Modeling extreme wave heights from laboratory experiments with the nonlinear Schrödinger equation (original) (raw)
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Physical Review Letters, 2009
We discuss two independent, large scale experiments performed in two wave basins of different dimensions in which the statistics of the surface wave elevation are addressed. Both facilities are equipped with a wave maker capable of generating waves with prescribed frequency and directional properties. The experimental results show that the probability of the formation of large amplitude waves strongly depends on the directional properties of the waves. Sea states characterized by long-crested and steep waves are more likely to be populated by freak waves with respect to those characterized by a large directional spreading.
Deep-Water Waves: on the Nonlinear Schrödinger Equation and its Solutions
Journal of Theoretical and Applied Mechanics, 2013
We present a brief discussion on the nonlinear Schrödinger equation for modelling the propagation of the deep-water wavetrains and a discussion on its doubly-localized breather solutions, that can be connected to the sudden formation of extreme waves, also known as rogue waves or freak waves.
Emergence of coherent wave groups in deep-water random sea
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.
A numerical study of water-wave modulation based on a higher-order nonlinear Schrödinger equation
Journal of Fluid Mechanics, 1985
In existing experiments it is known that the slow evolution of nonlinear deep-water waves exhibits certain asymmetric features. For example, an initially symmetric wave packet of sufficiently large wave slope will first lean forward and then split into new groups in an asymmetrical manner, and, in a long wavetrain, unstable sideband disturbances can grow unequally to cause an apparent downshift of carrier-wave frequency. These features lie beyond the realm of applicability of the celebrated cubic Schrodinger equation (CSE), but can be, and to some extent have been, predicted by weakly nonlinear theories that are not limited to slowly modulated waves (i.e. waves with a narrow spectral band). Alternatively, one may employ the fourth-order equations of Dysthe (1979), which are limited to narrow-banded waves but can nevertheless be solved more easily by a pseudospectral numerical method. Here we report the numerical simulation of three cases with a view to comparing with certain recent experiments and to complement the numerical results obtained by others from the more general equations.
For modelling a series of depth profiles covering the relative depth parameter interval 2>k(p)h>0.3, evolution of two-dimensional gravity wave spectra is calculated in the frame of three-wave quasi-kinetic approximation derived by Zaslavskii and Polnikov (1988). The relative impact of refraction and nonlinearity on a change of two-dimensional spectra shape for gravity waves is estimated. It is shown that on the background of refraction impact on the spectrum shape, the three-wave nonlinearity results in a remarkable change of angular and frequency distribution for a wave energy spectrum. Herewith, in the spectral peak domain the nonlinearity reduces the value of the angular narrowness parameter by 20-30%, counteracting the refraction during the wave propagation into a shoal zone. In contrast to the high frequency domain of the spectrum, the angular narrowness parameter is increased due to the nonlinearity. For this reason, the nonlinearity can result in more than 10% change of wave energy in a shallow water zone with respect to the linear wave evolution case. These conclusions were checked by using the SWAN model under the same conditions. It was found that the SWAN model describes some of the main peculiarities of nonlinear waves in shallow water. Some recommendations were made to elaborate the three-wave nonlinear term in the source function of the SWAN model. Keywords: shallow water waves; refraction; nonlinearity; three-wave quasi-kinetic approximation; wave spectrum; spectrum shape parameters
Journal of Fluid Mechanics, 2010
Nonlinear modulational instability of wavepackets is one of the mechanisms responsible for the formation of large-amplitude water waves. Here, mechanically generated waves in a three-dimensional basin and numerical simulations of nonlinear waves have been compared in order to assess the ability of numerical models to describe the evolution of weakly nonlinear waves and predict the probability of occurrence of extreme waves within a variety of random directional wave fields. Numerical simulations have been performed following two different approaches: numerical integration of a modified nonlinear Schrödinger equation and numerical integration of the potential Euler equations based on a higher-order spectral method. Whereas the first makes a narrow-banded approximation (both in frequency and direction), the latter is free from bandwidth constraints. Both models assume weakly nonlinear waves. On the whole, it has been found that the statistical properties of numerically simulated wave fields are in good quantitative agreement with laboratory observations. Moreover, this study shows that the modified nonlinear Schrödinger equation can also provide consistent results outside its narrow-banded domain of validity.
Journal of Geophysical Research: Oceans, 2010
The estimation of nonlinear wave-wave interactions is one of the central problems in the development of operational and research models for ocean wave prediction. In this paper, we present results obtained with a numerical model based on a quasi-exact computation of the nonlinear wave-wave interactions called the Gaussian quadrature method (GQM) that gives both precise and computationally efficient calculations of the four-wave interactions. Two situations are presented: a purely nonlinear evolution of the spectrum and a duration-limited case. Properties of the directional wave spectrum obtained using GQM and the Discrete Interaction Approximation Method (DIM) are compared. Different expressions for the wind input and dissipation terms are considered. Our results are consistent with theoretical predictions. In particular, they reproduce the self-similar evolution of the spectrum. The bimodality of the directional distribution of the spectrum at frequencies lower and greater than the peak frequency is shown to be a strong feature of the sea states, which is consistent with high-resolution field measurements. Results show that nonlinear interactions constitute the key mechanism responsible for bimodality, but forcing terms also have a quantitative effect on the directional distribution of the spectrum. The influence of wind and dissipation parameterizations on the high-frequency shape of the spectrum is also highlighted. The imposition of a parametric high-frequency tail has a significant effect not only on the high-frequency shape of the spectrum but also on the energy level and peak period and on the global directional distribution.
Nonlinear Parabolic Equation and Extreme Waves on the Sea Surface
Radiophysics and Quantum Electronics, 2003
UDC 551.46 Nearly 40 years have passed since V. I. Talanov discovered the nonlinear parabolic equation which played an important role in the nonlinear optics. It was very quickly understood that this equation could also be adapted for nonstationary wave packets of different physical nature and of any dimension. Under the later name of the nonlinear (cubic) Schrödinger equation, it became a fundamental equation in the theory of weakly nonlinear wave packets in media with strong dispersion. The article is devoted to only one application of the nonlinear Schrödinger equation in the theory of the so-called freak waves on the sea surface. In the last five years a great boom has occurred in the research of extreme waves on the water, for which the nonlinear parabolic equation played an important role in the understanding of physical mechanisms of the freak-wave phenomenon. More accurate, preferably numerical, models of waves on a water with more comprehensive account of the nonlinearity and dispersion come on the spot today, and many results of weakly nonlinear models are already corrected quantitatively. Nevertheless, sophisticated models do not bring new physical concepts. Hence, their description on the basis of the nonlinear parabolic equation (nonlinear Schrödinger equation), performed in this paper, seems very attractive in view of their possible applications in the wave-motion physics.
Focusing of nonlinear wave groups in deep water
Journal of Experimental and Theoretical Physics Letters, 2001
The freak wave phenomenon in the ocean is explained by the nonlinear dynamics of phase-modulated wave trains. It is shown that the preliminary quadratic phase modulation of wave packets leads to a significant amplification of the usual modulation (Benjamin-Feir) instability. Physically, the phase modulation of water waves may be due to a variable wind in storm areas. The well-known breather solutions of the cubic Schrödinger equation appear on the final stage of the nonlinear dynamics of wave packets when the phase modulation becomes more uniform. © 2001 MAIK "Nauka/Interperiodica".
On four highly nonlinear phenomena in wave theory and marine hydrodynamics
Applied Ocean Research, 2002
Some recent developments in the formation of extreme waves, kinematics of steep waves, the phenomenon of ringing and currents in the ocean induced by internal waves are reviewed. Formation of extreme waves are simulated by means of a rapid fully nonlinear model. A large wave event taking place in a wave group is characterized by an elevation being significantly larger than the initial amplitude of the group. Recurrence occurs. PIV measurements of Stokes waves exhibit an exponential velocity profile all the way up to the surface elevation (wave slope up to 0.16). The computed velocity profile under crest of an extreme wave corresponds also to an exponential profile. Experimental results of the horizontal force on a vertical circular cylinder in long and steep waves exhibit a secondary cycle of high frequency in the force history. This typically occurs for waves longer than about 10 times the cylinder diameter and a Froude number vh m = ffiffiffiffi gD p larger than about 0.4, v the wave frequency, h m the maximal elevation, g the acceleration of gravity, D the cylinder diameter. Properties of internal solitons and the induced fluid velocities are described in terms of weakly and fully nonlinear models supported by PIV measurements. A rapid scheme for fully nonlinear interfacial waves in three dimensions is derived, complementing the rapid model of free surface waves. q