Luigi Cavaleri,et al.  Modulational instability and non-Gaussian statistics in experimental random water-wave trains , 2005 .

[20]  K. Dysthe,et al.  Note on a modification to the nonlinear Schrödinger equation for application to deep water waves , 1979, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[21]  Karsten Trulsen,et al.  NOTE ON BREATHER TYPE SOLUTIONS OF THE NLS AS MODELS FOR FREAK-WAVES , 1999 .

[22]  V. Shrira,et al.  Can bottom friction suppress ‘freak wave’ formation? , 2008, Journal of Fluid Mechanics.

[23]  Yosuke Watanabe,et al.  Breather solutions to the focusing nonlinear Schrödinger equation , 1998 .

[24]  B. Jalali,et al.  Optical rogue waves , 2007, Nature.

[25]  Alexey Slunyaev,et al.  Numerical simulation of “limiting” envelope solitons of gravity waves on deep water , 2009 .

[26]  V. Shrira,et al.  Evolution of kurtosis for wind waves , 2009 .

[27]  D. Chalikov.  Freak waves: Their occurrence and probability , 2009 .

[28]  C. Raab,et al.  Symbolic Computation for Nonlinear Wave Resonances , 2008 .

[29]  N. Akhmediev,et al.  Modulation instability and periodic solutions of the nonlinear Schrödinger equation , 1986 .

[30]  P. Drazin,et al.  Solitons: An Introduction , 1989 .

[31]  L. Shemer,et al.  An experimental study of spatial evolution of statistical parameters in a unidirectional narrow‐banded random wavefield , 2009 .

[32]  John Grue,et al.  Long time interaction of envelope solitons and freak wave formations , 2006 .

[33]  K. Dysthe,et al.  Probability distributions of surface gravity waves during spectral changes , 2005, Journal of Fluid Mechanics.

[34]  Adrian Ankiewicz,et al.  Rogue waves and rational solutions of the Hirota equation. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[35]  Gregory Falkovich,et al.  Kolmogorov Spectra of Turbulence I , 1992 .

[36]  N. Akhmediev,et al.  Waves that appear from nowhere and disappear without a trace , 2009 .

[37]  A. Osborne,et al.  Freak waves in random oceanic sea states. , 2001, Physical review letters.

[38]  I. E. Alber,et al.  The effects of randomness on the stability of two-dimensional surface wavetrains , 1978, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[39]  S. Novikov,et al.  Theory of Solitons: The Inverse Scattering Method , 1984 .

[40]  N. Akhmediev,et al.  Exact first-order solutions of the nonlinear Schrödinger equation , 1987 .

[41]  C. Soares,et al.  Modeling freak waves from the North Sea , 2005 .

[42]  Adrian Ankiewicz,et al.  Solitons : nonlinear pulses and beams , 1997 .

[43]  Benjamin J. Eggleton,et al.  Photonics: Rogue waves surface in light , 2007, Nature.

[44]  R. S. Johnson,et al.  A Modern Introduction to the Mathematical Theory of Water Waves: Frontmatter , 1997 .

[45]  Yan‐Chow Ma,et al.  The Perturbed Plane‐Wave Solutions of the Cubic Schrödinger Equation , 1979 .

[46]  Stanisław Ryszard Massel,et al.  Ocean Surface Waves: Their Physics and Prediction , 1996 .

[47]  E. Pelinovsky,et al.  Nonlinear wave focusing on water of finite depth , 2002 .

[48]  Nobuhito Mori,et al.  On Kurtosis and Occurrence Probability of Freak Waves , 2006 .

[49]  J. Dold,et al.  Unsteady water wave modulations: fully nonlinear solutions and comparison with the nonlinear Schrodinger equation , 1999 .

[50]  V. Shrira,et al.  Role of non-resonant interactions in the evolution of nonlinear random water wave fields , 2006, Journal of Fluid Mechanics.

[51]  Miguel Onorato,et al.  The nonlinear dynamics of rogue waves and holes in deep-water gravity wave trains , 2000 .

[52]  L. Ostrovsky,et al.  Modulation instability: The beginning , 2009 .

[53]  Peter A. E. M. Janssen,et al.  Nonlinear Four-Wave Interactions and Freak Waves , 2003 .

[54]  M. Stiassnie,et al.  On the interaction of four water-waves , 2005 .