D. Kivotides et al- Quantum Signature of Superfluid Turbulence

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VOLUME 87, NUMBER 27 P H Y S I C A L R E V I E W L E T T E R S 31 DECEMBER 2001

Quantum Signature of Superfluid Turbulence

D. Kivotides,1 J. C. Vassilicos,2 C. F. Barenghi,1 M. A.I. Khan,3 and D. C. Samuels1

1 Mathematics Department, University of Newcastle, Newcastle NE1 7RU, United Kingdom2 Department of Aeronautics, Imperial College of Science, Technology and Medicine,

South Kensington, London SW7 2BY, United Kingdom3 DAMTP, University of Cambridge, Silver Street, Cambridge CB3 9EW, United Kingdom

(Received 18 July 2001; published 10 December 2001)

Using a numerical simulation backed up by physical arguments, we predict that the pressure spectrum

of superfluid turbulence has a k22 dependence on the wave number k, which represents a macroscopic

quantum signature not to be found in the classical Kolmogorov theory of turbulence.

DOI: 10.1103 /PhysRevLett.87.275302 PACS numbers: 67.40. w, 47.37. +q

Experiments performed in the last few years showed thatsome turbulent flows of the superfluid phase of 4He (he-

lium II) are similar to analogous turbulent flows in a clas-

sical fluid. Typically helium II was made turbulent using

a towed grid [1] or a rotating propeller [2], or pushing it

at high velocity along pipes and channels [3] or around

spheres [4]. The apparent classical results (in terms of

energy spectrum, decay rates, pressure drops, drag crisis,etc.) are surprising because helium II is a quantum fluid.

According to Landaus two-fluid theory, it consists of two

copenetrating fluid components, the normal fluid and the

superfluid, whose relative proportion depends on the abso-lute temperature T. The normal fluid, which is viscous, can

form eddies of any size and strength. On the contrary, rota-

tion in the superfluid component is constrained by quantum

mechanics, and is possible only in the form of quantized

vortex filaments. All superfluid vortex filaments have the

same quantized circulation G 9.97 3 1024 cm2secand the same microscopic core radiusa 1028 cm. Sincethe superfluids viscosity is zero, the issue raised by the

experiments is what causes the observed classical behav-

ior, which remarkably extends to temperatures so low

T 1.4 K that the normal fluid fraction is only a fewpercent. The problem has stimulated theoretical investiga-

tions of the coupling between superfluid vortices and nor-

mal fluid [5,6], as well as studies of vortex tangles [7] and

dissipation in the limit of absolute zero [8], including the

effects of Kelvin waves and reconnections [9].This Letter attempts to clarify the limits of validity of

the similarities between helium II turbulence and clas-

sical turbulence by drawing attention to the spectrum of

the pressure field p. Stimulated by the helium turbulence

experiments and by the recent work on pressure spectra inclassical turbulence [1012], we show that the spectrumof p in turbulent helium II must be very different from

the classical Kolmogorov spectrum of a classical turbulent

flow. Using results of numerical simulations confirmed by

theoretical arguments we show that the pressure spectrum

Epk, defined such that

Z`0

Epk dk 1

V

ZZZpr

2

dV , (1)

where r is helium IIs density, scales like Epp k22,

where k jkj is the magnitude of the three-dimensionalwave number k and V is volume. This result is in con-trast with what happens in the classical theory of Kol-mogorov turbulence. Following the classical argument ofKolmogorov, assuming that in the inertial range Epk de-pends only on the wave number k and the energy dissi-

pation rate e dy22dt where y is the velocity, it isfound by dimensional argument that Epk e

43k273,

a scaling [13] which corresponds to the celebrated k253

power law of the energy spectrum. The experimentalistsshould not have difficulty in distinguishing the two powerlaws: k22 in turbulent helium II and k273 k22.33 inclassical turbulence (e.g., helium I), hence in identifyingthe quantum signature of helium II turbulence.

Our numerical simulations are based on the approach ofSchwarz [14]. We represent a quantized vortex filamentas a space curve s sj, t where j is arclength and t istime, which moves with velocity vL dsdt given by

vL vs 1 as0 3 vn 2 vs 1 a0s0 3 s0 3 vn 2 vs ,

(2)

where s0 dsdj, a and a0 are mutual friction coeffi-cients [15], vn is the prescribed normal fluid velocity, andvs is the self-induced velocity at the point s given by

vs G

4p

Z r 2 s 3 drjr 2 sj3

. (3)

The calculation is performed in a periodic box of vol-ume V d3 0.13 cm3. The numerical technique isstandard [14] and the details of our algorithm, includinghow to perform vortex reconnections, have been publishedelsewhere [9]. The time step is chosen so that we can re-solve Kelvin waves corresponding to the minimum spatialscale, d d128 0.78 3 1023 cm.

In the current experiments helium II is made turbulentby standard classical techniques (imposed pressure gra-dients, grids, propellers, etc.), so it is natural to assumethat the normal fluid vn consists of a uniform mean flowand superimposed turbulent fluctuations, vn Un 1 un.Following Vassilicos and co-workers [16], we choose the

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fluctuations un so that they have a Kolmogorov energyspectrum Enk k

253, which is defined such thatZkmax0

Enk dk 1

V

ZZZ1

2jv2nj dV , (4)

where kmax 12h where h represents the Kolmogorovlength. Specifically we have

un

mM

Xm1

Am 3 km coskm ? x 1 vmt

1 Bm 3 km sinkm ? x 1 vmt , (5)

where M 64 is the number of modes, km is a randomunit vector km kmkm, and the directions and orien-tations of Am and Bm are chosen randomly under theassumption (with no loss of generality) that they are nor-mal to km, the random choice of directions for the mthwave mode being independent of the choices of the otherwave modes. Note that the velocity field un is incom-pressible by construction. The amplitudes Am and Bmof the vectors Am and Bm are determined by the Kol-

mogorov energy spectrum Enk k253 via the rela-tions 32A2m 32B

2m EnkmDkm where Dkm

km11 2 km212. Finally, the unsteadiness frequenciesvm are determined by the eddy turnover time of wavemode m, that is, vm

pk3mEnkm.

The results presented here correspond to the followingchoice of parameters: T 1.3 K, Un 2.36 cmsec,and un,rms 11.927 cmsec. The Kolmogorov lengthh 0.78 3 1023 cm is consistent with the minimumscale d used; h is estimated from h R234

where d2 is the integral scale, R un,rmsnn,nn mrn 2.34 3 10

23 cm2sec is the kinematicviscosity of the normal fluid, rn is the normal fluid den-

sity, and m is the viscosity of helium II. The calculationstarts with some initial superfluid vortex rings [14] (theresult is independent of the arbitrary initial configura-tion), which immediately interact with each other andwith the normal fluid, soon forming a tangle of vortexlines. We stop the calculation at time t 0.093 secwhen the tangle is so intense that the vortex line densityis L0 LV 14180 cm

22 where L is the totalvortex length; this value is large enough to be typical ofexperiments [17].

We now calculate the pressure spectrum Epk.Figure 1 shows that the pressure spectrum is visi-bly different from the Kolmogorov k273 pressurespectrum over a range of wave numbers smaller than1h. It exhibits a k22 shape even though the energyof the normal fluid follows the Kolmogorov k253

spectrum over the same range of wave numbers. In fact,as shown by Kivotides et al. [18], even the energy of thetotal fluid follows the Kolmogorov k253 spectrum overthis wave number range. The pressure spectrum of thesuperfluid turbulence stands out because of its nonclassicalbehavior over inertial range wave numbers.

FIG. 1. Compensated pressure spectrum Epk (in units ofcm5sec4) as a function of wave number k (in units of cm21).Epk is multiplied times k

2 to make the scaling Epk k22

apparent. The solid line with crosses shows the result of our cal-culation for a vortex tangle of density L0 14180 cm

22. Thesolid line without crosses shows the classical Kolmogorov pres-sure scaling k273. The difference between the quantum (k22)and classical (k273) dependence is clearly noticeable.

It is interesting to notice that we find that the scalingEp k

22 is valid in the region 12h $ k $ 10

119 cm21 where 0 L2120 8.4 3 10

23 cm is the av-erage spacing between the vortex lines. A similar k22

dependence of the spectrum is found in runs with smallervalues ofun,rms.

Why the k22 dependence? The k22 pressure spectrumis a consequence of the locality in physical space of thevorticity filaments in the superfluid turbulence, as the fol-lowing argument shows. Consider the equations of motionof the normal fluid and the superfluid [19]:

rn vn

t 1 rnvn ? =vn 2 rnr =

p 2 rsS=T

1 m=2un 1 Fns , (6)

rsvs

t1 rsvs ? =vs 2

rs

r=p 1 rsS=T 2 Fns ,

(7)

where S is the entropy per unit mass, rs r 2 rn thesuperfluid density, and Fns the mutual friction force. Bytaking the divergence of the sum of (6) and (7) and assum-ing incompressibility (= ? vs 0 and = ? vn 0) we ob-tain the governing Poisson equation for the pressure field

which we have used to calculate Fig. 1

=2p

r

rs

rv2s 2 s

2s 1

rn

rv2n 2 s

2n , (8)

where the source term on the right hand side is a functionof the vorticities v = 3 v and the strain rate tensors,sij 12 yixj 1 yjyi, of the normal fluid andthe superfluid velocity fields (indicated by the subscripts nand s, respectively). Unlike the normal fluid contribution,

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the superfluid part of this source term is extremely well lo-calized in physical space because the superfluid vorticity isconcentrated in thin vortex filaments. Fourier transforma-tion of (8) shows that k2pkr is equal to a function thatis very slowly varying in wave number space for k 1aplus contributions from the normal fluid which can be ex-pected to decay with increasing wave number k becausethe normal fluid vorticity and strain rate fields are not so

localized in physical space. This property is independentof the exact nature of the core size. Hence

jpkj2

r2 k24 (9)

for k 1a. The pressure spectrum Epk of the pressurefield is defined such that

Z`0

Epkdk ZZZ j pkj2

r2d3k , (10)

and therefore Epk k2j pkj2. The conclusion is that

Epk k22 (11)

for k 1a, which is confirmed by the numerical simu-lation. Note that although we used spherical coordinatesin Fourier space, we did not make any assumption aboutsymmetry. The result (11) is a consequence of the extremelocalization of superfluid vorticity in physical space, thatis to say of the quantum nature of helium II. The effect isincreasingly sharp when the temperature is made so smallthat rs is much larger than rn [see Poisson equation (8)above]. In the absence of superfluid vorticity the pressurespectrum is dominated by the normal fluid velocity field(5) and has a well defined Kolmogorov k273 scaling [20].

It must be said, however, that the exact Kolmogorov

scaling for the pressure field has not been verified in recentexperiments [12] and in direct numerical simulations ofclassical fluid turbulence either: Gotoh and Fukayama[10] claimed evidence of a k273 range in the spectrum,but also of a different dependence at higher wave numbers.More work is clearly required. It is, however, encouragingto remark that Abry et al. [12] found that the removalof the signal arising from strong pressure drops (intensevortices) affects the spectrum at small wave number k, notat high k. This observation, that it is the pressure spectrumat the large scale which is mostly affected by localizedvortices, is not inconsistent with our suggestion that thereshould be a macroscale effect of superfluid filaments.

Finally, it must be noticed that there is nothing compa-rable in classical fluid turbulence to the extreme localiza-tion of superfluid vortex filaments in superfluid turbulence.There is small-scale organized vorticity in classical fluidturbulence [21] which does indeed seem to affect the pres-sure spectrum [10,22] but only at the smallest scales andwithout the generation of a clear k22 pressure spectrum.

In conclusion, numerical simulations backed up byphysical arguments predict a k22 pressure spectrum for

superfluid turbulence which is a macroscale quantumsignature not to be found in the classical theory of fluidturbulence. Experimental attempts to measure spectrain turbulent helium II concentrated the attention on thevelocity field [2]. However, accurate pressure gauges forcryogenics fluid now exist [23], and work is in progress tobuild devices suitable for helium turbulence [24]. Thereare clearly experimental problems to solve regarding

the sensitivity and the size of the probe, as well as thefrequency response, but they do not seem impossible toovercome, more so if MEME technology is used. We hopetherefore that our prediction will add to the motivationsbehind this development of instrumentation and stimulatethe measurement of pressure spectra in turbulent heliumII. Ideally one should measure turbulent pressure spectrain the same apparatus at two different temperatures, justabove and below the lambda point, to compare predictionsin the classical (above the lambda point) and superfluid(below the lambda point) regimes. On the theoretical side,our results call for the investigation of the fully coupledmotion of turbulent normal fluid and superfluid vortices,

something which, until now, has been attempted only forvery simple vortex configurations [6].

This research is supported by the Leverhulme Trust.J. C. V. acknowledges support from the Royal Society andM. A. I. K. from the Cambridge Commonwealth Trust.

[1] M. R. Smith, R. J. Donnelly, N. Goldenfeld, and W. F. Vi-nen, Phys. Rev. Lett. 71, 2583 (1993); S. R. Stalp, L. Skr-bek, and R. J. Donnelly, Phys. Rev. Lett. 82, 4831 (1999);L. Skrbek, J. J. Niemela, and R. J. Donnelly, Phys. Rev.Lett. 85, 2973 (2000).

[2] J. Maurer and P. Tabeling, Europhys. Lett. 43, 29 (1998).[3] P. L. Walstrom, J. G. Weisend, J. R. Maddocks, and S. W.

VanSciver, Cryogenics 28, 101 (1988).[4] M. R. Smith, D. K. Hilton, and S. V. VanSciver, Phys. Fluids

11, 751 (1999); M. Niemetz, W. Schoepe, J. T. Simola, andJ. T. Tuoriniemi, Physica (Amsterdam) 280B, 559 (2000).

[5] D. Kivotides, C. F. Barenghi, and D. C. Samuels, Science290, 777 (2000); O. C. Idowu, A. Willis, C. F. Barenghi,and D. C. Samuels, Phys. Rev. B 62, 3409 (2000).

[6] D. Kivotides, C. F. Barenghi, and D. C. Samuels, Science290, 777 (2000).

[7] M. Tsubota, T. Araki, and S. K. Nemirowskii, J. Low Temp.Phys. 119, 337 (2000); C. Nore, M. Abid, and M. E. Bra-chet, Phys. Rev. Lett. 78, 3896 (1997).

[8] D. C. Samuels and C. F. Barenghi, Phys. Rev. Lett. 81, 4381(1998).

[9] D. Kivotides, J. C. Vassilicos, D. C. Samuels, and C. F.Barenghi, Phys. Rev. Lett. 86, 3080 (2001); M. Lead-beater, T. Winiecki, D. C. Samuels, C. F. Barenghi, andC. S. Adams, Phys. Rev. Lett. 86, 1410 (2001); B. V. Svis-tunov, Phys. Rev. B 52, 3647 (1995); W. F. Vinen, Phys.Rev. B 61, 1410 (2000).

[10] T. Gotoh and D. Fukayama, Phys. Rev. Lett. 86, 3775(2001).

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[11] O. Cadot, S. Douady, and Y. Couder, Phys. Fluids 7, 630(1995).

[12] P. Abry, S. Fauve, P. Flandrin, and C. Laroche, J. Phys. II(France) 4, 725 (1994).

[13] W. K. George, P. D. Beuther, and R. E. Arndt, J. FluidMech. 148, 155 (1984).

[14] K. W. Schwarz, Phys. Rev. B 31, 5782 (1985).[15] O. Idowu, D. Kivotides, C. F. Barenghi, and D. C. Samuels,

J. Low Temp. Phys. 120, 269 (2000).

[16] J. C. H. Fung and J. C. Vassilicos, Phys. Rev. E 57, 1677(1998); N. A. Malik and J. C. Vassilicos, Phys. Fluids 11,1572 (1999).

[17] R. J. Donnelly, in Quantized Vortex Dynamics and Su- perfluid Turbulence, edited by C. F. Barenghi, R. J. Don-nelly, and W. F. Vinen, Lecture Notes in Physics Vol. 571(Springer, Berlin, 2001), pp. 17 35.

[18] D. Kivotides, C. J. Vassilicos, C. F. Barenghi, and D. C.Samuels (to be published).

[19] R. J. Donnelly, Quantized Vortices in Helium II(CambridgeUniversity Press, Cambridge, U.K., 1991).

[20] J. C. Vassilicos, N. A. Malik, J. Davila, and Y. Sakai, in Advances in Turbulence VII, edited by U. Frisch (KluwerAcademic Publishers, Holland, 1998), p. 581.

[21] Z.-S. She and S. A. Orszag, Nature (London) 344, 226(1990); J. Jimenez and A. Wray, J. Fluid Mech. 373, 255

(1998).[22] T. Gotoh and R. S. Rogallo, J. Fluid Mech. 396, 257 (1999).[23] C. J. Swanson, K. Johnson, and R. J. Donnelly, Cryogenics

38, 673 (1998).[24] C. J. Swanson, R. J. Donnelly, and G. G. Ihas, Physica (Am-

sterdam) 284B, 77 (2000).

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D. Kivotides et al- Quantum Signature of Superfluid Turbulence