Joint statistics of amplitudes and phases in wave turbulence

Choi, Yeontaek; Lvov, Yuri V.; Nazarenko, Sergey

doi:10.1016/j.physd.2004.11.016

Cited by 60 publications

(94 citation statements)

References 22 publications

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“…Note that the wave turbulence approach permits the description of statistical objects which are more general than the spectra, in particular, the one-mode and multi-mode probability density functions (PDFs) [3,[74][75][76]. The evolution of the one-mode PDF is interesting when the initial statistics has random phases and amplitudes, but is not Gaussian.Also, it predicts solutions with fluxes in the probability space which describe intermittency -an anomalously high probability of strong waves [3,75].…”

Section: Long-time Statistical Evolution Of Weakly Non-linear Wave Symentioning

confidence: 99%

Wave turbulence

Nazarenko

2015

Contemporary Physics

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Wave turbulence is the statistical mechanics of random waves with a broadband spectrum interacting via non-linearity. To understand its difference from non-random well-tuned coherent waves, one could compare the sound of thunder to a piece of classical music. Wave turbulence is surprisingly common and important in a great variety of physical settings, starting with the most familiar ocean waves to waves at quantum scales or to much longer waves in astrophysics. We will provide a basic overview of the wave turbulence ideas, approaches and main results emphasising the physics of the phenomena and using qualitative descriptions avoiding, whenever possible, involved mathematical derivations. In particular, dimensional analysis will be used for obtaining the key scaling solutions in wave turbulence -Kolmogorov-Zakharov (KZ) spectra.

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Section: Long-time Statistical Evolution Of Weakly Non-linear Wave Symentioning

confidence: 99%

Wave turbulence

Nazarenko

2015

Contemporary Physics

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289

669

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“…More recently, this statistical framework has been nicely revisited using the diagrammatic technique [4] and performing analytical calculations, in the 3-wave case [22][23][24]. Interestingly, many experimental and theoretical results have shown that deviations from wave-turbulence predictions can be found for rare events, e.g.…”

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confidence: 99%

Wave-turbulence theory of four-wave nonlinear interactions

et al. 2017

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The Sagdeev-Zaslavski (SZ) equation for wave turbulence is analytically derived, both in terms of generating function and of multi-point pdf, for weakly interacting waves with initial random phases. When also initial amplitudes are random, the one-point pdf equation is derived. Such analytical calculations remarkably agree with results obtained in totally different fashions. Numerical investigations of the two-dimensional nonlinear Schrödinger equation (NLSE) and of a vibrating plate prove that: (i) generic Hamiltonian 4-wave systems rapidly attain a random distribution of phases independently of the slower dynamics of the amplitudes, vindicating the hypothesis of initially random phases; (ii) relaxation of the Fourier amplitudes to the predicted stationary distribution (exponential) happens on a faster timescale than relaxation of the spectrum (Rayleigh-Jeans distribution); (iii) the pdf equation correctly describes dynamics under different forcings: the NLSE has an exponential pdf corresponding to a quasi-gaussian solution, like the vibrating plates, that also show some intermittency at very strong forcings.Introduction Dispersive waves are ubiquitous in nature, and their nonlinear interactions make them intriguing and challenging [1,2]. Wave Turbulence is the theory that describes the statistical properties of large numbers of incoherent interacting waves, with tools such as the wave kinetic equation analytically derived in the late sixties. This equation describes the evolution of the wave spectrum in time, when homogeneity and weak nonlinearity are assumed [3][4][5]. It has been applied to numerous phenomena, including ocean waves [6][7][8], capillary waves [9, 10] Alfvén waves [11], optical waves [12] and solid oscillations [13][14][15][16][17][18]. It is the analogue of the Boltzmann equation for classical particles and it allows the RayleighJeans equilibrium state as well as non-equilibrium solutions, in terms of Kolmogorov-Zakharov (KZ) spectra [19].To characterise the invariant measure of the dynamics, that is to find the complete statistical description concerning all quantities of interest, an important step has been taken by Sagdeev and Zaslavski [20], who obtained the Brout-Prigogine equation for the probability density function (pdf) of wave turbulence [21]. More recently, this statistical framework has been nicely revisited using the diagrammatic technique [4] and performing analytical calculations, in the 3-wave case [22][23][24]. Interestingly, many experimental and theoretical results have shown that deviations from wave-turbulence predictions can be found for rare events, e.g. intermittency [8,[25][26][27][28][29]. This seems to be the case when a more general theoretical framework [30][31][32][33][34] is required, because the nonlinearities are not small [35,36].In this communication, the complete wave-turbulence theory is developed for a fully general 4-wave system, whose hamiltonian is expressed by the following canoni-

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“…Of course, to have a non-trivial description valid over the nonlinear evolution time, the fields must remain of the RPA type over the nonlinear time in the leading order in ǫ. The proof of this involves considering the full multi-particle PDF and will be published separately because it is rather lengthy and outside of the scope of the present paper [20]. Let us introduce a generating function Z(λ, t) = e λ|a k (t)| 2 , where λ is a real parameter.…”

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confidence: 99%

Probability densities and preservation of randomness in wave turbulence

2004

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Time evolution equation for the Probability Distribution Function (PDF) is derived for system of weakly interacting waves. It is shown that a steady state for such system may correspond to strong intermittency.Introduction -Wave Turbulence (WT) is a common name for the fields of dispersive waves which are engaged in stochastic weakly nonlinear interactions over a wide range of scales. Numerous examples of WT are found in oceans, atmospheres, plasmas and Bose-Einstein condensates [1][2][3][4][5][6][7][8][9]. For a long time, describing and predicting the energy spectra was the only concern in WT theory. More recently, some attention was given to the study of turbulence intermittency. WT intermittency, or "burstiness" of the turbulent signal, was observed experimentally and numerically and was attributed, as in most turbulent systems, to the presence of coherent structures. Examples include collapsing filaments in Bose-Einstein condensates with attractive potentials [9,10], condensate quasi-solitons in systems with repulsive potentials [9,11,12], white caps of sea waves at small scales [13], freak ocean waves at larger scales [14]. Often, such coherent structures are intense but quite sparse so that in most of the space waves remain weakly nonlinear and mostly unaffected by these structures.

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Joint statistics of amplitudes and phases in wave turbulence

Cited by 60 publications

References 22 publications

Wave turbulence

Wave turbulence

Wave-turbulence theory of four-wave nonlinear interactions

Probability densities and preservation of randomness in wave turbulence

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