Comment on "Dynamics of the Density of Quantized Vortex-Lines in Superfluid Turbulence"

In the paper by Khomenko et al. [Phys. Rev. B \textbf{91}, 180504 (2015)] the authors, analyzing numerically the steady counterflowing helium in inhomogeneous channel flow, concluded that the production term $\mathcal{P}$ in the Vinen equation is proportional to $\left\vert \mathbf{V}_{ns}\right\vert ^{3}\mathcal{L}^{1/2}$ (where $\mathcal{L}$ is vortex line density and $\mathbf{V}_{ns}$ is the counterflow velocity). In present comment we demonstrated that the procedure, implemented by the authors includes a number of questionable steps, such as a decomposition of velocity of line and interpretation of the flux term. Additionally, the overall strategy - extracting information on the temporal behavior from the stationary solution also remains questionable. Because of that the method of determination of the explicit shape of Vinen equation is very sensitive to the listed elements, the final conclusion of the authors cannot be considered as unambiguous.Comment: 3 page

Coarse-grained Hydrodynamics of turbulent superfluids: HVBK approach and the bundle structure of the vortex tangle

In the comment I develop a critical analysis of the use of the HVBK method for the study of three-dimensional turbulent flows of superfluids. The conception of the vortex bundles forming the structure of quantum turbulence is controversial and does not justify the use of the HVBK method. In addition, this conception is counterproductive, because it gives incorrect ideas about the structure of the vortex tangle as a set of bundles containing parallel lines. The only type of dynamics of vortex filaments inside these bundles is possible, namely, Kelvin waves running along the filaments. At the same time, as shown in numerous numerical simulations, a vortex tangle consists of a set of entangled vortex loops of different sizes and having a random walk structure. These loops are subject to large deformations (due to highly nonlinear dynamics), they reconnect with each other and with the wall, split and merge, creating a lot of daughter loops. They also bear Kelvin waves on them, but the latter have little impact. I also propose and discuss an alternative variant of study of three-dimensional turbulent flows, in which the vortex line density $\mathcal{L}(r,t)$ is not associated with $\nabla \times \mathbf{v}_{s}$, but it is an independent variable described by a separate equation.Comment: 6 pages, 40 ref. There are some changes after discussion with the referee

Reconnection of vortex filaments and Kolmogorov spectrum

The energy spectrum of the 3D velocity field, induced by collapsing vortex filaments is studied. One of the aims of this work is to clarify the appearance of the Kolmogorov type energy spectrum $E(k)\varpropto k^{-5/3}$, observed in many numerical works on discrete vortex tubes (quantized vortex filaments in quantum fluids). Usually, explaining classical turbulent properties of quantum turbulence, the model of vortex bundles, is used. This model is necessary to mimic the vortex stretching, which is responsible for the energy transfer in classical turbulence. In our consideration we do not appeal to the possible "bundle arrangement" but explore alternative idea that the turbulent spectra appear from singular solution, which describe the collapsing line at moments of reconnection. One more aim is related to an important and intensively discussed topic - a role of hydrodynamic collapse in the formation of turbulent spectra. We demonstrated that the specific vortex filament configuration generated the spectrum $E(k)$ close to the Kolmogorov dependence and discussed the reason for this as well as the reason for deviation. We also discuss the obtained results from point of view of the both classical and quantum turbulence.Comment: 4 pages,4 figure

Langevin dynamics of vortex lines in the counterflowing He II. Talk given at the Low Temperature Conference, Kazan, 2015

The problem of the statistics of a set of chaotic vortex lines in a counterflowing superfluid helium is studied. We introduced a Langevin-type force into the equation of motion of the vortex line in presence of relative velocity $\mathbf{v_{ns}}$. This random force is supposed to be Gaussian satisfying the fluctuation-dissipation theorem. The corresponding Fokker-Planck equation for probability functional in the vortex loop configuration space is shown to have a solution in the form of Gibbs distribution with the substitution E\{\mathbf{s\}\rightarrow }E(\{\mathbf{% s\}-P(v_{n}-v_{s})}, where $E\{\mathbf{s\}}$ is the energy of the vortex configuration $\{\mathbf{s\}}$, and $\mathbf{P}$ is the Lamb impulse. Some physical consequences of this fact are discussed.\\ \newline PACS numbers: 47.32.C- (Vortex dynamics) 47.32.cf (Vortex reconnection and rings), 47.37.+q (Hydrodynamic aspects of superfluidity)Comment: 4 pages, talk given at the Low Temperature Conference, Kazan, 201

On the Nonuniform Quantum Turbulence in Superfluids

The problem of quantum turbulence in a channel with an inhomogeneous counterflow of superfluid turbulent helium is studied. \ The counterflow velocity $V_{ns}^{x}(y)$ along the channel is supposed to have a parabolic profile in the transverse direction $y$. Such statement corresponds to the recent numerical simulation by Khomenko et al. [Phys. Rev. B \textbf{91}, 180504 (2015)]. The authors reported about a sophisticated behavior of the vortex line density (VLD) $\mathcal{L}(\mathbf{r},t)$, different from $\mathcal{L}\propto V_{ns}^{x}(y)^{2}$, which follows from the naive, straightforward application of the conventional Vinen theory. It is clear, that Vinen theory should be refined by taking into account transverse effects and the way it ought to be done is the subject of active discussion in the literature. In the work we discuss several possible mechanisms of the transverse flux of VLD $\mathcal{L}(\mathbf{r},t)$ which should be incorporated in the standard Vinen equation to describe adequately the inhomogeneous quantum turbulence (QT). It is shown that the most effective among these mechanisms is the one that is related to the phase slippage phenomenon. The use of this flux in the modernized Vinen equation corrects the situation with an unusual distribution of the vortex line density, and satisfactory describes the behavior $\mathcal{L}(\mathbf{r},t)$ both in stationary and nonstationary situations. The general problem of the phenomenological Vinen theory in the case of nonuniform and nonstationary quantum turbulence is thoroughly discussed.Comment: 7 pages, 3 figure

Chaotic Quantum Vortexes In A Weakly Non Ideal Bose Gas. Thermodynamic Equilibrium And Turbulence

We study the stochastic behavior of a set of chaotic vortex loops appeared in imperfect Bose gas. Dynamics of Bose-gas is supposed to obey Gross-Pitaevskii equation with additional noise satisfying fluctuation-dissipation relation. The corresponding Fokker-Planck equation for probability functional has solution ${\cal P}(\{{\psi}({\bf r})\})={\cal N}\exp (-H\{{\psi}({\bf r)}\} /T),$ where $H\{{\psi}({\bf r})\}$ is the Ginzburg-Landau free energy. Considering vortex filaments as topological defects of field ${\psi}({\bf r})$ we derive a Langevin-type equation of motion of the line with the correspondingly transformed stirring force. The respective Fokker-Planck equation for probability functional ${\cal P}(\{{\bf s}(\xi)\})$ in vortex loop configuration space is shown to have a solution in the form of ${\cal P}(\{{\bf s}(\xi)\})={\cal N}\exp (-H\{{\bf s}\} /T),$ where ${\cal N}$ is the normalizing factor and $H\{{\bf s}\}$ is energy of vortex line configurations. Analyzing this result we discuss possible reasons for destruction of the thermodynamic equilibrium and follow the mechanisms of transition to the turbulent stateComment: 10 pages, RevTeX, submitted to JLT

Diffusive Decay of the Vortex Tangle and Kolmogorov turbulence in quantum fluids

The idea that chaotic set of quantum vortices can mimic classical turbulence, or at least reproduce many main features, is currently actively being developed. Appreciating significance of the challenging problem of the classical turbulence it can be expressed that the idea to study it in terms of quantized line is indeed very important and may be regarded as a breakthrough. For this reason, this theory should be carefully scrutinized. One of the basic arguments supporting this point of view is the fact that vortex tangle decays at zero temperature, when the apparent mechanism of dissipation (mutual friction) is absent. Since the all possible mechanisms of dissipation of the vortex energy, discussed in the literature, are related to the small scales, it is natural to suggest that the Kolmogorov cascade takes the place with the flow of the energy, just as in the classical turbulence. In a series of recent experiment attenuation of vortex line density was observed and authors attribute this decay to the properties of the Kolmogorov turbulence. In the present work we discuss alternative possibility of decay of the vortex tangle, which is not related to dissipation at small scales. This mechanism is just the diffusive like spreading of the vortex tangle. We discuss a number of the key experiments, considering them both from the point of view of alternative explanation and of the theory of Kolmogorov turbulence in quantum fluids.Comment: The work was presented at SUR 2010, submitted in JLT

Applications of Gaussian model of the vortex tangle in the superfluid turbulent HeII

In spite of an appearance of some impressive recent results in understanding of the superfluid turbulence in HeII they fail to evaluate many characteristics of vortex tangle needed for both applications and fundamental study. Early we reported the Gaussian model of the vortex tangle in superfluid turbulent HeII. That model is just trial distribution functional in space of vortex loop configurations constructed on the basis of well established properties of vortex tangle. It is designed to calculate various averages taken over stochastic vortex loop configurations. In this paper we use this model to calculate some important characteristics of the vortex tangle. In particular we evaluate the average superfluid mass current J induced by vortices and the average energy E associated with the chaotic vortex filament.Comment: latex, 7 pages, 1 Postscript figure, uses subeqnar.sty, sprmindx.sty, cropmark.sty, physprbb.sty, svmult.cls, to be published in "Quantized vortex dynamics and superfluid turbulence", Ed. C. Barenghi (Springer Verlag, Berlin, 2001

Statistical signature of vortex filaments in classic turbulence: dog or tail?

The title of this paper echoes the title of a paragraph in the famous book by Frisch on classical turbulence. In the relevant chapter, the author discusses the role of the statistical dynamics of vortex filaments in the fascinating problem of turbulence and the possibility of a breakthrough in constructing an advanced theory. This aspect arose due to the large amount of evidence, both experimental and numerical, that the vorticity field in turbulent flows has a pronounced filamentary structure. In fact, there is unquestionably a strong relationship between the dynamics of chaotic vortex filaments and turbulent phenomena. However, the question arises as to whether the basic properties of turbulence (cascade, scaling laws. etc.) are a consequence of the dynamics of the vortex filaments (the `dog' concept), or whether the latter have only a marginal significance (the `tail' concept). Based on well-established results regarding the dynamics of quantized vortex filaments in superfluids, we illustrate how these dynamics can lead to the main elements of the theory of turbulence. We cover key topics such as the exchange of energy between different scales, the possible origin of Kolmogorov-type spectra and the free decay behavior.Comment: 19 pages, 4 figure

Energy spectrum of the 3D velocity field, induced by vortex tangle

A review of various exactly solvable models on the determination of the energy spectra $E (k)$ of 3D-velocity field, induced by chaotic vortex lines is proposed. This problem is closely related to the sacramental question whether a chaotic set of vortex filaments can mimic the real hydrodynamic turbulence. The quantity $$ can be exactly calculated, provided that we know the probability distribution functional $\mathcal{P}(\{\mathbf{s}(\xi,t)\})$ of vortex loops configurations. The knowledge of $\mathcal{P}(\{\mathbf{s}(\xi,t)\})$ is identical to the full solution of the problem of quantum turbulence and, in general, $\mathcal{P}$ is unknown. In the paper we discuss several models allowing to evaluate spectra in the explicit form. This cases include standard vortex configurations such as a straight line, vortex array and ring. Independent chaotic loops of various fractal dimension as well as interacting loops in the thermodynamic equilibrium also permit an analytical solution. We also describe the method of an obtaining the 3D velocity spectrum induced by the straight line perturbed with chaotic 1D Kelvin waves on it.Comment: 7 pages, 1 figure. Paper is submitted to JLTP, Proceedings of QFS 2012 (Lancaster 2012