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arXiv:1008.3050 · Physical Review B

Energy spectra of quantum turbulence: Large-scale simulation and modeling

Narimasa Sasa, Takuma Kano, Masahiko Machida, Victor S. L’vov, Oleksii Rudenko, Makoto Tsubota

Published Aug 11, 2011• 55 citations
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Abstract

In a ${2048}^{3}$ simulation of quantum turbulence within the Gross-Pitaevskii equation, it is demonstrated that the large-scale motions have a classical Kolmogorov-1941 energy spectrum $E(k)\ensuremath{\propto}{k}^{\ensuremath{-}5/3}$, followed by an energy accumulation with $E(k)\ensuremath{\simeq}$ const at $k$ about the reciprocal mean intervortex distance. This behavior was predicted by the L'vov-Nazarenko-Rudenko bottleneck model of gradual eddy-wave crossover [L'vov, Nazarenko, and Rudenko, J. Low Temp. Phys. 153, 140 (2008)], further developed in the paper.

DOI
10.1103/physrevb.84.054525

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In a 20483{2048}^{3}20483 simulation of quantum turbulence within the Gross-Pitaevskii equation, it is demonstrated that the large-scale motions have a classical Kolmogorov-1941 energy spectrum E(k)\ensuremath∝k\ensuremath−5/3E(k)\ensuremath{\propto}{k}^{\ensuremath{-}5/3}E(k)\ensuremath∝k\ensuremath−5/3, followed by an energy accumulation with E(k)\ensuremath≃E(k)\ensuremath{\simeq}E(k)\ensuremath≃ const at kkk about the reciprocal mean intervortex distance. This behavior was predicted by the L'vov-Nazarenko-Rudenko bottleneck model of gradual eddy-wave crossover [L'vov, Nazarenko, and Rudenko, J. Low Temp. Phys. 153, 140 (2008)], further developed in the paper.
We numerically study two-dimensional quantum turbulence with a Gross-Pitaevskii model. With the energy initially accumulated at large scale, quantum turbulence with many quantized vortex points is generated. Due to the lack of enstrophy conservation in this model, direct energy cascade with a Kolmogorov-Obukhov energy spectrum $E(k)\ensuremath{\propto}{k}^{\ensuremath{-}5/3}$ is observed, which is quite different from two-dimensional incompressible classical turbulence in the decaying case. A positive value for the energy flux guarantees a direct energy cascade in the inertial range (from large to small scales). After almost all the energy at the large scale cascades to the small scale, the compressible kinetic energy realizes the thermodynamic equilibrium state without quantized vortices.
There is a growing interest in the relation between classical turbulence and quantum turbulence. Classical turbulence arises from complicated dynamics of eddies in a classical fluid. In contrast, quantum turbulence consists of a tangle of stable topological defects called quantized vortices, and thus quantum turbulence provides a simpler prototype of turbulence than classical turbulence. In this paper, we investigate the dynamics and statistics of quantized vortices in quantum turbulence by numerically solving a modified Gross-Pitaevskii equation. First, to make decaying turbulence, we introduce a dissipation term that works only at scales below the healing length. Second, to obtain steady turbulence through the balance between injection and decay, we add energy injection at large scales. The energy spectrum is quantitatively consistent with the Kolmogorov law in both decaying and steady turbulence. Consequently, this is the first study that confirms the inertial range of quantum turbulence.
Abstract Quantum turbulence — the chaotic, non-equilibrium state of a superfluid threaded by a dense tangle of quantized vortex lines — provides a uniquely clean testbed for the universal statistical mechanics of turbulent energy transfer. We develop a theoretical account of the energy cascade across all relevant length scales: from macroscopic, quasi-classical Richardson–Kolmogorov eddies, through the dimensionally frustrated L’vov–Nazarenko–Rudenko (LNR) bottleneck at the mean inter-vortex spacing, down to the weakly nonlinear Kelvin-wave cascade on individual filaments, and finally to the zero-temperature dissipation cutoff. Starting from the Gross–Pitaevskii equation and its Madelung hydrodynamic decomposition, we separate the compressible and incompressible components of the flow to isolate the vortical and acoustic energy channels. Vortex reconnection is identified as the microscopic mechanism coupling these regimes, and the competing predictions of the Vortex Filament Model and the Gross–Pitaevskii equation for reconnection dynamics are compared. We review the theoretical controversy between the Kozik–Svistunov six-wave cascade and the L’vov–Nazarenko four-wave cascade, together with the associated tilt-symmetry debate, and present the finite-temperature two-fluid (HVBK) description, showing how mutual friction locks the normal and superfluid components at large scales while damping Kelvin waves at small scales. PACS numbers: 67.25.dk, 47.37.+q, 03.75.Kk, 47.32.C−
Large scale numerical simulation of quantum turbulence is performed by using 3-D time-dependent Gross-Pitaevskii equation.The energy spectrum obeying Kolmogorov law and large scale self-similar structure of quantum vortex tangle are found in a fully developed dumped turbulent state.Width of the inertial range becomes large depending on the system size of the simulation that is consistent with the result of the normal fluid turbulence.On the other hand, bottleneck effect near coherent length prevents the inertial range from extending to smaller scale.
Quantum turbulence is numerically studied by solving the Gross‐Pitaevskii equation. By introducing both energy dissipation at small scales and energy injection at large scales, we succeed in obtaining steady turbulence sustained by a balance between energy injection and energy dissipation. The energy spectrum in the inertial range takes the form of the Kolmogorov spectrum and the energy flux is consistent with the energy dissipation rate at small scales. These results reveal the properties of quantum turbulence in the inertial range, allowing us to propose a prototype of turbulence much simpler than conventional classical turbulence.