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arXiv:1207.7296 · Physical Review Letters

Vortex-Density Fluctuations, Energy Spectra, and Vortical Regions in Superfluid Turbulence

Andrew William Baggaley, Jason Laurie, Carlo F. Barenghi

Published Nov 14, 2012• 94 citations
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Abstract

Measurements of the energy spectrum and of the vortex-density fluctuation spectrum in superfluid turbulence seem to contradict each other. Using a numerical model, we show that at each instance of time the total vortex line density can be decomposed into two parts: one formed by metastable bundles of coherent vortices, and one in which the vortices are randomly oriented. We show that the former is responsible for the observed Kolmogorov energy spectrum, and the latter for the spectrum of the vortex line density fluctuations.

DOI
10.1103/physrevlett.109.205304

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Measurements of the energy spectrum and of the vortex-density fluctuation spectrum in superfluid turbulence seem to contradict each other. Using a numerical model, we show that at each instance of time the total vortex line density can be decomposed into two parts: one formed by metastable bundles of coherent vortices, and one in which the vortices are randomly oriented. We show that the former is responsible for the observed Kolmogorov energy spectrum, and the latter for the spectrum of the vortex line density fluctuations.
Measurements of the energy spectrum and of the vortex-density fluctuation spectrum in superfluid turbulence seem to contradict each other. Using a numerical model, we show that at each instance of time the total vortex line density can be decomposed into two parts: one formed by metastable bundles of coherent vortices, and one in which the vortices are randomly oriented. We show that the former is responsible for the observed Kolmogorov energy spectrum, and the latter for the spectrum of the vortex line density fluctuations.
We discuss a recent experiment in which the spectrum of the vortex line density fluctuations has been measured in superfluid turbulence. The observed frequency dependence of the spectrum, f − 5/3 , disagrees with classical vorticity spectra if, following the literature, the vortex line density is interpreted as a measure of the vorticity or enstrophy. We argue that the disagrement is solved if the vortex line density field is decomposed into a polarised field (which carries most of the energy) and an isotropic field (which is responsible for the spectrum).
We discuss a recent experiment in which the spectrum of the vortex line density fluctuations has been measured in superfluid turbulence. The observed frequency dependence of the spectrum, f- 5/3, disagrees with classical vorticity spectra if, following the literature, the vortex line density is interpreted as a measure of the vorticity or enstrophy. We argue that the disagrement is solved if the vortex line density field is decomposed into a polarised field (which carries most of the energy) and an isotropic field (which is responsible for the spectrum). © Europhysics Letters Association.
Superfluid turbulence has been one of the most important problems of superfluid hydrodynamics, in which quantized vortices play a significant role. The recent research on superfluid turbulence enters a new stage rather different from the old ones chiefly devoted to thermal counterflow. After describing the current motivation on these topics, we discuss our research which studies the energy spectrum both by the vortex filament model and the Gross-Pitaevskii model. Both energy spectra are consistent with the Kolmogorov law, which shows a close similarity between superfluid (quantum) and classical turbulence.
We study two dimensional superfluid turbulence by employing an effective description valid in the limit where the density of superfluid vortices is parametrically small. At sufficiently low temperatures the effective description yields an inverse cascade with Kolmogorov energy spectrum E(k)∼k−5/3E(k) \sim k^{-5/3}E(k)∼k−5/3. Denoting the number of vortices as a function of time by N(t)N(t)N(t), we find that the vortex annihilation rate scales like N˙∼N5/3\dot N \sim N^{5/3}N˙∼N5/3 in states with an inverse cascade and N˙∼N2\dot N \sim N^2N˙∼N2 for laminar flow.