paper

Energy spectra of vortex distributions in two-dimensional quantum turbulence

arXiv:1204.1103 · doi:10.1103/PhysRevX.2.041001

Abstract

We theoretically explore key concepts of two-dimensional turbulence in a homogeneous compressible superfluid described by a dissipative two-dimensional Gross-Pitaeveskii equation. Such a fluid supports quantized vortices that have a size characterized by the healing length . We show that for the divergence-free portion of the superfluid velocity field, the kinetic energy spectrum over wavenumber may be decomposed into an ultraviolet regime () having a universal scaling arising from the vortex core structure, and an infrared regime () with a spectrum that arises purely from the configuration of the vortices. The Novikov power-law distribution of intervortex distances with exponent -1/3 for vortices of the same sign of circulation leads to an infrared kinetic energy spectrum with a Kolmogorov power law, consistent with the existence of an inertial range. The presence of these and power laws, together with the constraint of continuity at the smallest configurational scale , allows us to derive a new analytical expression for the Kolmogorov constant that we test against a numerical simulation of a forced homogeneous compressible two-dimensional superfluid. The numerical simulation corroborates our analysis of the spectral features of the kinetic energy distribution, once we introduce the concept of a {\em clustered fraction} consisting of the fraction of vortices that have the same sign of circulation as their nearest neighboring vortices. Our analysis presents a new approach to understanding two-dimensional quantum turbulence and interpreting similarities and differences with classical two-dimensional turbulence, and suggests new methods to characterize vortex turbulence in two-dimensional quantum fluids via vortex position and circulation measurements.

arXiv comment19 pages, 8 figures

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