Spectral energy transport in two-dimensional quantum vortex dynamics

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Spectral energy transport in two-dimensional quantum vortex dynamics

T. P. Billam, M. T. Reeves, and A. S. Bradley

Phys. Rev. A 91, 023615 – Published 17 February 2015

Abstract

We explore the possible regimes of decaying two-dimensional quantum turbulence, and elucidate the nature of spectral energy transport by introducing a dissipative point-vortex model with phenomenological vortex-sound interactions. The model is valid for a large system with weak dissipation, and also for systems with strong dissipation, and allows us to extract a meaningful and unambiguous spectral energy flux associated with quantum vortex motion. For weak dissipation and large system size we find a regime of hydrodynamic vortex turbulence in which energy is transported to large spatial scales, resembling the phenomenology of the transient inverse cascade observed in decaying turbulence in classical incompressible fluids. For strong dissipation the vortex dynamics are dominated by dipole recombination and exhibit no appreciable spectral transport of energy.

Received 20 November 2014

DOI:https://doi.org/10.1103/PhysRevA.91.023615

Authors & Affiliations

T. P. Billam^1,2,*, M. T. Reeves¹, and A. S. Bradley^1,†

¹Jack Dodd Centre for Quantum Technology, Department of Physics, University of Otago, Dunedin 9016, New Zealand
²Joint Quantum Centre (JQC) Durham-Newcastle, Department of Physics, Durham University, Durham, DH1 3LE, United Kingdom

^*Author to whom correspondence should be addressed: thomas.billam@durham.ac.uk
^†Author to whom correspondence should be addressed: abradley@physics.otago.ac.nz

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Vol. 91, Iss. 2 — February 2015

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Images

Figure 1
Regimes of decaying 2DQT following the decay of an unstable lattice of charge $κ_{i} \pm 6$ vortices, computed using the dPGPE (see also Supplemental Material [20] for movies of the time evolution). We show schematically the effects of dimensionless dissipation rate $γ$ and box size $L$ (see axes, top left) on the dynamics: We show the density (gray scale, background) and vortex configuration analyzed with the recursive cluster algorithm [4] (see key) at both short (i) and long (ii) times after the break-up of the lattice, compared to the characteristic timescale of the dynamics. In (e) we show the initial lattice for $L = 175$ . The regimes can be characterised as dissipative dipole gas dynamics (a, b), hydrodynamic vortex turbulence with energy transport to large scales (c), and strong wave turbulence with close vortex-sound coupling (d) [indicated schematically on axes (top left) by shaded region].
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Figure 2
Spectral energy transport in the phenomenological point-vortex model (see also Supplemental Material [20] for movies of the time evolution): For the parameters given in Fig. 1, (a)–(c) show ensemble and time-averaged incompressible energy spectrum (top), flux (bottom), and transfer function (bottom, inset). Straight lines show the universal ultraviolet spectrum $\tilde{E} = C {(k ξ)}^{- 3}$ and the infrared spectrum for randomly distributed vortices $\tilde{E} = C {(k ξ)}^{- 1} / Λ^{2}$ (see text). A Kolmogorov scaling law $\tilde{E} = C {(k ξ)}^{- 5 / 3}$ is also shown for reference. Time derivatives are scaled to the turnover time $τ$ (see text) and time averaging is done over a window of duration $0.63 τ$ $[0.75 τ]$ in (a) [(b), (c)], centered as indicated. Transfer functions and fluxes are shown for time $t \approx 2.62 τ$ . Vertical lines in (c) indicate the transition from negative to positive flux. The spectrum and flux in case (c) are broadly consistent with results expected for classical decaying turbulence [19].
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Figure 3
Accidental Kolmogorov $k^{- 5 / 3}$ scaling close to the vortex-core crossover: In (a), ensemble averaging over quiescent, nonturbulent, low-energy configurations of 20 vortices consisting mostly of vortex dipoles ( $L = 175$ , example inset) yields an appreciable apparent $k^{- 5 / 3}$ -scaling region directly connected to the $k^{- 3}$ vortex-core scaling. In (b), analytically removing the vortex-core scaling destroys this $k^{- 5 / 3}$ region, which for $γ = 10^{- 1}$ is clearly not related to a lossless inertial range (see inset). Curves as in Fig. 2 of the main text.
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