Warm turbulence in the Boltzmann equation

Proment, Davide; Nazarenko, Sergey; Asinari, Pietro; Onorato, Miguel

doi:10.1209/0295-5075/96/24004

Cited by 3 publications

(2 citation statements)

References 18 publications

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“…The authors of [50] suggest that in such situations, we might expect to observe finite temperature cascade solutions. Such "warm cascades" were studied in the example of the Boltzmann kinetics in [130] and in the BEC WT context in [56]. These solutions are predominantly thermodynamic ones similar to (48), but with a correction which is small in the inertial range and which causes a sharp falloff at the dissipation scale.…”

Section: The Differential Approximation and The Cascade Directionsmentioning

confidence: 99%

One-dimensional optical wave turbulence: Experiment and theory

et al. 2012

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We present a review of the latest developments in one-dimensional (1D) optical wave turbulence (OWT). Based on an original experimental setup that allows for the implementation of 1D OWT, we are able to show that an inverse cascade occurs through the spontaneous evolution of the nonlinear field up to the point when modulational instability leads to soliton formation. After solitons are formed, further interaction of the solitons among themselves and with incoherent waves leads to a final condensate state dominated by a single strong soliton. Motivated by the observations, we develop a theoretical description, showing that the inverse cascade develops through six-wave interaction, and that this is the basic mechanism of nonlinear wave coupling for 1D OWT. We describe theory, numerics and experimental observations while trying to incorporate all the different aspects into a consistent context. The experimental system is described by two coupled nonlinear equations, which we explore within two wave limits allowing for the expression of the evolution of the complex amplitude in a single dynamical equation. The long-wave limit corresponds to waves with wave numbers smaller than the electrical coherence length of the liquid crystal, and the opposite limit, when wave numbers are larger. We show that both of these systems are of a dual cascade type, analogous to two-dimensional (2D) turbulence, which can be described by wave turbulence (WT) theory, and conclude that the cascades are induced by a six-wave resonant interaction process. WT predicts several stationary solutions (non-equilibrium and thermodynamic) to both the long-and short-wave systems, and we investigate the necessary conditions required for their realization. Interestingly, the long-wave system is close to the integrable 1D nonlinear Schrödinger equation (NLSE) (which contains exact nonlinear soliton solutions), and as a result during the inverse cascade, nonlinearity of the system at low wave numbers becomes strong. Subsequently, due to the focusing nature of the nonlinearity, this leads to modulational instability (MI) of the condensate and the formation of solitons. Finally, with the aid of the the probability density function (PDF) description of WT theory, we explain the coexistence and mutual interactions between solitons and the weakly nonlinear random wave background in the form of a wave turbulence life cycle (WTLC).

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Section: The Differential Approximation and The Cascade Directionsmentioning

confidence: 99%

One-dimensional optical wave turbulence: Experiment and theory

et al. 2012

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“…This imply checkig that the collision integral in the kinetic equation (13) converges on the KZ spectra n(k) = ck −α . As these solutions are scale-invariant, the integral is easily written in the ω space as takes into account the 3D average of δ(k 1 + k 2 − k 3 − k 4 ) over the solid angles Ω 1 , Ω 2 , Ω 3 and Ω 4 , see [3,33,51,52] for details. In this coordinate system the frequency δ-function can be easily used, for example as ω 2 = ω 3 +ω 4 −ω 1 .…”

Section: Appendix B Locality Of Interactionsmentioning

confidence: 99%

Sustained turbulence in the three-dimensional Gross–Pitaevskii model

Proment

Nazarenko

Onorato

2012

Physica D: Nonlinear Phenomena

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We study the 3D forced-dissipated Gross-Pitaevskii equation. We force at relatively low wave numbers, expecting to observe a direct energy cascade and a consequent power-law spectrum of the form k −α . Our numerical results show that the exponent α strongly depends on how the inverse particle cascade is attenuated at k's lower than the forcing wave number. If the inverse cascade is arrested by a friction at low k's, we observe an exponent which is in good agreement with the weak wave turbulence prediction k −1 . For a hypo-viscosity, a k −2 spectrum is observed which we explain using a critical balance argument. In simulations without any low-k dissipation, a condensate at k = 0 is growing and the system goes through a strongly-turbulent transition from a four-wave to a three-wave weak turbulence acoustic regime with k −3/2 Zakharov-Sagdeev spectrum. In this regime, we also observe a spectrum for the incompressible kinetic energy which formally resembles the Kolmogorov k −5/3 , but whose correct explanation should be in terms of the Kelvin wave turbulence. The probability density functions for the velocities and the densities are also discussed.

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