Breakdown of wave turbulence and the onset of intermittency

Biven, Laura; Nazarenko, Sergey; Newell, Alan C.

doi:10.1016/s0375-9601(01)00016-0

Cited by 44 publications

(78 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…If this is the case, Figure 10: Vortices in the (x, y) plane at different times: t = 2500, t = 3250, t = 5000, t = 7500. one can expand the GP equation about the new equilibrium state, uniform condensate, use Bogoljubov transform to find new normal modes and a dispersion relation for them, equation (15), and to obtain a new WKE for this system that would be characterised by by three-wave interactions, (17). However, as we saw in Figure 6 the peak at small k remains quite broad that is the coherent condensate, if present, remains somewhat non-uniform.…”

Section: Late Condensation Stage: Acoustic Turbulencementioning

confidence: 94%

Wave turbulence and vortices in Bose–Einstein condensation

Nazarenko

Onorato

2006

Physica D: Nonlinear Phenomena

109

142

View full text Add to dashboard Cite

We report a numerical study of turbulence and Bose-Einstein condensation within the two-dimmensional GrossPitaevski model with repulsive interaction. In presence of weak forcing localized around some wave number in the Fourier space, we observe three qualitatively different evolution stages. At the initial stage a thermodynamic energy equipartition spectrum forms at both smaller and larger scales with respect to the forcing scale. This agrees with predictions of the the four-wave kinetic equation of the Wave Turbulence (WT) theory. At the second stage, WT breaks down at large scales and the interactions become strongly nonlinear. Here, we observe formation of a gas of quantum vortices whose number decreases due to an annihilation process helped by the acoustic component. This process leads to formation of a coherent-phase Bose-Einstein condensate. After such a coherent-phase condensate forms, evolution enters a third stage characterised by three-wave interactions of acoustic waves that can be described again using the WT theory. Background and motivationFor dilute gases with large energy occupation numbers the Bose-Einstein condensation (BEC) can be described by the Gross-Pitaevsky (GP) equation [1,2]:where Ψ is the condensate "wave function" (i.e. the c-number part of the boson annihilation field) and γ is an operator which models possible forcing and dissipation mechanisms which will be discussed later. Renewed interest to the nonlinear dynamics described by GP equation is related to relatively recent experimental discoveries of BEC [3,4,5]. GP equation also describes light behaviour in media with Kerr nonlinearities. In the nonlinear optics context it is usually called the Nonlinear Schoedinger (NLS) equation. It is presently understood, in both the nonlinear optics and BEC contexts, that the nonlinear dynamics described by GP equation is typically chaotic and often non-equilibrium [6,7,8,9]. Thus, it is best characterised as "turbulence" emphasizing its resemblance to the classical Navier-Stokes (NS) turbulence. On the other hand, the GP model has an advantage over NS because it has a weakly nonlinear limit in which the stochastic field evolution can be represented as a large set of weakly interactive dispersive waves. A systematic statistical closure is possible for such systems and the corresponding theory is called Wave Turbulence (WT) [10]. For small perturbations about the zero state in the GP model, WT closure predicts that the main nonlinear process will be four-wave resonant interaction. This closure was used in [6,8,9] to describe the initial stage of BEC. In the present paper we will examine this description numerically. We report that our numerics agree with the predicted by WT spectra at the initial evolution stage.

show abstract

Section: Late Condensation Stage: Acoustic Turbulencementioning

confidence: 94%

Wave turbulence and vortices in Bose–Einstein condensation

Nazarenko

Onorato

2006

Physica D: Nonlinear Phenomena

109

142

View full text Add to dashboard Cite

show abstract

“…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- …”

mentioning

confidence: 99%

Wave-turbulence theory of four-wave nonlinear interactions

et al. 2017

View full text Add to dashboard Cite

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-

show abstract

“…Note that the deviations from Gaussianity at low k correspond to the physical process of building a coherent condensate state. The results of [18,33] obtained for homogeneous GP turbulence will hold for trapped turbulence too because inhomogeneity has a neutral effect on the phase correlations. Indeed, according to the linear WKB equations the phases propagate unchanged along the rays.…”

Section: Let Us Introduce Notationsmentioning

confidence: 86%

“…A similar method has been used to investigate optical turbulence [13]. Classical weak turbulence theory yields a closed kinetic equation for the long time behavior of the energy spectrum without having to make unjustifiable assumptions about the statistics of the processes [14][15][16]18,22,24]. Second, the kinetic equation admits classes of exact equilibrium solutions [14,19,20].…”

Section: Introductionmentioning

confidence: 99%

“…Note that the first two terms on the RHS of this equation can be obtained one from another by exchanging k For homogeneous turbulence, the validity of this assumption was examined by Newell et al [18,33] who showed that initially Gaussian turbulence (characterized by random independent phases) remains Gaussian for the energy cascade range whereas in the particle cascade range deviations from Gaussianity grow toward low k values. However, these deviations remain small over a large range of k for small initial amplitudes and the random phase assumption can be used for these scales.…”

Section: Let Us Introduce Notationsmentioning

confidence: 99%

See 1 more Smart Citation

Wave turbulence in Bose–Einstein condensates

Lvov

Nazarenko

West

2003

Physica D: Nonlinear Phenomena

View full text Add to dashboard Cite

Abstract. The kinetics of nonequilibrium Bose-Einstein condensates are considered within the framework of the Gross-Pitaevskii equation. A systematic derivation is given for weak small-scale perturbations of a steady confined condensate state. This approach combines a wavepacket WKB description with the weak turbulence theory. The WKB theory derived in this paper describes the effect of the condensate on the short-wave excitations which appears to be different from a simple renormalization of the confining potential suggested in previous literature.

show abstract

Breakdown of wave turbulence and the onset of intermittency

Cited by 44 publications

References 8 publications

Wave turbulence and vortices in Bose–Einstein condensation

Wave turbulence and vortices in Bose–Einstein condensation

Wave-turbulence theory of four-wave nonlinear interactions

Wave turbulence in Bose–Einstein condensates

Contact Info

Product

Resources

About