Dissipation in quantum turbulence in superfluid 4 He above 1K
aa r X i v : . [ c ond - m a t . o t h e r] A p r Dissipation in quantum turbulence in superfluid He above 1K
J. Gao and W. Guo
National High Magnetic Field Laboratory, 1800 East Paul Dirac Drive, Tallahassee, FL 32310, USA andMechanical Engineering Department, Florida State University, Tallahassee, FL 32310, USA
S. Yui and M. Tsubota
Department of Physics, Osaka City University, 3-3-138 Sugimoto, Sumiyoshi-Ku, Osaka 558-8585, Japan
W.F. Vinen
School of Physics and Astronomy, University of Birmingham, Birmingham B15 2TT, United Kingdom (Dated: November 9, 2018)There are two commonly discussed forms of quantum turbulence in superfluid He above 1K: inone there is a random tangle of quantizes vortex lines, existing in the presence of a non-turbulentnormal fluid; in the second there is a coupled turbulent motion of the two fluids, often exhibitingquasi-classical characteristics on scales larger than the separation between the quantized vortexlines in the superfluid component. The decay of vortex line density, L , in the former case is oftendescribed by the equation dL/dt = − χ ( κ/ π ) L , where κ is the quantum of circulation, and χ is adimensionless parameter of order unity. The decay of total turbulent energy, E , in the second case isoften characterized by an effective kinematic viscosity, ν ′ , such that dE/dt = − ν ′ κ L . We presentnew values of χ derived from numerical simulations and from experiment, which we compare withthose derived from a theory developed by Vinen and Niemela. We summarise what is presentlyknown about the values of ν ′ from experiment, and we present a brief introductory discussion of therelationship between χ and ν ′ , leaving a more detailed discussion to a later paper. PACS numbers: 67.25.dg, 67.25.dk, 67.25.dm
I. INTRODUCTION
Below about 2.17 K, liquid He becomes a superfluid,in which an inviscid irrotational superfluid componentcoexists with a viscous normal-fluid component [1]. Anyvorticity in the superfluid component is confined to quan-tized vortex lines, each of which carries a single quantumof circulation κ = h/m , where h is Planck’s constant and m is the mass of a He atom [2]. Flow in each of thetwo fluids can be turbulent. Turbulence in the superfluidcomponent (quantum turbulence) takes the form of anirregular tangle of vortex lines which interact with eachother and, through a force of “mutual friction”, with thenormal fluid [3]. Turbulence in the normal fluid is similarto that in a classical fluid, but modified by the mutualfriction. Dissipation, associated with viscosity, plays animportant role in classical turbulence; notably, in pro-viding a sink where the energy flux in a high Reynoldsnumber Richardson cascade can be absorbed at smalllength scales. It must play a similarly important rolein quantum turbulence, although, as we shall see, dissi-pative mechanisms are then more complex than in theclassical case.Except at temperatures well below 1K, where the nor-mal fluid has disappeared, dissipation in the turbulent su-perfluid component is due, as we shall see, to the mutualfriction. If we ignore a small transverse (non-dissipative)component, the force of mutual friction per unit length ofvortex line can be expressed in terms of a dimensionlessparameter α [3]. Except at temperatures very close tothe superfluid transition temperature, α is significantly less than unity, with the result that vortex line motionis determined largely by vortex-vortex interactions, themutual friction leading to only a relatively slow shrink-age in the total length, L , of vortex line per unit volume.Dissipation in the normal fluid is due to both mutualfriction and viscosity.It is the aim of this paper to discuss these forms ofdissipation for two commonly studied types of quantumturbulence (QT), the dissipation being observed in thefree decay of the turbulence.QT can be most easily produced by a heat current,which is carried in superfluid helium by a counterflowof the two fluids, and this is the form of QT that wasfirst subject to detailed experimental study [4–6]. It wasthought for many years that this thermal counterflow tur-bulence (TCT) involved only the superfluid component,and took the form of a more or less random vortex tangle,for which the turbulent energy is confined to scales com-parable with or less than the average spacing, ℓ = L − / ,between the vortex lines. The corresponding energy spec-trum, E Q ( k ), has the form E Q ( k ) = ρ s κ πρℓ k f (cid:16) kℓ π (cid:17) , (1)where the function f ( x ) depends on the precise form ofthe “random tangle”, but tends to unity for large x , andtends rapidly to zero for x < ρ s /ρ is the superfluidfraction. It was suggested, on dimensional and physicalgrounds [6], that, when the heat current is switched off,the line density might decay as dLdt = − χ κ π L , (2)where χ is a dimensionless parameter of order unity.Noting that the energy per unit mass associated with arandom tangle of vortex lines is given by E Q = Z ∞ E Q ( k ) dk ≈ ρ s κ πρ L ln ℓξ , (3)where ξ is the vortex core parameter, we see that theturbulent energy per unit mass would then decay as dE r dt = − ν ′ v κ L , ν ′ v κ = χ ρ s π ρ ln ℓξ , (4)where ν ′ v is an effective kinematic viscosity.Recent experiments [8, 9], based on the use of He ∗ ex-cimer molecules as tracers of the normal-fluid flow, haveshown that this form of QT, involving only what we shallcall a random vortex tangle , exists in TCT only at suffi-ciently small heat fluxes; at larger heat fluxes the tangleis accompanied by turbulence in both fluids on scales upto the size of the containing channel. We shall write theresulting energy spectrum as E ( k ) = E Q ( k ) + E Cs ( k ) + E Cn ( k ) , (5)where E Q ( k ) is still given by Eq.(1), E Cs ( k ) is pro-duced by partial polarization of the vortex lines, and E Cn ( k ) relates to the turbulent energy in the normalfluid. In the steady state this large-scale turbulence inthe two fluids is partially coupled and has an energy spec-trum, E ( k ) ∝ k − n on scales significantly larger than ℓ ,where the exponent n varies with the heat flux but is al-ways larger that the Kolmogorov value [10], 5 / n > / k − n energy on large scales. Finally, over a fur-ther period of typically 1-10 s, the energy spectrum onlarge scales evolves into the form expected for in a classi-cal inertial-range Richardson cascade; i.e. a Kolmogorovspectrum, E ( k ) ∝ k − / [10].We emphasize three points relating to the fully cou-pled turbulence: as long as full coupling is maintained,there is no dissipation due to mutual friction; the largescale non-dissipative motion in the superfluid componentis generated by a partial polarization of the vortex lines;and large scale motion in the normal component is non-dissipative because the viscosity of the normal fluid issufficiently small. As we shall see more clearly later, dis-sipation can occur in both fluids on scales comparablewith or less than ℓ , that in the superfluid componentbeing due to mutual friction, partial decoupling having occurred, and that in the normal component being duea combination of viscosity and mutual friction. Becausedissipation on scales of order ℓ is now much more com-plicated than is the case if the turbulence is confined tothe superfluid component and to scales of order ℓ , Eq.(4)need no longer apply.The decay of line density associated with large-scalecoupled turbulence was first studied by Stalp et al [11],the coupled turbulence having been generated in thewake of a moving grid. These authors showed that theirexperimental results could be explained in purely classi-cal terms, if it was assumed that there was at all timesa Richardson-Kolmogorov cascade ( E ( k ) ∝ k − / ), ter-minated at small scales by dissipation described by theequation dE C dt = − ǫ = − ν ′ κ L , (6)where ν ′ is another effective kinematic viscosity; E C isthe total quasi-classical turbulent energy, given by in-tegrating E Cs ( k ) + E Cn ( k ) over k (the contribution of E Q ( k ) to the total energy is small and can be neglected).Stalp et al argued that Eq.(6) is the analogue of the ex-pression ν h ω i for dissipation in classical homogeneousturbulence, where h ω i is the mean square classical vor-ticity. We emphasize that, although the expressions (4)and (6) for the rate of decay of turbulent energy are sim-ilar in form, they relate to different physical situations,and in neither case has there been any really rigorousdiscussion of their validity. Furthermore, as we shall dis-cuss later, the two effective kinematic viscosities, ν ′ v and ν ′ need not have the same value. In future we shall referto large-scale coupled turbulence of the type produced byflow through grid, or in the decay of strongly excited TCTat large times, as quasi-classical quantum turbulence .We remark here that a Kolmogorov energy spectrumcan, strictly speaking, apply only to a steady state inwhich energy is fed in continuously at some large scale D at a rate ǫ ; there is then a constant energy flux, equal to ǫ , down an inertial sub-range, 2 π/D ≫ k ≫ π/ℓ , withinwhich the energy spectrum has the full Kolmogorov form E ( k ) ∼ ǫ / k − / (we are ignoring the effects of intermit-tency) . In decaying turbulence the energy flux, ǫ , cannotbe strictly independent of either time or wave number,so that the Kolmogorov dependence on wave number, k − / , cannot be strictly correct. In practice, however,most of the energy is often concentrated in the largest ed-dies (wave numbers close to 2 π/D ), so that ǫ is indepen-dent of k , for k > π/D , to a reasonable approximation;and the decay is sufficiently slow that the Kolmogorovspectrum holds with a slowly decreasing value of ǫ .Except perhaps for a simple theoretical calculation of χ , reviewed later in Section III, there has so far beenhardly any serious theoretical justification for the twoforms of dissipation, and for many years even experimen-tal justification was inadequate. Similarly it has proveddifficult to derive reliable values of the two effective kine-matic viscosities from experiment. In the case of ν ′ v (orequivalently χ ) there had been no careful study of thedecay of TCT at heat currents sufficiently small thatthere was no large-scale turbulence. In the other case val-ues of ν ′ were obtained from observations of the decay ofvortex line density combined with questionable assump-tions about the form of the large-scale energy spectrumas it relates to turbulence in a channel of finite cross-section. Only very recently has ν ′ been determined in amore satisfactory way for the case of decaying TCT [12],although the results have yet to be compared carefullywith those obtained solely from the decay of vortex linedensity. The general aims of this paper are, as far aspossible, to remedy these various shortcomings.The results of our new experiments on the decay of arandom vortex tangle and our measurements of χ aredescribed in Section II. In Section III we summarize anexisting theory of χ , assess its likely validity, and com-pare its predictions with experiment. In Section IV wedescribe the numerical simulations relating to a randomvortex tangle, and we compare the results with the exper-iment and with the theory of Section III. In Section V wepresent a critical summary of our present knowledge ofthe experimental values of the effective kinematic viscos-ity ν ′ , and in Section VI we present a brief introductorytheoretical discussion of the relationship between χ (or ν ′ v ) and ν ′ , leaving a more serious discussion of what isactually a difficult problem to a later paper. We presentan overall summary of our work in Section VII. II. DISSIPATION IN A RANDOM VORTEXTANGLE: THE EXPERIMENTALMEASUREMENT OF χ . Our new experiments on the decay of vortex line den-sity associated with TCT have been based on the ob-served attenuation of second sound, using what is now astandard technique, as described in, for example, refer-ences [5, 13]. The actual apparatus is identical with thatdescribed in reference [8].As we have explained, the form of decay of vortex linedensity given by Eq.(2) can be expected to be observedin the decay of TCT only if the steady heat flux is smallenough to ensure that there is no large-scale turbulence.This is indeed the case is evident in the decay shown bythe lowest line in Fig.1.In Fig.2 we show data for a decay from a small heatflux plotted in a form, (1 /L ) versus t , which serves todemonstrate more clearly that Eq.(2) is indeed obeyed.Values of χ deduced from decays of this type areshown as a function of temperature in Fig.3. III. DISSIPATION IN A RANDOM VORTEXTANGLE: A THEORY OF χ In this section we shall summarise a theory of χ thatwas proposed by Vinen and Niemela [3], and we shall ~ t −
426 mW/cm150 mW/cm50 mW/cm
10 2 ( ) (1 ) L L t L t χ κπ − = + t (s) L ( c m ) - FIG. 1. (color online) Observed decays of vortex line densityin decaying TCT (1.65K) . t (s) / L ( c m ) − T=1.65 K
FIG. 2. Observed decay in line density from a small heat flux. compare the results with experiment.We assume that the force of mutual friction per unitlength of vortex line is given by f = − ρ s κα ˆ κ × [ˆ κ × ( v n − v L )] , (7)where ˆ κ is a unit vector along the length of the vortex and v L is the velocity with which the vortex moves perpen-dicular to its length. We have neglected any transversecomponent of the mutual friction. We shall further as-sume that during the decay described by Eq.(2) the nor-mal fluid is at rest, apart from the local dragging by amovingˆaˆa vortex that is incorporated into the definitionof the mutual friction parameter α [14]. Dissipation isthen due entirely to mutual friction. Finally, we shall as-sume that the magnitude of v L is given to a good enough T (K) χ FIG. 3. (color online) Observed (open circles) and theoretical(filled circles) values of χ , the theoretical values being derivedfrom Eq.(12). approximation by the local induction approximation v L = κ πR ln (cid:16) Rξ (cid:17) , (8)where R is the local radius of curvature of the vortex,and ξ is the vortex core parameter. In other words, wehave neglected the effect of both long-range interactionsand the force of mutual friction itself on the motion of avortex. It follows that the rate of dissipation of energyper unit mass of helium is given by dE r dt = − ρ s ρ καL h v L i = − α (cid:16) ρ s κ π ρ (cid:17) h h R (cid:16) ln Rξ (cid:17) i i L, (9)where h ... i denotes an average over the vortex tangle. Weneglect the slow variation of the logarithmic term with L ,putting R = R ≈ ℓ in that term, and we follow Schwarz[15] by assuming that h h R i i = c L, (10)where c depends only on temperature. It follows that ν ′ v κ = αc ρ s π ρ h ln ℓξ i , (11)and therefore χ = αc ℓξ . (12)This derivation is based on three assumptions: that,as we have mentioned, there is no motion of the normalfluid; that the vortex lines form a random tangle; andthat use of the local induction approximation is justi-fied. We shall present an argument in favour of the first assumption in Section VI. The other assumptions seemreasonable.Values of χ derived from Eq.(12) are included in Fig.3.The required values of c are taken from the simulationsof the steady state described in Section IV, and values of α are taken from reference [16]. We see that within theerror bars there is agreement with experiment. IV. DISSIPATION IN A RANDOM VORTEXTANGLE: SIMULATIONS RELATING TO χ A brief report of simulations leading a verification ofthe form of the decay of line density and to values of χ ata temperature of 1.9 K has already been published [17].Here we present the results of a more detailed studies,covering a range of temperatures, first for the case ofspatially uniform flows, and then for flows between solidboundaries. A. The steady state for spatially uniform flows.
For a given temperature we must first simulate thesteady state counterflow, for two reaons. It is from thesestates that the decays must start, and we can deter-mine whether values of the parameter c , obtained forthe steady state, lead via Eq.(12) to agreement with ex-perimentally observed values of χ .Our numerical simulation is based on the vortex fila-ment model with the full Biot-Savart integral. We carryout simulations for spatially uniform flows in a cubicalbox, side 1 mm, with periodic boundary conditions in alldirections. We replace the vortex lines by a discrete set ofpoints with minimum spatial resolution ∆ ξ = 8 . × − cm. We integrate in time with a fourth-order Runge-Kutta scheme with time resolution ∆ t = 1 . × − s. The initial state is a set of randomly oriented vor-tex loops of radius 0 .
23 mm. The spatially uniform ap-plied velocities satisfy the condition of no net mass flow ρ n v n + ρ s v s,a = 0. We have checked that any contributionto the net superflow from the evolving vortex tangle isnegligible in comparison with v s,a . The parameters usedin the simulations are shown in Table I. TABLE I. Parameters used in numerical simulations.T α α ′ v n K mm s − We run the simulations for 20 s. The vortex line den-sity, L , is found to reach a steady average value, L , withfluctuations, in about 5 s. The parameter c , calculatedfrom the Eq.(10) and the equation h R i = 1Ω L Z dξR , (13)where Ω is the volume of the numerical box, togetherwith the values of χ derived from Eq.(12), are shown asa function of time for a temperature of 1.4 K in Fig. 4. Aswe see, they too reach steady states after a few seconds,but with significant fluctuations. The relatively largefluctuations have their origin in the relatively small com-putational box; a larger box would require prohibitivelylong computer runs. We have performed similar simula-tions for several temperatures, the results of which aresummarized in terms of time-averages in Table II. Thecomputed values of χ in Fig. 3 were taken from Ta-ble II.We emphasize that the theoretical/computational val-ues of χ plotted in Fig.3 were derived from Table II;the agreement with experiment was therefore evidencethat Eq.(12) is at least approximately valid if the valuesof c are taken from numerical simulations of the steadystate. We must now turn to numerical simulations of thedecaying turbulence, to check whether the simulated de-cays obey Eq.(2) with values of χ that agree with thosein Table II. TABLE II. Statistically steady values of the vortex line den-sity, L , the parameter c , the mean radius of curvature, R ,and the corresponding values of χ derived from Eq.(12). T v n L c R χ K cm/s 10 cm − − cm1 . . . ± .
03 2 . ± .
12 5 . ± .
25 2 . ± . . . . ± .
24 2 . ± .
13 5 . ± .
32 2 . ± . . . . ± .
25 2 . ± .
12 5 . ± .
31 2 . ± . . . . ± .
30 2 . ± .
11 6 . ± .
33 3 . ± . . . . ± .
29 2 . ± .
21 6 . ± .
59 2 . ± . . . . ± .
27 2 . ± .
08 4 . ± .
13 1 . ± . B. Decays from spatially uniform flows
In these simulations the applied velocities, v n and v s,a ,are turned off at time t = 0, and the way in which the linedensity decays with t is determined. Data are averagedover 30 decays at each temperature.Fig.5 shows the way in which the simulated line densitydecays with time at 1.4K, in the form of a plot of 1 /L against time.We see that, in contrast to the corresponding exper-imental decay (Fig.2), Eq.(2) is apparently not obeyed;the slope of the plotted line, which ought to be propor-tional to the constant χ increases markedly with time(the values of χ are also too large). The increase at timesgreater than about 1 s may be due in part to the vortexline density becoming too small (the ratio of line spac-ing to the spatial period has become greater than about c t (s)(a) χ t (s)(a)(b) FIG. 4. Value of c derived from simulations of the approachof counterflow to a steady state, and the corresponding valueof χ derived from Eq.(12). t (s) / L ( mm ) = 2.6 χ = 3.1 χ FIG. 5. (color online) (1 /L ) plotted against time from simu-lations at T = 1 .
4K and v n = 9 mm s − . c in Eq.(12) changesduring the simulated decay. That c does indeed changeduring the simulated decay is shown in Fig.6(a); further-more, as we see from Fig.6(b), this changing c leads via Eq.(12) to a changing value of χ that would lead, at leastqualitatively, to a decay curve with the shape shown inFig.5. Similar results emerge from simulations at othertemperatures. c t (s)(a) χ t (s)(a)(b) FIG. 6. a) The variation of the parameter c with time fromsimulations of the decaying line density at T = 1 .
4K and v n = 9 mm s − . (b) The variation of χ with time, obtainedby substituting c from Fig.6a into Eq.(12). That the variation with time of the slope of the linein Fig.5 is indeed due to the variation with time of theparameter c is shown more strikingly in Fig.7, where wecompare on the same graph the time dependence of thevalue χ derived both by differentiating 1 /L in Fig.5 withrespect to time and by substituting the value of c fromFig.6(a) into Eq.(12). Even the random fluctuations of c are reflected to a significant degree in fluctuations in χ derived from Fig.5. The situation at other temperaturesis similar. We conclude then that the theory underlyingEq.(12) is in reasonably good agreement with the resultsof the simulations, but not, to a significant extent, withexperiment. This suggests strongly that some factor rel-evant to the experiments is missing from both the theoryand the simulations. A possible candidate for this factoris the fact that, in contrast to the theory and the simula-tions, the experiments relate to flow in a channel of finitecross-section. We investigate this possibility in the nextsub-section. χ t (s) (2 π / κ ) (dL -1 /dt)Vinen-Niemela Eq. FIG. 7. (color online) Plots of χ against time derived asexplained in the text. C. Decays from flows in channels of finitecross-section
Simulations relating to decaying counterflow in twotypes of channel have been carried out: one is formedbetween two parallel solid boundaries, separated by 1mm; the other is a channel with square (1 mm × TABLE III. Parameters analogous to those in Table II, forflow between parallel plates.
T v n L c R χ K cm s − cm − − cm1 . . . ± .
64 2 . ± .
14 6 . ± .
37 1 . ± . . . . ± .
43 2 . ± .
14 6 . ± .
41 1 . ± . . . . ± .
70 1 . ± .
12 6 . ± .
39 2 . ± . . . . ± .
67 1 . ± .
11 6 . ± .
41 2 . ± . Before we proceed further we recall that the presence ofsolid boundaries in the steady state is known from simu-lations to lead to severe spatial inhomogeneity in the vor-tex line density [18–21]; the vortex line density is greatlyenhanced near the boundary (values of L , and other pa-rameters, in Table III are spatial averages). We musttherefore enquire whether there is also inhomogeneity inthe value of c . That there is indeed such inhomogeneityis shown in Figs 8 and 9, derived from the simulations.We see that the parameter c is strongly reduced in re-gions where the vortex line density is increased, and thatthis reduction persists in time during a decay. c y (mm) Parallel platesSpatially uniform flow
FIG. 8. (color online) Plots showing how c , averaged overtime in the steady state, varies with position across the chan-nel. Blue line: flow between parallel plates; red line: spatiallyuniform flow. Temperature = 1.4 K.
0 0.5 1 1.5 2 t (s) -0.5 0 0.5 y ( mm ) c FIG. 9. (color online) Diagram showing how c varies withposition across the channel and with time during a decay.Temperature = 1.4 K. The experimental observations of χ , reported in Sec-tion II, relate to the decay of spatially averaged vortexline densities. Our simulations of the decays between par-allel plates lead to the corresponding values of χ thatare displayed in Fig.10, where they are compared withthe predictions of Eq.(12), in which we have substitutedspatially averaged values of the parameter c taken fromour simulations. We see that the agreement between thesimulations and the predictions of Eq.(12) is still goodand provides further confirmation that the theory of Sec-tion III is valid. Furthermore, for times less than 1 s,the variation with time of χ has largely disappeared,and that the actual values of χ are in better agreementwith experiment. This improved agreement with exper-iment is comforting and suggests that boundary effectsare important in determining values of c and thereforethe precise form of the decays. However, reservationsmust be emphasized. It is now clear that values of c are quite sensitive to the precise form of the flows, and χ (a) 1.4 K (2 π / κ ) (dL -1 /dt) Vinen-Niemela Eq. χ (a) 1.4 K(b) 1.5 K χ (a) 1.4 K(b) 1.5 K(c) 1.6 K χ t (s) (a) 1.4 K(b) 1.5 K(c) 1.6 K(d) 1.7 K FIG. 10. (color online) Plots of χ against time for flow be-tween parallel plates. our simulations still relate to flows that are not exactlythe same as those in our experiments. The experiments[8] use wider channels; in practice the velocity profileof the normal fluid differs generally from the Poiseuilleform [8]; and in practice the vortex lines in the superfluidcomponent are likely to suffer drag or pinning at the solidboundaries. Simulations that take account of these dif-ferences are starting to be practicable (Yui, Tsubota andKobayashi, to be published), and could eventually allowmore satisfactory comparison with experiment. V. DISSIPATION IN QUASI-CLASSICALQUANTUM TURBULENCE: EXPERIMENTALVALUES OF ν ′ We turn now to the decay of large-scale coupled tur-bulence, as observed in the wake of flow through a gridand in the decay of TCT when the steady heat flux islarge. We shall not be concerned with the early stagesin these decays. In the case of grid turbulence, it hasbeen supposed [11] that a Kolmogorov spectrum is es-tablished quickly, with energy-containing eddies havinga size significantly smaller than the channel width; thenthe energy-containing eddies grow in size, essentially bya classical process (see ref. [22], pg.347), until their sizesaturates at a value comparable with the width of thechannel. Recent experiments have cast doubt on the sup-posed details of this evolution of the energy-containingeddies, but, as we shall see, there seems now to be lit-tle doubt that eventually the turbulence settles down toa quasi-steady state in which the energy-containing ed-dies have a fixed size, determined by the channel width,and in which there is an inertial sub-range characterizedby a Kolmogorov energy spectrum, terminated by dissi-pation described by Eq.(6). In the case of the decay ofTCT when the steady heat flux is large, the initial stagesare complicated, as we outlined in our introduction, butagain there is little doubt that eventually the turbulencesettles down to a state similar to that seen in the decayof grid turbulence.As was shown first by Stalp et al [11], the decay ofvortex line density in the state to which the turbulencesettles down is given by L ( t ) = (3 C ) / D πκν ′ / ( t − t ) − / , (14)where C is the Kolmogorov constant, D is the (time-independent) linear size of the energy-containing eddies,and t is a constant. Eq.(14) is based on an assumedenergy spectrum that has the Kolmogorov form with asimple cut-off for wave number less than 2 π/D . Untilrecently, all measurements of the effective kinematic vis-cosity, ν ′ , have been based on observations of L ( t ) andthe assumption that D is exactly equal to the width of thechannel in which the flow is taking place. The question-able assumptions underlying this work meant that thevalues of ν ′ were, at best, uncertain to within a factorof perhaps two or three. Furthermore, since the effec-tive size of the energy-containing eddies could depend onthe precise way in which the turbulence was generated,values of ν ′ from different experiments might not agree.This uncertainty can be circumvented if a measure-ment of L ( t ) is combined with a measurement of the wayin the total turbulent energy decays, since this decay oftotal energy yields the value of the energy flux, ǫ , inEq.(6). The time-dependence of the total energy can bededuced from the recently developed visualization tech-nique based on the use of He ∗ excimer molecules as trac-ers, provided that it is assumed that the turbulence isisotropic. The first such study, on the decay of TCT, wasreported recently [12], and the resulting values of ν ′ aredisplayed in Fig.11, along with values of ν ′ derived fromthe same measurements of the decay of line density, butusing Eq.(14) (all these measurements relate to a channelwith square cross section, 9.5 mm × D in T (K)10 -1 Based on both turbulent energy and vortex density decaysBased on vortex densitydecay only
FIG. 11. (color online) Values of ν ′ for decaying TCT derivedfrom measurements of both the decaying turbulent energy andthe decaying vortex line density. Values of ν ′ derived from thedecay of line density alone, based on Eq.(14) are included forcomparison. Eq.(14) was taken to be 9.5 mm). We see that the mea-surements based on the new technique are systematicallyslightly larger than those based on Eq.(14), but only bya factor that is barely outside the experimental error.
T (K)1.2 1.4 1.6 1.8 2.0 2.2
FIG. 12. (color online) Values of ν ′ for various types of de-caying coupled turbulence. N : ref [12]; ◦ : ref [11]; (cid:7) : ref[13] decay of superflow in channel D7; • : ref [13] decay ofsuperflow in channel D10; (cid:4) : ref [13] decay of counterflow inchannel D10; × : ref [23] no grid; ∗ : ref [23] with grid. In Fig.12 we collect together the results of measure-ments of ν ′ for various types of decaying coupled quan-tum turbulence, as described in the caption to the figure.Most of these data were derived from measurements online density only, and for this reason are subject to someuncertainty. There is a hint that the value of ν ′ may de-pend a little on the type of flow, but the relatively largeexperimental errors make it hard to be sure. All thatwe can say is that ν ′ /κ lies in the range 0.1 to 1, itsvalue increasing as the temperature increases from 1.3Kto 2.1K.These measurements of ν ′ have all been based on thedecay of the quantum turbulence. Some informationabout ν ′ has also been obtained from observations ofvortex line density in the steady flow of superfluid Hethrough a channel or through a grid [23]. In essence, itwas tentatively assumed that the steady flow led to thegeneration of large eddies, size D and characteristic ve-locity U . The velocity U is assumed to be proportional tothe average steady flow velocity U ( U = ζU , where theconstant ζ is a little less than unity), and D is assumedto be independent of U . The large eddies are assumedto decay through a cascade at a rate determined by theturnover time D/U , the energy being ultimately dissi-pated at a rate given by Eq.(6). These assumptions leadthen to a steady vortex line density given by L = ζ / ν ′ κD ) / U / . (15)That L is proportional to U / is confirmed by experi-ment. Eq.(15) can then be used to estimate ν ′ , subjectto reasonable guesses about the values of ζ and D . Theresults are not inconsistent with those described above,demonstrating that the concept of an effective kinematicviscosity is applicable to dissipation in both steady anddecaying turbulence; but reliable absolute values of ν ′ cannot be deduced. VI. DISSIPATION IN QUASI-CLASSICALQUANTUM TURBULENCE: THE RELATIONBETWEEN ν ′ v AND ν ′ A. Introduction
We devote this section to an introductory discussion ofthe relation between ν ′ v , derived from our values of χ ,and ν ′ . We have already emphasized that these two kine-matic viscosities relate to different physical situations,and that they may not therefore be equal.In the case of ν ′ v we are dealing with a situation whereturbulent energy in the superfluid component is confinedto scales of order ℓ or less, in the form of a random vor-tex tangle, and we assumed in our earlier discussion thatthere was no turbulent motion of the normal fluid. Aswe have seen, turbulent energy is then being dissipatedby mutual friction, at a rate that is given to a reason-able approximation by the prediction of Eq.(12). In thecase of ν ′ there is again turbulent energy in the super-fluid component on scales of order ℓ or less, but this is accompanied by turbulent energy in both fluids at largerscales. On sufficiently large scales there is strong couplingbetween the two fluids, and viscosity in the normal fluidcan be neglected. There is then a Kolmogorov (inertialrange) energy spectrum in this coupled motion, leadingto constant fluxes of energy in k -space in both the su-perfluid and normal components ( ǫ s and ǫ n ). We mustdiscuss how this situation changes as the scale of the tur-bulence moves towards the scale ℓ ; in other words howthe energy spectra for the two fluids behave as the wavenumber approaches 2 π/ℓ . In connection with dissipation,we need ultimately to answer several questions. How, andat what wave numbers, is turbulent energy in the normalfluid dissipated, remembering that such dissipation canbe due to both viscosity and mutual friction? Is there sig-nificant dissipation in the superfluid component due tomutual friction at wave number smaller than 2 π/ℓ ? Andhow is dissipation in the superfluid component modified,in comparison with that for a random vortex tangle, forwave numbers of order or greater than 2 π/ℓ , by any mo-tion on those scales of the normal fluid or by the polar-ization of the tangle required to generate the large-scaleturbulence. B. Guidance from the calculations of Bou´e et al
These questions can be answered to some degree by ap-pealing to the work of Bou´e et al [24]. These authors useda two-fluid Sabra shell-model, based on modified HVBKequations, to calculate the energy spectra for both the su-perfluid and normal components. The HVBK equationsare course-grained (continuum) equations of motion forthe two fluids, and Bou´e et al modify them by the ad-dition of an effective kinematic viscosity, equal to our ν ′ , to the equation for the superfluid component. Our ν ′ is indeed an effective kinematic viscosity in the sensethat the associated dissipation, equal to ν ′ κ L , appearsto be analogous to the classical dissipation ν h ω i , where h ω i is the mean square classical vorticity. However, thisanalogy is misleading because our ν ′ is actually due, atleast in part, to mutual friction, so that its effect oughtnot to be represented by a term of the form ν ′ ∇ v s , asassumed by Bou´e et al . Leaving aside this questionableaspect of the analysis by Bou´e et al , there is still the as-sumption that course-grained equations of motion can beused. This assumption is probably justified in describingturbulence on scales large compared with the vortex linespacing, ℓ , but Bou´e et al use it on scales as small asthe vortex line spacing. It must fail on such small scales,although the precise scale below which it fails noticeablyis not clear. We shall return to this point later.In spite of these shortcomings, it is interesting to ex-amine the conclusions to be drawn from the analysis ofBou´e et al , especially as they relate to the effect of thefinite viscosity of the normal fluid in the range of temper-atures in which we are interested (Fig.1(b) in ref [24]).We find that, at temperatures less than about 1.5 K, the0normal fluid is brought to rest by its viscosity on a lengthscale significantly larger than ℓ , but that the superfluid isthen brought to rest only on significantly smaller scales,comparable with ℓ . At first sight this is surprising, be-cause it might be thought that with the normal fluid atrest the superfluid would also be brought to rest by mu-tual friction. There is, however, a simple explanation.On scales larger than ℓ there is according to Bou´e et al aflow of energy in the turbulent superfluid to higher wavenumbers in a Richardson cascade. If the normal fluid isat rest, this cascade has associated with it two character-istic times: the turnover time for eddies centred on wavenumber k , which is of order τ t = ( ku ) − , where u is thecharacteristic velocity in these eddies; and the time takenfor the energy in these eddies to be dissipated by mutualfriction, which is of order τ γ = ℓ /ακ . If τ t ≪ τ γ , thenthe mutual friction has little effect. It is easy to showthat this condition is indeed satisfied in the cases we areconsidering.At temperatures above about 1.5K Bou´e et al showthat energy is lost from both the normal component andthe superfluid component only on length scales compara-ble with ℓ . It follows then that at all temperatures rele-vant to the present study turbulent energy is lost from thesuperfluid component only on length scales comparablewith the vortex line spacing, ℓ . We emphasize that thisconclusion is dependent on the questionable assumptionthat turbulence in the superfluid component is behavingquasi-classically on scales down to a value close to thevortex line spacing ℓ . C. Dissipation in the superfluid component
If we accept this assumption, then we can concludethat, even in quasi-classical quantum turbulence of thetype we are considering, energy is dissipated in the super-fluid component only on length scales comparable withthe vortex line spacing ℓ , as is the case when we haveonly a random vortex tangle. It is therefore temptingto conclude that the dissipation in the superfluid compo-nent is still given by the theory of Section III. However,two questions must still be addressed. The first relatesto the fact that, according to Bou´e et al , and in con-trast to the assumptions in Section III, there is motionof the normal fluid on scales of order ℓ , at least at thehigher temperatures. But it seems reasonable to assumethat on this scale the vortex line velocity given by thelocal induction approximation, which is dominated byquantum effects, is uncorrelated with the velocity fieldof the normal fluid, which is essentially classical. In thiscase the theory of Section III still holds. The secondrelates to the fact that in our quasi-classical quantumturbulence the vortex lines must be moving at a velocitythat includes a quasi-classical component, correspondingto a large scale quasi-classical velocity field arising froma partial polarization of the lines. This component isassociated with the long-range, non-local, interaction of the vortex lines, and the large-scale coupling between thetwo fluids ensures that this component is not subject toany dissipation by mutual friction. But the fact that theargument of Section III is based on the local inductionapproximation ensures that this component is automat-ically excluded from any contribution to the dissipation(although the existence of the large-scale motion may in-fluence the value of c ).We conclude then that the dissipation in the super-fluid component in quasi-classical quantum turbulencemay still given correctly by the theory of Section III, sub-ject, of course, to the assumption implicit in the work ofBou´e et al that turbulence in the superfluid componentcan be regarded as quasi-classical on all scales larger than ℓ . D. The total energy dissipation
To obtain the total energy dissipation we must addthe energy dissipated in the normal fluid. We note thatat small wave numbers, within the inertial range, wherethere is complete coupling between the two fluids, theenergy fluxes in the normal and superfluid componentsmust be given respectively by ǫ n = ( ρ n /ρ ) ǫ and ǫ s =( ρ s /ρ ) ǫ , where ǫ is the total energy flux. It follows thatthe effective kinematic viscosity ν ′ , describing the totaldissipation, is given by ν ′ κ = αc π h ln ℓξ i = χ π ln ℓξ (16)We emphasize that, as is the case with ν ′ v , there is astrong dependence on the parameter c , to which we shallreturn. E. Comparison with experiment
In principle Eq.(16) serves to predict both the value of ν ′ and the relation between ν ′ and χ (or ν ′ v ). The latestexperimental data displayed in Figs. 3 and 11 are, withinlarge experimental errors, more or less consistent withthe predicted relation between ν ′ and χ . However, suchagreement has in practice little real significance, becauseall three dissipative coefficients depend on the parameter c , the precise value of which depends on the details ofthe flow concerned. Strictly speaking, our demonstrationthat c depends on these details has been established bysimulations that relate only to particular flows in whichthe normal fluid is not turbulent, and for which the den-sity of vortex line is small. These flows rarely correspondto reality, especially when we are dealing with flows athigh velocities that involve turbulence in both fluids andhigh densities of vortex line. Although the developmentof simulations that relate to these more general condi-tions has started, we can be fairly certain that full de-velopment will many years. In the meantime we must1assume that the dependence of c on the details of theflow is quite general. The consequences are particularlyserious for the value of ν ′ , since the flows to which ν ′ is applicable are as yet the least accessible to realisticsimulation.In comparing experiment and theory relating to quasi-classical quantum turbulence we must also recognize, aswe have already emphasized, that the theory is based onthe questionable assumption made by Bou´e et al that theturbulence in the superfluid component behaves classi-cally (in effect that the discrete vortex structure is unim-portant) even when the length scale is comparably with ℓ . Perhaps fully classical behaviour may not be required,but at least there must still be something equivalent toa Richardson cascade, with ”eddies” that have lifetimessufficiently small that they are not damped significantlyby mutual friction with a stationary normal fluid. Weguess that justification of this assumption can come onlyfrom suitable numerical simulations. It is our plan toattempt such simulations in the near future. VII. SUMMARY AND CONCLUSIONS
We have summarised what we know from experimentabout dissipation in quantum turbulence in superfluid He above 1K, the dissipation being described by eitherthe parameter χ , applicable to turbulence existing inthe superfluid component only on scales comparable withthe spacing between the quantized vortex lines (“randomtangles”) , or the effective kinematic viscosity ν ′ applica-ble to quasi-classical quantum turbulence, such as that generated by flow through a grid. Theoretical predictionsfor these two parameters are discussed, and it is arguedthat both depend on the dissipative effects of mutual fric-tion, which are in turn dependent on the dimensionlessparameter c that relates the mean square curvature ofthe vortices to their mean square separation. This pa-rameter can in principle be obtained from simulations,but it is argued that simulations that are sufficiently re-alistic are for the most part not yet practicable. To thisextent our understanding of dissipation in quantum tur-bulence in He above 1K remains incomplete.We have also drawn attention to the need to inves-tigate more carefully than hitherto the extent to whichturbulence in the superfluid component can be treatedclassically on length scales larger than, but comparablewith, the spacing between the vortex lines.
ACKNOWLEDGMENTS
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