Phase-averaged equation for water waves

2013 ◽  
Vol 718 ◽  
pp. 280-303 ◽  
Author(s):  
Odin Gramstad ◽  
Michael Stiassnie
Keyword(s):  
Kinetic Equation ◽  
Time Scale ◽  
Water Waves ◽  
Order Effects ◽  
Two Dimensional ◽  
Spectral Evolution ◽  
Weakly Nonlinear ◽  
Averaged Equations ◽  
Numerical Solver ◽  
Fast Field

AbstractWe investigate phase-averaged equations describing the spectral evolution of dispersive water waves subject to weakly nonlinear quartet interactions. In contrast to Hasselmann’s kinetic equation, we include the effects of near-resonant quartet interaction, leading to spectral evolution on the ‘fast’ $O({\epsilon }^{- 2} )$ time scale, where $\epsilon $ is the wave steepness. Such a phase-averaged equation was proposed by Annenkov & Shrira (J. Fluid Mech., vol. 561, 2006b, pp. 181–207). In this paper we rederive their equation taking some additional higher-order effects related to the Stokes correction of the frequencies into account. We also derive invariants of motion for the phase-averaged equation. A numerical solver for the phase-averaged equation is developed and successfully tested with respect to convergence and conservation of invariants. Numerical simulations of one- and two-dimensional spectral evolution are performed. It is shown that the phase-averaged equation describes the ‘fast’ evolution of a spectrum on the $O({\epsilon }^{- 2} )$ time scale well, in good agreement with Monte-Carlo simulations using the Zakharov equation and in qualitative agreement with known features of one- and two-dimensional spectral evolution. We suggest that the phase-averaged equation may be a suitable replacement for the kinetic equation during the initial part of the evolution of a wave field, and in situations where ‘fast’ field evolution takes place.

scholarly journals Spectral evolution of weakly nonlinear random waves: kinetic description versus direct numerical simulations

2018 ◽  
Vol 844 ◽  
pp. 766-795 ◽  
Author(s):  
Sergei Y. Annenkov ◽  
Victor I. Shrira
Keyword(s):  
Time Scale ◽  
Kinetic Equations ◽  
Wave Spectra ◽  
Kinetic Description ◽  
Spectral Evolution ◽  
Short Term ◽  
Weakly Nonlinear ◽  
Term Evolution ◽  
Integral Characteristics

Kinetic equations are widely used in many branches of science to describe the evolution of random wave spectra. To examine the validity of these equations, we study numerically the long-term evolution of water wave spectra without wind input using three different models. The first model is the classical kinetic (Hasselmann) equation (KE). The second model is the generalised kinetic equation (gKE), derived employing the same statistical closure as the KE but without the assumption of quasistationarity. The third model, which we refer to as the DNS-ZE, is a direct numerical simulation algorithm based on the Zakharov integrodifferential equation, which plays the role of the primitive equation for a weakly nonlinear wave field. It does not employ any statistical assumptions. We perform a comparison of the spectral evolution of the same initial distributions without forcing, with/without a statistical closure and with/without the quasistationarity assumption. For the initial conditions, we choose two narrow-banded spectra with the same frequency distribution and different degrees of directionality. The short-term evolution ($O(10^{2})$ wave periods) of both spectra has been previously thoroughly studied experimentally and numerically using a variety of approaches. Our DNS-ZE results are validated both with existing short-term DNS by other methods and with available laboratory observations of higher-order moment (kurtosis) evolution. All three models demonstrate very close evolution of integral characteristics of the spectra, approaching with time the theoretical asymptotes of the self-similar stage of evolution. Both kinetic equations give almost identical spectral evolution, unless the spectrum is initially too narrow in angle. However, there are major differences between the DNS-ZE and gKE/KE predictions. First, the rate of angular broadening of initially narrow angular distributions is much larger for the gKE and KE than for the DNS-ZE, although the angular width does appear to tend to the same universal value at large times. Second, the shapes of the frequency spectra differ substantially (even when the nonlinearity is decreased), the DNS-ZE spectra being wider than the KE/gKE ones and having much lower spectral peaks. Third, the maximal rates of change of the spectra obtained with the DNS-ZE scale as the fourth power of nonlinearity, which corresponds to the dynamical time scale of evolution, rather than the sixth power of nonlinearity typical of the kinetic time scale exhibited by the KE. The gKE predictions fall in between. While the long-term DNS show excellent agreement with the KE predictions for integral characteristics of evolving wave spectra, the striking systematic discrepancies for a number of specific spectral characteristics call for revision of the fundamentals of the wave kinetic description.

Implementing New Nonlinear Term in Third Generation Wave Models

2014 ◽  
Author(s):  
Odin Gramstad ◽  
Alexander Babanin
Keyword(s):  
Kinetic Equation ◽  
Spread Spectrum ◽  
Time Scale ◽  
Source Term ◽  
Wave Spectrum ◽  
Wave Model ◽  
Third Generation ◽  
Spectral Evolution ◽  
Linear Interaction ◽  
Wavewatch Iii

The non-linear interaction term is one of the three key source functions in every third-generation spectral wave model. An update of physics of this term is discussed. The standard statistical/phase-averaged description of the nonlinear transfer of energy in the wave spectrum (wave-turbulence) is based on Hasselmann’s kinetic equation [1]. In the derivation of the kinetic equation (KE) it is assumed that the evolution takes place on the slow O(ε−4) time scale, where ε is the wave steepness. This excludes the effects of near-resonant quartet interactions that may lead to spectral evolution on the ‘fast’ O(ε−2) time scale. Generalizations of the KE (GKE) that enable description of spectral evolution on the O(ε−2) time scale [2–4] are discussed. The GKE, first solved numerically in [4], is implemented as a source term in the third generation wave model WAVEWATCH-III. The new source term (GKE) is tested and compared to the other nonlinear-interaction source terms in WAVEWATCH-III; the full KE (WRT method) and the approximate DIA method. It is shown that the GKE gives similar results to the KE in the case of a relatively broad banded and directional spread spectrum, while it shows somewhat larger difference in the case of a more narrow banded spectrum with narrower directional distribution. We suggest that the GKE may be a suitable replacement to the KE in situations where ‘fast’ spectral evolution takes place.

Elementary derivation of the kinetic equation for the two-dimensional guiding centre plasma

1978 ◽  
Vol 19 (1) ◽  
pp. 121-133 ◽  
Author(s):  
Michael Mond ◽  
Georg Knorr
Keyword(s):  
Kinetic Equation ◽  
Langevin Equation ◽  
Time Scale ◽  
Equations Of Motion ◽  
Planck Equation ◽  
Stochastic Equations ◽  
Iteration Scheme ◽  
Two Dimensional ◽  
Hopf Equation ◽  
Rigorous Solution

A kinetic equation for a two-dimensional inviscid hydrodynamic fluid is derived in two ways. First, the equations of motion for the modes of the fluid are interpreted as stochastic equations resembling the Langevin equation. To lowest order a Fokker–Planck equation can be derived which is the kinetic equation for one mode. Secondly, a suitable iteration scheme is applied to the Hopf equation which results in the same kinetic equation. A parameter describing the time scale is arbitrary and cannot be determined by the applied methods alone. It is shown that the kinetic equation satisfies the conservation requirements and relaxes to an equilibrium which is a rigorous solution of the Hopf equation.

SYMBOLIC COMPUTATION AND WEIERSTRASS ELLIPTIC FUNCTION SOLUTIONS OF THE DAVEY–STEWARTSON (DS) SYSTEM

2003 ◽  
Vol 14 (08) ◽  
pp. 1127-1134 ◽  
Author(s):  
ZHENYA YAN
Keyword(s):  
Symbolic Computation ◽  
Elliptic Function ◽  
Water Waves ◽  
Modulational Instability ◽  
Dimensional Space ◽  
Two Dimensional ◽  
Solitary Wave Solutions ◽  
Nonlinear Water Waves ◽  
Weakly Nonlinear ◽  
Weierstrass Elliptic Function

In this paper, with the symbolic computation, the doubly-periodic solutions of the Davey–Stewartson (DS) system, which describes the modulational instability of uniform train of weakly nonlinear water waves in the two-dimensional space, are investigated in terms of the Weierstrass elliptic function ℘(ξ; g2, g3). In particular, we also give new solitary wave solutions of this system.

scholarly journals Four-wave resonant interactions in the classical quadratic Boussinesq equations

2009 ◽  
Vol 618 ◽  
pp. 263-277 ◽  
Author(s):  
M. ONORATO ◽  
A. R. OSBORNE ◽  
P. A. E. M. JANSSEN ◽  
D. RESIO
Keyword(s):  
Energy Transfer ◽  
Kinetic Equation ◽  
Shallow Water ◽  
Water Waves ◽  
Wave Spectrum ◽  
Boussinesq Equations ◽  
Wave Model ◽  
Flat Bottom ◽  
Weakly Nonlinear ◽  
Resonant Interactions

We investigate theoretically the irreversibile energy transfer in flat bottom shallow water waves. Starting from the oldest weakly nonlinear dispersive wave model in shallow water, i.e. the original quadratic Boussinesq equations, and by developing a statistical theory (kinetic equation) of the aforementioned equations, we show that the four-wave resonant interactions are naturally part of the shallow water wave dynamics. These interactions are responsible for a constant flux of energy in the wave spectrum, i.e. an energy cascade towards high wavenumbers, leading to a power law in the wave spectrum of the form of k−3/4. The nonlinear time scale of the interaction is found to be of the order of (h/a)4 wave periods, with a the wave amplitude and h the water depth. We also compare the kinetic equation arising from the Boussinesq equations with the arbitrary-depth Hasselmann equation and show that, in the limit of shallow water, the two equations coincide. It is found that in the narrow band case, both in one-dimensional propagation and in the weakly two-dimensional case, there is no irreversible energy transfer because the coupling coefficient in the kinetic equation turns out to be identically zero on the resonant manifold.

scholarly journals SPECTRA AND BISPECTRA OF OCEAN WAVES

1974 ◽  
Vol 1 (14) ◽  
pp. 16
Author(s):  
Ole Gunnar Houmb
Keyword(s):  
Power Spectrum ◽  
Water Waves ◽  
Surface Displacement ◽  
Ocean Waves ◽  
Higher Order ◽  
Order Effects ◽  
Two Dimensional ◽  
Wave Surface ◽  
Linear Superposition ◽  
Breaking Energy

The two dimensional (directional) power spectrum gives an adequate description of water waves that may be regarded as a linear superposition of statistically independent waves. In such cases the sea surface is linear to the first order and the surface displacement is represented by CO n(t) = Z an sm(u> t + n) n=l where an are the amplitudes of individual waves and is a Tn randomly distributed phase angle, and the process is stationary. Under such circumstances the wave surface is Gaussian, which means that ordinates measured from MWL are normally distributed rf they are sampled at constant intervals of time. It is equally important that the wave heights are Rayleigh distributed. This formulation of the wave surface is widely used e.g. in wave forecasting. There are, however, phenomena such as wave breaking, energy transfer between wave components and surf beat which can only be described by higher order effects of wave motion (1, 2, 3, 4). In this case the two dimensional power spectrum fails to give an accurate description of the wave surface. This means that the first and second order moments (mean and covariance) no longer give all the probability information, and we have to consider higher order moments (5, 6, 7).

scholarly journals New exact relations for steady irrotational two-dimensional gravity and capillary surface waves

2017 ◽  
Vol 376 (2111) ◽  
pp. 20170220 ◽  
Author(s):  
Didier Clamond
Keyword(s):  
Free Surface ◽  
Gravity Waves ◽  
Water Waves ◽  
Two Dimensional ◽  
Physical Plane ◽  
Theme Issue ◽  
Nonlinear Water Waves ◽  
Dimensional Gravity ◽  
Irrotational Motion ◽  
Exact Relations

Steady two-dimensional surface capillary–gravity waves in irrotational motion are considered on constant depth. By exploiting the holomorphic properties in the physical plane and introducing some transformations of the boundary conditions at the free surface, new exact relations and equations for the free surface only are derived. In particular, a physical plane counterpart of the Babenko equation is obtained. This article is part of the theme issue ‘Nonlinear water waves’.

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