On optimal canonical variables in the theory of ideal fluid with free surface

Lushnikov, Pavel M.; Захаров, В. Е.

doi:10.1016/j.physd.2005.02.010

Cited by 12 publications

(8 citation statements)

References 15 publications

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“…(ω k − v 0 k) 2 one can see, that instability reappears only with fourth order nonlinearity terms, which means that it can influence only computations with relatively high steepness or with very high values of k. This instability can be eliminated by proper canonical change of variables, as it was shown in [32].…”

Section: Appendix B Matrix Elementsmentioning

confidence: 72%

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Numerical simulation of surface waves instability on a homogeneous grid

Korotkevich

Dyachenko

Захаров

2016

Physica D: Nonlinear Phenomena

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We performed full-scale numerical simulation of instability of weakly nonlinear waves on the surface of deep fluid. We show that the instability development leads to chaotization and formation of wave turbulence.Instability of both propagating and standing waves were studied. We studied separately pure capillary wave unstable due to three-wave interactions and pure gravity waves unstable due to four-wave interactions. The theoretical description of instabilities in all cases is included into the article. The numerical algorithm used in these and many other previous simulations performed by authors is described in details.

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Section: Appendix B Matrix Elementsmentioning

confidence: 72%

“…In details this question was considered in [32] but here we shall follow original consideration which was done by A. I. Dyachenko in 1995 (result was mentioned in [33]) with some changes for 3D hydrodynamics.…”

Section: Appendix B Matrix Elementsmentioning

confidence: 99%

Numerical simulation of surface waves instability on a homogeneous grid

Korotkevich

Dyachenko

Захаров

2016

Physica D: Nonlinear Phenomena

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“…For example, instead of making expansions in the evolution equations, one may make expansions of the Hamiltonian, truncate this series and then derive evolution equations from the truncated Hamiltonian [20][21][22][23]. In [24], the well-posedness of some such models is considered, and it is argued that by including the effect of surface tension, a wellposed model can be formed. Again, we point out that if the model really needs to have surface tension effects included for well-posedness, then it is unlikely to be a good model on geophysical spatial and temporal scales.…”

Section: Discussionmentioning

confidence: 99%

On ill-posedness of truncated series models for water waves

2014

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The evolution of surface gravity waves on a large body of water, such as an ocean, is reasonably well approximated by the Euler system for ideal, freesurface flow under the influence of gravity. The well-posedness theory for initial-value problems for these equations, which has a long and distinguished history, reveals that solutions exist, are unique, and depend continuously upon initial data in various function-space contexts. This theory is subtle, and the design of stable, accurate, numerical schemes is likewise challenging. Depending upon the wave regime in question, there are many different approximate models that can be formally derived from the Euler equations. As the Euler system is known to be well-posed, it seems appropriate that associated approximate models should also have this property. This study is directed to this issue. Evidence is presented calling into question the well-posedness of a well-known class of model equations which are widely used in simulations. A simplified version of these models is shown explicitly to be ill-posed and numerical simulations of quadratic-and cubicorder water-wave models, initiated with initial data predicted by the explicit analysis of the simplified model, lends credence to the general contention that these models are ill-posed.

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“…We will perform transformation to new variables R k , ξ k using the following generation function (see also [10]):…”

Section: How To Separate Resonant and Slave Harmonics?mentioning

confidence: 99%

Energy balance in a wind-driven sea

Захаров

2010

Phys. Scr.

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In this paper, we offer the answers to certain questions extremely important for the development of a self-consistent analytical theory for the wind-driven sea. (i) We discuss the separation into 'resonant' and 'slave' harmonics in an ensemble of weakly nonlinear gravity waves on the surface of deep water, and we construct an explicit form of the generation function for canonical transformation that eliminates the slave harmonics. (ii) When two waves compiling a quadruple are short in comparison with two others, we find an asymptotic form for the four-wave coupling coefficient. This result makes it possible to reduce the Hasselmann equation to the nonlinear diffusion equation, whose solution describes the well-known effect of angular spreading of wave spectra on its rear face. (iii) Studying the isotropic Kolmogorov-Zakharov solution of the Hasselmann equation, we find numerically the values of Kolmogorov constants. (iv) We calculate the nonlinear damping of surface waves appearing due to four-wave interaction and compare the damping with the growth rate of the instability of the wave surface induced by the wind. It is found that for all known models of wind input, the nonlinear damping surpasses the instability at least in order of magnitude. This result, supported by numerical simulation of the Hasselmann equation, leads to the conclusion: in a real sea, except for the case of very young waves, four-wave interaction is the dominant process. This statement opens the way for the development of a well-justified analytical theory for the wind-driven sea.

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On optimal canonical variables in the theory of ideal fluid with free surface

Cited by 12 publications

References 15 publications

Numerical simulation of surface waves instability on a homogeneous grid

Numerical simulation of surface waves instability on a homogeneous grid

On ill-posedness of truncated series models for water waves

Energy balance in a wind-driven sea

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