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arXiv:2003.05305 · Physical Review Letters

Abrupt Transition between Three-Dimensional and Two-Dimensional Quantum Turbulence

Nicolás P. Müller, Marc Brächet, Alexandros Alexakis, Pablo D. Mininni

Published Mar 31, 2020• 18 citations
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Abstract

We present numerical evidence of a critical-like transition in an out-of-equilibrium mean-field description of a quantum system. By numerically solving the Gross-Pitaevskii equation we show that quantum turbulence displays an abrupt change between three-dimensional (3D) and two-dimensional (2D) behavior. The transition is observed both in quasi-2D flows in cubic domains (controlled by the amplitude of a 3D perturbation to the flow), as well as in flows in thin domains (controlled by the domain aspect ratio) in a configuration that mimics systems realized in laboratory experiments. In one regime the system displays a transfer of the energy towards smaller scales, while in the other the system displays a transfer of the energy towards larger scales and a coherent self-organization of the quantized vortices.

DOI
10.1103/physrevlett.124.134501

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We present numerical evidence of a critical-like transition in an out-of-equilibrium mean-field description of a quantum system. By numerically solving the Gross-Pitaevskii equation we show that quantum turbulence displays an abrupt change between three-dimensional (3D) and two-dimensional (2D) behavior. The transition is observed both in quasi-2D flows in cubic domains (controlled by the amplitude of a 3D perturbation to the flow), as well as in flows in thin domains (controlled by the domain aspect ratio) in a configuration that mimics systems realized in laboratory experiments. In one regime the system displays a transfer of the energy towards smaller scales, while in the other the system displays a transfer of the energy towards larger scales and a coherent self-organization of the quantized vortices.
We investigate the dynamic transition of quantum turbulence (QT) in a confined potential field as the system evolves from purely two-dimensional (2D) to quasi-two-dimensional, and ultimately to three-dimensional (3D), by fixing the lateral dimensions of the trapping box while varying its height. In the 2DQT, distinct Onsager vortex cluster formation and inverse energy cascade are observed, while 3DQT exhibits a direct energy cascade consistent with the Vinen turbulence decay rate, which display striking differences. By systematically altering the system height, we explore how dimensionality drives the differentiation of turbulence types and find that this transition is closely related to the excitation of Kelvin waves. Kelvin waves not only introduce additional dissipation mechanisms but also serve as mediators for direct energy transfer across scales. When the wavelength of the permitted Kelvin waves exceeds the critical size of vortex clusters, turbulence begins transitioning to 3D type, culminating in fully developed 3DQT at the characteristic scale. In the transitional region, we observe continuous variations in the decay rate and vortex cluster correlation functions.
Finite-temperature quantum turbulence is often described in terms of two immiscible fluids that can flow with a nonzero-mean relative velocity. Such out-of-equilibrium state is known as counterflow superfluid turbulence. We report here the emergence of a counterflow-induced inverse energy cascade in three-dimensional superfluid flows by performing extensive numerical simulations of the Hall-Vinen-Bekarevich-Khalatnikov model. As the intensity of the mean counterflow is increased, an abrupt transition, from a fully three-dimensional turbulent flow to a quasi-two-dimensional system exhibiting a split cascade, is observed. The findings of this work could motivate new experimental settings to study quasi-two-dimensional superfluid turbulence in the bulk of three-dimensional experiments. They might also find applications beyond superfluids in systems described by more than one fluid component.
Transition to condensate state in the degenerate gas of bosons is studied using the Gross-Pitaevskii equation. It is shown that adiabatic invariance of an enstrophy (mean squared vorticity) based on a generalized vorticity curl(∣ψ∣2v)/∣ψ∣2curl (|ψ|^2 {\bf v})/|ψ|^2curl(∣ψ∣2v)/∣ψ∣2, controls dynamics of the transitional turbulence just before formation of the condensate with its tangles of quantized vortices (∣ψ∣2v|ψ|^2 {\bf v}∣ψ∣2v is the weighted velocity field defined on the macroscopic wave function ψψψ). Scaling of the angle-averaged occupation number spectrum, Nk∼k−1N_k \sim k^{-1}Nk​∼k−1, has been obtained for the free developing transitional turbulence for the weak nonlinear (and completely disordered) initial conditions. Results of the three-dimensional numerical simulations have been used to support the theoretical consideration.
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 investigate numerically the statistics of quantized vortices in two-dimensional quantum turbulence using the Gross-Pitaevskii equation. We find that a universal -5/3 scaling law in the turbulent energy spectrum is intimately connected with the vortex statistics, such as number fluctuations and vortex velocity, which is also characterized by a similar scaling behavior. The -5/3 scaling law appearing in the power spectrum of vortex number fluctuations is consistent with the scenario of passive advection of isolated vortices by a turbulent superfluid velocity generated by like-signed vortex clusters. The velocity probability distribution of clustered vortices is also sensitive to spatial configurations, and exhibits a power-law tail distribution with a -5/3 exponent.
We discuss the configurations of vortices in two-dimensional quantum turbulence, studying energy spectrum of superfluid velocity and correlation functions with the distance between two vortices. We apply the above method to quantum turbulence described by Gross-Pitaevskii equation in Bose-Einstein condensates. We make two-dimensional quantum turbulence from many dark solitons through the dynamical instability. A dark soliton is unstable and decays into vortices in two- and three-dimensional systems. In our work, we propose a method of discriminating between the uncorrelated turbulence and the correlated turbulence. We decompose the energy spectrum into two terms, namely the self-energy spectrum E self (k) made by individual vortices and the interactive energy spectrum E int (k) made by interference of two vortices. The uncorrelated turbulence is defined as turbulence with E int (k)≪E self (k), while the correlated turbulence is turbulence where E int (k) is not much smaller than E self (k). Our simulations show that in the decay of dark solitons, the vortices created consist of correlated pairs of opposite circulation vortices, leading to the correlated turbulence.