Table Of ContentDYNAMICS OF QUANTISED VORTICES
IN SUPERFLUIDS
A comprehensive overview of the basic principles of vortex dynamics in superfluids, this
bookaddressestheproblemsofvortexdynamicsinallthreesuperfluidsavailableinlabo-
ratories – 4He, 3He and Bose–Einstein condensate of cold atoms – alongside discussions
of the vortex elasticity, forces on vortices and vortex mass. Beginning with a summary
of classical hydrodynamics, the book guides the reader through examinations of vortex
dynamicsfromlargescalestothemicroscopicscale.Topicssuchasvortexarraysinrotating
superfluids, bound states in vortex cores and interaction of vortices with quasiparticles
are discussed. The final chapter of the book considers implications of vortex dynamics
for superfluid turbulence using simple scaling and symmetry arguments. Written from a
unifiedpointofviewthatavoidsacomplicatedmathematicalapproach,thistextisidealfor
students and researchers working with vortex dynamics in superfluids, superconductors,
magneticallyorderedmaterials,neutronstarsandcosmologicalmodels.
edouard b. sonin is an Emeritus Professor at the Hebrew University of Jerusalem,
Israel.Hisresearchinterestscentreonvortexdynamicsinsuperfluidsandsuperconductors
andmesoscopicphysics.
DYNAMICS OF QUANTISED
VORTICES IN SUPERFLUIDS
EDOUARD B. SONIN
HebrewUniversityofJerusalem
UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom
CambridgeUniversityPressispartoftheUniversityofCambridge.
ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof
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Informationonthistitle:www.cambridge.org/9781107006683
©EdouardB.Sonin2016
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Firstpublished2016
PrintedintheUnitedKingdombyTJInternationalLtd.PadstowCornwall
AcatalogrecordforthispublicationisavailablefromtheBritishLibrary
LibraryofCongressCataloguinginPublicationdata
Sonin,EdouardB.,author.
Dynamicsofquantisedvorticesinsuperfluids/EdouardB.Sonin,
HebrewUniversityofJerusalem.
pages cm
Includesbibliographicalreferencesandindex.
ISBN978-1-107-00668-3(Hardback:alk.paper)
1. Superfluidity. 2. Fluiddynamics. 3. Vortex-motion. I. Title.
QC175.4.S662016
(cid:2)
530.42–dc23 2015024943
ISBN978-1-107-00668-3Hardback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracy
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Contents
Preface pagexi
1 Hydrodynamicsofaone-componentclassicalfluid 1
1.1 Thermodynamicsofaone-componentperfectfluid 1
1.2 Hydrodynamicsofaone-componentperfectfluid 4
1.3 Motionofacylinderinanincompressibleperfectfluid:backflow 7
1.4 Motionofacylinderinanincompressibleperfectfluid:Magnusforce 10
1.5 Cylinderwithfluidcirculationarounditmovinginacompressiblefluid 12
1.6 Hydrodynamicalmodesofaperfectfluid 14
1.7 Hydrodynamicsofaviscousfluid 17
1.8 Motionofacylinderinaviscousfluid:StokesandOseenproblems 19
1.9 Longitudinalandtransverselocalforcesonthefluid 24
1.10 Hydrodynamicsofarotatingperfectfluid 28
1.11 Hydrodynamicalmodesofarotatingincompressiblefluid 29
1.12 Inertialwaveresonancesinaninviscidfluid 32
1.13 Inertialwaveresonancesinaviscousfluid 34
1.14 Hydrodynamicalmodesofarotatingcompressibleperfectfluid 36
1.15 Gross–Pitaevskiitheory 38
2 Dynamicsofasinglevortexline 43
2.1 Vortexlineinaperfectfluid 43
2.2 Linearandangularmomentaofavortexline 48
2.3 Motionofavortex:Magnusforce 51
2.4 Experimental detection of quantum circulation: vortex mass versus
Magnusforce 52
2.5 VortexmassinBosesuperfluids 55
2.6 Precessionofastraightvortexaroundanextremumofvortexenergy 56
2.7 Dynamicsofacurvedvortexline:Biot–Savartlawandlocalinduction
approximation 57
2.8 Vortexring 60
2.9 Kelvinwavesonanisolatedvortexline 63
v
vi Contents
2.10 Helicalvortex 66
2.11 Helicalvortexring 71
2.12 Precessionofasinglecurvedvortex 77
3 Vortexarrayinarotatingsuperfluid:elasticityandmacroscopichydrodynamics 82
3.1 Macroscopichydrodynamicsofrotatingsuperfluids 82
3.2 Symmetriesofperiodicvortextextures 90
3.3 Elasticmoduliandlinearequationsofmotion 92
3.4 Hall–Vinen–Bekarevich–Khalatnikovhydrodynamics 94
3.5 Tkachenkoshearrigidity 98
3.6 Spectrumofoscillationsinanincompressiblefluid 101
3.7 Axialmodesofvortexoscillations 102
3.8 Tkachenkowaves:elasticitytheoryofatwo-dimensionalvortexlattice 104
3.9 Slowmodeinanincompressibleperfectfluid 106
3.10 Glaberson–Johnson–Ostermeierinstability 107
3.11 Vortexoscillationsinacompressibleperfectfluid 108
3.12 Rapidly rotating Bose–Einstein condensate in the lowest Landau
levelstate 110
4 Oscillationoffinitevortexarrays:two-dimensionalboundaryproblems 115
4.1 Equilibriumfinitevortexarray 115
4.2 Distortionsofvortexlatticeproducedbyaboundary 117
4.3 AxisymmetricTkachenkomodesinafinitevortexbundle:comparison
ofcontinuumtheoryandnumericalexperiments 120
4.4 Chiraledgewaves 122
4.5 Groundstateofatwo-dimensionalBose–Einsteincondensatecloud 124
4.6 Ground state of a rotating two-dimensional Bose–Einstein
condensatecloud 127
4.7 TkachenkowavesinaBose–Einsteincondensatecloud 128
4.8 Observation of Tkachenko waves in a rotating Bose–Einstein
condensatecloud 132
5 Vortexoscillationsinfiniterotatingcontainers:three-dimensional
boundaryproblems 134
5.1 Torsionaloscillator(Andronikashvili)experiment 134
5.2 Boundaryconditionsonahorizontalsolidsurface:surfacepinning 135
5.3 Collectivesurfacepinning 139
5.4 Pile-of-disksoscillations:Hallresonanceversusinertialwaveresonance 142
5.5 Effectiveboundaryconditionforslowmotioninahorizontallayerof
rotatingfluid 146
5.6 Uniformlytwistedvortexbundle 151
5.7 Torsionaloscillationsofavortexbundle 155
5.8 Slowoscillationsofasuperfluidinafinitecylindricalcontainer 160
Contents vii
5.9 SearchforTkachenkowavesinsuperfluid4Heandpulsars:Tkachenko
waveversusinertialwave 163
6 Vortexdynamicsintwo-fluidhydrodynamics 167
6.1 Two-fluidmacroscopichydrodynamicsofarotatingsuperfluid 167
6.2 Longitudinalmodes:firstandsecondsound 175
6.3 Hydrodynamicalequationsforacompletelyincompressiblesuperfluid 177
6.4 Axialmodes 178
6.5 In-planemodes 180
6.6 Slowmodesinacompletelyincompressiblesuperfluid 182
6.7 Vortexdynamicsintheclampedregime 184
6.8 Oscillationsofanincompressiblefluidintheclampedregime 186
6.9 Phenomenologicaltheoryclosetothecriticaltemperature 188
7 Boundaryproblemsintwo-fluidhydrodynamics 194
7.1 Boundaryconditionsonahorizontalsolidsurface 194
7.2 Pile-of-disksoscillationsandeffectiveboundarycondition 194
7.3 Oscillationsintheclampedregime:dampedslowmode 197
7.4 Boundaryconditiononaverticalsolidsurface 199
7.5 Oscillationsofacylinderimmersedinarotatingsuperfluid 201
7.6 Singlevortexlineterminatingatalateralwall 202
7.7 Vortexbundleterminatingatawall:propagationofthevortexfront 207
8 Mutualfriction 213
8.1 Mutualfrictionandmacroscopichydrodynamics 213
8.2 Semiclassicalscatteringofquasiparticles(geometricoptics) 215
8.3 Scatteringofphononsbyavortex 223
8.4 Iordanskiiforce 226
8.5 Partial-waveanalysisandtheAharonov–Bohmeffect 230
8.6 TransverseforceandBerryphaseintwo-fluidhydrodynamics 235
8.7 Mutualfrictionnearthecriticalpoint 238
8.8 Comparisonwithexperimentsandothertheories 241
9 MutualfrictionandvortexmassinFermisuperfluids 244
9.1 Bardeen–Cooper–Schrieffer theory and Bogolyubov–de Gennes
equations 244
9.2 Mutual friction from scattering of free Bardeen–Cooper–Schrieffer
quasiparticlesbyavortex 247
9.3 Semiclassicaltheoryofpartialwavesversusgeometricoptics:accuracy 250
9.4 Semiclassical partial-wave theory for scattering of free Bardeen–
Cooper–Schriefferquasiparticlesbyavortex 252
9.5 BoundAndreevstatesinaplanarSNSjunction 255
9.6 Boundvortexcorestatesinanormalcore 259
viii Contents
9.7 Mutualfrictioninavortexcore:Kopnin–Kravtsovforce 262
9.8 VortexmassinFermisuperfluids 264
9.9 Spectralflowandvortexdynamics 267
10 Vortexdynamicsandhydrodynamicsofachiralsuperfluid 271
10.1 OrderparameterintheAphaseofsuperfluid3He 271
10.2 Gross–Pitaevskiitheoryforp +ip -wavesuperfluids 273
x y
10.3 Hydrodynamics of a chiral superfluid with an arbitrary intrinsic
angularmoment 276
10.4 Gaugewheel 283
10.5 VorticesandmacroscopichydrodynamicsofchiralsuperfluidAphase
of3He 284
10.6 MutualfrictionforcontinuousvorticesintheAphaseof3He 286
11 Nucleationofvortices 290
11.1 Thermalnucleationofvorticesinauniformflow 290
11.2 Thermalnucleationofvorticesinanon-uniformsuperflow 293
11.3 Nucleationofamasslessvortexviamacroscopicquantumtunnelling:
semiclassicaltheory 296
11.4 Quantumnucleationofavortexwithmassatathinfilmedge 299
11.5 Quantumnucleationofvortices:many-bodyapproach 301
12 Berezinskii–Kosterlitz–Thoulesstheoryandvortexdynamicsinthinfilms 308
12.1 Staticaltheory 308
12.2 Dynamicaltheory 314
12.3 Rateofpairdissociation 318
12.4 Coreless vortices in superfluid films on rotating porous substrates:
fromtwo-dimensionaltothree-dimensionalvortexdynamics 319
12.5 Torsionaloscillationsinfilmsonrotatingporoussubstrates:rotation
dissipationpeak 323
13 Vortexdynamicsinlatticesuperfluids 326
13.1 MagnusforceinJosephsonjunctionarrays 326
13.2 Vortexdynamicsincontinuousapproximationforalatticesuperfluid 329
13.3 VortexdynamicsfromBlochbandtheory 334
13.4 VortexdynamicsintheBose–Hubbardmodel 336
13.5 Magnusforce,Halleffectandtopology 341
14 Elementsofatheoryofquantumturbulence 343
14.1 A tour to classical turbulence: scaling arguments, cascade and
Kolmogorovspectrum 343
14.2 Vinen’stheoryofquantumvortextangle 345
14.3 Classicalversusquantumturbulence 348
14.4 Kelvinwavecascadeinthequantuminertialrange 351
Contents ix
14.5 CrossoverfromKolmogorovtoKelvinwavecascade 353
14.6 SymmetryofKelvinwavedynamicsandKelvinwavecascade 355
14.7 Short-wavelengthcut-offofKelvinwavecascade:soundemission 358
14.8 Beyond the scaling theory of developed homogeneous superfluid
turbulence 360
References 364
Index 383
