Spatial versions of the Zakharov and Dysthe evolution equations for deep-water gravity waves

2002 ◽  
Vol 450 ◽  
pp. 201-205 ◽  
Author(s):  
ELIEZER KIT ◽  
LEV SHEMER
Keyword(s):  
Gravity Waves ◽  
Fourth Order ◽  
Deep Water ◽  
Evolution Equations ◽  
Velocity Potential ◽  
Surface Elevation ◽  
Two Dimensional ◽  
Zakharov Equation ◽  
Dimensional Version

A spatial two-dimensional version of the Zakharov equation describing the evolution of deep-water gravity waves is used to derive two fourth-order evolution equations, for the amplitudes of the surface elevation and of the velocity potential. The scaled form of the equations is presented.

scholarly journals Five-Wave Resonances in Deep Water Gravity Waves: Integrability, Numerical Simulations and Experiments

Fluids ◽  
2021 ◽  
Vol 6 (6) ◽  
pp. 205
Author(s):  
Dan Lucas ◽  
Marc Perlin ◽  
Dian-Yong Liu ◽  
Shane Walsh ◽  
Rossen Ivanov ◽  
...  
Keyword(s):  
Normal Form ◽  
Numerical Simulations ◽  
Gravity Waves ◽  
Deep Water ◽  
Plane Waves ◽  
Surface Elevation ◽  
Wave Interactions ◽  
Frequency Space ◽  
Target Mode ◽  
The Hilbert Transform

In this work we consider the problem of finding the simplest arrangement of resonant deep-water gravity waves in one-dimensional propagation, from three perspectives: Theoretical, numerical and experimental. Theoretically this requires using a normal-form Hamiltonian that focuses on 5-wave resonances. The simplest arrangement is based on a triad of wavevectors K1+K2=K3 (satisfying specific ratios) along with their negatives, corresponding to a scenario of encountering wavepackets, amenable to experiments and numerical simulations. The normal-form equations for these encountering waves in resonance are shown to be non-integrable, but they admit an integrable reduction in a symmetric configuration. Numerical simulations of the governing equations in natural variables using pseudospectral methods require the inclusion of up to 6-wave interactions, which imposes a strong dealiasing cut-off in order to properly resolve the evolving waves. We study the resonance numerically by looking at a target mode in the base triad and showing that the energy transfer to this mode is more efficient when the system is close to satisfying the resonant conditions. We first look at encountering plane waves with base frequencies in the range 1.32–2.35 Hz and steepnesses below 0.1, and show that the time evolution of the target mode’s energy is dramatically changed at the resonance. We then look at a scenario that is closer to experiments: Encountering wavepackets in a 400-m long numerical tank, where the interaction time is reduced with respect to the plane-wave case but the resonance is still observed; by mimicking a probe measurement of surface elevation we obtain efficiencies of up to 10% in frequency space after including near-resonant contributions. Finally, we perform preliminary experiments of encountering wavepackets in a 35-m long tank, which seem to show that the resonance exists physically. The measured efficiencies via probe measurements of surface elevation are relatively small, indicating that a finer search is needed along with longer wave flumes with much larger amplitudes and lower frequency waves. A further analysis of phases generated from probe data via the analytic signal approach (using the Hilbert transform) shows a strong triad phase synchronisation at the resonance, thus providing independent experimental evidence of the resonance.

On the subharmonic instabilities of steady three-dimensional deep water waves

1994 ◽  
Vol 262 ◽  
pp. 265-291 ◽  
Author(s):  
Mansour Ioualalen ◽  
Christian Kharif
Keyword(s):  
Gravity Waves ◽  
Deep Water ◽  
Water Waves ◽  
Transverse Direction ◽  
Plane Waves ◽  
Three Dimensional ◽  
Numerical Procedure ◽  
Two Dimensional ◽  
Wave Steepness ◽  
Oblique Angle

A numerical procedure has been developed to study the linear stability of nonlinear three-dimensional progressive gravity waves on deep water. The three-dimensional patterns considered herein are short-crested waves which may be produced by two progressive plane waves propagating at an oblique angle, γ, to each other. It is shown that for moderate wave steepness the dominant resonances are sideband-type instabilities in the direction of propagation and, depending on the value of γ, also in the transverse direction. It is also shown that three-dimensional progressive gravity waves are less unstable than two-dimensional progressive gravity waves.

Wave Kinematics Computed With the Nonlinear Schro¨dinger Method for Deep Water

1999 ◽  
Vol 121 (2) ◽  
pp. 126-130 ◽  
Author(s):  
K. Trulsen
Keyword(s):  
Time Series ◽  
Gravity Waves ◽  
Deep Water ◽  
Water Wave ◽  
Surface Displacement ◽  
Horizontal Position ◽  
Two Dimensional ◽  
Limited Bandwidth ◽  
Wave Kinematics ◽  
The Waves

The nonlinear Schro¨dinger method for water wave kinematics under two-dimensional irregular deepwater gravity waves is developed. Its application is illustrated for computation of the velocity and acceleration fields from the time-series of the surface displacement measured at a fixed horizontal position. The method is based on the assumption that the waves have small steepness and limited bandwidth.

scholarly journals The effect of randomness on the stability of deep water surface gravity waves in the presence of a thin thermocline

1998 ◽  
Vol 40 (2) ◽  
pp. 190-206
Author(s):  
Sudebi Bhattacharyya ◽  
K. P. Das
Keyword(s):  
Growth Rate ◽  
Evolution Equation ◽  
Gravity Waves ◽  
Fourth Order ◽  
Deep Water ◽  
Water Surface ◽  
Spectral Width ◽  
Surface Gravity ◽  
Surface Gravity Waves ◽  
The Stability

AbstractThe effect of randomness on the stability of deep water surface gravity waves in the presence of a thin thermocline is studied. A previously derived fourth order nonlinear evolution equation is used to find a spectral transport equation for a narrow band of surface gravity wave trains. This equation is used to study the stability of an initially homogeneous Lorentz shape of spectrum to small long wave-length perturbations for a range of spectral widths. The growth rate of the instability is found to decrease with the increase of spectral widths. It is found that the fourth order term in the evolution equation produces a decrease in the growth rate of the instability. There is stability if the spectral width exceeds a certain critical value. For a vanishing bandwidth the deterministic growth rate of the instability is recovered. Graphs have been plotted showing the variations of the growth rate of the instability against the wavenumber of the perturbation for some different values of spectral width, thermocline depth, angle of perturbation and wave steepness.

scholarly journals Fourth order nonlinear evolution equations for gravity-capillary waves in the presence of a thin thermocline in deep water

2002 ◽  
Vol 43 (4) ◽  
pp. 513-524 ◽  
Author(s):  
Suma Debsarma ◽  
K.P. Das
Keyword(s):  
Fourth Order ◽  
Deep Water ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Capillary Wave ◽  
Capillary Waves ◽  
Nonlinear Evolution Equations ◽  
Marginal Stability ◽  
Stability And Instability ◽  
The Stability

AbstractFor a three-dimensional gravity capillary wave packet in the presence of a thin thermocline in deep water two coupled nonlinear evolution equations correct to fourth order in wave steepness are obtained. Reducing these two equations to a single equation for oblique plane wave perturbation, the stability of a uniform gravity-capillary wave train is investigated. The stability and instability regions are identified. Expressions for the maximum growth rate of instability and the wavenumber at marginal stability are obtained. The results are shown graphically.

Hamiltonian form of the modified nonlinear Schrödinger equation for gravity waves on arbitrary depth

2011 ◽  
Vol 670 ◽  
pp. 404-426 ◽  
Author(s):  
ODIN GRAMSTAD ◽  
KARSTEN TRULSEN
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Gravity Waves ◽  
Fourth Order ◽  
Schrodinger Equation ◽  
Wave Motion ◽  
Nonlinear Schrodinger Equation ◽  
Hamiltonian Form ◽  
Zakharov Equation ◽  
Nonlinear Schrödinger

The commonly used forms of the modified nonlinear Schrödinger equations for deep water (Dysthe, Proc. R. Soc. Lond. A, vol. 369, 1979, p. 105) and arbitrary depth (Brinch–Nielsen & Jonsson, Wave Motion, vol. 8, 1986, p. 455) do not conserve momentum and are not Hamiltonian. We show how these equations can be brought into Hamiltonian form, with the action, momentum and Hamiltonian being conserved. We derive the new fourth-order nonlinear Schrödinger equation for arbitrary depth, starting from the Zakharov equation enhanced with the new kernel of Krasitskii (J. Fluid Mech., vol. 272, 1994, p. 1).

A dynamic equation for 2D surface waves on deep water

2020 ◽  
Author(s):  
Dmitry Kachulin ◽  
Alexander Dyachenko ◽  
Vladimir Zakharov
Keyword(s):  
Fourth Order ◽  
Canonical Transformation ◽  
Three Dimensional ◽  
Fourier Method ◽  
Hamiltonian Formalism ◽  
Two Dimensional ◽  
Russian Science ◽  
Wave Interactions ◽  
Zakharov Equation ◽  
Limiting Case

<p>Using the Hamiltonian formalism and the theory of canonical transformations, we have constructed a model of the dynamics of two-dimensional waves on the surface of a three-dimensional fluid. We find and apply a canonical transformation to a water wave equation to remove all nonresonant cubic and fourth-order nonlinear terms. The found canonical transformation also allows us to significantly simplify the fourth-order terms in the Hamiltonian by replacing the coefficient of four-wave Zakharov interactions with a new simpler one. As a result, unlike the Zakharov equation (written in k-space), this equation can be written in x-space, which greatly simplifies its numerical simulation. In addition, our chosen form of a new coefficient of four-wave interactions allows us to generalize this equation to describe two-dimensional waves on the surface of a three-dimensional fluid. An effective numerical algorithm based on the pseudospectral Fourier method for solving the new 2D equation is developed. In the limiting case of plane (one-dimensional) waves, we found solutions in the form of breathers propagating in one direction. The dynamics of such nonlinear traveling waves perturbed in the transverse direction is numerically investigated.</p><p>The work was supported by the Russian Science Foundation (Grant No. 19-72-30028).</p>

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