Nonlinear Behaviour of Sea Surface Waves Based on Low-Gradient Phase-Only Scattering Effects (original) (raw)
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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.
Study on the behavior of weakly nonlinear water waves in the presence of random wind forcing
Nonlinear Dynamics, 2019
Specific solutions of the nonlinear Schrödinger equation, such as the Peregrine breather, are considered to be prototypes of extreme or freak waves in the oceans. An important question is, whether these solutions also exist in the presence of gusty wind. Using the method of multiple scales, a nonlinear Schrödinger equation is obtained for the case of wind forced weakly nonlinear deep water waves. Thereby, the wind forcing is modeled as a stochastic process. This leads to a stochastic nonlinear Schrödinger equation, which is calculated for different wind regimes. For the case of wind forcing which is either random in time or random in space, it is shown that breather type solutions such as the Peregrine breather occur even in strong gusty wind conditions.
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.
Surface wave predictions in weakly nonlinear directional seas
Applied Ocean Research, 2017
We have employed laboratory and numerical experiments in order to investigate propagation of waves in both long and short-crested wave fields in deep water. For long-crested waves with steepness, ǫ = k c a c = 0.1 (a fairly extreme case), reliable prediction can be performed with the modified nonlinear Schrödinger equation up to about 40 characteristic wavelengths. For short-crested waves the accuracy of prediction is strongly reduced with increasing directional spread.
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.
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".
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.
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
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.