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arXiv:1702.05229 · Physical Review Letters

Vortex Thermometry for Turbulent Two-Dimensional Fluids

Andrew J. Groszek, Matthew J. Davis, David M. Paganin, Kristian Helmerson, Tapio Simula

Published Jan 19, 2018• 30 citations
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Abstract

We introduce a new method of statistical analysis to characterize the dynamics of turbulent fluids in two dimensions. We establish that, in equilibrium, the vortex distributions can be uniquely connected to the temperature of the vortex gas, and we apply this vortex thermometry to characterize simulations of decaying superfluid turbulence. We confirm the hypothesis of vortex evaporative heating leading to Onsager vortices proposed in Phys. Rev. Lett. 113, 165302 (2014)PRLTAO0031-900710.1103/PhysRevLett.113.165302, and we find previously unidentified vortex power-law distributions that emerge from the dynamics.

DOI
10.1103/physrevlett.120.034504

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•Jan 22, 2018•3 citations
We introduce a new method of statistical analysis to characterize the dynamics of turbulent fluids in two dimensions. We establish that, in equilibrium, the vortex distributions can be uniquely connected to the temperature of the vortex gas, and we apply this vortex thermometry to characterize simulations of decaying superfluid turbulence. We confirm the hypothesis of vortex evaporative heating leading to Onsager vortices proposed in Phys. Rev. Lett. 113, 165302 (2014)PRLTAO0031-900710.1103/PhysRevLett.113.165302, and we find previously unidentified vortex power-law distributions that emerge from the dynamics.
We have theoretically [1-5] and experimentally [6] studied statistical mechanics of vortices in two-dimensional superfluids in confined geometries. We are particularly interested in the part of the phase diagram where negative absolute temperature states of the vortex degrees of freedom are realized. The vortex dynamics is driven by two opposing mechanisms: a vortex evaporative heating due to vortex-antivortex annihilations drives the system toward low entropy negative temperature states while the coupling to the  phonon bath results in cooling of the vortices driving the system toward vortex dipole dominated positiver temperature states. Furthermore, we have observed, both theoretically and experimentally, signatures of anomalous dynamical scaling and evolution toward non-thermal fixed points. For sufficiently high energies the Onsager vortex clusters are predicted to undergo Einstein-Bose condensation - a negative temperature counterpart to the Berezinskii-Kosterlitz-Thouless transition. [1] T Simula, MJ Davis, K Helmerson, Emergence of Order from Turbulence in an Isolated Planar Superfluid , Physical Review Letters 113, 165302, (2014). [2] AJ Groszek, TP Simula, DM Paganin, K Helmerson, Onsager vortex formation in Bose-Einstein condensates in two-dimensional power-law traps, Physical Review A 93, 043614, (2016). [3] RN Valani, AJ Groszek, TP Simula, Einstein–Bose condensation of Onsager vortices , New Journal of Physics 20, 053038, (2018). [4] AJ Groszek, MJ Davis, DM Paganin, K Helmerson, TP Simula, Vortex thermometry for turbulent two-dimensional fluids, Physical Review Letters 120, 034504, (2018). [5] AJ Groszek, MJ Davis, TP Simula, Decaying quantum turbulence in a two-dimensional Bose-Einstein condensate at finite temperature , arXiv:1903.05528. [6] Shaun P Johnstone, Andrew J Groszek, Philip T Starkey, Christopher J Billington, Tapio P Simula, Kristian Helmerson, Order from chaos: Observation of large-scale flow from turbulence in a two-dimensional superfluid, arXiv:1801.06952.
We have studied computationally turbulent non-equilibrium dynamics of quasi-two-dimensional superfluid Bose-Einstein condensates. Beginning with a random out-of-equilibrium distribution of vortices and antivortices in the condensate, we have observed emergence of ordered vortex structures corresponding to absolute negative temperature states in this conservative Hamiltonian system. We explain the spontaneous self-organization of the singly quantized vortices into two``Onsager vortex''clusters of like-signed vortices in terms of an``evaporative heating''mechanism of the vortex gas. These results provide a clear pathway to observing Onsager vortex structures and their unusual thermodynamics in a superfluid Bose gas
Critical dynamics of an ultracold Bose gas far from equilibrium is studied in two spatial dimensions. Superfluid turbulence is created by quenching the equilibrium state close to zero temperature. Instead of immediately rethermalizing, the system approaches a meta-stable transient state, characterized as a nonthermal fixed point. A focus is set on the vortex density and vortex-antivortex correlations which characterize the evolution towards the nonthermal fixed point and the departure to final (quasi-)condensation. Two distinct power-law regimes in the vortex-density decay are found and discussed in terms of a vortex unbinding process and a kinetic description of vortex scattering. A possible relation to decaying turbulence in classical fluids is pointed out. By comparing the results to equilibrium studies of a two-dimensional Bose gas, an intuitive understanding of the location of the nonthermal fixed point in a reduced phase space is developed.
Nonthermal fixed points of the dynamics of a dilute degenerate Bose gas far from thermal equilibrium are analyzed in two and three spatial dimensions. Universal power-law distributions, previously found within a nonperturbative quantum-field theoretical approach and recently shown to be related to vortical dynamics and superfluid turbulence [Phys. Rev. B 84, 020506(R) (2011)], are studied in detail. The results imply an interpretation of the scaling behavior in terms of independent vortex excitations of the superfluid and show that the statistics of topological excitations can be described in the framework of wave turbulence. The particular scaling exponents observed in the single-particle momentum distributions are found to be consistent with irreversibility as well as conservation laws obeyed by the wave interactions. Moreover, long-wavelength acoustic excitations of the vortex-bearing condensate, driven by vortex annihilations, are found to follow a nonthermal power law. Considering vortex correlations in a statistical model, the long-time departure from the nonthermal fixed point is related to vortex-antivortex pairing. The studied nonthermal fixed points are accessible in cold-gas experiments. The results shed light on fundamental aspects of superfluid turbulence and have strong potential implications for related phenomena, for example, in early universe inflation or quark-gluon plasma dynamics.
Despite the prominence of Onsager's point-vortex model as a statistical description of 2D classical turbulence, a first-principles development of the model for a realistic superfluid has remained an open problem. Here we develop a mapping of a system of quantum vortices described by the homogeneous 2D Gross-Pitaevskii equation (GPE) to the point-vortex model, enabling Monte Carlo sampling of the vortex microcanonical ensemble. We use this approach to survey the full range of vortex states in a 2D superfluid, from the vortex-dipole gas at positive temperature to negative-temperature states exhibiting both macroscopic vortex clustering and kinetic energy condensation, which we term an Onsager-Kraichnan condensate (OKC). Damped GPE simulations reveal that such OKC states can emerge dynamically, via aggregation of small-scale clusters into giant OKC clusters, as the end states of decaying 2D quantum turbulence in a compressible, finite-temperature superfluid. These statistical equilibrium states should be accessible in atomic Bose-Einstein condensate experiments.
Turbulence in classical fluids is a ubiquitous non-equilibrium phenomenon, yet a complete theoretical description for turbulent flow remains a challenging problem. A useful simplification for ideal two-dimensional (2D) fluids is to describe the turbulent flow with long-range-interacting point vortices, each possessing quantised circulation. In 1949, Onsager applied statistical mechanics to determine the equilibria of this model. He showed that at sufficiently high energies, like-circulation vortices preferentially aggregate into large-scale clusters, and are characterised by a negative absolute temperature. Onsager's theory has been highly influential, providing understanding of diverse quasi-2D systems such as turbulent soap films, guiding-centre plasmas, and self-gravitating systems. It also predicts the striking tendency of 2D turbulence to spontaneously form large-scale, long-lived vortices -- Jupiter's Great Red Spot is a well-known example. However, Onsager's theory doesn't quantitatively apply to classical fluids where vorticity is continuous, and experimental systems demonstrating Onsager's point-vortex statistical mechanics have remained elusive. Here we realise high energy, negative-temperature vortex clusters in a uniform superfluid Bose-Einstein condensate. Our results confirm Onsager's prediction of negative temperature clustered phases of quantum vortices, and demonstrate the utility of point-vortex statistical mechanics in 2D quantum fluids. This work opens future directions for the study of turbulent dynamics and we anticipate exploring the entire phase diagram of 2D quantum vortices, including the formation of clusters from 2D quantum turbulence.