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 wave number 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 for vortices of the same sign of circulation leads to an infrared kinetic-energy spectrum with a Kolmogorov power law, which is 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 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.
- Received 6 April 2012
DOI:https://doi.org/10.1103/PhysRevX.2.041001
This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Published by the American Physical Society
Popular Summary
One has only to watch the chaotic eddies in a rapidly flowing stream to realize that fluid turbulence is ubiquitous in nature, yet many of the underlying mechanisms remain mysterious. In quantum fluids, such as superfluid helium at low temperature, flow occurs only in discrete units called quantized vortices, which results in strong constraints on the allowed fluid behavior. This means that a much clearer understanding of turbulence may be achievable in quantum fluids, particularly if the dynamics of vortices are further limited by reducing the dimensionality. In our paper, we report a theoretical and computational study of energy flow in a two-dimensional superfluid and derive a method for calculating the so-called Kolmogorov constant that plays a central role in turbulence.
Compressibility in a superfluid such as a Bose-Einstein condensate allows vortices to emit and absorb sound waves, and the large size of the vortex cores raises the importance of understanding their structure. Our analysis of spatial vortex configurations reveals that ideal two-dimensional turbulence, associated with the transfer of energy between length scales, occurs when vortices of the same circulation sign cluster together with intervortex distances obeying a unique power law. This characteristic clustering is scale-free, a property shared by sand piles, earthquakes, forest fires, and other naturally occurring nonlinear dynamical processes, including turbulence. The kinetic energy of the quantum fluid can move to large length scales either through the spatial expansion of clusters, or through the accumulation of more vortices in clusters.
Surprisingly, the compressible nature of the superfluid also suggests a means to calculate the Kolmogorov constant, a parameter that determines how rapidly energy is transferred between scales. Simulation of a stirred superfluid supports the concept of vortex clustering into power-law configurations and the analytically obtained value of the Kolmogorov constant, offering a new quantum window into turbulence phenomena.
