Nonlinear Four-Wave Interactions and Freak Waves (original) (raw)
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Effects of high-order nonlinear wave-wave interactions on gravity waves
2000
Numerical simulations of gravity waves with high-order nonlinearities in two and three dimensional domain are performed by using pseudo spectral method. High-order nonlinearities more than thirdorder excite apparently chaotic evolutions of the Fourier energy in deepwater random waves. The high-order nonlinearities increase kurtosis, wave height distribution and Hmax/H 1/3 in deep-water and decrease these wave statistics in shallow water. They, moreover, can generate a single extreme high wave with an outstanding crest height in deep-water. The high-order nonlinearities more than third-order can be regarded as one of a reason of a cause of a freak wave in deep-water.
Journal of Physics: Conference Series, 2021
The nonlinear evolution equations of fourth-order have been established for two surface gravity waves in infinite depth of water including the effect of air flowing over water. In the present paper, we have applied a general approach depending upon the integral equation due to Zakharov. Based on these equations, the stability analysis has been studied in the appearance of a uniform gravity wave packet with another wave packet of the identical group velocity. Graphs have been drawn for the instability growth rate of the uniform wave packet with shorter wave number versus the perturbed wave number for several values of the amplitude of the wave packet of larger wave-number and several values of wind velocity. We have observed from the figures that the instability growth rate of the second wave packet enhances with the enhancement of nondimensional wind velocity for the settled value of the amplitude of the first wave packet.
Nonlinear Effects in Gravity Waves Propagating in Shallow Water
Coastal Engineering Journal, 2012
Nonlinear energy transfers due to triad interactions change the characteristics of the wave-field in the shoaling region. The degree of nonlinear coupling is examined using numerical simulations based on an accurate set of deterministic evolution equations for the propagation of fully dispersive weakly nonlinear waves. The model validation, using existing experimental measurements for wave transformation over a shoal, showed that it accurately predicts nonlinear energy transfer for irregular waves with large wave-numbers. The bound higher harmonics and nonlinear statistical measures, i.e. the wave skewness and asymmetry, are well simulated by the model in both the shoaling and deshoaling regions. Numerical simulation of steep waves in shallow water with the Ursell number O(1), showed that nonlinear dispersion and phase locking lead to triad interactions even on a horizontal bottom. Nonlinear energy transfers in monochromatic waves lead to rapid spatial recurrence of the primary wave amplitudes. This is in contrast to the case of irregular waves where the Fourier coefficients of the wave-field do not recur due to the presence of innumerable interactions, which are expected to cancel resulting in no spatial evolution of the wave spectrum.
Coupling of surface and internal gravity waves: a mode coupling model
Journal of Fluid Mechanics, 1976
A surface-wave/internal-wave mode coupled model is constructed to describe the energy transfer from a linear surface wave field on the ocean to a linear internal wave field. Expressed in terms of action-angle variables the dynamic equations have a particularly useful form and are solved both numerically and in some analytic approximations. The growth time for internal waves generated by the resonant interaction of surface waves is calculated for an equilibrium spectrum of surface waves and for both the Garrett-Munk and two-layer models of the undersea environment. We find energy transfer rates as a function of undersea parameters which are much faster than those based on the constant Brunt-ViiisSila model used by Kenyon (1968) and which are consistent with the experiments of Joyce (1974). The modulation of the surface-wave spectrum by internal waves is also calculated, yielding a ‘mottled’ appearance of the ocean surface similar to that observed in photographs taken from an ERTS1 sa...
Higher-order spectral analysis of nonlinear ocean surface gravity waves
Journal of Geophysical Research, 1995
Bispectral and trispectral analyses are used to detect secondary and tertiary wave components resulting from nonlinear interactions among largeamplitude ocean surface gravity waves in 8-and 13-m water depths. Bispectra of bottom-pressure measurements indicate forced secondary waves at frequencies 2fp about twice the primary power spectral peak frequency fp. However, the interpretation of the bispectrum at sum frequencies of approximately 3f• is ambiguous because contributions of both secondary and tertiary forced waves may be significant. Trispectral analysis confirms the presence of tertiary waves with frequency approximately 3f•. In 8 rn depth the tertiary bottom-pressure field is dominated by interactions between three colinearly propagating wind-wave components with frequencies close to f•. In 13 m depth these relatively shortwavelength forced waves are strongly attenuated at the seafloor and the tertiary wave field is driven by interactions between the dominant waves at f• and obliquely propagating higher-frequency wind waves. The phases of the higher-order spectra are consistent with weakly nonlinear wave theory (Hasselmann, 1962). scribed and then applied to the field data, followed by a summary of the results.
On the interaction of four water-waves
Wave Motion, 2005
The mathematical and statistical properties of the evolution of a system of four interacting surface gravity waves are investigated in detail. Any deterministic quartet of waves is shown to evolve recurrently, but the ensemble averages taken over many realizations with random initial conditions reach constant asymptotic values. The characteristic time-scale for which such asymptotic values are approached is extremely large when randomness is introduced through the initial phases. The characteristic time-scale becomes of an order comparable to that of the recurrence periods when beside the random initial phases, the initial amplitudes are taken to be Rayleigh-distributed. The ensemble-averaged results in the second case resemble, to a certain extent, those derived from the kinetic equation.
Interplay of Resonant and Quasi-Resonant Interaction of the Directional Ocean Waves
Journal of Physical Oceanography, 2009
Recent experimental study of the evolution of random directional gravity waves in deep water provides new insight into the nature of the spectral evolution of the ocean waves and the relative significance of resonant and quasi-resonant wave interaction. When the directional angle containing half the total energy is broader than ;208, the spectrum evolves following the energy transfer that can be described by the fourwave resonant interaction alone. In contrast, in the case of a directionally confined spectrum, the effect of quasi-resonant wave-wave interaction becomes important, and the wave system becomes unstable. When the temporal change of the spectral shape due to quasi resonance becomes irreversible owing to energetic breaking dissipation, the spectrum rapidly downshifts. Under such extreme conditions, the likelihood of a freak wave is high.
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.
Weak turbulent approach to the wind-generated gravity sea waves
Physica D: Nonlinear Phenomena, 2003
We performed numerical simulation of the kinetic equation describing behavior of an ensemble of random-phase, spatially homogeneous gravity waves on the surface of the infinitely deep ocean. Results of simulation support the theory of weak turbulence not only in its basic points, but also in many details. The weak turbulent theory predicts that the main physical processes taking place in the wave ensemble are down-shift of spectral peak and "leakage" of energy and momentum to the region of very small scales where they are lost due to local dissipative processes. Also, the spectrum of energy right behind the spectral peak should be close to the weak turbulent Kolmogorov spectrum which is the exact solution of the stationary kinetic (Hasselmann) equation. In a general case, this solution is anisotropic and is defined by two parameters-fluxes of energy and momentum to high wave numbers. Even in the anisotropic case the solution in the high wave number region is almost proportional to the universal form ω −4. This result should be robust with respect to change of the parameters of forcing and damping. In all our numerical experiments, the ω −4 Kolmogorov spectrum appears in very early stages and persists in both stationary and non-stationary stages of spectral development. A very important aspect of the simulations conducted here was the development of a quasi-stationary wave spectrum under wind forcing, in absence of any dissipation mechanism in the spectral peak region. This equilibrium is achieved in the spectral range behind the spectral peak due to compensation of wind forcing and leakage of energy and momentum to high wave numbers due to nonlinear four-wave interaction. Numerical simulation demonstrates slowing down of the shift of the spectral peak and formation of the bimodal angular distribution of energy in the agreement with field and laboratory experimental data. A more detailed comparison with the experiment can be done after developing of an upgraded code making possible to model a spatially inhomogeneous ocean.