On the Nonuniform Quantum Turbulence in Superfluids
aa r X i v : . [ c ond - m a t . o t h e r] A ug On the Nonuniform Quantum Turbulence in Superfluids
Sergey K. Nemirovskii
Institute of Thermophysics, Lavrentyev ave,1, 630090, Novosibirsk, Russiaand Novosibirsk State University, Novosibirsk (Dated: August 29, 2018)The problem of quantum turbulence in a channel with an inhomogeneous counterflow of superfluidturbulent helium is studied. The counterflow velocity V xns ( y ) along the channel is supposed to havea parabolic profile in the transverse direction y . Such statement corresponds to the recent numericalsimulation by Khomenko et al. [Phys. Rev. B , 180504 (2015)]. The authors reported about asophisticated behavior of the vortex line density (VLD) L ( r , t ), different from L ∝ V xns ( y ) , whichfollows from the naive, straightforward application of the conventional Vinen theory. It is clear, thatVinen theory should be refined by taking into account transverse effects and the way it ought tobe done is the subject of active discussion in the literature. In the work we discuss several possiblemechanisms of the transverse flux of VLD L ( r , t ) which should be incorporated in the standardVinen equation to describe adequately the inhomogeneous quantum turbulence (QT). It is shownthat the most effective among these mechanisms is the one that is related to the phase slippagephenomenon. The use of this flux in the modernized Vinen equation corrects the situation with anunusual distribution of the vortex line density, and satisfactory describes the behavior L ( r , t ) bothin stationary and nonstationary situations. The general problem of the phenomenological Vinentheory in the case of nonuniform and nonstationary quantum turbulence is thoroughly discussed. I. INTRODUCTION.
The question of evolution of the vortex line density(VLD) L ( r , t ) of the vortex tangle (VT) is the key issuein the macroscopic theory of quantum turbulence (QT).Although the VLD is a rough characteristic of the QT,it is responsible for many (mainly hydrodynamic) phe-nomena in superfluids and the knowledge of its exact dy-namics is very important for an adequate interpretationof various experiments.Long ago Vinen [2] suggested that the rate of changeof VLD ∂ L ( t ) /∂t can be described in terms of only thequantity L ( t ) itself (and also other, external parameters,such as the counterflow velocity V ns and the tempera-ture). He called this statement as a self-preservation as-sumption. The corresponding balance equation for thequantity L ( r, t ), the so called Vinen equation, reads: ∂ L ∂t = α V | v ns | L / − β V L . (1)Here α V and β V are the parameters of the theory, α V is close to the mutual friction coefficient α , β V is of theorder of the quantum of circulation κ . Throughout itslong history, the Vinen equation has undergone variousimprovements and modifications (see e.g. [3], [4],[5],[6],[7]although at present the form (1) is mainly used.One of serious problems, is the application of the Vinentheory to complicated situations, in particular to inhomo-geneous flows (for recent papers see e.g. [8],[9], [10], [11],[12]). In the cited papers the authors, analyzing numer-ically the steady counterflowing helium in an inhomoge-neous channel flow, obtained a very specific behavior ofthe VLD L ( r , t ), which cannot be interpreted in terms ofequation (1). Thus, Khomenko et al. [8] observed thatthe VLD field is concentrated near the side walls. Quite similar behavior was observed in the work by Yui et al.[10].Analyzing the obtained results, the authors of the pa-per [8] proposed, that the first term on the right handside of the Vinen equation (the so called production term)has the structure ∝ | V ns | L / , a combination that hasnever been discussed before. This conclusion was thesubject of a polemics between the authors of the article[8] and the author of this paper (see [13] and [14]).In the present paper, I would like to digress from thecontent of the mentioned polemics, and to present ourview on the macroscopic behavior of the VLD in inho-mogeneous flows (referring to numerical results of thework [8]). In the study I retain the conventional form ofthe production term in the Vinen equation (1).In short, the results of work [8] can be formulated asfollows. In a rectangular channel 2 × .
05 cm wide, aparabolic counterflow V xns ( y ) = V (1 − ( y/ . ) is ap-plied in x direction. The periodic conditions were as-sumed in all directions. The resulting distributions ofthe dimensional VLD, the normal and counterflow veloc-ities L ( y ) , V n ( y ) , V ns ( y ) are presented in Fig. 1.If one applies straightforwardly the well-known rela-tion L = γ V xns ( y ) ≈ ∗ V xns ( y ) (here γ = α V /β V ),which immediately arises from equation (1), then the di-mensionless L should be about 2 ∗ − V xns ( y )) , whichessentially exceeds the value obtained in [8]. Anotherstriking feature is that the profile L ( y ) is radically differ-ent from the quadratic velocity profile L ∝ ( V xns ( y )) .In the paper we develop an approach explaining thisunusual (from the point of view of the naive use of theVinen theory) behavior of the VLD L ( y ). In the inhomo-geneous situation the Vinen equation should be correctedto include the transverse spacial effects. In particular weoffer to incorporate into classic Vinen theory an addi- FIG. 1: (Color online) Prescribed parabolic normal velocityprofile V n (– · –), the resulting counterflow profile V ns (– –)and the resulting profile of L ( y ) (–) in dimensionless unites,T = 1.6 K. (from paper by Khomenko et al. [8]) tional space flux J ( r , t ) of the VLD, which redistributesthe quantity L ( y ) in the y direction. It is clear thata transverse gradient of the flux ∂J y ( y, t ) /∂y should beadded into the balance equation (1).In the next Sec. II we discuss several mechanismsof these possible fluxes, derive mathematical expressionsand compare contributions from them. In Sec. III wepresent numerical solutions for stationary and nonsta-tionary cases and compare the results with the numericaldata of paper [8]. In Sec. IV we discuss the problem ofnonuniform and unsteady quantum turbulence and theVinen phenomenological theory. The Conclusion is de-voted to a discussion of the results and probable gener-alizations of the presented approach. II. VORTEX-LINE DENSITY FLUX
Let’s describe various ideas on the transversevortex-line density flux J ( r , t ) in inhomogeneousflows/counterflows of superfluid helium. As it was men-tioned above, the first remark in this respect had beenmade by Vinen himself in the context of the possibleinfluence of the channel width [2]. Unfortunately, no ad-vanced theory had been supplemented. It is clear thatthe most general expression for the flux of quantity L is J ( r , t ) = L V L , where V L is the macroscopic local ve-locity of the vortex tangle (see explanations in papers[15],[16],[4]). However, unless we don’t have a generalexpression for V L as a function (functional) of quantity L , we can not ascertain a closure procedure, i.e. obtaina description of the vortex tangle dynamics in terms ofthe VLD itself. This procedure is not uniquely definedand admits different approaches.Thus, in the cited paper [8] the authors proceededfrom the following microscopic expression for the trans-verse flux J micro J micro = 1Ω Z | V ns ( y ) | s ′ z dξ = α Ω Z | V ns ( y ) | s ′ z dξ. (2)Here the integration is performed over the whole vortexline configuration, so it should be understood as an inte-gration along each vortex loops constituting the vortextangle and summation over all loops, i.e. Z dξ → X j L j Z dξ j . The quantity Ω is the total volume, α is the mutual fric-tion coefficient. The authors of work [8] calculated thequantity (2) in numerical simulation and concluded thatthe macroscopic expression J Kh ( r , t ) = α κ C flux ∂ V ns ∂y , (3)best corresponds to the microscopic flux (2). The quan-tity C flux is a constant, determined from numerical simu-lations. Another mechanism, frequently discussed in theproblems of nonuniform flow, is related to the diffusionflux [16],[17]. That mechanism is not connected with mu-tual friction, and realized by the emission of vortex loops,(see, e.g., [18],[19]). The diffusion flux can be written asfollows J dif ( r , t ) = D ∇L , (4)where the diffusion coefficient is estimated as D ≈ ∗ − cm /s.The next contribution, which we consider here, is re-lated to the so called phase slippage phenomenon. Thisphenomenon implies appearance of additional the chem-ical potential ∇ µ , and accordingly the mutual frictionwhen the crossing by the vortices of the main flow. Thiseffect is especially important for monitoring the quantiza-tion of vortices. We will use the corresponding techniqueto describe the transverse flux of VLD J y ( y, t ). To find ananalytical expression for J y ( y, t ), consider the followingequation (see [20], [21], [22]) A = Z ( ˙s ( ξ ) × s ′ ( ξ )) dξ. (5)The right-hand side of (5) is a net area, swept outby the motion of the line elements. Therefore, the x -component of vector A is simply the rate of phase slip-page (without the factor κ ) caused by the transverse mo-tion of the vortex lines (see [22]). It is important, how-ever, that the sign of the x -component of the vector A does not depend on the direction of motion of vortexline segments (either in the positive or in the negativedirections along axis y ). It makes no differences in thecalculation of the phase slippage, and accordingly theadditional drop in the chemical potential ∇ µ , but it isessential for our purposes to determine flux J ps ( y, t ) ofthe VLD L to the side wall. To overcome this problem weassume that all the vortex filaments are closed loops, sothe averaged fluxes in both directions are equal. There-fore, the required transverse flux J ps ( y, t ) of the VLD L isjust half of the x -component of the vector A . Taking ve-locity of elements ˙s ( ξ ) in the form of the local inductionapproximation (see e.g. [3]), we arrive at the followingexpression J ps ( r , t ) = 12 Z ([ α s ′ × ( V ns − β ( s ′ × s ′′ ))] × s ′ ( ξ )) dξ. (6)Here the combination ˙ s i = β ( s ′ × s ′′ ) is the self-inducedvelocity of the line elements in the the local inductionapproximation.To move further we have to introduce the closure pro-cedure and to express the right hand side of Eq. (6)via quantities L and V ns . It corresponds to the self-preservation assumption expressed by Vinen, that themacroscopic dynamics of the vortex tangle depends onlyon the VLD L ( t ). The other, more subtle characteristicsof the vortex structure, different from L , must adjust toit. In particular, the first contribution, containing the ex-ternal counterflow velocity can be written as αI k L | V ns | ,where I k is the structure parameter of the vortex tan-gle, introduced by Schwarz [3]. The last term in Eq.(6) with the self-induced velocity can be expressed as αβ L ( I l L / ). where I l is another structure parameter.Usually at this point the substitution L / = γ | V ns | isused, and both contributions are reduced to a combina-tion J ps, ( r , t ) = 12 α ( I k − γβI l ) L | V ns | . (7)Being multiplied by ρ s κ this expression (up to a factor1 /
2) coincides with the formula for mutual friction. Thisis not surprising, because it is well known from the vor-tex dynamics that a vortex crossing the channel transfersthe momentum to the main flow (see [23]). Therefore thefinal expression should be proportional to V ns and thewhole scheme becomes self-consistent. But this aboveconsideration concerns only homogeneous or near - ho-mogeneous cases. In the highly inhomogeneous situation,which we are interested in here, the simple relations suchas L / = γ | V ns | do not work and the question of deter-mining the transverse flux remains open. A very similarproblem of using the structure parameters of the vortextangle also arises for nonstationary situations (see a re-lated discussion in the review article [4]). This problemis very intriguing, and we decided to explore yet anotherversion of the closure procedure, which leads to the fol-lowing formula for the transverse flux J ps ( y, t ) = αI k L | V ns | − αβI l L / . (8)Thus, we have obtained two forms for the transverseflux associated with the phase slippage mechanism. They are identical in case of an uniform flow , when L / = γ | V ns | , however, in inhomogeneous situations they differand can result in different results.Our further goal is to analyze the results on the nonuni-form quantum turbulence obtained in the numerical workby Khomenko et al. [8], basing on supposition of thetransverse flux of VLD L ( y ). Using the conditions oftheir modeling and taking that | V ns | ∼ L ∼ , α ∼ . ∂/∂y ∼ / .
05, we conclude that themost effective mechanism among those considered above,is the one related to the phase slippage mechanism. It ex-ceeds other contributions almost by the order and furtherwe will concentrate on the only this effect.Beside the usual estimation and comparison of vari-ous fluxes written above we can appeal to the fact thatneither Khomenko et al. flux J Kh,cl no the diffusion flux J dif are effective enough to produce the complicated spa-cial distribution of the vortex line density which was ob-served in paper [8] and is shown in Fig 1. As far as theKhomenko et al. flux J Kh,cl this problem was discussedin details in the paper [13] (Sec. IV).The impact of the diffusion flux was studied in a re-cent work by Saluto et al. [11]. The authors observedthat the influence of the vortex diffusion is focused onthe local values of L ( y ) rather than on the form of thespatial distribution VLD. Thus the diffusion term (4) isalso small for this particular problem, although, being asecond-order derivative, it would be essential for othersituations. In this paper we will not consider this term. III. SOLUTIONS
Thus we introduced and discussed several mechanismsfor the transverse flux of VLD and concluded that themost effective of them is associated with the phase slip-page mechanism. A microscopic equation for this flux isgiven by Eq. ( 6), its macroscopic closure variants aregiven by the formulas (7),(8). Our goal now is to incor-porate these terms into the Vinen equation (1) ∂ L ∂t + ∂J ps ( y, t ) ∂y = α V | V ns | L / − β V L , (9)and to study its solutions under the conditions that iden-tical to those studied in the work by Khomenko et al. [8].Namely, we have selected the temperature of system, thegeometry and size of the of the channel, parabolic coun-terflow velocity V ns ( y ) coinciding with the ones acceptedin their work. We study two cases, a stationary situationand a completely unsteady problem. A. Stationary case, profile of VLD L ( y ) . In Fig. 2 we displayed the VLD L ( y ) profiles ob-tained through the numerical solution of equation (9)without the term ∂ L /∂t . The upper and lower images FIG. 2: (Color online) Profiles of VLD L ( y ) obtained in nu-merical solution of the equation (9) without the term ∂ L /∂t .The upper and lower pictures correspond to different expres-sions for transverse flux (7),(8). correspond to different expressions for the transverse flux(7),(8). We have chosen the system temperature T = 1 . ∗ .
05 cm, the parabolic counter-flow velocity V ns ( y ) = 1 . − ( y/ . ) cm/s, coincid-ing with the conditions adopted in the work [8]. Ad-ditionally, only half of the channel width is considered,namely 0 y .
05 cm. The boundary condition L ( y = 0) = 1000 1/cm had been taken from the resultof paper [8] and from the solution of the fully nonstation-ary problem (see below). It is noteworthy that they arevery close to each other.The most important (albeit expected) result is thatthe VLD profile does not really satisfy the standard Vi-nen relation L ( y ) = γ | V ns | . On the contrary, the vor-tex tangle is concentrated in the region closer to the sidewall, (but not directly on the wall). This behavior canbe understood qualitatively from the following considera-tions. The structure of flux expressed by the formula (7)is that its maximal value is at the central parts ( y = 0)of the channel (due to the large value of the counterflowvelocity V ns ) and the VLD L is intensively removed fromthis region. On the contrary, because of the vanishing ofthe counterflow velocity V ns on the side walls ( y = 0 . L does not penetrateinto this region. Clearly, to support a stationary solutionin the regions where L ( y ) = γ | V ns | , either the produc-tion or the decay (second) term on the right hand side ofequation (9) should prevail. Another remarkable resultis that there is a very good agreement, both qualitativeand quantitative, with the data of the paper [8] depicted in Fig. 1.One more important result concerns the fundamentalquestion of the use of the Schwarz’s relations for thestructure parameters of the nonuniform quantum tur-bulence In the lower picture of Fig. 2 we presentedthe quantity L ( y ) obtained in numerical solution of theequation (9) with the transverse flux expressed by Eq.(8), which includes an alternative variant of the struc-ture parameter. It is easy to see that qualitatively solu-tions are very similar, although they are a bit different.This fact confirms the widespread view that the Vinenequation can be a good tool for studying rough engineer-ing problems, although relevant approaches may requiresome fitting parameters. At the same time the wholeVinen macroscopic theory is not suitable for the investi-gation of the fine structure of the vortex tangle. B. Nonstationary case, development of quantumturbulence in the inhomogenious counterflow.
The rather elegant results are obtained when solvingthe full equation (9), with the term ∂ L /∂t . This pro-cedure faces the standard problem of initial conditions,typical for the Vinen theory. Equation ((9)) is a balancerelation between the growth and the disappearance ofvortex lines. The mechanism of spontaneous appearanceof vortices in the helium flow has not been built into thisequation.At present, there are various theories of the initial ap-pearance of vortex filament, which can be divided intotwo groups. The first group offers the different mech-anisms (tunnelling, fluctuation growth, etc.) of initialgeneration of vortices. Another group is based on theidea that in the helium permanently exists a backgroundof remnant vortices. From the point of view of the phe-nomenological theory the former group can be taken intoaccount by introducing the initiating term into the Vinenequation. In turn, the latter group should lead to someinitial value of VLD ( L ( t = 0) = L back ) in the Vinenequation. The better agreement between experimentaldata on the propagation of intense heat pulses (generat-ing vortices and interacting with these ”own” vortices)and the corresponding numerical solution, was obtainedwhen assuming the existence of an initial level of VLD L back , whereas the introduction of the initiating term ledto an unsatisfactory correlation with the experimentalobservations (see e.g. [24]). Thus it may be surmisedthat this is an argument in favour of the theory of rem-nant vortices. Usually, the level of the remnant vorticity L back is estimated approximately as 10 − .The spatio - temporal behavior of VLD L ( y, t ) ob-tained in the numerical solution of the equation (9) withthe nonstationary term ∂ L /∂t is shown in Fig. 3. Theupper and lower images the correspond to the differentexpressions for the flux (7),(8). We again have chosen allconditions of work [8]. As for initial conditions we as-sume that the background vorticity L back = 1000 1/cm . FIG. 3: (Color online) The spatio - temporal behavior of VLD L ( t, y ) obtained in numerical solution of the equation (9).Theupper and lower pictures correspond to different expressionsfor transverse flux (7),(8). The obtained picture confirms all the conclusions on thebehavior of the VLD L ( y, t ), made in the previous para-graph, and demonstrates how the according scenario isdeveloping in time. On a time slice of 2 s (It is probablesaturation and crossover to the steady-state regime), thesolution L ( y, t = 2 c ) agrees with the data found in Ref.[8] (see also Fig. 1). That is a remarkable fact becausein our study no fitting parameters have been used. IV. NONUNIFORM QUANTUM TURBULENCEAND THE VINEN PHENOMENOLOGICALTHEORY
In Sec. II we described the problems of the closureprocedure for the microscopic equation for the flux ( 6)and questions of the choice of the form for the structureparameters. Bearing in mind to compare various possi-bilities we have chosen two variants, leading to differentexpressions (7),(8). In this regard, it seems appropri-ate to return to the basics of Vinen’s phenomenologicaltheory as applied to the complex nonstationary and in-homogenious situations.The main idea of the Vinen approach was the assump-tion of self-preservation , i.e. the suggestion that themacroscopic vortex dynamics can be described in termsof the quantity L ( t ) only. Selecting a set of variables todescribe the macroscopic dynamics of statistical systems is, in general, a difficult and delicate step. For instance,the usual gas dynamics variables, such as density, mo-mentum and energy (per unit volume) are just the firstmoments of the distribution function of the Boltzmann’skinetic theory. Higher moments relax to approach equi-librium much faster than do the first listed variables.This circumstance allows one to truncate an infinite hi-erarchy of the moment equations and obtain a closed de-scription using the listed quantities.Unfortunately in case quantum turbulence, the as-sumption of self-preservation is not motivated, the re-striction to the only variable L ( t ) is not justified, and,in general, the Vinen equation is not valid. Indeed, letus consider a very simple counterexample. Assume thatthe velocity V ns ( s , t ) changes instantly to the opposite.Since the Vinen-type equation include the absolute valueof relative velocity | V ns ( s , t ) | magnitude, then formallythe system remains unaffected by the change. This iswrong, of course. The structure of the VT, mean cur-vature, anisotropy and polarization parameters will be-come reorganized. That implies the violation of the self-preservation assumption, and dynamics of the VLD L ( t )depends on other, more subtle characteristics of the vor-tex structure, different from L ( t ).To clarify the situation, let us consider a way of deriva-tion of VE from the dynamics of vortex filaments in thelocal induction approximation (see, e.g. [25]). It will suf-fice for the illustration sake. Integrating an equation forthe change of the length of line element over ξ inside avolume Ω, Schwarz concluded that in the counterflowinghelium II the quantity L ( t ) obeys the equation (see [3]) ∂ L ∂t = α V ns Ω Z h s ′ × s ′′ i dξ − αβ Ω Z (cid:10) | s ′′ | (cid:11) dξ . (10)The quantity L ( t ) is related to the first derivative s ′ ofthe function s ( ξ ), since L ( t ) ∝ R | s ′ | dξ . The rate ofchange of L ( t ) includes quantities involving the higher-order derivative s ′′ , namely h s ′ × s ′′ i and (cid:10) | s ′′ | (cid:11) . Ina steady-state, these higher-order quantities are are di-rectly expressed via the VLD L as h s ′ × s ′′ i ∝ I l L / and (cid:10) | s ′′ | (cid:11) ∝ c ( T ) L . Here the I l , c ( T ) are temperaturedependent parameters introduced by Schwarz [3]. Butin the nonstationary situation s ′′ is a new independentvariable, and one needs a new independent equation forit and for other quantities, related to curvature of line.This new equation, in turn, will involve higher deriva-tives s ′′′ , s IV and so on. This infinite hierarchy can betruncated if, for some reasons, the higher-order deriva-tives relax faster, than the low-order derivatives, and taketheir ”equilibrium” values (with respect to the momentsof low order).Strictly speaking, there are no theoretical grounds forassuming that the relaxation of higher moments is fasterthan that of the quantity L ( t ). Thus, in general, no equa-tion of the type ∂ L ( t ) /∂t = F ( L ) exists! At the sametime, in some (unclear) conditions, and with the use ofadditional arguments (see, [2]), the required equation canbe written down. The attempt was successful, this theoryexplained a large number of hydrodynamic experiments,including the main experiment by Gorter and Mellink[26] (see, for details, the review by [27]). It concerned, however, only stationary or near-stationary situations.In a strongly unsteady case, the region of applicabilityof this equation is unclear , see the above counterexamplewith a sudden inversion of the counterflow velocity.Meanwhile, it seems intuitively plausible that for slowchanges (both in space and time) the assumption of self-preservation is valid. That was the starting point in theconstruction of the so-called Hydrodynamics of Super-fluid Turbulence (HST), which was the unification of theVinen equation and the classical two-fluid hydrodynam-ics (see, e.g., [15],[28],[29]). The HST equations havebeen applied to study a large number of hydrodynamicand thermal problems, including heat transfer and boil-ing in He II (see, e.g., [30],[31], [32],[33], [34],[35],[24]).The numerical and analytic results were in very goodagreement with numerous experimental data. This factpointed out that the Vinen equation is robust and is, ingeneral, quite suitable for the unsteady hydrodynamicproblems.It follows from the results of this work that the situ-ation with inhomogenious flow is quite similar. This isconfirmed by the curves depicted on the upper and lowerimages in Figures 2 and 3. In these images we displaythe results obtained from solutions of the Vinen equation( 9) with different expressions (7),(8) for the transverseflux. The qualitative similarity and closeness of the quan-titative solutions indicates again that the Vinen equationis rather insensitive to a particular choice of the trans-verse flux and is robust to study various inhomogenioussituations. V. CONCLUSION
We conclude by saying that the study of the inhomoge-nious flow/counterflow of superfluids in the channel on the basis of the Vinen equation (1) requires the introduc-tion of additional terms describing the transverse flux ofthe VLD L towards the side walls. The analysis demon-strated that the most efficient mechanism is related to thephase slippage mechanism. The corresponding solutionsof the Vinen equation with the additional term in bothstationary and nonstationary cases agree with observa-tions obtained earlier in numerical simulations. Theyshowed that the VLD L ( y, t ), as function of y is concen-trated in the domain near the side walls. The reason forthis behavior is the special structure of the transverseflux. This construction forces the vortex filaments to es-cape from the central part, at the same time does notallow them to touch the walls.One of our results, important for the macroscopic the-ory of quantum turbulence concerns the structure func-tions of the vortex tangle, such as the parameters ofanisotropy and polarization. Just like in the unsteadysituation, the use of such parameters in the usual form,introduced by Schwarz, can only be done approximatelyand with reservations. This fact confirms the widespreadview that the Vinen equation can be used to explore therough, engineering problems (although the correspond-ing studies may require some fitting parameters), butit’s not suitable for the description of the fine structureof the vortex tangle.I would like to thank Prof. I. Procaccia for the veryfruitful discussion of questions touched in the paper. Thework was supported by Grant No. 14-29-00093 fromRSCF (Russian Scientific Foundation) [1][2] W. Vinen, Proc. R. Soc. Lond. Ser. A 242 , 493 (1957).[3] K. W. Schwarz, Phys. Rev. B , 2398 (1988).[4] S. K. Nemirovskii and W. Fiszdon, Rev. Mod. Phys. ,37 (1995).[5] D. Jou, M. Mongiovi, and M. Sciacca, Physica D ,249 (2011).[6] R. Donnelly, Quantized Vortices in Helium II (Cam-bridge University Press, Cambridge, UK, 1991).[7] S. K. Nemirovskii, Physics Reports , 85 (2013).[8] D. Khomenko, L. Kondaurova, V. L’vov, P. Mishra,A. Pomyalov, and I. Procaccia, Physical Review B ,180504 (2015).[9] A. W. Baggaley and J. Laurie, Journal of Low Temper-ature Physics , 35 (2015), ISSN 1573-7357.[10] S. Yui, K. Fujimoto, and M. Tsubota, Phys. Rev. B ,224513 (2015). [11] L. Saluto and M. S. Mongiovi, Communications in Ap-plied and Industrial Mathematics , 130 (2016).[12] D. Kivotides, Journal of Fluid Mechanics , 58 (2011).[13] S. K. Nemirovskii, Phys. Rev. B , 146501 (2016).[14] D. Khomenko, V. S. L’vov, A. Pomyalov, and I. Procac-cia, Phys. Rev. B , 146502 (2016).[15] S. Nemirovskii and V. Lebedev, Sov. Phys. JETP ,1009 (1983).[16] J. A. Geurst, Physica B: Condensed Matter , 327(1989).[17] M. Tsubota, T. Araki, and W. F. Vinen, Physica B: Con-densed Matter , 224 (2003).[18] C. F. Barenghi and D. C. Samuels, Phys. Rev. Lett. ,155302 (2002).[19] L. Kondaurova and S. K. Nemirovskii, Phys. Rev. B ,134506 (2012). [20] M. Rasetti and T. Regge, Physica A: Statistical Mechan-ics and its Applications , 217 (1975).[21] S. K. Nemirovskii, Phys. Rev. B , 5972 (1998).[22] C. Swanson and R. Donnelly, J. Low Temp. Phys. ,363 (1985).[23] S. K. Nemirovskii and M. Tsubota, Journal of Low Tem-perature Physics , 591 (1998).[24] L. Kondaurova, V. Efimov, and A. Tsoi, Journal of LowTemperature Physics , 80 (2017), ISSN 1573-7357.[25] K. W. Schwarz, Phys. Rev. B , 245 (1978).[26] C. J. Gorter and J. H. Mellink, Physica , 285 (1949).[27] J. Tough, Progress in Low Temperature Physics, Vol. 8 (North-Holland, Amsterdam, 1982).[28] K. Yamada, S. Kashiwamura, and K. Miyake, Physica B:Condensed Matter , 318 (1989), ISSN 0921-4526. [29] J. Geurst, Physica A: Statistical Mechanics and its Ap-plications , 279 (1992), ISSN 0378-4371.[30] W. Fiszdon, M. v. Schwerdtner, G. Stamm, andW. Poppe, Journal of Fluid Mechanics , 663 (1990).[31] M. Murakami, S. Gotoh, N. Koshizuka, S. Tanaka,T. Matsushita, S. Kambe, and K. Kitazawa, Cryogen-ics , 390 (1990).[32] M. Murakami and K. Iwashita, Computers & Fluids ,443 (1991).[33] W. Poppe, G. Stamm, and J. Pakleza, Physica B: Con-densed Matter , 247 (1992).[34] A. N. Tsoi and M. O. Lutset, Inzh. Fiz. Zh. , 5 (1986).[35] U. Ruppert, W. Z. Yang, and K. Luders, Jpn. J. Appl.Phys.26