Evolution of a random inhomogeneous field of nonlinear deep-water gravity waves

Crawford, Donald R.; Saffman, P. G.; Yuen, Henry C.

doi:10.1016/0165-2125(80)90029-3

Cited by 116 publications

(91 citation statements)

References 7 publications

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“…The interaction coefficients V^(k,ki,k 2 ) are lengthy algebraic expressions, given incorrectly in Yuen and Lake's Appendix A but correctly in Appendix A of Crawford et al [2], as…”

Section: General Formulationmentioning

confidence: 99%

“…This formulation was later adopted and expounded in greater detail by Crawford, Saffman & Yuen [2], Crawford, Lake, Saffman & Yuen [3] and Yuen & Lake [6]; see also Craik [1] for a brief account. Not surprisingly, solutions of Zakharov's equations are available only for special cases, but these known solutions often improve upon previous approximations.…”

Section: Introductionmentioning

confidence: 99%

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Interction of a short-wave field with a dominant long wave in deep water: derivation form Zakharov's spectral formulation

Craik

1988

J. Aust. Math. Soc. Series B, Appl. Math

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The leading-order interaction of short gravity waves with a dominant long-wave swell is calculated by means of Zakharov's [7] spectral formulation. Results are obtained both for a discrete train of short waves and for a localised wave packet comprising a spectrum of short waves.The results for a discrete wavetrain agree with previous work of LonguetHiggins Si Stewart [5], and general agreement is found with parallel work of Grimshaw [4] which employed a very different wave-action approach.

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“…The interaction coefficients V^(k,ki,k 2 ) are lengthy algebraic expressions, given incorrectly in Yuen and Lake's Appendix A but correctly in Appendix A of Crawford et al [2], as…”

Section: General Formulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Interction of a short-wave field with a dominant long wave in deep water: derivation form Zakharov's spectral formulation

Craik

1988

J. Aust. Math. Soc. Series B, Appl. Math

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“…The expressions for matrix elements 0 1 2 3 , , , M are well known (e.g., Hasselmann, 1962;Webb, 1968;Zakharov, 1968; see also Badulin et al, 2005), but we use the elements from Crawford et al (1980). In any representation, formulas for elements 0 1 2 3 , , , M are very cumbersome, so they are not given here.…”

Section: S( ) N( ) ( / G )N( )mentioning

confidence: 99%

Asymptotes of the nonlinear transfer and wave spectrum in the frame of the kinetic equation solution

Polnikov

Teng

2018

Preprint

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Abstract. The kinetic equation for a gravity wave spectrum is solved numerically to study the high frequencies asymptotes 10 for the one-dimensional nonlinear energy transfer and the variability of spectrum parameters that accompany the long-term evolution of nonlinear waves. The cases of initial two-dimensional spectra S(ω,θ) of modified JONSWAP type with the frequency decay-law S(ω) ~ ω -n (for n = 6, 5, 4 and 3.5) and various initial functions of the angular distribution are considered. It is shown that at the first step of the kinetic equation solution, the nonlinear energy transfer asymptote has the power-like decay-law, Nl(ω) ~ ω -p , with values p ≤ n-1, valid in cases when n ≥ 5, and the difference, n-p, changes 15 significantly when n approaches 4. On time scales of evolution greater than several thousands of initial wave periods, in every case, a self-similar spectrum S sf (ω,θ) is established with the frequency decay-law of form S(ω) ~ ω -4 . Herein, the asymptote of nonlinear energy transfer becomes negative in value and decreases according to the same law (i.e., Nl(ω)~-ω -4 ).The peak frequency of the spectrum, ω p (t), migrates in time t to the low-frequency region such that the angular and frequency characteristics of the two-dimensional spectrum S sf (ω,θ) remain constant. However, these characteristics depend 20 on the degree of angular anisotropy of the initial spectrum. The solutions obtained are interpreted, and their connection with the analytical solutions of the kinetic equation by Zakharov and co-authors for gravity waves in water is discussed.

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“…(4) is rather complicated (see, for example, Hasselmann, 1962, or, in more convenient form, Crawford et al, 1980). In addition to this, the multifold integration in Eq.…”

Section: Introductionmentioning

confidence: 99%

On the problem of optimal approximation of the four-wave kinetic integral

Polnikov

Farina

2002

Nonlin. Processes Geophys.

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Abstract. The problem of optimization of analytical and numerical approximations of Hasselmann's nonlinear kinetic integral is discussed in general form. Considering the general expression for the kinetic integral, a principle to obtain the optimal approximation is formulated. From this consideration it follows that the most well-accepted approximations, such as Discrete Interaction Approximation (DIA) (Hasselmann et al., 1985), Reduced Integration Approximation (RIA) (Lin and Perry, 1999), and the Diffusion Approximation proposed recently in Zakharov and Pushkarev (1999) (ZPA), have the same roots. The only difference among them is, essentially, the choice of the 4-wave configuration for the interacting waves. To evaluate a quality of any approximation for the 2-D nonlinear energy transfer, a mathematical measure of relative error is constructed and the meaning of approximation efficiency is postulated. By the use of these features it is shown that DIA has better accuracy and efficiency than ZPA. Following to the general idea of optimal approximation and by using the measures introduced, more efficient and faster versions of DIA are proposed.

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Evolution of a random inhomogeneous field of nonlinear deep-water gravity waves

Cited by 116 publications

References 7 publications

Interction of a short-wave field with a dominant long wave in deep water: derivation form Zakharov's spectral formulation

Interction of a short-wave field with a dominant long wave in deep water: derivation form Zakharov's spectral formulation

Asymptotes of the nonlinear transfer and wave spectrum in the frame of the kinetic equation solution

On the problem of optimal approximation of the four-wave kinetic integral

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