### 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 ξ.We show that, for the divergencefree portion of the superfluid velocity field, the kinetic-energy spectrum over wave number k may be decomposed into an ultraviolet regime (k ≫ ξ^{-1}) having a universal k^{-3} scaling arising from the vortex core structure, and an infrared regime (k ≪ ξ^{-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 ≈ ξ^{-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 twodimensional turbulence, and suggests new methods to characterize vortex turbulence in two-dimensional quantum fluids via vortex position and circulation measurements.

Original language | English (US) |
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Article number | 041001 |

Journal | Physical Review X |

Volume | 2 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2012 |

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### ASJC Scopus subject areas

- Physics and Astronomy(all)

### Cite this

*Physical Review X*,

*2*(4), [041001]. https://doi.org/10.1103/PhysRevX.2.041001