Freak Waves in Random Oceanic Sea States

We discuss a method for the determination of the shape of the ocean wave power spectrum that is based on the physics of the modulational instability for the nonlinear Schrödinger and the Zakharov equations. We find that the form of the spectrum includes an enhanced spectral peak and modulational channels that extend to both high and low frequency. Furthermore, this fundamental shape of the spectrum is found to also be contained in the kinetic equation commonly used for wind-wave models provided that the full Boltzmann four-wave interactions are included. We discuss a number of numerical simulations that demonstrate the modulational form of the power spectrum. We furthermore discuss how the enhanced spectral peak governs the formation of rogue wave packets. We provide ways to compute the properties of the rogue waves directly from the nonlinear spectrum of analyzed time series data or from wave forecasts and hindcasts.

2006

This report deals with extreme wave phenomena. Exploration of the classical wave theories are made, both on the theoratical approach and on the statistical one. The first one shows wave generation phenomenon using only Euler's equation for a perfect fluid and gravity. On the other one, the statistical approach provides us with more real observations. Both models fail to explain some rare (or not so rare ?) events: freak waves. Then we defined what is a freak wave and some of the explanations that are given. Exploration on the non linear Schrödinger equation, which is known to give birth to gigantic waves is then the path taken. This equation could be easily derived from Euler's equations. Numerical solution of this equation are provided in the last chapter. Finally, the third part deals with spectral methods and how they are used to compute very easily non linear interaction for waves. Last chapter provides also results on this. In fact, the last chapter is devoted to the results obtained, either on solving NLS, either on the computation of surface waves.

Coastal Engineering Proceedings, 2014

Nonlinear four-wave interactions amplify wave heights of deep-water generating extreme wave such as a freak wave. However, it is not clear the behavior of generated freak waves in deep-water shoaling to shallow water regions. In this study, a series of physical experiments and numerical simulations with several bathymetry configurations were conducted for unidirectional random waves from deep to shallow water regions. The maximum wave heights increase with an increase in kurtosis by third-order nonlinear interactions in deep water regions. The dependence of the kurtosis on the freak wave occurrence is weakened due to second-order nonlinear interactions associated with wave shoaling on the slope. Moreover, it is possible to understand the behavior of the high-order nonlinearity and the freak wave occurrence in shallow water regions if appropriate correction of the insufficient nonlinearity of more than O(ε 2 ) to the standard Boussinesq equation are considered analytically.

Physical Review E, 2013

Extreme surface waves in a deep-water long-crested sea are often interpreted as a manifestation in the real world of the so-called breathing solitons of the focusing nonlinear Schrödinger equation. While the spontaneous emergence of such coherent structures from nonlinear wave dynamics was demonstrated to take place in fiberoptics systems, the same point remains far more controversial in the hydrodynamic case. With the aim to shed further light on this matter, the emergence of breatherlike coherent wave groups in a long-crested random sea is investigated here by means of high-resolution spectral simulations of the fully nonlinear two-dimensional Euler equations. The primary focus of our study is to parametrize the structure of random wave fields with respect to the Benjamin-Feir index, which is a nondimensional measure of the energy localization in Fourier space. This choice is motivated by previous results, showing that extreme-wave activity in a long-crested sea is highly sensitive to such a parameter, which is varied here by changing both the characteristic spectral bandwidth and the average wave steepness. It is found that coherent wave groups, closely matching realizations of Kuznetsov-Ma breathers in Euler dynamics, develop within wave fields characterized by sufficiently narrow-banded spectra. The characteristic spatial and temporal scales of wave group dynamics, and the corresponding occurrence of extreme events, are quantified and discussed by means of space-time autocorrelations of the surface elevation envelope and extreme-event statistics.

Lately, strange waves originating from an unknown source even under mild weather conditions have been frequently reported along the coast of South Korea. These waves can be characterized by abnormally high run-up height and unpredictability, and have evoked the imagination of many people. However, how these waves are generated is a very controversial issue within the coastal community of South Korea. In 2006, Shukla numerically showed that extremely high waves of modulating amplitude can be generated when swell and locally generated wind waves cross each other with finite angle, by using a pair of nonlinear cubic Schrodinger Equations. Shukla (2006) also showed that these waves propagate along a line, that evenly dissects the angles formed by the propagating directions of swell and wind waves. Considering that cubic Schrodinger Equations are only applicable for a narrow banded wave train, which is very rare in the ocean field, Shukla (2006)'s work is subject to more severe testing. Based on this rationale, in this study, first we relax the narrow banded assumption, and numerically study the feasibility of the birth of freak waves due to the nonlinear interaction of swell and wind waves crossing each other with finite angle, by using a more robust wave model, the Navier-Stokes equation.

Engineering Sciences, 2021

Control signals with simultaneous modulation of periods and amplitudes were finetuned and fed to a wave flap for a generation of freak waves. The meshless Smoothed Particle Hydrodynamics method was used to predict the location and the amplitude of the maximum wave crest. The time series output from the DualSPHysics software was validated experimentally at the BSHC seakeeping wave basin. The experimental data is in good agreement with the simulations.

Coastal Engineering Proceedings, 2012

We investigate the dynamic and kinematic characteristics of freak waves using a direct phase-resolved nonlinear numerical method. The focus is on the understanding of the effects of different nonlinear wave-wave interactions on freak waves development and characteristics in the evolution process of modulated Stokes wave trains. Long time simulations of modulated Stokes wave trains, with different parameters, are obtained. Based on these simulations, we find that there are different kinds of freak waves in different time scales due to two kinds of different nonlinear mechanisms. One is the modulation instability and another related to the wave group interaction. Both the dynamic and kinematic characteristics of the different kinds of freak waves are distinct. Occurrence of freak waves (especially of large height) is usually correlated with broadband wave spectra.

Applied Ocean Research, 2005

Four freak events registered in the North Sea during a storm are presented and studied. The spatial evolution of the freak waves backward and forward wave propagation is simulated within the framework of the Dysthe equation. The lifetimes and travel distances of the freak waves are determined based on the results of the simulations. The wave evolution predicted by the Dysthe model is compared with the simulations of the nonlinear Schrödinger and kinematical equations. The contributions of the effects of the nonlinear self-focusing (Benjamin-Feir instability) and quasi-linear wave grouping are discovered with the help of the nonlinear Schrödinger approximation and the linear theory. It is found that though the Benjamin-Feir instability is important for the description of freak wave evolution, the significant wave enhancement by itself may be achieved even in the linear approximation. q (A. Slunyaev), [email protected] (C. Guedes Soares). Fig. 6. Maximum wave height versus distance for record 1: comparison of the simulations in the Dysthe model (thick solid line) with the linear limit (dashed line). The initial conditions: field at position XZK500, computed within the frameworks of the Dysthe equation.

Natural Hazards and Earth System Sciences, 2014

Spatial variation of nonlinear wave groups with different initial envelope shapes is theoretically studied first, confirming that the simplest nonlinear theoretical model is capable of describing the evolution of propagating wave packets in deep water. Moreover, three groups of laboratory experiments run in the wave basin of CEHIPAR (Canal de Experiencias Hidrodinámicas de El Pardo, known also as El Pardo Model Basin) was founded in 1928 by the Spanish Navy. are systematically compared with the numerical simulations of the nonlinear Schrödinger equation. Although a little overestimation is detected, especially in the set of experiments characterized by higher initial wave steepness, the numerical simulation still displays a high degree of agreement with the laboratory experiments. Therefore, the nonlinear Schrödinger equation catches the essential characteristics of the extreme waves and provides an important physical insight into their generation. The modulation instability, resulting from the quasi-resonant four-wave interaction in a unidirectional sea state, can be indicated by the coefficient of kurtosis, which shows an appreciable correlation with the extreme wave height and hence is used in the modified Edgeworth-Rayleigh distribution. Finally, some statistical properties on the maximum wave heights in different sea states have been related with the initial Benjamin-Feir index.

Journal of Physical Oceanography, 2003

Four-wave interactions are shown to play an important role in the evolution of the spectrum of surface gravity waves. This fact follows from direct simulations of an ensemble of ocean waves using the Zakharov equation. The theory of homogeneous four-wave interactions, extended to include effects of nonresonant transfer, compares favorably with the ensemble-averaged results of the Monte Carlo simulations. In particular, there is good agreement regarding spectral shape. Also, the kurtosis of the surface elevation probability distribution is determined well by theory even for waves with a narrow spectrum and large steepness. These extreme conditions are favorable for the occurrence of freak waves.