A Kolmogorov-type cascade of Kelvin waves--the distortion waves on vortex
lines--plays a key part in the relaxation of superfluid turbulence at low
temperatures. We propose an efficient numeric scheme for simulating the Kelvin
wave cascade on a single vortex line. The idea is likely to be generalizable
for a full-scale simulation of different regimes of superfluid turbulence. With
the new scheme, we are able to unambiguously resolve the cascade spectrum
exponent, and thus to settle the controversy between recent simulations [1] and
recently developed analytic theory [2].
  [1] W.F. Vinen, M. Tsubota and A. Mitani, Phys. Rev. Lett. 91, 135301 (2003).
  [2] E.V. Kozik and B.V. Svistunov, Phys. Rev. Lett. 92, 035301 (2004).Comment: 4 pages, RevTe

B. V. Svistunov

Boris Svistunov

Evgeny Kozik

R. J. Donnelly

English

arXiv

A Kolmogorov-type cascade of Kelvin waves—the distortion waves on vortex lines—plays a key part in the relaxation of superfluid turbulence at low temperatures. We propose an efficient numeric scheme for simulating the Kelvin-wave cascade on a single vortex line. This idea is likely to be generalizable for a full-scale simulation of different regimes of superfluid turbulence. With the new scheme, we are able to unambiguously resolve the cascade spectrum exponent, and thus to settle the controversy between recent simulations of Vinen, Tsubota, and Mitani [Phys. Rev. Lett. 91, 135301 (2003)] and recently developed analytic theory [Phys. Rev. Lett. 92, 035301 (2004)]

Kozik, E

Svistunov, Boris

ScholarWorks@UMass Amherst

University of Massachusetts AmherstScholarWorks@UMass AmherstPhysics Department Faculty Publication Series Physics2005Scale-Separation Scheme for Simulating SuperfluidTurbulence: Kelvin-Wave CascadeE KozikBoris SvistunovUniversity of Massachusetts - Amherst, svistunov@physics.umass.eduFollow this and additional works at: https://scholarworks.umass.edu/physics_faculty_pubsPart of the Physics CommonsThis Article is brought to you for free and open access by the Physics at ScholarWorks@UMass Amherst. It has been accepted for inclusion in PhysicsDepartment Faculty Publication Series by an authorized administrator of ScholarWorks@UMass Amherst. For more information, please contactscholarworks@library.umass.edu.Recommended CitationKozik, E and Svistunov, Boris, "Scale-Separation Scheme for Simulating Superfluid Turbulence: Kelvin-Wave Cascade" (2005). PhysicsReview Letters. 1201.Retrieved from https://scholarworks.umass.edu/physics_faculty_pubs/1201arXiv:cond-mat/0408241v5  [cond-mat.other]  8 Oct 2004Scale Separation Scheme for Simulating Superfluid Turbulence: Kelvin wave CascadeEvgeny Kozik1 and Boris Svistunov1,2,31 Department of Physics, University of Massachusetts, Amherst, MA 010032 Russian Research Center “Kurchatov Institute”, 123182 Moscow, Russia3 Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106A Kolmogorov-type cascade of Kelvin waves—the distortion waves on vortex lines—plays a keypart in the relaxation of superfluid turbulence at low temperatures. We propose an efficient numericscheme for simulating the Kelvin wave cascade on a single vortex line. The idea is likely to begeneralizable for a full-scale simulation of different regimes of superfluid turbulence. With the newscheme, we are able to unambiguously resolve the cascade spectrum exponent, and thus to settlethe controversy between recent simulations [1] and recently developed analytic theory [2].PACS numbers: 67.40.Vs, 03.75.Lm, 47.32.CcThe superfluid turbulence (for an introduction see[3, 4]) can be viewed as a tangle of quantized vortexlines. In the case of an essentially finite temperature, thesuperfluid turbulence is perfectly described by the localinduction approximation (LIA), as it has been shown inpioneering simulations by Schwarz [5, 6]. LIA reducesthe problem of finding the velocity of an element of thevortex line to its local differential characteristics:s˙ = β s′ × s′′ + friction , (1)β =κ4piln(R/a0) , (2)where s ≡ s(ξ, t) is the evolving in time radius-vectorof the vortex line element, the parameter ξ being thearc length, the dot and prime denote differentiation withrespect to time and ξ, respectively; κ is the quantumof velocity circulation; R is the typical curvature radius(which is of the same order that the typical interline spac-ing), and a0 is the vortex core radius. A necessary con-dition of applicability of LIA isln(R/a0)≫ 1 . (3)At zero temperature, a crucial role in the dynamicsof superfluid turbulence is played by the Kelvin wavecascade [1, 2, 7, 8, 9, 10]. LIA fails to properly describea considerable part of the cascade inertial range (see,e.g., [2] and references therein). In this case, one has touse the description in terms of full Biot-Savart equation(BSE):s˙ =κ4pi∫(s0 − s)× ds0/|s0 − s|3 . (4)Here the evolving in time radius-vector of the vortex lineelement, s ≡ s(ζ, t), is labelled by some parameter ζ(not necessarily the arc length), the vector s0 is of thesame physical meaning as s, understood as an integra-tion variable; the integration is over all the vortex lines;ds0 = (∂s0/∂ζ) dζ. The two conditions of applicabilityof Eq. (4) are: (i) the absence of the normal componentimplying the absence of the friction term, and (ii)R≫ a0 . (5)The need to use the full Biot-Savart law in dealingwith the Kelvin wave cascade had been recognized inRefs. [1, 10]. It is worth noting that the Kelvin wave cas-cade is not the only problem where non-local vortex lineinteractions play a crucial part. The other examples in-clude the conditions of stability of vortex knots revealedby Ricca et al. [11]: the vortex knots that are unstableunder LIA acquire a greatly increased lifetime when BSEin used. In the simulation by Samuels [12], the non-localinteraction of vortex rings in laminar flow of He II resultsin overall parabolic superfluid velocity profile matchingthe normal-fluid velocity profile. In the problem of su-perfluid turbulence decay at length scales greater thanthe interline spacing, the non-local interactions of vortexlines result in Kolmogorov cascade of eddies mimickingthe behavior of the classical turbulence even without thenormal component [13, 14, 15].In theoretical studies of turbulent vortex dynamics, ananalytic solution is normally not available, and numericsimulations are of crucial importance. Clearly, the com-putational price of a direct simulation of BSE is macro-scopically larger than that of LIA, since to perform onestep in time for a given element of a line one has to ad-dress all the line elements. In this Letter, we observe thatthis extensive addressing is actually extremely excessiveand can be easily eliminated at the expense of a control-lable systematic error. The physical reason for that is asfollows. The interactions at a distance r are significantonly for the Kelvin waves of the wavelength λ >∼ r. Thecharacteristic time of evolution of a harmonic of the wave-length λ scales like λ2, which means that during the timeinterval much shorter than λ2 the configuration of theseharmonics remains essentially unchanged. Correspond-ingly, the maximal reasonable frequency of addressingthe distances ∼ r from a given element of the vortex linescales like 1/r2. This leads to a numeric scheme wherethe long-range interactions of Kelvin waves do not reducethe simulation efficiency. We employ this scheme to sim-ulate the Kelvin wave cascade on a single vortex line atzero temperature. Our data for the cascade spectrum arein an excellent agreement with the predictions of analytictheory [2].Scale separation scheme.—To be specific, we consider2the zero-temperature dynamics of Kelvin waves on astraight vortex line. We separate different scales of dis-tance in the integral in Eq. (4) by writing BSE in theforms˙ = Q1 +Q2 +Q3 + . . .+Qn + . . . , (6)whereQn(s) =κ4pi∫2n−1a0<|s0−s|<2na0(s0 − s)× ds0|s0 − s|3. (7)Strictly speaking, Eqs. (6)-(7) are meaningful only forn ≫ 1, since at distances <∼ a0 from the vortex corethe velocity field is model-dependent. With the sameaccuracy, one may replace in Eq. (6) the n ∼ 1 non-localterms with a local term. (As we do it in our simulationscheme.—See below.)  Qn+1(t)Qn(t) Q   (arb. units)time  (arb. units)FIG. 1: Comparing functions Qn(t) and Qn+1(t) for a fixedvortex line element. The data are obtained by simulatingKelvin wave cascade with full BSE.The idea behind the separation of scales of distance inEq. (6) is clearly seen in Fig. 1, where we compare twofunctions, Qn(t) and Qn+1(t), in the regime of Kelvinwave cascade. The characteristic period of variation ofthe function Qn+1(t) is approximately 4 times larger thanthat of Qn. This is precisely what one would expect onthe basis of qualitative analysis, since the typical wave-length of harmonics contributing to Qn is λn = 2na0,which leads to the estimateτn ∝ λ2n (8)for the typical time of variation of Qn. The contributionsto Qn from the harmonics with λ≪ λn are suppressed bythe denominator in Eq. (7) in combination with a randominterference in the numerator.The behavior of Qn’s suggests the following numericprocedure. Each Qn(t) is approximated by a piecewisepolynomial function Q˜n(t) based on an appropriate mesh{t(n)j }, j = 1, 2, 3, . . ., and corresponding approximatevalues of Qn(t = t(n)j ) obtained by integrating BSE withQn(s, t) → Q˜n(s, t). In the present study, we take thevalues of Qn calculated at the time moments t(n)j−1 andt(n)j , and use the linear extrapolation to get the approxi-mation Q˜n(t) for Qn(t) at any t in the interval [t(n)j , t(n)j+1].The key point then is to properly choose the spacingof the mesh in accordance with the estimate (8):t(n)j+1 − t(n)j ∝ γλ2n , (9)where γ is a global (n-independent) small parameter con-trolling the accuracy of the scheme. (The scheme isasymptotically exact in the limit of γ → 0.) With such ascaling of t(n)j+1 − t(n)j the long-range interactions do notform a bottleneck for the efficiency of the numeric pro-cess. Indeed, to numerically evaluate the integral (7) onehas to perform ∝ λn operations, but the time intervalbetween two integrations scales like ∝ λ2n, which meansthat on average this integration costs only ∝ 1/λn opera-tions per unit time. To control the accuracy, we performa test simulation (at a smaller system size) and comparethe results to the simulation of full BSE. At a reasonablysmall γ, we see no difference (up to a statistical noise,like the one noticeable in Fig. 2) in the evolution of thedistribution of harmonics.Kelvin wave cascade.—The decay of superfluid turbu-lence at T → 0 leads to a cascade of Kelvin waves gener-ated by vortex line reconnections [7, 8]. As it was arguedby one of us [7], in the limit of R/a0 →∞, when the dy-namics at length scales ∼ R is well captured by LIA, inthe low-energy part of the inertial range there takes placea specific regime when the energy flux in the wavenumberspace is assisted by local self-crossings of fractionalizedvortex lines. Corresponding spectrum of Kelvin waves(Kelvons) has the formnk ∝ k−3/(ln k)ν , (10)where nk is the Kelvon occupation number (see below)and ν is some dimensionless constant of order unity. Thespecial role of local self-crossings follows from the inte-grability of Eq. (1) (the friction is absent at T = 0).—The existence of extra constants of motion excludes thepossibility of the purely non-linear cascade. Vortex lineself-crossings lift these constraints and push the cascadedown to smaller wavelengths.Vinen conjectured [8] that—apart from the logarith-mic denominator—the spectrum (10) can be derived juston the basis of dimensional considerations and is thusinsensitive to the particular cascade scenario. If true,this statement would be of crucial importance, since atlarge enough wavenumbers the cascade should radicallychange its microscopic character. From dimensional esti-mates it is readily seen that with increasing wavenumbernon-local interactions eventually become dominant, and3the cascade is driven by purely non-linear dynamics ofKelvons. According to Vinen, the change of the regimeshould not affect the Kelvon spectrum. Recent numericsimulations [1, 10] seem to support this idea. However,an inconsistency can be seen in a dimensional analysis.Indeed, the spectrum nk ∝ k−3 implies that the typicalamplitude bλ of the Kelvon turbulence with the wave-length λ is of the order of λ. This leads to a strong non-linearity of BSE, which allows one to estimate the kinetictime τkin as the dynamic time corresponding to bλ ∼ λ.It follows then that τkin(λ) ∝ λ2. With this estimate forthe kinetic time, the spectrum nk ∝ k−3 is inconsistentwith the λ-independent energy flux (see, e.g., Ref. [2] forthe relation between spectrum, kinetic time, and energyflux). In our recent paper [2], we argue that the regimeof the pure Kelvin wave cascade is characterized by theinequalityαλ ≡bλλ≪ 1 , (11)with the parameter αλ becoming progressively smallerwith decreasing λ. This small parameter allows one todevelop a quantitative kinetic theory that predicts thespectrumnk ∝ k−17/5 . (12)In terms of bλ, Eq. (12) is equivalent tobλ ∝ λ6/5 , (13)while the Vinen’s spectrum means bλ ∝ λ, that is αλ ∼ 1for any λ.Since the number 17/5 = 3.4 is rather close to 3, theaccuracy of the data of Refs. [1, 10] turns out to be in-sufficient to distinguish between the two exponents.Numeric simulation.—The main goal of our simulationis to observe the exponent 17/5, and thus to verify thetheory of Ref. [2].We consider a straight vortex line along the Cartesianz-direction with small (αλ ≪ 1) distortions in the (x, y)-plane; with the periodic boundary conditions. We use thez-coordinate as a parameter of the curve, and introducethe complex-valued function w(z, t) = x(z, t)+ iy(z, t) tospecify the configuration of the vortex line at a given timemoment t. To proceed numerically, we employ the Hamil-tonian approach in terms of the discrete complex canon-ical variable wj(t) = xj(t) + iyj(t), j = 1, 2, 3, . . . , N ,with the Hamiltonian function [mesh spacing is set equalto unity]H = CN∑j=1√1 + |wj+1 − wj |2 +N∑j,l=11+[(wj+1 − wj−1)(w∗l+1 − w∗l−1)/8 + c.c.]√(j − l)2 + |wj − wl|2, (14)and corresponding equation of motioniw˙j =∂H∂w∗j. (15)Here C is a constant of order unity. In the long-wavelimit, the discreteness is irrelevant and the model (14)-(15) is equivalent to the Hamiltonian form of BSE interms of the function w(z, t), with z ≡ j [7]. From thetheoretical point of view, the particular value of C, andthe very presence of the first term in (14), is not impor-tant. In practice, we introduce this local term to ren-der the model well-behaved at large momenta, where thespectrum of Kelvons is very sensitive to the short-rangestructure of the Hamiltonian. In our simulations, we usedC = 23.The Kelvon occupation number is given by the squareof the absolute value of the Fourier transform of wj [2]:nk = |wk|2 , (16)wk = N−1/2N∑j=1wj eikj , (17)where k = 2pim/N , m ∈ [−(N − 1)/2 , (N − 1)/2] is aninteger (we used odd N).There are different ways to generate a Kolmogorov-type cascade. Perhaps the most natural one is to cre-ate a steady-state regime with a source and drain, asit was done in the simulation of Ref. [1]. However, anexternal source (random low-frequency force) is not nec-essary, since the cascade, by its definition, is a quasi-steady-state phenomenon, where the behavior of short-wavelength harmonics is adjusted to the energy flux com-ing from the long-wave harmonics. In a quasi-steady-state regime, the role of a source can be played by justthe long-wavelength harmonics, as, e.g., in the simula-tion of Ref. [10]. In both cases—with or without an ex-ternal force—the cascade regime requires some time todevelop. We employ the following trick to circumvent theproblem of transient period. We start with occupationnumbers that progressively exceed the quasi-steady-stateones in the direction of shorter wavelengths. With suchan initial condition, the cascade develops in the form ofa wave in the momentum space propagating from the re-gion of higher wavenumbers towards the region of lowerwavenumbers, the distribution behind the wave corre-sponding to the quasi-steady-state cascade [16]. For ourpurpose of resolving the spectrum (12) from the spectrumnk ∝ k−3, it is convenient to use the latter distributionof occupation numbers as an initial condition.Introducing a drain—a decay mechanism for high-frequency harmonics—significantly improves the qual-ity of simulation, because the large-amplitude high-frequency modes distort the short-wavelength part of thecascade spectrum. We arrange the drain by periodicallyeliminating all the harmonics with |k| > kcutoff , withkcutoff = pi/5. Note that the momentum kcutoff is 5 timessmaller than the largest momentum in our system. Wefind it important to severely cut off the high-frequencyharmonics because their dispersion strongly differs fromthe parabolic dispersion of Kelvin waves in the continu-ous space.40.1 1 nk ~ k-3 nk ~ k-17/5nk  (arb. units)k/kcutoffFIG. 2: A snapshot of time-averaged distribution of Kelvonoccupation numbers. For time averaging we use time intervalwhich is considerably shorter (by a factor of ∼ 3) than thetypical time of evolution.We simulated a system of the size N = 16385. Ourmain results are presented in Fig. 2. We do not showa large low-frequency part of the distribution with thespectrum nk ∝ k−3. Having these ballast long-wave har-monics is important to guarantee that the inertial rangeis far away from the region where the wavelength is com-parable to the system size and the kinetics is stronglyaffected by the wavenumber discreteness.Fig. 2 reveals the wave converting the initial dis-tribution nk ∝ k−3 into a new power-law distributionnk ∝ k−β. Quantitative analysis shows that the expo-nent β coincides with 17/5 with an absolute error lessthan 0.1, in an excellent agreement with the predictionof Ref. [2].Unfortunately, we were unable to check the predictionof Ref. [2] for the relation between the energy flux andthe amplitude A of the distribution nk = Ak−17/5. Theproblem is in calculating the mean value of the energyflux. With our quasi-steady-state setup, we can use onlytime averaging—with an essentially limited averaging pe-riod. This turns out to be insufficient to suppress hugefluctuations of the instant values of the energy flux. Forthese purposes, the steady-state setup of Ref. [1] seemsto be more adequate.In conclusion, we have proposed an efficient numericapproach for simulating the dynamics of superfluid tur-bulence in the framework of Biot-Savart equation. Thegeneral principle is to use a less detailed addressing forlonger-range interactions. The idea is implemented in ascheme for simulating Kelvin wave cascade on a singlevortex line. With the new scheme we were able to ac-curately resolve the spectrum of the pure Kelvin wavecascade, observing an excellent agreement with the pre-diction of the kinetic theory of Ref. [2].The research was supported in part by the NationalScience Foundation under Grant No. PHY99-07949.[1] W.F. Vinen, M. Tsubota and A. Mitani, Phys. Rev. Lett.91, 135301 (2003).[2] E.V. Kozik and B.V. Svistunov, Phys. Rev. Lett. 92,035301 (2004).[3] R.J. Donnelly, Quantized Vortices in He II, CambridgeUniversity Press, Cambridge, 1991.[4] C.F. Barenghi, R.J. Donnelly, and W.F. Vinen (eds.),Quantized Vortex Dynamics and Superfluid Turbulence,Vol. 571 of Lecture Notes in Physics, Springer-Verlag,Berlin, 2001.[5] K.W. Schwarz, Phys. Rev. B 31, 5782 (1985).[6] K.W. Schwarz, Phys. Rev. B 38, 2398 (1988).[7] B.V. Svistunov, Phys. Rev. B 52, 3647 (1995).[8] W.F. Vinen, Phys. Rev. B 61, 1410 (2000).[9] S.I. Davis, P.C. Hendry, and P.V.E. McClintock, Physica(Amsterdam) 280B, 43 (2000).[10] D. Kivotides, J.C. Vassilicos, D.C. Samuels, and C.F.Barenghi, Phys. Rev. lett. 86, 3080 (2001).[11] R.L. Ricca, D.C. Samuels, C.F. Barenghi, J. Fluid Mech.391, 29 (1999).[12] D.C. Samuels, Phys. Rev. B 46, 11714 (1992).[13] C. Nore, M. Abid, and M.E. Brachet, Phys. Rev. Lett.78, 3896 (1997).[14] T. Araki, M. Tsubota, S.K. Nemirovskii, Phys. Rev. Lett.89, 145301 (2002).[15] C.F. Barenghi, S. Hulton, and D.C. Samuels, Phys. Rev.Lett. 89, 275301 (2002).[16] B.V. Svistunov, J. Moscow Phys. Soc. 1, 373 (1991).

Scale-Separation Scheme for Simulating Superfluid Turbulence: Kelvin-Wave Cascade

This is the pre-published version harvested from ArXiv. The published version is located at http://prl.aps.org/abstract/PRL/v94/i2/e025301A Kolmogorov-type cascade of Kelvin waves—the distortion waves on vortex lines—plays a key part in the relaxation of superfluid turbulence at low temperatures. We propose an efficient numeric scheme for simulating the Kelvin-wave cascade on a single vortex line. This idea is likely to be generalizable for a full-scale simulation of different regimes of superfluid turbulence. With the new scheme, we are able to unambiguously resolve the cascade spectrum exponent, and thus to settle the controversy between recent simulations of Vinen, Tsubota, and Mitani [Phys. Rev. Lett. 91, 135301 (2003)] and recently developed analytic theory [Phys. Rev. Lett. 92, 035301 (2004)]

Scale Separation Scheme for Simulating Superfluid Turbulence: Kelvin-Wave Cascade

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