Effects of high-order nonlinear interactions on unidirectional wave trains

Abstract Numerical simulations of gravity waves with high-order nonlinearities in two-dimensional domain are performed by using the pseudo spectral method. High-order nonlinearities more than third order excite apparently chaotic evolutions of the Fourier energy in deep water random waves. The high-order nonlinearities increase kurtosis, wave height distribution and H max / H 1/3 in deep water and decrease these wave statistics in shallow water. Moreover, they can generate a single extreme high wave with an outstanding crest height in deep water. High-order nonlinearities (more than third order) can be regarded as one cause of freak waves in deep water.

[1]  Roles of sideband instability and mode coupling in forming a water-wave chaos , 1997 .

[2]  Bruce M. Lake,et al.  Nonlinear Dynamics of Deep-Water Gravity Waves , 1982 .

[3]  R. G. Dean,et al.  FREAK WAVES: A POSSIBLE EXPLANATION , 1990 .

[4]  Nobuhito Mori,et al.  Characteristics of Giant Freak Waves Observed in the Sea of Japan , 1998 .

[5]  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.

[6]  K. Hasselmann.  On the non-linear energy transfer in a gravity-wave spectrum Part 1. General theory , 1962, Journal of Fluid Mechanics.

[7]  Ming-Yang Su,et al.  Three-dimensional deep-water waves. Part 1. Experimental measurement of skew and symmetric wave patterns , 1982, Journal of Fluid Mechanics.

[8]  Lev Shemer,et al.  Energy computations for evolution of class I and II instabilities of Stokes waves , 1987, Journal of Fluid Mechanics.

[9]  M. Longuet-Higgins.  The effect of non-linearities on statistical distributions in the theory of sea waves , 1963, Journal of Fluid Mechanics.

[10]  Ove T. Gudmestad,et al.  Water wave kinematics , 1990 .

[11]  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.

[12]  P. Saffman,et al.  Instability and confined chaos in a nonlinear dispersive wave system , 1982 .

[13]  Akira Masuda,et al.  Nonlinear Energy Transfer Between Wind Waves , 1980 .

[14]  Dick K. P. Yue,et al.  A high-order spectral method for the study of nonlinear gravity waves , 1987, Journal of Fluid Mechanics.

[15]  Nobuhito Mori,et al.  Statistical Properties of Freak Waves Observed In the Sea of Japan , 2000 .

[16]  K. Hasselmann.  On the non-linear energy transfer in a gravity wave spectrum Part 2. Conservation theorems; wave-particle analogy; irrevesibility , 1963, Journal of Fluid Mechanics.

[17]  Warren E. Ferguson,et al.  Relationship between Benjamin–Feir instability and recurrence in the nonlinear Schrödinger equation , 1978 .

[18]  T. Yasuda,et al.  Occurrence Properties of Giant Freak Waves in Sea Area around Japan , 1997 .

[19]  Nobuhito Mori,et al.  A weakly non-gaussian model of wave height distribution for random wave train , 2002 .

[20]  C. T. Stansberg,et al.  Extreme Waves in laboratory Generated Irregular Wave Trains , 1990 .

[21]  Vladimir E. Zakharov,et al.  Stability of periodic waves of finite amplitude on the surface of a deep fluid , 1968 .