Energy Spectra of Vortex Distributions in Two-Dimensional Quantum Turbulence

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 $\xi$. 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\gg \xi^{-1} $ having a universal $k^{-3}$ scaling arising from the vortex core structure, and an infrared regime $k\ll \xi^{-1}$ 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 $k\approx\xi^{-1}$, 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.

Publication
Physical Review X