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arXiv:1711.08614 · Physical Review A

Finite-temperature effects in helical quantum turbulence

Patricio Clark Di Leoni, Pablo D. Mininni, Marc Brächet

Published Apr 30, 2018• 7 citations
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Abstract

We perform a study of the evolution of helical quantum turbulence at different temperatures by solving numerically the Gross-Pitaevskii and the stochastic Ginzburg-Landau equations, using up to ${4096}^{3}$ grid points with a pseudospectral method. We show that for temperatures close to the critical one, the fluid described by these equations can act as a classical viscous flow, with the decay of the incompressible kinetic energy and the helicity becoming exponential. The transition from this behavior to the one observed at zero temperature is smooth as a function of temperature. Moreover, the presence of strong thermal effects can inhibit the development of a proper turbulent cascade. We provide Ans\"atze for the effective viscosity and friction as a function of the temperature.

DOI
10.1103/physreva.97.043629

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We perform a study of the evolution of helical quantum turbulence at different temperatures by solving numerically the Gross-Pitaevskii and the stochastic Ginzburg-Landau equations, using up to 40963{4096}^{3}40963 grid points with a pseudospectral method. We show that for temperatures close to the critical one, the fluid described by these equations can act as a classical viscous flow, with the decay of the incompressible kinetic energy and the helicity becoming exponential. The transition from this behavior to the one observed at zero temperature is smooth as a function of temperature. Moreover, the presence of strong thermal effects can inhibit the development of a proper turbulent cascade. We provide Ans"atze for the effective viscosity and friction as a function of the temperature.
We study freely decaying quantum turbulence by performing high-resolution numerical simulations of the Gross-Pitaevskii equation (GPE) in the Taylor-Green geometry. We use resolutions ranging from 10243{1024}^{3}10243 to 40963{4096}^{3}40963 grid points. The energy spectrum confirms the presence of both a Kolmogorov scaling range for scales larger than the intervortex scale $\ensuremath{\ell}$, and a second inertial range for scales smaller than $\ensuremath{\ell}$. Vortex line visualizations show the existence of substructures formed by a myriad of small-scale knotted vortices. Next, we study finite-temperature effects in decaying quantum turbulence by using the stochastic Ginzburg-Landau equation to generate thermal states, and then by evolving a combination of these thermal states with the Taylor-Green initial conditions under the GPE. We use finite-temperature GPE simulations to extract mean-free path by measuring the spectral broadening in the Bogoliubov dispersion relation that we obtain from the spatiotemporal spectra, and use it to quantify the effective viscosity as a function of the temperature. Finally, we perform low-Reynolds-number simulations of the Navier-Stokes equations, in order to compare the decay of high-temperature quantum flows with their classical counterparts, and to further calibrate the estimations of the effective viscosity (based on the mean-free-path computations).
While in classical turbulence helicity depletes nonlinearity and can alter the evolution of turbulent flows, in quantum turbulence its role is not fully understood. We present numerical simulations of the free decay of a helical quantum turbulent flow using the Gross-Pitaevskii equation at high spatial resolution. The evolution has remarkable similarities with classical flows, which go as far as displaying a dual transfer of incompressible kinetic energy and helicity to small scales. Spatiotemporal analysis indicates that both quantities are dissipated at small scales through nonlinear excitation of Kelvin waves and the subsequent emission of phonons. At the onset of the decay, the resulting turbulent flow displays polarized large scale structures and unpolarized patches of quiescence reminiscent of those observed in simulations of classical turbulence at very large Reynolds numbers.
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−
We numerically model decaying quantum turbulence in two-dimensional disk-shaped Bose--Einstein condensates, and investigate the effects of finite temperature on the turbulent dynamics. We prepare initial states with a range of condensate temperatures, and imprint equal numbers of vortices and antivortices at randomly chosen positions throughout the fluid. The initial states are then subjected to unitary time-evolution within the c-field methodology. For the lowest condensate temperatures, the results of the zero temperature Gross--Pitaevskii theory are reproduced, whereby vortex evaporative heating leads to the formation of Onsager vortex clusters characterised by a negative absolute vortex temperature. At higher condensate temperatures the dissipative effects due to vortex--phonon interactions tend to drive the vortex gas towards positive vortex temperatures dominated by the presence of vortex dipoles. We associate these two behaviours with the system evolving toward an anomalous non-thermal fixed point, or a Gaussian thermal fixed point, respectively.
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.
We examine the energy cascades and quantum vortex structures in two-dimensional quantum turbulence through a special unitary time evolution algorithm. An early attempt at using the Lattice Boltzmann Method proved successful in correctly representing some features of the Nonlinear Schrodinger System (NLS), such as the phase shift following the one-dimensional soliton-soliton collision, as well as the two-dimentional modulation instability. However, to accurately evaluate NLS, the implicit Euler method is required to resolve the time evolution, which is computationally expensive. A more accurate and efficient method, the Quantum Lattice Gas model is employed to simulate the quantum turbulence governed by the Gross-Pitaevskii equation, an equaiton that describes the evolution of the ground state wave function for a Bose-Einstein condensate (BEC). It is discovered that when the ratio of the internal energy to the kinetic energy is below 0.05, an unexpected short Poincare recurrence occurs independent of the initial profile of the wave function. It is demonstrated that this short recurrence is destroyed as the internal energy is strengthened. to compare the two-dimensional quantum turbulence with its classical counterpart, the incompressible energy spectra of quantum turbulence is analyzed. However, the result reveals no sign of dual cascades which is a hallmark of the classical incompressible two-dimensional fluid (inverse energy cascade to large scales with a direct cascade of enstrophy to small scales). It is the spectra of the compressible energy that can exhibits multiple cascades, but this is strongly dependent on the initial condition.