Numerical simulation of freak waves in random sea state
alireza tilkoo
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Abstract
The generation of freak waves in a 2-dimensional random sea state characterized by the JONSWAP spectrum are simulated employing a nonlinear fourth-order Schrödinger equation. The evolution of the freak waves in deep water are analyzed. We investigate the effect of initial wave parameters on kurtosis and occurrence of freak waves. The results show that Benjamin-Feir index (BFI) is an important parameter to identify the presence of instability. The kurtosis presents a similar spatial evolution trend with the occurrence probability of freak waves. Freak waves in a random sea state are more likely to occur for narrow spectrum and small values of significant wave height.
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Physical Review Letters, 2001
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 $\alpha$ and the enhancement coefficient $\gamma$. We discuss how freak waves in a random sea state are more likely to occur for large values of $\alpha$ and $\gamma$. Our results are supported by extensive numerical simulations of the NLS equation with random initial conditions. Comparison with linear simulations are also reported.
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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.
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The mechanism of the freak wave formation related to the spatial-temporal focusing is studied within the framework of the Korteweg-de Vries equation. A method to find the wave trains whose evolution leads to the freak wave formation is proposed. It is based on the solution of the Korteweg-de Vries equation with an initial condition corresponding to the expected freak wave. All solutions of this Cauchy problem by the reversal of abscissa represent the possible forms of wave trains which evolve into the freak wave. It is found that freak waves are almost linear waves, and their characteristic Ursell parameter is small. The freak wave formation is possible also from the random wave field and the numerical simulation describes the details of this phenomenon. It is shown that freak waves can be generated not only for specific conditions, but also for relative wide classes of the wave trains. This mechanism explains the rare and short-lived character of the freak wave.
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Physical Review Letters, 2006
Here we consider a simple weakly nonlinear model that describes the interaction of two-wave systems in deep water with two different directions of propagation. Under the hypothesis that both sea systems are narrow banded, we derive from the Zakharov equation two coupled nonlinear Schrödinger equations. Given a single unstable plane wave, here we show that the introduction of a second plane wave, propagating in a different direction, can result in an increase of the instability growth rates and enlargement of the instability region. We discuss these results in the context of the formation of rogue waves.
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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.
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Comptes Rendus de l Académie des Sciences - Series IIB - Mechanics
The influence of wind on extreme wave events in shallow water is investigated numerically. A series of numerical simulations using a pressure distribution over the steep crests given by the modified Jeffreys' sheltering theory shows that wind blowing over a strongly modulated wave group due to the dispersive focusing of a chirped long wave packet increases the time duration and maximal amplitude of the extreme wave event. These results are coherent with those obtained within the framework of deep water. However, steep wave events are less unstable to wind perturbation in shallow water than in deep water.
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Journal of Fluid Mechanics, 2007
A large number of simulations have been performed to reveal how the occurrence of freak waves on deep water depends on the group and crest lengths for fixed steepness. It is found that there is a sharp qualitative transition between short- and long-crested sea, for a crest length of approximately ten wavelengths. For short crest lengths the statistics of freak waves deviates little from Gaussian and their occurrence is independent of group length (or Benjamin–Feir index, BFI). For long crest lengths the statistics of freak waves is strongly non-Gaussian and the group length (or BFI) is a good indicator of increased freak wave activity.
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Physics of Fluids, 2008
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Journal of Physical Oceanography, 2003
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Physics of Fluids, 2005
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Systematic study of rogue wave probability distributions in a fourth‐order nonlinear Schrödinger equation
Nonlinear instability and refraction by ocean currents are both important mechanisms that go beyond the Rayleigh approximation and may be responsible for the formation of freak waves. In this paper, we quantitatively study nonlinear effects on the evolution of surface gravity waves on the ocean, to explore systematically the effects of various input parameters on the probability of freak wave formation. The fourth-order current-modified nonlinear Schrödinger equation (CNLS 4 ) is employed to describe the wave evolution. By solving CNLS 4 numerically, we are able to obtain quantitative predictions for the wave height distribution as a function of key environmental conditions such as average steepness, angular spread, and frequency spread of the local sea state. Additionally, we explore the spatial dependence of the wave height distribution, associated with the buildup of nonlinear development.
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