Ashton S. Bradley, Brian P. Anderson
We theoretically explore key concepts of two-dimensional turbulence in a homogeneous compressible super- fluid 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 wavenumber $k$ 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. A 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, 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 measurements.
View original:
http://arxiv.org/abs/1204.1103
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