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Typeset with jpsj2.cls <ver.1.2.1> Full Paper

Quantum Turbulence

Makoto Tsubota1 ∗ †

1 Department of Physics, Osaka City University, Japan

The present article reviews the recent developments in the physics of quantum turbulence.Quantum turbulence (QT) was discovered in superfluid 4He in the 1950s, and the researchhas tended toward a new direction since the mid 90s. The similarities and differences betweenquantum and classical turbulence have become an important area of research. QT is comprisedof quantized vortices that are definite topological defects, being expected to yield a model ofturbulence that is much simpler than the classical model. The general introduction of the issueand a brief review on classical turbulence are followed by a description of the dynamics ofquantized vortices. Then, we discuss the energy spectrum of QT at very low temperatures. Atlow wavenumbers, the energy is transferred through the Richardson cascade of quantized vor-tices, and the spectrum obeys the Kolmogorov law, which is the most important statistical lawin turbulence; this classical region shows the similarity to conventional turbulence. At higherwavenumbers, the energy is transferred by the Kelvin-wave cascade on each vortex. This quan-tum regime depends strongly on the nature of each quantized vortex. The possible dissipationmechanism is discussed. Finally, important new experimental studies, which include investi-gations into temperature-dependent transition to QT, dissipation at very low temperatures,QT created by vibrating structures, and visualization of QT, are reviewed. The present articleconcludes with a brief look at QT in atomic Bose-Einstein condensates.

KEYWORDS: Quantum turbulence, Quantized vortex, Superfluid, Bose-Einstein condensation

1. Introduction

For high-velocity or high Reynolds number flows, theflow is generally turbulent. Turbulence has been investi-gated not only in basic science research, such as physicsand mathematics research, but also in applied sciences,such as fluid engineering and aeronautics. Although tur-bulence has been studied intensely in a number of fields,it is still not yet well understood. This is chiefly becauseturbulence is a complicated dynamical phenomenon withstrong nonlinearity that is quite different from an equi-librium state. Vortices may be key in understanding tur-bulence. For example, Leonardo da Vinci observed theturbulent flow of water and drew many sketches show-ing that turbulence had a structure comprised of vorticesof different sizes. However, vortices are not well-definedfor a typical classical fluid, and the relationship betweenturbulence and vortices remains unclear.

Independently of these studies in classical fluid dynam-ics, turbulence in superfluid helium has been studied inthe field of low-temperature physics. Liquid 4He enters asuperfluid state below the λ point (2.17 K) with Bose–Einstein condensation of the 4He atoms.1) Characteristicphenomena of superfluidity were discovered experimen-tally in the 1930s by Kapitza et al. The hydrodynamicsof superfluid helium is well described by the two-fluidmodel, for which the system consists of an inviscid su-perfluid (density ρs) and a viscous normal fluid (densityρn) with two independent velocity fields vs and vn. Themixing ratio of the two fluids depends on temperature.As the temperature is reduced below the λ point, the ra-tio of the superfluid component increases, and the entire

∗E-mail: tsubota@sci.osaka-cu.ac.jp†Present address: Department of Physics, Osaka City University,Osaka 558-8585, Japan.

fluid becomes a superfluid below approximately 1 K. TheBose-condensed system exhibits the macroscopic wavefunction Ψ(x, t) = |Ψ(x, t)|eiθ(x,t) as an order parameter.The superfluid velocity field is given by vs = (~/m)∇θwith boson mass m, representing the potential flow. Sincethe macroscopic wave function should be single-valuedfor the space coordinate x, the circulation Γ =

∮

v · dℓfor an arbitrary closed loop in the fluid is quantized bythe quantum κ = h/m. A vortex with such quantizedcirculation is called a quantized vortex. Any rotationalmotion of a superfluid is sustained only by quantizedvortices.

A quantized vortex is a topological defect characteris-tic of a Bose–Einstein condensate and is different from avortex in a classical viscous fluid. First, the circulationis quantized, which is contrary to a classical vortex thatcan have any value of circulation. Second, a quantizedvortex is a vortex of inviscid superflow. Thus, it cannotdecay by the viscous diffusion of vorticity that occurs ina classical fluid. Third, the core of a quantized vortex isvery thin, on the order of the coherence length, which isonly a few angstroms in superfluid 4He. Since the vortexcore is very thin and does not decay by diffusion, it isalways possible to identify the position of a quantizedvortex in the fluid. These properties make a quantizedvortex more stable and definite than a classical vortex.

Early experimental studies on superfluid turbulencefocused primarily on thermal counterflow, in which thenormal fluid and superfluid flow in opposite directions.The flow is driven by an injected heat current, and it wasfound that the superflow becomes dissipative when therelative velocity between the two fluids exceeds a criticalvalue.2) Feynman proposed that this is a superfluid tur-bulent state consisting of a tangle of quantized vortices.3)

2 J. Phys. Soc. Jpn. Full Paper Author Name

Vinen later confirmed Feynman’s findings experimentallyby showing that the dissipation comes from the mutualfriction between vortices and the normal flow.4–6) Subse-quently, several experimental studies have examined su-perfluid turbulence (ST) in thermal counterflow systemsand have revealed a variety of physical phenomenon.7)

Since the dynamics of quantized vortices is nonlinearand non-local, it has not been easy to understand vor-tex dynamics observations quantitatively. Schwarz clar-ified the picture of ST consisting of tangled vortices bya numerical simulation of the quantized vortex filamentmodel in the thermal counterflow.8, 9) However, since thethermal counterflow has no analogy to conventional fluiddynamics, this study was not helpful in clarifying therelationship between ST and classical turbulence (CT).Superfluid turbulence is often called quantum turbulence(QT), which emphasizes the fact that it is comprised ofquantized vortices.

Comparing QT and CT reveals definite differences,which demonstrates the importance of studying QT. Tur-bulence in a classical viscous fluid appears to be com-prised of vortices, as pointed out by Da Vinci. However,these vortices are unstable, and appear and disappear re-peatedly. Moreover, the circulation is not conserved andis not identical for each vortex. Quantum turbulence con-sists of a tangle of quantized vortices that have the sameconserved circulation. Thus, QT is an easier system tostudy than CT and has a much simpler model of turbu-lence than CT.

Based on this consideration, QT research has tendedtoward a new direction since the mid 90s. A chief inter-est is to understand the relationship between QT andCT.10, 11) The energy spectrum of fully developed clas-sical turbulence is known to obey the Kolmogorov lawin an inertial range. The energy transfer in an inertialrange is believed to be sustained by the Richardson cas-cade process, in which large eddies are broken up self-similarly to smaller eddies. Recent experimental and nu-merical studies support the Kolmogorov spectrum in QT.Another important problem is the dissipative mechanismin QT. At a finite temperature the mutual friction worksas the dissipative mechanism. However, it is not so easyto understand what kind of mechanism causes dissipa-tion at very low temperatures in which the normal fluidcomponent is negligible.

This article reviews recent research on QT. After re-viewing classical turbulence in Section 2, Section 3 de-scribes the dynamics of quantized vortices, which is keyin understanding QT. Section 4 summarizes the moti-vation and the problems of recent research on quantumturbulence. In Section 5, we discuss the issue of the en-ergy spectra and a possible dissipation mechanism atvery low temperatures. Section 6 describes recent im-portant experimental studies, including research into thetemperature-dependent transition to QT, dissipation atvery low temperatures, QT created by vibrating struc-tures, and visualization of QT. Section 7 presents a sum-mary and briefly addresses QT in atomic Bose-Einsteincondensates.

2. Classical turbulence

Before considering QT, we briefly review classical fluiddynamics and the statistical properties of CT.12) Clas-sical viscous fluid dynamics is described by the Navier–Stokes equation:

∂

∂tv(x, t)+v(x, t) ·∇v(x, t) = −1

ρ∇P (x, t)+ν∇2v(x, t),

(1)where v(x, t) is the velocity of the fluid, P (x, t) is thepressure, ρ is the density of the fluid, and ν is the kine-matic viscosity. The flow of this fluid can be character-ized by the ratio of the second term of the left-hand sideof Eq. (1), hereinafter referred to as the inertial term,to the second term of the right-hand side, hereinaftercalled the viscous term. This ratio is the Reynolds num-ber R = vD/ν, where v and D are the characteristicvelocity of the flow and the characteristic scale, respec-tively. When v increases to allow the Reynolds number toexceed a critical value, the system changes from a lami-nar state to a turbulent state, in which the flow is highlycomplicated and contains many eddies.

Such turbulent flow is known to show characteristicstatistical behavior.13, 14) We assume a steady state offully developed turbulence of an incompressible classicalfluid. The energy is injected into the fluid at a rate of ε,the scale of which is comparable to the system size D inthe energy-containing range. In the inertial range, thisenergy is transferred to smaller scales without being dis-sipated. In this range, the system is locally homogeneousand isotropic, which leads to the statistics of the energyspectrum known as the Kolmogorov law:

E(k) = Cε2/3k−5/3. (2)

Here, the energy spectrum E(k) is defined as E =∫

dkE(k), where E is the kinetic energy per unit massand k is the wavenumber from the Fourier transforma-tion of the velocity field. The spectrum of Eq. (2) is eas-ily derived by assuming that E(k) is locally determinedby only the energy flux ε and k. The energy transferredto smaller scales in the energy-dissipative range is dis-sipated at the Kolmogorov wavenumber kK = (ǫ/ν3)1/4

through the viscosity of the fluid with dissipation rate εin Eq. (2), which is equal to the energy flux Π in the iner-tial range. The Kolmogorov constant C is a dimensionlessparameter of order unity. The Kolmogorov spectrum isconfirmed experimentally and numerically in turbulenceat high Reynolds numbers. The inertial range is thoughtto be sustained by the self-similar Richardson cascadein which large eddies are broken up into smaller eddiesthrough many vortex reconnections. In CT, however, theRichardson cascade is not completely understood be-cause it is impossible to definitely identify each eddy.The Kolmogorov spectrum is based on the assumptionthat the turbulence is homogeneous and isotropic. How-ever, actual turbulence is not necessarily homogeneousor isotropic, and so the energy spectrum deviates fromthe Kolmogorov form. This phenomenon is called inter-mittency, and is an important problem in modern fluiddynamics.12) Intermittency is closely related to the co-herent structure, which may be represented by vortices.

J. Phys. Soc. Jpn. Full Paper Author Name 3

Research into QT could also shed light on this issue.

3. Dynamics of quantized vortices

As described in Section 1, most experimental studieson superfluid turbulence have examined thermal coun-terflow. However, the nonlinear and non-local dynamicsof vortices have delayed progress in the microscopic un-derstanding of the vortex tangle. Schwarz overcame thesedifficulties8, 9) by developing a direct numerical simula-tion of vortex dynamics connected with dynamical scal-ing analysis, enabling the calculation of physical quanti-ties such as the vortex line density, anisotropic parame-ters, and mutual friction force. The observable quantitiesobtained by Schwarz agree well with some typical exper-imental results.

Two formulations are generally available for studyingthe dynamics of quantized vortices. One is the vortex fil-ament model and the other is the Gross–Pitaevskii (GP)model. We will briefly review these two formulations.

3.1 Vortex filament model

As described in Section 1, a quantized vortex has quan-tized circulation. The vortex core is extremely thin, usu-ally much smaller than other characteristic scales in vor-tex motion. These properties allow a quantized vortexto be represented as a vortex filament. In classical fluiddynamics,15) the vortex filament model is a convenientidealization. However, the vortex filament model is ac-curate and realistic for a quantized vortex in superfluidhelium.

The vortex filament formulation represents a quantizedvortex as a filament passing through the fluid, having adefinite direction corresponding to its vorticity. Exceptfor the thin core region, the superflow velocity field hasa classically well-defined meaning and can be describedby ideal fluid dynamics. The velocity at a point r due toa filament is given by the Biot–Savart expression:

vs(r) =κ

4π

∫

L

(s1 − r) × ds1

|s1 − r|3 , (3)

where κ is the quantum of circulation. The filament isrepresented by the parametric form s = s(ξ, t) with theone-dimensional coordinate ξ along the filament. Thevector s1 refers to a point on the filament, and the inte-gration is taken along the filament. Helmholtz’s theoremfor a perfect fluid states that the vortex moves with thesuperfluid velocity. Calculating the velocity vs at a pointr = s on the filament causes the integral to diverge ass1 → s. To avoid this divergence, we separate the velocitys of the filament at the point s into two components:8)

s =κ

4πs′ × s′′ ln

(

2(ℓ+ℓ−)1/2

e1/4a0

)

+κ

4π

∫′

L

(s1 − r) × ds1

|s1 − r|3 .

(4)The first term is the localized induction field arising froma curved line element acting on itself, and ℓ+ and ℓ− arethe lengths of the two adjacent line elements after dis-cretization, separated by the point s. The prime denotesdifferentiation with respect to the arc length ξ. The mu-tually perpendicular vectors s′, s′′, and s′×s′′ point alongthe tangent, the principal normal, and the binormal, re-

spectively, at the point s, and their respective magni-tudes are 1, R−1, and R−1, with the local radius R ofcurvature. The parameter a0 is the cutoff correspond-ing to the core radius. Thus, the first term representsthe tendency to move the local point s along the binor-mal direction with a velocity inversely proportional to R.The second term represents the non-local field obtainedby integrating the integral of Eq. (3) along the rest ofthe filament, except in the neighborhood of s.

Neglecting the non-local terms and replacing Eq. (4)by s = βs′ × s′′ is referred to as the localized induc-tion approximation (LIA). Here, the coefficient β is de-fined by β = (κ/4π) ln (c〈R〉/a0), where c is a constantof order 1 and (ℓ+ℓ−)1/2 is replaced by the mean ra-dius of curvature 〈R〉 along the length of the filament.This approximation is believed to be effective for analyz-ing isotropic dense tangles due to cancellations betweennon-local contributions.

A better understanding of vortices in a real systemresults is obtained when boundaries are included in theanalysis. For this purpose, a boundary-induced velocityfield vs,b is added to vs, so that the superflow can satisfythe boundary condition of an inviscid flow, that is, thatthe normal component of the velocity should disappearat the boundaries. To allow for another, presently un-specified, applied field, we include vs,a. Hence, the totalvelocity s0 of the vortex filament without dissipation is

s0 =κ

4πs′ × s′′ ln

(

2(ℓ+ℓ−)1/2

e1/4a0

)

+κ

4π

∫′

L

(s1 − r) × ds1

|s1 − r|3

+vs,b(s) + vs,a(s). (5)

At finite temperatures, it is necessary to take into ac-count the mutual friction between the vortex core andthe normal flow vn. Including this term, the velocity ofs is given by

s = s0 + αs′ × (vn − s0) − α′s′ × [s′ × (vn − s0)], (6)

where α and α′ are temperature-dependent friction coef-ficients,8) and s0 is calculated from Eq. (5).

The numerical simulation method based on this modelis described in detail elsewhere.8, 9, 16) A vortex filamentis represented by a single string of points separated bya distance ∆ξ. The vortex configuration at a given timedetermines the velocity field in the fluid, thus moving thevortex filaments according to Eqs. (5) and (6). Vortex re-connection should be properly included when simulatingvortex dynamics. A numerical study of a classical fluidshows that the close interaction of two vortices leads totheir reconnection, chiefly because of the viscous diffu-sion of the vorticity.17) Schwarz assumed that two vor-tex filaments reconnect when they come within a criticaldistance of one another and showed that statistical quan-tities such as vortex line density were not sensitive to howthe reconnections occur.8, 9) Even after Schwarz’s study,it remained unclear as to whether quantized vortices canactually reconnect. However, Koplik and Levine solveddirectly the GP equation to show that two closely quan-tized vortices reconnect even in an inviscid superfluid.18)

More recent simulations have shown that reconnectionsare accompanied by emissions of sound waves having

4 J. Phys. Soc. Jpn. Full Paper Author Name

wavelengths on the order of the healing length.19, 20)

Starting with several remnant vortices under thermalcounterflow, Schwarz studied numerically how these vor-tices developed into a statistical steady vortex tangle.9)

The tangle was self-sustained by the competition be-tween the excitation due to the applied flow and thedissipation through the mutual friction. The numericalresults were quantitatively consistent with typical experi-mental results. This was a significant accomplishment fornumerical research.

Here, we shall introduce some quantities that are char-acteristic of a vortex tangle. The line length density Lis defined as the total length of vortex cores in a unitvolume. The mean spacing ℓ between vortices is given byℓ = L−1/2.

3.2 The Gross-Pitaevskii (GP) model

In a weakly interacting Bose system, the macroscopicwave function Ψ(x, t) appears as the order parame-ter of Bose–Einstein condensation, obeying the Gross–Pitaevskii (GP) equation:21)

i~∂Ψ(x, t)

∂t=

(

− ~2

2m∇2 + g|Ψ(x, t)|2 − µ

)

Ψ(x, t). (7)

Here, g = 4π~2m/a represents the strength of the inter-

action characterized by the s-wave scattering length a, mis the mass of each particle, and µ is the chemical poten-tial. Writing Ψ = |Ψ| exp(iθ), the squared amplitude |Ψ|2is the condensate density and the gradient of the phaseθ gives the superfluid velocity vs = (~/m)∇θ, which is africtionless flow of the condensate. This relation causesquantized vortices to appear with quantized circulation.The only characteristic scale of the GP model is the co-herence length defined by ξ = ~/(

√2mg|Ψ|), which gives

the vortex core size. The GP model can explain not onlythe vortex dynamics but also phenomena related to vor-tex cores, such as reconnection and nucleation. However,the GP equation is not applicable quantitatively to su-perfluid 4He, which is not a weakly interacting Bose sys-tem. It is, however, applicable to Bose–Einstein conden-sation of a dilute atomic Bose gas.21)

4. Modern research trends in QT

Most older experimental studies on QT were devotedto thermal counterflow. Since this flow has no classicalanalogue, these studies do not contribute greatly to theunderstanding of the relationship between CT and QT.Since the mid 90s, important experimental studies onQT that did not focus on thermal counterflow have beenpublished.

4.1 New experiments on energy spectra

The first important contribution was made by Maurerand Tabeling,22) who confirmed experimentally the Kol-mogorov spectrum in superfluid 4He for the first time. Aturbulent flow was produced in a cylinder by driving twocounter-rotating disks. The authors observed the localpressure fluctuations to obtain the energy spectrum. Theexperiments were conducted at three different tempera-tures 2.3 K, 2.08 K, and 1.4 K. Both above and below theλ point, the Kolmogorov spectrum was confirmed. The

observed behavior above the λ point is not surprising be-cause the system is a classical viscous fluid. However, itis not trivial to understand the Kolmogorov spectrum attwo different temperatures below the λ point.

The next significant step was a series of experimentson grid turbulence performed for superfluid 4He above 1K by the Oregon group.23–27) The flow through a gridis usually used to generate turbulence in classical fluiddynamics.12) At a sufficient distance behind the grid,the flow displays a form of homogeneous isotropic turbu-lence. This method has also been applied to superfluidhelium. In the experiments by the Oregon group, thehelium was contained in a channel with a square crosssection, through which a grid was pulled at a constantvelocity. A pair of second-sound transducers was set intothe walls of the channel. When a vortex tangle appearedin a channel, it was detected by second-sound attenua-tion.10) The decay of the vorticity of the tangle createdbehind the towed grid was observed by the pair of trans-ducers. In combining the observations with the decay ofthe turbulence, the authors made some assumptions. Infully developed turbulence, the energy dissipation ratecan be shown to be given as ǫ = ν〈ω2〉, where 〈ω2〉 isthe mean square vorticity (rot v) in the flow.28) The au-thors assumed that a similar formula applies in superfluidhelium above 1 K. They noted that the quantity κ2L2

would be a measure of the mean square vorticity in thesuperfluid component. Hence, they assumed that in gridturbulence the dissipation rate is given by ǫ = ν′κ2L2

with an effective kinematic viscosity ν′. In order to com-bine this representation with the observations of second-sound attenuation for grid turbulence, the authors fur-thermore assumed that a quasi-classical flow appears atlength scales much greater than ℓ. The flow is thoughtto come from a mechanism coupling the superfluid andthe normal fluid by mutual friction, causing the fluid tobehave like a one-component fluid.29) By choosing suit-able values of ν′ as a function of temperature,27) it wasfound that κL decays as t−3/2.

This characteristic decay L ∝ t−3/2 is quite impor-tant, because it is related to the Kolmogorov spectrum.Thus, we herein present the simple argument given in aprevious review article.10) We assume that a turbulentfluid obeys the Kolmogorov spectrum of Eq. (2) in theinertial range of D−1 < k < kd with D−1 << kd. Thetotal energy is approximately given by

E =

∫ kd

D−1

Cε2/3k−5/3 ≈ 3

2Cε2/3D2/3. (8)

If the turbulence decays slowly with time and the dissi-pation rate ε is assumed to be time-dependent, we canwrite

ε = −dE

dt= −Cε−1/3D2/3 dε

dt. (9)

The solution gives the time-dependence of ε:

ε = 27C3D2(t + t0)−3, (10)

where t0 is a constant. Combining Eq. (10) with the

J. Phys. Soc. Jpn. Full Paper Author Name 5

above formula of ε gives the decay of L:

L =(3C)3/2D

κν′1/2(t + t0)

−3/2. (11)

This behavior has been observed, and a quantitativecomparison with observations at any temperature givesν′ as a function of temperature. The observation of thedecay L ∼ t−3/2 indicates that the Kolmogorov spectrumapplies in turbulence, although it is not necessarily directproof. Note that this simple analysis is applicable onlywhen the maximum length scale of the turbulent energysaturates at the size of the channel. For the complete dy-namics, a more complicated decay of vortices has beenobserved and has been found to be consistent with theclassical model of the Kolmogorov spectrum.24–26) Thistype of decay is also observed at very low temperaturesin the turbulence induced by an impulsive spin down forsuperfluid 4He30) and in grid turbulence for superfluid3He-B.31, 32)

4.2 Energy spectra at finite temperatures

These observations lead us to inquire as to the na-ture of the velocity field that gives rise to the observedenergy spectrum. Vinen considered the situation theoret-ically29) and proposed that it is likely that the superfluidand normal fluid are coupled by mutual friction at scaleslarger than the characteristic scale ℓ of the vortex tangle.If so, the two fluids behave as a one-component fluid atthese scales, where mutual friction does not cause dis-sipation. Since the normal fluid is viscous, the two cou-pled fluids can be turbulent and obey the Kolmogorovspectrum. The observation of the energy spectrum byMaurer et al. and Stalp et al. should support the ideaof the coupled dynamics of two fluids. This behavior hasbeen confirmed numerically as well.33) At small scales,the two fluids should be decoupled, so that both mutualfriction and normal fluid viscosity operate. What thenhappens to the energy spectrum? Although theoreticalconsideration has been given to this problem,34) the an-swer remains controversial and has not yet been clarified.While this is an important problem, it is not investigatedin the present study.

4.3 Modern research trends

The following three trends are currently the primaryresearch areas in QT. The first is the energy spectraand the dissipation mechanism at zero temperature.35)

The second is QT created by vibrating structures.36) Thethird is visualization of QT.37) The remainder of this ar-ticle is devoted to the review of these topics.

5. Energy spectra and dissipation at zero tem-

perature

What happens to QT at zero temperature is not sotrivial. The first problem is determining the nature ofthe energy spectrum of turbulence for the pure superfluidcomponent.35) If QT has a classical analogue, the energyspectrum is expected to obey the Kolmogorov law andhave an inertial range in which the energy is transferredself-similarly from large to small scales. In QT at zerotemperature, any rotational motion should be carried

by quantized vortices. Since quantized vortices are defi-nite topological defects, the cascade can be attributed di-rectly to their dynamics, which is different from the casefor CT. The second problem is determining how the en-ergy is transferred from large to small scales. There is nodissipative mechanism at large scales. However, some dis-sipative mechanism should operate at small scales, as de-scribed below. A vortex tangle has a characteristic scaleℓ, which is defined by the mean spacing between vor-tex lines. At scales greater than ℓ, a Richardson cascadeshould transfer the energy through the breakup of vor-tices. However, the Richardson cascade becomes ineffec-tive at small scales, especially below ℓ. What mechanismcascades the energy instead of the Richardson cascadeat these scales? The third problem is understanding thedissipation in the system. The first possibility is acousticemission at vortex reconnections. In classical fluid dy-namics, vortex reconnections cause acoustic emission. Inquantum fluids, numerical simulations of the GP modelshows acoustic emission at every reconnection event.19)

However, this mechanism is thought to be unimportantbecause of the very short coherence length. The secondpossible mechanism is the radiation of sound (phonons)by the oscillatory motion of vortex cores. We will returnto these mechanisms, as well as other possibilities.

5.1 Energy spectra

No experimental studies have addressed this issue di-rectly, although a few numerical studies have been con-ducted. The first study was performed by Nore et al.using the GP model.38, 39) They solved the GP equationnumerically starting from Taylor–Green vortices, and fol-lowed the time development. The quantized vortices be-come tangled and the energy spectra of the incompress-ible kinetic energy seemed to obey the Kolmogorov lawfor a short period, although the energy spectra even-tually deviated from the Kolmogorov law. The secondstudy was performed by the vortex filament model,40)

and the third study was performed by the modified GPmodel.41, 42)

5.1.1 Energy spectra by the vortex filament model

Using the vortex filament model, Araki et al. generateda vortex tangle arising from Taylor–Green vortices andobtained an energy spectrum consistent with the Kol-mogorov law.40) It would be informative to describe howto obtain the energy spectra under the vortex filamentmodel. The energy spectrum is calculated by Fouriertransform of the superfluid velocity vs(r), which is deter-mined by the configuration of quantized vortices. Thus,the energy spectrum can be calculated directly from theconfiguration of the vortices. Using the Fourier transformvs(k) = (2π)−3

∫

drvs(r) exp(−ik·r) and Parseval’s the-orem

∫

dk|vs(k)|2 = (2π)−3∫

dr|vs(r)|2, the kinetic en-ergy of the superfluid velocity per unit mass is

E =1

2

∫

dr|vs(r)|2 =(2π)3

2

∫

dk|vs(k)|2. (12)

The vorticity ω(r) = rotvs(r) is represented in Fourierspace as vs(k) = ik × ω(k)/|k|2, so that we have E =(

(2π)3/2) ∫

dk|ω(k)|2/|k|2. The vorticity is concentrated

6 J. Phys. Soc. Jpn. Full Paper Author Name

on the vortex filament, represented by

ω(r) = κ

∫

dξs′(ξ)δ(s(ξ) − r), (13)

which can be rewritten as

ω(k) =κ

(2π)2

∫

dξs′(ξ) exp(−is(ξ) · k). (14)

Using the definition of the energy spectrum E(k) fromE =

∫ ∞

0dkE(k), these relations yield

E(k) =κ2

2(2π)3

∫

dΩk

|k|2∫ ∫

dξ1dξ2s′(ξ1) · s′(ξ2)

× exp(−ik · (s(ξ1) − s(ξ2))), (15)

where dΩk = k2 sin θkdθkdφk is the volume element inspherical coordinates.

Starting from a Taylor–Green vortex and following thevortex motion under the full Biot–Savart law withoutmutual friction, Araki et al. obtained a roughly homoge-neous and isotropic vortex tangle.40) This was a decay-ing turbulence, dissipated by the cutoff of the smallestvortices, the size of which is comparable to the space res-olution. Initially, the energy spectrum has a large peakat the largest scale, where the energy is concentrated.The spectrum changes as the vortices become homoge-neous and isotropic. The energy spectrum of the vortextangle at some late stage was quantitatively consistentwith the Kolmogorov spectrum in the small k region. Bymonitoring the development of the vortex size distribu-tion, the decay of a tangle is found to be sustained bya Richardson cascade process. These results support thequasi-classical picture of the inertial range in QT at verylow temperatures.

5.1.2 Energy spectra by the GP model

As the third trial, the Kolmogorov spectra was con-firmed for both decaying41) and steady42) QT by themodified GP model. The normalized GP equation is

i∂

∂tΦ(x, t) = [−∇2 − µ + g|Φ(x, t)|2]Φ(x, t), (16)

which determines the dynamics of the macroscopic wavefunction Φ(x, t) = f(x, t) exp[iφ(x, t)]. The condensatedensity is |Φ(x, t)|2 = f(x, t)2, and the superfluid veloc-ity v(x, t) is given by v(x, t) = 2∇φ(x, t). The vortic-ity ω(x, t) = rotv(x, t) vanishes everywhere in a single-connected region of the fluid and thus all rotational flowis carried by quantized vortices. In the core of each vor-tex, Φ(x, t) vanishes so that the circulation around thecore is quantized by 4π. The vortex core size is given bythe healing length ξ = 1/f

√g.

Note that the hydrodynamics described by the GPmodel is compressible, which is different from the caseof the vortex filament model. The total energy of the GPmodel

E(t) =1

N

∫

dxΦ∗(x, t)[

−∇2 +g

2f(x, t)2

]

Φ(x, t), (17)

is represented by the sum of the interaction energyEint(t), the quantum energy Eq(t), and the kinetic en-

ergy Ekin(t):38, 39)

Eint(t) =g

2N

∫

dx f(x, t)4, Eq(t) =1

N

∫

dx [∇f(x, t)]2,

Ekin(t) =1

N

∫

dx [f(x, t)∇φ(x, t)]2. (18)

The kinetic energy is furthermore divided into a com-pressible part Ec

kin(t) due to compressible excitationsand an incompressible part Ei

kin(t) due to vortices. If theKolmogorov spectrum is observed, the spectrum shouldbe that for the incompressible kinetic energy Ei

kin(t).The failure to obtain the Kolmogorov law under the

pure GP model38, 39) would be attributable to the fol-lowing reasons. Note that the situation here is decay-ing turbulence. Although the total energy E(t) was con-served, Ei

kin(t) decreased with increasing Eckin(t). This

was because many compressible excitations were cre-ated through repeated vortex reconnections19, 20) anddisturbed the Richardson cascade of quantized vorticeseven at large scales.

Kobayashi and Tsubota overcame the difficulties ofNore et al. and obtained the Kolmogorov spectra in QTand clearly revealed the energy cascade.41, 42) They per-formed numerical calculation for the Fourier transformedGP equation with dissipation:

(i − γ(k))∂

∂tΦ(k, t) = [k2 − µ(t)]Φ(k, t)

+g

V 2

∑

k1,k2

Φ(k1, t)Φ∗(k2, t) × Φ(k − k1 + k2, t). (19)

Here, Φ(k, t) is the spatial Fourier component of Φ(x, t)and V is the system volume. The healing length is givenby ξ = 1/|Φ|√g. The dissipation should have the formγ(k) = γ0θ(k − 2π/ξ) with the step function θ, whichdissipates only the excitations smaller than ξ. This formof dissipation can be justified by the coupled analysis ofthe GP equation for the macroscopic wave function andthe Bogoliubov–de Gennes equations for thermal excita-tions.43)

First, Kobayashi et al. confirmed the Kolmogorovspectra for decaying turbulence.41) To obtain a turbu-lent state, they started the calculation from an initialconfiguration in which the density was uniform and thephase of the wave function had a random spatial distribu-tion. The initial wave function was dynamically unstableand soon developed into fully developed turbulence withmany quantized vortex loops. The spectrum Ei

kin(k, t)of the incompressible kinetic energy was then found toobey the Kolmogorov law.

A more elaborate analysis of steady QT was performedby introducing energy injection at large scales as wellas energy dissipation at small scales.42) Energy injectionat large scales was effected by moving a random poten-tial V (x, t). Numerically, Kobayashi et al. placed randomnumbers between 0 and V0 in space-time (x, t) at inter-vals of X0 for space and T0 for time and connected themsmoothly using a four-dimensional spline interpolation.The moving random potential exhibited a Gaussian two-

J. Phys. Soc. Jpn. Full Paper Author Name 7

point correlation:

〈V (x, t)V (x′, t′)〉 = V 20 exp

[

− (x − x′)2

2X20

− (t − t′)2

2T 20

]

.

(20)This moving random potential had a characteristic spa-tial scale of X0. Small vortex loops were first nucleatedby the random potential, growing to the scale of X0

by its motion subjected to Eq. (20). The vortex loopswere then cast into the Richardson cascade. If steadyQT is obtained by the balance between the energy in-jection and the dissipation, it should have an energy-containing range of k < 2π/X0, an inertial range of2π/X0 < k < 2π/ξ, and an energy-dissipative range of2π/ξ < k.

0

2

4

6

8

10

12

14

0 1 2 3 4 5

E(t)

Ekin (t)

Ekinc(t)

Ekini(t)

t

(a)

0

5

10

15

20

25

25 26 27 28 29 30

t

(b)

Fig. 1. Time development of E(t), Ekin(t), Ec

kin(t), and Ei

kin(t)

at (a) the initial stage 0 ≤ t ≤ 5 and (b) a later stage 25 ≤ t ≤

30.42)

Fig. 2. Flow of the incompressible kinetic energy Ei

kin(t) (upper

half of diagram) and compressible kinetic energy Ec

kin(t) (lower

half) in wavenumber space.42)

A typical simulation of steady turbulence was per-formed for V0 = 50, X0 = 4, and T0 = 6.4 × 10−2. Thedynamics started from the uniform wave function. Fig-ure 1 shows the time development of each energy com-ponent. The moving random potential nucleates soundwaves as well as vortices, but both figures show that the

incompressible kinetic energy Eikin(t) due to vortices is

dominant in the total kinetic energy Ekin(t). The fourenergies are almost constant for t & 25, and steady QTwas obtained.

Such a steady QT enables us to investigate the energycascade. Here, we expect an energy flow in wavenum-ber space similar to that in Fig. 2. The upper half ofthe diagram shows the kinetic energy Ei

kin(t) of quan-tized vortices, and the lower half shows the kinetic en-ergy Ec

kin(t) of compressible excitations. In the energy-containing range k < 2π/X0, the system receives in-compressible kinetic energy from the moving random po-tential. During the Richardson cascade process of quan-tized vortices, the energy flows from small to large k inthe inertial range 2π/X0 < k < 2π/ξ. In the energy-dissipative range 2π/ξ < k, the incompressible kineticenergy transforms to compressible kinetic energy throughreconnections of vortices or the disappearance of smallvortex loops. The moving random potential also createslong-wavelength compressible sound waves, which are an-other source of compressible kinetic energy and also pro-duce an interaction with vortices. However, the effect ofsound waves is weak because Ei

kin(t) is much larger thanEc

kin(t), as shown in Fig. 1.This cascade can be confirmed quantitatively by check-

ing whether the energy dissipation rate ε of Eikin(t) is

comparable to the flux of energy Π(k, t) through theRichardson cascade in the inertial range. Although thedetails are described in Reference 42, Π(k, t) is found tobe approximately independent of k and comparable toε. As shown in Fig. 3, the energy spectrum is quantita-tively consistent with the Kolmogorov law in the inertialrange 2π/X0 < k < 2π/ξ (b), which is equivalent to0.79 . k . 6.3.

(a)

1

2

3

4

5

6

7

8

-1 -0.5 0 0.5 1 1.5 2

Log Ekini(k)

Log Cε2/3k-5/3

(C~0.55)

log k

inertial

range

(b)

Fig. 3. (a) A typical vortex tangle. (b) Energy spectrumEi

kin(k, t) for QT. The plotted points are from an ensemble av-

erage of 50 randomly selected states at t > 25. The solid line isthe Kolmogorov law.42)

Kobayashi et al. studied the decay of QT under thesame formulation.44) After obtaining a steady tangle,they switched off the motion of the random potentialand found that L decayed as t−3/2.

8 J. Phys. Soc. Jpn. Full Paper Author Name

5.2 The Kelvin-wave cascadeThe arguments in the last section were chiefly limited

to the large scale, usually larger than the mean spacingℓ of a vortex tangle, in which the Richardson cascade iseffective for transferring the energy from large to smallscales. Here, we should consider what happens at smallerscales, for which the Richardson cascade should be lesseffective. The most probable scenario is the Kelvin-wavecascade. A Kelvin-wave is a deformation of a vortex lineinto a helix with the deformation propagating as a wavealong the vortex line.45) Kelvin-waves were first observedby making torsional oscillations in uniformly rotating su-perfluid 4He.46, 47) The approximate dispersion relationfor a rectilinear vortex is ωk = (κk2)/(4π)(ln(1/ka0)+c)with a constant c ∼ 1. Note that this k is the wavenum-ber of an excited Kelvin-wave and is different from thewavenumber used for the energy spectrum in the lastsubsection. At a finite temperature, a significant fractionof normal fluid damps Kelvin-waves through mutual fric-tion. At very low temperatures, however, mutual frictiondoes not occur, and the only possible mechanism of dis-sipation is the radiation of phonons.48) Phonon radia-tion becomes effective only when the frequency becomesvery high, typically on the order of GHz (k ∼ 10−1

nm−1), so a mechanism is required to transfer the en-ergy to such high wavenumbers in order for Kelvin-wavesto be damped. An early numerical simulation based onthe vortex filament model showed that Kelvin-waves areunstable to the buildup of side bands.49) This indicatesthe possibility that nonlinear interactions between differ-ent Kelvin-wave numbers can transfer energy from smallto large wavenumbers, namely the Kelvin-wave cascade.This idea was first suggested by Svistunov50) and waslater developed and confirmed through theoretical andnumerical analyses by Vinen, Tsubota and Mitani,51)

and Kozik and Svistunov.52–54)

Observation of the Kelvin-wave cascade is crucial.Such studies are not easy for a vortex tangle. The easiestapproach would be to consider rotation. In a rotating ves-sel, quantized vortices form a vortex lattice parallel to therotational axis. By oscillating the vortices, Kelvin-wavescan be excited. This approach was used in the pioneer-ing experiments of Hall.46, 47) The challenge is detectingthe Kelvin-wave cascade when it occurs. There are twopossible methods. The first is the direct visualization ofvortices. Recently, remarkable progress has been made invisualizing superflow and quantized vortices,55, 56) as de-scribed in Section 6.4. Bewley et al. visualized quantizedvortices by trapping micron-sized solid hydrogen parti-cles.56) They also observed a vortex array under rota-tion. Applying this technique to the Kelvin-wave cascadecould reveal important information directly. The secondmethod is to observe acoustic emission resulting froma Kelvin-wave cascade. Since the frequency of emittedphonons is estimated to be of GHz order, this observationmay not be easy and presents a challenging experimentalproblem.

5.3 Classical-quantum crossoverAn important trend is investigation of the nature of

the transition between Richardson (classical) and Kelvin-

wave (quantum) cascades. Several theoretical considera-tions on this topic have been investigated,57, 58) althoughfew numerical or experimental studies have been con-ducted.

L’vov et al. theoretically discussed a bottleneckcrossover between the two regions.57) Their investiga-tion was based on the idea that a vortex tangle obeyingthe Kolmogorov spectrum is polarized to some degree,and they viewed the vortex tangle as a set of vortexbundles. When they tried to connect the energy spec-tra between two cascades, a serious mismatch occurredat the crossover scale between the classical and quantumspectra. This mismatch prevented the energy flux frompropagating fully through the crossover region, whichis referred to as the bottleneck effect. In order to rem-edy this mismatch, L’vov et al. used warm cascade solu-tions59) and proposed a thermal-equilibrium type spec-trum between the classical and quantum spectra. Theanalysis of L’vov et al. was based on the assumption thatthe coarse-grained macroscopic picture of quantized vor-tices remains valid down to a scale of ℓ. Without thisassumption, Kozik and Svistunov theoretically investi-gated the details of the structure of the vortex bundle inthe crossover region.58) Depending on the types of vor-tex reconnections, the crossover range near ℓ was furtherdivided into three subranges, resolving the mismatch be-tween two cascades. However, this topic is not yet fixed,and remains controversial.

5.4 Possible dissipation mechanism at zero temperature

Here, we should investigate the possible dissipationmechanisms of QT at very low temperatures, in whichthe normal fluid component is so negligible that mutualfriction does not occur. In this case, there is no dissipa-tion at relatively large scales, and dissipation can onlyoccur at small scales. Energy at large scales should betransferred to smaller scales by a Richardson cascadeand then a Kelvin-wave cascade until the dissipation be-comes effective. The description presented herein is onlyschematic. Please refer to other papers for a more de-tailed discussion.10, 29)

Dissipation of QT at zero temperature was first pro-posed by Feynman.3) Feynman suggested that a largevortex loop should be broken up into smaller loopsthrough repeated vortex reconnections, which is essen-tially identical to the Richardson cascade. Feynmanthought that the smallest vortex ring having a radiuscomparable to the atomic scale must be a roton, al-though this belief is not currently accepted universally.Later, Vinen considered the decay of superfluid turbu-lence in order to understand his experimental results forthermal counterflow,4, 5) leading him to propose Vinen’sequation.6) Although the experiments were performed atfinite temperatures, the physics is important in orderto study the decay of QT, as described briefly herein.Vortices in turbulence are assumed to be approximatelyevenly spaced with a mean spacing of ℓ = L−1/2. Theenergy of the vortices then spreads from the vortices ofwavenumber 1/ℓ to a wide range of wavenumbers. Theoverall decay of the energy is governed by the character-istic velocity vs = κ/2πℓ and the time constant ℓ/vs of

J. Phys. Soc. Jpn. Full Paper Author Name 9

the vortices of size ℓ, giving

dv2s

dt= −χ

v2s

ℓ/vs= −χ

v3s

ℓ, (21)

where χ is a dimensionless parameter that is generallydependent on temperature. Rewriting this equation usingL, we obtain

dL

dt= −χ

κ

2πL2. (22)

This is called Vinen’s equation. The original Vinen’sequation also includes a term of vortex amplification,which is not shown here. This equation describes the de-cay of L, the solution of which is

1

L=

1

L0+ χ

κ

2πt, (23)

where L0 is L at t = 0. The thermal counterflow obser-vations are well described by this solution, which givesthe values of χ as a function of temperature.6) The de-cay at finite temperatures is due to mutual friction. Thevalue of χ at zero temperature is obtained by a numericalsimulation of the vortex filament model.16)

What causes the different types of decay of L ∼ t−3/2

of Eq. (11) and L ∼ t−1 of Eq. (23)? The different typesof decay originate from the different structures of vor-tex tangles in QT. When the energy spectrum of a vor-tex tangle obeys the Kolmogorov law, the tangle is self-similar in the inertial range, and most energy is in thelargest vortex. The line length density then decays asL ∼ t−3/2, as described in Section 4.1. If a vortex tangleis random and has little correlation, however, the onlylength scale is ℓ = L−1/2, yielding a decay of L ∼ t−1,as described herein. Therefore, the observation of how Ldecays is helpful in understanding the structure of vortextangles in QT.

However, these studies do not explain what happens toQT at zero temperature at small scales. There are severalpossibilities. The first is acoustic emission at vortex re-connections. In classical fluid dynamics it is known thatvortex reconnections cause acoustic emission. In quan-tum fluids, numerical simulations of the GP model showthat acoustic emission occurs at every reconnection.19, 20)

The dissipated energy during each reconnection is ap-proximately 3ξ times the vortex line energy per unitlength. We can estimate how L decays by this mecha-nism. The number of reconnection events per volume pertime is on the order of κL5/2.16) Since each reconnectionshould reduce the vortex length by the order of ξ, thedecay of L can be described as dL/dt = −χ1ξκL5/2 witha constant χ1 of order unity. The solution is

1

L3/2=

1

L3/20

+3

2χ1ξκt. (24)

In superfluid helium, especially 4He, ξ is so small thatdecay due to this mechanism should be negligible.

The second possible mechanism is the radiation ofsound (phonons) by the oscillatory motion of a vortexcore. The characteristic frequencies of vortex motion ona scale ℓ are of order κ/ℓ2, which is too small to causeeffective radiation.29) In order to make the radiation ef-

fective, the vortices should form small-scale structures,which would be realized by two consecutive processes.The first process is due to vortex reconnections.50) Ina dense isotropic tangle, vortex reconnections occur re-peatedly at a rate of order κ/ℓ5 per unit volume.16) Whentwo vortices approach one another, they twist so thatthey become locally antiparallel, and reconnect.8) Theythen separate leaving a local small-scale structure. How-ever, even such local structures created by single recon-nection events are insufficient to cause effective acousticradiation. The next process occurs due to a Kelvin-wavecascade described in Section 5.2. The Kelvin-wave cas-cade could transfer the energy into the small scales inwhich the radiation of sound becomes effective.

In actual experiments, we may have to consider theeffects of vortex diffusion, even though these effects arenot due to dissipation. Using the vortex filament model,Tsubota et al. numerically studied how an inhomoge-neous vortex tangle diffuses.60) The obtained diffusion ofthe line length density L(x, t) was well described by

dL(x, t)

dt= −χ

κ

2πL(x, t)2 + D∇2L(x, t). (25)

Without the second term on the right-hand side, this isVinen’s equation (22).6) The second term represents thediffusion of a vortex tangle. The numerical results showthat the diffusion constant D is approximately 0.1κ. Thedimensional argument indicates that D must be on theorder of κ.60)

6. New QT experiments

This section reviews briefly recent important experi-ments on QT, although additional research is needed forall of these experiments.

6.1 Temperature-dependent transition to QT

Krusius et al. conducted a series of experimental stud-ies on QT in superfluid 3He -B under rotation61–66) andobserved the strong temperature-dependent transition ofa few seed vortices to turbulence. A seed vortex can de-velop into turbulence when the temperature is lower thanthe onset temperature Ton, whereas the seed vortex doesnot lead to turbulence above Ton. The onset temperatureTon is approximately 0.6Tc with the superfluid transitiontemperature Tc being independent of flow velocity.64)

The key characteristics of using 3He -B for research onQT are as follows. First, the normal fluid component istoo viscous to become turbulent, which is very differentfrom the case of superfluid 4He. Secondly, monitoring theNMR absorption spectra enables us to count the numberof quantized vortices, which is not possible in 4He.

Figure 4 shows schematically the vortex instability andturbulence in rotating 3He-B in the turbulent tempera-ture region studied by the Helsinki group. A seed vortexis injected into 3He-B in a cylindrical vessel rotating withthe angular velocity Ω. In the laboratory frame, the nor-mal fluid component causes solid body rotation, while thesuperfluid component remains at rest. When the tem-perature is higher than Ton, the vortex just orientatesalong the rotational axis because of the mutual friction.If the temperature is lower than Ton, however, the vortex

10 J. Phys. Soc. Jpn. Full Paper Author Name

becomes unstable, through several vortex reconnectionsthat occur chiefly near the boundary,65) splitting intolots of vortices, and eventually reaching an equilibriumvortex lattice state. While the turbulent front propagatesto the vortex-free region, the front takes on the beauti-ful ”twisted vortex state”.66) The onset temperature isdetermined by considering the dynamic mutual frictionparameter ζ = (1−α′)/α, where α and α′ are the mutualfriction coefficients appearing in Eq. (6).61) This param-eter ζ works in this system like the Reynolds number ina usual fluid.

Fig. 4. Schematic vortex instability and turbulence in rotat-ing 3He-B in the turbulent temperature region studied by theHelsinki group.62)

6.2 Dissipation at very low temperatures

Recently a few experimental studies on these topicshave been conducted, showing the reduction of the dis-sipation at low temperatures. Eltsov et al. studied thevortex front propagation into a region of vortex-free flowin rotating superfluid 3He-B by NMR measurements fol-lowing a series of investigations.67) The observed frontvelocity as a function of temperature shows the tran-sition from laminar through quasiclassical turbulent toquantum turbulent. The front velocity is related to theeffective dissipation, which exhibits a peculiar reductionat very low temperatures below approximately 0.25Tc

with the critical temperature Tc. Eltsov et al. claim thatthis is attributable to the bottleneck effect. Walmsleyet al. made another observation of the effective viscos-ity of turbulence in superfluid 4He.30) Turbulence wasproduced by an impulsive spin down from an angular ve-locity to rest for a cube-shaped container, and the linelength density was measured by scattering negative ions.The observed effective kinematic viscosity showed thestriking reduction at low temperatures below approxi-mately 0.8 K. In this case, the bottleneck effect may notbe so significant. The authors believe this may be due toanother mechanism, namely, the difficulty in transferringenergy through wavenumbers from the three-dimensionalRichardson cascade to the one-dimensional Kelvin-wavecascade.

6.3 QT created by vibrating structures

Recently, vibrating structures, such as discs, spheres,grids, and wires, have been widely used for researchinto QT.36) Despite detailed differences between the used

structures, the experiments show some surprisingly com-mon phenomena. This trend started with the pioneeringobservation of QT on an oscillating microsphere by Jageret al.68)

The sphere used by Jager et al. had a radius of approx-imately 100 µm. The sphere was made from a stronglyferromagnetic material and had a very rough surface.The sphere was magnetically levitated in superfluid 4He,and its response with respect to the alternating drivewas observed. At low drives, the velocity response v wasproportional to the drive FD, taking the ”laminar” formFD = λ(T )v, with the temperature-dependent coefficientλ(T ). At high drives, the response changed to the ”tur-bulent” form FD = γ(T )(v2 − v2

0) above the critical ve-locity v0. At relatively low temperatures the transitionfrom laminar to turbulent response was accompanied bysignificant hysteresis.

Subsequently, several groups have experimentally in-vestigated the transition to turbulence in superfluids 4Heand 3He-B by using grids,31, 69–72) wires,73–77) and tun-ing forks.78) The details of the observations are describedin a review article.36) Although a detailed discussion ofthe cases of 4He and 3He at both low and relatively hightemperatures is required, we shall next briefly describe afew important points.

These experimental studies reported some commonbehaviors independent of the details of the structures,such as the type, shape, and surface roughness. The ob-served critical velocities are in the range from 1 mm/sto approximately 200 mm/s. Since the velocity is usu-ally much lower than the Landau critical velocity of ap-proximately 50 m/s, the transition to turbulence shouldcome from not intrinsic nucleation of vortices but exten-sion or amplification of remnant vortices. Such behavioris shown in the numerical simulation by the vortex fila-ment model.79) Figure 5 shows how the remnant vorticesthat are initially attached to a sphere develop into tur-bulence under an oscillating flow. Such behavior mustbe related to the essence of the observations, many unre-solved problems remain, such as the nature of the criticalvelocity and the origin of the hysteresis for the transitionbetween laminar and turbulent response.

6.4 Visualization of QT

There has been little direct experimental informationabout the flow in superfluid 4He. This is mainly becauseusual flow visualization techniques are not applicable tosuperfluid. However, the situation is rapidly changing.37)

For QT, we should seed the fluid with tracer particles inorder to visualize the flow field and, if possible, quan-tized vortices, which are observable by appropriate opti-cal techniques.

A significant contribution was made by Zhang and VanSciver.55) Using the particle image velocimetry (PIV)technique with 1.7-µm-diameter polymer particles, theyvisualized a large-scale turbulent flow both in front ofand behind a cylinder in a counterflow in superfluid 4Heat finite temperatures. In classical fluids, such large-scaleturbulent structures are seen downstream of objects suchas cylinders, with the structures periodically detaching toform a vortex street.80) In the present case of 4He coun-

J. Phys. Soc. Jpn. Full Paper Author Name 11

Fig. 5. Evolution of the vortex line near a sphere of radius 100 µmin an oscillating superflow of 150 mm−1 at 200 Hz [Hanninen,Tsubota and Vinen: Phys. Rev. B 75 (2007) 064502, reproducedwith permission. Copyright 2007 by the American Physical So-ciety].

terflow, the locations of the large-scale turbulent struc-tures was relatively stable, and they did not detach andmove downstream, although local fluctuations in the tur-bulence were evident.

Here, it is necessary to understand whether such tracerparticles follow the normal flow or the superflow, or evena more complex flow. Poole et al. studied these problemstheoretically and numerically to show that the situationchanges depending on the size and mass of the tracerparticles.81)

Another significant contribution was the visualizationof quantized vortices by Bewley et al.56) In their ex-periments, the liquid helium was seeded with solid hy-drogen particles smaller than 2.7 µm at a temperatureslightly above Tλ after which the fluid was cooled be-low Tλ. When the temperature is above Tλ, the particleswere seen to form a homogeneous cloud that dispersesthroughout the fluid. However, on passing through Tλ,the particles coalesced into web-like structures. Bewleyet al. suggested that these structures represent decorated

quantized vortex lines. They reported that the vortexlines appear to form connections rather than remainingseparated, and were homogeneously distributed through-out the fluid. The fork-like structures may indicate thatseveral vortices are attached to the same particle, as indi-cated by numerical simulations of vortex pinning.82) Thisexperimental work on the visualization of quantized vor-tices is described in detail elsewhere.83)

7. Summary and discussions: QT in atomic

Bose-Einstein condensates

In this article, we have reviewed recent research onQT in superfluid helium. Research on QT is currentlyone of the most important branches in low-temperaturephysics, attracting the attention of many scientists. QTis comprised of quantized vortices as definite elements,which differs greatly from conventional turbulence. Thus,investigation of QT may lead to a breakthrough in un-derstanding one of the great mysteries of nature. Thislast section addresses the topic of QT in atomic Bose-Einstein condensates (BECs).

The realization of Bose–Einstein condensation intrapped atomic gases in 1995 has stimulated intense ex-perimental and theoretical activity in modern physics.21)

As proof of the existence of superfluidity, quantized vor-tices were created and observed in atomic BECs, andnumerous efforts have been devoted to solving a num-ber of fascinating problems.84) Atomic BECs have severaladvantages over superfluid helium. The most importantpoint is that modern optical techniques enable us to di-rectly visualize quantized vortices. The long history ofresearch into superfluid helium reveals two main cooper-ative phenomena of quantized vortices. One is a vortexlattice in a rotating superfluid, and the other is a vor-tex tangle in QT. To date, most studies of quantizedvortices in atomic BECs have been devoted to the for-mer, and few have examined the latter. Recently, it hasbeen shown theoretically that QT can also be created intrapped BECs and that the energy spectrum obeys theKolmogorov law.85, 86) Considering the advantages of thissystem, it would be possible to perform new studies onQT beyond those in superfluid helium.

Acknowledgment

The present study was supported in part by a Grant-in-Aid for Scientific Research from JSPS (No. 18340109)and by a Grant-in-Aid for Scientific Research on PriorityAreas from MEXT (No. 17071008).

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