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Quantum Turbulence

Matthew S. Paoletti

Department of Physics, Institute for Research in Electronics and

Applied Physics, University of Maryland, College Park,

Maryland 20742-4111; email: [email protected]

Daniel P. Lathrop

Department of Physics, Department of Geology, Institute for Research in

Electronics and Applied Physics, Institute for Physical Sciences and Technology,

University of Maryland, College Park, Maryland 20742-4111;

email: [email protected]

Annu. Rev. Condens. Matter Phys. 2011. 2:213–34

First published online as a Review in Advance on

December 17, 2010

The Annual Review of Condensed Matter Physics is

online at conmatphys.annualreviews.org

This article’s doi:

10.1146/annurev-conmatphys-062910-140533

Copyright © 2011 by Annual Reviews.

All rights reserved

1947-5454/11/0310-0213$20.00

Keywords

superfluid, superconductor, reconnection, quantum fluids,

vortex rings

Abstract

We examine developments in the study of quantum turbulence with a

special focus on clearly defining many of the terms used in the field.

We critically review the diverse theoretical, computational, and

experimental approaches from the point of view of experimental

observers. Similarities and differences between the general properties

of classical and quantum turbulence are elucidated. The dynamics

and interactions of quantized vortices and their role in quantum

turbulence are discussed with particular emphasis on reconnection

and vortex ring collapse. A stark distinction between the velocity

statistics of quantum and classical turbulence is exhibited and used

to highlight a potential analogy between quantum turbulence and

magnetohydrodynamic (MHD) turbulence in astrophysical plasmas.

Although much of this review pertains to superfluid 4He (He II), the

underlying science is broadly applicable to other quantum fluids

such as 3He-B, type-II superconductors, Bose-Einstein condensates,

Weinberg-Salam fields, and grand-unified-theory (GUT) Higgs fields.

213

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1. INTRODUCTION

Turbulent processes are ubiquitous on Earth and throughout the universe. Turbulence provides

crucial mechanisms necessary to sustain life on Earth by, for example, transporting spores of

plants across land, nutrients, heat and salinity in the oceans, and various gases in the atmo-

sphere. Turbulent motions of molten iron in Earth’s core produce its magnetic field (1), which is

used for navigation as well as providing a barrier to potentially harmful charged particles

emitted by the sun. More familiar experiences with turbulence might be found on airplane

flights, the patterns and motions of clouds, or the flow of a rushing river. In addition to these

naturally occurring phenomena, turbulence plays a vital role in the design and operation of

most industrial processes.

Despite the abundant examples of turbulence, there is no consensus definition of the term.

Here, we define turbulence as a dynamic field that is spatially complex, aperiodic in time, and

involves processes spanning several orders of magnitude in spatial extent and temporal fre-

quency. Therefore, turbulence by this definition is not restricted to fluid motions alone. The

human heart, for example, undergoes filament turbulence immediately before fibrillation (2);

magnetohydrodynamic (MHD) turbulence dominates the behavior of many astrophysical bod-

ies (3–5); electromagnetic waves in laboratory plasmas may also be turbulent (6); and related

states can occur in optical systems (ring laser turbulence) (7), opto-electronic systems such as a

Mach-Zehnder feedback loop (8), and neuronal systems (9).

The interplay between large and small scales in turbulent fields has made their study difficult

owing to the need to resolve several orders of magnitude in spatial and temporal extent. For the

experimentalist this requires large experiments and the need to observe fast fluctuations on micro-

scopic or mesoscopic scales. For theoreticians the nonlinear equations of motion are difficult

owing to the vast span of scales involved in turbulence. As such, terms in the equations cannot

be neglected because the contribution from each may vary over the relevant scales. This requires

numerical simulations of experimental and naturally occurring systems to have extremely large

domains that are also capable of resolving the small scales dominated by dissipation.

Quantum fluids, such as superfluids, superconductors, the Higgs field, and Bose-Einstein

condensates, may exhibit turbulent states that are quite distinct from their classical counter-

parts owing to long-range quantum order. Such order places quantum-mechanical constraints

on their dynamics, as measured by a non-zero order parameter. Specifically, all vorticity (mag-

netic field) in the case of superfluids and Bose-Einstein condensates (superconductors) is

restricted to topological defects in the order parameter of the system. These line-like structures

are referred to as quantized vortices because continuity in the order parameter quantizes the

circulating flow around each topological defect. Turbulence in a quantum fluid, then, exhibits a

tangle of interacting quantized vortices as first envisioned by Feynman (10), which is quite

distinct from the continuous distributions of vorticity present in classical fluid turbulence.

In this review, we focus on the dynamics and interactions of quantized vortices in turbulent

quantum fluids. In particular, we place emphasis on the role of reconnection between quantized

vortices and the collapse of vortex rings for quantum fluids driven from equilibrium. Although

much of the discussion centers on superfluid 4He (He II), many of the concepts are broadly

applicable to turbulence in other quantum fluids. In Section 2 we discuss turbulence in classical

and quantum fluids. Classical and quantum turbulence are compared and contrasted in Section 3.

Quantized vortices and complex order parameters used to describe quantum fluids are intro-

duced in Section 4. In Section 5 we describe impurity pinning of quantized vortices.

Reconnection and ring collapse are detailed in Section 6, followed by a discussion of quantum

turbulence in Section 7. We conclude in Section 8.

Turbulence: Spatially

extended field with a

large, continuousrange of aperiodic

spatial and temporal

dynamics

Order parameter:

Spatio-temporal field

that serves as ameasure of an ordered

thermodynamic phase

Topological defect:

Topological structure

in the order parameterthat cannot be

removed by simple

symmetry operations

(e.g., magnetic domainwalls or quantized

vortices)

Reconnection:

Dynamical, topology-

changing process bywhich two line-like

objects cross,

exchange tails, and

separate

Vortex ring: Classicalor quantized flow field

consisting of a vortex

in a closed loop

Quantum turbulence:

Turbulent complex

field or interacting setof complex fields

Impurity pinning:

Trapping of a topolog-

ical defect in the order

parameter by animpurity or boundary

roughness

Ring collapse: Dynam-

ical process by which a

vortex ring shrinksand disappears

through dissipation

214 Paoletti � Lathrop

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2. FLUID TURBULENCE

In this section we review properties of turbulence in both classical and quantum fluids. The

description of classical, or Newtonian fluid, turbulence stems from Kolmogorov’s description

(11, 12) given in 1941, while the details of quantum turbulence are derived from the two-fluid

model initially proposed by Tisza (13) and Landau (14).

2.1. Classical Fluid Turbulence

The pioneering work of Kolmogorov (11, 12) remains the cornerstone of the statistical theory

of turbulence in classical fluids. Kolmogorov made two key assumptions: (a) local isotropy and

homogeneity and (b) an inertial range in which turbulent energy is transferred from large to

small scales independent of viscosity and generation mechanisms. Dimensional arguments then

yield the spectral density

EðkÞ ¼ cKE�2=3k�5=3, 1:

where �E is the average energy dissipation rate per unit mass, cK is the Kolmogorov constant,

and k is the wavenumber. Such turbulence in classical fluids is often referred to as Kolmogorov

turbulence. Although experiments have found the effects of intermittency to be important for

high-order moments, the correction to the spectral form is small.

Kolmogorov also hypothesized that for sufficiently intense turbulence the statistics of

small-scale motions are universally and uniquely determined by the mean energy dissipation

rate �E and the kinematic viscosity of the fluid n. Dimensional arguments then yield a single

length scale

� ¼ n3

�E

� �1=4, 2:

which is referred to as the Kolmogorov length. On scales comparable or smaller than � viscosity

homogenizes the flow, and the dissipation of shear converts mechanical energy into heat.

2.2. Quantum Fluid Turbulence

Quantum fluids are typically described as two interpenetrating fluids: a viscous normal fluid

akin to water and an inviscid superfluid exhibiting long-range quantum order. Each component

has a distinct velocity field {vn, vs} and temperature-dependent density {rn, rs} for the normal

and superfluid components, respectively. There is no conventional viscous dissipation in the

superfluid component; rather the flow of a superfluid is similar to the resistance-free motion of

electrons in a superconductor. This so-called two-fluid model was initially proposed for the

description of phenomena related to He II by Tisza (13) in 1938 and refined by Landau (14) in

1941 to include a hydrodynamic description.

Turbulence can arise in either or both of the fluid components. Motions of the normal fluid

appear to be indistinguishable from those of classical fluids. Turbulence in the superfluid

component, though, is dominated by quantum-mechanical constraints. Specifically, vorticity is

constrained to the cores of quantized vortices that can be as thin as a single fluid atom.

Quantum mechanics requires the circulation around each vortex filament to be given by an

integer multiple of the quantum of circulation k ¼ h=m, where h is Planck’s constant, and m is

the mass of a fluid atom. Turbulence, then, in the superfluid component is dominated by a

complex, interacting tangle of quantized vortices (10), as shown in Figure 1.

Newtonian fluid

turbulence:

Turbulence in aclassical, Newtonian

viscous fluid

Kolmogorov

turbulence:

Turbulence exhibiting

spatial power spectrawith EðkÞ / k�5=3

www.annualreviews.org � Quantum Turbulence 215

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The relevant length scales of a quantum turbulent state have a lower threshold given by the

diameter of a quantized vortex core (�10�8 cm in He II, except very near Tl) and an upper

bound of the system size, or at least many times the typical intervortex spacing (�1 cm). The

slowest timescales are produced by long-range vortex-vortex interactions, which may be on

the order of 1 s, whereas the fastest are wave motions along the quantized vortices with periods

less than 10�9 s (16). These waves are transverse, circularly polarized displacements that are

restored by vortex tension produced by the kinetic energy per unit length of a quantized vortex.

Such Kelvin waves along a rectilinear vortex have an approximate dispersion relation given by (16)

o ¼ kk2

4pln

1

ka0

� �þ c

� �, 3:

where a0 is a vortex cutoff parameter, and c� 1. Evidently, a quantized vortex tangle involves

the interaction of a wide breadth of spatial and temporal scales, as required for the system to be

called turbulent.

Additional complications arise from interactions between the normal and superfluid com-

ponents mediated by the quantized vortices. The two fluids couple; thus, turbulence in one is

capable of triggering turbulence in the other. A consensus on the coupled equations of motion

for the two fluid components still evades the scientific community; although, several proposi-

tions have been made. One approach is to describe the normal fluid with the Navier-Stokes

equation (17), which is the standard equation of motion used for Newtonian fluids, and the

Gross-Pitaevskii equation for the superfluid component (18, 19), which involves a complex

field and is applicable for superfluids at T¼ 0 K. To each equation, coupling terms can be added

to describe the mutual friction between the two fluids, as initially observed and described

phenomenologically by Vinen (20–22). Another approach to two-fluid dynamics uses only the

Gross-Pitaevski equation (23), or the Gross-Pitaevskii equation coupled with the Bogoliubov-de

Gennes equation (24), where the phonon field acts as a dissipative normal component. A

shortcoming of such approaches is the overall Hamiltonian structure that only allows energy

Figure 1

A snapshot of an evolving quantized vortex tangle from the simulations of Tsubota et al. (15).


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to be transferred from the vortex field to the phonon field. Thermostating the phonon field in

such models might allow isothermal modeling.

3. COMPARING CLASSICAL AND QUANTUM TURBULENCE

Despite the fundamental differences between classical fluids and quantum fluids, there have

been notable studies since 1992 demonstrating similarities between quantum and classical

turbulence (25–43). Experiments by Maurer & Tabeling (29) on turbulence generated in 4He

by two counter-rotating disks observed Kolmogorov energy spectra typical of classical fluids

(see Equation 1) that were indistinguishable above and below the superfluid transition. The

Kolmogorov energy spectrum was also seen by Kobayashi & Tsubota (37) in numerical simu-

lations of the Gross-Pitaevskii equation with small-scale dissipation added to the otherwise

energy-conserving dynamics. The classical decay of vorticity (26) has been observed in several

experiments in He II, independent of the mechanism used for driving the turbulence, as shown

in Figure 2. The experiments by Smith et al. (26) and Stalp et al. (31) generated turbulence using

towed grids, Skrbek et al. (36) drove the system with thermal counterflows, and Walmsley et al.

(42) produced turbulence via impulsive spin down. The expected exponent of �3=2 for the

2.2

2.0

1.8

1.6

1.4

1.2

1.0

m

4.2 4.4 4.6 4.8 5.0 5.2 5.4Log10(ReM)

1,000

100

10

1

0.1 1 10

κL (

Hz)

Time (s)

T = 1.6 K

t–3/2

a

b

c

0.88 W cm–2

0.57 W cm–2

0.31 W cm–2

0.22 W cm–2

0.14 W cm–2

0.08 W cm–2

0.08 W cm–2

10–1 100 101101

102

103

104

105

102 103 104

Ωt

LΩ–

3/2

(cm

–2 s

3/2

)

T = 0.15 K

0.05 rad s–1

0.15 rad s–1

0.5 rad s–1

1.5 rad s–1

Figure 2

(a) Measured decay exponentm as a function of the grid Reynolds number ReM (adapted from Reference 26).

The average value of hmi ¼ 1.50�0.2 agrees with the predicted value of 1.5 for classical turbulence.

(b) Vorticity defined as o ¼ kL, where L is the vortex line-length density, for decaying thermal counterflowturbulence (adapted from Reference 36). (c) Rescaled vorticity decay for turbulence generated by impulsive

spin down (adapted from Reference 42). In all cases, o � t�3/2.


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decay of vorticity in classical fluids is shown for comparison as the dashed line in Figure 2a and

the solid lines in parts b and c. Indeed, the late-time decay of vorticity in quantum fluids closely

resembles the behavior expected of classical fluids.

In general, one should expect quantum-mechanical systems to closely approximate their

classical counterparts when the system is composed of many quanta. One might argue this is

required by the correspondence principle (44). In such cases, the disparity between discrete values

of the relevant quantized parameter begins to resemble the continuity of the classical field. In the

case of fluids, the large-scale flow produced by many quantized vortices bears strong resemblance

to motions typical of classical fluids. On length scales comparable to the typical spacing between

quantized vortices, though, quantum turbulence is quite distinct from its classical analog.

A comparison of classical and quantum turbulence is further detailed in the following subsections.

3.1. Energy Cascades

Turbulence in both quantum and classical fluids is often described as an energy cascade medi-

ated by nonlinear interactions from large scales to small scales where dissipation prevails

(11, 12). In classical fluids this takes the form of a Richardson cascade (45), whereby energy is

injected at small wavenumbers k, which spawn larger wavenumber structures through the

nonlinearities of inertia. When k grows sufficiently large such that 1=k � �, where � is the

Kolmogorov length defined in Equation 2, energy is lost to viscous heating. However, one

should not picture large vortices or eddies spawning smaller ones, given that this has never been

observed. Rather, the correct picture is that large eddies (small k) interact with one another,

producing small-scale structures (large k) with high strain and shear that dissipate energy.

The cascade process is quite different for quantum fluids because all of the vortex cores are

topologically constrained, and there are no direct viscous losses (46). The conventional picture

for the quantum turbulent energy cascade is as follows. (a) Bundles of nearly parallel quantized

vortices form through interactions with the normal fluid that tend to align them. These bundles

produce large-scale motions similar to small wavenumber (large spatial extent) eddies in classi-

cal fluids (23, 43, 47). (b) Energy is transmitted to larger k via reconnection between individual

vortices, which is described in Section 6. (c) Reconnection events trigger polychromatic helical

Kelvin waves on the vortex lines. (d) The Kelvin waves interact nonlinearly, producing even

larger wavenumbers until they lose energy to phonon emission, which radiates energy to the

boundaries (16). Aspects of the quantum turbulent cascade remain controversial, and this is an

active area of research and debate (15, 16, 48–59).

3.2. Large-Scale Motions

Large-scale motions in classical fluid turbulence typically refer to eddies and other coherent

structures that are much larger than length scales in which molecular diffusion converts me-

chanical energy into heat. The nature of large-scale motions in quantum fluids would appear to

bear no analogy to classic fluids because vorticity in the superfluid component does not diffuse

and is incapable of forming such coherent structures. However, it is possible for groups of

quantized vortices to mimic large-scale circulation common in classical fluids.

As a simple example, we consider the case of a fluid contained in an infinitely long, cylindri-

cal vessel that rotates at a constant angular velocity about its vertical axis. If the fluid inside the

vessel is classical, then it will eventually reach a state of solid-body rotation in which every fluid

element rotates about the rotation axis at a frequency O. Vorticity in a quantum fluid is

quantized; thus, it is clear that many quantized vortices aligned parallel to the rotation axis are


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required to produce a similar coarse-grained velocity field to the classical case. The flow around

each quantized vortex, though, is given by v ¼ fk=ð2psÞ, where f is the azimuthal coordinate

and s the cylindrical radius; this is at odds with the velocity field for solid-body rotation, which

grows linearly with s rather than decaying. These complications can be overcome if the quan-

tized vortices are arranged in a regular lattice, with a triangular lattice producing the lowest

energy state. Feynman first calculated this triangular arrangement as the lowest energy approxi-

mation to solid-body rotation (10). This lattice structure has been observed experimentally in

superfluid 4He (60) and analogously in superconductors (61) and Bose-Einstein condensates (62).

It is therefore possible for quantized vortices to closely approximate the large-scale flows of

classical fluids. This approximation is only applicable, though, on length scales much larger

than the typical spacing between quantized vortices. In all of the quantum turbulence studies

mentioned above (25–43) that observed similarities to classical turbulence, the flow scales

observed were considerably larger than typical intervortex spacings. These results can reason-

ably be attributed to the fact that on such scales the pairwise interactions of quantized vortices

are insignificant. Many authors also posit that the normal and superfluid components become

locked as a result of mutual friction.

3.3. Small-Scale Motions

Small-scale motions in classical turbulence are drastically different from those in quantum

fluids. In classical fluids, small-scale motions take place over length scales comparable with the

Kolmogorov length � (11, 12), below which viscosity smooths the flow. Even though the

vorticity and strain (not the velocity field itself) undergo local, intermittent bursts and are

nonuniform in both space and time, viscosity diffuses momentum and homogenizes mean

quantities on dissipative length scales.

We define small scales in a quantum fluid as lengths smaller than the typical spacing between

quantized vortices, l. As mentioned in the previous section, it is assumed that the interactions

between quantized vortices are smoothed over for motions coarse-grained over scales much

greater than l. Vorticity in the superfluid component cannot diffuse, rendering it incapable of

homogenizing even on atomic length scales. For scales below l, vorticity in a quantum fluid

undergoes extreme fluctuations because it is topologically constrained to dynamic, line-like

filaments, thereby bearing no resemblance to a classical fluid observed on scales comparable to �.

The local interactions between quantized vortices, such as reconnection and ring collapse, can

produce very large velocities bounded only by the speed of sound of the fluid. Consequently, it is

now generally accepted that the small-scale dynamics of quantum turbulence and classical

turbulence are quite different.

4. SUPERFLUID ORDER PARAMETER AND QUANTIZED VORTICES

Transitions into superfluid, superconducting, or Bose-Einstein condensed states are symmetry-

breaking phase transitions. As such, these systems may be described by an order parameter. In

the context of the two-fluid model for He II (see chapter XVI of Reference 17 for more details),

only the superfluid component is described by the order parameter, which takes the form of a

complex field analogous to a wave function

c ¼ feif, 4:

for real fields f and f. As the quantum fluid exhibits long-range quantum order, the order

parameter indicates the level of synchronization of the atomic wave functions at long range.


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Atomic wave functions are complex fields; thus, the order parameter quantifies the amplitude and

phase of the ordered (synchronized) part of these wave functions. The superfluid density is given by

rs ¼ mc�c ¼ mf 2, 5:

wherem is the mass of a fluid atom. In analogy with the increase in net magnetization as T! 0 K

in simple ferromagnets, the order in quantum fluids, measured by the superfluid fraction rs=r,also increases for decreasing temperature. The superfluid fraction is zero at the transition temper-

ature (complete disorder) and unity at T ¼ 0 K (complete order).

The quantum-mechanical mass current js takes the form

js ¼ħ2iðc�rc� crc�Þ ¼ ħf 2rf: 6:

Recognizing that a mass current is given by the product of a density and a velocity, we can

combine these two equations to obtain the superfluid velocity vs ¼ js=rs. This yields

vs ¼ ðħf 2rfÞ=ðmf 2Þ ¼ ðħ=mÞrf, 7:

which is well defined where f 6¼ 0. Evidently the flow of the superfluid component is potential

flow (r � vs ¼ 0) in any simply connected region of the fluid because r � rf ¼ 0 for any

scalar f. A region in space is (informally) defined as simply connected if every closed contour in

the space can be contracted to a point without leaving the space. If a hole or other topological

object is present within the space, though, it ceases to be simply connected. Spaces with such

holes are called multiply connected.

Multiply connected regions in quantum fluids are caused by the cores of topological defects.

These line-like defects must either end on boundaries or form closed loops. Topological defects

in quantum fluids produce gradients in the phase f. Continuity of the order parameter cdefined in Equation 4 requires that the change in phase along any closed contour C be a

multiple of 2p, yielding

rCrf d‘ ¼ 2pn, 8:

where n is an integer. The standard definition for circulation around a classical vortex with

velocity field v is

G ¼ rCv d‘, 9:

for any contour C containing the vortex. Analogous arguments can be made for the magnetic

field in type-II superconductors, where the flux through a loop is quantized and the velocity

field is replaced by electrical current. Recognizing that we can replace rf in Equation 8 with

mvs=ħ using Equation 7, we can define the circulation around a topological defect in quantum

fluids as

G ¼ rCvs d‘ ¼ nk, 10:

where k h=m (9.97 � 10�4 cm2 s�1 in He II) is the quantum of circulation. Therefore, the

circulation around a topological defect in a quantum fluid is quantized in units of k. For a

rectilinear quantized vortex, we have

rCvs d‘ ¼ 2psvs ¼ nk, 11:

vs ¼ nk2ps

. 12:


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This velocity field is analogous to the flow field around a classical, ideal vortex with circula-

tion nk. Multiply quantized vortices (n 6¼ 1) are energetically unfavorable and are expected to

break up into multiple singly quantized (n ¼ 1) vortices. For this reason, it is often assumed that

a vortex state in a quantum fluid is composed of identical, singly quantized vortices.

The dynamics of complex fields and their respective topological defects has seen much use in

pattern-forming systems. The complex Ginzburg-Laudau equations are often used as long-

wavelength models for diverse systems including nonlinear waves and liquid crystals. These

equations have received a great deal of attention owing to their application as a model system

for pattern formation in biological and non-equilibrium systems. We refer the reader to the

review by Aranson & Kramer (63) for a detailed discussion.

4.1. Quantized Vortex Dynamics

Landau’s initial formulation of the two-fluid model (14) stated that the superfluid and normal

fluid components pass through one another without exchanging momentum. The mutual inter-

actions between the two fluids in this context are restricted to those required to satisfy the

typical conservation laws (e.g., mass, momentum, energy, entropy, etc.). Quantized vortices are

topologically constrained and do not diffuse; thus, they are locked to the superfluid, again

assuming there is no momentum exchanged with the normal fluid component. Determining

the dynamics of the quantized vortices, then, amounts to computing the velocity field induced

by the vortices themselves and the background superfluid velocity vs produced by other means

such as heat fluxes or boundary conditions.

Following the pioneering work of Schwarz (64–66), we consider a single quantized vortex in

an infinite fluid with a position parameterized by the curve s ¼ s(x,t). The velocity produced at

a position r by the vortex filament vo, in the context of line-vortex models, is given by a Biot-

Savart expression of the form

voðrÞ ¼ k4p

ZL

ðs1 � rÞ � ds1

js1 � rj3 , 13:

where the integral is taken along the filament and s1 is a point on the vortex. This integral

diverges as r approaches a point on the vortex filament. To control this divergence, theories

typically divide the velocity of the vortex filament itself, for example s, at the point s into two

components (65)

voðsÞ ¼ s ¼ vsðsÞ þ k4p

s0 � s00 ln2ðlþl�Þ1=2e1=4a0

!þ k4p

Z~Lðs1 � rÞ � ds1

js1 � rj3 , 14:

where primes denote differentiation with respect to the arc length x, vs is the background

superfluid velocity field, a0 is a cutoff parameter corresponding to the radius of the vortex

filament, lþ and l� are the lengths of the two adjacent line elements connected to the point s,

and ~L represents integration along the vortex line outside the region specified by lþ and l�.Schematic representations of lþ, l�, s0, s00 and s0 � s00 are given in figures 2 and 3 in Schwarz’s

initial formulation in Reference 65.

Equation 14 would approximately describe the dynamics of quantized vortices in the

absence of any exchange of momentum between the superfluid and normal fluid components.

However, in 1957 Vinen (20–22) discovered that momentum is exchanged between vs and vnthrough a mutual friction that acts on the vortices. This description yields the following velocity

for a point s on a vortex filament (65)


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s ¼ s0 þ as0 � ðvn � s0Þ � a0s0 � ½s0 � ðvn � s0Þ�, 15:

where a and a0 are temperature-dependent coefficients (67).

The dynamics of quantized vortices may also be described by the Gross-Pitaevskii equation

(18, 19),

iħ@c@t

¼ � ħ2

2mr2 þ VðrÞ þ g jc j2

� �c, 16:

where c is the order parameter field, V(r) is an external potential, and g is a coupling constant.

This description is most applicable in the limit of T ! 0 K, where the normal fluid is absent. In

this context, a quantized vortex core is defined by contours where c ¼ 0. The solution for a

straight vortex centered along the z-axis in cylindrical coordinates {s,f, z} is given by c ¼ R(s)eif,

which, upon substitution into Equation 16, yields

ħ2

2m

d2R

ds2þ 1

s

dR

ds� 1

s2R

� �� VðrÞR� gR3 ¼ 0: 17:

4.2. Theoretical and Computational Challenges

Performing numerical simulations of any turbulent system is a great challenge owing to the

extent of spatial and temporal scales of relevance, as discussed in Section 1. The two-fluid

nature of quantum fluids requires simulations to take into account the complex motions of the

normal fluid component, as in any simulation of a classic fluid, while also incorporating the

dynamics of the superfluid component and coupling between the two fluids. Unlike classical

fluid turbulence, there is no consensus on well-tested equations of motion for He II, resulting in

many simulations of distinct equations.

The study of turbulence in quantum fluids provides challenges in addition to those present

for studying classical fluid turbulence; thus, many numerical simulations are restricted to

turbulence in the superfluid component only. Numerical simulations (15, 64–66, 68) typically

begin with an initial distribution of quantized vortices within the simulation domain and a

prescribed normal fluid velocity field that affects the quantized vortices. These studies are

referred to as line-vortex simulations because the problem is reduced to following only the

dynamics of the quantized vortices, which are treated as one-dimensional.

Although this is a great reduction in complexity, several important difficulties remain. The

first is that the resulting superfluid velocity field from a given spatial distribution of quantized

vortices is given by a Biot-Savart-type integral (see Equation 13) in analogy to the magnetic

field derived from a distribution of electrical current, which has nonlocal effects. Fully com-

puting these integrals greatly increases the computational cost of the simulation. Many

numericists prefer to assume the localized-induction approximation (LIA) (15, 65, 66, 68),

which ignores the nonlocal contributions from Equation 14 and greatly reduces the computa-

tional workload. This assumption allows for simulations that contain a greater density of

quantized vortices at the cost of a restricted range of validity. Indeed, this approach has

recently been questioned in studies by Tsubota and his collaborators (69, 70) that compute

the full Biot-Savart integral.

The second shortcoming of line-vortex simulations stems from quantized vortex reconnec-

tion (10). Reconnection occurs when two quantized vortices are driven by their mutual velocity

field to cross at a point and exchange tails, which results in a different velocity field that drives

them apart, as discussed in detail in Section 6 below. In the naive implementation of vortex-line

simulations, reconnections cannot occur naturally. To compensate for this defect, numerical


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schemes implement special algorithms to induce reconnection. One method is to always

reconnect any two vortices that are separated by less than a prescribed distance (65, 66,

69, 70), whereas another only reconnects vortices that appear to cross (68). These distinct

methods of reconnecting vortices introduce systematic effects that are not yet clearly under-

stood. Furthermore, reconnection has a different impact on the turbulent state depending

upon whether the simulation assumes the LIA, as recently discussed by Adachi et al. (69).

A few numerical studies, which are discussed in Section 6 below, have investigated reconnection

in He II.

5. PINNING BY IMPURITIES

Impurities in any quantum fluid can become dynamically significant to the motions of the

quantized vortices. Owing to the inviscid nature of superfluids, impurities do not orbit about

the quantized vortices. Instead, as initially discussed by Parks & Donnelly (71), impurities

reduce the total kinetic energy of the system by displacing circulating superfluid. The reduction

in energy is maximized when the impurity is centered on the vortex core (where the kinetic

energy density is largest). As an impurity, such as an electron, ion, or solid particle, approaches

a vortex (or the vortex approaches the impurity), a gradient in energy causes an attractive force;

although, some dissipative mechanism is required to prevent radial oscillations. Drag between

the impurity and the normal fluid likely serves as the dissipative mechanism in the case of He II.

Impurity pinning in He II has been used in the development of several important experimen-

tal techniques. Reppy (72) used porous media to pin and immobilize the quantized vortices,

which allowed for the demonstration of pure superflow in the absence of a pressure gradient.

The trapping of ions or electrons by quantized vortices was experimentally confirmed by

Schwarz & Donnelly (73) in 1966. The same technique was successfully used to observe

vortices in a rotating container by Williams & Packard (74) in 1974 and Yarmchuk et al. (60)

in 1979.

These techniques have recently been extended to the use of micron-sized hydrogen parti-

cles to observe the dynamics of quantized vortices in the bulk of He II. Several previous

attempts to implement this technique failed given that the typical particle size was too large

for vortex trapping to occur. In 2006, Bewley et al. (75) developed a technique for producing

hydrogen particles with diameters on the order of 1 mm. This technique directly injects

a room-temperature mixture of approximately 2% H2 and 98% 4He into the bulk of the

fluid; the helium liquefies while the hydrogen solidifies, forming viable tracer particles that

may be imaged and tracked. Further developments of this technique have been made by

Bewley et al., Paoletti et al., and Xu et al. to observe vortex reconnection (76–78), a vortex

ring (79), thermal counterflows (80), He II forced flow (81), and quantum turbulence (78,

80). Guo et al. (82) recently implemented a novel technique to image metastable, excited

helium molecules using laser-induced fluorescence. This technique has the potential to serve

as a visualization method at lower temperatures (T <1.6 K), which is presently inaccessible

to standard optical cryostats.

The pinning of magnetic vortices is an important component of the theory of high-temperature

superconductors. M.P.A. Fisher (83) attributed the observation of superconductivity in highly

disordered type-II superconductors to the pinning of magnetic vortices on crystal impurities.

In the case of He II, quantized vortices remain mobile even when they trap impurities on

their cores, although with modified dynamics. Crystal impurities, however, are highly immobile

and can therefore greatly restrict the motion and dynamics of magnetic vortices in type-II

superconductors. Along these lines, M.P.A. Fisher (83) predicted the possibility of both a vortex


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glass state with immobilized magnetic vortices and a vortex liquid state in which the vortices

would be mobile in analogy with those in He II.

An opportunity in current theoretical work involves understanding how impurities might

pin vortices in Higgs models. These might need separate treatment for Weinberg-Salam models

(the standard model) (84) and for GUT Higgs theories involved in the diverse literature on

cosmic strings thought to be important to the early universe (85). By analogy, one might expect

that pinning may occur between these topological defects and matter.

6. RECONNECTION AND VORTEX RINGS

The topological nature of quantized vortices dominates the dynamics of quantum turbulence,

particularly on small scales. The pairwise interactions of quantized vortices are essential to

various dissipative mechanisms. Feynman initially envisioned (10) that the topologically

constrained vorticity could be removed from the system by producing vortex rings through

quantized vortex reconnection, which would eventually decay (see Figure 3). We outline recent

advancements in the understanding of reconnection and ring collapse dynamics in the forth-

coming sections.

6.1. Reconnection

Reconnection, shown in Figures 3 and 4, has been considered to play an important dissipative

role in quantum turbulence since it was first proposed by Feynman in 1955 (10). Vinen (20–22)

described how the balance of reconnection and mutual friction leads to saturated vortex-line

lengths in counterflow turbulence, which is analogous to the saturation of dynamo action (3) or

the magnetorotational instability (4) produced by magnetic reconnection in astrophysical

plasmas (5, 86–93). Reconnection has also been studied in liquid crystals (94), superconductors

(95–97), cosmic strings (98), viscous (99–101) and Euler vortices (102), and Bose-Einstein

condensates (103).

Reconnection is an essential mechanism for removing vortex-line length and dissipation in

He II as T ! 0 K. One means of removing vortex-line length via reconnection is shown in

Figure 3, which is taken from Feynman’s initial prediction in Reference 10. Feynman argued

that if reconnection could occur, then larger quantized vortex loops could form smaller ones,

a b c d

Figure 3

In 1955, Feynman (10) predicted that vortex rings (a) could reconnect (b to c here), which could lead to the

generation of smaller vortex rings (d). We now know that reconnection and vortex ring dynamics dominate

the small-scale behavior of quantum turbulence.


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and so on, until the loops became sufficiently small to decay from friction with the normal fluid

or interactions with the boundaries. Reconnection is also thought to directly emit sound

(104–107) and produce Kelvin waves on the vortex lines that are underdamped and interact

nonlinearly. The nonlinear coupling between waves will produce higher-frequency oscillations

until they become high enough to emit phonons into the surrounding fluid that are eventually

absorbed by the boundary (46, 48). It is important to stress, though, that dissipation and

Kelvin-wave cascades in He II for temperatures near absolute zero remains an active area of

research (15, 16, 48–59).

Quantized vortex reconnection in He II has been studied numerically and analytically by

employing vortex-line methods (64–66, 108–110) and by integrating the Gross-Pitaevskii equa-

tion (104, 105, 111, 112). Schwarz (65, 66) forced reconnection upon any two vortices that

were separated by less than some threshold distance in his line-vortex simulations of counter-

flow turbulence. He invoked reconnection, even though it had not yet been proven to occur in

He II, to attain an otherwise elusive statistical steady state for the vortex-line density L in his

pioneering simulations. Koplik & Levine (111) were the first to show that two quantized

vortices will reconnect using the Gross-Pitaevskii equation, even though there is no diffusion,

which was thought to play an essential role in reconnection.

To characterize the evolution of reconnecting vortices, de Waele & Aarts (108), followed by

Lipniacki (109) and Nazarenko&West (112), examined the dynamics of the minimum separation

Figure 4

After reconnection, quantized vortices rapidly retract while generating Kelvin waves, which may serve as an

important dissipative mechanism (redrawn from Reference 65).


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distance d(t) between the vortices. Assuming that the only relevant parameter in reconnection

dynamics is the quantum of circulation k, dimensional analysis yields the relation

dðtÞ ¼ Aðk j t � t0 jÞ1=2, 18:

where t0 is the moment of reconnection, and A is a dimensionless factor of the order unity. de

Waele & Aarts measured A � ffiffiffiffiffiffiffiffiffiffiffi1=2p

pprior to the reconnection of initially antiparallel vortices

in their numerical simulations. One may optimistically expect this scaling to be valid for length

scales between the vortex core size (�0.1 nm) and the typical intervortex spacing, which ranges

from �0.1 to 1 mm in typical experiments. Basically, however, Equation 18 should represent an

asymptotic expression, subject to corrections for longer times. Indeed, slight deviations were

observed in prior simulations implementing line-vortex models (65, 108).

Reconnection was directly visualized experimentally by Bewley et al. (76) in 2008 using the

hydrogen particle technique described above. These initial observations were supplemented by

the experimental characterization of approximately 20,000 reconnection events by Paoletti

et al. (78) in 2010. The major finding of both experiments was strong support for the asymp-

totic form given by Equation 18. Even though strong event-to-event fluctuations were observed,

the authors claimed that the quantum of circulation k is the dominant controlling quantity in

the dynamics of quantized vortex reconnection.

6.2. Vortex Ring Dynamics

A closed loop is the simplest topology for a quantized vortex. Quantized vortex rings were first

predicted by Feynman (10). In the prescient diagram shown in Figure 3, Feynman conjectured

that multiple reconnections could lead to the creation of vortex rings. This process, he sup-

posed, could provide an essential dissipative means of removing the otherwise topologically

constrained quantized vortices from the system. The first direct evidence for vortex rings was

provided by Rayfield & Rief (113), who in 1963 observed the motion of ions that were

presumably trapped on the vortex rings, as discussed in Section 5.

The generation of vortex rings has been studied theoretically since the work of Iordanski��

(114) in 1965. Subsequent research concerned with the theory of critical velocities in quantum

fluids was carried about by Langer et al. (115) and Donnelly (116). More recently, vortex rings

have been generated and used as a measurement tool in superfluid 3He-B (117–122).

Recent theoretical work has continued to examine quantized vortex ring dynamics and

interactions since earlier work by Schwarz (65) in 1985. Kiknadze & Mamaladze (123) found

that large amplitude nonlinear Kelvin waves lead to a slowing or reversal of the ring transla-

tional velocity. These studies assumed the LIA and were later confirmed by Barenghi et al.

(124), who computed the full Biot-Savart integral (see Figure 5). In 2007, Berloff & Youd

(125) found that the Gross-Pitaevskii equation could be used to examine the ring dissipative

dynamics in a background field of phonons. They noted that collisions between quantized

vortex rings and background excitations reduce the radius of the ring until it completely

disappears. Barenghi & Sergeev (126) defined criteria that quantify the effects of trapped

impurities on the motion of vortex rings.

Vortex rings of quantized fields have also been examined in the context of high-energy

theories. Huang & Tipton (127), Volkov (128), and Garaud & Volkov (129) have explored

the very intriguing possibility that the Weinberg-Salam theory (the standard model) (84) would

admit vortex solutions. Such a possible extension to the theory toward realizable vacuum

vortex defects is an interesting future research direction.


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7. QUANTUM TURBULENCE

A great deal of information regarding turbulent states in He II has been gleaned from

experiments, despite the previous inability to directly probe the local velocity field. Many

notable studies have observed similarities between quantum turbulence and classical turbulence

(25–43), as discussed in Section 3. These studies (a) observed either energy spectra in quantum

turbulence similar to the Kolmogorov prediction for classical fluid turbulence, given by Equa-

tion 1 (28, 29, 37, 39, 40); (b) observed the decay of root-mean-squared vorticity orms scaling

as orms � t�3/2 (26, 31, 33, 35, 36, 38, 41, 42); (c) made arguments for the normal fluid and

superfluid components to become locked and behave classically (25, 27, 43); or (d) reviewed a

combination of these phenomena (30, 32, 34).

The observations of classical behavior in quantum fluids pertains to length scales greater

than the typical intervortex spacing. However, the dynamics of quantum turbulence are distinct

from classical turbulence on length scales smaller than the typical intervortex spacing, which

can now be probed with the recently developed visualization techniques discussed in Section 5.

Paoletti et al. (77) claimed that the small-scale dynamics of turbulent states in He II greatly

differed from those observed in classical fluids (see the online movies in Reference 77).

0

–2

0 30

a b

c

y R–1

x R–1

Time (s)

z R–1

–1

–1

1

1

t = 40 s

t = 20 s

t = 0 s

5

1

z R–1

Figure 5

Vortex rings are capable of rich dynamical behavior. Large-amplitude Kelvin waves have been shownby Barenghi et al. (124) to modify or even reverse the translational velocity of the ring. (a) A comparison

of the translation along the z-direction of vortex rings of radius R with and without Kelvin waves.

(b) Examples of the in-plane deformations of vortex rings caused by Kelvin waves. (c) Measurements of

the propagation of vortex rings with varying Kelvin-wave amplitudes. Indeed, sufficiently strong Kelvinwaves are capable of reversing the direction of propagation.


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The authors attributed this discrepancy to reconnection events between quantized vortices

that produce anomalously large velocities in localized areas. The velocity field is not smoothed

by viscosity, thereby bearing little resemblance to the smooth, small-scale motions of a classical

fluid.

Kivotides et al. (130) used numerical simulations to predict that pressure spectra for turbu-

lent quantum fluids differ from those predicted by Kolmogorov’s arguments for classical fluid

turbulence (11, 12). Furthermore, White et al. (131) recently observed nonclassical velocity

statistics in their numerical simulations of turbulent Bose-Einstein condensates. Rather than

velocities obeying the near-Gaussian statistics observed in classical fluids (132–134), the distri-

butions were characterized by algebraic tails very similar to those observed experimentally by

Paoletti et al. (77, 78). We review the arguments for this distinction between quantum and

classical turbulence in the sections below, as well as discuss possible analogies between quan-

tum turbulence and MHD turbulence in astrophysical plasmas.

7.1. Velocity Statistics

Theoretical models and experimental observations discussed in Section 6 give significant

evidence that the quantum of circulation k is the relevant dimensional quantity for the dynam-

ics of quantized vortices. Assuming such, we expect length scales to evolve locally as

RðtÞ ¼ ~A j kðt � t0Þj1=2, as measured for reconnecting vortices. Thus, velocities scale as

vðtÞ ¼~A

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik

j t � t0 jr

, 19:

which far exceed typical fluid velocities when t ! t0. Note, however, that such velocities must

be cut-off, at least, by the speed of first sound.

By assuming that the high-velocity events in a turbulent quantum fluid are produced by the

dynamics of quantized vortices, we can model the tails of the probability distribution function

(pdf) of velocities by using the transformation

PrvðvÞ ¼ Prt½tðvÞ�j dt=dv j, 20:

where Prv(v)dv is the probability of observing a velocity between v and v þ dv at any time, and

Prt(t)dt is the uniform probability of taking a measurement at a time between t and t þ dt.

Hence, accepting the scaling relation (Equation 19), one expects for large v (small t) the behavior

PrvðvÞ /�� dtdv

�� /�� kv3

��: 21:

In fact, as pointed out by Min et al. (135), the velocity distribution around a straight,

singular vortex has the same power-law tails. The velocity field around such a vortex located

along the z-axis is given by

v ¼ G2ps

f, 22:

where G is the circulation and s is the cylindrical radius. To model the pdf of this velocity field,

we may use the following:

PrvðvÞ ¼ Prs½sðvÞ�j ds=dv j, 23:

where Prv(v)dv is again the probability of observing a velocity between v and v þ dv at any

radius, and Prs(s)ds is the probability of taking a measurement at a radius between s and s þ ds.


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The probability of taking a measurement at a radius between s and s þ ds is proportional to s,

or equivalently v�1. Assuming the velocity field given by Equation 22, the ratio jds=dvj isproportional to v�2. Thus we again arrive at

PrvðvÞ /�� Gv3

��, 24:

where G ¼ k in the case of a quantum fluid.

To test these predictions, Paoletti et al. (77, 78) measured the velocity statistics for decaying

turbulent He II. Example velocity distributions from one of their experiments are shown in

Figure 6. The distributions show close agreement with Equation 21 but strongly differ from the

near-Gaussian distributions typical of classical turbulence (132–134). The authors attributed

this distinction to the topological nature and interactions of the quantized vortices, as discussed

above.

Numerical simulations byWhite, Barenghi, and Proukakis (131) observed very similar velocity

distributions in simulations of turbulent Bose-Einstein condensates. The authors did not attribute

the algebraic tails to reconnection events and other topological interactions between quantized

vortices. Rather, they appealed to the singular nature and the 1=s velocity field of quantized

vortices, as initially described by Min et al. (135) to explain their observations.

The interpretations given by Paoletti et al. (71) and White, Barenghi, and Proukakis (131)

are not at odds with one another. The hydrogen tracers used experimentally in (77) do not

respond directly to the local superfluid velocity, unless they are trapped by a quantized vortex.

Therefore, to observe high velocities that only occur near the cores of quantized vortices, a

tracer must be trapped on a quantized vortex that is in close proximity to another vortex. In

many cases, these two quantized vortices would reconnect, leading to the interpretation given in

Reference 77. In the numerical studies in Reference 131, however, the velocity field was directly

Pr v(v

i)σv i

vi /σv

10–1

10–3

10–5–10 –5 0 5 10

vx vz

i

Figure 6

Quantum turbulence exhibits unusual velocity statistics, as shown by the decaying counterflow turbulence

experiments by Paoletti et al. (77). The probability distribution functions (pdfs) of vx and vz are shown byblack circles and red squares, respectively. The solid lines are fits to PrvðviÞ ¼ a j vi ��vi j�3, where i is eitherx (black) or z (red), and �vi is the mean value of vi. For comparison, the blue dashed line shows the

distribution for classical turbulence in water (136).


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computed, and the arguments given by Min et al. (135) and summarized above directly apply.

Independent of the measurement technique or details of the interpretation, however, all of these

studies attribute the nonclassical velocity pdfs to the singular, topological nature of the quan-

tized vortices.

7.2. Analogies with Magnetohydrodynamics

Previous studies have argued that the interactions of magnetic field lines can cause the velocity

statistics of MHD turbulence also to differ from classical turbulence, particularly in astrophys-

ical plasmas with extremely small resistivities. In such environments, magnetic field lines

become frozen into the underlying velocity field. If two field lines are driven sufficiently close

together, they may also reconnect by diffusive processes. The power-law tails in the distributions

of electron energies observed in astrophysical settings (figure 3 in Reference 89 and figure 2 in

Reference 5) have been attributed to magnetic reconnection by Drake and coworkers (92, 93).

Furthermore, theories for MHD turbulence developed by del-Castillo-Negrete et al. (137) pro-

pose that fractional diffusion may be the dominant transport mechanism. Such diffusion is

associated with power-law tails in velocity distribution functions.

By the same arguments made in Section 7.1, Paoletti et al. (77) argued that the tails of

the pdf for the kinetic energy per unit mass E ¼ ðv2x þ v2z Þ=2 in quantum fluids will be domi-

nated by motions of the quantized vortices. Accepting the relation (19), the energy evolves as

EðtÞ /j t � t0 j�1; thus, for large E they predicted

PrEðEÞ /�� dtdE

�� /�� kE2

��: 25:

The pdf of E, computed from the data in Figure 6 (which includes all particle trajectories and is

taken from Reference 77), is shown in Figure 7a. A single-parameter fit of the form PrE(E) ¼aE�2 is shown as a solid, red line for comparison. The departure from the predicted power-law

behavior for low energies may reasonably be attributed to effects from the boundaries and

nearby vortices, as well as to the background drift of the normal fluid. We compare this

a b

Pr E

(E)σ

E 10–2

100

10–4

10–29

10–27

10–25

10–23

10–21

10–19

10–17

10–15

f (e

lect

ron

s) (

s3 m

–3)

E/σE

10–1 100 101 101 102 103 104 105

Energy (eV)

07:26 k = 5.707:45 k = 5.407:52 k = 4.907:59 k = 4.8

Figure 7

Distributions of the kinetic energy for (a) decaying counterflow turbulence in He II from Paoletti et al. (77),

compared to (b) those of electrons (presumably) accelerated by magnetic reconnection in the diffusion

region of Earth’s magnetotail observed by �ieroset et al. (89). The solid lines are fits to (a) PrE ¼ aE�2 and(b) f ¼ E�k, with k denoted in the legend.


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distribution to the electron energy distribution observed by �ieroset et al. (89), which is shown

in Figure 7b.

It is clear that the two distributions in Figure 7 bear resemblance; however, several important

distinctions must be made. The distribution computed from the data in He II is the kinetic

energy per unit mass of the He II, with the tails dominated by the motions of the quantized

vortices themselves. In the MHD case taken from Reference 89, the distributions are computed

from electrons that are accelerated by the magnetic field lines, rather than from the magnetic

field lines themselves. The mechanism by which electrons are accelerated in regions near

magnetic reconnection is a process unto itself and is still actively studied (92, 93). Furthermore,

reconnection in astrophysical plasmas is a much more complicated process. In He II reconnection

occurs between identical, topologically constrained quantized vortices, whereas magnetic field

lines do not share the same constraints. Nevertheless, taking the reductionist approach, studying

reconnection-dominated quantum turbulence could aid in the study of MHD turbulence in low

resistivity environments.

8. CONCLUSIONS

Systems exhibiting long-range quantum order may exhibit turbulent behavior when driven

from equilibrium. The interaction of topological defects (quantized vortices) underlies quan-

tum turbulence. As one would expect from the correspondence principle, quantum turbulence

at large scales may exhibit course-grained behavior similar to classical turbulence in a New-

tonian fluid. At scales smaller than the typical intervortex spacing, the behavior is dominated

by vortex reconnection, the generation of Kelvin waves, and the formation of quantized

vortex rings.

Beyond the significant advances we have touched on here, there remains room for consid-

erable progress. Conducting experimental tests of individual terms in different proposed

equations of motion, as has been done for the Navier-Stokes equation, is an area of consid-

erable future promise and importance. Ongoing research will continue to work to understand

the causal connection between vortices, Kelvin waves, and phonons. Finally, understanding

the local dynamics and the interaction between topological defects in systems beyond 4He,

such as magnetic vortices, defects in 3He, or in other Higgs fields, may also warrant future

attention.

SUMMARY POINTS

1. Quantum turbulence is dominated by reconnection and vortex ring dynamics.

2. At large length scales, quantum turbulence shares some qualities with Newtonian

fluid turbulence.

3. Compared to classical turbulence, quantum turbulence has some unique properties in

the velocity distributions and time dynamics.

4. The pinning of defects by impurities or boundary roughness is important in the study

of quantum turbulence.

DISCLOSURE STATEMENT

The authors are not aware of any affiliations, memberships, funding, or financial holdings that

might be perceived as affecting the objectivity of this review.


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ACKNOWLEDGMENTS

We would like to acknowledge fruitful discussions with M.E. Fisher, K.R. Sreenivasan,

M. Tsubota, C. Barenghi, W.F. Vinen, G.P. Bewley, J.F. Drake, J.F. Pinton, N. Goldenfeld,

T. Antonsen, A. Pumir, E. Ott, and E. Fonda.

LITERATURE CITED

1. Merrill RT, McElhinny MW, McFadden PL. 1997. The Magnetic Field of the Earth: Paleomagnetism,

the Core, and the Deep Mantle. San Diego, CA: Academic

2. Winfree AT. 1994. Science 266:1003–6

3. Vainshtein SI, Cattaneo F. 1992. Astrophys. J. 393:165–71

4. Balbus SA, Hawley JF. 1998. Rev. Mod. Phys. 70(1):1–53

5. Holman GD, Sui L, Schwartz RA, Emslie AG. 2003. Astrophys. J. 595:L97–101

6. Goldman MV. 1984. Rev. Mod. Phys. 56:709–35

7. Williams QL, Garcıa-Ojalvo J, Roy R. 1997. Phys. Rev. A 55:2376–86

8. Callan KE, Illing L, Gao Z, Gauthier DJ, Scholl E. 2010. Phys. Rev. Lett. 104(11):113901

9. Giralt F, Arenas A, Ferre-Gine J, Rallo R, Kopp GA. 2000. Phys. Fluids 12(7):1826–35

10. Feynman RP. 1955. Progress in Low Temperature Physics, ed. CJ Gorter, 1:17–53. Amsterdam:

North-Holland

11. Kolmogorov A. 1941.Dokl. Akad. Nauk SSSR 30:301–5; 1991. Proc. R. Soc. Lond. A 434:9–13

12. Kolmogorov A. 1941.Dokl. Akad. Nauk SSSR 31:538–40; 1991. Proc. R. Soc. Lond. A 434:15–17

13. Tisza L. 1938.Nature 141:913

14. Landau L. 1941. Phys. Rev. 60:356–58

15. Tsubota M, Araki T, Nemirovskii SK. 2000. Phys. Rev. B 62:11751–62

16. Vinen WF, Tsubota M, Mitani A. 2003. Phys. Rev. Lett. 91(13):135301

17. Landau LD, Lifshitz EM. 1987. Fluid Mechanics. Oxford, UK: Pergamon. 2nd ed.

18. Pitaevskii LP. 1961. Sov. Phys. JETP 13:451–54

19. Gross EP. 1963. J. Math. Phys. 4:195–207

20. Vinen WF. 1957. Proc. R. Soc. A 240:114–27



23. Berloff NG, Youd AJ. 2007. Phys. Rev. Lett. 99(14):145301

24. Kobayashi M, Tsubota M. 2006. Phys. Rev. Lett. 97(14):145301

25. Samuels DC. 1992. Phys. Rev. B 46:11714–24

26. Smith MR, Donnelly RJ, Goldenfeld N, Vinen WF. 1993. Phys. Rev. Lett. 71:2583–86

27. Barenghi CF, Samuels DC, Bauer GH, Donnelly RJ. 1997. Phys. Fluids 9:2631–43

28. Nore C, Abid M, Brachet ME. 1997. Phys. Rev. Lett. 78:3896–99

29. Maurer J, Tabeling P. 1998. Europhys. Lett. 43:29–34

30. Barenghi CF. 1999. J. Phys. Condens. Matter 11:7751–59

31. Stalp SR, Skrbek L, Donnelly RJ. 1999. Phys. Rev. Lett. 82:4831–34

32. Vinen WF. 2000. Phys. Rev. B 61:1410–20

33. Skrbek L, Niemela JJ, Donnelly RJ. 2000. Phys. Rev. Lett. 85:2973–76

34. Vinen WF, Niemela JJ. 2002. J. Low Temp. Phys. 128(5–6):167–231

35. Barenghi CF, Hulton S, Samuels DC. 2002. Phys. Rev. Lett. 89(27):275301

36. Skrbek L, Gordeev AV, Soukup F. 2003. Phys. Rev. E 67(4):047302

37. Kobayashi M, Tsubota M. 2005. Phys. Rev. Lett. 94(6):065302

38. Kobayashi M, Tsubota M. 2006. J. Low Temp. Phys. 145:209–18

39. Kobayashi M, Tsubota M. 2007. Phys. Rev. A 76(4):045603

40. L’vov VS, Nazarenko SV, Rudenko O. 2007. Phys. Rev. B 76(2):024520

41. Chagovets TV, Gordeev AV, Skrbek L. 2007. Phys. Rev. E 76(2):027301

42. Walmsley PM, Golov AI, Hall HE, Levchenko AA, Vinen WF. 2007. Phys. Rev. Lett. 99(26):265302

10. Predicted

reconnection and ring

formation.


Ann

u. R

ev. C

onde

ns. M

atte

r Ph

ys. 2

011.

2:21

3-23

4. D

ownl

oade

d fr

om w

ww

.ann

ualr

evie

ws.

org

by U

nive

rsity

of

Tex

as -

Aus

tin o

n 10

/02/

11. F

or p

erso

nal u

se o

nly.

43. Morris K, Koplik J, Rouson DWI. 2008. Phys. Rev. Lett. 101(1):015301

44. Bohr N. 1920. Z. Phys. 2:423–69

45. Richardson LF. 1926. Proc. R. Soc. Lond. A 110:709–37

46. Vinen WF. 2001. Phys. Rev. B 64(13):134520

47. Alamri SZ, Youd AJ, Barenghi CF. 2008. Phys. Rev. Lett. 101(21):215302

48. Kivotides D, Vassilicos JC, Samuels DC, Barenghi CF. 2001. Phys. Rev. Lett. 86:3080–83

49. Kozik E, Svistunov B. 2004. Phys. Rev. Lett. 92(3):035301


51. Nazarenko S. 2007. Sov. JETP Lett. 84:585–87

52. Kozik E, Svistunov B. 2008. Phys. Rev. B 77(6):060502


54. Tsubota M. 2008. J. Phys. Soc. Jpn. 77(11):111006

55. Tsubota M. 2009. J. Phys. Condens. Matter 21(16):164207

56. Kozik EV, Svistunov BV. 2009. J. Low Temp. Phys. 156:215–67

57. Boffetta G, Celani A, Dezzani D, Laurie J, Nazarenko S. 2009. J. Low Temp. Phys. 156:193–214

58. Yepez J, Vahala G, Vahala L, Soe M. 2009. Phys. Rev. Lett. 103(8):084501

59. Laurie J, L’vov VS, Nazarenko S, Rudenko O. 2010. Phys. Rev. B 81(10):104526

60. Yarmchuk EJ, Gordon MJV, Packard RE. 1979. Phys. Rev. Lett. 43:214–17

61. Hess HF, Robinson RB, Dynes RC, Valles JM Jr, Waszczak JV. 1989. Phys. Rev. Lett. 62:214–16

62. Abo-Shaeer JR, Raman C, Vogels JM, Ketterle W. 2001. Science 292:476–79

63. Aranson IS, Kramer L. 2002. Rev. Mod. Phys. 74:99–143

64. Schwarz KW. 1978. Phys. Rev. B 18:245–62



67. Donnelly RJ, Barenghi CF. 1998. J. Phys. Chem. Ref. Data 27:1217–74

68. Kondaurova LP, Andryuschenko VA, Nemirovskii SK. 2008. J. Low Temp. Phys. 150:415–19

69. Adachi H, Fujiyama S, Tsubota M. 2010. Phys. Rev. B 81:104511

70. Adachi H, Tsubota M. 2010. J. Low Temp. Phys. 158:422–27

71. Parks PE, Donnelly RJ. 1966. Phys. Rev. Lett. 16:45–48

72. Reppy JD. 1965. Phys. Rev. Lett. 14:733–35

73. Schwarz KW, Donnelly RJ. 1966. Phys. Rev. Lett. 17:1088–90

74. Williams GA, Packard RE. 1974. Phys. Rev. Lett. 33:280–83

75. Bewley GP, Lathrop DP, Sreenivasan KR. 2006. Nature 441:588

76. Bewley GP, Paoletti MS, Sreenivasan KR, Lathrop DP. 2008. Proc. Natl. Acad. Sci. USA 105:13707

77. Paoletti MS, Fisher ME, Sreenivasan KR, Lathrop DP. 2008. Phys. Rev. Lett. 101:154501

78. Paoletti MS, Fisher ME, Lathrop DP. 2010. Physica D 239:1367–77

79. Bewley GP, Sreenivasan KR. 2009. J. Low Temp. Phys. 156:84–94

80. Paoletti MS, Fiorito RB, Sreenivasan KR, Lathrop DP. 2008. J. Phys. Soc. Jpn. 77(11):111007

81. Xu T, van Sciver SW. 2007. Phys. Fluids 19(7):071703

82. Guo W, Wright JD, Cahn SB, Nikkel JA, McKinsey DN. 2009. Phys. Rev. Lett. 102(23):235301

83. Fisher MPA. 1989. Phys. Rev. Lett. 62:1415–18

84. Weinberg S. 1995. The Quantum Theory of Fields. Cambridge, UK: Cambridge Univ. Press

85. Copeland EJ, Myers RC, Polchinski J. 2004. J. High Energy Phys. 6:13

86. Lin RP, Hudson HS. 1971. Sol. Phys. 17:412–35

87. Terasawa T, Nishida A. 1976. Planet. Space Sci. 24:855–66

88. Baker DN, Stone EC. 1976. Geophys. Res. Lett. 3:557–60

89. �ieroset M, Lin RP, Phan TD, Larson DE, Bale SD. 2002. Phys. Rev. Lett. 89(19):195001

88. Dmitruk P, Matthaeus WH, Seenu N, Brown MR. 2003. Astrophys. J. 597:L81–84

91. Lin RP, Krucker S, Hurford GJ, Smith DM, Hudson HS, et al. 2003. Astrophys. J. 595:L69–76

92. Drake JF, Shay MA, Thongthai W, Swisdak M. 2005. Phys. Rev. Lett. 94(9):095001

93. Drake JF, Swisdak M, Che H, Shay MA. 2006. Nature 443:553–56

94. Chuang I, Yurke B, Durrer R, Turok N. 1991. Science 251:1336–42

60. Berkeley group

imaged the array of

quantized vortices in

small groups up to

11 vortices.

65. Developed the

line-vortex method in

important ways that

include reconnection.

67. Details the

properties of helium.

78. Details the

reconnection

dynamics of quantized

vortices.


Ann

u. R

ev. C

onde

ns. M

atte

r Ph

ys. 2

011.

2:21

3-23

4. D

ownl

oade

d fr

om w

ww

.ann

ualr

evie

ws.

org

by U

nive

rsity

of

Tex

as -

Aus

tin o

n 10

/02/

11. F

or p

erso

nal u

se o

nly.

95. Brandt EH. 1991. Int. J. Mod. Phys. B 5:751–95

96. Blatter G. 1994. Rev. Mod. Phys. 66:1125–388

97. Bou-Diab M, Dodgson MJ, Blatter G. 2001. Phys. Rev. Lett. 86:5132–35

98. Hindmarsh MB, Kibble TWB. 1995. Rep. Prog. Phys. 58:477–562

99. Fohl T, Turner JS. 1975. Phys. Fluids 18(4):433–36

100. Ashurst WT, Meiron DI. 1987. Phys. Rev. Lett. 58:1632–35

101. Kerr RM, Hussain F. 1989. Physica D 37:474–84

102. Siggia ED. 1985. Phys. Fluids 28:794–805

103. Caradoc-Davies BM, Ballagh RJ, Blakie PB. 2000. Phys. Rev. A 62(1):011602

104. Leadbeater M, Winiecki T, Samuels DC, Barenghi CF, Adams CS. 2001. Phys. Rev. Lett. 86:1410–13

105. Ogawa S-i, Tsubota M, Hattori Y. 2002. J. Phys. Soc. Jpn. 71:813–21

106. Barenghi CF, Samuels DC, Kivotides D. 2002. J. Low Temp. Phys. 126:271–79

107. Kuz’min PA. 2006. JETP Lett. 84:204–8

108. de Waele ATAM, Aarts RGKM. 1994. Phys. Rev. Lett. 72:482–85

109. Lipniacki T. 2000. Eur. J. Mech. B 19:361–78

110. Kivotides D, Barenghi CF, Samuels DC. 2001. Europhys. Lett. 54:774–78

111. Koplik J, Levine H. 1993. Phys. Rev. Lett. 71:1375–78

112. Nazarenko S, West RJ. 2003. J. Low Temp. Phys. 132:1–10

113. Rayfield GW, Reif F. 1963. Phys. Rev. Lett. 11:305–8

114. Iordanski�� SV. 1965. Sov. JETP 21:467

115. Langer JS, Fisher ME. 1967. Phys. Rev. Lett. 19:560–63

116. Donnelly RJ. 1971. R. Soc. Lond. Philos. Trans. A 271:41–100

117. Bradley DI. 2000. Phys. Rev. Lett. 84:1252–55

118. Bradley DI, Clubb DO, Fisher SN, Guenault AM, Haley RP, et al. 2005. Phys. Rev. Lett. 95(3):

035302

119. Bradley DI, Clubb DO, Fisher SN, Guenault AM, Haley RP, et al. 2006. Phys. Rev. Lett. 96(3):035301

120. Hanninen R, Tsubota M, Vinen WF. 2007. Phys. Rev. B 75(6):064502

121. Bradley DI, Fisher SN, Guenault AM, Haley RP, Matthews CJ, et al. 2007. J. Low Temp. Phys. 148:

235–43

122. Bradley DI, Fisher SN, Guenault AM, Haley RP, Holmes M, et al. 2008. J. Low Temp. Phys. 150:364–72

123. Kiknadze L, Mamaladze Yu. 2002. J. Low Temp. Phys. 124:321

124. Barenghi CF, Hanninen R, Tsubota M. 2006. Phys. Rev. E 74(4):046303

125. Berloff NG, Youd AJ. 2007. Phys. Rev. Lett. 99(14):145301

126. Barenghi CF, Sergeev YA. 2009. Phys. Rev. B 80(2):024514

127. Huang K, Tipton R. 1981. Phys. Rev. D 23:3050–57

128. Volkov MS. 2007. Phys. Lett. B 644:203–7

129. Garaud J, Volkov MS. 2010.Nucl. Phys. B 826:174–216

130. Kivotides D, Vassilicos JC, Barenghi CF, Khan MAI, Samuels DC. 2001. Phys. Rev. Lett. 87(27):

275302

131. White AC, Barenghi CF, Proukakis NP, Youd AJ, Wacks DH. 2010. Phys. Rev. Lett. 104(7):075301

132. Noullez A, Wallace G, Lempert W, Miles RB, Frisch U. 1997. J. Fluid Mech. 339:287–307

133. Vincent A, Meneguzzi M. 1991. J. Fluid Mech. 225:1–20

134. Gotoh T, Fukayama D, Nakano T. 2002. Phys. Fluids 14:1065–81

135. Min IA, Mezic I, Leonard A. 1996. Phys. Fluids 8:1169–80

136. Zeff BW, Lanterman DD, McAllister R, Roy R, Kostelich EJ, Lathrop DP. 2003. Nature 421:146–49

137. del-Castillo-Negrete D, Carreras BA, Lynch VE. 2004. Phys. Plasmas 11:3854–64

111. Realized

reconnection of

quantized vortices via

the Gross-Pitaevskii

equation.


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Annual Review of

Condensed Matter

Physics

Contents

Reflections on My Career in Condensed Matter Physics

Mildred S. Dresselhaus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

The Ubiquity of Superconductivity

Anthony J. Leggett . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

The Quantum Spin Hall Effect

Joseph Maciejko, Taylor L. Hughes, and Shou-Cheng Zhang . . . . . . . . . . . 31

Three-Dimensional Topological Insulators

M. Zahid Hasan and Joel E. Moore . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Unconventional Quantum Criticality in Heavy-Fermion Compounds

O. Stockert and F. Steglich . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

Electronic Transport in Graphene Heterostructures

Andrea F. Young and Philip Kim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

Materials and Novel Superconductivity in Iron Pnictide Superconductors

Hai-Hu Wen and Shiliang Li . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

Interface Physics in Complex Oxide Heterostructures

Pavlo Zubko, Stefano Gariglio, Marc Gabay, Philippe Ghosez,

and Jean-Marc Triscone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

Mott Physics in Organic Conductors with Triangular Lattices

Kazushi Kanoda and Reizo Kato . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

Hybrid Solid-State Qubits: The Powerful Role of Electron Spins

John J.L. Morton and Brendon W. Lovett . . . . . . . . . . . . . . . . . . . . . . . . 189

Quantum Turbulence

Matthew S. Paoletti and Daniel P. Lathrop . . . . . . . . . . . . . . . . . . . . . . . 213

Electron Glass Dynamics

Ariel Amir, Yuval Oreg, and Yoseph Imry . . . . . . . . . . . . . . . . . . . . . . . . 235

Volume 2, 2011

vi

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Characterizing Structure Through Shape Matching and Applications

to Self-Assembly

Aaron S. Keys, Christopher R. Iacovella, and Sharon C. Glotzer . . . . . . . 263

Controlling the Functionality of Materials for Sustainable Energy

George Crabtree and John Sarrao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

Energy Conversion in Photosynthesis: A Paradigm for Solar Fuel

Production

Gary F. Moore and Gary W. Brudvig . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

Equalities and Inequalities: Irreversibility and the Second Law of

Thermodynamics at the Nanoscale

Christopher Jarzynski. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

Deformation and Failure of Amorphous, Solidlike Materials

Michael L. Falk and J.S. Langer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

Life is Physics: Evolution as a Collective Phenomenon Far from

Equilibrium

Nigel Goldenfeld and Carl Woese . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375

Errata

An online log of corrections to Annual Review of Condensed Matter Physics

articles may be found at http://conmatphys.annualreviews.org/errata.shtml

Contents vii

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