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arXiv:1309.1797 · Physical Review B

Nonlocality in homogeneous superfluid turbulence

Owen Dix, Rena J. Zieve

Published Oct 21, 2014• 3 citations
arXivHTMLPDF

Abstract

Simulating superfluid turbulence using the localized induction approximation allows neighboring parallel vortices to proliferate. In many circumstances a turbulent tangle becomes unsustainable, degenerating into a series of parallel, noninteracting vortex lines. Calculating with the fully nonlocal Biot-Savart law prevents this difficulty but also increases computation time. Here we use a truncated Biot-Savart integral to investigate the effects of nonlocality on homogeneous turbulence. We find that including the nonlocal interaction up to roughly the spacing between nearest-neighbor vortex segments prevents the parallel alignment from developing, yielding an accurate model of homogeneous superfluid turbulence with less computation time.

DOI
10.1103/physrevb.90.144511

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Simulating superfluid turbulence using the localized induction approximation allows neighboring parallel vortices to proliferate. In many circumstances a turbulent tangle becomes unsustainable, degenerating into a series of parallel, noninteracting vortex lines. Calculating with the fully nonlocal Biot-Savart law prevents this difficulty but also increases computation time. Here we use a truncated Biot-Savart integral to investigate the effects of nonlocality on homogeneous turbulence. We find that including the nonlocal interaction up to roughly the spacing between nearest-neighbor vortex segments prevents the parallel alignment from developing, yielding an accurate model of homogeneous superfluid turbulence with less computation time.
A zero-temperature superfluid is arguably the simplest system in which to study complex fluid dynamics, such as turbulence. We describe computer simulations of such turbulence and compare the results directly with recent experiments in superfluid 3He-B^{3}\text{H}\text{e-B}3He-B. We are able to follow the entire process of the production, evolution, and decay of quantum turbulence. We find striking agreement between simulation and experiment and gain insights into the mechanisms involved.
Abstract We present a gauge-field extension of the Bose condensate model that describes T ≈ 0 superfluid turbulence generated by the macroscopic motion of the superfluid. We first establish that the condensate model is dual to the short-range interacting loop gas model, wherein the loops represent quantum vortex lines. Vortex lines form, interact and proliferate as a result of the superfluid motion. Our extension is based on incorporating the Biot–Savart interaction between vortex lines, which is lacking in the loop gas model. We show that the extended loop gas is dual to a Ginzburg–Landau model, wherein the gauge coupling is between the macroscopic velocity field of the superfluid and the condensate. Applying the model to cylindrical and pipe flows, we describe how turbulence transitions with and without intermediate vortex flow, respectively.
We argue that the physics of interacting Kelvin Waves (KWs) is highly nontrivial and cannot be understood on the basis of pure dimensional reasoning. A consistent theory of KW turbulence in superfluids should be based upon explicit knowledge of their interactions. To achieve this, we present a detailed calculation and comprehensive analysis of the interaction coefficients for KW turbuelence, thereby, resolving previous mistakes stemming from unaccounted contributions. As a first application of this analysis, we derive a local nonlinear (partial differential) equation. This equation is much simpler for analysis and numerical simulations of KWs than the Biot-Savart equation, and in contrast to the completely integrable local induction approximation (in which the energy exchange between KWs is absent), describes the nonlinear dynamics of KWs. Second, we show that the previously suggested Kozik-Svistunov energy spectrum for KWs, which has often been used in the analysis of experimental and numerical data in superfluid turbulence, is irrelevant, because it is based upon an erroneous assumption of the locality of the energy transfer through scales. Moreover, we demonstrate the weak nonlocality of the inverse cascade spectrum with a constant particle-number flux and find resulting logarithmic corrections to this spectrum.
Large scale numerical simulation of quantum turbulence is performed by using 3-D time-dependent Gross-Pitaevskii equation.The energy spectrum obeying Kolmogorov law and large scale self-similar structure of quantum vortex tangle are found in a fully developed dumped turbulent state.Width of the inertial range becomes large depending on the system size of the simulation that is consistent with the result of the normal fluid turbulence.On the other hand, bottleneck effect near coherent length prevents the inertial range from extending to smaller scale.
The quantization of vortex lines in superfluids requires the introduction of their density $\mathcal{L}(\mathbit{r},t)$ in the description of quantum turbulence. The space homogeneous balance equation for L(t)\mathcal{L}(t)L(t), proposed by Vinen on the basis of dimensional and physical considerations, allows a number of competing forms for the production term P\mathcal{P}P. Attempts to choose the correct one on the basis of time-dependent homogeneous experiments ended inconclusively. To overcome this difficulty we announce here an approach that employs an inhomogeneous channel flow which is very suitable to distinguish the implications of the various possible forms of the desired equation. We demonstrate that the originally selected form which was extensively used in the literature is in strong contradiction with our data. We therefore present a new form of an inhomogeneous equation for $\mathcal{L}(\mathbit{r},t)$ that is in agreement with our data and propose that it should be considered for further studies of superfluid turbulence.