Study on the behavior of weakly nonlinear water waves in the presence of random wind forcing (original) (raw)
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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.
Role of non-resonant interactions in the evolution of nonlinear random water wave fields
Journal of Fluid Mechanics, 2006
We present the results of direct numerical simulations (DNS) of the evolution of nonlinear random water wave fields. The aim of the work is to validate the hypotheses underlying the statistical theory of nonlinear dispersive waves and to clarify the role of exactly resonant, nearly resonant and non-resonant wave interactions. These basic questions are addressed by examining relatively simple wave systems consisting of a finite number of wave packets localized in Fourier space. For simulation of the longterm evolution of random water wave fields we employ an efficient DNS approach based on the integrodifferential Zakharov equation. The non-resonant cubic terms in the Hamiltonian are excluded by the canonical transformation. The proposed approach does not use a regular grid of harmonics in Fourier space. Instead, wave packets are represented by clusters of discrete Fourier harmonics.
Modeling extreme wave heights from laboratory experiments with the nonlinear Schrödinger 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 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.
Linear and Weakly Nonlinear Models of Wind Generated Surface Waves in Finite Depth
Journal of Applied Fluid Mechanics
This work regards the extension of the Miles' and Jeffreys' theories of growth of wind-waves in water of finite depth. It is divided in two major sections. The first one corresponds to the surface water waves in a linear regimes and the second one to the surface water waver considered in a weak nonlinear, dispersive and antidissipative regime. In the linear regime, we extend the Miles' theory of wind wave amplification to finite depth. The dispersion relation provides a wave growth rate depending to depth. A dimensionless water depth parameter depending to depth and a characteristic wind speed, induces a family of curves representing the wave growth as a function of the wave phase velocity and the wind speed. We obtain a good agreement between our theoretical results and the data from the Australian Shallow Water Experiment as well as the data from the Lake George experiment. In a weakly nonlinear regime the evolution of wind waves in finite depth is reduced to an anti-dissipative Kortewegde Vries-Burgers equation and its solitary wave solution is exhibited. Anti-dissipation phenomenon accelerates the solitary wave and increases its amplitude which leads to its blow-up and breaking. Blow-up is a nonlinear, dispersive and anti-dissipative phenomenon which occurs in finite time. A consequence of anti-dissipation is that any solitary waves' adjacent planes of constants phases acquire different velocities and accelerations and ends to breaking which occurs in finite space and in a finite time prior to the blow-up. It worth remarking that the theoretical amplitude growth breaking time are both testable in the usual experimental facilities. At the end, in the context of wind forced waves in finite depth, the nonlinear Schrödinger equation is derived and for weak wind inputs, the Akhmediev, Peregrine and Kuznetsov-Ma breather solutions are obtained.
Instability and Evolution of Nonlinearly Interacting Water Waves
Physical Review Letters, 2006
We consider the modulational instability of nonlinearly interacting two-dimensional waves in deep water, which are described by a pair of two-dimensional coupled nonlinear Schrödinger equations. We derive a nonlinear dispersion relation. The latter is numerically analyzed to obtain the regions and the associated growth rates of the modulational instability. Furthermore, we follow the long term evolution of the latter by means of computer simulations of the governing nonlinear equations and demonstrate the formation of localized coherent wave envelopes. Our results should be useful for understanding the formation and nonlinear propagation characteristics of large-amplitude freak waves in deep water.
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.
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".
Physics Letters A, 2012
We consider the effect of the wind and the dissipation on the nonlinear stages of the modulational instability. By applying a suitable transformation, we map the forced/damped Nonlinear Schrödinger (NLS) equation into the standard NLS with constant coefficients. The transformation is valid as long as |Γt| 1, with Γ the growth/damping rate of the waves due to the wind/dissipation. Approximate rogue wave solutions of the equation are presented and discussed. The results shed some lights on the effects of wind and dissipation on the formation of rogue waves.