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

TL;DR

This paper critically examines the HVBK approach for superfluid turbulence, arguing it misrepresents vortex structures and proposing an alternative model where vortex line density is treated independently.

Contribution

It challenges the vortex bundle concept in superfluid turbulence and introduces a new framework with independent vortex line density dynamics.

Findings

HVBK method misrepresents vortex tangle structure.

Vortex tangle consists of entangled loops, not bundles.

Proposes an alternative model with independent vortex line density.

Abstract

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.

Equations

$${\rho }_n\frac{\partial {\mathbf{v}}_n}{\partial t}+{\rho }_n({\mathbf{v}}_n\cdot \nabla ){\mathbf{v}}_n=-\frac{{\rho }_n}{\rho }\nabla p_n-{\rho }_{s}S\nabla T+{\mathbf{F}}_{mf}+\eta {\nabla }^2{\mathbf{v}}_n,$$

$${\rho }_s\frac{\partial {\mathbf{v}}_s}{\partial t}+{\rho }_s({\mathbf{v}}_s\cdot \nabla ){\mathbf{v}}_s=-\frac{{\rho }_s}{\rho }\nabla p_s+{\rho }_{s}S\nabla T-{\mathbf{F}}_{mf}$$

$${\mathbf{F}}_{mf}=\mathcal{L}<{\mathbf{f}}_{MF}>=\alpha {\rho }_s\kappa \mathcal{L}⟨{\mathbf{s}}^{\prime }\times [{\mathbf{s}}^{\prime }\times ({\mathbf{v}}_{ns}-{\mathbf{v}}_{\mathbf{i}})]⟩+{\alpha }^{\prime }{\rho }_s\kappa \mathcal{L}⟨{\mathbf{s}}^{\prime }\times ({\mathbf{v}}_{ns}-{\mathbf{v}}_{\mathbf{i}})⟩.$$

$$\nabla \times {\mathbf{v}}_s=\kappa \mathcal{L}$$

$${\mathbf{F}}_{mf}^{(HVBK)}={\rho }_s\alpha \hat{\omega }\times [\omega \times (\mathbf{v}-\tilde{\beta }\nabla \times \hat{\omega })]+{\rho }_s{\alpha }^{\prime }\omega \times (\mathbf{v}-\tilde{\beta }\nabla \times \hat{\omega }),$$

$${\mathbf{F}}_{mf}\propto A(T)(v_n-v_s{)}^2({\mathbf{v}}_n-{\mathbf{v}}_s).$$

Full text

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 equipollent variable described by a separate equation.

I Introduction

The concept of quantum turbulence began with the well-known experiment by Gorter and Mellink (GM)on heat exchange in superfluid helium and Feynman’s theory, which explains the GM experiment on the basis of the dynamics of a set of chaotic quantized vortices, or vortex tangle Gorter and Mellink (1949),Feynman (1955). Since then, the concepts of quantum turbulence as a chaotic tangle of vortex filaments have expanded and appeared in many fields of physics, ranging from superfluids and cold atoms to heavy ions, and neutron stars (see e.g. INT Program INT-19-1a, URL http://www.int.washington.edu/PROGRAMS/19-1a/).

The experimental studies of superfluid turbulence is mainly based on hydrodynamic methods. These include, for example, the study of the vortex tangle using the first and second sounds, the measurement of the temperature drop or the pressure field, the investigation of fluctuating velocity fields (via studying pressure fluctuation), etc. On the other hand, most of the known methods of creating superfluid turbulence also have the hydrodynamic origin. These are the generation of a vortex tangle by a flow (counterflow), by powerful sound fields, using oscillating devices etc. Therefore, knowledge how to describe hydrodynamic processes in superfluids in the presence of the vortex tangle is very actual and important problem.

The article discusses the problem of construction of the coarse-grained hydrodynamics of turbulent flows of superfluids. In particular, I conduct a critical analysis of the use of the HVBK (Hall-Vinen-Bekarevich-Khalatnikov) method for the study of three-dimensional flows of superfluids. Indeed, this approach, which relates the vortex , line density $\mathcal{L}(r,t)$ with the coarse-garined vorticity $\nabla\times\mathbf{v}_{s}$ via the favor Feynman rule, was initially elaborated and is only suitable for rotating stationary cases. Meanwhile, at present, the use of this method for three-dimensional unsteady flows is a widespread practice. Sometime this is done without justification, sometimes they refer to the so-called vortex bundle structure of quantum turbulence. The conception of the vortex bundles forming the structure of quantum turbulence is also critically discussed in the paper. I also propose an alternative variant in which the vortex line density $\mathcal{L}(r,t)$ is not associated with $\nabla\times\mathbf{v}_{s}$, but it is an independent and equipollent variable described by a separate equation. The structure of paper is following. The next, second section is devoted to general formulation of coarse-grained hydrodynamics of superfluids in presence of the vortex tangle. In the third Section the problem of the use of the HVBK methods for rotating superfluids and in the three -dimensional flows are discussed. An alternative variant of study of three-dimensional turbulent flow is described in fourth Section.

II Coarse-grained Hydrodynamics of turbulent superfluids.

In presence of the vortex filaments the two-fluid hydrodynamics of the superfluid helium should be modernized and be represented as follows:

[TABLE]

[TABLE]

We assume that the motion of both components is incompressible, $\nabla\cdot\mathbf{v}_{n}=0$, $\nabla\cdot\mathbf{v}_{s}=0$. and where $\mathbf{v}_{n}$ and $\mathbf{v}_{s}$ are the coarse-grained velocity of the normal and superfluid component (averaged over a small volume $\mathcal{V}$), $p_{n}$ and $p_{s}$ are the effective pressures acting on the normal and the superfluid component ($\nabla p_{n}=\nabla p+(\rho_{s}/2)\nabla v_{ns}^{2}$ and $\nabla p_{s}=\nabla p-(\rho_{n}/2)\nabla v_{ns}^{2}$), $p$ is the total pressure, $S$ is the entropy, $T$ is the absolute temperature, $\eta$ is the dynamic viscosity of the normal component, and $\mathbf{v}_{ns}=\mathbf{v}_{n}-\mathbf{v}_{s}$.The effects of the vortices on the two components (normal and superfluid) are described by the friction force exerted by the superfluid component on the normal component $\mathbf{F}_{mf}$ .When these forces are averaged over all vortices inside the small volume $\mathcal{V}$, then the following expression of $\mathbf{F}_{mf}$ is obtained (see e.g. Schwarz (1988))

[TABLE]

In this equation $\mathbf{s^{\prime}(}\xi\mathbf{)}$ is the tangent vector along the vortex filaments $\mathbf{s(}\xi\mathbf{)}$, composing the vortex tangle, $\alpha,\alpha^{\prime}$ are temperature-dependent dimensionless mutual friction parameters, $\mathbf{v_{i}}$.is the self-induced velocity of the vortex filament. Quantity $\mathcal{L}$ is the vortex line density, averaging $\left\langle\cdot\right\rangle$ is performed over various configurations of vortex filaments. Equations (1) - (3) are coarse-grained equations, hence the inclusion of the effects of the vortex lines requires a high vortex line density per unit volume. These equations had been written and discussed for a long time, a classic variant, close to the stated above can be found in the book Donnelly (1991).

In the written form these equations are common to any type of flow, such as counterflow, flow past obstacles, acoustic waves, etc. The main difference for different types of flows is the determination of the vortex line density $\mathcal{L}$ entering into the expression for the macroscopic mutual friction force (3), and the choice of averaging method $\left\langle\cdot\right\rangle$.

Another question concerns the temperature range for applying equations (1) - (3), In general, the quantum turbulence is the chaotic dynamics of three strongly interacting nonlinear fields. These are the motion of the normal and superfluid component and stochastic evolution of a set of vortex filaments (vortex tangle). The type of interaction depends of the temperature $T$ (via mutual friction). For large $T$ the coupling is strong, both components are clumped and move together. As a result, we got almost the one-fluid hydrodynamics. The case of very low temperature is very interesting because it is ideal for testing the idea of whether the dynamics of discrete vortices (quantized vortex lines) can imitate classical turbulence. The study of superfluid turbulence for intermediate temperatures is not suitable for this purpose (due to the presence of a normal component). Quantum turbulence in superfluid helium (for intermediate temperatures) is rather the separate problem, not identical to classical turbulence. And it is precisely this case that requires the study of two coupled Navier-Stokes Equations. Thus, we can say that we study the intermediate temperature range.

The above equations (1) - (3) are a point of consensus among physicists. Disagreements begin with the question how to fulfil averaging and to treat variable $\mathcal{L}$. Here we discuss two main ways to perform these procedures. They are the Hall-Vinen-Bekarevich-Khalatnikov (HVBK) model (see e.g., book by Khalatnikov, Khalatnikov (1965) 1965) and the Hydrodynamics of superfluid turbulence model (Nemirovskii & Lebedev, 1983 Nemirovskii and Lebedev (1983),Nemirovskii and Fiszdon (1995))

III HVBK approach

III.1 HVBK approach for rotating superfluids

The Hall-Weinen-Bekarevich-Khalatnikov (HVBK) model (see, for example, the book Khalatnikov Khalatnikov (1965) became the basis for the mathematical formalism of the hydrodynamics of rotating superfluids. As is well known (see Feynman (1955) , in a vessel rotating with an angular velocity $\Omega$, appears a regular array of vortex filaments with a density $n=2\Omega/\kappa$. Such a distribution of vortices creates an average coarse-grained superfluid velocity $\left\langle\mathbf{v}_{s}\right\rangle$, which satisfies the condition of solid body rotation $\left\langle\mathbf{v}_{s}\right\rangle=\mathbf{\Omega\times r}$. The vorticity filed $\mathbf{\omega}$ is $\mathbf{\omega}=2\mathbf{\Omega}$. Therefore, the density of the vortex filaments in this case can be related to the vorticity field by the following relation:

[TABLE]

Due to the smallness of the quantum of circulation $\kappa$, even a relatively weak rotational speed produces a high density of vortex lines. Thus, it is possible to construct  a coarse-grained hydrodynamics of hydrodynamic equations, which the average contribution of many individual vortex lines and incorporate their contribution to the macroscopic evolutionary equations for superfluid and normal He II velocities.

Combining (3) with (4) one get expression for average coarse-garined mutual friction $\mathbf{F}_{mf}^{(HVBK)}$

[TABLE]

where $\mathbf{\omega}=\nabla\times\mathbf{v}_{s}$ is the averaged superfluid vorticity, $\hat{\omega}=\mathbf{\omega}/|\omega|$.  Thus, the question of elimination of the vortex line density is resolved with the use of the Feynman rule, it allows to study various problems of coarse-grained dynamics of rotating supefluids (see, e.g. book be Sonin Sonin (2016)).

III.2 HVBK approach for three -dimensional flows

The HVBK model is a fruitfull and elegant approach, however it is principally assigned for rotating superfluids. Nevertheless this approach, which uses ansatz $\nabla\times\mathbf{v}_{s}=\kappa\mathcal{L}$ (4),  and the force $\mathbf{F}_{mf}^{(HVBK)}$ (5) is widely used for numerical and analytical studies of coarse-grained hydrodynamic problems of turbulent superfluid in three -dimensional situations.

This approach seems to be unfounded. In my opinion, there is no way to apply it to three-dimensional hydrodynamics. Anticipating the objections of readers, I would like to discuss the usual arguments for using the HVBK method. Probably the only argument is the conviction that the vortex tangle consists of so-called vortex bundles, unifying many near parallel vortex filaments.

There are few papers (see, e.g. Baggaley et al. (2012), Sasa et al. (2011)), where the authors claim of the existence of the bundles. In fact, they only demonstrated how, with the help of statistical analysis, one can get a small polarization (prevailing of one direction over the other) in the vortex tangle. But, firstly, it is just a statistical effect and, secondly, under no circumstances this small polarization allows to use the Feynman rule (4), which is crucially needed in the HVBK equations. To use this ansatz, it is necessary that all filaments in the vortex tangle are involved in the rotation. But if the polarization is partial, then there is a lot of free (randomly orientated) vortex filaments that contribute to mutual friction and do not contribute to the Feynman rule (4). In addition, these chaotic lines interact with polarized lines, thereby destroying the polarization and, correspondingly, the quasi-bundle structure.

There are other examples of observation of vortex bundles (in numerical works), when they are artificially prepared structure, or are initiated by eddies of normal component (see, e.g. Samuels (1993),Samuels (1993)). However, there are works in the literature, in which it is stated that even if the vortex bundles are artificially created, they can be destroyed rather soon. For instance, G. Volovik Volovik (2004)) have shown that at low temperatures, where the mutual friction is small, the existence of the bundles is impossible. They should melt, changing into a highly irregular structure. Other example is a series of numerical simulations by Kivotides Kivotides (2011), Kivotides (2012),Kivotides (2014),Kivotides (2018)), who studied the exact (not HVBK) dynamics of quantum vortices in the turbulent flows (at finite temperature) and concluded ”that the results do not show that a turbulent normal-fluid with a Kolmogorov energy spectrum induces superfluid vortex bundles in the superfluid”. In paper Kivotides (2014) Kivotides reported about observation of clusters with weakly polarized vortex lines and associated Kolmogorov - type spectra $E(k)\propto k^{5/3}$. To some measure this is expected result since the appearance of the Kolmogorov spectrum requires formation of coherent structures. However, the partial polarization also prevents the use of closure $\nabla\times v_{s}=\kappa L$ in the mutual friction, since ALL the lines contribute into friction, and partial polarization (if any) includes A SMALL FRACTION of the total vortex line density.

In this regard, it seems appropriate to discuss question of relation of the vortex bundle arrangement and the Kolmogorov-type spectrum. For the uniform array of vortex filaments the coarse grained velocity field is $\mathbf{v(r)=\Omega\times r}$. Accordingly the Fourier transform $\mathbf{v(k)}$ scales as $1/k^{-3}$. This implies that two-dimensional spectrum $E(\mathbf{k})=$ $dE/d^{2}\mathbf{k}$ should behave as $1/k^{-6}$  and the isotropic spectrum $E(k)$ depends on absolute value of the wave vector $k$ as $E(k)\varpropto 1/k^{5}$. Thus, we state that the uniform vortex bundles do not generate the Kolmogorov type spectra. It can be shown that the nonuniform vortex array with the distribution of density of vortex filaments $n(r)=\Delta N/\Delta r\varpropto 1/r^{-2/3}$ does generate Kolmogorov type spectra $E(k)\varpropto 1/k^{5/3}$ (see Kozik and Svistunov (2009), Nemirovskii (2015)).

Of course, for very strong mutual friction vortex filaments can be completely trapped by the eddies of normal component and follow the dynamics of normal fluid. In fact, in this situation the coarse- grained hydrodynamics become the one-fluid dynamics, when both components move together.

There are many physical mechanisms that result in the destruction of the regular vortex bundle structure. The apparent source of the destruction of the bundle structures is the various reconnections. Thus, as demonstrated in the paper by Kursa, Bajer & Lipniacki, Kursa et al. (2011), and in work by Kerr Kerr (2011) even a single reconnection results in a cascade of vortex loops of various sizes being chaotically radiated from the reconnection point. Clearly these propagating loops collide with the lines composing bundles, triggering new reconnections, and developing an avalanche-like randomization.

Some authors(see e.g. Alamri et al. (2008),Baggaley (2012)), based on their numerical results, claim that the vortex bundles are the robust structures with respect to reconnection of two adjacent bundles. This conclusion, however, concerns the situation when the different bundles have the same structure (N strands in each bundle). In quantum fluids full reconnection between bundles carrying different numbers of threads is not possible for topological reasons, and the residual structure, analogous to the ”bridging” in classical hydrodynamics, should accompany the collision of such bundles (see e.g. papers Melander and Hussain (1989), Zabusky and Melander (1989), Kida et al. (1991), Boratav et al. (1992)). The analog of classical ”bridging” leads to the randomization and violation of the structure of the bundles and to the creation of vortex loops. Also, the long-range interaction between vortex filaments in the bundles and the ”external” vortices also destroy regular array due to the action of tidal forces.

As for the filaments inside bundle, the direct reconnection event for them is impossible, since for the reconnection the approaching vortices must be antiparallel. Reorganization of lines destroys the parallel array of vortex filaments.

Similarly, the processes of emission and re-absorption of the ring by the vortices (the ”anti-bottleneck”, proposed by Svistunov Svistunov (1995); Kozik and Svistunov (2009)) should also lead to the fragmentation of the regular arrays and the appearance of chaotic loops.

There are also experimental results which do questionable the idea of bundles. Thus, in experiments by Roche et al. Roche et al. (2007),  and by Bradley et al.Bradley et al. (2008), it was observed that the spectrum of the fluctuation of the VLD $\mathcal{L}$ is compatible with a $-5/3$ power law. This contradicts the idea of the bundles structure, since the spectrum of the vorticity (and, correspondingly, of the VLD $\mathcal{L}$ (via Eq. (4))) should scale as $1/3$ power law, which,indeed was observed in paper by Baggaley Baggaley et al. (2012).

One more objection to using HVBK approach is that the Feynman rule (4) is applicable only for stationary situations. It is not clear its validity for transient processes that take place in highly fluctuating turbulent flows.

Resuming this subsection we would like to stress, that the bundle structure of the quasi-classical quantum turbulence, although clear and transparent, is not strongly confirmed . The regular vortex bundles, even if they spontaneously appear, are extremely unstable structures,which can be easily destroyed.

III.3  Where is it from?

This is a somewhat mysterious question: how did at all the HVBK method, designed to a rotational or two-dimensional case, become used for three-dimensional turbulent flows? Trying to find the origin I analyzed a large mass of literary sources. The most frequent references are to the papers by Sonin Sonin (1987) and Hills and Roberts (1977). But these links are absolutely irrelevant, since the authors definitely wrote that they work with rotating helium. Probably, the one of the first papers in which the use of this method for three-dimensional turbulent flows is discussed is the work of Holm Holm (2001). It is interesting, however, that he started paper with the text ”Recent experiments establish the Hall-Vinen-Bekarevich-Khalatnikov (HVBK) equations as a leading model for describing superfluid Helium turbulence. See Nemirovskii and Fiszdon [1995] and Donnelly [1999] for authoritative reviews.”

But R. Donnelly Donnelly (1999) discussed HVBK approach namely for rotating helium. As for my (with W. Fiszdon) paper Nemirovskii and Fiszdon (1995) on superfluid turbulence, then, firstly, there was no mention on HVBK theory at all, and, secondly, I generally opposed to this method to treat three-dimensional quantum turbulence.

Thus, the origin of the idea of the using a pure rotational HVBK approximation for a three-dimensional turbulent flows is rather vague.

Resuming this Section, I can state the HVBK ansatz $\nabla\times\mathbf{v}_{s}=\kappa\mathcal{L}$ is in general unfounded in the three-dimensional case, therefore, works using this approach are questionable, and the corresponding results are not reliable.

IV Other ways to treat the vortex line density

The main attractive advantage in HVBK approach was to rid out of vortex line density $\mathcal{L}$ in equations of motions (1) - (3). It seems,however, that the question of ”elimination” the vortex line density $\mathcal{L}(\mathbf{r},t)$ should be solved in a fundamentally different way. We have not to ”eliminate” the quantity $\mathcal{L}(\mathbf{r},t)$, but on the contrary, to include it into consideration as an independent and equipollent variable. Correspondingly we have to consider the problem, in which there are three independent variables

• velocities $\mathbf{v}_{n}(\mathbf{r},t)$ and $\mathbf{v}_{s}(\mathbf{r},t)$, and vortex line density $\mathcal{L}(\mathbf{r},t)$. We also can add the density field $\rho(\mathbf{r},t),$ as well as the entropy field $S(\mathbf{r},t)$ if it is pertinent. In this way, however, we need an additional independent equation for the temporal and spacial evolution of quantity $\mathcal{L}(\mathbf{r},t)$.

This is a particular task that requires a lot of efforts. Derivation of such an equation, certainly, depends on the type of flow, such as counterflow, co-flow, flow past objects, unsteady rotation, etc. In fact, so far the corresponding equation exists only for the case of counterflow, this is the famous Vinen equation (or some modernized versions of this equation). Although, there are some problems with this equation (see e.g. Sec. IV in Nemirovskii (2018)), it works well for hydrodynamic e.g. for acoustic or engineering problems. It is important to stress that the construction of a theory of the evolution of three fields ($\mathbf{v}_{n}$, $\mathbf{v}_{s}$ and $\mathcal{L}(\mathbf{r},t)$) is not an automatic addition of Vinen equation to the Landau two-fluid hydrodynamics. It is a more involved procedure, since all variables (energy, entropy, etc.) change in the presence of the vortex tangle. This self-consistent procedure was implemented in Nemirovskii and Lebedev (1983), it is called Hydrodynamics of Superfluid Turbulence (HST).

This theory was successful, it explained a lot of experimental results on the nonlinear acoustics of the first and second sounds, the evolution of strong heat pulses, the formation of shock waves etc. It was also successfully applied to the problems of unsteady heat transfer and boiling of He II (Nemirovskii and Fiszdon (1995),Jou and Mongiovì (2005),Kondaurova et al. (2017)). These examples demonstrate that the way to treat the vortex line density $\mathcal{L}(\mathbf{r},t)$  as an additional and equipollent variable seems to be productive and fruitful.

Unfortunately, for other types of flows there is no theory describing temporal - spatial evolution of the vortex line density, and there is no ideas (similar to the Feynman qualitative scenario) how to obtain the according equation. Vinen’s equation reflects the fact that the vortex line density $\mathcal{L}(\mathbf{r},t)$ grows due to the relative velocity $\mathbf{v}_{n}-\mathbf{v}_{s}$ and attenuates, probably, due to the cascade like breaking down of vortex loops, described by Feynman Feynman (1955). That is good guideline how to develop an appropriate theory for any flow, involving, of course, some auxiliary speculations.

Beside of introduction of the vortex line density into the coarse-grained hydrodynamics of superfluid turbulence, there was one more, crude and simplified way, which had been applied on the early stages of research on the superfluid turbulence. It was the use the Gorter - Mellink formula for mutual friction which immediately follows, from the Vinen theory

[TABLE]

Here $A(T)$ is the Gorter - Mellink. In fact, in this formula it was used the ansatz $\mathcal{L}\propto(v_{n}-v_{s})^{2}$, well known in the theory of quantum turbulence The use of equation (6) also ”resolves” the problem of elimination the vortex line density $\mathcal{L}(r,t)$.

V Conclusion

The two approaches how to investigate turbulent flow in superfluids - HVBK and  HST have been described in the paper. In the first one, the vortex line density $\mathcal{L}(\mathbf{r},t)$, crucial for the whole dynamics is straightforwardly excluded from equations of motion (1) and (2) with the use of  Feynman rule (4). In the second approach, the variable $\mathcal{L}(r,t)$ is considered as an additional independent variable obeying the according evolution equation.

In the present paper it had been argued that the HVBK ansatz $\nabla\times\mathbf{v}_{s}=\kappa\mathcal{L}$ is suitable only for rotating cases and fails in three-dimensional situations. The attempts to justify this procedure give rise to a whole scientific direction (trend), which asserts that the vortex tangle in quantum turbulence is composed of the so called vortex bundles containing a set of near parallel lines. In the paper, I put a number of arguments criticizing the conception of the vortex bundle structure. References to the fact that in some numerical works a partial polarization of vortex filaments had been observed cannot be considered as justification for using the ansatz $\nabla\times\mathbf{v}_{s}=\kappa\mathcal{L}$, since ALL the lines contribute into friction, and polarization (if any) includes a small fraction of the total vortex line density.

Furthermore, it is of great concern that the concept of vortex bundles has gone beyond the coarse-grained hydrodynamics of superfluid turbulence and often serves as the basis for other (more subtle) aspects of the theory of quantum turbulence. This seems counterproductive, since after the pioneering works of Feynman, Vinen, Donnelly, Schwartz and others, it was customary to present a vortex tangle as a set of stochastic loops with rich and diverse dynamics. 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. This vision was observed in numerous numerical simulations (see e.g. Schwarz (1988),Aarts and de Waele (1994),Tsubota et al. (2000),Berloff and Svistunov (2002), Kondaurova et al. (2008),Kivotides (2014)). This, let’s say, Feynman-Vinen model is very different. from the vortex bundle model, where almost the only possible dynamics of the vortex filaments is the Kelvin waves evolution along the lines composing the bundles.

In summary, the use of ansatz $\nabla\times\mathbf{v}_{s}=\kappa\mathcal{L}$, for the closure procedure for coupled Navier-Stokes equations (1) - (3) in the 3D turbulent flow is not motivated and would lead to unreliable results. And the commonly used the vortex bundle model, which justifies the use of this method, is questionable and unfounded. Moreover, the vortex bundle concept disavows the real structure of the vortex tangle as a set of vortex loops, and prevents developing of an adequate theory. We assert that the introduction of an additional independent field $\mathcal{L}(r,t)$ to the classical two-fluid hydrodynamics of supefluids is the only correct way to construct the coarse-garined hydrodynamics of turbulent flows.

VI Acknowledgements.

The study on the Hall-Vinen-Bekarevich-Khalatnikov (HVBK) approach was carried out under state contract with IT SB RAS (No. 17-117022850027-5), the study on the Hydrodynamics of Superfluid Turbulence (HST) method was financially supported by RFBR Russian Science Foundation (Project No. 18-08-00576).