Energy Spectra of Vortex Distributions in Two-Dimensional Quantum Turbulence

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 xi. We show that, for the divergence-free portion of the superfluid velocity field, the kinetic-energy spectrum over wave number k may be decomposed into an ultraviolet regime (k >> xi(-1)) having a universal k(-3) scaling arising from the vortex core structure, and an infrared regime (k >> xi(-1)) 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 k(-5/3) power law, which is consistent with the existence of an inertial range. The presence of these k(-3) and k(-5/3) power laws, together with the constraint of continuity at the smallest configurational scale k approximate to xi(-1), 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 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.