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arXiv:1709.00671 · Physical review. B./Physical review. B

Dynamics of the vortex line density in superfluid counterflow turbulence

Dmytro Khomenko, Victor S. L’vov, Anna Pomyalov, Itamar Procaccia

Published Jan 16, 2018• 5 citations
arXivHTMLPDF

Abstract

Describing superfluid turbulence at intermediate scales between the intervortex distance and the macroscale requires an acceptable equation of motion for the density of quantized vortex lines $\mathcal{L}$. The closure of such an equation for superfluid inhomogeneous flows requires additional inputs besides $\mathcal{L}$ and the normal and superfluid velocity fields. In this paper, we offer a minimal closure using one additional anisotropy parameter ${I}_{l0}$. Using the example of counterflow superfluid turbulence, we derive two coupled closure equations for the vortex line density and the anisotropy parameter ${I}_{l0}$ with an input of the normal and superfluid velocity fields. The various closure assumptions and the predictions of the resulting theory are tested against numerical simulations.

DOI
10.1103/physrevb.97.014508

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Describing superfluid turbulence at intermediate scales between the intervortex distance and the macroscale requires an acceptable equation of motion for the density of quantized vortex lines L\mathcal{L}L. The closure of such an equation for superfluid inhomogeneous flows requires additional inputs besides L\mathcal{L}L and the normal and superfluid velocity fields. In this paper, we offer a minimal closure using one additional anisotropy parameter Il0{I}_{l0}Il0​. Using the example of counterflow superfluid turbulence, we derive two coupled closure equations for the vortex line density and the anisotropy parameter Il0{I}_{l0}Il0​ with an input of the normal and superfluid velocity fields. The various closure assumptions and the predictions of the resulting theory are tested against numerical simulations.
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.
Direct numerical simulations are used to examine the locking of quantized superfluid vortices and normal fluid vorticity in evolving turbulent flows. The superfluid is driven by the normal fluid, which undergoes either a decaying Taylor-Green flow or a linearly forced homogeneous isotropic turbulent flow, although the back reaction of the superfluid on the normal fluid flow is omitted. Using correlation functions and wavelet transforms, we present numerical and visual evidence for vortex locking on length scales above the intervortex spacing.
We consider superfluid turbulence near absolute zero of temperature generated by classical means, e.g., towed grid or rotation but not by counterflow. We argue that such turbulence consists of a polarized tangle of mutually interacting vortex filaments with quantized vorticity. For this system, we predict and describe a bottleneck accumulation of the energy spectrum at the classical-quantum crossover scale $\ensuremath{\ell}$. Demanding the same energy flux through scales, the value of the energy at the crossover scale should exceed the Kolmogorov-41 (K41) spectrum by a large factor ${\mathrm{ln}}^{10∕3}(\ensuremath{\ell}∕{a}_{0})$ ($\ensuremath{\ell}$ is the mean intervortex distance and a0{a}_{0}a0​ is the vortex core radius) for the classical and quantum spectra to be matched in value. One of the important consequences of the bottleneck is that it causes the mean vortex line density to be considerably higher than that based on K41 alone, and this should be taken into account in (re)interpretation of new (and old) experiments as well as in further theoretical studies.
Abstract Based on the theory of the thermodynamic equilibrium in a system of quantum vortices in superfluid liquids and in a Bose – Einstein condensate in the presence of a counterflow of normal and superfluid components, we investigate the structure of a chaotic tangle of quantum vortices in turbulent superfluid liquids. Using the characteristic functional method, the properties of hydrodynamic vortex filaments are examined. It is shown that the average curvature of the vortex lines is on the order of the intervortex distance, with the proportionality coefficient being independent of the counterflow velocity. It is found that the degree of anisotropy of the vortex loops does not depend on the applied counterflow velocity. The obtained results explain the origin of the anisotropy as well as the relationship between the curvature of the lines and the intervortex space and their dependence on the parameters of the problem.
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.
ABSTRACT We theoretically study the line density of quantum vortices in a superfluid turbulent wake based on the conjecture of quantum turbulence of a pure superfluid in the zero temperature limit. The conjecture states that a collection of many quantum vortices can mimic a classical eddy in the continuum approximation. By straightforwardly relating the Kolmogorov's similarity to this conjecture, we can estimate that the effective viscosity of a pure superfluid turbulence is comparable to the circulation quantum , the minimum circulation of quantum vortices. This result naturally leads to a definition of the superfluid Reynolds number, with the characteristic size and velocity , supported by the fact that the equation of motion of a pure superfluid reduces to the Euler equation for high . By using the classical description of the rate of energy injection or dissipation, we quantitatively evaluate a characteristic density of vortex lines in a superfluid turbulent wake and its spatial profile of grid turbulence in terms of the superfluid Reynolds number.