Dynamics of the vortex line density in superfluid counterflow turbulence
aa r X i v : . [ c ond - m a t . o t h e r] J a n Dynamics of the vortex line density in superfluid counterflow turbulence
D. Khomenko , V. S. L’vov , A. Pomyalov and I. Procaccia , The Department of Chemical Physics, The Weizmann Institute of Science, Rehovot 76100, Israel Niels Bohr International Academy, University of Copenhagen Blegdamsvej 17, DK-2100 Copenhagen, Denmark
Describing superfluid turbulence at intermediate scales between the inter-vortex distance and themacroscale requires an acceptable equation of motion for the density of quantized vortex lines L . Theclosure of such an equation for superfluid inhomogeneous flows requires additional inputs besides L and the normal and superfluid velocity fields. In this paper we offer a minimal closure using oneadditional anisotropy parameter I l . Using the example of counterflow superfluid turbulence wederive two coupled closure equations for the vortex line density and the anisotropy parameter I l with an input of the normal and superfluid velocity fields. The various closure assumptions and thepredictions of the resulting theory are tested against numerical simulations. I. INTRODUCTION
The experimental study of the statistical propertiesof mechanically excited He superfluid turbulence wassomewhat stalled by a realization that these statistics,when observed on large spatial scales, do not differ muchfrom the statistics of the classical counterpart . Theseresults could mean that the two-fluid model of superfluidturbulence may suffice to predict the statistical charac-teristics of superfluid turbulence. On the other hand,even earlier work on counterflow turbulence , showedvery definitely that on scales that are intermediate be-tween the inter-vortex distances ℓ and the outer scale L the observed physics is influenced dramatically by vor-tex dynamics, vortex reconnections, Kelvin waves on theindividual vortex lines, or in short, those effects thatexist due to the quantization of vorticity in quantumfluids. These findings underline the difference between“co-flowing” situations (in which the normal and supercomponents average flow is in the same direction) from“counter-flowing” situations. Evidently, a fuller under-standing of the statistical theory of “counterflow” super-fluid turbulence on these interesting scales calls for de-riving an equation for the relevant characteristics of thetangle of quantized vortices which is ubiquitous in thistype of turbulence. Indeed, the search for an equationof one of these characteristics, i.e. the density of vortexlines L , has been long, starting with the seminal papersof Vinen from the nineteen fifties . The context in whichthis equation was studied was that of “counterflow” tur-bulence in which He at temperatures below the λ -pointis put in a channel with a temperature gradient, suchthat the mean normal velocity V n is directed from thehot to the cold end while the mean superfluid velocity V s is directed oppositely. The difference between thesetwo mean velocities was denoted as V ns and the equationthat was proposed by Vinen may be written as: d L dt = αC | V ns |L / − αC κ L , (1)where C and C are dimensionless coefficients, α isknown as the “mutual friction parameter” and κ ≈ . / s is the quantum of circulation. Remarkably, this equation which was not derived from first princi-ples, and which was explicitly assumed to apply to ho-mogeneous isotropic situations, has been employed foralmost 60 years, becoming the only widely accepted andused equation in the field. Admittedly, Vinen himselfexpressed concerns whether this equation may apply toinhomogeneous situations , and in subsequent work at-tempts have been made to generalize this equation, butno conclusive results were obtained. To find out moreabout the history of the problem see the review paper and references therein.The next important step was the work of Schwarz whoapplied vortex filament method to quantum turbulenceto derive an equation for the vortex line density frommicroscopic principles, but it also underlined the closureproblem that results from this approach. Additional andmore complicated characteristics of the vortex tangle popup from the derivation. The two most important objectsthat appear naturally are the mean curvature and theanisotropy of the vortex tangle. Accordingly, finding anacceptable equation for L for general flows becomes achallenge of finding a reasonable closure in terms of vari-ables that can be measured to compare observations withtheory.A serious hindrance to the completion of this programwas the lack of sufficient data to provide verification ofpossible theories. This hindrance began to lift recentlywith the advent of numerical simulations. Lipniacki pro-posed to complement the equation for vortex line densitywith a coupled equation for the anisotropy parameter inthe context of homogeneous but time dependent counter-flow turbulence. Further improvement of numerical simu-lations made it possible to study inhomogeneous counter-flow turbulence in a channel . In our own work wesuggested a new form of the equation for L in a channelwhich started some discussion in the literature . Thepresent work is a continuation and an extension of Ref. .Here we present a new derivation of an equation for theanisotropy parameter which generalizes and complementsour previous results. In our work we opt to focus onsteady inhomogeneous flow, in contrast to Ref. whichconsidered homogeneous unsteady flow. We should stateat this point that the derivations presented below arenot mathematically rigorous. They are based on argu-ments and estimates. Whenever we can we supplementthe analytic arguments with numerical tests. The stateof the art of numerical simulations of quantum fluids isnot sufficient to nail all the assumption made with cer-tainty. So the reader is advised to consider the presentpaper as a statement of the state of the art which is possi-bly not final. More careful work is needed to reach finalconclusions. Nevertheless the use of numerical simula-tions allows us to directly check and verify some crucialassumptions; this fundamentally distinguishes our workfrom some of the previous papers in this field .The structure of this paper is as follows: in Sect. IIwe review the derivation of the equation of the density ofvortex lines and explain how closure bring up the neces-sity to study new objects like the anisotropy parameter.In Sect. III we derive the equation for anisotropy parame-ter. A discussion and conclusions are offered in Sect. IV.The appendix provides those technical details that arenot given explicitly in the text. II. EQUATION FOR THE DENSITY OFVORTEX LINES
The starting point for the derivation of an equation ofmotion of the density of vortex lines is the microscopicequation for a single vortex line . Denote by s ( ξ, t ) agiven point in space that belongs to a vortex line whichis parameterized by the arc-length ξ . The one writes theequation d s ( ξ, t ) dt = V s + V BS ( s , t ) + ( α − ˜ α s ′ × (cid:1) s ′ × V ns ( s , t ) . (2)Here V BS ( s , t ) is the velocity generated by the entire vor-tex tangle, the prime means a derivative along an arclength d/dξ and α and ˜ α are mutual friction parameters.From this equation we can calculate rate of elongation ofthe vortex line segment δξ :1 δξ dδξdt = s ′ · d s ′ dt . (3)Integrating Eq. (3) over the vortex tangle provides thechange in the density of vortex lines L . Note that ininhomogeneous condition this wanted equation has thegeneral form: ∂ L ∂t + ∇ · J L = P L − D L . (4)where J L is the flux of the vortex line density, P L isthe rate of production and D L the rate of decay of thesaid density. In the appendix and in Refs. it is shownthat in a channel geometry where the mean flow is in the x -direction and the wall normal direction is y one can derive the approximate equations J L = Z (cid:20) d s dt (cid:21) y dξ ≈ − ακ V ns dV s dy , (5a) P L ≈ α V ns · I l h κ i L , (5b) D L = αβ (cid:10) κ (cid:11) L . (5c)Here κ ≡ | s ′′ | is the curvature of the vortex line and thedimensionless vector I l is defined by: I l = h b i , b = s ′ × s ′′ / κ . (5d)The derivation of these equations is based on the fol-lowing assumptions:1. Separation of scales; we assume that the “macro-scopic” variable V ns changes slowly in space in com-parison to rate of spatial change of s ′ and b . Ac-cordingly the average of their product can be esti-mated as a product of the averages.2. We neglect the term in Eq. (2) proportional to ˜ α .The reason is that in counterflow conditions thiscoefficient is much smaller than α .3. We assume that the derivative dV ns /dξ along vortexlines is negligible on the average.Equations (5) expose the appearance of new fields, i.e.the objects h κ i , (cid:10) κ (cid:11) and I l . To be able to close the sys-tem of equations these objects should be either expressedin terms of known variables ( L , V ns , ...) or supplementedby equations for the new objects.To proceed, another fundamental assumption is calledfor. We assert that there exists only one typical length-scale in the problem which is inter-vortex distance.Therefore the radius of curvature should be proportionalto it. This is a closure relation that reads h κ i ≃ c L / .This assumption is expected to be reasonable for rela-tively low gradients of counterflow velocity. We tested itand found that it works well for parabolic profiles of thenormal velocity in the case of T1 turbulence in a channel,cf. Ref. .The closure of I l is more complicated. In Ref. , inanalogy with the Vinen equation, we assume that I l canbe expressed in terms of V ns and L only. Being dimen-sionless, it must be a function of the dimensionless vari-able ζ = V ns / ( κ L / ). Based on numerical experimentswe concluded that taking I l ∝ ζ gives a good matchfor all the available data obtained by simulations in achannel. Below we generalize this to other geometries.With these explicit assumptions one obtains the closurefor Eq. (5c): J L ≈ − ακ V ns dV s dy , (6a) P L ≈ αC prod V √L /κ , (6b) D L ≈ αβC dec L . (6c) −1 −0,5 0 0,5 1 −0,5 0 0,5−0,500,5 y/Lx/L z / L V n FIG. 1: Sketch of the computational setup. ζ I x l −0.5 0 0.500.20.40.60.811.2 y/L V n FIG. 2: Upper panel: examples of imposed normal veloc-ity profiles in the simulation channel. Lower panel: para-metric plot [ ζ ( y ) , I xl ( y )] of the stream-wise projection of theanisotropy parameter. III. EQUATION FOR THE ANISOTROPYPARAMETER
Admittedly, while the closure for I l matches well withthe available numerical results for T1 counterflow turbu-lence in the channel, its derivation did not provide ad-equate physical intuition or an explanation why it gavea good agreement with the data. Accordingly it is notclear how general this closure is. To test the limits of its applicability we did a series of additional numericalsimulations with various different profiles of the normalvelocity shown in the upper panel of Fig. 1. The simu-lations were performed using the full Biot-Savart calcu-lation, employing vortex filament methods as explainedin detail in Ref. . Using periodic boundary conditions, we have a greater freedom of choice for the normalvelocity profile. The simulations were carried out in acomputational box of size 2 L × L × L, L = 0 . T = 1 . V n = (0 . ÷ .
0) cm/s. In all these simulations wemeasured the anisotropy parameter I l and plotted theresults in the lower panel of Fig. 2 as a parametric plot[ I l ( y ) , ζ ( y )]. The results convey the clear message that I l is not a function of ζ only. Additional dimensionlessvariables in addition to ζ appear to be necessary. Can-didates are for example ζ = 1 κ L dV s dy , ζ = 1 κ L / d V s dy , . . . (7)As a test of this conclusion we show in Fig. 3 the re-sults of allowing the anisotropy parameter to depend ontwo variables, i.e. ζ and ζ . An improvement in thedata representation is evident. To find the two-variableparametric surface, we used general polynomial and Pad´eapproximants. The resulting mean square deviation wasreduced by 50 % compared to the fit that depended on ζ only. Accordingly we turn now to a derivation based onmicroscopic relations. A. Derivation based on microscopic relations
To find an expression for the anisotropy parameter I l in a systematic way we will derive its equation of motion.We start again from Eq. (2). In the appendix we showthat applying this equation to the unit vector b we get: d b dt = α κ V − α κ (cid:0) b · V (cid:1) b + βτ s ′′ / κ , (8)where τ is a torsion of the vortex line. This equation canbe rewritten in the following form: d b dt = α κ b × (cid:0) V × b (cid:1) + βτ s ′′ / κ . (9)Note that Eq. (9) is very similar to the equation for vor-tex rings that was used in Ref. for I l in a homogeneousflow. In an inhomogeneous and anisotropic flow the equa-tion satisfied by I l will again have a production termdenoted as P I l , a decay term, denoted as D I l , and aflux term denoted as ˆ J I l . Two of these can be obtainedeasily. Integrating Eq. (8) over the tangle provides theproduction term for I l . The flux term can also be foundanalytically, ˆ J I l = D d s dt ⊗ b E . (10) ζζ I x l ζ I x l ζ = 0 . ζ = 0 . ζ = 0 . FIG. 3: Parametric plot of [ ζ ( y ) , ζ ( y ) , I l ( y )] for all the nor-mal velocity profiles shown in Fig. 2. Upper Panel: The sur-face embedded in 3-dimensions which spans all the availabledata points. Lower Panel: I l ( y ) as a function of ζ for differ-ent values of ζ , as marked in the plot. Points are data fromall the simulations for ζ = 0 . ± . . ± . . ± . The last ingredient, the decay, is governed by the effectof vortex reconnection. This term is difficult to derivefrom the microscopic equation and at this point it canonly be written down phenomenologically. The effect ofvortex reconnection is to locally form sharp tips thatcan be oriented in any spatial direction which is deter-mined by the relative orientation of the two reconnectingvortex lines. This tends to destroy any pre-orientationof the vortex lines. Based on this picture we can as-sume that in each event of reconnection a region whichis affected by it will loose its anisotropy I l . Phenomeno-logically we can write the decay due to reconnections inthe following way: D I l ≈ C h κ i L dN rec dt I l . (11)Here dN rec /dt is the reconnection rate, 1 / κ is the radiusof curvature which defines the region affected by the re-connection event and I l is the value of anisotropy thatwas lost during the reconnection. −0.5 0 0.500.511.522.5 y/L τ τ L −0.5 0 0.500.10.2 y/L h b i i h b x ih b y ih b z ih b x i FIG. 4: Upper panel: the profiles of τ ( y ) and the normal-ized density of vortex lines L ( y ) / hLi , plotted for the sine-likenormal velocity profile. Lower panel: the corresponding pro-files of (cid:10) b i (cid:11) and h b i i . Now that all the necessary objects are expressed interms of macroscopic fields, we recognize that ones morethere appeared new quantities, i.e. h b x i , τ and dN rec /dt that should be modeled. We will start with torsion. Tor-sion is a second curvature and we expect that the radiusof torsion would be proportional to the inter-vortex dis-tance: (cid:10) τ (cid:11) ≈ C τ L . (12)Note that this assumption is similar to the previouslyperformed closure for the curvature κ and is expected tohave similar validity. Unfortunately at present we can nottest this assumption numerically due to the insufficientaccuracy of our simulations. Nevertheless preliminaryresults shown in the upper panel of Fig. 3 look promising.Regarding the object (cid:10) b x (cid:11) there are two limiting casesthat could be considered: (i) strongly polarized tangle:in this case (cid:10) b x (cid:11) ≈ h b x i ; (ii) a fully isotropic case: (cid:10) b x (cid:11) ≈ /
3. Based on the results of many numerical simulationwe can say that in the case of counterflow turbulence(T1 regime) we are closer to the second case, and eventhough (cid:10) b x (cid:11) is slightly larger than ( (cid:10) b y (cid:11) + (cid:10) b y (cid:11) ) /
2, it canbe considered as a constant with reasonable accuracy, (cid:10) b x (cid:11) ≈ const. (13) −0.5 0 0.5345678 x 10 y/L d N r ec d t d N rec dt C rec κ L / FIG. 5: The profile of the reconnection rate and its estimatevia the closure discussed in the text. In this example theimposed normal velocity profile is sine-like.
Evidence is provided in the lower panel of Fig. 4.The last object is the vortex reconnection rate. Moti-vated by dimensional reasoning and supported by resultsfrom homogeneous turbulence , we can model the re-connection rate profile by: dN rec dt ≈ C rec κ L / . (14)The numerical data presented in Fig. 5 support this con-clusion and see also the results in Ref. . B. Putting things together
Summarizing together all the discussed results we endup with a system of equations: ∂I l ∂t + ∂ J I l ∂y = P I l − D I l , (15a) J I l ≈ L (cid:16) − ακ V ns dV s dy I l + Cβ dV s dy (cid:17) , (15b) P I l ≈ αC V ns L / + C β L d V s dy , (15c) D I l ≈ C dec κ L I l . (15d)Note that in Eq. (15b) we used results from Ref. tomodel h s ′′ i . IV. DISCUSSION AND CONCLUSIONS
The central result of this paper is the equation of mo-tion for anisotropy parameter I l ; we showed that it ispossible to close it in terms of known variables. Togetherwith the equation for the density of vortex lines we pos-sess now a set of equations that allows a calculation ofthe profile of the anisotropy parameter and the vortex line density self-consistently, relaxing the assumption onthe production term made in Ref. . The approach de-scribed above has pros and cons. On the negative side, werecognize that a number of new assumptions were madewhose verification is currently beyond our computationalcapabilities. Of particular concern is the phenomenologi-cal model for the anisotropy decay due to reconnections,and the precise modeling of the torsion and the flux of I l . On the positive side the work described above pre-pares a theoretical background for future numerical sim-ulations. It underlines the missing links in our currentunderstanding of the subject. Moreover, even on the ba-sis of what is known now we have enough informationfrom numerical simulations to afford making simplifica-tions that are relevant for the study of counterflow tur-bulence. Finally, possible future experiments in inhomo-geneous and anisotropic superfluid flows should also becarefully designed and analyzed to shed further light onthe issues discussed above. Indeed, a proper descriptionof the physics of vortex lines requires knowledge of boththe vortex line density and the anisotropy of the vortextangle; the measurement of at least two different fields isneeded. For example, one can use the fact that the at-tenuation of second-sound depends on the angle betweenits propagation direction and the vortex line direction .Thus, measuring the attenuation along two orthogonaldirections one can extract the total line density as wellas information about the the anisotropy of the vortextangle. Probably even more informative would be ex-periments of a counterflow and pure superflow in a rect-angular channel with a large aspect ratio measuring thesecond sound attenuation, propagating in all three direc-tions. This type of experiments, possibly combined withvisualization techniques, may expand our understandingof counterflows in superfluid He, and in particular, shedlight on the importance of the vortex tangle anisotropy.
Acknowledgments
IP is grateful for the hospitality of Prof. Poul HenrikDamgaard at the Neils Bohr International Academy.
Appendix A: Microscopic Derivations1. Equation for the density of vortex lines.
We will start from Eqs. (2) and Eq. (3):1 δξ dδξdt = α V ns ( s , t ) · ( s ′ × s ′′ )+ s ′ · V s nl ′ − ˜ α s ′′ · V ns . (A1)The first simplification that we make is neglecting theterm proportional to ˜ α . This simplification is justifi-able for temperatures higher than T=1.3 K when ˜ α muchsmaller than α , but might be reconsidered for lower tem-peratures. Second, we can argue that the second term onthe RHS of Eq. (A1) is in fact negligible. To see this notethat V nl is a macroscopic field that changes slowly onthe length-scale of inter-vortex distance. For the presentflow which is inhomogeneous in the y -direction, we notethat due to symmetry all the average values depend onlyon the y-coordinate, and their only non-zero componentis in the x-direction. In contrast, (cid:10) s ′ y (cid:11) = h s ′ x i = 0. To-gether this is a strong reason to assert that this term isnegligible compared to the first term in Eq. (A1). In-deed in our numerical simulations we could confirm thisassertion, cf. Ref. .Next we can decompose the counterflow velocity V ns into a local − β s ′ × s ′′ and slowly-varying nonlocal V components:1 δξ dδξdt ≈ αV · ( s ′ × s ′′ ) − αβ | s ′′ | . (A2)Integrating this equation over the tangle results in anequation for the production and decay of the vortex linedensity: P L = α Ω Z V · ( s ′ × s ′′ ) dξ ≈ α L h κ i V ns · h b i , (A3) D L = 1Ω Z αβ | s ′′ | dξ ≈ αβ L (cid:10) κ (cid:11) . (A4)
2. The flux of the vortex lines density
The flux of the density of vortex lines is defined by: J L = D d s dt E . (A5)In the considered geometry only the y -component sur-vives:[ J L ] y ≈ h αs ′ z V ns i ≈ α h s ′ z i V ns = − ακ V ns dV s dy . (A6)Note that in general the vortex flux can also contain otherterms , for example a diffusive component ; howeverif the tangle is polarised, as in the case of counterflowor rotating turbulence, these terms are expected to benegligible. Nevertheless, in particular geometries theymay become important.
3. The equation for the anisotropy parameter
We start from Eq. (2) after discarding the term pro-portional to ˜ α : d s dt = V s + α s ′ × V ns . (A7)The derivative for binormal vector s ′ × s ′′ reads ddt ( s ′ × s ′′ ) = d s ′ dt × s ′′ + s ′ × d s ′′ dt . (A8) Using these equations together gives: ddt ( s ′ × s ′′ ) = V ′ s × s ′′ + α ( s ′′ × V ns ) × s ′′ + α ( s ′ × V ′ ns ) × s ′′ + s ′ × V ′′ s + α s ′ × ( s ′′′ × V ns )+2 α s ′ × ( s ′′ × V ′ ns ) + α s ′ × ( s ′ × V ′′ ns ) . (A9)To express higher derivatives in terms of s ′ and s ′′ wecan use the Frenet-Serret formula : s ′′′ = − κ s ′ + τ s ′ × s ′′ + κ ′ κ s ′′ , (A10)where κ is the curvature and τ is the torsion. At thispoint it is advantageous to split V ns into a local term − β s ′ × s ′′ and a non local term V : V ns = V − β s ′ × s ′′ . (A11)If we neglect terms with derivatives of the nonlocal ve-locity component, we get: ddt ( s ′ × s ′′ ) = 2 α κ V + α s ′ (cid:2) − β κ τ − κ ( s ′ · V ) (cid:3) + s ′′ (cid:2) βτ − α ( s ′′ · V ) (cid:3) − αβ (cid:2) κ + τ (cid:3) ( s ′ × s ′′ )(A12)The next step of simplification is to neglect terms pro-portional to τ . The torsion, in contrast to the curvature,is not positive definite and we can assume that the meanvalue of τ is negligible. Our current numerical simula-tions are not sufficiently precise to test this assertion, al-though preliminary results indicate that this assumptionis correct.Now having an equation for s ′ × s ′′ we can write anequation for the unit vector b : d b dt = 1 κ (cid:18) d s ′ × s ′′ dt − b d s ′ × s ′′ dt · b (cid:19) . (A13)Substituting d ( s ′ × s ′′ ) /dt : d b dt = α κ V − α κ (cid:0) b · V (cid:1) b + βτ s ′′ / κ (A14)
4. Flux of the anisotropy parameter
The flux of the anisotropy parameter I l can be writtenas: ˆ J I l = D d s dt ⊗ b E . (A15)In the channel geometry only the xy component of thetensor survives:[ J I l x ] y = Dh d s dt i y b x E ≈ h β κ b x b y i + h αs ′ z V ns b x i . (A16)The first term is given by local velocity and with a goodaccuracy can be expressed as: h b x b y i = * s ′ y s ′ z s ′′ z s ′′ x − s ′ x s ′ y s ′′ z − s ′ z s ′′ z s ′′ x + s ′ x s ′ z s ′′ y s ′′ z κ + ≈ − (cid:10) s ′ x s ′ y (cid:11) / , (A17)while for second term we make the uncontrolled approx-imation: h αs ′ z V ns b x i ≈ (cid:10) αs ′ z V (cid:11) h b x i − αβ h s ′ z i (cid:10) κ b x (cid:11) . This approximation needs to be verified in future numer-ical simulations. J. Maurer and P. Tabeling,
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