Dynamics of Quantised Vortices in Superfluids 9781139047616

Table of contents :
Contents......Page 5
Preface......Page 11
1.1 Thermodynamics of a one-component perfect fluid......Page 20
1.2 Hydrodynamics of a one-component perfect fluid......Page 23
1.3 Motion of a cylinder in an incompressible perfect fluid: backflow......Page 26
1.4 Motion of a cylinder in an incompressible perfect fluid: Magnus force......Page 29
1.5 Cylinder with fluid circulation around it moving in a compressible fluid......Page 31
1.6 Hydrodynamical modes of a perfect fluid......Page 33
1.7 Hydrodynamics of a viscous fluid......Page 36
1.8 Motion of a cylinder in a viscous fluid: Stokes and Oseen problems......Page 38
1.9 Longitudinal and transverse local forces on the fluid......Page 43
1.10 Hydrodynamics of a rotating perfect fluid......Page 47
1.11 Hydrodynamical modes of a rotating incompressible fluid......Page 48
1.12 Inertial wave resonances in an inviscid fluid......Page 51
1.13 Inertial wave resonances in a viscous fluid......Page 53
1.14 Hydrodynamical modes of a rotating compressible perfect fluid......Page 55
1.15 Gross–Pitaevskii theory......Page 57
2.1 Vortex line in a perfect fluid......Page 62
2.2 Linear and angular momenta of a vortex line......Page 67
2.3 Motion of a vortex: Magnus force......Page 70
2.4 Experimental detection of quantum circulation: vortex mass versus Magnus force......Page 71
2.5 Vortex mass in Bose superfluids......Page 74
2.6 Precession of a straight vortex around an extremum of vortex energy......Page 75
2.7 Dynamics of a curved vortex line: Biot–Savart law and local induction approximation......Page 76
2.8 Vortex ring......Page 79
2.9 Kelvin waves on an isolated vortex line......Page 82
2.10 Helical vortex......Page 85
2.11 Helical vortex ring......Page 90
2.12 Precession of a single curved vortex......Page 96
3.1 Macroscopic hydrodynamics of rotating superfluids......Page 101
3.2 Symmetries of periodic vortex textures......Page 109
3.3 Elastic moduli and linear equations of motion......Page 111
3.4 Hall–Vinen–Bekarevich–Khalatnikov hydrodynamics......Page 113
3.5 Tkachenko shear rigidity......Page 117
3.6 Spectrum of oscillations in an incompressible fluid......Page 120
3.7 Axial modes of vortex oscillations......Page 121
3.8 Tkachenko waves: elasticity theory of a two-dimensional vortex lattice......Page 123
3.9 Slow mode in an incompressible perfect fluid......Page 125
3.10 Glaberson–Johnson–Ostermeier instability......Page 126
3.11 Vortex oscillations in a compressible perfect fluid......Page 127
3.12 Rapidly rotating Bose–Einstein condensate in the lowest Landau level state......Page 129
4.1 Equilibrium finite vortex array......Page 134
4.2 Distortions of vortex lattice produced by a boundary......Page 136
4.3 Axisymmetric Tkachenko modes in a finite vortex bundle: comparison of continuum theory and numerical experiments......Page 139
4.4 Chiral edge waves......Page 141
4.5 Ground state of a two-dimensional Bose–Einstein condensate cloud......Page 143
4.6 Ground state of a rotating two-dimensional Bose–Einstein condensate cloud......Page 146
4.7 Tkachenko waves in a Bose–Einstein condensate cloud......Page 147
4.8 Observation of Tkachenko waves in a rotating Bose–Einstein condensate cloud......Page 151
5.1 Torsional oscillator (Andronikashvili) experiment......Page 153
5.2 Boundary conditions on a horizontal solid surface: surface pinning......Page 154
5.3 Collective surface pinning......Page 158
5.4 Pile-of-disks oscillations: Hall resonance versus inertial wave resonance......Page 161
5.5 Effective boundary condition for slow motion in a horizontal layer of rotating fluid......Page 165
5.6 Uniformly twisted vortex bundle......Page 170
5.7 Torsional oscillations of a vortex bundle......Page 174
5.8 Slow oscillations of a superfluid in a finite cylindrical container......Page 179
5.9 Search for Tkachenko waves in superfluid 4He and pulsars: Tkachenko wave versus inertial wave......Page 182
6.1 Two-fluid macroscopic hydrodynamics of a rotating superfluid......Page 186
6.2 Longitudinal modes: first and second sound......Page 194
6.3 Hydrodynamical equations for a completely incompressible superfluid......Page 196
6.4 Axial modes......Page 197
6.5 In-plane modes......Page 199
6.6 Slow modes in a completely incompressible superfluid......Page 201
6.7 Vortex dynamics in the clamped regime......Page 203
6.8 Oscillations of an incompressible fluid in the clamped regime......Page 205
6.9 Phenomenological theory close to the critical temperature......Page 207
7.2 Pile-of-disks oscillations and effective boundary condition......Page 213
7.3 Oscillations in the clamped regime: damped slow mode......Page 216
7.4 Boundary condition on a vertical solid surface......Page 218
7.5 Oscillations of a cylinder immersed in a rotating superfluid......Page 220
7.6 Single vortex line terminating at a lateral wall......Page 221
7.7 Vortex bundle terminating at a wall: propagation of the vortex front......Page 226
8.1 Mutual friction and macroscopic hydrodynamics......Page 232
8.2 Semiclassical scattering of quasiparticles (geometric optics)......Page 234
8.3 Scattering of phonons by a vortex......Page 242
8.4 Iordanskii force......Page 245
8.5 Partial-wave analysis and the Aharonov–Bohm effect......Page 249
8.6 Transverse force and Berry phase in two-fluid hydrodynamics......Page 254
8.7 Mutual friction near the critical point......Page 257
8.8 Comparison with experiments and other theories......Page 260
9.1 Bardeen–Cooper–Schrieffer theory and Bogolyubov–de Gennes equations......Page 263
9.2 Mutual friction from scattering of free Bardeen–Cooper–Schrieffer quasiparticles by a vortex......Page 266
9.3 Semiclassical theory of partial waves versus geometric optics: accuracy......Page 269
9.4 Semiclassical partial-wave theory for scattering of free Bardeen– Cooper–Schrieffer quasiparticles by a vortex......Page 271
9.5 Bound Andreev states in a planar SNS junction......Page 274
9.6 Bound vortex core states in a normal core......Page 278
9.7 Mutual friction in a vortex core: Kopnin–Kravtsov force......Page 281
9.8 Vortex mass in Fermi superfluids......Page 283
9.9 Spectral flow and vortex dynamics......Page 286
10.1 Order parameter in the A phase of superfluid 3He......Page 290
10.2 Gross–Pitaevskii theory for px + ipy-wave superfluids......Page 292
10.3 Hydrodynamics of a chiral superfluid with an arbitrary intrinsic angular moment......Page 295
10.4 Gauge wheel......Page 302
10.5 Vortices and macroscopic hydrodynamics of chiral superfluid A phase of 3He......Page 303
10.6 Mutual friction for continuous vortices in the A phase of 3He......Page 305
11.1 Thermal nucleation of vortices in a uniform flow......Page 309
11.2 Thermal nucleation of vortices in a non-uniform superflow......Page 312
11.3 Nucleation of a massless vortex via macroscopic quantum tunnelling: semiclassical theory......Page 315
11.4 Quantum nucleation of a vortex with mass at a thin film edge......Page 318
11.5 Quantum nucleation of vortices: many-body approach......Page 320
12.1 Statical theory......Page 327
12.2 Dynamical theory......Page 333
12.3 Rate of pair dissociation......Page 337
12.4 Coreless vortices in superfluid films on rotating porous substrates: from two-dimensional to three-dimensional vortex dynamics......Page 338
12.5 Torsional oscillations in films on rotating porous substrates: rotation dissipation peak......Page 342
13.1 Magnus force in Josephson junction arrays......Page 345
13.2 Vortex dynamics in continuous approximation for a lattice superfluid......Page 348
13.3 Vortex dynamics from Bloch band theory......Page 353
13.4 Vortex dynamics in the Bose–Hubbard model......Page 355
13.5 Magnus force, Hall effect and topology......Page 360
14.1 A tour to classical turbulence: scaling arguments, cascade and Kolmogorov spectrum......Page 362
14.2 Vinen’s theory of quantum vortex tangle......Page 364
14.3 Classical versus quantum turbulence......Page 367
14.4 Kelvin wave cascade in the quantum inertial range......Page 370
14.5 Crossover from Kolmogorov to Kelvin wave cascade......Page 372
14.6 Symmetry of Kelvin wave dynamics and Kelvin wave cascade......Page 374
14.7 Short-wavelength cut-off of Kelvin wave cascade: sound emission......Page 377
14.8 Beyond the scaling theory of developed homogeneous superfluid turbulence......Page 379
References......Page 383
Index......Page 402

Citation preview

DY NA M I C S O F QUA N T I S E D VO RT I C E S IN SUPERFLUIDS

A comprehensive overview of the basic principles of vortex dynamics in superfluids, this book addresses the problems of vortex dynamics in all three superfluids available in laboratories – 4 He, 3 He 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 dynamics from large scales to the microscopic scale. Topics such as vortex arrays in rotating 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 unified point of view that avoids a complicated mathematical approach, this text is ideal for students and researchers working with vortex dynamics in superfluids, superconductors, magnetically ordered materials, neutron stars and cosmological models. e d o ua r d b. s o n i n is an Emeritus Professor at the Hebrew University of Jerusalem, Israel. His research interests centre on vortex dynamics in superfluids and superconductors and mesoscopic physics.

DY NA M I C S O F QUA N T I S E D VO RT I C E S I N S U P E R F L U I D S E D O UA R D B . S O N I N Hebrew University of Jerusalem

University Printing House, Cambridge CB2 8BS, United Kingdom Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107006683 © Edouard B. Sonin 2016 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2016 Printed in the United Kingdom by TJ International Ltd. Padstow Cornwall A catalog record for this publication is available from the British Library Library of Congress Cataloguing in Publication data Sonin, Edouard B., author. Dynamics of quantised vortices in superfluids / Edouard B. Sonin, Hebrew University of Jerusalem. pages cm Includes bibliographical references and index. ISBN 978-1-107-00668-3 (Hardback : alk. paper) 1. Superfluidity. 2. Fluid dynamics. 3. Vortex-motion. I. Title. QC175.4.S66 2016 530.4 2–dc23 2015024943 ISBN 978-1-107-00668-3 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Contents

Preface

page xi

1

Hydrodynamics of a one-component classical fluid 1.1 Thermodynamics of a one-component perfect fluid 1.2 Hydrodynamics of a one-component perfect fluid 1.3 Motion of a cylinder in an incompressible perfect fluid: backflow 1.4 Motion of a cylinder in an incompressible perfect fluid: Magnus force 1.5 Cylinder with fluid circulation around it moving in a compressible fluid 1.6 Hydrodynamical modes of a perfect fluid 1.7 Hydrodynamics of a viscous fluid 1.8 Motion of a cylinder in a viscous fluid: Stokes and Oseen problems 1.9 Longitudinal and transverse local forces on the fluid 1.10 Hydrodynamics of a rotating perfect fluid 1.11 Hydrodynamical modes of a rotating incompressible fluid 1.12 Inertial wave resonances in an inviscid fluid 1.13 Inertial wave resonances in a viscous fluid 1.14 Hydrodynamical modes of a rotating compressible perfect fluid 1.15 Gross–Pitaevskii theory

1 1 4 7 10 12 14 17 19 24 28 29 32 34 36 38

2

Dynamics of a single vortex line 2.1 Vortex line in a perfect fluid 2.2 Linear and angular momenta of a vortex line 2.3 Motion of a vortex: Magnus force 2.4 Experimental detection of quantum circulation: vortex mass versus Magnus force 2.5 Vortex mass in Bose superfluids 2.6 Precession of a straight vortex around an extremum of vortex energy 2.7 Dynamics of a curved vortex line: Biot–Savart law and local induction approximation 2.8 Vortex ring 2.9 Kelvin waves on an isolated vortex line

43 43 48 51 52 55 56 57 60 63 v

vi

Contents

2.10 Helical vortex 2.11 Helical vortex ring 2.12 Precession of a single curved vortex 3

4

5

66 71 77

Vortex array in a rotating superfluid: elasticity and macroscopic hydrodynamics 3.1 Macroscopic hydrodynamics of rotating superfluids 3.2 Symmetries of periodic vortex textures 3.3 Elastic moduli and linear equations of motion 3.4 Hall–Vinen–Bekarevich–Khalatnikov hydrodynamics 3.5 Tkachenko shear rigidity 3.6 Spectrum of oscillations in an incompressible fluid 3.7 Axial modes of vortex oscillations 3.8 Tkachenko waves: elasticity theory of a two-dimensional vortex lattice 3.9 Slow mode in an incompressible perfect fluid 3.10 Glaberson–Johnson–Ostermeier instability 3.11 Vortex oscillations in a compressible perfect fluid 3.12 Rapidly rotating Bose–Einstein condensate in the lowest Landau level state

82 82 90 92 94 98 101 102 104 106 107 108

Oscillation of finite vortex arrays: two-dimensional boundary problems 4.1 Equilibrium finite vortex array 4.2 Distortions of vortex lattice produced by a boundary 4.3 Axisymmetric Tkachenko modes in a finite vortex bundle: comparison of continuum theory and numerical experiments 4.4 Chiral edge waves 4.5 Ground state of a two-dimensional Bose–Einstein condensate cloud 4.6 Ground state of a rotating two-dimensional Bose–Einstein condensate cloud 4.7 Tkachenko waves in a Bose–Einstein condensate cloud 4.8 Observation of Tkachenko waves in a rotating Bose–Einstein condensate cloud

115 115 117

Vortex oscillations in finite rotating containers: three-dimensional boundary problems 5.1 Torsional oscillator (Andronikashvili) experiment 5.2 Boundary conditions on a horizontal solid surface: surface pinning 5.3 Collective surface pinning 5.4 Pile-of-disks oscillations: Hall resonance versus inertial wave resonance 5.5 Effective boundary condition for slow motion in a horizontal layer of rotating fluid 5.6 Uniformly twisted vortex bundle 5.7 Torsional oscillations of a vortex bundle 5.8 Slow oscillations of a superfluid in a finite cylindrical container

110

120 122 124 127 128 132 134 134 135 139 142 146 151 155 160

Contents

vii

Search for Tkachenko waves in superfluid 4 He and pulsars: Tkachenko wave versus inertial wave

163

6

Vortex dynamics in two-fluid hydrodynamics 6.1 Two-fluid macroscopic hydrodynamics of a rotating superfluid 6.2 Longitudinal modes: first and second sound 6.3 Hydrodynamical equations for a completely incompressible superfluid 6.4 Axial modes 6.5 In-plane modes 6.6 Slow modes in a completely incompressible superfluid 6.7 Vortex dynamics in the clamped regime 6.8 Oscillations of an incompressible fluid in the clamped regime 6.9 Phenomenological theory close to the critical temperature

167 167 175 177 178 180 182 184 186 188

7

Boundary problems in two-fluid hydrodynamics 7.1 Boundary conditions on a horizontal solid surface 7.2 Pile-of-disks oscillations and effective boundary condition 7.3 Oscillations in the clamped regime: damped slow mode 7.4 Boundary condition on a vertical solid surface 7.5 Oscillations of a cylinder immersed in a rotating superfluid 7.6 Single vortex line terminating at a lateral wall 7.7 Vortex bundle terminating at a wall: propagation of the vortex front

194 194 194 197 199 201 202 207

8

Mutual friction 8.1 Mutual friction and macroscopic hydrodynamics 8.2 Semiclassical scattering of quasiparticles (geometric optics) 8.3 Scattering of phonons by a vortex 8.4 Iordanskii force 8.5 Partial-wave analysis and the Aharonov–Bohm effect 8.6 Transverse force and Berry phase in two-fluid hydrodynamics 8.7 Mutual friction near the critical point 8.8 Comparison with experiments and other theories

213 213 215 223 226 230 235 238 241

9

Mutual friction and vortex mass in Fermi superfluids 9.1 Bardeen–Cooper–Schrieffer theory and Bogolyubov–de Gennes equations 9.2 Mutual friction from scattering of free Bardeen–Cooper–Schrieffer quasiparticles by a vortex 9.3 Semiclassical theory of partial waves versus geometric optics: accuracy 9.4 Semiclassical partial-wave theory for scattering of free Bardeen– Cooper–Schrieffer quasiparticles by a vortex 9.5 Bound Andreev states in a planar SNS junction 9.6 Bound vortex core states in a normal core

244

5.9

244 247 250 252 255 259

viii

Contents

9.7 9.8 9.9

Mutual friction in a vortex core: Kopnin–Kravtsov force Vortex mass in Fermi superfluids Spectral flow and vortex dynamics

10 Vortex dynamics and hydrodynamics of a chiral superfluid 10.1 Order parameter in the A phase of superfluid 3 He 10.2 Gross–Pitaevskii theory for px + ipy -wave superfluids 10.3 Hydrodynamics of a chiral superfluid with an arbitrary intrinsic angular moment 10.4 Gauge wheel 10.5 Vortices and macroscopic hydrodynamics of chiral superfluid A phase of 3 He 10.6 Mutual friction for continuous vortices in the A phase of 3 He

262 264 267 271 271 273 276 283 284 286

11 Nucleation of vortices 11.1 Thermal nucleation of vortices in a uniform flow 11.2 Thermal nucleation of vortices in a non-uniform superflow 11.3 Nucleation of a massless vortex via macroscopic quantum tunnelling: semiclassical theory 11.4 Quantum nucleation of a vortex with mass at a thin film edge 11.5 Quantum nucleation of vortices: many-body approach

290 290 293

12 Berezinskii–Kosterlitz–Thouless theory and vortex dynamics in thin films 12.1 Statical theory 12.2 Dynamical theory 12.3 Rate of pair dissociation 12.4 Coreless vortices in superfluid films on rotating porous substrates: from two-dimensional to three-dimensional vortex dynamics 12.5 Torsional oscillations in films on rotating porous substrates: rotation dissipation peak

308 308 314 318

13 Vortex dynamics in lattice superfluids 13.1 Magnus force in Josephson junction arrays 13.2 Vortex dynamics in continuous approximation for a lattice superfluid 13.3 Vortex dynamics from Bloch band theory 13.4 Vortex dynamics in the Bose–Hubbard model 13.5 Magnus force, Hall effect and topology

326 326 329 334 336 341

14 Elements of a theory of quantum turbulence 14.1 A tour to classical turbulence: scaling arguments, cascade and Kolmogorov spectrum 14.2 Vinen’s theory of quantum vortex tangle 14.3 Classical versus quantum turbulence 14.4 Kelvin wave cascade in the quantum inertial range

343

296 299 301

319 323

343 345 348 351

Contents

14.5 14.6 14.7 14.8

Crossover from Kolmogorov to Kelvin wave cascade Symmetry of Kelvin wave dynamics and Kelvin wave cascade Short-wavelength cut-off of Kelvin wave cascade: sound emission Beyond the scaling theory of developed homogeneous superfluid turbulence

References Index

ix

353 355 358 360 364 383

Preface

The motion of vortices has been an area of study for more than a century. During the classical period of vortex dynamics, from the late 1800s, many interesting properties of vortices were discovered, beginning with the notable Kelvin waves propagating along an isolated vortex line (Thompson, 1880). The main object of theoretical studies at that time was a dissipationless perfect fluid (Lamb, 1997). It was difficult for the theory to find a common ground with experiment since any classical fluid exhibits viscous effects. The situation changed after the works of Onsager (1949) and Feynman (1955) who revealed that rotating superfluids are threaded by an array of vortex lines with quantised circulation. With this discovery, the quantum period of vortex dynamics began. Rotating superfluid 4 He provided the testing ground for the theories of vortex motion developed for the perfect fluid. At the same time, some effects needed an extension of the theory to include twofluid effects, and the quantum period of vortex studies was marked by progress in the understanding of vortex dynamics in the framework of the two-fluid theory. The first step in this direction was taken by Hall and Vinen (1956a), who introduced the concept of mutual friction between vortices and the normal part of the superfluid and derived the law of vortex motion in two-fluid hydrodynamics. Hall (1958) and Andronikashvili et al. (1961) were the first to study experimentally the elastic properties of vortex lines using torsional oscillators. This made it possible to observe Kelvin waves with a spectrum modified by the interaction between vortices. Elastic deformations of vortex lines were caused by pinning of vortices at solid surfaces confining the superfluid. Vortex pinning was another important concept, which emerged during the study of dynamics of quantised vortices. The third important theoretical framework, invented to describe vortex motion in rotating superfluids, was so-called macroscopic hydrodynamics. This relied on a coarsegraining procedure of averaging hydrodynamical equations over scales much larger than the intervortex spacing. Such hydrodynamics was used in the pioneering work on dynamics of superfluid vortices by Hall and Vinen (1956a) and further developed by Hall (1960) and Bekarevich and Khalatnikov (1961). It was a continuum theory similar to the elasticity theory. However, it only included bending deformations of vortex lines and ignored the

xi

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Preface

crystalline order of the vortex array. This theory, called the Hall–Vinen–Bekarevich– Khalatnikov (HVBK) theory (Khalatnikov, 2000), was successful in explaining a variety of experiments on rotating superfluids. In the late 1960s attention was attracted by phenomena connected with crystalline order in a vortex array predicted by Tkachenko (1965). He also predicted (Tkachenko, 1966) that the vortex lattice sustains collective elastic waves, later called ‘Tkachenko waves’, in which vortex lines undergo displacements which are homogeneous along the vortex lines and transverse to the wave vector. This type of wave is a transverse-sound mode of the vortex lattice and is derived from the elasticity theory of the two-dimensional vortex lattice when the wavelength is much larger than the distance between vortices (Tkachenko, 1969). Tkachenko modes could not be described within HVBK hydrodynamics, but later the continuum theory was developed which incorporated the effects of vortex lattice rigidity (Sonin, 1976; Williams and Fetter, 1977; Baym and Chandler, 1983). Only 14 years after the paper by Tkachenko (1965), the existence of a regular vortex lattice in rotating superfluid 4 He was demonstrated experimentally, although for a rather small number of vortices (Gordon et al., 1978; Yarmchuk et al., 1979). Indirect evidence was obtained by Tsakadze (1978), who deduced a value of the vortex-lattice shear rigidity by observing a slow mode of vortex oscillations (the Tkachenko mode modified by vortex bending) in a free-spinning container with superfluid 4 He. New problems challenged vortex dynamics after the discovery of superfluid phases of 3 He. The A phase turned out to be especially unusual. This is an example of a chiral p-wave superfluid with an intrinsic angular momentum. It possessed a remarkable property unknown before: the rotating A phase, while remaining a superfluid, sustains a continuous vorticity. The latter is not homogeneous in space as in a rotating classical fluid, but forms a two-dimensional periodic texture sometimes with more intricate symmetry compared with the simple hexagonal symmetry of the triangular vortex array in superfluid 4 He (Volovik and Kopnin, 1977). One can call this quantum continuous vorticity. Quantum continuous vorticity in the rotating A phase was detected by Hakonen et al. (1982) using the NMR technique. A dynamical theory of quantum continuous vorticity was developed by Kopnin (1978a) who derived the law of motion for a continuous axisymmetric vortex in the A phase. However, vortex dynamics in a chiral superfluid is rather far from being understood completely because of problems with the hydrodynamical theory at T = 0. The family of two helium superfluids available in laboratories (putting aside superconducting electron fluids in metals) was extended significantly after the discovery of Bose– Einstein Condensation (BEC) of cold-atom gases. This brought new possibilities and new challenges. In contrast to strong interaction in 4 He and 3 He superfluids, in cold-atom gases the interaction is weak, and weak-coupling theories (the Gross–Pitaevskii theory, for example) became both qualitatively and quantitatively reliable for the prediction and interpretation of experiments. An important feature of cold-atom superfluids is a much higher compressibility than in 4 He and 3 He superfluids and non-uniform density. This required revision of the theory of vortex motion, especially for slow Tkachenko modes. Very effective optical methods which were not accessible in old superfluids allowed clear

Preface

xiii

visualisations of equilibrium and oscillating vortex arrays (Abo-Shaeer et al., 2001; Coddington et al., 2003). Cold-atom superfluids are confined by potential traps formed by laser beams. They have no contacts with rough solid surfaces like the old superfluids. This excludes pinning, which was the main hurdle for the observation of pure Tkachenko waves. While evidence of the Tkachenko mode in the old superfluids was rather circumstantial and did not allow a decisive quantitative comparison with the theory, experiments with cold-atom superfluids provided the first unambiguous observation of Tkachenko waves (Coddington et al., 2003). Superfluids available in laboratories do not exhaust all applications of superfluid vortex dynamics. Long ago it was supposed that the interior matter of neutron stars is in the superfluid state and is threaded by quantised vortices because of rotation (Ginzburg and Kirzhnitz, 1964). A rich variety of phenomena in pulsars can be interpreted using the concept of quantum vortices. In the past, astrophysical applications greatly stimulated vortex dynamics studies. For example, experimental studies of Tkachenko modes by Tsakadze and Tsakadze (1973) were encouraged by the theory of Ruderman (1970) associating periodic variations observed in the pulse period of pulsars with Tkachenko waves. The study of vortex dynamics in rotating superfluids is advantageous because rotation creates vortices with well controlled form and density. But vortices can appear chaotically after the transition to the turbulent regime at high fluid velocities. The phenomenon of turbulence was studied intensively in classical fluids for centuries. In superfluids, turbulence can also emerge, but it is strongly affected by the fact that only vortices with quantised circulation are possible, hence superfluid turbulence is also called quantum turbulence. Quantum turbulence has been studied for more than a half-century, starting with the pioneering theoretical and experimental investigations by Vinen (1957c, 1961b). The interest in this field has not subsided. In principle, one might expect that the theory of quantum turbulence could be derived from the dynamics of quantised vortices, and that the non-linear Navier–Stokes equation should describe all the features of classical turbulence. However, such development of the theory from first principles is not feasible practically and is hardly useful. Inevitably the theory of turbulence requires introduction of essential assumptions and concepts. This transforms it into a special area of research, which does not reduce to hydrodynamics of laminar flows or to pure vortex dynamics, in full analogy with the fact that solid state physics does not reduce to atomic physics. But the theory of turbulence, nevertheless, is an important application of vortex dynamics. Earlier theoretical and experimental investigations of vortex dynamics were treated in a number of comprehensive reviews and books (Hall, 1960; Andronikashvili et al., 1961; Andronikashvili and Mamaladze, 1966, 1967; Putterman, 1974; Donnelly, 1991; Khalatnikov, 2000) addressing mostly vortices in superfluid 4 He. Phenomena associated with crystalline order in the vortex lattice were considered only fragmentarily. There were books on superfluid 3 He with some chapters addressing vortex dynamics (Vollhardt and W¨olfle, 1990; Volovik, 2003b) and there was an extensive review by Salomaa and Volovik (1985) who focused on the topological analysis of various vortex textures. A review focusing more on the effects of crystalline ordering of vortices on vortex dynamics in 4 He and 3 He was

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also published (Sonin, 1987). Investigations of vortex dynamics in cold-atom BEC are still at an early stage, but one can already find reviews or chapters of books addressing vortex dynamics in BEC of cold atoms (Pitaevskii and Stringari, 2003; Fetter, 2009; Ueda, 2010). In the present book I have made an attempt to bring together vortex dynamics and its applications to all known neutral superfluids available now in laboratories (4 He, 3 He, and cold-atom BEC), addressing them from a unified position, which allows us to see the generality and differences of vortex dynamics in various media. A natural question is why vortices in charged superfluids (superconductors) were excluded from the list. An honest answer is that it would make the project too ambitious and the chance of completing it in a reasonable time bleak. The same applies to vortices in exciton and polariton BEC. Consideration of vortices in superconductors and excitonic superfluids would require addressing two important effects which are beyond the scope of the book: Meissner screening in superconductors and non-conservation of the particle number in excitonic superfluids. On the other hand, vortex dynamics in neutral and charged superfluids are definitely closely interrelated and have general problems and approaches. Therefore I hope that the content of the book will also be useful for people interested in vortex dynamics in superconductors. One can find an extensive theoretical investigation of vortex dynamics in superconductors in the book by Kopnin (2001), and some problems and approaches in Kopnin’s book are also included in the present book. The present book includes the material of the old review (Sonin, 1987) except for some outdated parts. A huge work on vortex dynamics done after publication of the review required a new analysis, which this book is supposed to present. This book mostly addresses the theory, it is written by a theorist, and from the position of a theorist. This does not mean that no attention is paid to the connection with experimental investigations, which have been done or can be done. However, only the main experimental results, and not techniques, are discussed. The experiments deal mostly with effects produced by a large number of vortices. So the book widely exploits macroscopic hydrodynamics referring to infinite vortex arrays. Even though a lot of attention is devoted to boundary problems for finite vortex arrays (because of their great importance for contact of the theory with experiments on vortex oscillations) they are assumed to be large enough to be treated within macroscopic hydrodynamics. In order to make the book self-contained we must inevitably address problems of classical and superfluid hydrodynamics as a whole, but only to an extent which is necessary for understanding vortex dynamics. In this book the principle ‘from particular to general’ is preferred to the principle ‘from general to particular’. Though the latter makes the text more compact and helps to avoid repetition, the former is more convenient for readers who have no intention to enter deeply into the theory and want to stop at some level. Chapter 1 addresses problems of classical hydrodynamics relevant for the main content of the book. The chapter deals with the hydrodynamical approach in general and with hydrodynamical modes in perfect and viscous fluids, either resting or rotating. The chapter analyses the motion of a cylinder with and without circulation of velocity around it through a perfect and a viscous fluid in the light of the key importance of this classical problem for the motion of quantised vortices. Only the last section of the chapter, Section 1.15,

Preface

xv

turns from classical theory to quantum theory, addressing the Gross–Pitaevskii theory of a weakly interacting Bose gas. This section demonstrates how under some conditions the quantum mechanical Gross–Pitaevskii theory justifies the application of classical hydrodynamics to quantum problems. In Chapter 2, the dynamics of a single quantised vortex is considered at zero temperature when the normal component is totally absent. Apart from quantisation of circulation around the vortex, this problem can also be analysed within classical hydrodynamics of a perfect fluid. Basic principles of vortex dynamics are introduced for rectilinear vortices. A key feature of vortex dynamics is that an external force on a vortex is balanced by the Magnus force proportional to the vortex velocity, while the inertia force proportional to acceleration usually (but not always) can be neglected. This is supported by calculation of the vortex mass. The dynamical theory for curved vortex lines starts from the Biot–Savart law for a vortex line moving in a perfect fluid. Then the local induction approximation is introduced, which reduces the integral Biot–Savart equation of vortex motion to a differential equation. The latter is mostly used later in the chapter, which addresses various examples of vortex dynamics applications: Kelvin waves, vortex rings, helical vortices and helical vortex rings. Chapter 3 formulates macroscopic hydrodynamics of rotating superfluids derived from coarse-graining over scales exceeding the intervortex distance. The analysis addresses the HVBK theory, which neglects shear rigidity of the vortex array, and its extension which takes into account shear rigidity. Elastic moduli of the vortex lattice are introduced, and the Tkachenko theory, which provides an exact value of the shear modulus, is surveyed. Then macroscopic hydrodynamics is used for investigation of collective modes in a rotating superfluid. These are Kelvin modes modified by long-range intervortex interaction, which introduces a gap into the originally gapless spectrum. Another important mode is the slow mode, which is a hybrid of an inertial wave in a classical rotating fluid and of an elastic shear Tkachenko mode in a lattice of quantised vortex lines. The analysis extends to a compressible fluid describing the Tkachenko mode in a rotating BEC of cold atoms. Most of the chapter assumes the existence of an array of singular vortex lines with core radius very small compared with the intervortex distance. This is called the Vortex Line Lattice (VLL) state. The last section, of the chapter, Section 3.12, considers the opposite case of a rapidly rotating Bose–Einstein condensate, when the vortex array is very dense and vortex cores strongly overlap. This is called the Lowest Landau Level (LLL) state. The state is an analogue of the mixed state of type II superconductors in strong magnetic field close to the second critical field. Chapter 4 addresses two-dimensional boundary problems of macroscopic hydrodynamics. This requires the formulation of boundary conditions at lateral walls or borders of vertically uniform vortex bundles. After this, edge waves are considered. Edge modes on the boundary of the vortex bundle are chiral unidirectional modes similar to edge modes in the quantum Hall effect. But the focus of the chapter is pure Tkachenko modes in an incompressible and in a compressible fluid. The latter case is relevant for rotating BEC of cold atoms, where visual observation of Tkachenko modes becomes possible. The theory is compared with these observations.

xvi

Preface

In Chapter 5, three-dimensional boundary problems of macroscopic hydrodynamics are considered. This requires boundary conditions at the tops and bottoms of containers, which take into account pinning of vortices at rough solid surfaces. The concept of collective surface pinning is discussed. The theory is applied for the analysis of pile-of-disks torsional oscillations, which have been intensively investigated experimentally during many decades of study of superfluids. The analysis of the slow mode in a finite container provides an explanation of why it was so difficult to observe the Tkachenko mode in helium superfluids: even rather weak pinning transforms it to a classical inertial wave with the spectrum independent of the shear rigidity of the lattice of quantised vortices. Chapter 6 turns from one-fluid vortex dynamics to vortex dynamics in two-fluid macroscopic hydrodynamics. The equations of macroscopic hydrodynamics are reformulated, including the effect of mutual friction and other dissipative processes. These equations allow us to investigate the temperature dependences and damping of collective modes discussed in previous chapter in a perfect fluid at T = 0. The chapter also considers the clamped regime, when the normal component is rigidly connected with a container and rotates as a solid body together with it. The regime is realised in rotating superfluid 3 He because of its very large normal viscosity. The last section of the chapter, Section 6.9, surveys the phenomenological Ginzburg–Pitaevskii theory and its extensions, which must be used to study vortex motion close to the critical temperature. Chapter 7 reconsiders boundary problems investigated in previous chapters in the framework of two-fluid hydrodynamics. The last sections of the chapter, Sections 7.6 and 7.7, address a single vortex line and a vortex tangle terminated at the lateral walls of a container. This geometry became an object of experimental investigations of superfluid turbulence emerging at the propagation of a vortex front separating the vortex bundle from a vortexfree area (Eltsov et al., 2014). Chapter 8 descends from macroscopic hydrodynamics down to scales much less than the intervortex distance. This is necessary for derivation of mutual friction parameters, which entered macroscopic hydrodynamics as phenomenological parameters. This issue was a matter of decades long dispute and controversy. Mutual friction originates from scattering by a vortex of quasiparticles (phonons and rotons) forming the normal component. The presence of a long-range velocity field ∼ 1/r around the vortex complicates the scattering theory of quasiparticles. This field leads to an effect similar to the Aharonov–Bohm effect for electrons interacting with an electromagnetic vector potential ∼ 1/r. The Aharonov– Bohm effect for scattering of quasiparticles leads to the emergence to a transverse force on a vortex (the Lifshitz–Pitaevskii force from rotons and the Iordanskii force from phonons), which must be added to the Magnus force produced by the superfluid component. In the past, the Iordanskii force was rejected on the basis of topological arguments using the Berry phase of a moving vortex. These topological arguments ignored the contribution of the normal component to the Berry phase. The chapter also considers mutual friction near the critical temperature, where the analysis based on the scattering theory of non-interacting quasiparticles becomes invalid. Instead of this, mutual friction parameters are derived from the phenomenological time-dependent Ginzburg–Pitaevskii theory.

Preface

xvii

Peculiar features of vortex dynamics in Fermi superfluids are considered in Chapter 9, which starts with a short overview of some aspects of the Bardeen–Cooper–Schrieffer (BCS) theory which are important for vortex dynamics. There are peculiarities of the Aharonov–Bohm effect for BCS quasiparticles, which are important for determination of the contribution of high energy quasiparticles to the transverse force on the vortex. However, the main difference between vortex dynamics in the Bose and Fermi superfluids arises from the existence of core bound states in Fermi superfluids. Interaction of quasiparticles bound in the vortex core with free quasiparticles or with impurities leads to the transverse Kopnin–Kravtsov force. The force has opposite direction with respect to the Magnus force from the superfluid component and sometimes can fully cancel the latter. Bound states also make an essential contribution to the vortex mass. In the past the effects of bound states on mutual friction and vortex mass were interpreted as resulting from spectral flow of bound states across the gap in the BCS quasiparticle spectrum (Volovik, 2003b). The analysis presented in Chapter 9 shows that vortex motion does not produce spectral flow, and the spectral flow interpretation must be abandoned. Chapter 10 focuses on hydrodynamics of chiral superfluids like the A phase of 3 He. Chirality is related to a spontaneous orbital moment of Cooper pairs (orbital ferromagnetism in the terminology of P. W. Anderson). The order parameter of chiral superfluids allows superfluid velocity fields with continuous vorticity. This has a dramatic effect on hydrodynamics. There is a fundamental still unresolved problem in the formulation of hydrodynamics of chiral superfluids at zero temperature. This is related to an intrinsic angular momentum connected with the orbital moment of Cooper pairs. The magnitude of the intrinsic angular momentum is debated and, as explained in Chapter 10, is ambiguous because there are different definitions of the intrinsic angular momentum. However, leaving the magnitude as an arbitrary phenomenological parameter, one can derive mutual friction parameters from two-fluid hydrodynamics at finite temperature. Chapter 11 considers possible mechanisms of vortex nucleation. Vortex nucleation is impeded by a potential barrier for vortex creation. The Iordanskii–Langer–Fisher theory assumes that the barrier is overcome due to thermal fluctuations, and the nucleation rate is given by the Arrhenius law with the barrier height in the exponent. Approaching zero temperature, thermal nucleation becomes ineffective, and it was expected that vortices could be nucleated via quantum tunnelling through the potential barrier. A vortex is a macroscopic excitation of the fluid, and quantum tunnelling of vortices is an example of macroscopic quantum tunnelling. Thermal and quantum nucleation is discussed for various geometries of superflows. The chapter deals only with simple cases of vortex nucleation and does not address more realistic situations, which the theory of critical velocities in superfluids encounters. Chapter 12 analyses vortex dynamics in superfluid films. It starts with a survey of the Berezinskii–Kosterlitz–Thouless theory and then turns to superfluid films on porous substrates, when one can observe an interesting interplay of two-dimensional and threedimensional physics. The theory provides an explanation of a dissipation peak revealed in experiments on torsional oscillations of rotating porous substrates with superfluid 4 He film

xviii

Preface

which is proportional to the angular velocity. This was absent in films on rotating plane substrates. Chapter 13 deals with vortex dynamics in lattice superfluids. The best known example of a lattice superfluid is the Josephson junction array, where suppression of the Magnus force responsible for the Hall effect was revealed theoretically and experimentally. Nowadays, interest in the vortex dynamics in lattice superfluid has been boosted by investigations of BEC of cold atoms in optical lattices. The chapter analyses the phenomenological model decribing lattice superfluids in the continuous approximation. The model follows from the Bloch band theory for particles in periodic potentials. Within this model, forces on the vortex are derived from the balances of quasimomentum and true momentum. The analysis addresses the case of BEC close to the superfluid–insulator transition where the Magnus force changes its sign. The last chapter of the book is devoted to superfluid turbulence, which emerges as a random tangle of quantised vortex lines. The discussion addresses only a restricted list of topics, which are chosen as examples of the role of vortex dynamics and illustrations of applications of scaling arguments and laws of symmetry for studying turbulence. The theory and experiments on superfluid turbulence in a broader sense are beyond the scope of the present book, and in order to study this field, the reader should use other reviews and books. During writing the book I benefited greatly from discussions of topics of the book with Assa Auerbach, Michael Berry, Vladimir Eltsov, Bill Glaberson, Andrei Golov, Risto H¨anninen, Nikolai Kopnin, Evgeny Kozik, Matti Krusius, Victor L’vov, Netanel Lindner, Sergey Nazarenko, Sergei Nemirovskii, Richard Packard, Lev Pitaevskii, Michael Stone, Boris Svistunov, Erkki Thuneberg, Joe Vinen and Grigori Volovik. Edouard B. Sonin

1 Hydrodynamics of a one-component classical fluid

1.1 Thermodynamics of a one-component perfect fluid In the strict sense of the word, hydrodynamics describes the dynamical behaviour of a fluid. But sometimes the hydrodynamical approach refers to phenomenological theories dealing with various types of condensed media, such as solids, liquid crystals, superconductors, magnetically ordered systems and so on. Two important and interconnected features characterise the hydrodynamical description. • It refers to spatial and temporal scales much longer than any relevant microscopical scale of the medium under consideration. • It does not need the microscopical theory for derivation of dynamical equations but uses as a starting point a set of conservation laws and thermodynamical and symmetry properties of the medium under consideration. The latter feature gives us the possibility to study condensed matter without waiting for the moment when a closed self-consistent microscopical theory is developed. Sometimes it can be a long time to wait for such a moment. For example, one may recall the microscopical theory of fluid with strong interactions, or as the latest example the microscopical theory of high-Tc superconductivity. In fact, the cases when the hydrodynamical description can be derived rigorously from the ‘first-principle’ theory are more the exceptions rather than the rule. Such exceptions include, for example, weakly non-ideal gases and weak-coupling superconductors. Even if it is possible to derive the hydrodynamical description from the microscopical theory, the former as based on the most global properties (conservation laws and symmetry) is a reliable check of the microscopical theory. If hydrodynamics does not follow from a microscopical theory this is an alarming signal of potential problems with the microscopical theory. Impressive evidence of the fruitfulness of the hydrodynamical (phenomenological) approach to condensed matter physics is provided by the volumes of Landau and Lifshitz’s course addressing continuous media: Electrodynamics of Continuous Media, Theory of Elasticity, and Fluid Mechanics (Landau and Lifshitz, 1984, 1986, 1987). The hydrodynamical approach was very fruitful also for studying properties of rotating superfluids, as will be demonstrated in this book. The hydrodynamical description always deals with the 1

2

Hydrodynamics of a one-component classical fluid

continuous medium even if the medium under consideration is a lattice (an atomic lattice in elasticity theory, for example). Indeed, the lattice constant is a microscopical scale which should be ignored in accordance with the nature of the hydrodynamical description. As a result, the hydrodynamical theories reduce to continuous field theories, and this similarity stimulated and is stimulating a useful exchange of ideas between condensed matter physics and field theory. The basic idea of the hydrodynamical description is that any small volume of the fluid (small with respect to the hydrodynamical scales, but large with respect to any microscopical scale!) is in a state near to equilibrium described by thermodynamics, and we start from a discussion of the thermodynamics of the fluid. The equilibrium state of any condensed medium, however complicated its microscopical properties, can be described by a few thermodynamical variables. Their number depends on the nature and symmetry of the condensed matter under consideration. The simplest are one-component fluids and gases, characterised by the highest symmetry and the minimal number of variables. The state of a resting one-component fluid (gas) in a very large volume V is completely defined by the mass and entropy density ρ and S. The energy density E0 (the subscript 0 points out that a resting fluid is considered) as well as other thermodynamical parameters are functions of ρ and S. The surface effects are neglected, and the total mass M = ρV , the total entropy S = SV , and the total energy E0 = E0 V are proportional to the volume V . For the energy density differential one has the Gibbs relation dE0 = μ0 dρ + T dS,

(1.1)

where the chemical potential μ0 and the temperature T are partial derivatives of the energy density: μ0 =

∂E0 , ∂ρ

T =

∂E0 . ∂S

(1.2)

Two contacting fluids are in equilibrium if their chemical potential and temperature are equal. It is often more convenient to use other pairs of variables rather than ρ and S. Correspondingly, a different thermodynamical potential other than the energy should be used to describe the equilibrium state (ensemble). This arises from the Legendre transformation. Choosing the mass density ρ and the temperature T as thermodynamical variables, the thermodynamical potential is the Helmholtz free energy for the canonical ensemble with density F0 (ρ, T ) = E0 − T S, which has the differential dF0 = d(E0 − T S) = μ0 dρ − SdT .

(1.3)

The great canonical ensemble is defined by μ0 and T and the thermodynamical potential is the pressure P = −E0 + μ0 ρ + T S.

(1.4)

1.1 Thermodynamics of a one-component perfect fluid

3

Equation (1.4) together with Eq. (1.1) gives the Gibbs–Duhem relation for the pressure differential: dP = ρdμ0 + SdT .

(1.5)

The pressure is directly connected with the mechanical work performed by an expanding fluid. Small work for a small adiabatic variation δV of the volume is δA = −

dE0 δV . dV

(1.6)

On adiabatic expansion the total mass and entropy are conserved, and the derivative dE0 /dV is calculated at fixed total mass ρV and total entropy SV . Therefore ∂ρ/∂V = −ρ/V ,

∂S/∂V = −S/V .

(1.7)

Then using Eq. (1.1) one obtains: dE0 d(E0 V ) dE0 = = E0 + V dV dV dV   dS dρ = E0 + V μ0 +T = E0 − μ0 ρ − T S. dV dV

(1.8)

Comparing this with Eq. (1.4) we find that the pressure P =−

∂E0 ∂V

(1.9)

determines the adiabatic work δA = P δV indeed. If the fluid is moving with the centre-ofmass velocity v, its momentum density or mass current is j = ρv while the fluid energy density is E = E0 + ρv 2 /2 = E0 + j 2 /2ρ.

(1.10)

The differential of the energy density is dE = μj dρ + T dS + v · dj

(1.11)

if the mass current j = ρv is used as a hydrodynamical variable. Here the chemical potential μj at fixed j is connected with the chemical potential μ0 of the resting fluid by the relation μj = μ0 − v 2 /2.

(1.12)

Calculating the pressure from the energy of the moving fluid, the total momentum j V of the fluid is kept fixed and ∂j /∂V = −j /V .

(1.13)

P = −∂E/∂V = −E + μj ρ + T S + v · j = −E0 + μ0 ρ + T S,

(1.14)

Then

4

Hydrodynamics of a one-component classical fluid

and the Gibbs–Duhem relation becomes dP = ρdμj + SdT + j · dv = ρdμ0 + SdT .

(1.15)

So the pressure and the Gibbs–Duhem relation are not affected by the fluid motion. Instead of j one may also choose the velocity v as a thermodynamic variable describing fluid motion. Then dE = μdρ + T dS + j · dv,

(1.16)

and the chemical potential is μ = μ0 +

v2 . 2

(1.17)

The pressure and the Gibbs–Duhem relation in this case also reduce to those in the coordinate frame where the fluid is at rest [Eqs. (1.14) and (1.17)].

1.2 Hydrodynamics of a one-component perfect fluid In the hydrodynamical theory the thermodynamical variables (the chemical potential and the temperature) are not constant in general but vary smoothly in space and time. Instead of a tremendous number of microscopical variables (coordinates and velocities of atoms or their wave functions in the quantum mechanical description) one is dealing with continuous classical fields of a few thermodynamical variables describing the dynamical behaviour of the condensed medium. For a one-component perfect fluid, these are the mass density ρ(R, t), the entropy density S(R, t), and the mass current j (R, t). Since they are slowly varying functions of the three-dimensional position vector R and the time t, for their local values one may use the thermodynamical relations given in the previous section. Hydrodynamical equations are derived from conservation laws. We follow the procedure described by Landau and Lifshitz (1987). A conservation law for some quantity means that the variation of its density in time at a given point in space is due only to flows from other parts of space. The mass conservation law gives the mass continuity equation ∂ρ + ∇ · j = 0. ∂t

(1.18)

There are conservation laws for the linear momentum, ∂ji + ∇j ij = 0, ∂t

(1.19)

∂E + ∇ · Q = 0, ∂t

(1.20)

and for the energy,

1.2 Hydrodynamics of a one-component perfect fluid

5

where ij is the momentum flux tensor and Q is the energy flux. For a perfect (inviscid) fluid the total entropy is also conserved. This leads to the continuity equation for entropy ∂S + ∇ · J S = 0. ∂t

(1.21)

Here J S is the entropy flux. All equations, the conservation laws included, must be invariant for any inertial coordinate frame (Galilean invariance). Suppose that there are two inertial coordinate systems R, t and R  , t  , the latter moving with the relative velocity w: R = R  + wt  ,

t = t ,

v = v  + w.

(1.22)

Then the time and space derivatives in the two frames are connected by the relations:   ∂ ∂ ∂ ∂ ∂ . (1.23) = =  − w· , ∂R ∂R  ∂t ∂t ∂R  Transforming the conservation laws (1.18)–(1.21) to the coordinate frame (R  , t  ) with the help of Eqs. (1.22) and (1.23), we see that the equations are invariant (do not change their form) if the flows are transformed as j = j  + ρw,

(1.24)

J S = J S + Sw,

(1.25)

ij = ij + ji wj + wi jj + ρwi wj ,

(1.26)

w2  (j + ρwi ). (1.27) 2 i For the one-component isotropic fluid the only vector at our disposal is the velocity v (or the current j parallel to v) and there are no tensors besides δij and vi vj . One can check that the system of equations is Galilean invariant and all conservation laws are satisfied if the fluxes are given by the following expressions (Landau and Lifshitz, 1987): Qi = Qi + E  wi + wj (j i + jj wi ) +

ij = P δij + ρvi vj ,

(1.28)

J S = Sv,     v2 2 Q = [ρ(μj + v ) + T S]v = ρ μ0 + + T S v. 2

(1.29) (1.30)

The Euler equation follows directly from the mass continuity equation and the momentum conservation law: ∇P ∂v + (v∇)v = − . (1.31) ∂t ρ The vector identity (v∇)v = ∇

v2 + [[∇ × v] × v] 2

(1.32)

6

Hydrodynamics of a one-component classical fluid

transforms the Euler equation into another form: ∇P v2 ∂v + [ω˜ × v] = − −∇ . ∂t ρ 2

(1.33)

Here ω˜ = [∇ × v] is the vorticity of the velocity. Later on in this chapter we neglect the temperature variation or consider the case T = 0. Then using the Gibbs–Duhem relation (1.5) at zero temperature, the Euler equation becomes   v2 ∂v + [ω˜ × v] = −∇ μ0 + . (1.34) ∂t 2 On the right-hand side one sees the chemical potential μ [see Eq. (1.17)] for the fluid moving with velocity v. Let us discuss now the angular momentum (moment) conservation law. For an isotropic fluid, which consists of particles without internal degrees of freedom, i.e., without intrinsic angular momentum, the moment density M is not an independent variable, but is directly determined by the linear momentum density j : M = [j × R].

(1.35)

Then ∂jj ∂Mi = ij k Rk = −ij k ∇n j n Rk = −∇n (ij k j n Rk ) + ij k j k . (1.36) ∂t ∂t For any symmetric momentum flux tensor ij this equation takes the form of the conservation law, ∂Mi + ∇n Gin = 0 ∂t with the angular momentum flux tensor Gin = ij k j n Rk .

(1.37)

(1.38)

For an isotropic fluid the tensor ij is symmetric indeed, as seen from Eq. (1.28). Usually when deriving hydrodynamics one starts from the Euler equation and checks the momentum conservation law afterwards. We have shown here the opposite direction since it will be helpful for understanding two-fluid hydrodynamics. But it is useful to reproduce here the usual derivation also. One starts from Newton’s second law for the unit volume of the fluid, which is dv = f. dt The force f per unit volume (the force density) is defined by the stress tensor, ρ

fi = −∇j σij ,

(1.39)

(1.40)

which is a scalar for the isotropic fluid: σij = P δij .

(1.41)

1.3 Motion of a cylinder in perfect fluid: backflow

7

The time derivative dv/dt in Eq. (1.39) is a substantial derivative in the Lagrange description of fluid dynamics (Batchelor, 1970). The velocity is considered as a Lagrange variable: taking the Lagrange (substantial) derivative one looks for a difference of two velocities of the same fluid particle in two instants of time at which, because of its motion, the particle is located in different points in space. On the other hand, the partial time derivative ∂v/∂t in the Euler equation (1.31) is a derivative in the Euler description: taking the time Euler derivative one compares velocities of two different particles, which at two instants of time were located at the same point of space. The two derivatives are connected by the relation ∂v dv = + (v · ∇)v. dt ∂t As a result Eqs. (1.39)–(1.42) give the Euler equation Eq. (1.31).

(1.42)

1.3 Motion of a cylinder in an incompressible perfect fluid: backflow We address an elementary problem of classical hydrodynamics considered in numerous textbooks: motion of a cylinder immersed in a perfect incompressible fluid. The problem is important for the analysis of vortex dynamics later in the book. Classic hydrodynamics tells us that a moving cylinder of radius R0 induces a dipole velocity field around it (backflow):   2(v bf · r)r 2 v bf . (1.43) V bf (r) = ∇bf = −R0 − r2 r4 Here bf = −R02 v bf · r/r 2 is the scalar velocity potential, the velocity v bf is a constant which determines the backflow intensity, and r is a two-dimensional position vector normal to the cylinder axis (the axis z) with an origin at this axis. The velocity field of the backflow is shown in Fig. 1.1a. The presence of backflow makes it possible to satisfy a natural condition that in the coordinate frame with velocity v L of the cylinder, a current normal to the cylinder boundary must vanish. Suppose that far from the cylinder the fluid moves with velocity v ∞ . Then the total fluid velocity field around the cylinder is v ∞ + V bf (r), and the radial current ρ[v ∞ + V bf (r) − v L ] · r/r vanishes at r = R0 if v bf = v L − v ∞ . The kinetic energy of the backflow in the coordinate frame moving with fluid velocity v is given by     2  ρR04 (v L − v ∞ ) · r 2 (v L − v ∞ )2 2  2 (v L − v ∞ ) = . (1.44) μa dr ∇ = π R ρ 0  2 2 2 r2 r>R0

This determines the mass μa = π R02 ρ per unit length, which is equal to the mass per unit length of the fluid inside a cylinder of radius R0 . It is called the associated mass since this mass must be added to the mass of the cylinder itself when determining the kinetic energy of a moving cylinder. The associated mass must be connected with an additional momentum of the fluid dragged by the moving object. The calculation of this momentum has a subtlety which is

8

Hydrodynamics of a one-component classical fluid (a)

(b)

Figure 1.1 Cylinder of radius R0 moving through a resting fluid with the velocity v L . (a) Velocity field of the backflow around the cylinder. (b) The cylinder moves in a straight channel with periodic boundary conditions at its ends.

well known in classical hydrodynamics. The momentum of the perfect fluid with potential velocity field v(R) = ∇(R) is determined by an integral, which after integrating by parts reduces to a surface integral over a surface S confining the fluid:  P =ρ

 v(R) dR = ρ

 ∇(R) dR = ρ

dS.

(1.45)

S

Here dS is a vector with magnitude equal to the element dS of the surface area and directed normally to the surface outside the bulk occupied by the fluid. In the case of backflow, the surface S consists of the surface of the cylinder of radius R0 and the surface S∞ confining

1.3 Motion of a cylinder in perfect fluid: backflow

9

the fluid at large distance from the cylinder. The momentum per unit length of the cylinder is  P =ρ

⎛ ⎜ ∇bf (r) dr = ρ ⎝





r=R0

bf dS +

⎞ ⎟ bf dS ⎠ .

(1.46)

S∞

This expression cannot define the momentum as long as the integral over S∞ is undefined. Classical hydrodynamics (Lamb, 1997) tells us that the integral over S∞ must simply be deleted from the expression. The remaining integral over the surface of the cylinder is called the Kelvin impulse:  PK = ρ

bf dS = μa v bf .

(1.47)

r=R0

In classical hydrodynamics the Kelvin impulse is justified by considering the momentum transferred to the object when making it move from rest (see Lamb, 1997, Sec. 119). Forces which make the object move from the state of rest are applied locally and do not affect the velocity and the velocity potential in the fluid far away from the object. This justifies ignoring the integral over S∞ in the expression (1.46) for the momentum. The derivation in fact assumes that there is finite compressibility of the fluid and the momentum transferred to the fluid locally is not distributed over the whole fluid instantly. In superfluid hydrodynamics one can justify using the Kelvin impulse in a simpler way (Sonin, 1973). In superfluids the velocity potential  = hθ/m is determined by the ¯ quantum mechanical phase θ of the condensate wave function (see Section 1.15). Let us consider a cylinder moving in a straight channel with periodic boundary conditions at its ends (Fig. 1.1b). The surface S∞ contains lateral walls of the channel and cross-section planes at channel ends. Since there is no flow normal to the lateral walls, only the channel ends contribute to the momentum component along the channel. The periodic boundary condition requires that the phase difference at the channel ends is an integer number of 2π . In the absence of an essential transport current along the channel, the phase difference must vanish. Then the whole contribution of the surface S∞ to the momentum also vanishes. This example of rather simple geometry illustrates the general rule that local perturbations of the velocity field cannot change the phase at infinity. On the other hand, motion of an object through an incompressible perfect fluid inevitably leads to a very small but still finite velocity at infinity. The finite momentum of the fluid dragged by the moving object means that that the whole fluid moves with average velocity P /ρV inversely proportional to the volume V . The fluid moves with this velocity at the entrance to and the exit from the channel shown in Fig. 1.1b. This tiny velocity integrated over the whole volume gives a contribution of the same order as the momentum P . According to Fig. 1.1a, the fluid in the backflow area around the cylinder moves in the direction opposite to the direction of the cylinder velocity v L . Nevertheless, at large distance from the cylinder the fluid moves with small average velocity in the same direction as the cylinder.

10

Hydrodynamics of a one-component classical fluid

Although external forces are necessary in order to make an object move through a perfect fluid, no force is needed to support steady translational motion of the object. The latter moves without any resistance. In classical hydrodynamics this was called d’Alembert’s paradox (Landau and Lifshitz, 1987), since it was in evident contradiction to real observations. Resolution of this paradox in classical hydrodynamics is quite evident: in a real fluid one cannot neglect viscosity (see Section 1.8 addressing motion of a cylinder through a viscous fluid). But for superfluids, d’Alembert’s paradox ceases to be a paradox. It is a real observed phenomenon: at low velocities objects move through a superfluid without dissipation and without generation of vorticity. So in a sense d’Alembert’s paradox is a precursor of the phenomenon of superfluidity.

1.4 Motion of a cylinder in an incompressible perfect fluid: Magnus force A transverse force normal to the velocity of a moving cylinder appears even in a perfect fluid if there is a circular flow around the cylinder. The velocity field of the circular flow is [κ × r] , (1.48) 2π r 2 where the vector κ is parallel to the  z axis. Its magnitude is the circulation of the velocity given by the linear integral κ = v v · dl over any closed path around the cylinder. In superfluids the circulation can only have special quantised values, whereas in classical fluids κ is arbitrary. In addition to the circular flow around the cylinder, there is a fluid current past the cylinder with constant velocity v ∞ (Fig. 1.2a). Then the net velocity field is vv =

v(r) = v v (r) + v ∞ .

(1.49)

Streamlines of this velocity field are shown in Fig. 1.2b. Here we ignore the backflow around the moving cylinder, which has no contribution to the transverse force. The cylinder moves with a constant velocity v L , and one should replace the position vector r in the coordinate frame moving together with the cylinder by r − v L t. Then the time derivative of the velocity is ∂v = −(v L · ∇)v. (1.50) ∂t In the coordinate frame moving with the cylinder, the velocity field does not vary in time, and the Euler equation (1.33) yields the Bernoulli law, which determines the pressure variation around the moving cylinder: P = P0 −

ρ[v(r) − v L ]2 ρvv (r)2 = P0 − − ρv v (r) · (v ∞ − v L ). 2 2

(1.51)

Here P0 and P0 = P0 − 12 ρ(v ∞ − v L )2 are constants, which are of no importance for the further derivation. Figure 1.2b shows that due to superposition of two fluid motions given by Eq. (1.49), the velocity above the cylinder is higher than that below the cylinder. According to the Bernoulli law, the pressure is higher in the area where the velocity is

1.4 Motion of a cylinder in perfect fluid: Magnus force (a)

(b)

11

high velocity, low pressure lift force

low velocity, high pressure

(c)

(d)

(e)

Figure 1.2 Magnus (lift) force. This is derived from the momentum balance inside an area restricted by the surface S. (a) Superposition of the circular flow with velocity v v and the flow with constant velocity v ∞ . (b) Streamlines of the net velocity field v v (r) + v ∞ . (c) The momentum balance is considered in the area restricted by the cylindrical surface. (d) The momentum balance is considered in the strip oriented normally to the velocity v 0 . (e) The momentum balance is considered in the strip oriented parallel to the velocity v 0 .

lower. As a result of this, a fluid produces a force on the cylinder normal to the relative velocity of the fluid with respect to the cylinder. This is part of the transverse force. The whole force is determined from the momentum conservation law, which requires that the momentum transferred to the fluid, i.e., to the external force F , is equal to the momentum flux through a distant surface S surrounding the cylinder in the coordinate frame moving with the cylinder velocity v L . The momentum flux tensor is given by Eq. (1.28). In the reference frame moving with the velocity v L of the cylinder, the velocity v should be replaced by the relative velocity v − v L : ij = P δij + ρ(vi − vLi )(vj − vLj ),

(1.52)

where the pressure P is determined from the Bernoulli law (1.51). The momentum flux through the surface surrounding the cylinder is given by the integral dSj ij , where dSj are the components of the vector dS directed along the outer normal to the surface and equal to the elementary area of the surface in magnitude. Only cross-terms ∼ v v · (v∞ − v L ) contribute to this integral. Let us choose S to be a cylindrical surface coaxial with the

12

Hydrodynamics of a one-component classical fluid

cylinder of radius R0 (Fig. 1.2c). Then the momentum balance yields the relation ρ[(v L − v) × κ] = F ,

(1.53)

where the fluid velocity v replaces the velocity v ∞ far from the cylinder. The left-hand side of the relation is the Magnus force as it is defined in classical hydrodynamics. The Magnus force balances the resultant force F applied to the cylinder. In the absence of such a force the cylinder moves with the transport velocity of the fluid: v L = v (Helmholtz’s theorem). The Magnus force is a lift force which follows from the more general Kutta–Joukowski theorem. It is the same force which keeps planes in the air. At steady motion of the cylinder, the total momentum flux through any surface surrounding the cylinder does not depend on its form or distance from the cylinder. Choosing a cylindrical surface S far from the cylinder, one can neglect the backflow velocity field, which decays as 1/r 2 . In the momentum flux through this surface, half of the flux was the Bernoulli contribution to the pressure, Eq. (1.51), while the other half was due to the convection term ∝ vi vj in the momentum flux tensor. However, because of the long-range circular velocity field v v , such decomposition of the resultant momentum flux into the Bernoulli and the convection parts is not universal and depends on the shape of the area for which we are considering the momentum balance. We can surround the cylinder with a strip, oriented normally to the transport flow (see Fig. 1.2d). This again yields Eq. (1.53), but now the pressure (Bernoulli) term does not contribute to the transverse force at all, and only convection is responsible for the Magnus force. The fluid enters the strip, which contains the cylinder, with one value of transverse velocity v v and exits from the strip with another value of transverse velocity. The transverse force is the total variation of the transverse momentum of the fluid per unit time resulting from the fluid flow crossing the strip. One can also consider the momentum balance in a strip oriented parallel to the fluid velocity v ∞ as shown in Fig. 1.2e. Then only the pressure (Bernoulli) term on the righthand side of Eq. (1.52) contributes to the total momentum flux through the boundaries of the strip. The Bernoulli term is dominant if one considers the momentum flux through the interface between the solid cylinder and the fluid, i.e., through the surface of the cylinder itself. There is no convection of the momentum through this surface simply because, in the coordinate frame moving with the velocity v L of the cylinder, the fluid velocity normal to the cylinder surface vanishes. But when calculating the total momentum flux one must take into account the backflow velocity field, which provides the absence of the fluid velocity normal to the cylinder surface. One can check that eventually one arrives at the same equation (1.53) of motion.

1.5 Cylinder with fluid circulation around it moving in a compressible fluid For future discussion of vortex dynamics in trapped BEC clouds of cold atoms, it is useful to analyse the motion of a cylinder with fluid circulation around it in a compressible fluid. One effect of compressibility is an additional contribution to the associated mass

1.5 Cylinder with fluid circulation around it

13

of the moving cylinder proportional to the velocity circulation κ around the cylinder. In a compressible fluid the pressure variation around the moving cylinder, which is determined by the Bernoulli law (1.51), produces the mass density variation:   ρ vv (r)2 − v v (r) · (v ∞ − v L ) , (1.54) δρ = − 2 2 cs where cs is the sound velocity and the thermodynamic relation ∂ρ/∂P = 1/cs2 was used. Assuming that the compressibility effect is weak, ρ is a constant mass density far from the cylinder. The cross-term in mass density variation linear in vv gives a contribution to the energy:   εc (v L − v)2 (v L − v)2 ∂ 2 E δρ 2 c2 δρ 2 = = = 2 , (1.55) d 2r 2 d 2r s μc 2 2ρ 2 ∂ρ 2 cs r>R0

r>R0

where the thermodynamic relation ∂ 2 E/∂ρ 2 = cs2 /ρ was used, and  ρκ 2 rm ρvv (r)2 = ln εc = d 2r 2 4π R0

(1.56)

r>R0

is the kinetic energy per unit length of the circular velocity field around the cylinder. The energy εc is logarithmically divergent and must be cut off by some appropriate scale rm , distance from solid walls, for example. The contribution (1.55) to the kinetic energy defines the compressibility associated mass εc (1.57) μc = 2 , cs which must be added to the associated mass μa related to the backflow [Eq. (1.44)]. Now let us put the fluid in a smoothly varying potential U (R). The potential determines a force −∇U (r) on any particle of the fluid and a force −n∇U (r) per unit volume occupied by the fluid with particle density n = ρ/m. For a potential flow (ω˜ = 0) the Euler equation (1.33) transforms to ∇P v2 1 ∂v =− − ∇ − ∇U (r). ∂t ρ 2 m

(1.58)

Now there is no translational invariance and no momentum conservation law. Instead, the balance of momentum takes into account the momentum transferred to the fluid by the external force: ∂ji ρ + ∇j ij = − ∇i U (r). (1.59) ∂t m In an incompressible fluid with constant ρ = mn, the source of the momentum on the right-hand side is a gradient ∇i [nU (r)], which can be included as an additional term to the momentum flux tensor ij . One can unite this additional term with the pressure P , introducing a renormalised pressure P˜ = P + nU (r). After this the external potential U (r)

14

Hydrodynamics of a one-component classical fluid

disappears from the equations and has no effect on the derived equation of motion (1.53) of the cylinder. But if the fluid is compressible and the particle density varies in space, the integral of the momentum balance equation for the fluid confined by the distant surface of radius R cannot be reduced to the surface integral only and contains also the bulk integral. The bulk integral is a force  δρ(r) ∇U dr, (1.60) F =− m R0 0. It is sufficient to consider a single bulk inertial mode which is a plane wave in the xy plane with the in-plane wave vector k. The boundary layer is the Ekman layer, where a superposition of three waves is generated: the inertial wave and the two viscous modes, all of them with the same k and ω. Taking into account the relations between velocity components found in Section 1.11, the three velocity components of the three-wave superposition for slow motion ω   are  K z   iK0 z iK z √ − √0 − √0 ipz 2 2 e 2 eik·r−iωt , + v− e vt = vI e + v+ e  K z   iK0 z iK z iω √ − √0 − √0 ipz 2 2 e 2 eik·r−iωt , vI e − i v+ e − v− e (1.155) vk = 2     iK0 z iK z K z k iω k √ − √0 − √0 (i + 1)v+ e 2 + (i − 1)v− e 2 e 2 eik·r−iωt . vI eipz + √ vz = − p 2 2K0 Here vI is the amplitude of the inertial wave, and v± are amplitudes of the two axial viscous modes. We assumed that viscous effects are weak and neglected viscosity corrections to relations between velocity components. At z = 0 all three velocity components must vanish. Excluding the two viscous mode amplitudes v± from these three boundary conditions and neglecting terms of the order of ω/  or k/K0 , one obtains the effective boundary condition imposed on the inertial wave amplitude: √ 2K0 ω vI + vI = 0. (1.156) p 2 The physical meaning of this condition becomes clear if one rewrites it using relations between velocity components in the inertial and the viscous modes: ±∞ ik vvk (z) dz = vz ± √ (v+ + v− ) = 0. vz ∓ ik 2K0

(1.157)

0

Here vz is the axial velocity component of the bulk inertia wave close to the solid surface z = 0 and vvk (z) is the longitudinal in-plane velocity component induced by the viscous

1.13 Inertial wave resonances in a viscous fluid

35

modes in the Ekman layer. This condition follows from mass conservation: the axial mass current into the Ekman layer is compensated for by the in-plane horizontal mass current supported by viscous modes inside the Ekman layer. The upper sign refers to the case of fluid above a solid surface (the container bottom) while the lower sign must be chosen for fluid below a solid surface (the container top). Equation (1.156) is a Fourier transform of the effective boundary condition in the configurational space. The inverse Fourier transformation of Eq. (1.156) yields the effective boundary condition in the configurational space: √ i 2K0 2 ∂vI (1.158) ± (v I − v B ) = 0, − ω ∂z k2 where v I is the in-plane velocity in the bulk inertial mode. We generalised the effective boundary condition for the case when the solid surface moves with in-plane velocity v B . Turning back to free oscillations (vB = 0), for even inertial wave resonances vt ∝ cos pz the effective boundary conditions (1.158) on both solid surfaces are satisfied at the condition √ i 2K0 tan(pL) + 1 = 0, (1.159) k while the condition for odd resonances vt ∝ sin pz is √ i 2K0 cot(pL) − 1 = 0. k

(1.160)

Solving these equations with respect to p yields imaginary corrections to the discrete values of p given by Eq. (1.152):  √  π i 2k pn = . (1.161) n− 2L π K0 This leads to a corresponding imaginary correction to the resonance frequencies and broadening of resonances. Our analysis is valid as long as the distance 2L between solid surfaces confining the fluid is much larger than the Ekman depth δE . This requires small values of the Ekman number Ek =

ν 2 = 2 . 2 L K0 L 2

(1.162)

Assuming that k is of the order of the inverse radius 1/R of the container, the imaginary corrections to frequencies of resonances with n ≥ 1 are relatively small if the parameter √ EkL/R is small. Apart from imaginary corrections to resonance frequencies, the viscosity adds a new dynamical mode. This corresponds to the value pn at n = 0. Neglecting the viscosity, p0 = 0 and the corresponding frequency vanishes, i.e., the value n = 0 does not correspond

36

Hydrodynamics of a one-component classical fluid

to any dynamical mode. In a viscous fluid the frequency for n = 0 does not vanish but is purely imaginary: √ k2 p0 i 2 2 − νk = − − i2 2 . ω0 = 2 k K0 L K0

(1.163)

This overdamped mode is the slowest mode in the rotating fluid between two horizontal solid surface. The fluid performs columnar motion without any nodes in the velocity field. The time 1/|ω0 | is the relaxation time after spin-up or spin-down of rotating solid surfaces confining the fluid layer. A similar effective boundary condition can be derived at vertical walls parallel to the rotation axis. In this case the width of the viscous boundary layer is determined not by the √ √ Ekman depth δE = ν/2 but by the viscous penetration depth δ = ν/ω, which can exceed the Ekman depth significantly at slow motion ω  .

1.14 Hydrodynamical modes of a rotating compressible perfect fluid In order to find the modes of a rotating compressible perfect fluid, let us refer to the system of equations including the mass continuity equation and the Euler equation for a rotating fluid. The linearised equations look like ∂ρ  + ρ∇ · v = 0, ∂t

(1.164)

∂v c2 + [2 × v] = − s ∇ρ  , ∂t ρ

(1.165)

or for a plane wave with frequency ω and wave vector K(k, p): −iωρ  + iρ(kvk + pvz ) = 0, −iωvz = −

ipcs2  ρ, ρ

−iωvk − 2vt = −

(1.166) (1.167)

cs2 ikρ  , ρ

−iωvt + 2vk = 0.

(1.168) (1.169)

Solving Eqs. (1.166) and (1.167) with respect to ρ  and vz , ρ =

ω2

ωk ρvk , − cs2 p2

vz =

ω2

cs2 pk vk , − cs2 p2

(1.170)

1.14 Hydrodynamical modes of a rotating compressible perfect fluid

37

and substituting these equations into the remaining two equations, we obtain two equations for vk and vt : −iω

ω2 − cs2 K 2 vk − 2vt = 0, ω2 − cs2 p2

(1.171)

−iωvt + 2vk = 0. These equations give the dispersion law ω2 = 42

ω2 − cs2 p2 . ω2 − cs2 K 2

(1.172)

In the limit of infinite sound velocity cs , Eq. (1.172) yields the spectrum of the inertial wave, Eq. (1.137), in an incompressible fluid. Solving Eq. (1.172) as an equation for ω2 one obtains the expression  1 1 2 2 2 2 (42 + cs2 K 2 )2 − (2cs p)2 . (1.173) ω = (4 + cs K ) ± 2 4 For a fluid at rest ( = 0) this gives two frequencies, ω = cs K and ω = 0, corresponding to the longitudinal sound and the zero-frequency transverse mode of a fluid at rest (see Section 1.6). In a rotating fluid both modes have finite frequency. This occurs due to the Coriolis force: rotation makes a fluid rigid in the direction normal to the rotation axis. At the same time the Coriolis force mixes the longitudinal and transverse modes: both have longitudinal and transverse components of the velocity field. When analysing the effect of rotation on the hydrodynamical modes, it is important that the natural frequency and wave vector scales for a rotating fluid are  and /cs respectively. Thus the expansion in K is an expansion in the dimensionless parameter cs K/, and the limit of an incompressible fluid cs → ∞ is approached at large wave vectors when cs K/ → ∞. Then one branch of the spectrum, Eq. (1.173), yields the inertial wave with the spectrum Eq. (1.137) and the other branch is a sound wave modified by rotation: k2 + cs2 K 2 . (1.174) K2 So due to rotation, a gap appears in the sound spectrum when a sound wave propagates at some angle to the rotation axis. In the inverse limit of small wave vectors cs K/  1, expansion of Eq. (1.173) yields two frequencies: ω2 = 42

ω2 = 42 + cs2 K 2 , ω2 = cs2 p2 .

(1.175)

We see that the model of the incompressible fluid is invalid when the wavelength is long enough (K → 0). This conclusion holds in a fluid with quantised vorticity as well. Therefore one should be careful with the term ‘hydrodynamics of an incompressible rotating

38

Hydrodynamics of a one-component classical fluid

fluid’. On the one hand, the hydrodynamical theory of an incompressible fluid should describe the long-wavelength behaviour of a fluid; on the other hand, it is just in the extreme long wavelength limit that the hydrodynamics of an incompressible fluid fails. One should also remember that at the distance cs /  from the rotation axis the fluid velocity approaches the sound velocity cs , and the centrifugal force should not be ignored. Then the fluid density ceases to be uniform, and the theory presented here should be modified. However, in classical fluids with high sound velocity it would be difficult to reach such rapid rotation (see further discussion in Section 3.11).

1.15 Gross–Pitaevskii theory The hydrodynamical equations were derived from the conservation laws and symmetry. For such a derivation, the coefficients in the equations remain undetermined. One can find them either from experiment or from the microscopic theory. Real classical fluids are strongly interacting systems, and an accurate quantitative microscopic analysis is not available for them. The situation is more favourable for quantum fluids, where the hydrodynamical equations of a perfect fluid can be derived for a weakly non-ideal Bose gas using the Gross– Pitaevskii theory (Gross, 1961; Pitaevskii, 1961). In this theory the ground state and weakly excited states of a Bose gas are described by the condensate wave function ψ = a exp(iθ ) which is a solution of the non-linear Schr¨odinger equation i h¯

h¯ 2 2 ∂ψ =− ∇ ψ + U (R)ψ + V |ψ|2 ψ. ∂t 2m

(1.176)

Here U (R) is an external single-particle potential (for example, the potential trapping the cold-atom BEC) and V is the amplitude of two-particle interaction. The derivation of the non-linear Schr¨odinger equation is straightforward (Pitaevskii and Stringari, 2003). The Hamiltonian of a Bose gas weakly interacting via δ-function potential in the second quantisation formalism is    V † h¯ 2 † † † (1.177) ∇ ψˆ · ∇ ψˆ + U (R)ψˆ ψˆ + ψˆ ψˆ ψˆ ψˆ . H = dR 2m 2 The Ansatz of the theory is that all particles are in the same quantum state, as one expects for the BEC, and the many-body wave function for N particles is N = N −N/2

N 

ψ(R i , t).

(1.178)

i

ˆ Following the idea of Bogolyubov (1947), one may replace the operators ψ(R, t) and † ˆ ψ (R, t) of annihilation and creation of particles by the classic fields ψ(R, t) and ψ ∗ (R, t). This yields the Hamiltonian for the Gross–Pitaevskii theory from which the non-linear Schr¨odinger equation (1.176) follows. This approach is justified if there is a large number

1.15 Gross–Pitaevskii theory

39

N of Bose particles in the same quantum state and non-commutivity of the field operators ˆ ψ(R, t) and ψˆ † (R, t) is not important. Later on in this section we omit the external potential. The non-linear Schr¨odinger equation is the Euler–Lagrange equation for the Lagrangian   h¯ 2 ∂ψ ∗ ∂ψ V i h¯ ψ∗ −ψ − |∇ψ|2 − |ψ|4 . (1.179) L= 2 ∂t ∂t 2m 2 The energy density (Hamiltonian) of the Gross–Pitaevskii theory is E=

h2 ∂L ∗ ∂L V ˙ − L = ¯ |∇ψ|2 + |ψ|4 . ψ˙ + ψ ∗ 2m 2 ∂ ψ˙ ∂ ψ˙

(1.180)

Noether’s theorem yields the mass conservation law, which leads to the mass continuity equation (1.18) and the momentum conservation law (1.19). In these equations the mass density is ρ = m|ψ|2 , the mass current is j =−

! i h¯ ∂L ∂L ψ ∗ ∇ψ − ψ∇ψ ∗ , ∇ψ ∗ = − ∇ψ − ∗ ˙ ˙ 2 ∂ψ ∂ψ

(1.181)

and the momentum flux tensor is given by ∂L ∂L ∇i ψ ∗ + Lδij ∇i ψ − ∂∇j ψ ∂∇j ψ ∗ ! h¯ 2 ∇i ψ∇j ψ ∗ + ∇i ψ ∗ ∇j ψ + δij P , = 2m

ij = −

(1.182)

with the pressure P equal to P =

V h¯ 2 2 2 |ψ|4 − ∇ |ψ| . 2 4m

(1.183)

An equivalent expression for the momentum flux tensor is   ! h¯ 2 V h2 ∇i ψ∇j ψ ∗ + ∇i ψ ∗ ∇j ψ + δij |ψ|4 − (2|∇ψ|2 + ψ∇ 2 ψ ∗ + ψ ∗ ∇ 2 ψ) . ij = 2m 2 4m (1.184) The Madelung transformation (Madelung, 1927) of the non-linear Schr¨odinger equation (1.176) helps to show the connection of the Gross–Pitaevskii theory with hydrodynamics. The transformation replaces the complex wave function ψ = a exp(iθ ) by two real functions, the fluid density ρ = ma 2 and the phase θ , the latter determining the fluid velocity v = (κ/2π )∇θ , where κ = h/m is the circulation quantum. After the Madelung transformation the energy density is E=

ρv 2 Vρ 2 h¯ 2 (∇ρ)2 + + , 2 8m2 ρ 2m2

(1.185)

40

Hydrodynamics of a one-component classical fluid

and the Schr¨odinger equation reduces to two hydrodynamical equations for a perfect fluid, one of them being the mass continuity equation (1.18), while the second one is the Josephson equation (Anderson, 1966) h¯ ∂θ = −μ. m ∂t

(1.186)

Here μ is the chemical potential in the laboratory coordinate frame (see Section 1.1). Taking the gradient of this equation one obtains the Euler equation (1.34) for the curlfree potential flow (ω˜ = 0). The chemical potential is determined by the thermodynamic expression generalised to the case when the energy depends not only on the fluid density but also on density gradients. This means that it is not a partial derivative but a functional derivative of the energy which defines the chemical potential:   ∂E ∂E h¯ 2 (∇ρ)2 v2 Vρ ∇ 2ρ δE = −∇· = + + 2. − (1.187) μ= δρ ∂ρ ∂∇ρ 2ρ 2 2m2 4ρ 2 m The pressure P in the expression (1.182) for the momentum flux given by Eq. (1.183) is connected with this chemical potential by the usual thermodynamic relation P = μρ − E =

h¯ 2 2 Vρ 2 − ∇ ρ. 2m2 4m2

(1.188)

Equations (1.18) and (1.186) are the Hamilton equations for the pair of canonically conjugated variables ‘phase–particle density’. They differ from the equations of hydrodynamics for a perfect fluid only by the presence of terms depending on the density gradients. Such a term in the expression Eq. (1.188) for the pressure is called the quantum pressure, because it brings the Planck constant into an otherwise purely classical hydrodynamical expression. The quantum pressure and the gradient terms in the energy density (1.185) are important only if thermodynamical parameters vary significantly at the scale of the coherence length (also called the healing length) h¯ . ξ0 = √ 2Vρ

(1.189)

Let us discuss the spectrum of plane waves in the Gross–Pitaevskii theory. In hydrodynamics the only mode in the perfect fluid is the sound mode ω = cs K (Section 1.6). In √ the Gross–Pitaevskii theory the sound velocity is cs = V n/m. But the sound spectrum is valid only in the long wavelength (low frequency) limit. In general, keeping all terms in Eq. (1.187) and linearising the mass continuity equation and the Euler equation with respect to small oscillation of the particle density n = n − n0 and the phase θ  , one obtains for the plane wave n ∝ θ  ∝ eiK·R−iωt : hK ¯ 2 n0 θ  , m h¯ K 2 n V n , + iωθ  = h¯ 4m n0 −iωn =

(1.190)

1.15 Gross–Pitaevskii theory

41

where n0 is the constant particle density in the ground state. This yields the spectrum of oscillations in a weakly non-ideal Bose gas h¯ K 2 ω = m 2



hK V n0 ¯ 2 + 4m h¯

 = cs2 K 2 +

h¯ 2 K 4 . 4m2

(1.191)

The dispersion relation shows how the sound-like spectrum at small wave vectors K transforms at large K to the spectrum of free particles with energy hω ¯ = h¯ 2 K 2 /2m. The crossover occurs at wave numbers K of the order of the inverse coherence length given by Eq. (1.189). The spectrum (1.191) was obtained in the seminal paper by Bogolyubov (1947) in the secondary quantisation formalism. Here it was obtained as the spectrum of plane waves in a classical field, namely the macroscopic wave function ψ (see Pitaevskii and Stringari, 2003). This clearly demonstrates the connection of the Gross–Pitaevskii theory with hydrodynamics. Later on we shall also use the grand canonical ensemble with the Hamiltonian defined via the grand thermodynamic potential with the density G = E − μm|ψ|2 =

h¯ 2 V ∇ψ ∗ · ∇ψ − μm|ψ|2 + U (R)|ψ|2 + |ψ|4 . 2m 2

(1.192)

In the grand canonical ensemble the non-linear Schr¨odinger equation becomes i h¯

h¯ 2 2 ∂ψ =− ∇ ψ − μmψ + U (R)ψ + V |ψ|2 ψ. ∂t 2m

(1.193)

In the rotating coordinate frame the grand thermodynamic potential transforms to   2  2π h¯ 2  i∇ + G = E − μm|ψ| = v 0 ψ  + U (R)|ψ|2  2m κ   v2 V + |ψ|4 − μ + 0 m|ψ|2 2 2 2       2π h¯ 2  V m2 r 2 4  i∇ + v 0 ψ  + |ψ| + U (R) − |ψ|2 − mμ|ψ|2 . = 2m  κ 2 2 (1.194) 2

Here E = E −  · M is the energy in the rotating coordinate frame determined by Eq. (1.131). If one ignores the trapping potential U (R) and the centrifugal energy −m2 r 2 /2, the grand potential is invariant with respect to the gauge transformation ψ → ψeiφ , v 0 → v 0 + (κ/2π )∇φ. The velocity of solid body rotation v 0 plays the role of gauge field, like the electromagnetic vector potential A in the Schr¨odinger equation for

42

Hydrodynamics of a one-component classical fluid

a charged particle. The Schr¨odinger equation (1.193) after transformation to the rotating coordinate frame becomes 2    h¯ 2 2π m2 r 2 ∂ψ = i∇ + v 0 ψ − mμψ + U (R) − ψ + V |ψ|2 ψ. (1.195) i h¯ ∂t 2m κ 2 The mass current in the rotating coordinate frame is       i h¯ 2π i 2π i ∗ j =− ψ ∇− v0 ψ − ψ ∇ + v0 ψ ∗ 2 κ κ ! i h¯ ψ ∗ ∇ψ − ψ∇ψ ∗ − m|ψ|2 v 0 . (1.196) 2 The Coriolis force invalidates the momentum conservation law even if the trapping potential and the centrifugal energy are neglected and translational invariance is restored. In the latter case there is the equation of momentum balance =−

∂ji + [2 × j ]i + ∇j ij = 0 ∂t with the momentum flux tensor      h¯ 2 2π i 2π i Re ∇i − v0i ψ ∇j − v0j ψ ∗ + δij P . ij = m κ κ

(1.197)

(1.198)

2 Dynamics of a single vortex line

2.1 Vortex line in a perfect fluid The concept of a vortex line was invented in classical hydrodynamics (Lamb, 1997) and has been intensively studied for centuries. One can define it as a line whose direction is parallel everywhere to the vorticity vector [∇ × v]. Now suppose that a bunch of such vortex lines forms a tube, and motion of the fluid is irrotational (curl-free) everywhere except for the space within the tube. This is known as a ‘vortex tube’, or a ‘vortex filament’, or simply a ‘vortex’. Circulation around a vortex filament is a measure of its ‘strength’:  κ = v · dl. (2.1) Here one integrates over any closed path around the filament. Suppose that at a given circulation the diameter of the vortex filament decreases and becomes much smaller than any relevant hydrodynamical scale (this may be the radius of curvature of the vortex filament or the distance from other filaments and solid surfaces). Such an infinitely thin vortex filament is also widely known as a ‘vortex line’. But in contrast to vortex lines in a fluid with continuously distributed vorticity, we are now dealing with a singular vortex line along which all vorticity is concentrated.1 In superfluids the circulation κ is quantised and must be a multiple of the circulation quantum h/m, but mostly we shall address single-quantum vortices with κ = h/m. In a multiply connected region around a vortex line the velocity field is curl-free, and if the fluid is incompressible it is also divergence-free: ∇ · v = 0,

∇ × v = 0.

(2.2)

We come to a standard classical field problem like that in electrostatics or magnetostatics. In complete analogy with the magnetic field around a filament with an electrical current, the velocity field v(R) around the vortex line is determined by the Biot–Savart law:   dR L × (R − R L ) κ(R L ) × (R − R L ) 1 κ = dl, (2.3) v(R) = 4π 4π |R − R L |3 |R − R L |3 L

L

1 In the literature on classical hydrodynamics they are also called concentrated vortices (Alekseenko et al., 2007).

43

44

Dynamics of a single vortex line

where the integral is taken over the whole vortex line, R is the three-dimensional position vector for the point where one determines the velocity, R L is the three-dimensional position vector for points on the vortex line, dl is the element of the vortex line length, and κ(R L ) = κ dR L /dl is a vector of magnitude equal to the circulation κ and tangent to the vortex line in the point with the position vector R L . The velocity v(R) = ∇×(R) can be determined via the vector potential   1 dR L κ(R L ) dl κ = . (2.4) (R) = 4π |R − R L | 4π |R − R L | L

L

Around a straight vortex line coinciding with the z axis, the velocity field v v is given by Eq. (1.48), which describes the circular flow around a cylinder. But now it is valid at any distance r from the axis. The velocity field satisfies the equation ∇ × v v = κδ(r),

(2.5)

where δ(r) is the two-dimensional δ-function in the xy plane transverse to the vortex axis. In classical hydrodynamics a singular vortex line was thought of as being an idealisation very distant from a real fluid, since viscosity leads to diffusion of vorticity, initially concentrated along singular lines, over the entire bulk of the fluid. But in superfluids vorticity is not compatible with the existence of a scalar complex order parameter; therefore the superfluid endeavours to confine the region of vorticity, making the vortex lines stable. Nevertheless, in superfluid hydrodynamics many properties of the vortex line can be discussed without referring to quantum theory except for the restrictive condition that circulation of the vortex line is quantised (Fig. 2.1). The energy of a vortex line is equal to the kinetic energy of the velocity field induced by it. For the velocity field around a straight vortex line given by Eq. (1.48), the energy per unit length is   ρκ 2 rm dr ρκ 2 rm v(r)2 = = ln . (2.6) ε = ρ dr 2 4π rc r 4π rc The vortex energy per unit length of the vortex line is responsible for the vortex line tension like that of an elastic string. The effect of the line tension force on vortex dynamics of a curved vortex line will be considered in Section 2.7. The upper cut-off rm of the logarithmically divergent integral depends on the particular hydrodynamical problem under consideration. It is the distance from the vortex line at which the velocity begins to decrease faster than 1/r. For example, for a vortex ring formed by a circularly curved vortex line (Section 2.8), the cut-off rm is of the order of the ring radius. In the case of the Kelvin wave discussed in Section 2.9, rm is of the order of the wavelength. A straight vortex in a cylinder of radius R is an example when rm can be found exactly. If the vortex is located at the axis of the cylinder, rm = R. For a vortex at distance u off the axis (Fig. 2.2) the velocity field must satisfy the boundary condition that there is no fluid velocity normal to the lateral wall. It is well known that this condition is satisfied

2.1 Vortex line in a perfect fluid

45

Figure 2.1 Vortex filament with quantised circulation and vortex core of radius rc (top). The radial distribution of the density ρ and the fluid velocity v, which diverges at r → 0 (bottom). Figure from Sonin and Krusius (1994).

by adding a velocity induced by an image vortex at distance u = R 2 /u from the axis to the velocity field induced by the vortex itself: v˜ v =

[κ × (r − u)] [κ × (r − u )] − , 2π|r − u|2 2π|r − u |2

(2.7)

where u and u = (R 2 /u2 )u are the position vectors for the vortex line and its image. Calculation of the kinetic energy for this velocity field inside the cylinder yields the energy per unit length of the off-centre vortex: ε˜ =

ρκ 2 R 2 − u2 ln . 4π Rrc

(2.8)

This expression is identical to Eq. (2.6) if rm = (R 2 − u2 )/R. The lower cut-off rc is a vortex core radius. The vortex core is a region around the vortex line where the hydrodynamics of an incompressible fluid fails and the fluid density is suppressed (see Fig. 2.1). This is expected when the magnitude of the velocity v = κ/2π r given by Eq. (1.48) exceeds the sound velocity cs , i.e., when the so-called Mach number M = v/cs exceeds unity. Then an approximate estimation of the core radius is rc ∼ κ/cs . Exact determination of rc requires an analysis of the vortex core structure beyond the

46

Dynamics of a single vortex line

Figure 2.2 Streamlines of the velocity field induced by an off-axis vortex in a cylinder of radius R. The vortex is at a distance u from the cylinder axis, while its image is at a distance u = R 2 /u from the axis.

hydrodynamical approach and the concept of infinitely thin vortex lines. In classical hydrodynamics a number of models were proposed for the vortex core, for example, a hollow (empty) core or a solid core with uniform distribution of the vorticity within it. In the Bose superfluid the core radius rc ∼ κ/cs is of the order of the coherence length ξ0 . In the Fermi superfluid in the weak-coupling limit (the BCS theory), the coherence length exceeds the scale κ/cs and just the coherence length is a proper estimation for the vortex core. In the Bose superfluid an accurate relation between rc and the coherence length ξ0 can be found from the Gross–Pitaevskii theory (Section 1.15), where ξ0 is given by Eq. (1.189). In the state with a vortex line along the z axis, the order parameter phase θ coincides with the azimuthal angle ϕ in the cylindrical system of coordinates (r, ϕ, z), and the non-linear Schr¨odinger equation (1.193) for the order parameter ψ = aeiθ , reduces to the equation for the stationary order parameter modulus a:     a 1 da a2 h¯ 2 d 2 a 2 − 2 − mcs a 1 − = 0, (2.9) + − 2m dr 2 r dr n r where n = a(∞)2 and a(∞) are the equilibrium particle density and the order parameter amplitude far from the vortex line. The expression μ = mcs2 connecting the chemical potential with the sound velocity cs in a fluid at rest was used. Ginzburg and Pitaevskii (1958) solved this equation numerically and calculated the vortex energy, Eq. (2.6). Their result corresponds to the vortex core radius equal to rc =

κ ξ0 = 0.077 . 1.46 cs

(2.10)

There are various definitions of the vortex core radius in the literature (Donnelly, 1991). Depending on the definition, some constants of order unity may be added to the logarithm

2.1 Vortex line in a perfect fluid

47

in the expression (2.6) for the energy per unit length. Our definition of the core radius rc is that a vortex with an empty core of radius rc and with strictly constant density ρ outside the core has exactly the same energy as the vortex with real core, whatever the structure of the core is. We call this an effective empty core. This definition assumes that all constants are incorporated in the radius rc . Since the vortex energy is positive, there are no vortices in the ground state of a resting fluid. But if a container with the fluid rotates with the angular velocity , the ground state corresponds to a minimum of the energy in the rotating coordinate frame given by Eq. (1.131). For the energy per unit length of the straight vortex the latter equation yields ε = ε − mz , where



R0

mz = 2π

vv (r)r 2 dr =

0

(2.11) ρκR02 2

(2.12)

is the angular momentum per unit length about the rotation axis z in the cylinder of radius R0 . Choosing the upper cut-off rm in Eq. (2.6) for ε equal to R0 , one obtains that the energy  becomes negative if the angular velocity  exceeds the critical value (Arkhipov, 1957) c1 =

κ R0 ln . 2 rc 2π R0

(2.13)

So at  > c1 vortices appear in the rotating cylinder, similarly to the appearance of vortices in type II superconductors at magnetic fields exceeding the first critical field Hc1 . A general expression for the energy of a curved vortex line can be obtained by partial integration of the volume integral for the kinetic energy    ρ v(R)2 ρ dR = E =ρ v(R) · [∇ × (R)] dR = [∇ × v(R)] · (R) dR. 2 2 2 (2.14) Since the vorticity ∇ × v(R) is non-zero only on the vortex line itself, this expression with the help of Eq. (2.4) transforms to a double integral over the vortex line:     dR L · dR L dR L dR L dd ρκ 2 ρκ 2 E= = · . (2.15)  8π |R L − R L | 8π d d |R L − R L | L L

L L

Equation (2.15) is an analogue of the expression for the magnetic energy of electrical currents flowing along thin filaments and for the electrostatic energy of thin charged filaments. The position and the shape of an arbitrary curved vortex line can be determined not only by the set of three-dimensional position vectors R L originating from some fixed origin, but also by two-dimensional displacements u(z) from some chosen axis z in the xy plane. In this presentation R L = zˆz + u(z) and Eq. (2.15) transforms to   du(z2 ) 1) 1 + du(z ρκ 2 dz · dz E= . (2.16) dz1 dz2 " 8π (z1 − z2 )2 + [u(z1 ) − u(z2 )]2

48

Dynamics of a single vortex line

Deriving Eq. (2.15) from Eq. (2.14) we integrate the kinetic energy over the whole space, excepting points occupied by the singular vortex line. The integral is divergent at R L → R L . Qualitatively it is clear that one must cut off the divergence at the scale of the order of rc . A more accurate treatment of the divergence is to abandon the concept of a singular infinitely thin line and to take into account the real structure of the vortex core for calculation of the energy. As long as the Biot–Savart law is valid, i.e., as long as the curvature radius is much longer than the core radius rc , the vortex line curvature does not affect the structure and the energy of the vortex derived for a straight vortex. Using the effective empty core radius rc defined above, one must exclude the core volume from the energy integral (2.14). Outside the core the vorticity ∇ × v(R) vanishes and integration by parts reduces the bulk integral for the energy to the surface integral over the surface Sc surrounding the core:  ρ [ × v(R)] · dS c . (2.17) E= 2 Sc

This approach was used by Van Vijfeijken et al. (1969) to calculate the vortex ring energy (see Section 2.8).

2.2 Linear and angular momenta of a vortex line We need to know the linear momentum connected with a vortex line. As in the case of a moving cylinder considered in Section 1.3, we define the momentum via an integral over the velocity field induced by the vortex line. In the presence of the vortex line the order parameter phase θ , which determines the velocity potential  = κθ/2π, is multivalued. In order to make the velocity potential (phase) single-valued one should introduce a cut at the surface SL attached to the vortex line. At the surface SL the phase θ jumps by 2π. Then the surface S, which confines the fluid and is present in the general expression (1.45) for the momentum, must include not only the surface S∞ restricting the fluid at infinity but also the surface SL of the cut. Applying the concept of the Kelvin impulse (Section 1.3), which requires that the integral over the distant surface S∞ is ignored, one obtains the general expression for the momentum of the closed vortex line:  (2.18) P = κρ dS, SL

where the direction of the vector element dS of the cut surface coincides with the direction of fluid flow through the cut. The surface integral is a vector with components equal to areas of projections of the surface SL on planes normal to the axes corresponding to those components. Strictly speaking one cannot apply this expression to a straight vortex line, which is not a closed one. We address first the case of a vortex dipole, i.e., two parallel straight vortices of opposite circulation, which form a closed loop connected at infinity. In this case the surface

2.2 Linear and angular momenta of a vortex line

49

SL is a strip restricted by two straight vortex lines, and the general expression (2.18) yields the following momentum per unit length of two lines: P = ρ[(u+ − u− ) × κ],

(2.19)

where u+ and u− are the position vectors in the xy plane of the vortices with positive +κ and negative −κ circulations respectively, and κ is the vector of magnitude κ directed along the axis z. On the basis of this expression one can formally introduce the linear momentum for the straight vortex as P = ρ[u × κ],

(2.20)

where u is the position vector of a single straight vortex in the xy plane. Of course, this expression leaves the momentum undefined since it depends on the point of origin chosen for the position vector (in analogy with an undefined dipole moment for a single electron). However, one can use the expression to calculate the variation of the momentum produced by displacements of the vortex. For a three-dimensional curved vortex line the surface integral for the momentum in Eq. (2.18) can be transformed to a linear integral over the vortex line (Saffman, 1995): P =

ρκ 2

 [R L (l) × dl].

(2.21)

L

The factor 12 , which is present in the expression (2.21) for the three-dimensional curved vortex line but is absent in the two-dimensional expression [Eq. (2.20)], originates from a different presentation of the surface area via a linear one-dimensional integral. This is demonstrated in Fig. 2.3, which illustrates the calculation of the area attached to a closed curved vortex line located at finite distance and to a vortex dipole which extends to infinite distance. Describing the position and the shape of the curved vortex line by two-dimensional displacements u(z) from the z axis as shown in Fig. 2.3 for a vortex dipole, the z component of the momentum contains the factor 12 , but the momentum component in the xy plane does not:     ∂u(z) ρκ u(z) × Pz = dz, P ⊥ = ρ [u(z) × κ] dz. (2.22) 2 ∂z z Later in the book we also need the general expression for the angular momentum of the velocity field induced by the vortex line:  M=ρ

[R × v(R)] dR =

ρκ 2π



 [R × dS].

[R × ∇θ (R)] dR = ρκ SL

(2.23)

50

Dynamics of a single vortex line (a)

(b)

Figure 2.3 Determination of the vortex line momentum via the area of the surface attached to the line. (a) Closed curved vortex line, which does not extend to infinite distance. An element of area 1 [R (l) × dl] contains the factor 1 . (b) Vortex dipole, which forms a closed line only at infinite 2 L 2 distance. Using Eq. (2.21) with the factor 12 present and the position vectors R L originating from a single point, in the line integral one should not ignore the contribution from horizontal short segments at infinity. This contribution is exactly equal to the contribution from vertical long segments. Alternatively, if one does not want to bother about what is going on at infinity, one can use Eq. (2.20) where the area element [u × dl] does not contain the factor 12 and is determined by the twodimensional position vectors u with origins on the line but not at a single point.

One can use an extension of the Stokes theorem reducing the surface integral to a linear integral along the vortex line (Saffman, 1995):     ρκ R2 × dS = − ∇ R 2 dl. (2.24) M = ρκ 2 2 SL

L

As in the case of the linear momentum, this expression cannot define the angular momentum of an unclosed vortex line because of the arbitrary choice of the origin for the threedimensional position vector R. But the expression does define variation of the angular momentum due to a shift of the vortex line. One can illustrate this by considering a simple example of an originally straight vortex line deformed and shifted off an axis z. We choose the origin of the position vectors R in Eq. (2.24) at the point where z = 0 and r = 0. Then R changes from R 1 = zˆz to R 2 = zˆz + u(z), where u is the z dependent two-dimensional displacement vector in the xy plane. The z component of the angular momentum of the

2.3 Motion of a vortex: Magnus force

51

vortex line is determined as the difference between angular momenta before and after deformation and shifting and is equal to    ρκ ρκ ρκ 2 2 (2.25) R2 dz + R1 dz = − u(z)2 dz. Mz = − 2 2 2 2.3 Motion of a vortex: Magnus force Deriving the Magnus force on a cylinder with circular flow around it in Section 1.4, we used the hydrodynamical equations only at large distance from the vortex line, and the radius R0 of the cylinder did not appear in the final result at all. Therefore Eq. (1.53) is valid even for a singular vortex line when the circular flow with circulation κ occurs without any cylinder at all. The fact that hydrodynamics is invalid inside the vortex core does not invalidate the derivation of the Magnus force for a Galilean invariant fluid. It is sufficient that the momentum is a well defined conserved quantity even inside the vortex core. An example of the force F in the Magnus relation Eq. (1.53) is the electrical force on ions captured by the vortex core. When the force F is absent, the vortex moves with the velocity v of the fluid current past the vortex. This is the content of Helmholtz’s theorem, which states that vortex lines in a perfect fluid move with the fluid, or are ‘frozen’ into the fluid. But the fluid velocity should be properly defined: the self-action, i.e., the velocity ∼ 1/r induced by the vortex itself, must be excluded. In the theory of superconductivity the term ‘Magnus force’ usually refers to the term proportional to v L and not to the relative velocity v L − v (Kopnin, 2001). We shall use this nomenclature later in the book and write the equation of motion leaving only the term ∝ v L on the left-hand side: −ρ[κ × v L ] = F˜ .

(2.26)

The force F˜ = F + F L on the right-hand side includes the force F L = −ρκ × v,

(2.27)

which in the theory of superconductivity is called the Lorentz force. In accordance with the definition of force in mechanics, the force F˜ is determined by a derivative of the energy ε˜ of the vortex with respect to its displacement. Here u is the two-dimensional position vector for a vortex line, and ε˜ = ε(u) + P · v

(2.28)

is the vortex energy per unit length in the laboratory frame, keeping in mind that the line tension ε was determined in Eq. (2.6) for the fluid at rest. The vortex momentum P is determined by Eq. (2.20) and depends linearly on u. The force F = −∂ε/∂u on the vortex appears if the energy ε(u) depends on the position vector u of the vortex. On the other hand, the derivative of the Doppler term P ·v with respect to the displacement is the Lorentz force.

52

Dynamics of a single vortex line

In the basic equation of vortex motion (2.26) we encounter a noteworthy feature of vortex dynamics: the resultant of forces on the vortex is balanced not by the absent inertia force, proportional to an acceleration (as in Newton’s second law), but by the gyrotropic Magnus force, proportional to a velocity. Any force acting upon the vortex can be described by its contribution to the net vortex velocity, and vice versa, any contribution to the vortex velocity can be presented as some force acting upon the vortex. Such a ‘force versus velocity’ relation (instead of the ‘force versus acceleration’ relation in Newton’s second law) is widely exploited in vortex dynamics. Despite all this, the vector equation of motion (2.26) has a canonical form and presents the Hamilton equations for the pair of canonical variables ‘coordinate–momentum’. If one chooses one of the two components ux of the position vector u(ux , uy ) as a coordinate, then according to Eq. (2.20) the canonically conjugated momentum Px = ρκuy is proportional to the other component uy . Eventually one can reduce the equation of motion (2.26) to the standard Hamilton equations: ∂ ε˜ dux = , dt ∂Px

∂ ε˜ dPx =− . dt ∂ux

(2.29)

Since the force on the vortex fixes its velocity, the latter cannot be an independent variable determined by initial conditions as in the case of a particle. So the vortex has no kinetic energy. A particle in the two-dimensional space has twice the degrees of freedom of a straight vortex performing two-dimensional motion. As a result, the dynamical properties of the vortex and of the particle are essentially different. Suppose both are located in a twodimensional well. The particle in such a well would have two linearly polarised oscillation modes, but the vortex would have one elliptically polarised mode reducing to a circularly polarised mode when the well is axisymmetric. We shall follow the crossover from the twomode dynamics governed by the inertia force to the one-mode dynamics governed by the Magnus force in the following section.

2.4 Experimental detection of quantum circulation: vortex mass versus Magnus force When we derived the Magnus relation we considered a vortex in steady motion at a constant velocity v L . If the velocity v L varies in time, the inertia force of the fluid inside the balance region can contribute to the momentum balance. Taking into account both the inertia and the Magnus forces discussed above, the equation of vortex motion is μv

dv L − ρ[κ × v L ] = F˜ , dt

(2.30)

where F˜ is the total force on the vortex including the Lorentz force. This equation is analogous to the equation of motion of a charged particle in a magnetic (the term ∝ κ) and an electric (the total force F˜ ) field. We shall see (Section 2.5) that within the framework of hydrodynamics the effect of the vortex mass is normally very weak, especially in superfluid 4 He, where the core radius rc

2.4 Experimental detection of quantum circulation

53

Figure 2.4 Vinen’s experiment for the detection of quantum circulation in superfluid 4 He.

does not exceed a few angstroms. But an important exclusion from this rule is a vortex line trapped by a flexible wire. This case was realised in the famous experiment by Vinen (1961a), which provided the first experimental confirmation of quantisation of circulation of a single vortex line. The experimental setup is shown schematically in Fig. 2.4. Superfluid 4 He fills a cylindrical container with a coaxial thin wire along the container axis. The device is placed in a magnetic field, and an oscillating electric current through the wire can excite transverse oscillations of the wire. The fluid is cooled down past the critical temperature while the apparatus is rotated around its axis. Rotation of the fluid must lead to a circular flow around the wire with quantised velocity circulation, which must affect the dynamics of the wire oscillations. The flow can remain even when rotation of the container stops. The wire together with the fluid can be considered as a complex vortex line (the vortex line trapped by the wire) with the wire playing the role of a vortex core. The radius of the core is now the radius of the wire rw , which even for a thin wire is larger by many orders than the microscopically small core radius rc of a free vortex. The mass μv = π rw2 (ρw + ρ) of the solid core includes the mass of the wire itself with density ρw and the associated mass of the fluid dragged by the wire (the term ∝ ρ). This mass is much larger than the vortex mass without the wire since rw rc . So in Eq. (2.30) both the Magnus force proportional to the velocity v L and the inertia force proportional to the acceleration dv L /∂t are important. The force F˜ = −Ku on the wire is an elastic force restoring the original axial location of the wire, where K is the elastic constant and u is the two-dimensional vector of displacement in the middle of the wire (see Fig. 2.4), and v L = du/dt. Then one can rewrite the vector equation (2.30) as two equations for Cartesian variables ux and uy for a monochromatic oscillation mode u ∝ e−iωt : −ω2 μv ux − iωρκuy = −Kux , −ω2 μv uy + iωρκux = −Kuy .

(2.31)

54

Dynamics of a single vortex line

The dispersion relation for oscillation of the wire is (ω2 − ω02 )2 −

ρ2κ 2 2 ω = 0, μ2v

(2.32)

√ where ω0 = K/μv is the oscillation frequency of the wire without the Magnus force. In Vinen’s experiment the Magnus force was weaker than the inertia force. Then the dispersion relation yields two frequencies: ρκ . (2.33) ω = ω0 ± 2μv In the absence of the Magnus force the wire has two degenerate modes with frequency ω0 , which can be chosen as either two linearly or two circularly polarised modes. The weak Magnus force lifts the degeneracy and leads to two circularly polarised modes with two close but different frequencies. The presence of two close modes means that free oscillations of the wire are accompanied by beats with frequencies ω = ρκ/μv . Measurements of the beat frequency yielded values of velocity circulation. The histogram in Fig. 2.5 shows that most of Vinen’s measurements gave circulation values close to the circulation quantum. Vinen’s experiment on circulation measurement was later repeated and developed for superfluid 4 He (see Karn et al., 1980, and references therein) and superfluid 3 He-B (Davis et al., 1991). Although the quantised value of circulation was measured most frequently, nonquantised values were also observed. Vinen (1961a) interpreted the non-quantised values as arising from vortices which were not completely trapped by the wire. They detached (unzipped) from the wire at some height and reattached to the cylinder wall.

Figure 2.5 Histogram showing the number of measurements of various values of circulation (Vinen, 1961a). The horizontal axis shows the ratio of circulation to the circulation quantum.

2.5 Vortex mass in Bose superfluids

55

The phenomenon was a nuisance, impairing the accuracy of the circulation quantum measurement. But later Zieve et al. (1992) demonstrated that the phenomenon has its own value, since it leads to observable precession of a single vortex line, which manifests important features of vortex dynamics. The single vortex line precession will be addressed later in this chapter, in Section 2.12. The equations (2.31) describing oscillations of the wire with quantum circulation around it illustrate the crossover from dynamics governed by the inertia force to dynamics governed by the Magnus force, which was formulated in Section 2.3. If the vortex mass μv decreases, the ratio ρκ/μv becomes very large, and the dispersion relation (2.32) yields two frequencies: ω1 =

K , ρκ

ω2 =

ρκ . μv

(2.34)

At μv → 0 the frequency of the second mode grows to infinity and cannot be treated within the hydrodynamical approach. Only a single circularly polarised mode remains, which is governed by the Magnus force.

2.5 Vortex mass in Bose superfluids Without an artificial solid core, as in Vinen’s experiment, a free vortex line normally has a rather small vortex mass. A naive estimation for the vortex mass is to deduce it from the picture of a cylinder without a mass of its own moving through a perfect fluid, assuming that the cylinder has a radius equal to the core radius rc . Then the vortex mass is simply equal to the associated mass of the cylinder of radius R0 = rc found in Section 1.3. This yields the vortex mass μv equal to μcore = π rc2 ρ, which is a mass per unit length of the fluid inside a cylinder of radius equal to the core radius rc (Baym and Chandler, 1983). Later we shall call μcore the core mass (in contrast to the more general term vortex mass which takes into account all possible contributions to the mass of the vortex). This estimation of the vortex mass assumed that the moving core is impenetrable for the fluid, as a real rigid cylinder. In reality the vortex core is not empty and the superfluid will flow through the core, thus producing a reduced backflow field (Sonin et al., 1998). So our simple calculation only provides an upper bound on the vortex mass related to the core. One can perform a more realistic calculation of the vortex core mass using the Gross–Pitaevskii theory. But in the Bose superfluid there is a more important contribution to the vortex mass, which is connected with finite compressibility of the fluid. The compressibility associated mass μc of the cylinder with velocity circulation κ around it was calculated in Section 1.5 and is given by Eq. (1.57). The derivation of the compressibility vortex mass is similar. This yields the compressibility vortex mass (Duan and Leggett, 1992; Duan, 1994) μc =

ε ρκ 2 rm = ln , 2 rc cs 4π cs2

(2.35)

56

Dynamics of a single vortex line

which differs from the compressibility associated mass of the cylinder by another value of lower cut-off of the logarithmic divergence: the core radius rc replaces the cylinder radius R0 . In the Bose superfluid, according to the Gross–Pitaevskii theory, the core radius rc ∼ κ/cs is also determined by the sound velocity cs . As a consequence, the compressibility mass is larger by the logarithmic factor than the core mass μcore = π rc2 ρ ≈ ρκ 2 /cs2 . In Fermi superfluids the problem of vortex mass is different from that in Bose superfluids. The vortex mass in Fermi superfluids will be discussed in Section 9.8. Despite this difference, eventually a crude estimation of the vortex mass as the core mass π rc2 ρ, ignoring a possible logarithmic factor, also remains valid for Fermi superfluids. The inertia force ∼ μv dvL /dt ∼ ωμv vL with such a mass can compete with the Magnus force ∼ ρκvL only at very high frequencies ω ∼ ρκ/μv ∼ κ/rc2 . At such frequencies the hydrodynamical framework, on which our derivation of the equation of motion rests, fails. In conclusion, except for the vortex with heavy solid core in Vinen’s experiment, usually the vortex mass does not play an essential role in vortex dynamics, and in most of the book we assume that the Magnus relation (2.26) without the inertia force is exact enough.

2.6 Precession of a straight vortex around an extremum of vortex energy Let us consider a simple example of vortex dynamics: a straight vortex close to an axisymmetric extremum of the vortex energy ε˜ as a function of the position vector u. The force F˜ = −∂ ε˜ /∂u on the vortex only has a radial component. At small displacements u the equation of motion (2.26) has a solution which corresponds to a circularly polarised mode u ∝ e−iωt with the frequency ω=

1 d 2 ε˜ . ρκ du2

(2.36)

We chose a positive sign of frequency for the energy minimum, i.e., when the energy of the mode is positive. For the mode with positive frequency the vortex precesses around the extremum centre in the direction opposite to circulation of the fluid around the vortex. In the coordinate frame rotating with the angular velocity , the force F˜ is determined by a derivative of the energy in the rotating frame ∂(˜ε − mz ) , F˜ = − ∂u

(2.37)

where mz = −ρκu2 /2 is the variation of the angular momentum per unit length of the straight vortex after its shift to the distance u from the extremum [see Eq. (2.25)]. Then the frequency of vortex precession around the energy extremum becomes ω =+

1 d 2 ε˜ . ρκ du2

(2.38)

Of course, the shift  of the precession frequency is a self-evident consequence of transformation to the rotating coordinate frame: if the energy ε˜ does not depend on u, the vortex

2.7 Dynamics of a curved vortex line: Biot–Savart law

57

is at rest in the laboratory coordinate frame, but rotates with the angular velocity  in the rotating frame. We presented a simple formal derivation of Eq. (2.38) from the Hamilton equations only for demonstration of consistency of the analysis. In an incompressible fluid filling a cylinder of radius R, dependence of the energy on displacement u off the cylinder axis is given by Eq. (2.8). Expanding Eq. (2.8) in u2 one obtains the frequency of vortex precession: ω =−

κ . 4π R 2

(2.39)

A negative sign of the precession frequency means that the position of the vortex at the cylinder axis is unstable without rotation. The axial position of the vortex becomes metastable at the rotation angular velocity m = κ/4π R 2 . This angular velocity is less by a large logarithmic factor than the first critical angular velocity c1 [see Eq. (2.13)] at which the state with the axial vortex becomes the ground state of the fluid. If a compressible fluid is in a potential U (r) there is a force on the vortex similar to the force on a cylinder with velocity circulation around it, which was defined by Eq. (1.61). The extremum of U (u) is an extremum of the vortex energy, and according to Eq. (2.38) the vortex precesses around the extremum with the frequency ω =+

κ rm ∂ 2 U 1 μc ∂ 2 U =  + ln . ρκ m ∂u2 rc ∂u2 4π mcs2

(2.40)

This expression will be used for vortex precession in a pancake shaped BEC cloud in a symmetric trap in Section 4.5.

2.7 Dynamics of a curved vortex line: Biot–Savart law and local induction approximation The equation of motion (2.26) with the Magnus force, which was derived for a straight vortex, can be extended to cases when the vortex line is not straight. Without an external force F , in accordance with Helmholtz’s theorem, the velocity v L (R L ) = dR L /dt of the point R L on the line is equal to the fluid velocity induced by the whole vortex line in this point. This velocity will be called the local velocity. It is determined from the Biot–Savart law (2.3) in the limit R → R L . In this limit the integral over the vortex line diverges, and the Neumann series near R L starts from the singular term ∼ 1/|R − R L |. One should ignore this term since it presents non-physical self-action. However, it does not remove all singularities in the local velocity: there remains the logarithmic singularity ∼ ln |R − R L | (Batchelor, 1970). The singularity is cut off at distances of order rc from the point R L for which the velocity v L (R L ) is determined (see Figure 2.6). Finally one has the closed integral equations to determine motion of the vortex line:  κ  dR L × (R L − R L ) dR L = v L (R L ) = , (2.41) dt 4π |R L − R L |3 L

58

Dynamics of a single vortex line (a)

(b)

Figure 2.6 Curved vortex line. (a) The position vectors R and R L used in Eqs. (2.3) and (2.41) together with the curvature radius R are shown. (b) It is outlined how the line tension force in Eq. (2.47) is obtained due to variation of the line tension vector ε as a function of the arc length dl. The magnitude of the line tension is constant, ε(R L + dR L ) = ε(R L ), while the orientation variation results in a line tension force dε, which is directed towards the centre of curvature. Its magnitude is obtained from the congruent triangles: dε/dl = ε/R. Then the force per unit length, dε/dl, in the limit dl → 0 corresponds to the force on the right-hand side of Eq. (2.47). Figure from Sonin and Krusius (1994).

 where means regularisation of the integral as explained above. The right-hand side of this equation is the local velocity. One can rewrite Eq. (2.41) in the following form: −ρκ(R L ) ×

δE dR L =− , dt δR L

(2.42)

where the energy E is given by Eq. (2.15). The functional derivative δE/δR L is a force produced by all other elements of the same curved vortex line upon a unit length of the element dl. Calculating the functional derivative one takes into account that the energy depends not only on R L but also on its derivative dR L /d. Then      κ(R L ) · κ(R L ) δE ρ  κ(R L ) · (R L − R L ) = dl  −(R L − R L ) + κ(R ) L δR L 4π L |R L − R L |3 |R L − R L |3   κ(R L ) × [κ(R L ) × (R L − R L )]  ρ = dl . 4π L |R L − R L |3 (2.43) Equation (2.42) is a generalisation of the equation of motion (2.26) [or Eq. (2.29)] for a straight vortex line to curved vortex lines. It has a canonical form of the Hamilton equations (Fetter, 1967; Sonin, 1987). Using two-dimensional displacement vectors u(z) to describe

2.7 Dynamics of a curved vortex line: Biot–Savart law

dynamics of the vortex line, the equation of vortex line motion is   δE du(z) =− , −ρκ zˆ × dt δu(z)

59

(2.44)

where the energy E is given by Eq. (2.16). The velocity of any element of the vortex line has only two components normal to the vortex line. As in the case of the straight vortex line, one may choose one of them as a canonical coordinate. Then the other one will be a canonically conjugate momentum. Both will be governed by the Hamilton equations for the pair of canonical variables ‘coordinate–momentum’. The canonical equations of vortex line motion were the basis for the canonical formalism developed by Kozik and Svistunov (2005, 2009) for studying vortex-phonon interaction (see also Svistunov et al., 2015). In general, the canonical equations of motion describe motion of the continuum of the canonical pairs, each for some element of the vortex line. But if the shape of the vortex line is known and does not vary during dynamical evolution, one can use a lower number of canonical pairs, or even a single pair, which become collective variables describing the dynamical process. This contracted description will be used later in the book. The equation of motion of the vortex line in the form of Eq. (2.41) or Eq. (2.42) is an integral equation with non-local contributions from distant parts of the line being essential. One can approximate it by an equation which contains only local parameters of the line in the vicinity of the given vortex line element. At distances which are short compared to the radius of curvature of the vortex line (but still long with respect to the core radius), one can treat any segment of the vortex line as straight and use Eq. (2.6) for the energy per unit length (line tension) ε of a straight vortex line. Describing the vortex line by twodimensional z dependent displacements u(z) from the z axis in the xy plane and assuming that the line tension ε does not vary along the vortex line, the energy of the vortex line is #     du(z) 2 1+ dz, (2.45) E = ε dl = ε dz  i.e., the energy of the vortex line is proportional to its length L = dl. Then the force on an element of the vortex line on the right-hand side of Eq. (2.42) is equal to F =−

R δE = 2 ε. δR L R

(2.46)

This transforms Eq. (2.42) to R dR L = 2 ε. (2.47) dt R Here R is the vector directed along the curvature radius and equal to it in magnitude. The upper cut-off rm in the line tension ε given by Eq. (2.6) must be of the order of the curvature radius R. The curvature radius R varies along the vortex line. So strictly speaking the line tension ε varies also as a result. But within logarithmic accuracy of the local induction approximation (see below), one may choose in the logarithm argument some constant value of R typical for the problem under consideration. −ρκ(R L ) ×

60

Dynamics of a single vortex line

Differential geometry tells that the inverse curvature radius is given by R d sˆ = = (ˆs · ∇)ˆs , 2 dl R

(2.48)

where sˆ = κ/κ is the unit vector parallel to the circulation vector κ tangent to the vortex line. This yields another form of the equation of motion of a curved vortex line in the local induction approximation d sˆ , dl

(2.49)

κ rm ε = ln ρκ 4π rc

(2.50)

[ˆs × v L ] = −νs where νs =

is the line tension parameter. In terms of displacements u(z)  2   d u d sˆ 1 du d 2 u . = · − zˆ dl dz dz2 [1 + (du/dz)2 ]2 dz2

(2.51)

So the integral equation of the vortex dynamics (Biot–Savart law) has been transformed to a differential equation, which makes the analysis much easier. In classical hydrodynamics this approach was called local induction approximation (Arms and Hama, 1965). In superfluid hydrodynamics the concept of the elastic vortex line tension was first used in the works of Hall (1960), Vinen (1961b), Andronikashvili et al. (1961) and Bekarevich and Khalatnikov (1961). The line tension force tries to keep the vortex line straight, restoring its minimum length. The local induction approximation has a logarithmic accuracy: it is valid if the logarithm of the ratio of the curvature radius R to the vortex core radius rc is large everywhere. It is worth mentioning that a large ratio R/rc is also required for the Biot–Savart approach to be accurate. The difference is that in the latter case the relative error is rc /R while in the local induction approximation the relative error is 1/ ln(R/rc ). A remarkable feature of the equation of motion of the vortex line in the local induction approximation is that the equation is integrable. Hasimoto (1972) showed that the equation of motion can be mapped onto the non-linear one-dimensional Schr¨odinger equation, which has an exact solution. This allowed exact solutions for various non-linear perturbations of the vortex line like solitons (Hasimoto, 1972), or breathers (Salman, 2013). On the other hand, a payoff for integrability of the local induction approximation was that one cannot use it for the description of non-trivial relaxation processes, in particular those responsible for the Kelvin wave cascade in superfluid turbulence, which is discussed in Section 14.4

2.8 Vortex ring The scheme of vortex dynamics based on the Biot–Savart law for very thin vortex lines, as described above, can be used for analytical calculations only for a restricted number

2.8 Vortex ring

61

of simple cases. One of these is the vortex ring. But we start from the local induction approximation. The radius R of the vortex ring is a constant curvature radius, and according to Eqs. (2.48) and (2.49) in the local induction approximation, all points of the vortex ring move with the same magnitude of velocity vL =

νs R

(2.52)

directed normally to the plane of the ring. The upper cut-off rm in the expression (2.50) for the line tension parameter νs must be of the order of R. The expression (2.52) also follows from the canonical relation vL =

∂E ∂E/∂R = , ∂P ∂P /∂R

(2.53)

which takes into account that the coordinate along the ring axis and the linear momentum along the same axis constitute a pair of canonically conjugate Hamiltonian variables. Deriving Eq. (2.52) from this relation, one should bear in mind that in the local induction approximation the energy of the vortex ring is E = 2πRε, while its momentum according to Eq. (2.18) is proportional to the ring area S = 2πR2 : P = π κρR2 .

(2.54)

Let us turn to a more accurate analysis using the Biot–Savart law. In the cylindrical system of coordinates (r, ϕ, z) with the polar axis z coinciding with the axis of the ring, the vector potential  of the velocity field given by Eq. (2.4) has only an azimuthal component independent of the angle ϕ:  cos φ dφ κR π ϕ = " 2 2 2π 0 R + r − 2rR cos φ + z2     k2 κ R 1− K(k) − E(k) . (2.55) = πk r 2 Here  K(k) = 0

π/2

"

dα 1 − k 2 sin2 α

 ,

E(k) =

π/2

" dα 1 − k 2 sin2 α

(2.56)

0

are complete elliptic integrals of the first and the second kinds (Gradstein and Ryzhik, 1965) with the argument # Rr . (2.57) k=2 (R + r)2 + z2 To determine the energy of the vortex ring we shall use the expression (2.17) for the vortex line energy (Van Vijfeijken et al., 1969; Putterman, 1974). This requires only values of the vector potential at the surface Sc restricting the core. At this surface, r = R+rc cos φ

62

Dynamics of a single vortex line

and z = rc sin φ. Keeping in mind that rc  R and introducing the angle φ˜ which is much larger than rc /R but still very small compared to unity, Eq. (2.55) yields:  cos φ dφ κR π ϕ ≈ " 2π 0 2R2 (1 − cos φ) + rc2  ˜   π φ dφ cos φ dφ κ ≈ + " √ 2π 2(1 − cos φ) φ 2 + rc2 /R2 φ˜ 0   κ 8R = ln −2 . (2.58) 2π rc The only component of fluid velocity at the surface Sc is tangent to the surface, normal to the vortex line, and is equal to κ/2π rc . Using the calculated values of the velocity and the vector potential on the core surface Sc with the area 4π 2 Rrc , from Eq. (2.17) one obtains the energy of the vortex ring:   8R ρκ 2 R ln −2 . (2.59) E= 2 rc After replacing rc by the coherence length ξ0 with the help of Eq. (2.10) this expression coincides with Eq. (5.36) of Pitaevskii and Stringari (2003). There was some disagreement concerning the constant −2 in Eq. (2.59) (Saffman, 1995). For example, one can find − 74 instead of −2, which follows from the analogy with the calculation of the self-energy of the current loop done, for example, by Landau and Lifshitz (1984). According to Van Vijfeijken et al. (1969) the disagreement arises from confusion in the definition of the vortex core. Landau and Lifshitz (1984) obtained − 74 taking into account the energy stored inside the wire forming the loop. Neglecting this energy, their calculation also yields −2. The energy inside the wire corresponds to the kinetic energy inside the core. Our definition of the core radius rc as an effective empty core size leads to the constant −2. The calculated energy can be presented as E = 2πRε with the expression (2.6) for the line tension ε, in which the upper cut-off is given by rm = 8e−2 R = 1.08R.

(2.60)

Knowing the energy one can determine the velocity of the ring with the help of the canonical relation (2.53)   1 dE κ 8R dE = = ln −1 . (2.61) vL = dP 2πρκR dR 4πR rc This expression demonstrates a remarkable feature of the vortex ring dynamics: the vortex ring velocity does not increase but decrease, with growing energy. This feature was confirmed in the very first experiments on quantised vortex rings by Rayfield and Reif (1964). They studied transport of ions, which produce vortex rings and then become trapped

2.9 Kelvin waves on an isolated vortex line

63

by vortex cores of the rings. One can find a discussion of these experiments and further references in the book by Donnelly (1991). Vortex rings play an important role in the study of superfluid turbulence. We shall return to the dynamics of deformed vortex rings later, in Section 2.11.

2.9 Kelvin waves on an isolated vortex line Circularly polarised waves propagating on an isolated vortex line were investigated by Lord Kelvin more than a hundred years ago (Thompson, 1880). Let us start from the local induction approximation. Linearising the equations given in Section 2.7 with respect to small displacements u(z), we obtain the canonical Hamilton equation governing small oscillations of the vortex line around the axis z:   d 2 u(z) du(z) =ε . (2.62) −ρκ zˆ × dt dz2 This equation reminds us of the equation describing linear oscillations of an elastic string. However, the force of line tension F lt on the right-hand side of Eq. (2.62) is balanced on the left-hand side not by the inertia force proportional to acceleration but by the Magnus force F M proportional to velocity (see Fig. 2.7). Equation (2.62) has plane-wave solutions ∝ exp(ipz − iωt) with the dispersion law ω = ±νs p2 ,

(2.63)

where the line tension parameter νs is determined by Eq. (2.50). The upper cut-off rm in the logarithm argument in this equation must be the distance which is penetrated by perturbations produced by oscillations. For a single vortex line in an unbound fluid, such a distance is expected to be of order of the wavelength 2π/p. This means that our approach based on the differential equation (2.62) is not quite rigorous, since the coefficient of this equation depends on the wave number p. But within the logarithmic accuracy of the local induction approximation, one may ignore this complication, estimating the wave number p in the logarithm argument by an order of magnitude. It is instructive to develop a more rigorous approach based on the Biot–Savart law, which is possible in the regime of weak oscillations. A general solution of the linear problem is a superposition of normal-mode solutions, each being a propagating plane wave (the Fourier expansion):  u(p, t)eipz dp. (2.64) u(z, t) = p

Expanding Eq. (2.16) in u(z, t), the energy of the vortex line is  du(z ) du(z )    1 2 2 · 1 |u(z ) − u(z )| ρκ 2 1 2 dz dz − E ≈ E0 + . dz1 dz2 4π |z1 − z2 | 2 |z1 − z2 |2

(2.65)

64

Dynamics of a single vortex line

Figure 2.7 A circularly polarised Kelvin wave, which propagates along a solitary vortex line, deforms the line to a helix. Points of the vortex line move along circular trajectories in transverse planes normal to κ, such that κ, v L , and F M form an orthogonal triad with the line tension force F lt balancing the Magnus force F M . Figure from Sonin and Krusius (1994).

The Fourier transformation yields E = E 0 + L0

ρκ 2  K(p)u(p)∗ · u(p). 4π p

Here E0 and L0 are the energy and the length of the straight vortex line and    ∞ dz 2 ipz 1 − eipz . p e − K(p) = z2 −∞ |z|

(2.66)

(2.67)

2.9 Kelvin waves on an isolated vortex line

65

Divergence at small z is cut-off by the core radius rc . Then after integration by parts of the second term in the brackets one obtains  ∞    1 3 dz 3 cos pz − ≈ p2 ln , (2.68) −γ − K(p) = p2 z 2 prc 2 rc where γ = 0.577 is Euler’s constant. The equation of motion in the Fourier representation takes the form   ρκ 2 K(p) ρκ 2 p2 1 3 du(p) =− u(p) = − ln u(p). (2.69) −γ − −ρκ × dt 4π 4π prc 2 Comparison with Eq. (2.62) shows that this equation describes the Kelvin oscillations with the frequency given by Eq. (2.63), but now the upper cut-off rm = e−γ −3/2 /p ≈ 0.125/p in Eq. (2.50) is exactly determined. In the Kelvin wave all vectors are confined to the xy plane. It is convenient to use the complex representation for two-dimensional vectors, widely applied in classical hydrodynamics (Milne-Thompson, 1968; Landau and Lifshitz, 1987). In this representation any two-component vector is represented by a complex number. The components of our vectors are already complex in the Fourier representation as a result of the presence of iω in the equations. In order to distinguish between complexity due to the Fourier transformation and that connected with the representation of two-dimensional vectors we introduce a new imaginary unit j , assuming j 2 = −1 as usual. Any vector a in the xy plane is represented by a complex number, a˜ = ax + j ay .

(2.70)

Separation of a j -complex number a˜ into its real and imaginary parts yields x and y components of the vector a: ˜ ax = Rej (a),

ay = Imj (a). ˜

(2.71)

In performing these operations, one should treat the other complex unit i as real. That is why we have introduced the second imaginary unit j . In the j -complex representation the vector product zˆ × u is j u, ˜ and Eq. (2.62) in the Fourier representation becomes ˜ iωj u˜ = −νs p2 u.

(2.72)

Eigenfrequencies of axial modes correspond to zeros of the complex determinant of Eq. (2.72)): D(j ) = j iω + νs p2 .

(2.73)

Then the condition D(j ) = 0 yields ω = −ij νs p2 ,

(2.74)

while relations between components of the displacement u(ux , uy ) are uy = −j ux .

(2.75)

66

Dynamics of a single vortex line

One obtains explicit formulas by replacing j by ±i. The two signs correspond to the two possible senses of circular polarisation. We considered Kelvin waves along a massless vortex since the effect of vortex mass is rather weak in most cases of practical interest. As was discussed in Section 2.4, the vortex mass doubles the number of vortex degrees of freedom, adding oscillation modes with high frequencies growing with decreasing vortex mass. Such a mode supplementary to the soft Kelvin mode also exists and was already considered by Lord Kelvin himself (Thompson, 1880). The mode is also discussed in Section 1.7.1 in the book by Donnelly (1991). Kelvin modes have been observed in experiments on torsion oscillations of a pile of disks in rotating superfluid 4 He. But in these experiments the frequencies were rather low, and the Kelvin oscillations were strongly modified by collective effects due to long-range interaction of vortices. Pile-of-disks experiments will be discussed later, in Section 5.4, after the analysis of boundary problems that is necessary for their interpretation. Ashton and Glaberson (1979) tested the dispersion law of the Kelvin mode at high frequencies, when collective effects are not important and the spectrum is given by Eq. (2.63). They investigated the motions of ions along vortex lines in the presence of an rf electric field transverse to vortex lines. It had been suggested earlier (Halley and Cheung, 1968; Halley and Ostermeier, 1977) that such a field would be strongly coupled to Kelvin waves. Resonant generation of Kelvin waves occurs under the following conditions: ωrf = ω(p) − vion p,

vion =

dω(p) . dp

(2.76)

Here ωrf is the rf frequency of the field, which was 107 Hz in the experiment, and vion is the ion velocity. The first condition means that the Kelvin wave frequency in the frame of reference of the moving ion is the same as the frequency ωrf . The second condition ensures that the ion, pumping the energy into the Kelvin wave, remains in the vicinity of the Kelvin wave packet moving with the group velocity dω(p)/dp. In addition to these two conditions, the sense of circular polarisation of the rf field should be the same as that of the Kelvin wave in the frame of reference of the ion. Ashton and Glaberson (1979) measured the ion velocity as a function of dc electric field, driving ions along the vortex line. They observed anomalies on the plot when all conditions of the resonance were satisfied. Despite a small discrepancy with the theory, explained by Ashton and Glaberson in terms of the field inhomogeneity, the experiment provides rather convincing evidence of the existence of propagating Kelvin waves. Recently Fonda et al. (2014) directly observed Kelvin waves emitted in the wake of vortex reconnection by visualising submicrometre tracer particles dispersed in superfluid 4 He. Evidence of Kelvin modes was reported by Bretin et al. (2003) studying damping of the collective quadrupole mode of a cigar-shaped Bose–Einstein condensate with a vortex.

2.10 Helical vortex Helical vortices were intensively studied in classical hydrodynamics (Saffman, 1995; Alekseenko et al., 2007). They appear in wakes of propellers and other spinning bodies. A helical

2.10 Helical vortex

67

vortex is a straight vortex line with a circularly polarised Kelvin wave of arbitrary amplitude propagating along it. In classical hydrodynamics an analytical solution of the Biot–Savart equation for helical vortices was obtained in the form of infinite series of Bessel functions [Kapteyn series, see Alekseenko et al. (2007)]. But here we restrict ourselves to a simpler approach using the local induction approximation (Sonin, 2012a). Let us consider a Kelvin wave of arbitrary amplitude propagating along a straight vortex line. In Cartesian coordinates for the vortex line displacements u(ux , uy ) with the z axis coinciding with the unperturbed vortex line, the circularly polarised Kelvin wave is described by ux = a cos(pz − ωt),

uy = a sin(pz − ωt).

(2.77)

So in the Kelvin mode the vortex line forms a helix with the pitch equal to the Kelvin wavelength λ = 2π/p (Fig. 2.8) moving along the z axis. We shall use the local induction approximation and start from the equation of motion (2.47). The curvature radius vector at any point of the helix is directed along the radius of the cylinder while its absolute value is R = (1 + p2 a 2 )/p2 a. In the cylindrical coordinate " tangent vector sˆ = κ/κ has components sr = 0, sϕ = " system (r, ϕ, z) the unit pa/ 1 + p2 a 2 , and sz = 1/ 1 + p2 a 2 . We look for an automodel solution of the equation of motion, which can be presented in two forms: ϕ(z, t) = pz − ωt = p(z − vz t),

(2.78)

or z(ϕ, t) =

ϕ − z t . p

(2.79)

Thus the helix motion may be described either as pure vertical translation with the velocity vz , or pure rotation with the angular velocity z . According to the equation of motion (2.47) νs p ω , (2.80) vz = = " p 1 + p2 a 2 z = −ω = − "

νs p2 1 + p2 a 2

.

(2.81)

In the limit of small a this yields the dispersion relation ω = νs p2 for the linear Kelvin wave. But the frequency decreases with increasing amplitude a of the Kelvin wave. When reducing the helix vortex dynamics to pure translation or pure rotation, one should remember that the displacement of points on the vortex line along the line itself has no physical meaning and does not lead to any physical consequence. So the choice of vortex velocity v  along the vortex line is fully arbitrary and is a matter of convention or convenience. Only the vortex velocity v L normal to the vortex line is well defined. For the helical vortex it is directed along a binormal to the vortex line, and Ricca (1994)

68

Dynamics of a single vortex line

(a)

(b)

Figure 2.8 Helical vortex. (a) Helix of high pitch 2π/p a. (b) Helix of low pitch 2π/p  a. Figure from Sonin (2012a).

called it the binormal velocity. Figure 2.9 illustrates how the velocity v  should be added to the velocity vL with azimuthal and axial components vLϕ and vLz in order to present motion of the helical vortex in terms of pure translation with velocity vz or pure rotation with angular velocity z and linear azimuthal velocity vϕ = z a. The azimuthal and the axial components vLϕ and vLz of the velocity vL are connected with the translational and rotational velocities by the relations: vLz = vz sin2 γ = z a sin γ cos γ ,

vLϕ = vz sin γ cos γ = z a cos2 γ ,

(2.82)

where γ = arctan pa is the angle between the vortex line and the vertical axis z. One can rewrite Eq. (2.82) as

2.10 Helical vortex (a)

69

(b)

Figure 2.9 Motion of the helical vortex line (thick solid line) on the surface of the cylinder of radius a. (a) Pure translation with the velocity vz , which is the z component of the velocity v L + v  . (b) Pure rotation with the velocity vϕ = z a, which is the azimuthal component of the velocity v L + v  .

vLz =

vz p2 a 2 z pa 2 = , 1 + p2 a 2 1 + p2 a 2

vLϕ =

vz pa z a = . 2 2 1+p a 1 + p2 a 2

(2.83)

Expressions (2.80) and (2.81) can be derived from the general canonical Hamilton equations vz =

∂E ∂E/∂a , = ∂Pz ∂Pz /∂a

where ρκ 2 L E= 4π

$

z =

∂E ∂E/∂a , = ∂Mz ∂Mz /∂a

1 pR 1 + p2 a 2 ln + ln " prc 1 + p2 a 2

(2.84)

 (2.85)

is the energy of the helical vortex, L and R are the height and the radius of the container with the helical vortex, and Pz and Mz are the components of the linear and angular momenta along the z axis, which can be calculated with the help of Eqs. (2.22) and (2.25):  #  2    ρκL dz a2p , (2.86) r 2 dϕ  = ρκL Pz = sϕ 1 + 4π r dϕ 2  r=a

Mz = −

ρκL 2 a . 2

(2.87)

Partial derivatives in Eq. (2.84) take into account that the only varying parameter of the helix is its radius a. Indeed, imposing the periodic boundary conditions in the box of height L, the pitch 2π/p of the helix becomes a topological " invariant. The first term in the expression (2.85) is the kinetic energy at distances r < 1 + p2 a 2 /k, where the helical

70

Dynamics of a single vortex line

" deformation increases the length of the vortex line by the factor 1 + p2 a 2 , while the sec" 2 2 ond term is the kinetic energy at distances r > 1 + p a /p, where helical deformation does not affect the velocity v = κ/2π r determined only by the circulation quantum. The momentum Pz is just the momentum of the constant vertical velocity field κp/2π inside the cylinder of radius a around which the vortex line is winding. The winding vortex line with tilted circulation vector produces a jump of the vertical velocity equal to κp/2π on average, and the latter is equal to the vertical velocity inside the helix under the natural assumption that there is no velocity along the axis z outside the helix. Although the vertical velocity determines the whole momentum, the kinetic energy of the vertical flow is ignored in the local induction approximation since it does not have a large logarithmic factor. But one cannot ignore the kinetic energy of the vertical flow at low pitch 2π/p  a [Fig. 2.8(b)], when the local induction approximation becomes inaccurate. At low pitch 2π/p  a distant parts of the same vortex line (distant in the sense of distance measured along the vortex line) approach a given point of the vortex line to distances much shorter than the curvature radius. Then the local induction approximation in its strict meaning becomes invalid: one must take into account velocity induced by other turns of the spiral. This urges use of the Biot–Savart law, as was done in classical hydrodynamics (Alekseenko et al., 2007). On the other hand, one can suggest a simpler but still reasonably accurate approach, which takes into account the velocity induced by distant parts of the vortex line by modification or extension of the local induction approximation. The surface of the cylinder, around which the vortex line is winding [Fig. 2.8(b)], can be considered as a continuous vortex sheet with a jump of the vertical velocity component κp/2π. Assuming that there is no velocity outside the helix, but only inside, one obtains the kinetic energy of axial (vertical) motion per unit length: ρκ 2 p2 a 2 /8π. Adding this energy to the energy in the local induction approximation given by Eq. (2.85), one obtains ρκ 2 L E= 4π

$

1 pR p2 a 2 1 + p2 a 2 ln + ln " + prc 2 1 + p2 a 2

 .

(2.88)

The added kinetic energy (the last term in the parentheses) is related to the velocity induced by all turns of the vortex line around the cylinder of radius a. It is the most essential term in the energy at pa | ln(prc )|. In this limit the approach coincides with that obtained from the Biot–Savart law. So the expression (2.88) is a reliable interpolation between two limits pa  | ln(prc )| (local induction limit) and ap | ln(prc )| (continuous vortex sheet limit). The extended local induction approximation does not change the expressions for the linear momentum Pz [Eq. (2.86)] and the angular momentum Mz [Eq. (2.87)] since the local induction approximation was not used for their derivation. Using the expression (2.88) for the energy in the canonic expressions for the translational and angular velocities [Eq. (2.84)] one obtains: vz =

νs p ω κp =" , + p 4π 1 + p2 a 2

z = − "

νs p2 1 + p2 a 2

−

κp2 . 4π

(2.89)

2.11 Helical vortex ring

71

In the limit pa | ln(prc )| (very small pitch) the helix vortex line reduces to a continuous cylindrical vortex sheet since the quantum term ∝ νs becomes unimportant. The axial and the azimuthal components of the velocity v L in this limit are vLz = vz = κp/4π and vLϕ = κ/4π a. They are averages of the corresponding fluid velocity components on two sides of the cylindrical vortex sheet, as in the case of plane vortex sheets well known in hydrodynamics. The principle on which the extended local induction approximation is based is not new. The Hall–Vinen–Bekarevich–Khalatnikov theory for rotating superfluids, which will be discussed in Section 3.1, was based on the same principle: the logarithmically large line tension contribution to the vortex line velocity is combined with the velocity induced by other vortices in the vortex array, which is approximated by continuous vorticity. In our case of a single helical vortex there are no other vortices, but the continuous vorticity represents the effect of other turns of the same vortex line.

2.11 Helical vortex ring One can roll up the helical vortex into a ring, keeping its helical deformation. Then it will be a helical vortex ring (Fig. 2.10), which recently became an object of analytical and numerical investigations (Kiknadze and Mamaladze, 2002; Barenghi et al., 2006; Helm et al., 2011). This problem, which is interesting in itself, may also have important implications for superfluid turbulence. An interesting outcome of studies of this object was that for Kelvin wave amplitudes large enough, the vortex ring may move in the direction opposite to its moment. This phenomenon was first revealed by Kiknadze and Mamaladze (2002) in numerical calculations and the perturbation theory within the local induction

Figure 2.10 Helical vortex ring. R1 and R2 are radii of the points of the vortex line closest to and furthest from the ring axis. R0 is the radius where the coordinate z(ϕ) has extrema.

72

Dynamics of a single vortex line

approximation. Later it was confirmed by numerical calculations based on the Biot–Savart law (Barenghi et al., 2006) and the Gross–Pitaevskii equation (Helm et al., 2011). The effect seemed to be mysterious and was called anomalous vortex-ring velocity (Barenghi et al., 2006; Krstulovic and Brachet, 2011). In this section we address the dynamics of helical vortex rings in the local induction approximation either solving the equations of motions of a vortex line directly or using simple canonical relations following from the Hamiltonian equations of motion (Sonin, 2012a). It is assumed that the helix makes n turns around the vortex ring and the radius a of the helix is much smaller than the radius R of the ring but may be of the same order or larger than the pitch 2πR/n of the helix (the Kelvin wavelength). The momentum P = πρκR2 of the vortex ring is not affected by helical deformation of the vortex line and does not differ from that given by Eq. (2.54) for an undeformed circular ring. The energy of the helical vortex ring can obtained from the expression (2.85) for the energy of a straight helical vortex by replacing the container radius R by the ring radius R, the height L by 2πR, and the wave number p by n/R:   R nR ρκ 2 " 2 . (2.90) R + n2 a 2 ln + R ln √ E= 2 nrc R2 + n2 a 2 The system has two degrees of freedom: translation along and rotation around the axis of the ring. We consider translation, deriving the velocity of the ring using the canonical relation (2.53) for the vortex ring velocity. We need to find a proper procedure for varying the ring radius in the canonical relation. The number n of turns is a topological invariant and cannot vary. Another invariant is the angular momentum along the ring axis. For large ring radius R a, it can be estimated from the component Pz of the straight helical vortex, Eq. (2.86): Max = πρκRa 2 n.

(2.91)

The angular momentum does not vary if the variation of R is accompanied by variation of the helix radius a: δa = −aδR/2R. Then retaining only the most important logarithmic terms, the canonical relation (2.53) yields the following expression for the helical ring velocity:     1 − n2 a 2 /2R2 ∂R ∂E  a ∂E  = νs √ − , (2.92) vL =   ∂P ∂R a 2R ∂a R R2 + n2 a 2 where the upper cut-off rm in the expression (2.50) for the line tension parameter νs must be chosen to be the ring radius R at very small na/R  1 or the helix pitch (Kelvin wave wavelength) R/n at na/R ∼ 1. At a large number of helix turns n = 2R/a the sign of the velocity changes, and the ring starts to move in the direction opposite to its momentum. This is the phenomenon of anomalous vortex-ring velocity, which here is directly explained by the canonical equations of motion and the angular momentum conservation law. Expansion of Eq. (2.92) in na/R yields the perturbation theory result of Kiknadze and Mamaladze √ (2002). The critical value of na/R = 2 at which the vortex ring stops is larger than

2.11 Helical vortex ring

73

na/R = 1 following from perturbation theory and is in reasonable agreement with the critical value na/R = 1.7 obtained in numerical calculations on the basis of the Biot– Savart law at large n (Barenghi et al., 2006). The angular velocity of the helical vortex ring rotation also follows from the canonical relation: νs n ∂E = √ . (2.93) = ∂Max R R2 + n2 a 2 Now taking the partial derivative, the linear momentum P (i.e., the average radius R) must be fixed. Rotation with angular velocity  corresponds to translation of the straight helical vortex, so  = vz /R, where vz is given by Eq. (2.80) but with p replaced with n/R. As in the case of a straight helical vortex, at very short helix pitch one should use the extended local induction approximation. The added continuous velocity is now an azimuthal velocity inside the helix formed along the ring. This leads to modification of the expression Eq. (2.92) for the ring translational velocity: 1 − n2 a 2 /2R2 κn2 a 2 − . vL = νs √ 8πR3 R2 + n2 a 2

(2.94)

The added non-logarithmic term is negative and increases the velocity of reverse motion of the helical vortex ring. Our analysis addressed the case of a single Kelvin mode. Krstulovic and Brachet (2011) investigated numerically the slowdown effect for the ensemble of Kelvin modes, assuming that the effect of various modes is additive. They discussed the role of the effect in the theory of superfluid turbulence. Slowdown of a vortex ring or even inversion of direction of its motion by an azimuthal fluid flow around the ring (swirl) are also known in classical hydrodynamics (see Cheng et al., 2010, and references therein). In classical hydrodynamics a swirl is produced by vortices piercing the ring, while in our case azimuthal flow was induced by helical deformation of the vortex ring. In the local induction approximation one can find the velocity and the shape of the helical vortex ring analytically, without the assumption of a small ratio a/R. The shape of the helical vortex ring is determined in cylindrical coordinates by the two functions r(ϕ, t) and z(ϕ, t). We look for an automodel solution r(ϕ, t) = r(ϕ − t) and z(ϕ, t) = z(ϕ − t) + vL t in the cylindrical system of coordinates (r, ϕ, z) with the polar axis z coinciding with the axis of the ring (Fig. 2.10). The solution describes stationary translation with the linear velocity vL and rotation of the helical ring with the angular velocity . The vector equation (2.42) of motion reduces to two equations for two time-independent functions r(ϕ) and z(ϕ):   νs dz/dϕ d dr = 0, (2.95) + r " dϕ dϕ r 1 + (dr/r dϕ)2 + (dz/r dϕ)2 −vL r

d dr + dϕ dϕ



 νs r = 0. [1 + (dr/r dϕ)2 + (dz/r dϕ)2 ]1/2

(2.96)

74

Dynamics of a single vortex line

The first integration of the equations is straightforward: 

dr dϕ 

2 =

dz dϕ

2

r 4 [4νs2 − 2 (r 2 − R20 )2 ] − r2 , $ 2 2 2 2 2 2 2 2 [vL (r − R1 ) + R1 4νs −  (R1 − R0 ) ]

(2.97)

r 4 2 (r 2 − R20 )2 = . $ [vL (r 2 − R21 ) + R1 4νs2 − 2 (R21 − R20 )2 ]2

The solution was obtained with the boundary conditions (i) the coordinate z(ϕ) has extrema (dz/dϕ = 0) at r = R0 , and (ii) the point of the vortex line with r = R1 < R0 is closest to the ring axis [the minimum of the function r(ϕ)]. The maximum of the function r(ϕ) (the vortex line point most distant from the ring axis) is at the distance r = R2 > R0 given by  ⎡  2    2 v vL2 vL2 1 1 2 2 2 2 2 L + R1 − 2 (R20 − R21 ) R2 = R 0 − R1 + 2 + ⎣ R1 + 2 2 4    4vL R1 νs 4ν 2 + 2s −  2

#

⎤1/2 2 (R21 − R20 )2 ⎦ . 1− 4νs2 (2.98)

The radii R0 , R1 , and R2 are shown in Fig. 2.10. The second integration yields expressions for functions ϕ(r) and z(r):    vL (R21 + R23 ) R23 2 ϕ(r) = F (φ, k) −  φ, − 2 k , k $ R1 R23 R22 + R23 $     4νs2 − 2 (R21 − R20 )2 R23 2 2 2 2 + (R1 + R3 ) φ, − 2 k , k − R1 F (φ, k) , $ R1 R1 R23 R22 + R23 (2.99)

z(r) =

(R21 + R23 )(φ, k 2 , k) − (R20 + R23 )F (φ, k) . $ R22 + R23

(2.100)

Here k = 2

R22 − R21

, 2

R22 + R3

# φ = arcsin

r 2 − R21 k 2 (r 2 + R23 )

v2 R33 = R21 + R22 − 2R20 + L2 , 

, (2.101)

2.11 Helical vortex ring

75

1

0

0.5 0.7

–1

0.9

–2

–3

0

1

2

3

4

5

Figure 2.11 The ring velocity versus the ratio na/R. The dotted line shows the velocity of an unperturbed vortex ring. The dashed line shows the velocity obtained from the simple approach assuming small a/R [Eq. (2.92)]. The thick lines show the velocities for R1 /R = 0.9, 0.7, and 0.5. Figure from Sonin (2012a).

and

 F (φ, k) =  (φ, l, k) = 0

φ

"

dα

, 1 − k 2 sin2 α dα " 2 (1 − l sin α) 1 − k 2 sin2 α 0

φ

(2.102)

are incomplete elliptic integrals of the first and the third kind (Gradstein and Ryzhik, 1965). Not all necessary boundary conditions are still satisfied. The first condition is that the vortex line forms a closed continuous curve spiralling along the toroidal surface only once. This is satisfied if at the maximum r = R2 the coordinate z returns to the same value as at the minimum r = R1 . Choosing the latter to be zero, this yields the condition z(R2 ) = 0. The second condition is dictated by the chosen period of the helix around the ring and is given by ϕ(R2 ) = π/n, where n is the number of turns of the vortex line along the ring. From these two conditions one can find dependences of the translational and the angular velocities of the helical vortex ring on the average radius R of the ring.2 In the limit of a weak Kelvin wave a = R − R1 = R − R2 → 0 our solution yields the same translational velocity vL = νs /R as that of a circular vortex ring [Eq. (2.52)]. Figure 2.11 compares the velocity vL of the helical vortex ring following from the general solution with vL obtained from the canonical relation for the case of small a/R. At small a/R one can expand the elliptic integrals Eq. (2.102), keeping terms of the second 2 The average radius R = (R + R )/2 is not equal to R in general, although it is very close to it if a/R is small. 1 2 0

76

Dynamics of a single vortex line

order in a 2 ∝ k 4 . In Fig. 2.11 the dependences of vL R/νs on na/R at fixed a/R are shown by continuous curves drawn over discrete values of na/R corresponding to integer n. The curves start from the value n = 1, which corresponds to a pure translation of the vortex ring without deformation. Then the velocity vL coincides with the velocity vL = νs /R of a circular vortex shown by the dotted line. The accuracy of the simple approach for small a/R drops not only at low n but also at very high n. If the ratio a/R is low enough, there √ is a rather long interval 1  na/R  R/a where the simple approach does not differ from the exact solution. Figure 2.12 shows helical vortex rings for n = 4 calculated without expansion of elliptic integrals and for various ratios R1 /R. The vortex line is winding around the surface of a toroid, which changes from an ideal torus at R1 → R with circular cross-section to a toroid obtained by revolution of a figure of eight at R1 → 0 (Fig. 2.13).

Figure 2.12 Helical vortex rings of four-fold symmetry n = 4 for various ratios R1 /R. The left column shows projections of rings on the plane normal to the ring axis (the axis z). Dashed lines show unperturbed ideally circular rings of radius R. The right column shows three-dimensional images of rings. Figure from Sonin (2012a).

2.12 Precession of a single curved vortex 0.4

0 0.6

0.2 1.5

1.0

77

0.9

0.5

0.5

1.0

1.5

– 0.2 – 0.4

Figure 2.13 Axial cross-section of toroids over which vortex lines wind for the case of four-fold symmetry n = 4 and ratios R1 /R = 0.9, 0.6, and 0 (shown near the curves). The cross-section varies from two ideal circles at R1 → R to the figure of eight at R1 → 0. Figure from Sonin (2012a).

2.12 Precession of a single curved vortex The local induction approximation approach can be used to determine the shape of the vortex line partially trapped by and precessing around the wire coaxial to the cylindrical container. This precession was investigated in experiments on observation of the circulation quantum in 3 He (Zieve et al., 1992). The geometry of the experiment is shown in Fig. 2.14(a). The z axis is the axis of the container of radius R and of the wire of radius rw . This is the same geometry as in Vinen’s experiment shown in Fig. 2.4, but now the vortex line is trapped by the wire only partially. The vortex filament with the circulation κ trapped on the wire at z < z0 , is ‘unzipped’ from the wire at z > z0 stretching to the container wall. Only the ‘unzipped’ (free) part of the vortex filament participates in the precession. If dissipation is negligible (this is a good approximation for the experiment performed at low temperatures), the vortex line is at the equilibrium state in the coordinate frame rotating with the angular velocity ωL of the vortex precession. In the rotating coordinate frame the vortex line shape in cylindrical coordinates is determined by the function z(r), which can be found by minimisation of the energy E = E − ωL Mz .

(2.103)

Here E and Mz are the energy and the angular momentum around the z axis of the fluid in the laboratory (resting) coordinate frame: #   R 2 R R ρκ 2 ρκ dz(r) 2 ln ln E = z0 + 1+ dr, (2.104) 4π rw 4π rc dr rw

Mz =

ρκ 2 (R − rw2 )z0 + ρκ 2



R

[z(r) − z0 ]r dr.

(2.105)

rw

The vortex line is not twisted, i.e., it lies inside an axial plane, and the azimuthal angle ϕ does not vary along the line. The first terms of these expressions are related to the segment

78

Dynamics of a single vortex line

(a)

(b)

Figure 2.14 Vortex line attached to a thin wire of radius rw coaxial to a cylindrical container of radius R (Zieve et al., 1992). The profile of the vortex line can be examined on two different length scales. (a) At large scales r rw the local induction approximation becomes applicable, in which the wire is treated as an enhanced vortex core. The vortex line meets the wire at the finite contact angle θ. (b) At small scales r ∼ rw the vortex line goes smoothly over into the image vortex (dashed continuation of the vortex line), with the wire surface perpendicular at the connection point. Figure from Sonin and Krusius (1994).

of the vortex line trapped by the wire, while the second integral terms are connected with the free vortex segment stretched between the wire and the container wall. The angular velocity ωL is determined by the canonical relation ωL =

∂E . ∂Mz

(2.106)

The derivative must be taken at fixed shape of the free vortex segment, variations of E and Mz being produced by variation of z0 . This yields the exact value of the precession frequency, independent of the shape of the free vortex segment (Misirpashaev and Volovik, 1992; Zieve et al., 1992; Schwarz, 1993; Sonin, 1994): ωL =

κ ∂E/∂z0 κ ln (R/rw ) R ≈ = ln . ∂Mz /∂z0 2π(R 2 − rw2 ) 2π R 2 rw

(2.107)

Minimisation of the energy (2.103) with respect to the function z(r) yields the Euler– Lagrange equation, which determines the shape of the free vortex segment:   ε dz/dr d , (2.108) ρκωL r = − " dr 1 + (dz/dr)2 where ε = (ρκ 2 /4π ) ln(R/rc ) is the line tension. Equation (2.108) presents the balance of forces on an element of the precessing vortex line: the Magnus force on the left-hand side is balanced by the line tension force ∝ ε on the right-hand side. The equation follows

2.12 Precession of a single curved vortex

79

also from the dynamical equation (2.47) for a single vortex line, bearing in mind that the curvature radius of the vortex line is given by   2 −3/2 d 2z dz 1 = 2 1+ . (2.109) R dr dr For solution of Eq. (2.108) one needs the boundary conditions on the lateral wall of the cylinder and the wire. The first one is the condition of transversality (the vortex line meets the lateral wall at a right angle):  dz  = 0. (2.110) dr r=R The second boundary condition follows from the balance of line tension forces on the element of the vortex filament at the ‘unzipping’ point z = z0 , r = rw ≈ 0. The small wire radius rw can be treated as a core radius for the trapped vortex segment. The line tension of the trapped segment is the energy per unit length εw =

ρκ 2 R ln . 2 rw 4π

(2.111)

Since a small rw , however, is larger than the core radius of the free vortex segment, the line tension of the trapped segment is less than that of the free segment. The balance of the components of the line tension forces parallel to the wire and applied to the unzipping point is illustrated in Fig. 2.14a and is given by ε cos θ = εw , where ln rRw ε = = cos θ = εw ln rRc



   "  2 1 + (dz/dr) 

(2.112)

dz/dr

.

(2.113)

r=rw

So the free vortex segment is not normal to the wire, except for the limit ln rRc ln rRw . The boundary condition at the wire can also be derived by minimisation of the energy in the rotating coordinate frame with respect to variation of the unzipping coordinate z0 (Sonin, 1994). It is valid only for rather thin wires with radius rw much less than the radius R of the container. Integration of Eq. (2.108) over r with the transversality condition Eq. (2.110) yields the first integral: dz/dr ρκ 2 R ρκ ωL (r 2 − rw2 ) = ln " . 2 4π rc 1 + (dz/dr)2

(2.114)

At the unzipping point this equation together with the boundary condition Eq. (2.112) gives that e = e − ωL mz = 0,

(2.115)

80

Dynamics of a single vortex line

where e and e = εw are the energies in the rotating and the laboratory coordinate frames per unit length, and mz = ρκωL (r 2 − rw2 )/2 is the angular momentum per unit length of the vortex line below the unzipping point. Emergence of the condition of zero energy density e , below the unzipping point, from the minimisation of the total energy in the rotating coordinate frame is quite natural. Much above the vortex line the energy density e vanishes, and if the energy density e below the unzipping point were non-zero, then there would be a force driving the untrapped segment of the vortex line upward or downward. The condition (2.115) leads to the precession angular velocity given by Eq. (2.107). The second integration of Eq. (2.114) with the boundary condition Eq. (2.113) yields the shape z(r) of the vortex line: #    2 R 2 + λ(R 2 − r 2 ) 1 E(φ, k) + r . (2.116) z(r) = R √ F (φ, k) − λ R 2 − λ(R 2 − r 2 ) 2λ Here λ=

ln rRw ln rRc

,

and

#

1+λ k = , 2

sin φ = r

2

 F (φ, k) = 0

φ

"

dα 1 − k 2 sin2 α

2λ , (1 + λ)[R 2 − λ(R 2 − r 2 )] 

,

E(φ, k) =

φ

" 1 − k 2 sin2 α dα

(2.117)

(2.118)

0

are incomplete elliptic integrals of the first and second kinds (Gradstein and Ryzhik, 1965). In the limit of small λ, Eq. (2.116) gives z(r) = λ(r − r 3 /3R 2 ). The analysis can be applied to a free axial vortex terminating on the wall (Sonin and Nemirovskii, 2011). The case is the limit of an extremely thin wire when the wire radius must be replaced by the vortex core radius in all expressions. In this limit the unzipping point goes to z → −∞, and the curved vortex line smoothly approaches the vertical axis of rotation in accordance with the boundary condition Eq. (2.113), which now tells us that  dz  → ∞. (2.119) dr r→0 The shape of the free vortex can be obtained analytically after the second integration of Eq. (2.114), taking into account the expression Eq. (2.107) for the precession frequency ωL : √ √ " R 2 + 2R 2 − r 2 R 2 2 . (2.120) z(r) = 2R − r − R − √ ln √ 2 r( 2 + 1) The expression demonstrates that the vortex line approaches the axis r = 0 exponentially: √ r ≈ Re−|z| 2/R . While the frequency of the single vortex precession follows straightforwardly from commonly accepted thermodynamic arguments, the shape of the vortex line was a matter of dispute focusing on the proper boundary condition at the unzipping point in the local induction approximation [see a more detailed discussion by Sonin (1994) and Sonin and

2.12 Precession of a single curved vortex

81

Krusius (1994)]. The condition (2.112) seems to contradict the well known rule that a vortex line meets any solid surface at a right angle. Instead of the boundary condition (2.112) based on the balance of forces following from the variational principle, Schwarz (1993) used the condition that the vortex line is normal to the wire (θ = π/2). At the very surface of the wire the latter boundary condition is definitely correct. But at small scales of the order of the wire radius rw there are forces which lead to fast deviation from the normal direction. These forces were neglected by Schwarz and all others addressing this problem. This is a legitimate approximation when one is looking for the vortex shape at scales exceeding the wire radius rw , but only if one uses the boundary condition (2.112) based on the balance of line tension forces. Factually this means that the boundary condition is imposed not exactly at the radius rw of the wire but at some distance from the wire larger than rw , but still much smaller than R, as illustrated in Fig. 2.14b. The analysis of the shape of a free precessing vortex gives one more justification of the force balance boundary condition. The latter provides a natural transition from a vortex partially trapped by the wire to a free vortex changing its direction smoothly from vertical to horizontal. Schwarz’s boundary condition becomes senseless for the free vortex since it requires that the vortex meets the axis at a right angle.

3 Vortex array in a rotating superfluid: elasticity and macroscopic hydrodynamics

3.1 Macroscopic hydrodynamics of rotating superfluids In a rotating container a superfluid without vortices cannot co-rotate together with the container because the superfluid velocity field is potential. However, at a rather low critical angular velocity c1 given by Eq. (2.13), a vortex must appear in the fluid. As the angular velocity is increased further, more and more vortices enter the container, and eventually they form a dense vortex array filling the whole container with constant areal density determined by the formula of Feynman (1955): nv =

2 . κ

(3.1)

This density provides solid body rotation of the superfluid on average. Tkachenko (1965) showed that vortex lines in a rotating superfluid form a stable triangular lattice like in a type II superconductor (Abrikosov, 1957). Observation of the regular vortex lattice turned out to be much more difficult in rotating superfluid 4 He than in type II superconductors because vortices in superfluid 4 He do not create magnetic or electric fields which make them easier to identify. Only 14 years after the paper by Tkachenko (1965), the existence of the regular vortex lattice in rotating superfluid 4 He was demonstrated experimentally by Yarmchuk et al. (1979), although for a rather small number of vortices. Their photograph of the vortex array is shown in Fig. 3.1a. New effective methods of visualisation of vortices became available in BEC clouds of cold atoms. Soon after the first reports of vortex images in rotating BEC clouds (Matthews et al., 1999; Madison et al., 2000), good photographs of well ordered vortex arrays were published by Abo-Shaeer et al. (2001) (Fig. 3.1b). The dynamical equations, which were formulated on the basis of the Biot–Savart law for a single curved vortex line in Chapter 2, can easily be generalised for an ensemble of many vortex lines. The net velocity field at the point with position vector R is equal to the sum of contributions of all vortex lines [cf. Eq. (2.3)]: v(R) =

 κj  [dR j × (R − R j )] . 4π |R − R j |3 j

82

(3.2)

3.1 Macroscopic hydrodynamics of rotating superfluids (a)

83

(b)

Figure 3.1 Photographs of vortex arrays. (a) Rotating superfluid 4 He [from Yarmchuk et al. (1979)]. (b) Rotating Bose–Einstein condensate of cold atoms [from Abo-Shaeer et al. (2001)].

Here κj is the circulation around the j th vortex line, and R j is the position vector of a point on the same line. Integration is performed over the whole length of all the vortex lines. Generalisation of the single-vortex formulas for the energy [Eq. (2.15)] and the equation of motion [Eq. (2.42)] yields:   dR i · dR j ρ  κi κj , (3.3) E= 4π |R i − R j | ij

−ρκ i ×

δE dR i =− , dt δR i

(3.4)

where κ i is the circulation vector of magnitude κi tangent to the ith vortex line in any point. Any term i  = j in the sum Eq. (3.3) is the energy of interaction between two vortex lines. Out of equilibrium, one can describe the positions and shapes of vortex lines by a set of two-dimensional vectors of the displacement ui (z) from equilibrium positions in a regular periodic array of vortices parallel to the rotation axis (the z axis). The displacement vectors ui (z) are parallel to the xy plane and depend on the coordinate z and the index i of the vortex. The original three-dimensional position vector R i of the ith vortex line is connected to ui (z) by R i = zˆz + r i + ui (z).

(3.5)

Here zˆ is the unit vector along the z axis and r i is the two-dimensional equilibrium position vector of the ith vortex line. Then we can rewrite Eq. (3.4) as follows (Fetter, 1967):   δE dui (z) =− . (3.6) −ρ κ × dt δui (z) This is a generalisation of the canonical equation (2.44) of motion for a single straight vortex on an array of vortices with the same circulation κ. In a perfect incompressible fluid, the canonical equations of motion given by Eq. (3.4) or (3.6) form a closed system which is sufficient to describe the whole dynamics of the

84

Vortex array in a rotating superfluid

fluid together with the vortex lines. The continuous fluid velocity field v(R) at any point of the space is determined from the Biot–Savart law. On the other hand, time evolution of the velocity field and the distribution of thermodynamic variables such as the chemical potential and the pressure is determined by the Euler equation. In the presence of singular vortex lines, the vorticity is the sum of singular contributions from all vortex lines:  κj  (3.7) dR j δ(R − R j ). ω(R) ˜ = 4π j

Here δ(R − R j ) is a three-dimensional δ-function. Substituting Eq. (3.7) into the Euler equation (1.34), we obtain  dv(R)  κj + (3.8) [dR j × v(R)]δ(R − R j ) = −∇μ. dt 4π j

Singular terms in Eq. (3.8) extend the Euler equation to points of vortex lines where the hydrodynamical theory is invalid in a strict sense. At these points the fluid velocity must be regularised in the manner described in Section 2.7. Our equations describe the dynamics of a perfect fluid with singular vortex lines, assuming that the vortex core radius is less than the distances between vortex lines. We call this the Vortex Line Lattice (VLL) state. Most experiments deal with fluids containing a large number of vortices constituting a very dense array, and only averaged parameters are available for experimental observation. We follow the approach used for the continuum elasticity of solids, and approximate equations for discrete vortex lines by equations for continuous fields of averaged parameters such as vortex density and deformations of the vortex array. It is assumed that these parameters vary slowly over the distance between vortices, and one can use an expansion in gradients or in wave vectors in the Fourier representation. The initial equations governing motion of the fluid with singular vortex lines play the role of microscopic equations for atoms in a solid. In this sense they can be called ‘microscopic’. But one must remember that they are formulated within the framework of the phenomenological hydrodynamical theory and have nothing to do with truly microscopic equations of fluids at the atomic level. Macroscopic hydrodynamical equations can be derived ab initio from microscopic hydrodynamics by a coarse-graining procedure, or they can be formulated on a phenomenological basis using conservation laws and requirements imposed by symmetry. Macroscopic hydrodynamics was used for the description of rotating superfluids, beginning with the pioneering works of Hall and Vinen (1956b), Hall (1958), Mamaladze and Matinyan (1960), and Bekarevich and Khalatnikov (1961). The theory is called the Hall–Vinen– Bekarevich–Khalatnikov (HVBK) theory. However, this theory neglected shear rigidity of the vortex lattice. A continuum hydrodynamical theory allowing for the crystalline order in the vortex lattice and its shear rigidity was developed by Tkachenko (1969), but his theory was purely two-dimensional and did not consider possible flexure of vortices. The extension of the hydrodynamical continuum theory to include the effects of shear rigidity and vortex flexure was carried out by Sonin (1976) and Williams and Fetter

3.1 Macroscopic hydrodynamics of rotating superfluids

85

(1977). Both papers started from the linear equations for quantised vorticity in the Fourier representation, and then expanded them in the wave vectors, retaining only first terms of the expansion. After inverse Fourier transformation this yielded the equations of motion within the continuum theory. An equivalent coarse-graining procedure directly in the configurational space was suggested by Baym and Chandler (1983). They also restricted themselves to a linear theory. A non-linear hydrodynamics of a rapidly rotating superfluid involving the effects of both vortex tension and Tkachenko shear rigidity of the vortex lattice was formulated by Andreev and Kagan (1984), see also Kagan (2013). Before focusing on the linear theory of vortex oscillations, we formulate the general nonlinear macroscopic hydrodynamics in order to give an overview of the problem as a whole (Sonin, 1987). First let us see what form the Euler equation (3.8) takes in macroscopic hydrodynamics. From now on we assume that all vortex lines bear one quantum of circulation κ = h/m. We shall average Eq. (3.8) over a vortex lattice cell. Because of the singular character of the vorticity field, the velocity of the fluid in the vector product is not affected by the procedure of averaging; it remains a regularised local velocity at the point on the vortex line, which differs in general from the average fluid velocity v and will be denoted as v l . In the absence of external forces, the vortex velocity v L is equal to v l (Helmholtz’s theorem), and averaging of Eq. (3.8) yields the Euler equation in macroscopic hydrodynamics: ∂v + ω˜ × v L = −∇μ. (3.9) ∂t Here ω˜ = ∇ × v is the averaged vorticity with magnitude equal to the number density of vortex lines per unit area multiplied by the circulation quantum κ. Symbols denoting averages are dropped. The scalar function μ on the right-hand side of Eq. (3.9) must be the chemical potential in order to satisfy the energy conservation law discussed below. Although the Euler equation (3.9) was derived in the absence of external forces when v L = v l , it also holds if external forces make the vortices move with a velocity v L different from v l . This follows from close connection of the Euler equation with the continuity equation for vorticity. The latter is obtained by taking the curl of both parts of Eq. (3.9), ∂ ω˜ + ∇ × [ω˜ × v L ] = 0, ∂t

(3.10)

or ∂ ω˜ ˜ + (v L · ∇)ω˜ + ω(∇ · v L ) − (ω˜ · ∇)v L = 0. (3.11) ∂t Because ω˜ is proportional to the density of vortex lines, Eq. (3.11) is the conservation law for vortex lines. Its validity is not restricted to the VLL state; it holds even when the vorticity is distributed continuously over the entire space. One may consider the Euler equation (3.9) as a consequence of the evident continuity equation for vorticity in the form of Eq. (3.10) or (3.11). Integration of Eq. (3.10) yields Eq. (3.9), but with an indeterminate scalar function μ. Another insight into the physical meaning of the Euler equation (3.9) is achieved by recalling the connection between the velocity v and the phase θ of the order parameter of

86

Vortex array in a rotating superfluid

Figure 3.2 Phase slippage. Phases at two points P1 and P2 in a channel change as a vortex moves between them. From (a) to (b) to (c) the vortex moves across the channel from left to right. As it moves from one wall to the other wall the relative phase changes by 2π . From Anderson (1966); Anderson’s notation ϕ for the phase differs from our notation θ.

the superfluid: v = (h/m)∇θ . A phase difference along the path around the vortex line is ¯ equal to 2π . Suppose that we integrate Eq. (3.9) along a path between some points P1 and P2 . This yields an expression for the temporal derivative of the phase difference:   dv d(θ2 − θ1 ) 2π 2 m 2 2π dl · dl · [ω˜ × v L ]. (3.12) = = (μ1 − μ2 ) − h¯ 1 dt dt κ κ 1 Here θ1,2 and μ1,2 are the phases and the chemical potentials at the points P1 and P2 (Fig. 3.2). The vector product in Eq. (3.9) gives a contribution to the phase difference variation related to the flow of vortex lines across the path between points P1 and P2 . Indeed, any passage of a vortex line across the path produces a change 2π in the phase

3.1 Macroscopic hydrodynamics of rotating superfluids

87

difference. The number of vortices crossing a unit length of this path per second is nv vL = ωv ˜ L /κ, and variation of the phase difference per second is given by the vector product term in Eq. (3.12). This is the ‘phase slippage’, a concept invented to explain Josephsontype phenomena in superconductors and superfluids (Anderson, 1966). In rotating superfluids the presence of the vortex array requires us to consider displacements u of vortices as additional hydrodynamical variables. At moderate rotation speeds when one can ignore the effect of centrifugal forces, the fluid is translationally invariant (see Section 1.10), and the energy cannot depend on constant translation u. If u varies in space, the energy depends on space gradients of u, and the total energy density is E = E0 (ρ) +

ρv 2 + Ev (∇i u). 2

(3.13)

Here E0 is the energy density of the fluid at rest, which depends only on the mass density ρ. The second term in Eq. (3.13) is the kinetic energy of the averaged velocity field v(R). The last u-dependent term takes into account the energy connected with spatial fluctuations of the velocity around the averaged velocity within a unit cell of the vortex array. The Gibbs thermodynamic relation for the energy Es = E0 (ρ) + Ev (∇i u)

(3.14)

in the coordinate frame moving with average velocity v is dEs = μs dρ +

∂Ev · d∇i u, ∂∇i u

(3.15)

where μs =

∂Ev ∂E0 + ∂ρ ∂ρ

(3.16)

is the chemical potential in the moving coordinate frame. The variation of the total energy density in the laboratory frame is dE = μdρ + ρv · dv +

∂Ev · d∇i u, ∂∇i u

(3.17)

where μ = μs +

v2 2

(3.18)

is the total chemical potential in the laboratory coordinate frame. Then the pressure P = −E + μρ = −Es + μs ρ

(3.19)

is the same in the two coordinate frames, and the Gibbs–Duhem relation is dP = ρdμs −

∂Ev · d∇i u. ∂∇i u

(3.20)

88

Vortex array in a rotating superfluid

One can rewrite Eq. (3.9) as ∂v f + ω˜ × v + ∇μ = , ∂t ρ

(3.21)

introducing a force, which emerges if the vortex moves with respect to the fluid with the relative velocity v L − v: f = −ρ ω˜ × (v L − v).

(3.22)

Comparing Eq. (3.22) with Eq. (1.53) for the force F on a cylinder with circulation of a fluid around it, and recalling that ω˜ is equal to the circulation quantum κ multiplied by the two-dimensional density of vortex lines, we see that f is a force acting upon vortex lines in a unit volume of the fluid moving with velocity v. In the absence of external forces, the vortex velocity v L is equal to the local fluid velocity v l , and the total force f reduces to the elastic force produced by deformations of the vortex lines and connected with the fluid velocities by the relation f el = −ρ ω˜ × (v l − v).

(3.23)

In general the force f also includes other forces, and v L  = v l . In accordance with the definition of force in mechanics, the elastic force f el should be determined as a derivative of the energy with respect to displacements u. Variation of the energy with respect to variations du of displacements is (for the sake of generality we keep the dependence on u for a while)        ∂Ev ∂Ev ∂Ev ∂Ev du + d∇j u = dR − ∇j du. (3.24) δu E = dR ∂u ∂∇j u ∂u ∂∇j u However, the functional derivative ∂Ev δE = − ∇j δu ∂u



∂Ev ∂∇j u

 (3.25)

is not a derivative that determines the elastic force f el . The displacement u(R, t) was considered as an Euler variable (see the end of Section 1.2), and the variation du was the difference between displacements of two different points on vortex lines, which were located at the same point in space with the position vector R after and before variation of positions of all vortex lines. Meanwhile, the definition of force on a vortex refers to the Lagrange description. If the displacement is a Lagrange variable, its variation Du is a difference between positions after and before variation of some fixed point on some fixed vortex line. The relation between the Lagrange and the Euler variations is analogous to the relation between the substantial (Lagrange) and the Euler time derivative of the velocity given by Eq. (1.42): Du = du + (Du · ∇)u.

(3.26)

3.1 Macroscopic hydrodynamics of rotating superfluids

Expressing du in Eq. (3.24) via Du with the help of Eq. (3.26), we obtain    ∂Ev ∂Ev [Du − (Du · ∇)u] + ∇j [Du − (Du · ∇)u] . δu E = dR ∂u ∂∇j u Integrating by parts and allowing for ∂Ev /∂u = 0, we obtain      ∂Ev ∂Ev ∇ui Du. + ∇j δEu = dR −∇j ∂∇j u ∂∇j ui

89

(3.27)

(3.28)

The quantity in brackets is the functional derivative in the Lagrange description, which determines the elastic force f el :   ∂Ev ∂Ev ∇ui . − ∇j f el = ∇j (3.29) ∂∇j u ∂∇j ui In the absence of external forces f = f el , and inserting this formula into Eq. (3.22) one obtains the general equation of vortex motion:   ∂Ev ∂Ev ∇ui . − ∇j (3.30) −ρ ω˜ × (v L − v) = ∇j ∂∇j u ∂∇j ui For known dependence of the energy on all variables, Eqs. (3.9) and (3.30) together with the mass continuity equation (1.18) constitute a closed Galilean invariant system of macroscopic hydrodynamical equations. One can check that our system of equations obeys the conservation laws for the momentum, ∂ji + ∇j ij = 0, ∂t

(3.31)

∂E + ∇ · Q = 0. ∂t

(3.32)

ij = P δij + ρvi vj + σij .

(3.33)

and for the energy,

The momentum flux tensor is

Here σij is the elastic stress tensor1 of the vortex lattice given by σij = −

∂Ev ∂Ev + ∇i uk . ∂∇j ui ∂∇j uk

(3.34)

The energy flux vector Q has components Qi = μρvi + σij vLj .

(3.35)

The energy conservation law is obeyed provided that μ in the Euler equation (3.9) is the true thermodynamical chemical potential defined by Eq. (3.18). 1 Note that the sign of the stress tensor is opposite to that given in the book by Landau and Lifshitz (1986) but coincides with the

sign chosen by Khalatnikov (2000).

90

Vortex array in a rotating superfluid

3.2 Symmetries of periodic vortex textures As in elasticity theory, symmetry analysis of the periodic vortex arrays in rotating superfluids is an important tool for studying their dynamics. A crucial feature of vortex arrays is that they are periodic only in the plane normal to the rotation axis, and are translationally invariant along the rotation axis. Symmetry classification of vortex structures in rotating superfluids should follow along the same lines as in solids. Macroscopic properties of periodic structures, on scales which are large compared to their period, are entirely determined by the symmetry of point groups, or the ‘symmetry of directions’ (Landau and Lifshitz, 1982b, Chapter XIII). With respect to such symmetry, all periodic structures are divided into crystal classes. Each class corresponds to some point group, which is compatible with the periodic crystal lattice, and belongs to some crystal system, or syngony, characterised by one or a few Bravais lattices. The Bravais lattice is obtained from some point (node) of the structure by periodic translations with respect to which the periodic structure is invariant. If two Bravais lattices are compatible with the same crystal classes (point groups), they belong to the same syngony. We start first from crystal classes in two-dimensional space, ignoring for a while time reversal as a possible element of symmetry. In two-dimensional space there are 4 syngonies and 5 Bravais lattices (instead of 7 syngonies and 14 Bravais lattices in three-dimensional space). Following Landau and Lifshitz (1982b, 1984, 1986) we use Schoenflies notation for crystal classes. Only classes Cn and Cnv with n = 1, 2, 3, 4 and 6 are possible in twodimensional space. The class Cn is the point group of the n-fold rotation axis (always the vertical z axis). In the class Cnv , the group Cn is supplemented by n vertical planes (lines in the plane) of reflection denoted by σv . The class C1v includes only the identity element E and the reflection σv and is also denoted as Cs . In total there are 10 crystal classes, instead of 32 classes in three-dimensional space (Buerger, 1963). Syngonies, Bravais lattices, and crystal classes in two-dimensional space are shown in Fig. 3.3. However, vortex structures are distributions of currents and moments, just like the magnetic structures in solids. Therefore their symmetry should be described by magnetic crystal classes that supplement the elements of the point groups with time reversal (Landau and Lifshitz, 1984). Rotation of the fluid breaks time-reversal symmetry, and only magnetic crystal classes, which allow angular momentum along the vertical rotation axis, are relevant for vortex structures. First let us consider the case when all currents are in the plane (as in the VLL state introduced in the previous section). The crystal classes Cn without the timereversal symmetry added can also be considered as possible magnetic classes for vortex structures, but the classes Cnv cannot because they include reflection symmetry in the vertical plane denoted as σv . A reflection plane allows currents only in this plane, whereas in vortex structures currents must appear in the horizontal plane. But there are possible nontrivial magnetic classes, in which the time-reversal symmetry element is absent alone, but still possible in combination with other symmetry elements (Landau and Lifshitz, 1984). The notation A(B) for such classes means that B is a subgroup of the group A. The subgroup B does not contain the time-inversion symmetry element R at all, whereas all

3.2 Symmetries of periodic vortex textures

Syngonies

Bravais lattices

Crystal classes

91

Magnetic classes

centred

Rectangular

primitive

Oblique

Quadratic

Hexagonal

Figure 3.3 Symmetry of vortex structures (see the text).

other elements of the group A outside B are combined with R. In two-dimensional space, possible non-trivial magnetic classes are Cnv (Cn ). In these classes the reflection element σv is always combined with the time-inversion symmetry element R, and the combined reflection–time-inversion symmetry σv R does not forbid the existence of currents in the horizontal plane (angular momentum along the vertical axis). All magnetic classes in the two-dimensional case, trivial and non-trivial, are shown in Fig. 3.3 in the column ‘Magnetic classes’ on the left of the vertical dashed line. Although all the vortex structures are periodic only in the horizontal plane, not all of them have currents fully confined to the plane. Some examples (continuous vortices in superfluid 3 He-A) will be considered later. This extends the list of possible point symmetry groups of vortex structures to groups Sn , Dn and Dnd . The group Sn corresponds to the n-fold axis of improper rotation (or rotoreflection), which is accompanied by reflection in the horizontal plane. In particular, the group S2 contains only two elements, the identity operation E and the 180◦ rotation around the vertical axis accompanied by reflection in the horizontal plane, which reduces to inversion in three-dimensional space. Therefore this group is also denoted by Ci . The group Dn contains, besides Cn , n twofold axes U2 in the horizontal plane. The group Dnd is the group Dn supplemented by n vertical reflection planes located between the horizontal axes U2 and called diagonal reflection planes σd . Note that if currents are confined to the horizontal plane, the symmetry operations σv and

92

Vortex array in a rotating superfluid

U2 are equivalent and the reflection in the horizontal plane σh is the identity operation. Then we return to the classes listed in Fig. 3.3 in the column ‘Magnetic classes’ on the right of the vertical dashed line. In total, we arrive at 8 trivial magnetic classes (Cn and Sn ) and 13 non-trivial classes [Cnv (Cn ), Dn (Cn ), and Dnd (S2n )], which are relevant to vortex structures in rotating superfluids (Sonin and Fomin, 1985). The 13 non-trivial classes are among the 18 non-trivial magnetic classes allowing ferromagnetism, i.e., spontaneous magnetic moment, which behaves on symmetry transformation as the angular momentum in rotating superfluids listed by Landau and Lifshitz (1984) (see the problem to Section 38 there). Our list does not include 5 ferromagnetic classes with subgroups Cnh , which contain reflection σh in the horizontal plane which is not compatible with the existence of vertical components of currents. It is necessary to stress that a definition of a plane as vertical or horizontal is a matter of convention in three-dimensional solids, while in vortex structures only the rotation axis should be defined as vertical. Therefore, some classes listed by Landau and Lifshitz (1984) as ferromagnetic are present in our list under other notation. The ferromagnetic class Cs (C1 ) = C1h (C1 ), which contains two elements E and σh R and allows the spontaneous moment in the horizontal plane, corresponds in our list to the class C1v (C1 ) with two elements E and σv R, which allows angular momentum in the vertical plane. Similarly, the ferromagnetic classes C2 (C1 ) and C2h (S2 ) are identical to our classes D1 (C1 ) and D1d (S2 ), respectively, after renaming the vertical axis as horizontal and vice versa.

3.3 Elastic moduli and linear equations of motion Now we descend from the general non-linear theory to the linear theory, which will be used further in the book. In the harmonic approximation, the elastic energy of a deformed crystal is a quadratic function of components of the deformation tensor uij =

1 (∇i uj + ∇j ui ), 2

(3.36)

where indices i and j take values from 1 to 3, corresponding to the three space coordinates x, y, and z respectively. In general the expression for the elastic energy of a crystal is Ev =

1 λik,lm uik ulm , 2

(3.37)

where λik,lm is the symmetric tensor of elastic moduli. Since the symmetric deformation tensor has 6 components, the tensor λik,lm has in general 21 components. In the theory of superconductivity (Brandt, 1995), Voigt’s system of notation is used for elastic moduli (Love, 1944), and we shall also use this later in the book. In this system the 6 components of the symmetric tensor of deformations are numbered from 1 to 6: U1 = uxx , U2 = uyy , U3 = uzz , U4 = 2uyz , U5 = 2uxz , and U6 = 2uxy . Then the elastic energy of a crystal is

3.3 Elastic moduli and linear equations of motion

Ev =

93

1 Cαβ Uα Uβ , 2

(3.38)

where α and β take values from 1 to 6. For vortex lattices, only displacements in the xy plane normal to the vortex lines have a physical meaning. This excludes the deformation uzz and, correspondingly, Voigt’s elastic moduli with at least one index 3 in the subscript vanish. As a result, the number of elastic moduli decreases from 21 to 15. A further decrease in the number of independent elastic moduli occurs for special symmetries of the vortex lattice. The most interesting case is a triangular vortex lattice, which belongs to the hexagonal crystal system (singony). There are only 5 independent elastic moduli in this system (Landau and Lifshitz, 1986), which for the vortex lattices without deformation uzz reduce to 3 moduli. One can choose these moduli to be the in-plane compression modulus C11 , the tilt modulus C44 , and the shear modulus C66 . Then the most general expression for the elastic energy of the triangular lattice is Ev = 2C44 (u2xz + u2yz ) +

C11 − C66 C66 (uxx + uyy )2 + [(uxx − uyy )2 + 4u2xy ], (3.39) 2 2

or in terms of two-dimensional displacements u in the xy plane: C44 Ev = 2



∂u ∂z

2

C11 C66 (∇ · u)2 + + 2 2



∂uy ∂ux + ∂y ∂x

2

∂ux ∂uy −4 ∂x ∂y

 .

(3.40)

All the other 12 Voigt elastic moduli are determined by these 3 moduli or they vanish: C22 = C11 , C55 = C44 , C12 = C11 − 2C66 , C14 = C15 = C16 = C24 = C25 = C26 = C45 = C46 = C56 = 0.

(3.41)

In terms of elastic moduli, the expression (3.29) for the elastic force becomes f el = C44

∂ 2u + (C11 − C66 )∇ ⊥ (∇ · u) + C66 ∇ 2⊥ u. ∂z2

(3.42)

Here ∇ ⊥ (∇x , ∇y ) is the two-dimensional vector of the gradient in the xy plane. According to Eq. (3.23), this means that the local fluid velocity is vl = v +

∂ 2 u C11 − C66 C44 C66 zˆ × 2 − zˆ × ∇ ⊥ (∇ · u) + zˆ × ∇ 2⊥ u. 2ρ 2ρ 2ρ ∂z

(3.43)

After introducing the elastic moduli, the linearised equation (3.30) of vortex motion, which is valid if the elastic force f el is the only force on the vortices, becomes −ρ [2 × (v L − v)] = C44

∂ 2u + (C11 − C66 )∇ ⊥ (∇ · u) + C66 ∇ 2⊥ u. ∂z2

(3.44)

94

Vortex array in a rotating superfluid

In terms of the elastic moduli one can rewrite Eq. (3.34) for the elastic stress tensor as follows: ∂ui δj z − [(C11 − 2C66 )(∇ · u)δij ∂z + C66 (∇i uj + ∇j ui )](1 − δiz )(1 − δj z ).

σij = −C44

(3.45)

Equation (3.44) together with the continuity and the Euler equations, ∂ρ  + ρ(∇ · v) = 0, ∂t

(3.46)

∂v ∇P  + 2 × v L = − , ∂t ρ

(3.47)

constitute the complete system of linearised equations of motion for macroscopic hydrodynamics. Here ∇P  = cs2 ∇ρ  , where cs = (∂P /∂ρ)1/2 , is the sound velocity. Equations are written in the rotating coordinate frame in which the velocities v and v L are small. Later we shall need the linear equations of macroscopic hydrodynamics for inhomogeneous fluids where the density, the sound velocity, and the elastic moduli vary in space: ∂ρ  + ∇(ρ · v) = 0, ∂t

(3.48)

∂v c2 + 2 × v L = − s ∇ρ  , ∂t ρ

(3.49)

  ∂u ∂ C44 + ∇ ⊥ [(C11 − C66 )(∇ · u)] −ρ [2 × (v L − v)] = ∂z ∂z + ∇ ⊥ (C66 ∇ ⊥ u).

(3.50)

3.4 Hall–Vinen–Bekarevich–Khalatnikov hydrodynamics The Hall–Vinen–Bekarevich–Khalatnikov (HVBK) hydrodynamics ignored the dependence of the energy on shear deformation of the vortex array connected with the crystalline ˜ is equal order in the vortex array. The theory assumed that the vortex energy density Ev (ω) to the energy ε of a single vortex line per unit length [see Eq. (2.6)] multiplied by the two-dimensional density of vortex lines ω/κ: ˜ Ev = ε

rm ρκ ω˜ = ρνs ω˜ = ω˜ ln . κ 4π rc

(3.51)

The upper cut-off rm in the logarithm argument must be of the order of the intervortex distance  √ 1/2 . (3.52) rv = 2κ/ 3ω˜

3.4 Hall–Vinen–Bekarevich–Khalatnikov hydrodynamics

95

The variation of the energy produced by variation of the vorticity with logarithmic accuracy ˜ is given by (i.e., neglecting the dependence of rv on ω)   ∂Ev ω˜ · d ω˜ d ω˜ = dRρνs , (3.53) δE = dR ∂ ω˜ ω˜ The variation d ω˜ is connected with the Lagrange variation of the displacement Du with the exact kinematic relation, which is a generalisation of Eq. (3.11): ˜ d ω˜ = −(Du · ∇)ω˜ − ω(∇ · Du) + (ω˜ · ∇)Du.

(3.54)

Equation (3.11) follows from this relation after the substitution d → ∂/∂t and Du → du/dt = v L . We substitute the expression (3.54) for d ω˜ into Eq. (3.53). After some integrations by parts and comparison with Eq. (3.28), one obtains the expression for the elastic force on the vortex in the HVBK theory: ) * (3.55) f el = −ω˜ × ∇ × (ρνs sˆ ) , ˜ ω˜ is a unit vector tangent to the vortex lines and parallel to the vorticity ω. ˜ where sˆ = ω/ The elastic force f el is the force per unit volume equal to the product of the line tension force per unit length of a single vortex and the area vortex density ω/κ. ˜ Equations (3.22) and (3.55) yield the expression for the vortex velocity in the absence of external forces when f = fel : vL = vl = v +

1 ∇ × (ρνs sˆ ) = v + ∇ × (νs sˆ ) + ∇ ln ρ × sˆ . ρ

(3.56)

The term ∝ ln ρ is connected with a force on the vortex arising in an inhomogeneous compressible superfluid, which will be discussed at the end of Section 4.5. In a uniform incompressible fluid the elastic force on vortices is ) * R ˜ s · ∇)ˆs = ρνs ω˜ 2 , f el = −ρνs ω˜ × ∇ × sˆ = ρνs ω(ˆ R

(3.57)

while the vortex velocity is given by v L = v + νs ∇ × sˆ .

(3.58)

The Gibbs relation in the HVBK theory for the energy in the coordinate frame moving with the average fluid velocity v is ˜ dEs = μs dρ + ρνs d ω,

(3.59)

while the Gibbs–Duhem relation for the pressure P = −Es + μs ρ is ˜ dP = ρdμs − ρνs d ω.

(3.60)

Our equations satisfy the conservation laws for the momentum [Eq. (3.31)] and the energy [Eq. (3.31)]. In the HVBK theory the stress tensor σij , which appears in the momentum

96

Vortex array in a rotating superfluid

flux tensor and the energy flux [Eqs. (3.33) and (3.35)], is given by   ω˜ i ω˜ j ˜ ij − . σij = ρνs ωδ ω˜ So the expression for the whole momentum flux tensor in the HVBK theory is   ω˜ i ω˜ j ij = P δij + ρvi vj + ρνs ωδ ˜ ij − . ω˜

(3.61)

(3.62)

Our hydrodynamical equations are identical with those of Bekarevich and Khalatnikov (1961) [see also Khalatnikov (2000)] at T = 0. The presence of vortex lines makes the fluid anisotropic. This leads to ambiguity in the definition of pressure. In contrast to an isotropic fluid, the standard thermodynamic definition of pressure as a derivative of the energy with respect to the volume now depends on how the form of the volume element varies during variation of the volume. Another way to define pressure is via the force (per unit area) from the fluid on a solid wall confining the fluid. It is determined by the diagonal element of the momentum flux tensor ii assuming that the ith coordinate axis is normal to the wall. Near a solid surface the velocity normal to the surface vanishes, and in an isotropic fluid the force per unit area is exactly the pressure. ˜ If the solid In the HVBK hydrodynamics, ij contains the term related to the vorticity ω. surface is normal to the vortex lines, the vortex stress tensor [the last term in Eq. (3.62)] vanishes and the force on the wall coincides with the pressure, as it was defined above. But ˜ As a matter if the solid surface is parallel to the vortex lines, the force becomes P + ρνs ω. of choice one may redefine the pressure, including the vorticity term ρνs ω˜ in it. Let us find the ground state of a rotating superfluid in the HVBK theory. One looks for the ground state by minimisation of the energy E = E −  · M,

(3.63)

in the rotating coordinate frame [see Eq. (1.131)]. Here E is obtained by integration of the energy density E given by Eq. (3.13) with the energy Ev determined by Eq. (3.51) and  M = ρ [r × v(r)] dr (3.64) is the angular momentum of the fluid. Choosing the z axis along the rotation axis and neglecting the energy E0 of the resting fluid, the energy is    1 2 ˜ . ρv − ρv · [ × r] + Ev (ω) (3.65) E = dr 2 Varying E with respect to the velocity v and keeping in mind that ω˜ = ∇ × v, one obtains the Euler–Lagrange equation ρ(v − [ × r]) + νs ∇ × (ρ sˆ ) = 0.

(3.66)

Bearing in mind the expression (3.56) for the vortex velocity, this condition means that in the ground state the vortex lattice rotates as a solid body with v L = [ × r]. If the

3.4 Hall–Vinen–Bekarevich–Khalatnikov hydrodynamics

97

vortex lattice did not rotate as a solid body, a mutual friction force would appear in the presence of a normal component. This is impossible in the ground state. In a rotating incompressible fluid with constant mass density, vortex lines are uniformly spaced at a constant two-dimensional density nv given by the Feynman formula (3.1), and the fluid also rotates on average as a solid body: v = v L . But the Feynman formula is exact only in a uniform fluid with constant mass density. In a non-uniform fluid, according to Eq. (3.66), v = [ × r] − νs [∇ ln ρ × sˆ ] = 0.

(3.67)

So the average fluid velocity does not imitate solid body rotation, although the vortex lattice does. Taking the curl of the expression (3.67) one obtains nv =

 2 1 rv  |∇ × v| = − ln ∇ × [∇ ln ρ × sˆ ] . κ κ 4π rc

(3.68)

The correction to the Feynman formula for the vortex density was discussed for a rotating non-uniform BEC cloud in a trapping potential (Anglin and Crescimanno, 2002; Sheehy and Radzihovsky, 2004). We shall address the role of this correction in Section 4.5. Now let us turn to the linear HVBK theory and derive elastic moduli. The HVBK theory ignores the shear rigidity of the vortex array, and the term in Eq. (3.40) proportional to the shear modulus C66 is absent. The expansion of the vortex energy Ev for a small deviation ω˜  = ω˜ − 2

(3.69)

of the vorticity ω˜ from the equilibrium value 2 includes terms of first order in ω˜  . But the variation of vorticity is not independent, being coupled with the variation of the velocity of the fluid. In the correct theory, first-order terms in ω˜  should be cancelled out by other first-order terms, so in the expansion of the HVBK vortex energy, Eq. (3.51), we retain only terms of the second order in ω˜  : 1 ∂ 2 Ev   ω˜ ω˜ 2 ∂ ω˜ i ∂ ω˜ j i j   1 ∂ 2 Ev  1 ∂Ev (ω˜  · )2 2 + ω˜ − = Ev0 + (ω˜ · )2 . 4 ∂ ω˜ 2 22 ∂ ω˜ 2

Ev ≈ Ev0 +

(3.70)

The deviation of vorticity ω˜  is connected with small displacements u of vortex lines by the linearised version of Eq. (3.54) after replacing d ω˜ by ω˜  and Du by u: ω˜  = −2(∇ · u) + 2( · ∇)u.

(3.71)

Now we can calculate the derivatives ∂Ev /∂ ω˜ and ∂ 2 Ev /∂ ω˜ 2 using Eq. (3.51), substitute them together with ω˜  given by Eq. (3.71) into Eq. (3.70), and compare the obtained expression with the first two terms in Eq. (3.40). This enables us to deduce values of two elastic moduli: C44 = 2ρνs ,

C11 = −

ρκ , 8π

(3.72)

98

Vortex array in a rotating superfluid

where the line tension parameter νs is given by Eq. (2.50) for a single vortex with the only difference that the upper cut-off rm is now of the order of the intervortex distance rv (instead of the Kelvin wavelength for a single line). The negative compression modulus C11 does not lead to instability since, as we have already mentioned, the displacements are not completely independent variables: the longitudinal part of the displacement field is connected with the variation of vorticity and therefore with the fluid velocity (Baym and Chandler, 1983). The force related with the tilt modulus C44 is a line tension force obtained in the local induction approximation. All other contributions to the force or to the local velocity given by Eq. (3.43) are contributions beyond the local induction approximation and take into account interaction with other vortices. The average fluid velocity v in Eq. (3.43) is a mean-field approximation for this interaction, similar to the self-consistent electric field on a charge induced by long-range interaction with other charges. This mean-field approach has already been used in the dynamics of helical vortices and rings (Sections 2.10 and 2.11), but there the velocity v was induced by distant parts of the same vortex line. Simple calculation of two elastic moduli in the HVBK theory was possible because the theory used the logarithmic approximation, neglecting numbers of order unity as small compared to the large logarithm. But these numbers, which are different for different types of vortex lattice, determine the shear rigidity. The elastic energy is in fact the kinetic energy of the velocity field induced by the vortices, and scaling estimation shows that the shear modulus should be of the order of C66 ∼ ρκ, i.e., of the same order as C11 . Exact determination of the shear elastic modulus C66 involves more ingenious calculations done by Tkachenko which are discussed in the next section.

3.5 Tkachenko shear rigidity Tkachenko found exact values of the energy and the oscillation spectrum for an arbitrary vortex lattice using the theory of elliptic functions on the complex plane (Tkachenko, 1965, 1966). His analysis, which was called by Dyson (1971) ‘a tour de force of powerful mathematics’, provided the exact value of the shear modulus of the vortex lattice (Sonin, 2013a). Tkachenko’s theory was two-dimensional and used the complex representation for twodimensional vectors. We have already discussed the complex representation with the imaginary unit j (j 2 = −1) at the end of Section 2.9. A complex variable r˜ = x + jy represents a two-dimensional position vector r(x, y) in the xy plane.2 Then the velocity field v˜ = vx + j vy induced by vortices located at nodes of a vortex lattice with position vectors r˜kl = 2k ω˜ 1 + 2l ω˜ 2 (k and l are arbitrary integers) is given by κ ∗ [ζ (˜r ) − λ˜r ∗ ], (3.73) v(˜ ˜ r) = 2π

2 The more frequent notation z for a complex variable is not used since throughout this book the letter z denotes the third

coordinate, which is absent in Tkachenko’s theory.

99

b

3.5 Tkachenko shear rigidity

a

a Figure 3.4 Vortex lattice before (solid lines) and after (dashed lines) shear deformation. Figure from Sonin (2005b).

where λ is a constant, which will be defined below,   1 1  1 r˜ ζ (˜r ) = + + + 2 r˜ r˜ − rkl rkl rkl k,l

(3.74)

is the quasiperiodic Weierstrass zeta function (Abramowitz and Stegun, 1972) with two complex semi-periods ω˜ 1 and ω˜ 2 , and a prime means exclusion of the term k = l = 0 from the sum. The quasiperiodicity conditions are ζ (˜r + 2k ω˜ 1 ) = ζ (˜r ) + 2k ω˜ 1 , ζ (˜r + 2l ω˜ 2 ) = ζ (˜r ) + 2l ω˜ 2 .

(3.75)

The lattice is shown in Fig. 3.4 for the semi-periods ω˜ 1 = a/2 and ω˜ 2 = bej α /2. The area of a unit cell of the lattice is A = 4Im(ω˜ 1∗ ω˜ 2 ) = ab sin α.

(3.76)

Tkachenko showed that a lattice with arbitrary semi-periods rotates as a solid body with the angular velocity  = κ/2A , if λ satisfies the condition ζ (ω˜ 1 ) + λω˜ 1 = ω˜ 1∗ .

(3.77)

100

Vortex array in a rotating superfluid

Taking into account the exact relation for the Weierstrass zeta function, ω˜ 2 ζ (ω˜ 1 ) − ω˜ 1 ζ (ω˜ 2 ) =

iπ , 2

(3.78)

another condition necessary for solid body rotation is also satisfied: ζ (ω˜ 2 ) + λω˜ 2 = ω˜ 2∗ .

(3.79)

For symmetric triangular and quadratic lattices λ = 0. The velocity field being known, after ingenious manipulations with integrals over elliptic functions Tkachenko (1965) found the exact value of the energy density E = E − Mz in the rotating coordinate frame for an arbitrary vortex lattice:     1  2|ω˜ 1 ω˜ 2 |1/2 ρκ ln 2    θ 0, − ln − (0, τ )θ E = 1 2π π rc 3  1 τ  (3.80)    √   ρκ 1 A|τ | ln 2   = θ (0, τ )θ1 0, −  , ln √ − 2π π rc τI 3  1 τ where the complex parameter τ = τR + j τI = ω˜ 2 /ω˜ 1 =

b jα e a

(3.81)

determines the type of lattice, θ1 (˜z, q) = −i

∞ 

2

(−1)n q (n+1/2) ej (2n+1)˜z

(3.82)

n=−∞

is one of the elliptic theta functions (Abramowitz and Stegun, 1972), and θ1 (˜z, q) is its derivative $ with respect to the first argument z˜ . The energy has a minimum at τ = ej π/3 √ (a = b = κ/ 3, α = π/3), which corresponds to the triangular lattice with the energy density   √  √  A κ/2 ρκ ρκ ln E = − 1.321 = − 1.321 . (3.83) ln 2π rc 2π rc Comparing this with Eq. (3.51) with ω˜ = 2 and the intervortex distance given by Eq. (3.52), one may conclude that the exact value of the upper cut-off rm in the expression for the energy of the triangular lattice must be #√ 3 −1.321 e rv = 0.248rv . (3.84) rm = 2 In order to find the shear modulus, let us deform the triangular lattice without varying the vortex density as shown in Fig. 3.4. Then only the real part of τ varies proportionally to the shear deformation uxy = 12 (∇y ux + ∇x uy ): δτ = δτR = 2uxy sin α. Expanding

3.6 Spectrum of oscillations in an incompressible fluid

101

the energy density Eq. (3.80) with respect to τR and comparing it with the elastic energy (3.40), one obtains the exact value of the shear modulus: C66 =

ρκ . 8π

(3.85)

3.6 Spectrum of oscillations in an incompressible fluid In an incompressible fluid the velocity field is purely transverse in the sense explained in Section 1.6. Therefore we keep only the transverse part of the Euler equation (3.47): ∂v + [2 × v L ]⊥ = 0. ∂t

(3.86)

The gradient term on the right-hand side of Eq. (3.47), which is longitudinal, cancels exactly with the longitudinal part of the vector product term on the left-hand side. Integration of this equation yields a linear relation between the fluid velocity and the displacements u: v = −[2 × u]⊥ = −[2 × u ].

(3.87)

This equation must be solved together with the equation of vortex motion (3.44). We look for plane-wave solutions ∝ eipz+ik·r−iωt and for them Eqs. (3.44) and (3.86) are −iωv + 2 × v L − [K · (2 × v L )]

K = 0, K2

ρ [2 × (v L − v)] = C44 p2 u + C11 k(k · u) + C66 [k 2 u − k(k · u)].

(3.88) (3.89)

Equations (3.88) and (3.89) look like three-dimensional vector equations. They are, however, effectively two-dimensional, since the vectors u and v L = −iωu have components only in the xy plane, and the velocity component vz can be excluded with the help of the incompressibility condition K · v = 0. As in the continuous vorticity model (Chapter 1), in the xy plane we choose axes parallel and normal to the vector k and denote the corresponding components by the subscripts k and t. In terms of k and t components, Eqs. (3.88) and (3.89) take the form −iωvk − 2

p2 vLt = 0, K2

(3.90)

−iωvt + 2vLk = 0, −iω(vLt − vt ) = − iω(vLk

C44 p2 C11 k 2 vLk − vLk , 2ρ 2ρ

C44 p2 C66 k 2 vLt − vLt . − vk ) = − 2ρ 2ρ

(3.91)

102

Vortex array in a rotating superfluid

These equations have a solution if the following dispersion relation holds:    p2 C44 p2 C44 p2 C11 k 2 C66 k 2 2 2 2 + . ω = 2 + + + 2ρ 2ρ 2ρ 2ρ K

(3.92)

Within macroscopic hydrodynamics the wavelength must essentially exceed the intervortex distance. Bearing in mind the value of the compression modulus C11 given by Eq. (3.72), the term ∝ C11 in the first multiplier in Eq. (3.92) is always small compared with the term 2 and can be ignored. But the term ∝ C66 in the second multiplier must be retained since there the term 2 is multiplied by the factor p2 /K 2 , which can be small. On the other hand, the term ∝ C44 must be retained in both multipliers since according to Eqs. (3.72) and (2.50) the tilt modulus C44 exceeds the compression and the shear moduli C11 and C66 by a large logarithmic factor. This leads to the conclusion that the compression modulus C11 has no essential effect on vortex dynamics and the term ∝ C11 in the equation of vortex motion (3.89) can be ignored without any loss of accuracy. Later, we shall use a simpler version of the equation of motion (3.44) with only two elastic moduli C44 and C66 : −ρ [2 × (v L − v)] = C44

∂ 2u + C66 [∇ 2⊥ u − ∇ ⊥ (∇ ⊥ · u)], ∂z2

(3.93)

which in the Fourier representation is −ρ [2 × (v L − v)] = −C44 p2 u + C66 [k 2 u − k(k · u)].

(3.94)

This equation of vortex motion together with Eq. (3.87) describes oscillations with the spectrum    p2 C66 k 2 C44 p2 C44 p2 2 2 + + . (3.95) ω2 = 2 + 2ρ 2ρ 2ρ K Further analysis of oscillation modes in an incompressible fluid will be carried out for different particular cases separately.

3.7 Axial modes of vortex oscillations We call waves axial when their wave vectors are directed along the z axis (k = 0, K = p). According to Eqs. (3.43) and (3.72), the local fluid velocity at the points on vortex lines is ∂ 2u . (3.96) ∂z2 Since v L = v l , an axial wave is described by two equations in the Fourier representation, v l = v + νs zˆ ×

v = −2 × u,

(3.97)

zˆ × (v L − v) = νs p2 u,

(3.98)

following from Eqs. (3.88) and (3.89). They have a solution if ω2 = (2 + νs p2 )2 .

(3.99)

3.7 Axial modes of vortex oscillations

103

These are circularly polarised Kelvin waves in a vortex array, which were studied extensively from the 1950s onward. The waves involve motion only within the xy plane, and all displacements and velocities are two-dimensional. The spectrum of axial modes in macroscopic hydrodynamics differs from the spectrum of Kelvin modes of a single vortex line (Section 2.9) by the gap 2 and by another choice of the upper cut-off rm in Eq. (2.50) for νs (the intervortex distance rv instead of rm of the order of the wavelength 2π/p for the single-vortex Kelvin mode). The gap is connected with the presence of the average velocity v in the expression (3.96) for the local velocity, which is absent in the case of a Kelvin wave along a single vortex line. This results partially from transformation of the equation for the vector field u to the rotating coordinate frame and partially from long-range interaction between vortices, in analogy with the gap in the plasma-oscillation spectrum which is a result of Coulomb interaction. Our derivation of the √ axial vortex mode from macroscopic hydrodynamics is valid as long as prv ∼ p κ/  1. Despite this condition restricting the line tension term νs p2 from above, it can be of the same order or larger than the gap 2 because the line tension parameter νs exceeds by a large logarithm factor the circulation quantum κ. The analysis of Kelvin waves along a single isolated vortex line in Section 2.9 demonstrated that a Kelvin wave perturbs fluid motion at distances from the line of the order of the Kelvin wavelength λ = 2π/p. If prv 1 this distance is much shorter than the intervortex distance. So oscillatory motion of any vortex line is fully detached from oscillations of neighbouring lines in the vortex array. However, we are looking for the spectrum of vortex oscillation in the rotating coordinate frame. The equations (2.62) of motion for the singlevortex Kelvin mode written in the laboratory coordinate frame assumed that the fluid does not oscillate on average, i.e., v = 0. Then in the rotating coordinate frame v = − × u.

(3.100)

This differs from Eq. (3.97) by a factor 2 because Eq. (3.97) also takes into account coupling with oscillations of other vortices. Eventually the dispersion relation becomes ω2 = ( + νs p2 )2 .

(3.101)

The difference between the Kelvin wave spectra for prv  1 and prv 1 was noticed by Raja Gopal (1964) many decades ago (see also Andereck and Glaberson, 1982).3 A consequence of the gapped spectrum (3.99) is that the axial mode in a rotating fluid does not propagate as a wave at frequencies lower than the gap 2. The mode is an √ evanescent wave with an imaginary wave number p, which goes to the value ±i 2/νs in the zero-frequency limit. So slow oscillations cannot penetrate into the fluid farther than the layer of width  νs E = , (3.102) 2 3 In the past it was stated (Sonin, 1987, the end of Section IV.D) that the Kelvin wave spectrum is the same independently of

whether prv  1 or prv 1. This statement was erroneous.

104

Vortex array in a rotating superfluid

which in analogy with the Ekman layer width δE in a viscous fluid [see Eq. (1.144)] can be called the superfluid Ekman layer width, or the superfluid Ekman depth. The superfluid Ekman depth plays a very impotent role in vortex dynamics in superfluids.4 In the j -complex representation introduced in Section 2.9, our hydrodynamical equations (3.90) and (3.91) for axial modes become more compact: −iωv˜ + 2j v˜L = 0,

(3.103)

νs p2 j v˜L = 0. (3.104) iω Eigenfrequencies of axial modes correspond to zeros of the complex determinant of Eqs. (3.103) and (3.104): v˜L = v˜ +

D(j ) = iω − j (2 + νs p2 ).

(3.105)

ω = −ij (2 + νs p2 ),

(3.106)

Then

and relations between the velocity components are vy = −j vx , vLy = −j vLx .

(3.107)

We arrive at explicit formulas for axial modes by replacing j by ±i. The two signs correspond to the two possible senses of circular polarisation.

3.8 Tkachenko waves: elasticity theory of a two-dimensional vortex lattice Let us now consider in-plane modes when p = 0 and the wave propagates in the xy plane normal to the rotation axis. Since displacements also lie in the xy plane, the problem becomes purely two-dimensional, and after omitting the vortex bending ∂u/∂z in Eq. (3.93) it reduces to C66 ∂u = vL = v + [ˆz × [∇ 2⊥ u − ∇ ⊥ (∇ ⊥ · u)]. (3.108) ∂t 2ρ Dividing the field of vortex displacements u into longitudinal and transverse parts, u = u + u⊥ (∇ · u⊥ = 0, ∇ × u = 0), and using Eq. (3.87) for exclusion of the fluid velocity v, we obtain in the long wavelength limit two equations for displacements: ∂u C66 = [ˆz × ∇ 2⊥ u⊥ ], ∂t 2ρ

(3.109)

∂u⊥ = −[2 × u ]. ∂t

(3.110)

4 The term ‘superfluid Ekman layer’ was suggested by Alpar (1978) while analysing relaxation of superfluid 4 He after spin up

of the container. However, in Eq. (3.102) Alpar used the circulation quantum κ instead of the line tension parameter νs , which differs from κ by a logarithmic factor.

3.8 Tkachenko waves: two-dimensional vortex lattice

105

Excluding the small longitudinal displacement u from the equations one obtains the wave equation ∂ 2 u⊥ = cT2 ∇ 2⊥ u⊥ ∂t 2

(3.111)

describing the Tkachenko wave. Plane Tkachenko waves ∝ eik·r−iωt have the sound-like spectrum ω = cT k with the Tkachenko velocity # cT =

C66 = ρ



κ . 8π

(3.112)

One can rewrite Eq. (3.111) as (the subscript ⊥ is omitted) ρ

∂ 2 ui = −∇j σij . ∂t 2

(3.113)

Here σij = −C66 (∇i uj + ∇j ui ) = −ρcT2 (∇i uj + ∇j ui )

(3.114)

is the elastic stress tensor, as follows from Eq. (3.45) assuming that ∂u/∂z = 0 and ∇ · u = 0. Subscripts i and j take only two values corresponding to the two axes in the xy plane. Vortices in the Tkachenko wave move on elliptical paths, but the longitudinal component u parallel to the wave vector k is proportional to a small factor ω/  [see Eq. (3.110)]. Thus it is fairly accurate to consider the Tkachenko wave as a transverse sound wave in a two-dimensional lattice of rectilinear vortices (Tkachenko, 1969). Comparing Eqs. (3.87) and (3.110), one can see that in our approximation the fluid and the vortices move with the same velocity. We see that the elasticity theory of the vortex lattice contains only a single elastic modulus, the shear modulus C66 = ρcT2 . Formally one arrives at this version of the elasticity theory (Landau and Lifshitz, 1986) in the limit of the infinite compression modulus that rules out longitudinal displacements (Ignatiev and Sonin, 1981). Strictly speaking, longitudinal displacements are not ruled out, but are excluded from the equations. However small, they remain finite and are not independent variables, as in the atomic crystal. This is a peculiar feature of vortex dynamics discussed earlier: vortices have two times less degrees of freedom than particles. Small longitudinal displacements of vortices generate flow of the fluid with the averaged velocity v, because they are coupled with the vorticity field. The motion of the fluid is responsible for the presence of an effective force of inertia on the lefthand side of the dynamical equation Eq. (3.111) despite the fact that vortices themselves have no mass. In the stationary case, longitudinal displacements are exactly ruled out by Eq. (3.110).

106

Vortex array in a rotating superfluid

3.9 Slow mode in an incompressible perfect fluid The general dispersion law for an incompressible fluid, Eq. (3.95), shows that low frequency oscillations ω   are possible only if p  k. This means that the Taylor– Proudman theorem (Chapter 1) holds in a perfect fluid with quantised vorticity too: the fluid in a state of slow motion is homogeneous along the rotation axis. Slow oscillation with small but finite p will be called the slow mode.5 The slow mode is described by the equations of motion (3.90) and (3.91) where all elastic moduli except for the shear modulus C66 = ρcT2 are neglected: p2 vLt = 0, K2 −iωvt + 2vLk = 0,

−iωvk − 2

(3.115)

vLt = vt , (3.116) cT2 k 2 vLt . 2 The solution of Eqs. (3.115) and (3.116), together with the incompressibility condition pvz + kvk = 0, yields relations between velocity components in the slow mode: iω(vLk − vk ) = −

vt ≈ vLt ,

ω2 − cT2 k 2 vk 2 p2 , =− = − vt iω K 2 iω2 k 2 pk vz = − vk = vt . p iω K 2

vLk iω , = vLt 2

(3.117)

When the quantum Tkachenko contribution cT2 k 2 increases from zero, the oscillatory motion of the fluid transforms from circularly polarised motion vz ≈ ivt , as in the classical inertial wave, to motion with transverse linear polarisation corresponding to the Tkachenko wave (vt vk , vz ). As for vortices, they move in the xy plane along an elliptical path with its major axis perpendicular to k. The ratio of the axes of the ellipse vLk /vLt is small at ω  , so one can neglect small longitudinal components vk and vLk and consider the slow motion in the xy plane to be transverse with nearly coincident vortex and averaged fluid velocities (vt ≈ vLt ). Excluding from Eqs. (3.115) and (3.116) all the velocity components except for vLt , we obtain   p2 ω2 − 42 2 − cT2 k 2 vLt = 0. (3.118) K This yields the dispersion relation for the slow mode (Sonin, 1976; Williams and Fetter, 1977): ω2 = 42

p2 + cT2 k 2 . K2

5 Earlier it was named the transverse vortex mode (Sonin, 1976).

(3.119)

3.10 Glaberson–Johnson–Ostermeier instability

107

The first term on the right-hand side is of classical origin and yields the frequency of the inertial wave. The second term is due to quantisation of the vorticity and is responsible for the Tkachenko waves. The frequency ω as a function of k at given non-zero p has a minimum (Williams and Fetter, 1977). The values of k and ω in the minimum at prv  1 are given by the expressions √ 2p 2 2 = = 4 2πp/κ, ωm = 4cT p. (3.120) km cT As usual, the minimum on the dispersion curve ω versus k should correspond to a peak of the density of states. This peak will be discussed later (Section 5.9) in connection with attempts to observe Tkachenko waves experimentally. The inverse Fourier transformation of Eq. (3.118) yields the equation for the vortex velocity v L in the configurational space: ⊥

2 ∂ 2vL 2 ∂ vL = −4 + cT2 4⊥ v L , ∂t 2 ∂z2

(3.121)

where ⊥ = ∂ 2 /∂x 2 + ∂ 2 /∂y 2 is the Laplace operator in the xy plane. The vortex velocity v L lies in the xy plane and is approximately divergence-free because its longitudinal component vLk is small compared with the transverse component vLt [see Eq. (3.117)]. Variation of v L along the z axis, which is slow in accordance with the Taylor–Proudman theorem, should not be ignored, since the small derivative ∂ 2 v L /∂z2 in Eq. (3.121) is multiplied by the large factor 42 . This derivative is responsible for the gap in the oscillation spectrum, which is important for the observation of Tkachenko waves in finite containers and will be discussed later (Sections 5.8 and 5.9).

3.10 Glaberson–Johnson–Ostermeier instability Glaberson et al. (1974) (see also Ostermeier and Glaberson, 1975) showed that a mass flow along a vortex line can cause instability with respect to excitation of vortex waves. Let us discuss the threshold for the Glaberson–Johnson–Ostermeier instability in a uniform superfluid rotating with the angular velocity . If an oscillation mode excited in a resting fluid has energy E and momentum P , its energy in a fluid moving with velocity v will be E˜ = E − P · v. This follows from Galilean invariance. The fluid becomes unstable with respect to excitation of the mode if E˜ can be negative. Instability starts from the mode with the momentum P antiparallel to v, and the critical velocity for instability is vcr = min

ω(p) E = min . P p

(3.122)

The equality E/P = ω/p is universal as one can check for any mode. The critical velocity was estimated using the same arguments as for derivation of the Landau critical velocity. Landau (1941) referred to quantum mechanical excitation (quasiparticles) with energy E = hω ¯ and momentum P = hp, ¯ but the derivation is also valid for any classical

108

Vortex array in a rotating superfluid

oscillation mode. One can consider vcr as the Landau critical velocity for the superflow along the vortex line when the latter becomes unstable with respect to generation of kelvons (quanta of Kelvin waves). Ostermeier and Glaberson (1975) analysed the stability of a single vortex in a rotating coordinate frame using the spectrum ω =  + νs p2 [see Eq. (3.101)] which is valid for an intervortex distance exceeding the Kelvin wavelength 2π/p. This yields the critical √ velocity vcr = 2 νs . It is interesting also to estimate the stability of the axial superflow for a dense vortex array. Neglecting shear (Tkachenko) rigidity in the general dispersion relation (3.95) of plane waves in a rotating superfluid and using the expression (3.72) for the tilt modulus C44 , the phase velocity of the plane wave is #   ω 2 (3.123) = (2 + νs p2 ) 2 + ν s . p k + p2 √ √ The minimum of ω/p is at small p  2/νs and large k 2/νs , and the critical velocity for the Glaberson–Johnson–Ostermeier instability is ω " (3.124) vcr = min = 2νs . p So the instability starts not from mostly axial waves (p k) but from waves with p  k propagating all but normally to the rotation axis. The critical velocity is smaller by a factor √ 2 than the value obtained for a single vortex. Glaberson et al. (1974) suggested that instability of vortex lines in axial flow was responsible for the influence of thermal counterflow on ion-trapping cross-sections reported by Cheng et al. (1973). Swanson et al. (1983) observed excess attenuation of the second sound in rotating 4 He in the presence of axial thermal counterflow, which the authors explained by the Glaberson–Johnson–Ostermeier instability. The instability was also considered as a possible trigger of superfluid turbulence (Section 14.8).

3.11 Vortex oscillations in a compressible perfect fluid For a discussion of the effect of compressibility on vortex oscillations, we need to return to the general linear equations of motion of Section 3.3. Performing the Fourier transformation of Eqs. (3.46) and (3.47), we obtain −iωρ  + iρK · v = 0, −iωv + 2 × v L +

cs2  iρ K = 0. ρ

(3.125) (3.126)

The third equation is the equation of vortex motion, which does not differ from Eq. (3.94) derived for the incompressible fluid. As in the case of the fluid with continuous vorticity (Chapter 1), we first determine ρ  and vz from the mass continuity equation (3.125) and the z component of the vector

3.11 Vortex oscillations in a compressible perfect fluid

109

equation (3.126). We obtain again Eq. (1.170) connecting ρ  and vz with vk . Exclusion of ρ  and vz from the equation for the in-plane components of Eq. (3.126) yields two equations for vk and vt : −iωvk − 2

ω2 − cs2 p2 vLt = 0, ω2 − cs2 K 2

(3.127)

−iωvt + 2vLk = 0. These equations transform into the equations of motion in the continuous vorticity model, Eq. (1.171), when v L = v, and into equations of motion for an incompressible fluid, Eq. (3.90), when the sound velocity cs → ∞. Equation (3.127) together with Eq. (3.91) leads to the dispersion relation   cT2 k 2 ω2 − cs2 p2 2 2 2 ω = (2 + νs p ) 2 2 + νs p + . (3.128) 2 ω − cs2 K 2 This equation has two solutions for ω2 at any given wave vector K. We restrict ourselves to the case p = 0 when the wave vector K lies in the xy plane (in-plane modes) and use the inequality cs cT . Then the first solution of Eq. (3.128), ω2 = 42 + cs2 k 2 ,

(3.129)

corresponds to the usual sound wave modified by rotation [cf. Eq. (1.174)]. The second solution, ω2 =

cs2 cT2 k 4 , cs2 k 2 + 42

(3.130)

yields the Tkachenko wave in the limit cs → ∞. Compressibility strongly alters the spectrum of this wave at small k  2/cs , making it parabolic: ω=

cs cT 2 k . 2

(3.131)

In superfluid 4 He and 3 He, the effect of compressibility on inertial waves was rather academic, because the space scale cs /  at which the effect becomes important is extremely large (of the order of hundreds of metres) and is not relevant to any real laboratory experiment. So at the time when this effect was first analysed (Sonin, 1976, 1987), it was considered as a theoretical curiosity, or belonging to some astrophysical applications. The situation became essentially different after discovery of the BEC of cold atoms. In contrast to strongly interacting Bose superfluids like both helium superfluids, the BEC of cold atoms is a weakly interacting Bose gas with very low sound speed and very high compressibility. The importance of the high fluid compressibility for Tkachenko waves in BEC was pointed out by Baym (2003), who discussed the observation of Tkachenko waves by

110

Vortex array in a rotating superfluid

Coddington et al. (2003).6 Haskell (2011) discussed a strong effect of compressibility on Tkachenko waves in rotating neutron stars (pulsars). Another important feature of a compressible fluid is that centrifugal forces make the fluid density essentially inhomogeneous at the scale cs /  (see the end of Section 1.14). The scale is less than the wavelength at small k  2/cs . Therefore, the analysis of this section cannot be directly applied to practical problems arising in observations of Tkachenko waves in BEC of cold atoms. In particular, there are no plane-wave solutions in an essentially inhomogeneous fluid. This will be taken into account in Section 4.7 when studying Tkachenko waves in rotating BEC.

3.12 Rapidly rotating Bose–Einstein condensate in the lowest Landau level state Up to now we have dealt with the VLL state, when the vortex core radius rc , which is of the order of$ the coherence length ξ0 , was very small compared with the intervortex √ distance rv = κ/ 3 (Section 3.1). With increasing , the vortex lattice becomes more and more dense, and vortex cores start to overlap, i.e., ξ0 becomes of the same order as or more than rv , like in the mixed state of a type II superconductor close to the second critical magnetic field Hc2 ∼ 0 /ξ02 (0 is the magnetic flux quantum), at which the transition to the normal state takes place. However, in a rotating BEC there is no phase transition at the second critical angular velocity c2 ∼ κ/ξ02 . Instead the crossover to a new state takes place. At  c2 all atoms condense into a state which is a coherent superposition of single-particle states in the lowest Landau level, similar to that of a charged particle in a magnetic field. It was called the Lowest Landau Level (LLL) state. This interesting state, which is also called the mean-field quantum Hall regime, was the subject of intensive experimental and theoretical investigations (Ho, 2001; Sinova et al., 2002; Coddington et al., 2003; Baym, 2004; Gifford and Baym, 2004; Cooper et al., 2004; Schweikhard et al., 2004; Watanabe et al., 2004; Aftalion et al., 2005; Ueda, 2010). The analysis of a uniform infinite rotating BEC in the LLL state must be based on the Gross–Pitaevskii theory (Section 1.15). As for type II superconductors close to Hc2 , in the LLL state ξ0 rv , one can neglect interaction (non-inear term ∝ |ψ|4 ). Then the non-linear Schr¨odinger equation (1.195) in the rotating coordinate frame transforms to the linear equation      2π v0y 2 ∂ 2π v0x 2 ∂ h¯ 2 −i −i + ψ. (3.132) μmψ = − 2m ∂x κ ∂y κ After the gauge transformation ψ → ψeiφ , v 0 → v 0 + (κ/2π )∇φ, with φ = 2π xy/κ, the linear Schr¨odinger equation becomes similar to that for a charged particle in a magnetic

6 The effect of compressibility on Tkachenko waves in BEC was also discussed using the compressibility sum rule by Cozzini

et al. (2004).

3.12 Rapidly rotating Bose–Einstein condensate

field in the Landau gauge (Landau and Lifshitz, 1982a):    h¯ 2 ∂ 4π y 2 ∂2 +i + 2 ψ. μmψ = − 2m ∂x κ ∂y At μ = h¯ /m it has a solution, which corresponds to the lowest Landau level:   (y − yk )2 , ψk ∝ exp ikx − 2l 2

111

(3.133)

(3.134)

where l 2 = κ/4π  and yk = −l 2 k. The frequency 2 is the analogue of the cyclotron frequency ωc = eH /mc for a charged particle in a magnetic field. If we consider a square L × L with periodic boundary conditions, then k = −2π s/L with integer s. Because of the evident requirement that 0 < yk < L, the integer s should vary from zero to the integer closest to L2 /2π l 2 . This is the total number of LLL states, and the area density of the LLL states is 1/2π l 2 . All these states are orthogonal to each other and have the same energy. Degeneracy is lifted by taking into account the interaction energy. The solution, which corresponds to the periodic vortex lattice, with one quantum per lattice unit cell, is (Abrikosov, 1957; Saint-James et al., 1969)    (y + sl 2 k)2 Cs exp iskx − , (3.135) ψ= 2l 2 s where Cs+1 = Cs exp(2π ib cos α/a), and a, b, and the angle α are the parameters of the unit lattice cell (see Fig. 3.4). The vortex density nv = 2/κ = 1/2π l 2 is equal to the density of LLL states. After averaging over the vortex lattice cell, i.e., after turning to a description in terms of macroscopic hydrodynamics, one obtains the grand thermodynamic potential of the infinite rotating BEC in the LLL state: + G = (−μm + h)n ¯

V 2 βn , 2

(3.136)

where n = |ψ|2 is the average particle density and the parameter (Saint-James et al., 1969)  2  2  √ |ψ|4

  2π iτ  2π iτ  (3.137) = τ (0, e ) + (0, e ) β=  θ2  I θ3 |ψ|2 2 depends on the lattice parameters a, b, and α via the complex parameter τ determined by Eq. (3.81). Here θ2 (z, q) =

∞ 

2

q (n+1/2) ei(2n+1)z ,

n=−∞ ∞ 

θ3 (z, q) =

n=−∞

(3.138) n2 i2nz

q e

,

112

Vortex array in a rotating superfluid

are theta functions (Abramowitz and Stegun, 1972). The minimum of the interaction $energy √ corresponds to the triangular vortex lattice with β = 1.1596, a = b = rv = 2l π/ 3, α = π/3. According to Eq. (3.136) the Gibbs potential has a minimum at the particle density n = (mμ − h¯ )/βV . This allows us to determine the sound velocity: #  βV n ∂μ = . (3.139) cs = ρ ∂ρ m Although derived for the LLL state, the expression (3.136) for the grand potential in the rotating coordinate frame is general enough and is valid in the whole interval of ratios ξ0 /rv . Going from the VLL state (the limit ξ0 /rv → 0) to the LLL state (the limit ξ0 /rv → ∞) the chemical potential varies from μ ≈ V n/m h/m to μ ≈ h/m

V n/m and ¯ ¯ the parameter β increases from 1 to 1.1596. Close to the VLL state, β − 1 is proportional to the small parameter (h/V n) ln(rv /rc ) ∼ (ξ02 /rv2 ) ln(rv /ξ0 ). So β differs insignificantly ¯ from unity in the whole interval of ξ0 /rv . However, just a small β − 1 is sensitive to shear deformation of the vortex lattice and is responsible for the finite shear modulus C66 . Calculation of the shear elastic modulus C66 in the LLL state is similar to that in the VLL state. Deforming the triangular lattice as shown in Fig. 3.4, the real part τR of the complex parameter τ varies proportionally to the shear deformation uxy . Expanding the expression Eq. (3.137) for β and comparing the term ∝ (δτR )2 = 4u2xy in the thermodynamic potential, Eq. (3.136), with the elastic energy, Eq. (3.40), one obtains the value of the shear modulus: C66 =

V n2 ∂ 2 β sin2 α = 0.2054ρcs2 . 2 ∂τR2

(3.140)

This agrees with the value of the shear modulus known for type II superconductors close to the critical field Hc2 (Brandt, 1969; Labusch, 1969)7 and with the value obtained by Sinova et al. (2002) (after taking into account the different definition of the elastic modulus c66 used by Sinova et al.: c66 = 2C66 ). Cozzini et al. (2006) have numerically calculated the shear modulus C66 in the whole interval of the parameter rv /ξ0 . Their results agree with analytical calculations in the limits of the VLL (rv /ξ0 → ∞) and the LLL (rv /ξ0 → 0) states. According to Eq. (3.140), in the LLL state the Tkachenko velocity differs from the sound velocity only by a numerical factor: # C66 = 0.453cs . (3.141) cT = ρ To derive the equation of vortex motion in the LLL state one should use the momentum balance equation (1.197) in the coordinate frame, which rotates and at the same time moves with the vortex velocity v L . Only in this coordinate system is the state stationary and the

7 In order to find Eq. (3.140) from these papers one should use the relation V n2 = (H − H )2 /8π κ 2 β , which follows from the c2 Ginzburg–Landau theory in the limit κ = λ/ξ0 → ∞.

3.12 Rapidly rotating Bose–Einstein condensate

113

time derivative vanishes. In this coordinate frame j = ρ(v − v L ), and the balance of momentum (balance of forces) is ρ[2 × (v − v L )]i = −∇j ij .

(3.142)

Here ρ is the mass density averaged over the vortex lattice cell. Note that in the rotating frame the circulation around the lattice unit cell vanishes, and in the linear theory the momentum flux tensor reduces to the stress tensor: ij = σij . We arrive at the equation of motion for the vortex lattice, which does not differ from that derived in the VLL state. But the Magnus and the Lorentz forces on the right-hand side of Eq. (3.142) arise from the Coriolis term in the equation of momentum balance (instead of the Bernoulli term and the term ρvi vj in the momentum flux tensor in the VLL state). Eventually the whole system of equations of linear macroscopic hydrodynamics derived for the VLL state, is valid in the LLL state, with the mass density ρ being the density averaged over the unit cell and with the shear modulus C66 given by Eq. (3.140). Let us consider restrictions on the existence of the LLL state. The crossover from the VLL to the LLL state occurs at ξ0 ∼ rv . In the LLL state the energy of the lowest Landau level, h, ¯ should exceed the interaction energy V n. This yields the inequality  V n/h. ¯ On the other hand, the very existence of the periodic vortex structure requires that the so-called filling factor n/nv (the number of atoms per vortex) exceeds unity (see below). This restricts the angular velocity from above:   κn = hn/m. The two restrictions on the angular velocity in the LLL state are compatible for a weakly interacting Bose gas with V  h2 /m. Since the two-dimensional interaction parameter is V ∼ h2 as /mlz , it is necessary that the oscillator length lz for the trapping potential localising the BEC cloud along the rotation axis exceeds the scattering length as . The Tkachenko mode can be considered as a mode in an incompressible fluid if   cs k. This inequality is compatible with the condition  V n for the LLL state if the wave number k V n/cs ∼ ξ0 /rv2 . Since in the LLL state ξ0 /rv 1, this leads to the inequality k 1/rv , i.e., the wavelength of the mode is much less than the intervortex distance. Then macroscopic hydrodynamics becomes invalid; so within its framework one cannot ignore compressibility when considering Tkachenko waves in the LLL state. What should happen with the LLL state when the filling factor n/nv approaches unity? This problem was intensively studied by theoreticians, both numerically and analytically (Cooper et al., 2001; Sinova et al., 2002; Baym, 2004; Cooper, 2008). A probable scenario can be derived from the quantum uncertainty principle for two coordinates of the vortex in the plane (Cooper, 2008). Earlier in the book (Section 2.2) it was pointed out that two coordinates of the vortex in the plane correspond to a pair of canonically conjugate variables ‘coordinate–momentum’. For example, after choosing the coordinate ux of the vortex position vector u as a canonical coordinate, the second coordinate uy determines the conjugate momentum Px = hny = ρκy [see Eq. (2.20)]. The Heisenberg uncertainty relation for these variables is ux Px = hnux uy = h¯ .

(3.143)

114

Vortex array in a rotating superfluid

This shows that the uncertainty of the vortex position in the plane is of the order of the √ interatomic distance 1/ n, and is of the same order as the intervortex distance rv when the filling factor approaches unity. It is interesting that although this conclusion emerges from the quantum mechanical principle, the Planck constant does not appear in the final estimation. Anyway, according to the Lindemann criterion, fluctuations in crystal node positions larger than the lattice constant (the intervortex distance in our case) make the existence of the crystal impossible. Since we are discussing quantum fluctuations at zero temperature, this is the phenomenon of quantum vortex lattice melting. After melting, the Tkachenko mode, which is connected with the crystalline order, is ruled out. One can also estimate amplitudes of quantum fluctuations from zero quantum oscillations of Tkachenko modes (Sinova et al., 2002). The plane Tkachenko mode u(k)eikx−iωt , where u(k) is the amplitude of vortex displacements, has the energy ∼ L2 |u(k)|2 C66 k 2 in the square L × L. This should be equal to the energy h¯ ω(k)/2 of zero quantum fluctuation. The average vortex displacement squared is obtained by integration of |u(k)|2 over the Brillouin zone of the vortex lattice:  1/rv  1/rv hω(k) 1 ¯ 2 2 2 dk ∼ . (3.144) |u(k)| k dk ∼ u ∼ L C k n 66 0 0 This yields the same uncertainty of the vortex position as obtained from the Heisenberg uncertainty relation. In addition to the prediction of quantum melting, it was also suggested that at low filling factor n/nv , Bose condensation is probably destroyed and a single macroscopic wave function cannot describe all atoms (Sinova et al., 2002). This could occur either together with quantum melting or after quantum melting, allowing the existence of the vortex liquid phase.

4 Oscillation of finite vortex arrays: two-dimensional boundary problems

4.1 Equilibrium finite vortex array For many experiments dealing with vortex oscillations, the finite dimensions of the containers play an important role. The theory can only be related to these experiments after an analysis of boundary problems: formulation of the boundary conditions for the equations of macroscopic hydrodynamics and their solution for a particular geometry. The symmetry and local properties of a finite array in the bulk are assumed to be the same as those of an infinite array. This approach is common for theories of continuum media, for example elasticity theory. We restrict ourselves in the present chapter to two-dimensional problems in which vortices move in the xy plane, remaining rectilinear and parallel to the rotation axis (the z axis). As was discussed in Section 3.4, in a rotating container a fluid in the ground state rotates as a solid body due to the presence of the vortex array with constant vortex density determined by the Feynman formula nv = 2/κ. In incompressible fluids the Feynman formula is exactly true even for finite vortex arrays. If it were not true, vortex lines would rotate with a velocity different from that of the container, and mutual friction between vortex lines and the normal part of the fluid would lead to energy dissipation, but this is impossible in the equilibrium state. On the other hand, it was noticed by Hall (1960) that vortices do not fill the container completely, and a vortex-free (irrotational) region should exist near the lateral walls of the container. We shall demonstrate the existence of the vortex-free region in the simplest geometry of an axisymmetric cylinder of radius R (Fig. 4.1). The vortex array is confined to a cylinder of radius R0 < R, where the fluid rotates as a solid body. We call this the vortex bundle. The cylindrical layer R0 < r < R is vortex-free. In polar coordinates (r, ϕ) the velocity distribution is (Fig. 4.1)  r r < R0 , vϕ = v = (4.1) vr = 0, R02 R > r > R0 . r

115

Oscillation of finite vortex arrays: boundary problems

vortex bundle

vortex-free region

116

Figure 4.1 Vortex bundle and vortex-free region in a rotating cylinder (top) and the radial distribution of the velocity (bottom). The dashed line shows the velocity v = r of solid body rotation. The solid line shows the velocity induced by the vortex bundle including fluctuation around the average velocity. From Sonin and Krusius (1994).

The angular momentum of the equilibrium vortex bundle per unit length is R vϕ (r)r 2 dr =

M = 2π

πρR04 + πρR02 (R 2 − R02 ). 2

(4.2)

0

The energy density of the fluid inside the vortex bundle in the rotating coordinate frame is determined by Eq. (3.65). After integration over the whole bundle cross-section and adding the energy of the fluid in the vortex-free region one obtains the total energy per unit length along the rotation axis in the rotating coordinate frame: πρ2 R04 R + π R02 Ev + πρ2 R04 ln 4 R0 − πρ2 R02 (R 2 − R02 ),

E = E − M = −

(4.3)

where the energy density Ev is determined by Eq. (3.51) with ω˜ equal to 2 and rm equal to the intervortex distance rv . The angular momentum M satisfies the canonical relation M = −∂E /∂. We shall also need the moment of inertia I of the fluid: I=

πρR04 ∂M = . ∂ 2

(4.4)

4.2 Distortions of vortex lattice produced by a boundary

117

Partial derivatives in the formulas for M and I must be taken at a fixed number of vortices, i.e., at fixed value of R02 . At equilibrium the radius R0 of the vortex bundle is determined from the condition of the minimum of E , which after expansion in the small difference R − R0 becomes ∂E ≈ −4πρ2 R(R − R0 )2 + 2π REv = 0. ∂R0

(4.5)

This yields the width of the vortex-free region:  d = R − R0 =

Ev 2ρ2

1/2

 =

rv κ ln 4π  rc

1/2 .

(4.6)

√ The width d is larger than the intervortex distance rv by the factor ln(rv /rc ). The large logarithmic factor justifies the determination of d within the scope of the continuum theory. Experimental evidence for the vortex-free region in superfluid 4 He was obtained by Tsakadze (1964), although the quantitative discrepancy with theory was considerable [see also the discussions by Andronikashvili and Mamaladze (1966)]. In superfluid 3 He the vortex-free region was investigated by Ruutu et al. (1997). We considered vortices in a cylindrical container, but the vortex-free region is formed near any solid surface bounding a superfluid and parallel to the rotation axis (Bendt and Oliphant, 1961; Kemoklidze and Khalatnikov, 1964; Stauffer and Fetter, 1968). The expression obtained for the width of the vortex-free region d is valid for an arbitrary shape of container cross-section as long as the lateral walls of the container are parallel to the rotation axis and the radius of curvature of the walls is large compared with d. The width of the vortex-free region can be different from the equilibrium value given by Eq. (4.6). Suppose that the rotation speed of the container with superfluid changes in value. Then the number of vortices in the container has to change too. But the nucleation of new vortices and their annihilation at the boundary are much slower than other relaxation processes. Therefore a metastable state of restricted equilibrium at a fixed number of vortices is possible. If the number of vortices is smaller than the equilibrium value, then the width d of the vortex-free region is larger than that given by Eq. (4.6). The case of infinite d corresponds to a bundle of a fixed number of vortices in an unbound fluid.

4.2 Distortions of vortex lattice produced by a boundary Of utmost importance is the question: what reliance can be placed on the continuum hydrodynamics approach developed for the infinite vortex lattice when dealing with a finite vortex array? In the past such an approach was brought into question on the basis of numerical calculations revealing strong boundary effects on the vortex lattice even deep within the interior of the vortex bundle (Campbell, 1981). Therefore our analysis of dynamics of finite vortex bundles is preceded by a discussion of boundary effects on the equilibrium properties of a finite vortex array.

118

Oscillation of finite vortex arrays: boundary problems

Let us consider a small distortion of the boundary (edge) of the finite vortex bundle, which occupies a cylindrical region of radius R0 in a cylindrical container of radius R. Because of distortion, the distance of the edge of the vortex bundle from the rotation axis is R0 + δu. We consider small radial displacements δu of the edge forming the nth cylindrical wave δu = u0 einϕ .

(4.7)

Here ϕ is the azimuthal angle of a point on the vortex bundle edge and n is an integer. Due to distortion, the velocity field differs from the velocity field v given by Eq. (4.1) with small correction v  . The vorticity ω˜ = ∇ × v remains equal to 2 everywhere inside the vortex bundle and is equal to zero outside this, but changes at the very edge of the bundle. For small displacements δu the vorticity deviation due to distortion is ω˜  = ∇ × v  = 2u0 δ(r − R0 )einϕ ,

(4.8)

where δ(r −R0 ) is a δ-function. This deviation produces a jump of the azimuthal velocity vϕ at r = R0 equal to 2u0 . At the same time, the particle number conservation law requires that the radial velocity vr is continuous at r = R0 and vanishes at the container wall r = R. The distortion-induced velocity field v  must satisfy these boundary conditions and must be divergence-free (∇ · v  = 0) everywhere and curl-free (∇ × v  = 0) everywhere except at the very bundle edge r = R0 . This field is given by (Ignatiev and Sonin, 1981)  n−1   2n  r R0  vr = iu0 1− einϕ , R0 R (4.9)  n−1   2n  r R0  inϕ vϕ = −u0 1− e , R0 R for r < R0 , and



n+1  n−1  2n  R R0 r 0 − vr = iu0 einϕ , r R0 R    n−1  2n  R0 n+1 R0 r  vϕ = u0 + einϕ , r R0 R

(4.10)

for R0 < r < R. Using Eq. (3.65) to calculate the energy of the distorted bundle in the rotating coordinate frame (ignoring the line tension contribution Ev ), one finds that distortion of the edge of the vortex bundle increases the energy by an amount   2n  2 R0 2 2 u0 δEd (n) = π R0 ρ . (4.11) 1− n R Now let us take into account the fact that vortices form a triangular lattice. Because of the lack of correspondence between the symmetry of a cylinder, which is the shape that

4.2 Distortions of vortex lattice produced by a boundary

119

the vortex bundle tends to take at equilibrium, and the hexagonal symmetry of the infinite vortex lattice, the bundle shape should deviate slightly from cylindrical. This is a source of the tendency to faceting which is well known for atomic crystals. Indeed, if we cut out a cylindrical region from an infinite vortex lattice, its boundary vortices cannot be located on one circumference; some of them will be at a distance of the order of the vortex spacing. One can consider this as a distortion of the edge of the vortex bundle, which results in deviation of the velocity from the solid body rotation velocity inside the vortex bundle. This velocity deviation was found by numerical calculations and was called the destabilising velocity (Campbell and Ziff, 1979). The net destabilising velocity is given by a sum over distortion harmonics labelled by an integer n, each given by Eqs. (4.9) and (4.10). The sum should include n = 6, 12, 18, . . . . allowed by hexagonal symmetry. Deep in the interior of the vortex bundle the contribution of the fundamental harmonic n = 6 is most important, and the destabilising velocity is proportional to r 5 . The destabilising velocity produces a force on the vortices, which deforms the triangular lattice. Deformation tends to decrease distortion of the edge and the energy associated with distortion. But at the same time the elastic energy of the vortex lattice increases. The competition between these energies determines the equilibrium structure of the vortex bundle. In the equilibrium state the solid body rotation of vortices must be restored, otherwise mutual friction with the normal part of the fluid will appear. So the velocity induced by lattice deformation must exactly cancel the destabilising velocity. It was shown (Ignatiev and Sonin, 1981) that shear deformation diminishes distortion of the vortex bundle edge and the energy of this distortion, but the latter still remains of the same order as given by Eq. (4.11) with u0 scaled by the intervortex distance rv . Deep in the interior of the vortex bundle, displacements of vortices from sites of the regular triangular lattice fall as r 5 in agreement with the numerical calculations of Campbell and Ziff (1979). Campbell and Ziff (1979) noticed that the ratio of the surface energy of the vortex bundle (the difference between its energy and the energy of the same number of vortices in an infinite lattice) to its total energy decreases too slowly when the number of vortices in the bundle increases, and probably does not approach zero. This is fully supported by Eq. (4.11) for vortex bundles in an unbound fluid (R → ∞), which were investigated numerically by Campbell and Ziff (1979). According to Eq. (4.11) the distortion energy (which is in fact a surface energy) grows proportionally to R02 , i.e., proportionally to the two-dimensional ‘bulk’. This means the absence of the thermodynamical limit when the ratio of the surface energy to the bulk energy goes to zero with increasing size of the system. However, if the vortex bundle is in a container and the width of the vortex-free region d = R − R0 is kept constant at growing radii of the vortex bundle and the cylinder, the distortion energy is smaller by a factor [1 − (R0 /R)2n ] ≈ 2nd/R0 than the same energy in an unbound fluid, and the energy grows proportionally to the two-dimensional ‘surface area’ ∼ R0 , as expected from the surface energy in the thermodynamical limit. In summary, both numerical calculations and the analytical estimations based on the continuum theory in the bulk show that in vortex crystals the boundary distortion affects the structure in the bulk more strongly than in atomic crystals. But the fact that the continuum theory was able to

120

Oscillation of finite vortex arrays: boundary problems

explain peculiar features of vortex crystals which had been revealed numerically proves that the analysis of finite vortex bundles can rely on this theory.

4.3 Axisymmetric Tkachenko modes in a finite vortex bundle: comparison of continuum theory and numerical experiments When looking for axisymmetric Tkachenko eigenmodes in a cylindrical container of finite radius, we can refer directly to the conservation law for angular momentum. Such an approach was taken by Ruderman (1970) for Tkachenko modes in pulsars (see the end of Section 5.9). In the elasticity theory of the two-dimensional vortex crystal (Section 3.8) the field of transverse displacements is determined by a vector potential  =  zˆ : u⊥ = ∇ ×  = −ˆz × ∇.

(4.12)

The potential  must satisfy the wave equation ∂ 2 − cT2  = 0. ∂t 2

(4.13)

Axisymmetric modes with the sound-like spectrum ω = cT k correspond to a cylindrical wave

ur ≈ 0,

 = 0 J0 (kr)e−iωt , ∂ = k0 J1 (kr)e−iωt , uϕ = − ∂r

(4.14)

where the subscripts r and ϕ denote radial and azimuthal components in the polar coordinate frame (r, ϕ). Since in the Tkachenko wave the fluid and vortices move together with nearly the same velocity, the angular momentum per unit length is 

R

M = 2πρ 0

 vϕ r 2 dr = −2iωπρ

R

uϕ r 2 dr = −2iωπρ0 R 2 J2 (kR)e−iωt .

(4.15)

0

Here we consider a vortex bundle filling the whole container and do not distinguish between the radius R of the container and the radius R0 of the vortex bundle. The conservation law for angular momentum requires that the angular momentum does not oscillate, i.e., M = 0. This condition yields eigenfrequencies of Tkachenko modes (Ruderman, 1970): ωR (s) = j2,s

cT . R

(4.16)

Here jn,s denotes the sth zero of the Bessel function Jn (z). For the fundamental frequency j2,1 = 5.14. The same spectrum follows from the boundary condition that the azimuthal component of the momentum flux through the fluid boundary r = R vanishes. The momentum flux is

4.3 Axisymmetric Tkachenko modes in a finite vortex bundle

121

equal to the stress tensor given by Eq. (3.114). In cylindrical coordinates the relevant stress tensor component is   ∂uϕ uϕ 1 ∂ur 2 . (4.17) σϕr = −ρcT − + ∂r r r ∂ϕ For axisymmetric modes there is no dependence on the azimuthal angle ϕ, and the condition σϕr (R) = 0 requires that ∂uϕ (R) uϕ (R) − = 0. (4.18) ∂r R This yields the same spectrum Eq. (4.16) as the condition M = 0. Williams and Fetter (1977) suggested another boundary condition. They assumed that the vortex bundle fills the whole bulk of a cylindrical container and suggested that in the rotating coordinate frame the vortices and the fluid are at rest near the wall of the container, i.e., uϕ ∝ J1 (kR) = 0. This leads to the eigenfrequencies cT (4.19) ωW F (s) = j1,s . R These two boundary conditions are the two opposite limits of the general boundary condition, which assumes that some external force is applied to the fluid boundary, restoring it to its initial state and proportional to the azimuthal displacement at the boundary: 2π Rσϕr (R) = −2π RKuϕ (R),

(4.20)

where K is the ratio of the force to the displacement. The right-hand side of this equation is the torque exerted on the fluid, and the left-hand side is the flux of the angular momentum through the circular boundary. For axisymmetric modes, substitution of Eq. (4.17) in Eq. (4.20) yields   ∂uϕ (R) uϕ (R) − + uϕ (R) = 0. (4.21) αb ∂r r Eq. (4.21), together with Eq. (4.14), gives the condition for determination of the eigenfrequencies ω = cT k of axisymmetric modes: αb kJ2 (kR) − J1 (kR) = 0.

(4.22)

Here αb = ρcT2 /K. The force sticking vortices to the wall, which is at rest in the rotating frame, can arise due to mutual friction between the vortices and the normal part of the fluid, which sticks to the wall. This requires analysis using two-fluid hydrodynamics, as done in Section 6.5. The parameter αb is imaginary and inversely proportional to iω, because the external force in this case is not elastic but frictional. We shall see that coupling between vortices and the wall is considerable even at rather low temperatures, and αb k can be quite small. But vortices can interact with the wall even at zero temperature because any real wall is not smooth. Effective friction arises as a result of the averaging of vortex motion over the irregular relief of the wall, similar to the residual resistance for electrons in dirty solids.

122

Oscillation of finite vortex arrays: boundary problems

0

3.18

5.14

7.02

R T

Figure 4.2 Eigenfrequencies ω of axisymmetric Tkachenko modes for a vortex bundle of radius R subject to action of a surface restoring force. The frequencies in units cT /R are shown for the two lowest modes. The thick solid lines with arrows show how the eigenfrequencies increase when the force increases from zero to infinity. From Sonin (1987).

The dispersion equation Eq. (4.22) yields Ruderman’s spectrum [Eq. (4.16)] when αb → ∞ and the spectrum of Williams and Fetter when αb = 0. Let us consider now the variation of αb from ∞ to 0. The eigenfrequencies of axisymmetric modes increase, as shown by the arrows in Fig. 4.2. Two lower eigenfrequencies map the two intervals on the frequency scale indicated by solid lines. It is important that the fundamental frequency of Ruderman, ωR (1) = 5.14cT /R, is not the lowest one. The spectrum also includes the zero frequency corresponding to the Goldstone mode related to rotational invariance. The mode with zero frequency is of no physical interest. But when rotational invariance is broken, this mode becomes an observable mode with finite frequency. Such a character of the spectrum of axisymmetric modes should be taken into account when comparing predictions of the continuum elasticity theory with results of numerical calculations for finite vortex bundles. In order to simulate the boundary condition uϕ = 0 used by Williams and Fetter (1977), Campbell (1981) calculated eigenfrequencies of the vortex bundle when the outermost ring of vortices was constrained to be strictly fixed. The lowest calculated frequency turned out to be a factor of 2 smaller than the frequency given by Eq. (4.19) and considerably smaller than the frequencies of Ruderman’s spectrum, Eq. (4.16). Campbell considered this as evidence of a soft oscillation mode that could not be predicted by the continuum elasticity theory for a triangular lattice. Another interpretation of these numerical results was proposed (Sonin, 1987), which was not so discreditable for continuum theory. All vortices in the bundle except those on the fixed outermost ring constitute their own inner bundle. Interaction between inner vortices and the outermost ring of vortices produces a force applied to the inner bundle, so the latter can sustain oscillation modes with a frequency corresponding to some finite αb in Eq. (4.22) and falling somewhere in the first interval of the scale of eigenfrequencies shown in Fig. 4.2. Therefore one should compare Ruderman’s fundamental frequency 5.14cT /R not with the lowest but with the next to lowest frequency calculated numerically. This comparison shows that the numerical frequency agrees with Ruderman’s frequency with 10% accuracy (Sonin, 1987).

4.4 Chiral edge waves Semi-infinite atomic crystals sustain surface waves, called Rayleigh waves (Landau and Lifshitz, 1986). Surface modes localised near the boundary of the vortex bundle were

4.4 Chiral edge waves

123

also found in the numerical calculations and were called edge waves (Campbell, 1981). Campbell and Krasnov (1981) developed a theory of edge waves in the framework of the continuous vorticity model. In classical hydrodynamics, edge waves were already known to Kelvin (Thompson, 1880), who discovered them while studying the stability of the columnar vortex tube in an unbound fluid [see Section 158 in the book of Lamb (1997) and Section 7.3 in the book by Batchelor (1970)]. Here we present a theory of edge waves in the continuous vorticity model. We consider an edge wave propagating along the edge of a vortex bundle of radius R0 in a cylindrical container of radius R. The velocity and the displacement perturbations induced by the edge wave are the same as given by Eqs. (4.7)–(4.10) for n-fold cylindrical distortion, but now the time evolution (rotation) of distortion is considered and the time-dependent factor e−iωt must be added. According to Helmholtz’s theorem, vortices at the bundle edge must move together with the fluid, and variation in time of the radial displacement δu of the edge is determined by the radial fluid velocity: dδu = vr . dt

(4.23)

Substitution of δu from Eq. (4.7) and vr from Eq. (4.9) or (4.10) readily yields the dispersion law of the edge wave in a rotating coordinate frame (Campbell and Krasnov, 1981):   2n  R0 2 . (4.24) ω = − 1 − =− R 1 + coth[n ln(R/R0 )] In the laboratory coordinate frame the dispersion law is   2n  R0 ω = n−1+ . R

(4.25)

The edge wave n = 1 is a translation of the bundle as a whole [displacement mode in the classification of Campbell (1981)]. Its frequency vanishes in the laboratory frame when R → ∞ because in an unbound fluid translation of the vortex bundle does not change the energy. If the width of the vortex-free region d = R −R0 is much smaller than R, we can rewrite the dispersion law, introducing the wave number k = n/R: ω=−

2 . 1 + coth kd

(4.26)

In the long wavelength limit kd  1 ω = −cE k,

cE = 2d.

(4.27)

If d takes its equilibrium value from Eq. (4.6), the velocity of the edge waves is equal to   κ rv 1/2 ln . (4.28) cE = π rc

124

Oscillation of finite vortex arrays: boundary problems

Since this velocity is larger than the Tkachenko wave velocity cT = (κ/8π )1/2 , the edge wave can emit volume Tkachenko waves and lose energy. The analysis of this phenomenon requires going beyond the scope of the continuous-vorticity model and including effects of Tkachenko shear rigidity. The analysis (Sonin, 1987, Section V.E) has shown that attenuation due to emission of Tkachenko waves by edge waves is quite weak. Edge modes are chiral: they are unidirectional and can propagate only in the direction opposite to the velocity of solid body rotation. Therefore reflection of edge waves is impossible. Here one can notice an analogy with edge states in two-dimensional topological insulators or in quantum Hall systems with broken time invariance. Cazalilla (2003) calculated chiral edge modes in a rapidly rotating trapped BEC cloud with methods widely used for quantum Hall systems. He obtained the dispersion law ω = n, which one can compare with ω = (n − 1) following from Eq. (4.25) for the vortex bundle in an infinite fluid (R → ∞).1 In contrast to the vortex bundle in an infinite fluid, in the case of the trapped BEC cloud, translational invariance is broken. Therefore the frequency of the edge mode of Cazalilla (2003) with orbital moment number n = 1, which corresponds to a translation of the cloud as a whole, does not vanish.

4.5 Ground state of a two-dimensional Bose–Einstein condensate cloud While the confines of He superfluids are the walls of containers, BEC clouds of cold atoms are confined by optical traps. In contrast to the practically incompressible He superfluids, the density of the BEC cloud depends strongly on the distance from the trap centre. We restrict our analysis to harmonic axisymmetric trapping potentials 2 r2 mωz2 z2 mω⊥ + , (4.29) 2 2 where ωz and ω⊥ are the trapping frequencies, which characterise the curvature of the trapping parabolic potential. Neglecting interaction in the Schr¨odinger equation (1.193) for the BEC wave function, i.e., considering an ideal Bose gas, the ground state in such a trapping potential is the harmonic oscillator state of the lowest energy. The density profile in this state is Gaussian: N 2 2 2 2 e−z / lz −r / l⊥ , (4.30) |ψ|2 = 2 3/2 π lz l⊥

U (R) =

where N is the number of particles and # h¯ lz = , mωz

# l⊥ =

h¯ , mω⊥

(4.31)

are the harmonic oscillator lengths along the z axis and in the in-plane radial direction respectively. 1 Strictly speaking Cazalilla (2003) obtained ω = ω n where ω is the trapping frequency. But we shall see in Section 4.6 that ⊥ ⊥ at rapid rotation of a trapped BEC cloud the angular velocity  approaches ω⊥ .

4.5 Ground state of a two-dimensional BEC cloud

125

Repulsive interaction leads to expansion of the cloud. Eventually with growing interaction, the size of the cloud becomes so large and the density profile becomes so smooth that one can ignore the kinetic energy in the Schr¨odinger equation, retaining only the interaction term. This approach is called the Thomas–Fermi approximation (Pitaevskii and Stringari, 2003). The density profile in the ground state in the Thomas–Fermi approximation is determined from the condition following from the Schr¨odinger equation (1.193): mμ − U (R) − V |ψ|2 = 0.

(4.32)

Assuming that the trapping potential U (R) vanishes at R = 0, one obtains the density varying in space as     U (R) U (R) mμ 2 , (4.33) 1− = n(0) 1 − n(R) = |ψ(R)| = V mμ mcs (0)2 √ where n(0) = mμ/V and cs (0) = V n(0)/m are the particle density and the sound velocity in the trap centre R = 0. For the axisymmetric parabolic potential (4.29) the density profile is   r2 z2 n(R) = n(0) 1 − 2 − 2 , (4.34) Rz R⊥ where

√

2μ = Rz = ωz

√

2cs (0) , ωz

√ 2cs (0) R⊥ = , ω⊥

(4.35)

are the semi-width of the cloud in the z direction and its radius in the xy plane respectively. The Thomas–Fermi approximation is valid if the kinetic energy, which is of the order 2 , is smaller than the interaction energy V n. This leads to the of nh¯ 2 /mRz2 + nh¯ 2 /mR⊥ √ condition that the coherence length ξ0 = h¯ / 2mV n is much shorter than the two oscillator lengths lz and l⊥ . The Gaussian and the Thomas–Fermi (inverted parabola) density profiles in the z direction are shown in Fig. 4.3. Later we shall focus on the case when l⊥ ξ lz . Then the density profile of the cloud has a Gaussian shape in the z direction but Thomas–Fermi shape in the xy plane. Integrating over the z coordinate, the original three-dimensional Schr¨odinger equation (1.193) is reduced to a two-dimensional equation (Dalfovo et al., 1999). All variables and the interaction parameter in the new equation now refer to the two-dimensional space, but we retain the former notation of the wave function ψ and V .2 The Thomas–Fermi equation (4.32) becomes mμ −

2 r2 mω⊥ − V |ψ|2 = 0, 2

(4.36)

2 Relations between the original three-dimensional quantities and the new two-dimensional quantities are given by Pitaevskii and

Stringari (2003).

126

Oscillation of finite vortex arrays: boundary problems

Figure 4.3 Density distribution in a BEC cloud. Left: Gaussian profile of an ideal Bose gas. Right: Inverted parabola profile in the Thomas–Fermi approximation.

and the BEC cloud has the shape of a disk (pancake geometry) with the inverted parabola density profile   r2 (4.37) n(r) = n(0) 1 − 2 . R⊥ Let us consider a single vortex in the centre of the two-dimensional BEC cloud. With logarithmic accuracy its energy is given by Eq. (2.6) with rm = R⊥ and the mass density ρ equal to the mass density ρ(0) = mn(0) in the cloud centre. The angular momentum mz of the vortex at the centre is two times less than that given by Eq. (2.12) (with ρ replaced by ρ(0)) because of suppressed density at the cloud periphery. Correspondingly, the first critical angular velocity c1 for penetration of the first vortex into the cloud is two times larger than the value given by Eq. (2.13) for an incompressible fluid. Because of variation of the density, the energy of the vortex depends on the displacement u of the vortex from the trap centre. If the vortex is shifted off the trap centre, there is a force which makes the vortex precess around the trap centre with the frequency given by Eq. (2.40). For the Thomas–Fermi density profile (4.37) (with r equal to the vortex shift u off the centre), the precession frequency of the off-centre vortex is ω =+

2 κω⊥ R⊥ d 2 U R⊥ κ ln =  − ln . 2 2 2 rc du rc 4π mcs 4π cs

(4.38)

Using the relation between the trapping frequency and the sound velocity in Eq. (4.35) one obtains that R⊥ κ ln . (4.39) ω =− 2 rc 2π R⊥ Without rotation this frequency is longer by a logarithmic factor than the precession frequency (2.39) in a cylinder filled by an incompressible fluid. Equation (4.39) agrees with the frequency derived by Svidzinsky and Fetter (2000) apart from another rotation shift 2 in their Eq. (47). Apparently the factor of 2 appeared

4.6 Ground state of a rotating two-dimensional BEC cloud

127

because Svidzinsky and Fetter (2000) considered a rotating fluid with a vortex array. The difference between rotation shifts in axial modes for a single vortex and for a vortex array was discussed in Section 3.7. Deriving the force on a vortex and the vortex precession frequency, Svidzinsky and Fetter (2000) (see also Fetter, 2009) looked for a solution of the time-dependent Gross– Pitaevskii equation by the method of matched asymptotic expansions.3 In this method (Pismen, 1999) the force on the vortex is obtained from matching of the solution at short distances inside the vortex core with the solution at long distances far outside the core. Meanwhile the force follows from the momentum balance in hydrodynamics and is not sensitive to what is going on at short distances inside the vortex core. The only relevant information about the core is its size, which appears in the argument of the large logarithm. This size can be the radius of a cylinder with fluid circulation around it, which moves through a perfect fluid (Section 1.5). Thus at least within logarithmic accuracy, solution of the time-dependent Gross–Pitaevskii equation inside the core is not necessary. Agreement between results obtained in hydrodynamics and from the explicit solution of the Gross– Pitaevskii equation only confirms this. Precession of the off-centre vortex was observed experimentally by Anderson et al. (2000). 4.6 Ground state of a rotating two-dimensional Bose–Einstein condensate cloud Let us now consider a pancake shaped BEC cloud rotating with angular velocity . The angular velocity is high enough to allow large numbers of vortices to be created in the cloud. The grand thermodynamic potential G of the infinite rotating BEC cloud was given by Eq. (3.136). When applying it to a trapped finite BEC cloud one should add to it the energies of centrifugal forces and of the trapping potential. This leads to the following Thomas–Fermi equation: mμ − h ¯ −

2 − 2 )r 2 m(ω⊥ − V βn = 0, 2

(4.40)

where β = |ψ|4 / |ψ|2 2 varies from 1 in the VLL state to 1.1596 in the LLL state, i.e., differs insignificantly from 1 (see Section 3.12). Eventually the inverted parabola density profile appears in the rotating BEC cloud if one$replaces the trapping frequency ω⊥ in the expression for R⊥ by the effective frequency pancake shaped cloud is # R = cs (0)

2 − 2 . So the radius of the rotating ω⊥

2 . 2 − 2 ω⊥

(4.41)

One can see that the centrifugal force reduces the trapping potential, spreading the cloud out. The trapped cloud loses its stability when the angular velocity  approaches the trapping frequency ω⊥ . 3 The vortex precession frequency was also found by numerical solution of the Gross–Pitaevskii equation (Jackson et al., 1999).

128

Oscillation of finite vortex arrays: boundary problems

In the past, the density profile of the BEC cloud in the LLL state was debated. The starting point of our analysis was an infinite rotating BEC cloud, taking into account interaction but neglecting first the trapping potential and the centrifugal force. They were added afterwards. This approach becomes more and more accurate on approaching the instability threshold  = ω⊥ , where the trapping potential and the centrifugal force cancel each other. The exact wave function for the infinite uniform state is well known from analogy with the Abrikosov solution for type II superconductors close to Hc2 (Abrikosov, 1957). After averaging over the vortex lattice cell, one can see that in macroscopic hydrodynamics the only effect of rotation on the BEC cloud is renormalisation of the trapping potential by the centrifugal forces. This leads to the natural conclusion that the rotating BEC cloud has the Thomas–Fermi density profile, which differs from that in the resting cloud only by renormalisation. However, Ho (1978) suggested another picture. He started from the LLL wave functions for non-interacting particles in a trapping potential and switched on the interaction afterwards. For a regular vortex lattice he obtained the Gaussian density profile, which was retained even after adding interaction. Later in a number of publications it was shown that Ho’s solution was unstable with respect to small distortions of the triangular lattice (Baym and Pethick, 2004; Cooper et al., 2004; Gifford and Baym, 2004; Watanabe et al., 2004; Aftalion et al., 2005). The instability transforms the Gaussian profile to an inverted parabola obtained from the Thomas–Fermi approach. So our further analysis assumes the Thomas–Fermi density profile in the ground state. Another assumption of our analysis was solid body rotation of the BEC cloud with constant vortex density given by the Feynman formula nv = 2/κ. In an inhomogeneous fluid this formula is not exact. Correction to this formula due to density gradients in the BEC cloud was considered by Anglin and Crescimanno (2002) and Sheehy and Radzihovsky (2004) and was discussed in Section 3.4 in the framework of the HVBK theory. According to Eq. (3.68), the corrected Feynman formula for the inverted parabola density profile (4.37) in the BEC cloud is nv =

R2 2 2 rv 1 rv 1 ≈ ln . − ln − 2 2 2 2 κ π rc (R − r ) κ rc 4π(R − r)

(4.42)

The correction to the constant Feynman vortex density becomes important at distances R − r from the cloud boundary of the order of rv and therefore is not essential as long as R rv , i.e., the number N of vortices in the cloud is very large.

4.7 Tkachenko waves in a Bose–Einstein condensate cloud As was shown in Section 3.11, at small wave vectors k  2/cs , compressibility of the fluid significantly affects the spectrum of Tkachenko waves. This transforms the spectrum from sound-like to parabolic [Eq. (3.131)]. It was also mentioned that when the compressibility becomes important, one cannot ignore inhomogeneity of the density distribution. The effect of density inhomogeneity on Tkachenko waves was taken into account in the theory of Anglin and Crescimanno (2002) within the framework of the continuum theory,

4.7 Tkachenko waves in a BEC cloud

129

which replaces the discrete vortex lattice by a continuous medium, i.e., in the framework of macroscopic hydrodynamics. But they neglected fluid compressibility, while a proper comparison with experiment requires a theory taking into account both features, compressibility and inhomogeneity. The Tkachenko mode in a rotating BEC cloud was investigated numerically, solving the equations of Gross–Pitaevskii theory (mean-field theory) (Baksmaty et al., 2004; Mizushima et al., 2004). The numerical results agreed well with experiment. It would also be useful to develop an analytical approach since this could provide a deeper insight into the physics of the phenomenon. In the experiment there are good conditions for application of a continuum theory (macroscopic hydrodynamics) since usually the relevant length scale (the BEC cloud size) essentially exceeds the intervortex distance. This allows us to use the continuum theory of Tkachenko modes but taking into account fluid compressibility and inhomogeneity. We consider an axisymmetric rotating BEC pancake shaped cloud in a parabolic trapping potential characterised by the frequency ω⊥ . Linear equations of macroscopic hydrodynamics for an inhomogeneous fluid are given by Eqs. (3.48)–(3.50). Applying them for the monochromatic Tkachenko mode ∝ e−iωt , neglecting all elastic moduli except for the shear modulus C66 , and using the polar system of coordinates, one obtains the equations   v  ∂ 1 t 2 3 ∂ 2iωvr = −ω vt − C66 r , (4.43) ∂r r ρ(r)r 2 ∂r   cs2 (r) ∂ 1 ∂(ρ(r)rvr ) 2iωvt = . (4.44) ρ(r) ∂r r ∂r We excluded all variables except for the tangential (azimuthal in polar coordinates) component of the velocity vt ≈ vLt and the radial component of the fluid velocity vr . The latter, though much smaller than vt , is crucial for the compressibility effect. Now ρ, cs and C66 = ρcT2 depend on the distance r from the rotation axis. For a weakly interacting Bose gas, cs2 is proportional to the density ρ. Therefore the ratio cs2 /ρ is a constant equal to its value cs2 (0)/ρ(0) in the cloud centre r = 0. We should solve the system of two coupled second-order differential equations (4.43) and (4.44) with proper boundary conditions. The square brackets on the right-hand side of Eq. (4.43) give the flux of the angular momentum σϕr r 2 through the cloud boundary r = R integrated over the whole boundary. The stress tensor component σϕr is given by Eq. (4.17). As in an incompressible fluid, the flux must vanish. Since the stress tensor (momentum flux) is proportional to ρ and the latter vanishes at r = R, it looks as though the angular momentum flux through the boundary vanishes independently of whether the boundary condition Eq. (4.18) is satisfied or not. But this is not true. Solving the equation of motion (4.43) close to r = R by expansion in small (R − r)/R, one obtains vt ≈ r[C1 + C2 ln(R − r)], where C1 and C2 are arbitrary constants. The component ∝ C2 diverges at r → R and gives a finite contribution to the stress tensor despite the factor ρ ∝ R − r. So this component should be absent. Then the angular momentum conservation law holds only if the boundary condition Eq. (4.18) holds.

130

Oscillation of finite vortex arrays: boundary problems

One also needs the second boundary condition imposed on the radial fluid velocity vr . The total mass balance requires that the radial mass current ρ(r)vr (r) at the BEC border r = R vanishes. Indeed, even though the BEC border is mobile because of oscillation, the total mass, which can be transferred through the equilibrium border r = R, is a secondorder quantity with respect to the oscillation amplitude. The radial mass current vanishes for any finite radial velocity vr (R) since ρ(r) → 0 at r → R. We use arguments similar to those used for the derivation of Eq. (4.18). In order to derive the condition imposed on vr (r) we solve Eq. (4.44) at r ≈ R by series expansion [neglecting terms ∼ (R − r)2 ]:   R−r C2 iωvt (R)R R − r + C1 1 + + . (4.45) vr (r) = 2 R R−r cs (0)2 The divergent component ∝ C2 gives a finite mass current into the border and must be deleted. Taking a derivative from this expression and excluding the constant C1 we find the boundary condition imposed on vr (r): iωR dvr (R) vr (R) + =− vt (R). dr R 2cs (0)2

(4.46)

Since compressibility effect becomes important at wave numbers k ∼ /cs , and the eigenvalues of k are expected to be of the order of 1/R, the compressibility effect is essential if the parameter s=√

R 2cs (0)

=$



(4.47)

2 − 2 ω⊥

is of order unity or more. Thus fluid compressibility becomes important at rapid rotation of the BEC cloud with angular velocity  close to the trapping frequency ω⊥ . It is useful to introduce dimensionless variables for Eqs. (4.43) and (4.44): # ωR 2 ωR r ivr vt ω˜ = . (4.48) z= , =2 √ , v˜r = , v˜t = R cT cT cs (0) 3 rv Then the equations become purely real:

   v˜t (z) d 1 2 3 d (1 − z , )z dz z z2 (1 − z2 ) dz   √ d 1 d[(1 − z2 )zv˜r (z)] 2 2s ω˜ v˜t (z) = . dz z dz

√ 2 2s ω˜ v˜r (z) = ω˜ 2 v˜t (z) +

(4.49) (4.50)

In the dimensionless form the boundary condition (4.45) is d v˜r (1)/dz + v˜r (1) = √ −s ω˜ v˜t (1)/ 2. Equations (4.49) and (4.50) depend only on the compressibility parameter s given by Eq. (4.47). They have been solved numerically (Sonin, 2005a). The reduced eigenfrequencies ω˜ i = fi (s) are functions of s. At s → 0 the functions fi (s) go to some s independent values, but at large s the eigenfrequencies ω˜ i = γi /s are inversely proportional to s.

4.7 Tkachenko waves in a BEC cloud

131

1.4

1.2

1

0.8

0.6

0.4

0.2

1

2

3

4

5

6

7

Figure 4.4 Comparison between the theory (solid $ √line) and experiment (black squares). Here ω1 is the first Tkachenko eigenfrequency and rv = κ/ 3 is the intervortex distance. Figure from Sonin (2005a) with experimental points plotted by Ian Coddington.

The first two eigenfrequencies correspond to γ1 = 9.66 and γ2 = 22.8. Returning back to dimensional frequencies at rapid rotation (ω⊥ −   ω⊥ ), the values are ωi =

√ cT γi cT ≈ 2γi (ω⊥ − ). s R cs (0)

(4.51)

Qualitatively this simple expression (aside from a numerical factor) follows from the dispersion relation (3.131) for Tkachenko plane waves, taking into account that the eigenmodes of the cloud correspond to wave numbers k ∼ 1/R. Figure 4.4 shows the first$reduced eigenfrequency ω1 R/rv found numerically, plotted as a function of

2 − 2 (solid line). s = / ω⊥ The expression (4.51) for Tkachenko eigenfrequencies at rapid rotation is valid for both √ VLL and LLL states. But in the VLL state cT = κ/8π [Eq. (3.112)], while in the LLL state the Tkachenko wave velocity is of the same order as the sound velocity: cT = 0.453cs

132

Oscillation of finite vortex arrays: boundary problems

[see Eq. (3.141)]. Substituting this into Eq. (4.51) one obtains a very simple expression for Tkachenko eigenfrequencies in the LLL state: ωi = 0.641γi (ω⊥ − ).

(4.52)

At the crossover from the VLL to the LLL state, the eigenfrequencies increase together with the increasing ratio cT /cs . They saturate in the LLL state, reaching their maximum values given by Eq. (4.52). For the lowest eigenfrequency i = 1 with γ1 = 9.66 [see the paragraph before Eq. (4.51)], Eq. (4.52) yields ω1 = 6.19(ω⊥ − ). 4.8 Observation of Tkachenko waves in a rotating Bose–Einstein condensate cloud Attempts to detect Tkachenko waves in rotating superfluid 4 He were undertaken soon after publication of the Tkachenko theory. But their analysis requires consideration of the effects of vortex bending and surface pinning in the framework of the three-dimensional theory. Therefore we postpone a discussion until Section 5.9. The first unambiguous visual experimental observation of Tkachenko waves was realised in a rotating Bose–Einstein condensate of cold atoms. Figure 4.5 shows the image of Tkachenko waves obtained by Coddington et al. (2003) in a rotating Bose–Einstein condensate of 87 Rb atoms. In Fig. 4.4 the black squares show the experimental points of Coddington et al. (2003) plotted in our dimensionless variables by Ian Coddington. They were obtained for various parameters, but

Figure 4.5 Tkachenko mode excited in a rotating BEC of 87 Rb atoms (Coddington et al., 2003). Lines are sine fits to distortions of the vortex lattice by the Tkachenko mode.

4.8 Tkachenko waves in a rotating BEC cloud

133

collapse on the same curve, as expected from the analysis of the previous section. Quantitative agreement between the theory for the VLL state (solid line) and experiment looks quite good. Coddington et al. (2003) also measured the ratio of the first two frequencies ω2 /ω1 = 1.8 at /ω⊥ = 0.95, which corresponds to s = 3.04. The present $ theory predicts

2 − 2 . This the ratio ω2 /ω1 = 2.09. The agreement becomes worse at larger s = / ω⊥ can be connected with the beginning of the crossover from the VLL to the LLL state. Schweikhard et al. (2004) extended observations of the Tkachenko mode to higher rotation speeds in an attempt to reach the LLL state. They observed linear dependence of the Tkachenko eigenfrequency on small ω⊥ − , as was predicted by the theory. The observed Tkachenko modes were softer than predicted by the theory for the VLL state (called the Thomas–Fermi regime by Schweikhard, et al.). On the basis of good quantitative agreement with the theoretical calculation for the LLL state by Baym (2004), Schweikhard et al. concluded that they had already reached the LLL state. However, the correct value of the shear modulus C66 in Eq. (3.140) is 10 times larger than the value (81/80π 4 )ρcs2 obtained by Baym (2004). Therefore this comparison was revised (Sonin, 2005b). In reality the frequencies of the observed Tkachenko mode were about 4 times less than expected for the LLL state [Eq. (4.52)]. This is evidence that the experiment had not yet reached the LLL limit. Since experimental values of (ω⊥ − )/ω⊥ look small enough, apparently in order to reach the LLL limit the experiment should be done with a smaller number of 2 − 2 ), the order of atoms N. Bearing in mind that N = π nR 2 and R 2 = 2βV n/m(ω⊥ $ 2 − 2 )/V . The LLL state occurs magnitude estimation for the density the is n ∼ mN (ω⊥ if V n  h. ¯ This condition requires an inequality

N

h2 2 . 2 − 2 V m ω⊥

(4.53)

Another condition is the presence of many vortices in the cloud: π nv R 2 1. This yields the inequality 2 − 2 ω⊥ h2 . (4.54) Vm 2 This is compatible with the previous inequality for rapid rotation when ω⊥ −   .

N

5 Vortex oscillations in finite rotating containers: three-dimensional boundary problems

5.1 Torsional oscillator (Andronikashvili) experiment The experiment with a torsional oscillator immersed in a rotating or resting superfluid is an important source of information on vortex dynamics in superfluids. Originally this was performed by Andronikashvili (1946) to measure the normal density of the superfluid, and sometimes it is called the Andronikashvili experiment. The body of the torsional oscillator in the Andronikashvili experiment is a pile of disks with distance 2L between disks. The experiment plays a key role for many issues addressed in this book, and here we discuss its principal scheme and the information available from it. Small oscillation of a torsional oscillator is described by an equation for the angle φ(t) of the oscillator rotation: It φ¨ + Gφ = 0,

(5.1)

where G is the elastic spring constant and It =

ρt R 4 4

(5.2)

is the moment of inertia per unit length along the oscillator axis, R is the radius of the oscillator body, and ρt is the mass density of the material of the oscillator averaged over the whole oscillator volume including the empty space between disks. It is also possible to present the oscillation equation in terms of the linear displacement ut = Rφ, normalising the equation on the unit volume: ρt u¨ t + Kut = 0,

(5.3)

where the effective spring constant is K = 4G/R 4 . The frequency of the bare oscillator is √ ωt = K/ρt . Now let us consider a pile of disks immersed in a viscous fluid. We need to solve the Navier–Stokes equation (1.89) for a viscous fluid between two oscillating solid plates. The solution of the problem is a superposition of two plane-wave viscous modes ∝ eiK·R−iωt with the spectrum ω = iνK 2 (Section 1.7), which is determined by the no-slip boundary 134

5.2 Boundary conditions on a horizontal solid surface

135

condition that the fluid velocity at z = ±L is equal to the velocity vt = u˙ t = iωut of the plate: ! √ cos iω/νz !. v(z) = vt (5.4) √ cos iω/νL The average velocity over the fluid layer is    1 ν iω v¯ = vt tan L , L iω ν

(5.5)

and the force on the oscillator disks is determined by the time derivative of the momentum 2Lρ v¯ of the oscillating fluid. This force modifies the oscillation equation (5.3): (ρt + ρ  )u¨ t + kut = 0, where the complex mass density ρ ρ = L



ν tan iω



iω L ν

(5.6)  (5.7)

describes the effect of the dragged fluid on the torsional oscillation. In the limit when the √ distance between disks is small compared to the viscous penetration depth δ = ν/ω, the dragging becomes 100% effective, and the whole fluid participates in the oscillation (the clamped regime). In the two-fluid model of superfluids, only the normal part of the fluid has viscosity, and the total fluid density ρ in Eq. (5.7) must be replaced by the normal density ρn . It is impossible to drag the superfluid part of the fluid, if the container is at rest, and there is no vorticity in the superfluid. But in a rotating container, vortices in the superfluid can be pinned at the rough surfaces of the disks, and the superfluid can be dragged by oscillating disks. Then the term proportional to the superfluid density ρs must appear in the expression for ρ  . So the torsional-oscillator experiment is an effective technique not only for measurement of the normal component (the first application of this technique) but also for investigation of vortex dynamics. If the dragged mass is much less than the mass of the bare oscillator, its effect on the oscillation period and the quality factor Q = −ωt /Imωt can be found using perturbation theory. The shift of the oscillation period T = 2π/Reω and of the inverse quality factor are determined by the real and the imaginary parts of the complex density ratio ρ  /ρt (Hall, 1958): ωt T 1 ρ =− = Re , T ωt 2 ρt

Q−1 =

1 ρ Im . 2 ρt

(5.8)

5.2 Boundary conditions on a horizontal solid surface: surface pinning Now we are going to formulate the boundary conditions imposed on a rotating perfect fluid at a horizontal solid surface normal to the vortex lines. The first condition is evident

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Vortex oscillations in finite rotating containers

and is imposed on the fluid velocity: the fluid velocity at the boundary should not have a component normal to the wall, i.e., vz = 0. The other boundary condition arises from constraints on the vortex displacement or the vortex velocity imposed by surface pinning. If the surface is ideally smooth, the vortex line meets the top or the bottom of the container normally. In experiments this condition holds only on a free surface of the fluid. Any real solid surface is rough. Suppose that the surface profile is given by the distribution of heights h(r) of points on the surface measured from some average level along the z direction (Sonin, 1992; Krusius et al., 1993; Sonin and Krusius, 1994). Here r is a position vector in the xy plane and h is negative for a bulge on the bottom surface. Let us consider a single vortex line (Fig. 5.1). How does the energy of the vortex vary with its position on the surface? If pinning is weak, a moving vortex remains approximately straight and its energy varies proportional to the line tension ε: E = εh(r).

(5.9)

The force on the vortex (pinning force) is f p = −ε

dh(r) . dr

(5.10)

Figure 5.1 Vortex line pinned to a shallow bulge on the boundary of the superfluid. In the linear regime of pinning, surface roughness is characterised by an asperity with a radius of curvature R = 1/b and a pinning force f p = −bu, which depends on the displacement u(z) from the summit of the bulge. The result is a harmonic pinning potential well. In the limit of weak pinning (applicable to 3 He-B, for example) the pinning strength b is small, the radius of curvature of the pinning site is large, and the bending of the vortex line is weak. Figure from Krusius et al. (1993).

5.2 Boundary conditions on a horizontal solid surface

137

The pinning force attracts the end point of the vortex to the summits of surface bulges. If the fluid flow dragging the vortex is too small, the vortex cannot escape from the bulge, and stationary motion of the vortex along the surface is impossible. The phenomenon of vortex pinning is well known for type II superconductors in the mixed state (Tinkham, 1975) although, in contrast to superfluids, in superconductors vortices can be pinned not only by surface defects (surface pinning) but also by impurities and defects in the bulk (bulk pinning). Coercivity affecting domain-wall and Bloch-line motion in ferromagnetic materials yields a similar effect. All these effects can be united under the title ‘dry friction’: steady motion is allowed only when a driving force exceeds some critical value. The pinning force at the surface is resisted by forces on the vortex line in the bulk (the Lorentz force, for example). The momentum transferred by the bulk forces to the vortex is then transported to the surface by the elastic momentum flux −εdu/dz proportional to the line tension ε. Here u is the displacement of the vortex line, measured in the xy plane. The balance of forces at the top and bottom surfaces is f± p = ±ε

du dz

at z = ±L/2.

(5.11)

The signs ± correspond to the top (z = +L/2, the upper sign) and the bottom (z = −L/2, the lower sign) of the container, respectively. With the help of Eq. (5.10), this gives the boundary conditions −

du dh(r) =± . dr dz

(5.12)

Suppose that the end of the vortex line is not very far from the summit of a bulge at r = r 0 in Fig. 5.1, and one can expand h(r) = h(r 0 + u). For simplicity we assume that the bulge is axisymmetric. Then h(r)  h(r 0 ) +

1 d 2h 2 u , 2 dr 2

(5.13)

and the linearised version of the boundary condition Eq. (5.12) is du = ∓bu dz

(5.14)

with the pinning parameter b = d 2 h/dr 2 being equal to the inverse curvature radius of an axisymmetric bulge (Sonin, 1992), as sketched in Fig. 5.1. Equation (5.14) is the elastic pinning condition. Pinning can be considered as weak if it does not significantly bend vortices. The vortices are assumed to be straight if the difference between vortex displacements in the middle of the fluid layer and at the fluid–solid interface is small compared to the displacement itself. For a thin layer L  E , this condition is L(du/dz) ∼ Lbu  u, i.e., b  1/L. But at low frequencies ω   in a thick fluid layer L E , vortices are bent only within the

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Vortex oscillations in finite rotating containers

superfluid Ekman layer of width E , and the condition of weak pinning becomes b

1 . E

(5.15)

A more general form of Eq. (5.14), which is allowed by symmetry, is ∂u = b(u − uB ) − b zˆ × (u − uB ), ∂z

(5.16)

or in the equivalent form u − uB = a

∂u ∂u + a  zˆ × . ∂z ∂z

(5.17)

Here uB is the displacement of the solid surface. According to Eq. (5.11), the boundary condition (5.17) corresponds to the pinning force f p = −ε[b(u − uB ) − b zˆ × (u − uB )] = −ρκνs [b(u − uB ) − b zˆ × (u − uB )]. (5.18) When the force dragging the vortices reaches the critical value for depinning, the pinning site cannot hold the vortex, and it will tend to jump from one surface bulge to another, thus producing an irregular and non-linear form of slip (Hall, 1958). After depinning, one can impose a boundary condition on the relative velocity of the vortex instead of its displacement. Such a condition was proposed by Hall (1958) and in a more general form by Bekarevich and Khalatnikov (1961): 1 1 ˜ ˜ + ζ  [ˆz × ω]. ω˜ × [ˆz × ω] ω˜ ω˜ 2 Here v B is the velocity of the solid surface in the xy plane. In the linear theory the vorticity is given by v L − v B = −ζ

(5.19)

∂u , ∂z

(5.20)

∂u ∂u + ζ  zˆ × . ∂z ∂z

(5.21)

ω˜ = 2 + 2 and one can rewrite Eq. (5.19) as v L − v B = −ζ

The parameters in Eqs. (5.16)–(5.21) are connected in the Fourier representation by a j -complex relation (see Section 2.9) b − j b =

1 −iω = . a + j a ζ + jζ

(5.22)

Explicit relations between the pairs of parameters are obtained by separation of the real and imaginary parts with respect to the imaginary unit j (assuming i to be ‘real’ in this operation). The phenomenological condition (5.19) of Bekarevich and Khalatnikov assumes that a surface dissipative force (surface friction) acts upon the vortices. In connection with

5.3 Collective surface pinning

139

this a question was raised by Campbell and Krasnov (1982) and Adams et al. (1985): ‘how can the alternate attachment and release (after some stretching) of a vortex line on surface irregularities result in a dissipative force proportional to the relative velocities’. A natural suggestion is that the dissipative force given by the Bekarevich–Khalatnikov condition has the same origin as the residual resistance in dirty metals at T = 0: it arises as a result of averaging of vortex motion over random irregularities of the solid surface. Such a dissipative force may arise even if there is no direct contact between a rough wall and the vortices, as in the case of a vertical wall parallel to the vortices [see discussion after Eq. (4.22) in Section 4.3]. In this case, however, the force is expected to be rather weak. The complete solution of the problem of surface pinning would be a theory connecting the force on the vortices with the amplitude and space scale of the surface irregularities. Some steps in this direction were taken by Schwarz (1985) for the motion of isolated vortices. Data on the magnitude of empirical parameters in the Bekarevich–Khalatnikov condition are available from experiments on vortex oscillations that will be discussed later. Unfortunately, these data are not sufficient to determine both parameters ζ and ζ  simultaneously. Therefore, beginning with Hall (1958), researchers in the field have been forced to assume the simplest scenario: ζ  = 0. The parameter ζ was measured by Gamtsemlidze et al. (1966) studying stationary flows. The values of Gamtsemlidze et al. (1966) and those of Hall (1958) roughly agree and fall in the interval ζ = 10−1 − 10−2 cm/sec. The data are discussed in greater detail by Andronikashvili and Mamaladze (1966). Surface pinning of vortices in experiments on stationary counterflow in superfluid 4 He was also investigated by Yarmchuk and Glaberson (1978, 1979) and more systematically by Hegde and Glaberson (1980). Data on vortex–surface interaction were also available from studies of transient phenomena such as the spin-up process, in which a freely rotating container with superfluid is impulsively spun up and allowed to relax back to solid–body rotation [for discussion and references see the papers of Campbell and Krasnov (1982); Adams et al. (1985); Sonin (1987)]. 5.3 Collective surface pinning Only at very low vortex density can isolated vortex lines be pinned independently. When the vortex density increases, pinning becomes collective: correlated groups of vortices move coherently over a random distribution of abundant pinning sites. The correlated motion is brought about by shear rigidity of the vortex array in the transverse xy plane. Shear rigidity causes coherence extending over a correlation length c . This suppresses the pinning strength. Collective pinning was invented for the bulk pinning of quantised flux lines in type II superconductors, where pinning at defects of the crystalline lattice is similarly reduced with increasing density of the flux line lattice (Larkin and Ovchinnikov, 1979). Collective surface pinning in type II superconductors was discussed in connection with the peak effect (Plac¸ais et al., 2004). The presence of collective surface pinning of vortices in superfluid 4 He was suggested in the reports on observation of the thermorotation effect by Yarmchuk and Glaberson (1978, 1979) and Hegde and Glaberson (1980).

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Vortex oscillations in finite rotating containers

Let us now present some qualitative scaling arguments to explain the origin of collective surface pinning (Krusius et al., 1993). At scales of the order of the correlation length c , vortices move retaining more or less the crystalline order. The number of vortices within a correlated group is given by the areal ratio Nc ≈ (lc /rv )2 . Assuming that the density of pinning sites exceeds the vortex density nv ∼ rv−2 , each vortex interacts mainly with its own nearest pinning site. Thus, Nc coherently moving vortices interact with approximately Nc pinning sites. Because of the random directions of pinning forces at these sites, the √ resultant random force on the Nc correlated vortices is of order Nc and not Nc times larger than the typical pinning force acting on a single vortex. This is illustrated in Fig. 5.2 √ and means that pinning per vortex is reduced by the factor 1/ Nc . The pinning parameter b is now estimated as rv bs ∼ bs , b∼ √ c Nc

(5.23)

where bs is the pinning parameter for a single isolated vortex line discussed in the previous section. The correlation length c is defined by the balance between the energy gain from selecting the best position for the correlated group of vortices with respect to random pinning sites and the energy loss incurred from the inevitable deformations of the vortex lattice. The pinning energy per vortex is proportional to the increase in vortex length and is of order rv (5.24) Ep ∼ bεu20 ∼ bs εu20 . c In contrast to the dynamic displacement u(t) discussed earlier, here u0 measures the static displacement of the end of the vortex line from its position in the bulk regular lattice inside the correlated group of vortices. The location of the group is determined by minimisation of the pinning energy for the whole group (see Fig. 5.2). The elastic energy density of the vortex lattice per unit volume with respect to shear deformation in the xy plane   ∂u0y 1 ∂u0x + , (5.25) uxy = 2 ∂y ∂x is given by Eel ∼ ρκu2xy .

(5.26)

Only shear deformation defined by Eq. (5.25) is relevant for collective pinning, since compression of the vortex lattice is energetically very expensive (Larkin and Ovchinnikov, 1979). This is a consequence of the extremely low compressibility of the vortex lattice discussed at the end of Section 3.8. In the correlated vortex group the single vortex displacement u0 is smeared out over the coherent region and thus the scale over which the displacement varies becomes ∼ c and uxy ≈ u0 /c . Shear deformation of the vortex lattice near the top and bottom surfaces is inevitably accompanied by vortex bending, which heals the distortions of the vortex lattice. The healing length is the width of the superfluid Ekman

5.3 Collective surface pinning

141

(a)

(b)

(c)

Figure 5.2 Collective pinning of vortex lines. (a) Viewed from the side, three vortex lines are pinned to the top surface of the cell, shown with dashed lines in their displaced positions during slow motion and with solid lines in their stationary positions determined by minimisation of the pinning energy for the whole correlated group of vortices. In the weak pinning limit the bending of vortex lines is much less than illustrated here. (b) Viewed from above, the pinning forces act randomly in a correlated group of Nc vortices. (c) The randomly oriented pinning forces can be united to give a resultant random pinning force, which acts on a group of vortices glued together by the shear elasticity of the vortex lattice over the correlation length c . Here u(t) is the dynamic displacement of the vortex from its equilibrium position, while u0 measures the displacement of the end of the vortex line from its stationary position in the bulk regular lattice. Figure from Krusius et al. (1993).

√ layer E = νs /2 [see (Eq. (3.102))] as shown in Fig. 5.2. Thus elastic deformation is confined within the superfluid Ekman layer and the elastic deformation energy per vortex line can be written as Eel ∼ Eel E rv2 ∼ ρκ 2

u20 E ε u2 . E ∼ 2 2c c ln ( rrvc ) 0

(5.27)

The sum of the pinning energy −Ep (negative) and the elastic energy Eel (positive) has a minimum when Ep ∼ Eel , and from this condition one obtains the correlation length

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Vortex oscillations in finite rotating containers

c =

1 E , rv bs ln ( rrvc )

(5.28)

which depends weakly (logarithmically) on  via the intervortex distance rv . The collective pinning concept is valid if the number of correlated vortices is sufficiently large, i.e., when c rv . This leads to the inequality b  1/E , which at the same time is the condition of weak pinning given by Eq. (5.15). Thus weak surface pinning is always collective.

5.4 Pile-of-disks oscillations: Hall resonance versus inertial wave resonance The experimental study of vortex oscillations in the pile-of-disks geometry began in the 1950s (Hall, 1958; Andronikashvili et al., 1961). Remarkable resonance effects were discovered when the disk separation was an odd number of half-wavelengths of the Kelvin wave propagating along the vortex line (Section 3.7). The simple explanation of these resonances called Hall resonances, is that the ends of the vortices are completely pinned to the rough surfaces of oscillating disks, which generate oscillations of vortex lines resembling oscillations of elastic strings. Motion of the vortices is coupled with motion of the fluid as a whole, and the fluid is dragged by oscillating disks. This affects the period and the quality factor of the pile-of-disks oscillations. A number of earlier comprehensive reviews (Hall, 1960; Andronikashvili et al., 1961; Andronikashvili and Mamaladze, 1966, 1967) addressed theoretical and experimental investigations of this phenomenon. We consider a layer of perfect fluid of width 2L between two horizontal solid surfaces located at z = L and z = −L. The surfaces perform harmonic torsional oscillations around the z axis. One could treat this problem as one-dimensional, looking for the dependence of fluid variables on z at a given velocity v B of the solid surfaces. Solid surfaces oscillate as a solid body. But because of the singular character of the oscillation spectrum, it is preferable to start from a solid surface velocity field v B eik·r−iωt varying in the xy plane. At the end of the analysis we shall take the limit k → 0. The solution of our hydrodynamical problem must be a linear superposition of plane waves eiK·R−iωt = eik·r+ipz−iωt . Any wave in the superposition corresponds to a value of p satisfying the dispersion equation (3.95) for waves in an incompressible perfect fluid, in which we neglect the Tkachenko contribution ∝ cT2 k 2 . This equation has three solutions for p2 . Two of them are much larger than small k 2 and correspond to axial Kelvin waves with the spectrum Eq. (3.106), p(j )2 = −

2 + j iω , νs

(5.29)

where the substitution j = ±i gives the value of p for two circularly polarised waves. The third value of p2 , pI2 =

ω2 k 2 , 42 − ω2

(5.30)

yields the inertial wave with velocity homogeneous in space in the limit k → 0, but varying in time.

5.4 Pile-of-disks oscillations: Hall resonance

143

The three possible values of p2 correspond to three standing waves in the wave superposition. For the fluid velocity in the xy plane we have in the j -complex representation:     iω (5.31) v(z, ˜ r, t) = v˜K (j ) cos p(j )z + vI + j cos pI z eik·r−iωt . 2 The real vI (with respect to j ) is the amplitude of the inertial wave, and the complex  determines the amplitudes of two axial Kelvin modes. Explicit expressions v˜K = vK +j vK for the velocity components vk and vt along and normal to k are obtained by separation of the real and imaginary parts with respect to the imaginary unit j :   iω vI cos pI z eik·r−iωt , vk = Rej v˜ = v+ cos p+ z + v− cos p− z + 2 (5.32) vt = Imj v˜ = [−iv+ cos p+ z + iv− cos p− z + vI cos pI z] eik·r−iωt .  ), and p are values of p(j ) at j = ±i. The third component of Here v± = 12 (vK ± ivK ± velocity,     v˜K (j ) sin p(j )z iω sin pI z , (5.33) + vI vz (z, r, t) = −k Rej p(j ) 2 pI

is found from the incompressibility condition ∂vz + ikvk = 0. ∂z

(5.34)

Now we take into account that pI is proportional to k [see Eq. (5.30)] and therefore small, and expand in pI z. Using the relations between velocity components given by Eqs. (3.107) and (3.117), we can find the vortex velocity v L and substitute it into the Bekarevich–Khalatnikov condition, Eq. (5.17) or (5.21). We limit ourselves to the simple case of perfect pinning when the vortex velocity v L is equal to the solid surface velocity and the two boundary conditions at the surfaces z = ±L are given by the complex relation   iω iω v˜K cos p(j )L + + j vI . (5.35) j vB = 2j 2 The third boundary condition is vz = 0 at z = ±L. According to Eq. (5.33), this yields   v˜K (j ) sin p(j )L iω + LvI = 0. Rej (5.36) p(j ) 2 Now one can see the reason why we retained the small wave vector k in the xy plane. If we did not, we would have lost the boundary condition, Eq. (5.36), which follows from Eq. (5.33) when k  = 0. But there is another way to deduce Eq. (5.36). In an incompressible fluid the mass current integrated over the whole width of the fluid layer should not have a

144

Vortex oscillations in finite rotating containers

component along the wave vector k in the xy plane. This restriction yields Eq. (5.36). In the pile-of-disks geometry k is directed along the radius, and absence of the radial mass current is required by the incompressibility of the fluid. The solution of Eqs. (5.35) and (5.36) yields the amplitudes of all waves in the space between disks. The fluid velocity averaged over the fluid layer has only a transverse component with subscript t [the k component vanishes as a result of Eq. (5.36)], and its ratio to the solid surface velocity vB is equal to the density ratio ρ  /ρ where ρ  is the mass density dragged by oscillating disks (Section 5.1):  L ρ 1 vt (z) dz = ρ 2LvB −L ) * 42 Z(j )Z(−j ) − Rej {Z(j )} + 2iωImj {Z(j )} +  , = iω Z(j ) ω2 − 42 Rej 1 − j2 =

(ω − 2)Z+ − (ω + 2)Z− + 42 Z+ Z− . ω2 − (ω + 2)Z+ + (ω − 2)Z−

(5.37)

Here Z(j ) = tan[p(j )L]/p(j )L and Z± = Z(±i) = tan[p± L]/p± L. This expression was obtained by Hall (1958). Knowing also the ratio ρ/ρt , one obtains the ratio ρ  /ρt introduced in Section 5.1. The poles of ρ  /ρ as a function of ω determine the frequencies of the resonances. Looking for resonance frequencies, Hall (1958) simplified Eq. (5.37), assuming that ω 2:    tan(p+ L) tan(p− L) ρ = − . (5.38) ρ ω p+ L p− L Since p− are imaginary, only poles of tan(p+ L) at real frequencies ω correspond to physical resonances. Resonance frequencies are linear functions of the angular velocity :   π 2n − 1 2 . (5.39) ωn = 2 + νs 2 L Here the integer n > 0 is the number of a branch of the spectrum. Under the assumption ω 2 used by Hall, the oscillating disks generate only Kelvin modes in the fluid. Thus one can obtain the resultant formulas of this approximation, Eqs. (5.38) and (5.39), by deleting the inertial wave contribution and the boundary condition vz = 0 at the fluid–solid interface from the very beginning. Any nth resonance corresponds to the condition that an √ odd number 2n − 1 of the Kelvin half-wavelengths π/p ≈ π νs /ωn becomes equal to the distance 2L between disks. Resonances with an even number of half-wavelengths are not observable, since in this case oscillating disks generate fluid motion with a velocity which vanishes after averaging over the fluid layer. The early experiments were performed at comparatively low rotation speeds. Later Andereck et al. (1980) and Andereck and Glaberson (1982) [see also Glaberson (1982)]

5.4 Pile-of-disks oscillations: Hall resonance

145

were able to extend the experiments to considerably higher rotation speeds (from ∼ 1 to ∼ 10 rad/sec). They discovered that at high rotation speed the observed resonance frequencies were considerably lower than given by Eq. (5.39) for the lowest branch n = 1. This was explained by the generation of Tkachenko waves. Later we shall discuss the arguments on which this claim was based (Section 5.9). Another interpretation of the results of Andereck et al. has been offered in the framework of Hall’s original theory, abandoning however the assumption ω , which simplified Eq. (5.37) to Eq. (5.38) (Sonin, 1983). Andereck et al. (1980) observed resonances at large ratio /ω when resonances on the Kelvin modes are impossible, since both values p(j ) are imaginary: 2 =− p(j )2 = −kE

2 . νs

(5.40)

In this case the Kelvin waves are evanescent and penetrate into the fluid only within the superfluid Ekman layer of width E = 1/kE (see Section 3.7). In the limit  ω, Eq. (5.37) reduces to [see Eq. (4.35) in the review paper by Andronikashvili et al. (1961)]: 1 − tanh kE L/kE L ρ = . ρ 1 − (ω2 /42 )kE L/ tanh kE L

(5.41)

The zero of the denominator in this formula, in the limit kE L → ∞, yields the resonance frequency equal to    ν 1/4 2 κ rv 1/4 s 3/4 ωL = √ = (2) = ln (2)3/4 . L2 4π L2 rc kE L

(5.42)

The resonance with frequency ωL is an inertial wave resonance. The resonance frequencies, which correspond to zeros of the denominator in the general Hall formula (5.37), were calculated numerically (Sonin, 1983). The results are shown in ¯ = L2 /νs and ω¯ = ωL2 /νs . On the Fig. 5.3 by solid lines in dimensionless variables  same plot the dispersion curves given by Eq. (5.39) are shown by dashed lines for n = 1 and 2. One can see that the numerically calculated curve n = 1 entirely differs from that defined by Eq. (5.39). The new curve goes into the region 2 > ω, where Hall resonances are impossible, and the resonance frequency approaches asymptotically the inertial wave resonance frequency given by Eq. (5.42). Thus the lowest resonance n = 1 is a Hall resonance only for   ω. At  ∼ ω the crossover from Hall resonance to inertial wave resonance takes place. The crossed dashed lines in Fig. 5.3 are drawn through resonance frequencies observed by Andereck and Glaberson (1982) for different distances d = 2L between disks. The observed frequencies lie quite close to the numerically calculated curve n = 1. The experimental data for d = 0.269 cm, which fell beyond the scale of the plot, at ¯ > 170, also agree well with theory. large  ¯ used in Fig. 5.3 is the inverse of the superfluid Ekman number The parameter  Eks =

νs , L2

(5.43)

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Vortex oscillations in finite rotating containers

Figure 5.3 Low frequency branches n = 1 and n = 2 of the spectrum of oscillations in the superfluid between two horizontal solid surfaces. Solid lines show eigenfrequencies corresponding to poles of the function ρ  /ρ given by Eq. (5.37). Dashed lines are eigenfrequencies determined by Hall’s approximate formula Eq. (5.39). The dot-dashed line is the asymptotic curve given by Eq. (5.42), to which the n = 1 branch approaches asymptotically at  ω. Crossed dashed lines are drawn through the experimental points of Andereck and Glaberson (1982) obtained for different distances 2L between disks: 1, 2L = 0.0208 cm; 2, 2L = 0.0366 cm; 3, 2L = 0.0508 cm; 4, 2L = 0.0762 cm. It was assumed that νs = 10−3 cm2 /sec. Figure from Sonin (1983).

which can be introduced by analogy with the Ekman number Ek = ν/ L2 for a classical fluid with kinematic viscosity ν. Inertial wave resonances were observed at low Eks , when there are narrow superfluid Ekman layers of width E  L near the horizontal solid surfaces confining the superfluid [see Eq. (3.102)]. The presence of superfluid Ekman layers can be taken into account by the effective boundary condition. This condition is derived in the following section. The vortex mode before and after the crossover between Hall resonance and inertial wave resonance is illustrated in Fig. 5.4.

5.5 Effective boundary condition for slow motion in a horizontal layer of rotating fluid Low frequency oscillations (ω  ) in a horizontal layer of a rotating fluid can be described in the framework of slow motion hydrodynamics developed in Section 3.9, but we need to derive an effective boundary condition for Eq. (3.121) governing slow motion in

5.5 Effective boundary condition for slow motion (a)

147

(b)

Figure 5.4 Hall resonance versus inertial wave resonance. (a) Standing Kelvin wave for a solitary vortex line. The oscillation takes place along a circular trajectory (dashed circumference) with respect to the equilibrium position (shown as a straight dashed line). The cosine z dependent displacement u is determined from the balance of the line tension force and the surface pinning force f p . Such an oscillation mode corresponds to the lowest Hall resonance n = 1 at low angular velocity   ω when L  E . (b) Inertial wave mode of a dense vortex array where the screening of deformations becomes important. The transverse solid surface on the bottom depicts one of two plates in a pile of parallel disks, which is driven to execute torsional oscillations around the vertical axis inside a rotating container filled with a superfluid. The vortices in the array between two disks are oscillating along elliptical trajectories, which are highly eccentric, and slow motion is primarily azimuthal (see Section 3.9). This is columnar motion when the vortex displacement is constant over most of the length of a vortex line as the bending from the pinning forces f p , which act at the transverse surfaces, is confined to the superfluid Ekman layers with a width E given by Eq. (3.102). Figures from Sonin and Krusius (1994).

the horizontal layer of the rotating superfluid. The derivation is based on the concept of the boundary layer, which was used for the analysis of the inertial wave resonance in a rotating viscous classical fluid (Section 1.13). At a solid surface normal to vortices, one must satisfy three boundary conditions formulated in Section 5.2: zero mass current into a solid surface and two pinning boundary conditions (for two components of the two-dimensional pinning force). This is possible by choosing a superposition of the slow wave (a hybrid of an inertial wave and a Tkachenko wave) and two Kelvin modes. At low frequencies the Kelvin modes

148

Vortex oscillations in finite rotating containers

are evanescent. In analogy with a viscous classical fluid, one can introduce a boundary layer near a solid surface where the Kelvin modes are confined. Outside the boundary layer only the slow mode is excited, which satisfies the effective boundary condition derived from the analysis of the boundary layer. While in a classical viscous fluid the boundary layer was √ the Ekman layer with width δE = ν/2, in a superfluid it is the superfluid Ekman layer √ with width E = νs /2, where the kinetic viscosity ν is replaced by the line tension parameter νs . We consider a fluid occupying semi-infinite space z > 0. This is the limit of a thick fluid layer L E between oscillating disks. The velocity of the solid surface z = 0 is given by the field v B eik·r−iωt , where v B is normal to k and both lie in the xy plane, as was assumed in the previous section. The three waves generated near the solid surface have the same k and ω. Unlike the previous section, the slow wave is not standing, but propagating along the z axis. The vortex velocity in the xy plane in the j -complex representation is given by   (5.44) v˜L (z, r, t) = v˜K (j )e−kE z + j vS eipz eik·r−iωt . Here p is the z component of the wave vector of the slow wave and vS is the amplitude of the transverse component vLt generated by the slow wave. The longitudinal in-plane component is proportional to the small parameter ω/  and is neglected. The amplitudes of the vortex velocity generated by two the Kelvin modes are determined by the real and the imaginary parts of the complex quantity v˜K (j ). The wave numbers of the Kelvin modes are imaginary and are determined by Eq. (5.40). The choice of sign provides attenuation of the Kelvin wave at z > 0. The net vortex velocity field v˜L is substituted into the boundary condition Eq. (5.17) on the surface z = 0. This yields the expression for the j -complex amplitude of the Kelvin modes, v˜K = −

j (vS − vB ) . 1 + ak ˜ E

(5.45)

˜ and neglected the contribution of Here we used the relations a˜ = a + j a  and v˜L = −iωu, the slow mode to ∂ u/∂z, ˜ since p is small at ω  . Using the relations between the components of the vortex and the fluid velocities in the Kelvin wave [Eq. (3.103)], and in the slow mode [Eq. (3.117)], one obtains the expression for the longitudinal in-plane component of the fluid velocity:     2j 2 p2 ipz ik·r−iωt e v˜K e−kE z + vS e . (5.46) vk (z, r, t) = − Rej iω iω K 2 The component vz of the fluid velocity, found from the incompressibility condition (5.34) as in the previous section, is substituted into the boundary condition vz = 0 at z = 0. This yields the equation   2j 2 p k v˜K + vS = 0. Rej (5.47) ikE iω iω k

5.5 Effective boundary condition for slow motion

149

Exclusion of the j -complex amplitude v˜K of the Kelvin waves with the help of Eq. (5.45) gives the equation for the slow mode amplitude: −A

ip vS + vS − vB = 0, k2

where 1 1 Rej = A kE



1 1 + ak ˜ E

(5.48)

 ,

(5.49)

or in an explicit form [see Eq. (5.22)] A = kE

(1 + akE )2 + (a  kE )2 (b + kE )2 + b2 = kE . 1 + akE b(b + kE ) + b2

(5.50)

For perfect pinning (b, b → ∞) A = kE , otherwise A > kE , since the dissipation parameter b is always positive. For weak pinning (small b and b ) A=

2 kE 2 = . b νs b

(5.51)

We have considered a single plane slow wave near the solid surface. A general velocity field in slow motion hydrodynamics is a superposition of such plane waves. Then Eq. (5.48) is a Fourier transform of the boundary condition we are looking for, and vS is a Fourier transform of the vortex velocity v L in the fluid bulk. The inverse Fourier transformation of Eq. (5.48) yields the effective boundary condition imposed on the vortex velocity v L : A ∂v L ∓ (v L − v B ) = 0, k 2 ∂z

(5.52)

where the upper and lower signs refer to the bottom and the top of the layer respectively. In deriving Eq. (5.52), the correspondence principle ip → ∂/∂z was used. Equation (5.52) differs from the effective boundary condition (1.158) for a viscous classical fluid by an extra parameter before the derivative. Now we use the derived boundary condition for an analysis of oscillations of the fluid layer between two horizontal solid surfaces located at z = ±L. In contrast to the previous section, we do take into account the Tkachenko rigidity, which contributes the term ∝ cT2 to the slow wave frequency given by Eq. (3.119). On the other hand, we restrict ourselves to the case of slow motion ω   and large spacing L E . The latter condition corresponds to small superfluid Ekman number Eks = νs / L2 [see Eq. (5.43)]. As in the analysis of inertial wave resonances in a classical fluid (Sections 1.12 and 1.13), there are even eigenmodes with the velocity field v L ∝ cos pz eik·r−iωt and odd eigenmodes with the velocity field v L ∝ sin pz eik·r−iωt . The effective boundary condition (5.52) yields the equation determining the set of discrete p at given ω and k. This equation is Ap tan pL − 1 = 0 k2

(5.53)

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Vortex oscillations in finite rotating containers

for even modes, and Ap cot pL + 1 = 0 k2

(5.54)

for odd modes. Odd modes do not involve oscillations of the container, because the total angular momentum of the fluid does not oscillate. They can be revealed by observation of fluid motion inside the resting container. For large A when Lk 2  A,

(5.55)

Eqs. (5.53) and (5.54) are satisfied at discrete values of p close to pn = π n/2L with n ≥ 1. Substitution of p = pn into Eq. (3.119) yields the eigenfrequencies   πn 2 + cT2 k 2 . (5.56) ωn2 = kL The lowest resonance at n = 0 corresponds to the smallest value of p obtained by expanding the tangent in Eq. (5.53): p02 =

k2 . AL

(5.57)

For p = p0 , Eq. (3.119) readily yields the dispersion law 2 + cT2 k 2 , ω2 = ωL

(5.58)

2 ωL = √ AL

(5.59)

where

is the frequency of the inertial wave resonance, which does not depend on k although p0 does. For weak pinning  2νs b , (5.60) ωL = L while if pinning is strong (A = kE ) this frequency is given by Eq. (5.42). In contrast to Hall resonances at ω 2 [Eq. (5.39)], all slow mode resonances, which we now discuss, are possible at rapid rotation 2 ω. In the eigenmode n = 0 under the condition p0 L  1, vortices and the fluid perform columnar motion: moving vortices remain nearly straight except for the superfluid Ekman layers. Averaging the equation of motion (3.121) for the slow mode over the fluid layer one obtains:     ∂ 2vL 42 ∂v L  ∂v L  − (5.61) + cT2 2⊥ v L . ⊥ 2 = − 2L ∂z z=+L ∂z z=−L ∂t

5.6 Uniformly twisted vortex bundle

151

Using the effective boundary condition (5.52) and the correspondence principle k 2 → −⊥ yields ∂ 2vL 2 2 − cT2 ⊥ v L + ωL v L = ωL vB . ∂t 2

(5.62)

One can use the effective boundary condition if the width of the superfluid Ekman layer is mush smaller than the thickness L of the fluid layer: 2E 1 = ≈ Eks  1, 2 2 2 L L kE

(5.63)

i.e., for small superfluid Ekman number Eks . However, if pinning is weak, the vortices remain straight everywhere, including the superfluid Ekman layers. Then Eq. (5.62) holds at any superfluid Ekman number with ωL given by Eq. (5.60). With growing L in the limit Lk 2 A opposite to the limit defined by Eq. (5.55), the wave number p0 approaches the value p0 = π/2L. In this limit vortex motion ceases to be columnar since vortex displacements near the top and bottom solid surfaces, even outside the Ekman layers, are much less than in the middle of the container. However, at the crossover between the two limits, the inertial wave frequency 2p/k ∼ /kL becomes smaller than the Tkachenko wave frequency cT k ∼ rv k, and the slow mode transforms to a pure Tkachenko mode with negligible effect of vortex bending. Although one cannot calculate ωL assuming columnar motion in such long containers, the frequency ωL itself becomes unimportant for the analysis of vortex oscillations in rotating containers (Section 5.8). Thus we have reduced the problem of fluid motion between two oscillating horizontal solid surfaces to the 2+1 partial differential inhomogeneous equation for the twodimensional velocity v ≈ v L . Though variation of the velocity is slow along the z axis and was taken into account with the perturbation theory, just a weak dependence on z together with surface pinning of vortices is responsible for the gap ωL in the oscillation spectrum Eq. (5.58).

5.6 Uniformly twisted vortex bundle The interest in twisted vortex bundles arose in connection with experimental studies of the transient process of establishing stable vorticity in rotating superfluid 3 He-B (Eltsov et al., 2006). A twisted vortex bundle appears in an originally vortex-free rotating container with a superfluid when vortex lines injected at the bottom of the container expand into the rest of the container. Experimental and theoretical studies of the twisted vortex bundle addressed possible turbulence of the vortex front and the transition from the laminar to the turbulent regime (Finne et al., 2003; Volovik, 2003a; Eltsov et al., 2007, 2009, 2010; Hosio et al., 2011). The velocity field in an untwisted vortex bundle of radius R0 in a container of radius R is determined by Eq. (4.1) (see Fig. 4.1). A uniformly twisted vortex bundle is a collection of helical vortices. Each of them forms a helix of pitch 2π/Q moving along a

152

Vortex oscillations in finite rotating containers

cylinder coaxial with the z axis, where Q=

ω˜ ϕ = ∇z ϕ ω˜ z r

(5.64)

is the wave number of the twist (called later the twist), ω˜ ϕ and ω˜ z are the azimuthal and the axial components of the vorticity ω˜ = ∇ × v, and ϕ(z) is the azimuthal angle of a point ˜ ω˜ on the vortex line. In cylindrical coordinates, the components of the unit vector sˆ = ω/ tangent to vortex lines are sr = 0,

Qr sϕ = " , 1 + Q2 r 2

sz = "

1 1 + Q2 r 2

.

(5.65)

If one ignores Tkachenko shear rigidity, the twist Q(r) can be an arbitrary function of the distance r from the axis. However, Tkachenko rigidity is important, and later we shall assume that the twist Q is r independent. We are looking for the stationary twisted state in the coordinate frame rotating with the angular velocity . The vortex lines are at rest if the vortex velocity in the rotating frame has no component normal to the vortex line, i.e., (v L − [ × r]) × sˆ = 0.

(5.66)

In the HVBK theory the vortex line moves with the velocity v L given by Eq. (3.58). In cylindrical coordinates this reduces the vector equation (5.66) to a single scalar equation:     ∂sz νs ∂(rsϕ ) sz − vz (r) + sϕ = 0. (5.67) vϕ (r) − r − νs ∂r r ∂r Another relation between the two components of the velocity v (in an incompressible fluid the radial component vr vanishes) follows from the connection between velocity and vorticity: 1 vϕ = r

r

r ω˜ z r dr,

vz = v0 −

0

r ω˜ ϕ dr = v0 − Q

0

ω˜ z r dr = v0 − Qrvϕ . (5.68) 0

The cylindrical components of the velocity and the vorticity satisfying the equations (5.67) and (5.68) are ( + Qv0 )r νs Q2 r + , 2 2 1+Q r (1 + Q2 r 2 )3/2 v0 − Qr 2 νs Q3 r 2 vz = v0 − Qrvϕ = − , 2 2 1+Q r (1 + Q2 r 2 )3/2 1 ∂(rvϕ ) 2( + Qv0 ) 2νs Q2 (1 − Q2 r 2 /2) ω˜ z = + , = r ∂r (1 + Q2 r 2 )2 (1 + Q2 r 2 )5/2 ∂vz 2Qr( + Qv0 ) νs Q3 r(2 − Q2 r 2 ) ω˜ ϕ = Qr ω˜ z = − + . = ∂r (1 + Q2 r 2 )2 (1 + Q2 r 2 )5/2 vϕ =

(5.69)

5.6 Uniformly twisted vortex bundle

153

This solution without the line tension terms ∝ νs was found by Eltsov et al. (2006). The fluid velocity field in the vortex-free region R0 < r < R is determined from the conditions of velocity continuity at r = R0 : vϕ =

vϕ (R0 )R0 , r

vz = vz (R0 ),

(5.70)

where vϕ (R0 ) and vz (R0 ) are determined from Eq. (5.69) at r = R0 . The constant v0 in these equations must be determined from the condition that the R total axial mass current 2π 0 vz (r)r dr vanishes in a closed container. This connects the constant velocity Qmz πρR 2

v0 =

(5.71)

with the angular momentum around the z axis per unit length,  mz = 2πρ

R

vϕ r 2 dr.

(5.72)

0

If R R0 , the axial velocity vz in the vortex-free region R > r > R0 must be very small and v0 = QR02 +

νs Q3 R02 (1 + Q2 R02 )1/2

.

(5.73)

In the opposite limit R0 ≈ R, the bundle occupies the whole cross-section of the container, and the axial mass current vanishes if   Q2 R 2 2˜νs Q  − 1 + , (5.74) v0 = Q ln(1 + Q2 R 2 ) ln(1 + Q2 R 2 ) where  ν˜ s = νs

 2 + Q2 R 2 −2 . " 1 + Q2 R 2

(5.75)

Knowing the velocity field, one can find the energy per unit length, 

R

e = 2πρ 0

vϕ2 + vz2 2

 r dr + νs

R0

 ωr ˜ dr ,

(5.76)

0

and the angular momentum mz per unit length [Eq. (5.72)]. For the bundle filling the whole container (R ≈ R0 ) this yields the energy per unit length in the rotating coordinate frame:

154

Vortex oscillations in finite rotating containers

v02 πρ( + Qv0 )2 2 2 − [Q R − ln(1 + Q2 R 2 )] 2 2Q4   1  + Qv0 + 4πρνs 1− Q2 (1 + Q2 R 2 )1/2   5Q2 R 2 πρνs2 2 2 + − ln(1 + Q R ) 2 1 + Q2 R 2   Q2 R 2 πρ2 R 2 −1 =− 2Q2 ln(1 + Q2 R 2 )   1 4π νs (R 2 + 2˜νs ) 1− + ln(1 + Q2 R 2 ) (1 + Q2 R 2 )1/2   5Q2 R 2 π νs2 2π ν˜ s2 2 2 . + − ln(1 + Q R ) + 2 1 + Q2 R 2 ln(1 + Q2 R 2 )

e = e − mz = πρR 2

(5.77)

Twisting of the bundle leads to the flux of the angular momentum along the vertical axis. Using the expressions (3.62) for the momentum flux tensor in the HVBK theory and (1.38) R for the angular momentum flux tensor Gin , the axial flux Jm = 2πρ 0 Gzz r dr of the angular momentum around the z axis integrated over the cross-section of the container is   ω˜ ϕ ω˜ z 2 vϕ vz − νs r dr ω˜ 0 0   Q4 R 4 π 2 R 2 − 1 =− Q3 (1 + Q2 R 2 ) ln2 (1 + Q2 R 2 ) 

Jm = 2πρ

− −

R



ϕz r 2 dr = 2πρ

R

4π ˜νs QR 4 (1 + Q2 R 2 ) ln (1 + Q2 R 2 ) 2

−

4π ν˜ s2 QR 2 (1 + Q2 R 2 ) ln2 (1 + Q2 R 2 )

π νs2 Q3 R 4 . (1 + Q2 R 2 )2

(5.78)

The expression for the flux of the angular momentum along the z axis can also be derived from the canonic relation Jm = −∂e /∂∇z ϕ. A twisted vortex bundle transports along its axis not only angular momentum but also linear momentum. The flux of the z component of the linear momentum has the dimensionality of force and is determined by the diagonal component of the momentum flux tensor (3.62) in the HVBK theory integrated over the bundle cross-section (as before, we consider the case R0 = R, i.e., the bundle fills the whole container): R Fz = 2π 0

 R  2 ρν ω ˜ s ϕ zz r dr = 2π P + ρvz2 + r dr. ω˜ 0

(5.79)

5.7 Torsional oscillations of a vortex bundle

155

Taking into account the absence of the current along the z axis, one can connect the part of the force which is not connected with the pressure with the angular momentum flux Jm :   R  R  νs ω˜ ϕ2 νs Qr ω˜ z ω˜ ϕ 2 2 vz (v0 − Qrvϕ ) + 2πρ r dr vz + r dr = 2πρ ω˜ ω˜ 0

0



R

= −2πρQ

ϕz r 2 dr = −QJm .

(5.80)

0

To determine the pressure P , one should first find the chemical potential distribution over the bundle cross-section using the Euler equation: μs (r) = μs (0) −

v(r)2 + rvϕ (r), 2

(5.81)

where μs (0) is the chemical potential at the bundle axis (r = 0). After this one can find the pressure distribution over the bundle cross-section from the Gibbs–Duhem relation (3.60) in the HVBK theory: P =−

ρv 2 ρv 2 ˜ + P0 , + ρrvϕ − ρνs ω˜ + 0 + ρνs ω(0) 2 2

(5.82)

˜ and P0 , are the velocity, the vorticity and the pressure at r = 0. Apart from where v0 , ω(0) constants, the right-hand side is the energy density in the rotation coordinate frame. Collecting all contributions, the whole linear momentum flux integrated over the crosssection of the bundle is Fz = −e − QJm + C = −e + mz − ∇z ϕJm + C.

(5.83)

The constant C depends on conditions at bundle ends. For a vortex bundle terminating at a lateral wall, this will be found in Section 7.7.

5.7 Torsional oscillations of a vortex bundle Now we consider the dynamics of an axisymmetric vortex bundle. Because of rotational symmetry there must be a Goldstone mode, which can be derived phenomenologically as a torsional mode of the vortex bundle. This mode is related to a degree of freedom described by a pair of canonically conjugated variables ‘mz –ϕ’, where mz is the z component of the moment and ϕ is the angle of rotation around the z axis. The phenomenological Hamiltonian is    2 mz K(∇z ϕ)2 + dz, (5.84) H= 2I 2

156

Vortex oscillations in finite rotating containers

where mz is a small correction to the equilibrium value of the moment mz , I is the moment of inertia, and K is torsion stiffness. The Hamilton equations are m (z) δH ∂ϕ = = z , ∂t δmz I

(5.85)

∂mz δH =− = ∇z (K∇z ϕ) . ∂t δϕ

(5.86)

According to these equations, a plane wave ∝ eipz−iωt propagating along the z axis has the spectrum 2 2 p = ω2 = vw

K 2 p , I

(5.87)

√ where vw = K/I is the velocity of the torsional mode. The integrand of the Hamiltonian (5.84) must be obtained from expansion of e [Eq. (5.77)] in small deviations from the equilibrium untwisted bundle. The second Hamilton equation (5.86) shows that the quantity Jm = −∂e /∂∇z ϕ = −K∇z ϕ is the angular momentum flux along the z axis. The fluid moment of inertia I = πρR04 /2 was found in Section 4.1 and is given by Eq. (4.4). It does not depend on the radius R. Expansion of e with respect to Q in the continuous vorticity approximation yields the torsion stiffness K = ∂ 2 e /∂Q2 = πρ2 R06 /3 for the thin bundle R0  R and K = πρ2 R 6 /12 for the thick bundle R ≈ R0 . Then Eq. (5.87) yields the dispersion relation R ω= √ p 6

(5.88)

for the vortex bundle filling the whole container (R0 ≈ R), and the dispersion relation  ω=

2 R0 p 3

(5.89)

for the thin vortex bundle (R0  R). Our simple phenomenological analysis relied on the assumption of a purely torsional mode: any cross-section of the vortex bundle rotates as a solid body without deformation, although the rotation angle varies from one cross-section to another. The assumption is not self-evident, and now we are going to derive the spectrum of the torsional Goldstone mode from the vortex dynamics equations without relying on this assumption. The slow motion is described in the Fourier representation by Eqs. (3.115) and (3.116) and the plane wave ∝ eik·r+ipz−iωt has the spectrum given by Eq. (3.119). After inverse Fourier transformation to the configurational space with the cylindrical system of coordinates, the equations of motion for the axisymmetric slow mode (no dependence on the azimuthal angle ϕ) are

5.7 Torsional oscillations of a vortex bundle

∂ 2 uϕ ∂ 2 vr − 2 + ∂z2 ∂z2



vr ∂ 2 vr 1 ∂vr − 2 + 2 r ∂r ∂r r

157

 = 0,

∂uϕ + 2ur = 0, ∂t   cT2 ∂ 2 uϕ uϕ ∂ur 1 ∂uϕ = vr − − 2 , + ∂t 2 ∂r 2 r ∂r r

(5.90)

where radial (subscript r) and azimuthal (subscript ϕ) components correspond to the longitudinal (subscript k) and transverse (subscript t) components in Eqs. (3.115) and Eqs. (3.116). In Eq. (5.90) the vortex velocity v L = −∂u/∂t was replaced by the vortex displacement u with the radial and azimuthal components ur and uϕ . We shall consider an oscillation mode, which is a plane wave along the z axis and an axisymmetric cylindrical wave in the xy plane. A solution of these equations finite at the axis of the container is uϕ = u0 J1 (kr)eipz−iωt ,

vr =

2p2 u0 J1 (kr)eipz−iωt . k 2 + p2

(5.91)

The axial velocity vz =

2kp iu0 J0 (kr)eipz−iωt k 2 + p2

(5.92)

is determined from the incompressibility condition ∂vz ∂vr vr + + = 0. dz dr r The stress tensor component σϕr (r) for axisymmetric modes is [see Eq. (4.17)]:   uϕ (r) 2 ∂uϕ (r) − = ρcT2 ku0 J2 (kr)eipz−iωt . σϕr (r) = −ρcT ∂r r

(5.93)

(5.94)

First let us neglect the Tkachenko shear rigidity (cT → 0). If R0 ≈ R this returns us back to inertial wave resonances in an inviscid classical fluid where only one boundary condition vr (R) = 0 was sufficient (Section 1.12). This boundary condition is satisfied at discrete values of k = j1,s /R, where j1,s are zeros of the Bessel function J1 (z). The lowest zero with finite k at s = 1 yields k = 3.83/R and the frequency1 p2 = 0.2722 R 2 p2 . (5.95) k2 If R0 < R, in order to derive the boundary condition one should analyse the vortexfree region, where the velocity is divergence-free and curl-free. The general axisymmetric velocity field satisfying these conditions is ω2 ≈ 42

vr (r) = [AI1 (pr) − BK1 (pr)]eipz−iωt ,

vϕ = 0,

vz (r) = i[AI0 (pr) + BK0 (pr)]eipz−iωt , 1 This mode was also known to Kelvin (Thompson, 1880).

(5.96)

158

Vortex oscillations in finite rotating containers

where I0 (pr) and K0 (pr) are modified Bessel functions (Abramowitz and Stegun, 1972). Constants A and B are determined by the boundary condition at r = R: vr (R) = [AI1 (pR) − BK1 (pR)]eipz−iωt = 0,

(5.97)

and by the continuity condition on the boundary of the vortex bundle r = R0 : vr (R0 ) = [AI1 (pR0 ) − BK1 (pR0 )]eipz−iωt , vz (R0 ) = i[AI0 (pR0 ) + BK0 (pR0 )]eipz−iωt ,

(5.98)

where vr (R0 ) and vz (R0 ) are the superfluid velocity components determined by Eqs. (5.91) and (5.92). At pR  1 this imposes the condition on the velocity at the vortex bundle boundary: ivr (R0 ) − vz (R0 )

p(R02 − R 2 ) = 0. 2R0

(5.99)

For a thin vortex bundle R0  R, Eq. (5.99) gives that at the bundle boundary vz (R0 ) = 0. According to Eq. (5.92) this boundary condition is satisfied at discrete k = j0,s /R0 . The lowest zero j0,1 = 2.4 yields the frequency ω2 ≈ 42

p2 = 0.6922 R02 p2 . k2

(5.100)

The spectra (5.95) and (5.100) differ from the spectra (5.88) and (5.89) of the torsional Goldstone modes by numerical factors. The difference appears because the derivation of (5.95) and (5.100) disagreed with the assumption of pure torsion. At finite values of radial wave numbers k ∼ 1/R0 , cross-sections of the bundle not only rotate but are also deformed. However, at low p and ω one cannot ignore Tkachenko rigidity, which resists deformation of the original structure of the vortex bundle. Taking into account Tkachenko rigidity, the dispersion relation (3.119) for the slow mode is a bi-quadratic equation for k with two solutions for k 2 :  .  2  2 . 1 ω2 1 ω 42 p2 / 2 2 2 k± = − p + p − . (5.101) ± 2 cT2 4 cT2 cT2 In the Goldstone mode k is small, and the term ω2 /cT2 exceeds p2 by a large factor 2 /cT2 k 2 . Since we are looking for modes with minimal p and ω ∝ p, the second term in the square-root argument in Eq. (5.101), which is proportional to p2 , is larger than the first one proportional to p4 . Altogether this allows us to simplify Eq. (5.101): 2 k± =

1 ω2 2 ±i p. 2 2 cT cT

(5.102)

5.7 Torsional oscillations of a vortex bundle

159

The general solution of the boundary problem is a superposition of two modes with k+ and k− : uϕ = [u+ J1 (k+ r) + u− J1 (k− r)]eipz−iωt ,   J1 (k+ r) J1 (k− r) ipz−iωt 2 + u− 2 , vr = 2p u+ 2 e k+ + p 2 k− + p 2 ω2 [u+ J1 (k+ r) + u− J1 (k− r)]eipz−iωt ,    k+ J0 (k+ r) k− J0 (k− r) ipz−iωt + u− 2 , vz = i2p u+ 2 e k+ + p 2 k− + p 2 vLr =

(5.103)

uϕ duϕ − = −[u+ k+ J2 (k+ r) + u− k− J2 (k− r)]eipz−iωt . dr r The presence of two modes in the solution requires two boundary conditions, from which one can find two constants u± . In addition to the condition imposed on the fluid velocity, we assume that there is no flux of the angular momentum through the vortex bundle boundary, and therefore σϕr (R) = 0. If the vortex bundle fills the whole container, the second boundary condition is vr (R) = 0. The system of two linear equations for u± has a solution if its determinant vanishes. This yields the equation for determination of the resonance frequencies: 3 J (k R) k 3 J2 (k− R) k+ 2 + − − = 0. J1 (k+ R) J1 (k− R)

(5.104)

It is important that |k± | is much larger than p, but still much smaller than the inverse radius 1/R. Because of the latter inequality one can expand Bessel functions in k± R:     2 R2 2 R2 k+ k− 2 R 2 p2 ω2 4 4 k+ 1 + = 0. (5.105) − k− 1 + ∝ 2 − 24 24 cT 6cT2 Equation (5.105) yields the spectrum (5.88) of the Goldstone mode. In the case of the thin vortex bundle, the second boundary condition is vz (R0 ) = 0, and the equation for determination of the resonance frequencies is 2 J (k R ) k 2 J2 (k− R0 ) k+ 2 + 0 − − = 0. J0 (k+ R0 ) J0 (k− R0 )

(5.106)

After expansion of Bessel functions this yields the Goldstone mode spectrum (5.89). It is interesting that the dispersion relations for the Goldstone modes do not contain the circulation quantum, which determines the Tkachenko rigidity. However, the rigidity is important since it suppresses deformations in cross-sections, justifying the assumption of pure torsional mode. As long as k± R  1, division of the torsional mode onto two modes with wave numbers k+ and k− is artificial and the oscillation mode effectively is a solid body rotation of any cross-section with a single k going to 0.

160

Vortex oscillations in finite rotating containers

5.8 Slow oscillations of a superfluid in a finite cylindrical container This section addresses the case of a slowly oscillating fluid restricted both along the z axis and in the xy plane. The fluid fills a rotating cylindrical container of height 2L and radius R, but the results of the analysis will also be valid for a fluid with a free surface, when the height of the fluid layer is L. For the torsional Goldstone mode with vanishing wave number k, the effective boundary condition at z = ±L gives that in columnar motion √ the ratio p/k = 1/ AL [see Eq. (5.57)] is finite and the axial wave number p is also vanishingly small. Thus the torsional mode of a vortex bundle in a rotating container of √ finite height is an inertial wave resonance with the frequency ωL = 2/ AL. Up to now we considered the fluid in containers which either were parts of torsional oscillators or were kept fixed and did not oscillate. Now we turn to the case of a freely rotating container, which participates in oscillatory motion (vB  = 0). The analysis of this case (Sonin, 1976, 1987) was motivated by experiments simulating pulsar rotation, discussed in the next section. The fluid in the container performs slow columnar motion governed by Eq. (5.62). In cylindrical coordinates Eq. (5.62) is an equation for an azimuthal velocity component:  2 − ω2 )vϕ − cT2 (ωL

d 2 vϕ vϕ 1 dvϕ − 2 + 2 r dr dr r

 2 = ωL vB .

(5.107)

When analysing oscillations of a vortex bundle in Section 5.7 we imposed two boundary conditions vr (R) = 0 and σϕr (R) = 0 at lateral walls (we assume here that the vortex bundle occupies the whole cylinder). The boundary conditions were satisfied with a superposition of two components in the velocity field corresponding to two possible values of k 2 . The two values of k 2 were obtained from a bi-quadratic equation with respect to k, following from the dispersion relation (3.119) for the slow mode. Now we consider the 2 + c2 k 2 [see Eq. (5.58)], which is linear columnar motion with the dispersion law ω2 = ωL T 2 in k . Mathematically we do not have enough degrees of freedom to satisfy two boundary conditions. So we can satisfy only one boundary condition, and this condition should be the absence of the angular momentum flux, from the fluid, σϕr (R) = 0. On the other hand, the boundary condition vr (R) = 0, which we are going to ignore, was based on the mass conservation law. The question arises whether our scheme ignores this law. One must take into account that the boundary condition vr (R) = 0 was imposed only on the columnar motion in the bulk outside the top and the bottom superfluid Ekman layers. Meanwhile, when deriving the effective boundary condition we used the condition of no normal mass current, vz = 0. In an incompressible fluid if there is no mass current to horizontal surfaces bordering the fluid, then there is no total radial mass current to the lateral wall. This means that a weak non-zero radial mass current in the main bulk, which is not ruled out by our scheme, must be cancelled by an opposite radial mass current in the boundary superfluid Ekman layers. Thus the effective boundary condition guarantees that globally the mass conservation law holds, although the details of radial current distribution along the z axis are beyond our approach neglecting any z dependence.

5.8 Slow oscillations of a superfluid

161

The boundary condition of no angular momentum flux from the fluid σϕr (R) = 0 assumed an ideal cylindrical lateral wall. If rotational symmetry is broken and some azimuthal force is applied to the fluid boundary a more general boundary, condition (4.21) must be used. Its extension to the case of a cylinder with oscillating rotation speed is obvious:   ∂vϕ (R) vϕ (R) − + [vϕ (R) − vB (R)] = 0. (5.108) αb ∂r r Here we wrote the boundary condition in terms of velocity instead of displacements, using the cylindrical coordinate frame. The velocity of the lateral walls and the velocity of the horizontal surfaces bounding the fluid are given by vB (r) =  re−iωt .

(5.109)

Here  is the amplitude of the angular velocity oscillations of the container. The equation determining the eigenfrequencies is derived from the condition that the net angular momentum of the fluid and of the container does not vary during the oscillation:  Ic + mz = 0,

(5.110)

where Ic is the moment of inertia of the container. Let us consider the simple case when the fluid oscillates as a solid body (vϕ ∝ r) and the lateral walls are ideal (αb → ∞). Then Eq. (5.110) is satisfied if the fluid velocity vϕ and the container linear velocity vB =  r are connected by the relation vϕ + βvB = 0,

(5.111)

where β = 2Ic /πρR 4 is the ratio of the moment of inertia of the container to the moment of inertia πρR 4 /2 (per unit length) of the fluid rotating as a solid body. Using this relation in Eq. (5.107) one obtains the oscillation frequency # 1+β . (5.112) ω = ωL β In the case of a very heavy container (β → ∞) the frequency ω = ωL is the frequency of the inertial wave resonance, in which the fluid oscillates with respect to the container rotating with nearly constant angular velocity. The frequency becomes very high for small β, when a light container oscillates with respect to the fluid which is mostly immobile in the rotating coordinate frame. The general solution of the inhomogeneous equation (5.107) with the boundary condition (5.108) is   2 ωL ω2 r J1 (kr) − 2 (5.113)  Re−iωt . vϕ (r, t) = 2 − ω2 R ωL ωL − ω2 J1 (kR) − αb kJ2 (kR)

162

Vortex oscillations in finite rotating containers

Here k = field is

$

2 /c . The angular momentum per unit height for such a velocity ω2 − ωL T

 mz = 2πρ

R

vϕ (r, t)r 2 dr 0   2 ω2 J2 (kR) 2π  ρR 4 ωL − = 2 e−iωt . 4 kR[J1 (kR) − αb kJ2 (kR)] ωL − ω2

(5.114)

Substitution of this expression into Eq. (5.110) yields the dispersion equation (Sonin, 1987):   1 J2 (kR) 1+β ω2 − − = 0. (5.115) 2 2 4 ω − ωL 4 kR[J1 (kR) − αb kJ2 (kR)] $ 2 /c : One can transform Eq. (5.115)) to the equation for k = ω2 − ωL T 2 − βc2 k 2 ωL ω2 + c2 k 2 T [J1 (kR) − αb kJ2 (kR)] − L 2 2T J2 (kR) = 0. 4kR k R

(5.116)

First let us consider an ideal lateral wall when αb → ∞ in the boundary condition (5.108). Then the eigenvalues of wave numbers k = j2,s /R are non-zero roots of the equation J2 (kr) = 0, and the eigenfrequencies are given by 2 ωs2 = ωL +

2 c2 j2,s T

R2

.

(5.117)

If pinning is absent or the container is very long, then ωL → 0 [see Eq. (5.59)], and Eq. (5.115) yields the Ruderman spectrum, Eq. (4.16), with the fundamental (lowest) frequency ωR =

5.14cT . R

(5.118)

In the opposite limit ωL ωR , the eigenfrequencies approach ωL . As will be shown in Section 6.5, mutual friction provides an effective force sticking an oscillating superfluid to lateral walls, and the assumption of ideal lateral walls αb → ∞ does not hold. In the case of the no-slip boundary condition when αb = 0 in Eq. (5.108), the eigenfrequencies at large ωL cT k ∼ cT /R are 2 ωs2 = ωL +

xs2 cT2 , R2

(5.119)

where xs > 0 are roots of the equation xJ1 (x) − J2 (x) = 0. 4

(5.120)

Thus oscillation frequencies of the freely rotating container with superfluid are bound from below and cannot be less than the frequency ωL of the lowest frequency inertial wave

5.9 Tkachenko waves in superfluid 4 He and pulsars

163

resonance. The oscillation is a pure Tkachenko mode with k ∼ 1/R only if ωL  cT k ∼ cT /R. In terms of container dimensions this requires the condition L 2  .

∼ 2 2 Aκ R AcT

(5.121)

Although in very long cylinders vortex motion is not columnar (Section 6.3) this affects only the frequency ωL , which can be ignored for calculation of the Tkachenko mode frequency in long cylinders. 5.9 Search for Tkachenko waves in superfluid 4 He and pulsars: Tkachenko wave versus inertial wave The first attempt to observe a Tkachenko wave was undertaken by Tkachenko (1974) himself in a study of torsional oscillations of a light cylinder immersed in rotating superfluid 4 He and suspended by a thin fibre. No conclusive data were obtained; nevertheless, we shall return to this experimental setup later, in Section 7.4, since a discussion requires knowledge of the two-fluid theory. Further efforts to discover Tkachenko waves experimentally were stimulated by Ruderman’s theory, which connected long-period oscillations of the pulsar rotation period with the Tkachenko mode (see the end of this section). In order to simulate the process in pulsars, Tsakadze and Tsakadze (1973, 1975) studied the free rotation of containers of various shapes, cylindrical included, filled with superfluid 4 He, and revealed rotationperiod oscillations superimposed on the steady deceleration of rotation. The oscillations disappeared above the critical temperature, which proved their superfluid nature. However, the oscillation frequencies observed for cylindrical containers were nearly eight times higher than the Ruderman fundamental frequency ωR . This disagreement was explained by the effect of pinning and bending of vortices (Sonin, 1976). We saw in the previous section that in cylinders of moderate ratio L/R 2 such effects transform a Tkachenko wave resonance into an inertial wave resonance. Tsakadze and Tsakadze (1973) observed not the Tkachenko mode with frequency ωR , but the inertial wave resonance with frequency ωL . This was proved by experimental investigation of the dependence on container dimensions L and R (Tsakadze, 1976). The experimental oscillation frequency depended on the height L of helium in the container (the length of vortices) and on the smoothness of the bottom, but did not depend on the container radius R. This agreed with the theoretical expression for the inertial wave resonance frequency [see Eqs. (5.59) and (Eq. (5.50))], but contradicted Eq. (5.118) for the Ruderman frequency. In a number of cases, beats arose in the oscillations. They are possible only in the region of inertial wave resonance, where intervals ∼ cT /R between neighbouring eigenfrequencies are much less than the fundamental eigenfrequency ωL [see a more detailed comparison and discussion by Sonin (1976); Tsakadze (1976); Tsakadze et al. (1980)]. In further experiments, Tsakadze (1978) used longer cylindrical containers in an effort to reach the region ωR ωL , where pure Tkachenko waves are possible. He could

164

Vortex oscillations in finite rotating containers

80

R/cT

60

40

20

0

0

20

40

60

80

LR/cT

Figure 5.5 Dependence of the eigenfrequencies ω of the slow mode on the frequency ωL characterising pinning in a cylindrical container of radius R filled by a superfluid up to height L. The solid lines were calculated from Eq. (5.115) at αb = 0 and β = 3. The experimental points and the theoretical curves are taken from Tsakadze (1978).

not do this completely, because it required impractical containers with too large ratio L/R 2 [see the inequality (5.121)]. In his experiments the Tkachenko contribution to the oscillation frequency was of the same order as the pinning contribution, but not more, and S. Tsakadze had to refer to the general theory allowing for both contributions. In Fig. 5.5, reproduced from the paper of Tsakadze (1978), the theoretical dependence of the oscillation frequencies on the frequency ωL is shown, calculated numerically with the help of Eq. (5.115). In order to draw the experimental points on the same plot, values of the parameter A were necessary. They were obtained by extrapolation of the plot of A versus the oscillation period obtained in the region of the inertial wave resonance ωL ωR [see Fig. 2 in the paper of Tsakadze (1978)]. The experimental points for the longest cylinder with the largest value of L/R 2 are the most important for determination of the Tkachenko rigidity of the vortex lattice. The points are rather close to the theoretical curve, also shown in the figure. The agreement looks satisfactory, bearing in mind possible inaccuracies in the experiment and in theoretical assumptions (see a more detailed discussion by Sonin, 1987). Experimental points in Fig. 5.5 for the longest container show at least a tendency to become independent of A, which means a crossover to the regime governed by Tkachenko rigidity and not by pinning. This was experimental evidence (although indirect) of the existence of Tkachenko rigidity and consequently of crystalline order in the vortex lattice.

5.9 Tkachenko waves in superfluid 4 He and pulsars

165

In Section 5.4 we have already mentioned the pile-of-disks experiments by Andereck et al. (1980) and Andereck and Glaberson (1982). They associated observed resonances with a peak in the density of state related with a minimum on the dispersion curve of the slow mode at given p. The position of the minimum is determined by Eq. (3.120) and depends on the Tkachenko wave velocity. The latter dependence was considered as evidence that the Tkachenko mode was obtained. Andereck et al. chose the value p = π/2L in Eq. (3.120), which assumed complete pinning of ends of vortices to disk surfaces. Then the frequency of the peak is given by  π cT 1  π κ 1/4 (2)3/4 . (5.122) = ωm = 2 2L 2 L2 Section 5.4 explained these experiments in terms of the inertial wave resonance, without any reference to Tkachenko rigidity. But remarkably, the frequency (5.122) differs from the inertial wave resonance frequency given by Eq. (5.42) only by a factor of about 0.7, and consequently it is not surprising that both interpretations are in reasonable agreement with the experiment. However, the scenario of Andereck et al. left unresolved a serious problem [by admission of the authors themselves; see p. 288 in the paper by Andereck and Glaberson (1982)]: how can the oscillations of disks, introducing perturbations with wavelengths of the order of the disk radius, generate Tkachenko waves (more exactly, slow waves with considerable Tkachenko contributions) whose wavelengths are an order of magnitude smaller than the radius of the disks? The density-of-state peak scenario does not provide an explicit mechanism for generation, but suggests that there should be one. On the other hand, the inertial wave resonance interpretation rests on a self-sufficient hydrodynamical derivation without any additional assumption. It is worth noting that the density-of-state peak interpretation does not distinguish between different branches and is extended to the branches n > 1 assuming that p = π(2n − 1)/2L in the expression for the frequency given by Eq. (3.120). Therefore it predicts that observable frequencies would be considerably lower than Hall resonance frequencies at large  ω for any n. In contrast, according to the theory of Section 5.4, only the lowest branch n = 1 differs essentially from the frequencies given by the Hall expression, Eq. (5.39) (compare the solid and dashed lines n = 2 in Fig. 5.3). Thus observation of branches n > 1 at high rotation speeds would be an important check of the competing theoretical pictures. As was already mentioned, Ruderman (1970) associated Tkachenko waves with very slow oscillations of the pulsar’s rotation period after glitches (sudden spin-ups of pulsars, or starquakes). Dyson (1971) argued that it is difficult to think of any other internal motion which would have a time scale as long as the period of these oscillations. Ruderman considered Tkachenko waves in a cylinder, ignoring the difference between cylindrical and spherical geometry. Inserting the data for the pulsar in the Crab nebula ( = 200 rad/sec, R = 106 cm, κ = 2 × 10 cm2 /sec) into Eq. (5.118), Ruderman found that the oscillation period should be T =

2π = 9.73 × 106 sec = 3.75 months, ωR

166

Vortex oscillations in finite rotating containers

in good agreement with the observed period of ∼3 months. Ruderman ignored possible pinning of vortices to the solid crust, confining the neutron matter in pulsars. Having no data on vortex pinning to the crust, Tsakadze et al. (1980), nevertheless, estimated the possible effect of pinning from above. Pinning can increase the oscillation frequency up to the frequency ωL of the inertial wave resonance. For strong pinning, this frequency is maximal and is given by Eq. (5.42). Following Ruderman, one can put L = R in Eq. (5.42). According to Dyson (1971), the vortex cores occupy a fraction 10−20 of the pulsar volume. This means that the ratio of the intervortex spacing to the core radius is rv /rc ∼ 1010 , and Eq. (5.42) yields for the Crab pulsar ωL = 1.3 × 10−2 rad/sec. This corresponds to a period of about 8 minutes. Thus pinning would lead to a strong decrease in the oscillation period. Later, interest in the interpretation of pulsar oscillations in terms of the Tkachenko mode declined to some extent and other interpretations were suggested. But recently some publications have urged returning to the Tkachenko mode interpretation of long-period pulsar oscillations (Noronha and Sedarkian, 2008; Popov, 2008; Shahabasyan, 2009; Haskell, 2011).

6 Vortex dynamics in two-fluid hydrodynamics

6.1 Two-fluid macroscopic hydrodynamics of a rotating superfluid Originally the two-fluid theory for superfluids was formulated for a vortex-free superfluid (Tisza, 1940; Landau, 1941). One can find a detailed discussion of various aspects of the two-fluid theory in a number of books on the superfluidity theory (Putterman, 1974; Landau and Lifshitz, 1987; Khalatnikov, 2000, as examples). A key assumption of the two-fluid theory is that fluid motion is described by two velocities: the superfluid velocity v s and the normal velocity v n . Correspondingly, there are two components of the fluid moving with these velocities. Superfluid motion is potential, i.e., curl-free, and is not accompanied by dissipation, while motion of the normal component leads to dissipation and is accompanied by transport of entropy. Choosing this key assumption as a starting point, one can formulate two-fluid hydrodynamics on the basis of phenomenology using the laws of conservation and symmetry (Landau and Lifshitz, 1987). If one deals with a superfluid threaded by vortex lines, the equations of the two-fluid theory hold in the multiply connected region outside the vortex lines; they should be supplemented by equations of motion for vortex lines. Together these equations constitute the theory governing the behaviour of the superfluid at finite temperatures. But, just as in the one-component perfect fluid, sometimes one can describe the motion of the vortex lines in terms of the averaged parameters of the vortex array. Such a theory is macroscopic hydrodynamics defined above (Section 3.1), extended to include finite-temperature two-fluid effects. Here we shall formulate the two-fluid macroscopic hydrodynamics, first omitting all dissipation terms except for mutual friction. The remaining dissipation effects can be introduced into the theory (and will be at the end of the section) in the same manner as in original ‘microscopic’ two-fluid hydrodynamics. The system of equations of two-fluid macroscopic hydrodynamics must include general thermodynamic relations. The Gibbs relation for the energy density in a reference frame moving with velocity v s is dEs = μs dρ + T dS + (v n − v s ) · dj 0 +

∂E d∇i uj . ∂∇i uj

(6.1)

167

168

Vortex dynamics in two-fluid hydrodynamics

Here u is the displacement of the vortex line, and j 0 is the mass current in the superfluid reference frame, j 0 = j − ρv s .

(6.2)

For the total energy density in the laboratory frame, 1 E = Es + j 0 · v s + ρvs2 , 2

(6.3)

the Gibbs relation is dE = μdρ + T dS + v n · dj 0 + j · dv s +

∂E d∇i uj , ∂∇i uj

(6.4)

where μ = μs +

vs2 2

(6.5)

is the chemical potential in the laboratory reference frame at a given current j 0 . The velocity v s and the vortex displacement u are not independent variables, since vortex displacements change velocities. The definition of the derivatives ∂E/∂v s and ∂E/∂∇i uj relies on the convention that the kinetic energy of the averaged superflow with velocity v s is a function of the velocity and involves the long-range interaction between vortices, but the remainder of the superfluid kinetic energy is a function of the displacements and involves the short-range interaction of vortices. The pressure is determined by the usual thermodynamic formula: P = −E + T S + μρ + j 0 · v n = −Es + T S + μs ρ + j 0 · (v n − v s ).

(6.6)

Differentiation of Eq. (6.6) yields the Gibbs–Duhem relation: ∂E d∇i uj ∂∇i uj ∂E = ρdμs + SdT + j 0 · d(v n − v s ) − d∇i uj . ∂∇i uj

dP = ρdμ + SdT + j 0 · dv n − j · dv s −

(6.7)

The two-fluid hydrodynamical theory includes the continuity equation for mass and the balance equation for entropy: ∂ρ + ∇ · j = 0, ∂t ∂S 2R + ∇ · (Sv n ) = . ∂t T

(6.8) (6.9)

Here j is the net mass current, S is the entropy per unit volume, and R is the dissipation function. Equation (6.9) takes into account that only the normal component transports the entropy with velocity v n .

6.1 Two-fluid macroscopic hydrodynamics

169

The Euler equation (3.9) retains its former form as deduced from purely kinematic arguments, but now it refers to the superfluid part of the fluid and contains the superfluid velocity v s :   vs2 ∂v s . (6.10) + ω˜ × v L = −∇μ = −∇ μs + ∂t 2 The next equation is provided by the momentum conservation law, which is ∂j i + ∇j ij = 0. ∂t

(6.11)

The momentum flux tensor is given by ij = P δij + j0i vnj + vsi jj + σij ,

(6.12)

where the stress tensor σij is defined by Eq. (3.34). Unlike the case of the one-fluid theory, in which the momentum conservation law is derived from the mass continuity equation and the Euler equation, in the two-fluid theory momentum conservation yields an independent equation. Now we are able to check the energy conservation law by calculating the time derivative of the energy density with the help of the Gibbs relation Eq. (6.4) and the dynamic equations (6.8)–(6.11). This yields ∂E + ∇ · Q = 0, ∂t

(6.13)

where the vector Q is the energy flux with components Qi = μji + [ST + (j 0 · v n )]vni + σik vLk   vs2 ji + [ST + (j 0 · v n )]vni + σik vLk . = μs + 2

(6.14)

The energy conservation law Eq. (6.13) holds if the dissipation function is equal to R=

1 (v L − v n ) · [f el − ω˜ × (j − ρv n )], 2

(6.15)

where the elastic force f el is defined by Eq. (3.29) as in a one-component perfect fluid. Our system of equations is closed if the dependence of the energy on all hydrodynamical variables is known. Later we shall consider the VLL state, in which vortex lines perturb a fluid considerably only in their immediate vicinity at scales of the order of the vortex core radius rc . Therefore on deriving macroscopic hydrodynamics the coarse-graining procedure yields values of hydrodynamical quantities that differ negligibly from those in the original ‘microscopic’ hydrodynamics, with the exception, of course, of quantities, which are completely absent in the original theory (mutual friction parameters and the elastic stress tensor of the vortex lattice). The accuracy of such an approach is determined by a small parameter rc /rv , which is extremely small in superfluid 4 He. Within this approach

170

Vortex dynamics in two-fluid hydrodynamics

and assuming linear relations between currents and velocities, we have explicit expressions for the currents, j = ρs v s + ρn v n , j 0 = ρn (v n − v s ),

(6.16)

and the momentum flux tensor, ij = P δij + ρs vsi vsj + ρn vni vnj + σij .

(6.17)

Here ρs and ρn are superfluid and normal mass densities, respectively, taken from microscopic two-fluid hydrodynamics. The energy density Es in the coordinate frame moving with velocity v s is Es = E0 (ρ) +

j02 , 2ρn

(6.18)

where E0 (ρ) is the energy density in the absence of any motion in the fluid (v s = v n = 0), which depends only on the total mass density ρ. Introducing the chemical potential μ0 = ∂E0 /∂ρ in a resting fluid, the chemical potential in the superfluid coordinate frame is   ∂ρn j02 ∂ρs (v n − v s )2 . (6.19) μs = μ0 − = μ0 − 1 − ∂ρ 2ρn2 ∂ρ 2 Our relations assumed that the energy was determined as a function of the hydrodynamical variables ρ, j 0 , v s , and ∇i uj . In this list of variables, it is possible to replace the current j 0 = j − ρv s in the coordinate frame moving with velocity v s by the current j in the laboratory frame. Then the Gibbs relation dE = μj dρ + T dS + v n · dj + j n · dv s +

∂E d∇i uj ∂∇i uj

(6.20)

must replace the Gibbs relation (6.4) while the Gibbs–Duhem relation transforms to dP = ρdμj + SdT + j · dv n − j n · dv s −

∂E d∇i uj . ∂∇i uj

(6.21)

The current j n = j − ρv n = ρs (v s − v n )

(6.22)

is the current in the coordinate frame moving with the normal velocity v n . With this choice of hydrodynamical variables the chemical potential is μj = μ − v n · v s ,

(6.23)

and the Euler equation becomes ∂v s (6.24) + ω˜ × v L = −∇(μj + v n · v s ). ∂t It is remarkable that the term v n · v s does not vanish in the limit T = 0. So although the normal component disappears in this limit, its velocity is still present in the equations.

6.1 Two-fluid macroscopic hydrodynamics

171

It looks as if an absent normal component can produce a force accelerating the superfluid component! A similar phenomenon was noticed by Liu and Cross (1979) in the A phase of superfluid 3 He and was called the gauge wheel. This will be discussed in Section 10.4. Ho and Mermin (1980) noticed that the gauge wheel is also possible in an isotropic superfluid like 4 He and argued that μj is a true chemical potential just because on using it the gauge effect appears. Meanwhile the definition of chemical potential is determined by the choice of the set of hydrodynamical variables. Any choice is legitimate in principle, and the chemical potential corresponding to some concrete choice cannot be true or untrue but only more or less convenient for some concrete problem. It is important also that in experiments, it is not the chemical potential itself which is measured, but the pressure or the temperature. So for final judgement on the gauge wheel it is better to express the chemical potential via the pressure and the temperature using the Gibbs–Duhem relation (6.21). Then the Euler equation can be rewritten as (neglecting the deformation energy)   ∇P vn2 S ρs vs2 ρn ∂v s . (6.25) + ω˜ × v L = − + ∇T − ∇ − ∇ (v n · v s ) − ∂t ρ ρ ρ 2 ρ 2 This form of the Euler equation demonstrates that the gauge wheel effect is impossible without a finite normal density ρn which is independent of the definition of the chemical potential. Any force in vortex dynamics is connected with a velocity by the Magnus relation, and in the hydrodynamics of a one-component perfect fluid Eq. (3.23) connected the elastic force f el with the difference between the average velocity v and the local velocity v l at points on vortex lines. In two-fluid hydrodynamics, vortices belong to the superfluid component, and the local and the average fluid velocities v l and v must be replaced by the local and the average superfluid velocities v sl and v s , while the fluid mass density ρ must be replaced by the superfluid mass density ρs . This transforms Eq. (3.23) to the relation f el = −ρs ω˜ × (v sl − v s ).

(6.26)

Then one can rewrite Eq. (6.15) as 1 R = − ρs (v L − v n ) · [ω˜ × (v sl − v n )]. (6.27) 2 If the vortices moved with the velocity v L equal to the local superfluid velocity v sl as Helmholtz’s theorem requires, the dissipation function would vanish. But in two-fluid hydrodynamics the theorem does not hold because of mutual friction with the normal component. The concept of mutual friction between superfluid and normal components was introduced by Hall and Vinen (1956a) in order to explain the effect of rotation on propagation of the second sound in rotating superfluids (see Section 6.2). In the general case v L  = v sl , the Euler equation (6.10) can be transformed to an equation similar to Eq. (3.21) for a one-component perfect fluid:   ∂v s v2 f + ∇ μ0 + s + ω˜ × v s = s . (6.28) ∂t 2 ρs

172

Vortex dynamics in two-fluid hydrodynamics

Here f s = f el + f f r = −ρs ω˜ × (v L − v s ),

(6.29)

is the net force on the vortices, which includes the elastic force f el and the mutual friction force f f r . The Magnus relation connecting f f r with the difference between v sl and v L is f f r = −ρs ω˜ × (v L − v sl ).

(6.30)

Both the forces f el and f f r are of quantum origin and vanish in the continuous vorticity limit. One can see from Eq. (6.27) that the dissipation function does not vanish only if there is a mutual friction force: 1 1 R = − (v L − v n ) · [f f r + ω˜ × (v L − v n )] = − (v L − v n ) · f f r 2 2 1 = − (v sl − v n ) · f f r . 2

(6.31)

Here Eqs. (6.26) and (6.30) were used. The most general linear phenomenological relation between the mutual friction force and the relative velocity of the two fluid components near the point on the vortex line is ) * (6.32) f f r = −ρs α sˆ × [ω˜ × (v n − v sl )] − ρs α  [ω˜ × (v n − v sl )]. The relation assumes that the fluid is isotropic but time invariance is broken. The latter explains the presence of the second term proportional to α  . The friction force has no component along vortex lines (see the discussion of this assumption in Section 8.1). Bearing in mind Eq. (6.30), one can rewrite Eq. (6.32) as a linear relation connecting three velocities v L , v sl , and v n : v L = v sl + α  (v n − v sl ) + α[ˆs × (v n − v sl )].

(6.33)

According to this expression, the velocity v L has a component along the vortex lines which does not produce any force. But this component will be automatically cancelled out on any further use of this expression. Alternatively one can exclude this component from the equation of vortex motion by rewriting it as v L = v sl − α  [ˆs × [ˆs × (v n − v sl )]] + α[ˆs × (v n − v sl )].

(6.34)

The two phenomenological mutual friction parameters α and α  , which were used by Schwarz (1985), are directly connected with the mutual friction parameters B and B  introduced by Hall and Vinen (1956a): α=

ρn B, 2ρ

α =

ρn  B. 2ρ

(6.35)

The mutual friction parameters will be analysed in Chapter 8. Sometimes it is more convenient to present the mutual friction force f f r as a linear function of the vortex

6.1 Two-fluid macroscopic hydrodynamics

velocity relative to the normal components (Bevan et al., 1997b): ) * f f r = ρs d sˆ × [ω˜ × (v L − v n )] − ρs d  [ω˜ × (v L − v n )].

173

(6.36)

The new friction parameters d and d  are connected with α and α  by the relations α=

d2

d , + (1 − d  )2

1 − α =

d2

1 − d . + (1 − d  )2

(6.37)

At weak mutual friction, α ≈ d and α  ≈ −d  . With the help of Eq. (6.30) one obtains from Eq. (6.36) the balance of forces (per unit volume) on vortices: ) * (6.38) −ρs ω˜ × (v L − v sl ) = ρs d sˆ × [ω˜ × (v L − v n )] − ρs d  [ω˜ × (v L − v n )]. The system of macroscopic hydrodynamical equations formulated above is invariant with respect to the Galilean transformation. The transformation to the rotation reference frame adds the Coriolis force to the momentum conservation law [Eq. (6.11)]. Other equations do not vary when ω˜ is defined as the absolute vorticity in the inertial reference frame (see Chapter 1) and the centrifugal force is neglected. In the linear theory the part of the superfluid kinetic energy, which depends on deformations of the vortex array and includes the short-range interaction between vortices, is defined by Eq. (3.40) as in the perfect fluid, but in Eqs. (3.72) and (3.85) for the elastic moduli C11 , C44 , and C66 the superfluid density ρs should replace ρ. Then the density Ev of this energy is given by    ρs κ rv du 2 Ev = − (∇ · u)2 2 ln 8π rc dz      ∂uy 2 1 ∂ux ∂uy 2 1 ∂ux − + + + . (6.39) 2 ∂x ∂y 2 ∂y ∂x Now let us bring together the linearised equations of macroscopic two-fluid hydrodynamics in the rotating coordinate frame including viscous terms omitted during our derivation1 : ∂ρ  + ρ(∇ · v) = 0, ∂t

(6.40)

∂S  χ + S(∇ · v n ) − ∇ 2 T = 0, ∂t T

(6.41)

∂v s + ∇μ + 2 × v L + ζ3 ∇[∇ · ρs (v n − v s )] + ζ4 ∇(∇ · v n ) = 0, ∂t ∇μ =

S ∇P − ∇T , ρ ρ

(6.42) (6.43)

1 Neglecting Tkachenko rigidity (c = 0) these equations agree with those in the books of Landau and Lifshitz (1987) and T

Khalatnikov (2000).

174

Vortex dynamics in two-fluid hydrodynamics

∇P ρs ∂v + + 2 × v + 2 × (v sl − v s ) ∂t ρ ρ + , 1 ρn ν∇ 2 v n + ζ1 ∇[∇ · ρs (v n − v s )] + ζ2 ∇(∇ · v n ) = 0, − ρ v sl = v s + νs zˆ ×

c2 ∂ 2u + T [ˆz × ∇ 2⊥ u − 2ˆz × ∇(∇ · u)]. 2 2 ∂z

Here v is the centre-of-mass fluid velocity, ρn ρs v = vs + vn, ρ ρ

(6.44) (6.45)

(6.46)

and the vortex velocity v L is given by Eq. (6.33). The equations include the thermal conductivity χ , which determines the heat flux −χ ∇T , the first viscosity coefficient ν (kinematic viscosity), and three independent second viscosity coefficients ζi (i = 1, 2, 3). The fourth second viscosity coefficient ζ4 is equal to ζ1 as required by the Onsager reciprocity principle. In this system of equations one can replace Eq. (6.44) for the centre-of-mass fluid velocity v by the Navier–Stokes equation for the normal velocity v n , which follows directly from Eqs. (6.42) and (6.44): ∇P ρs ρs ∂v n + − ∇μ + 2 × v n + 2 × (v sl − v L ) ∂t ρn ρn ρn ζ1 ζ2 − ν∇ 2 v n − ∇[∇ · ρs (v s − v n )] − ∇(∇ · v n ) ρn ρn ρs ρs − ζ3 ∇[∇ · ρs (v s − v n )] − ζ1 ∇(∇ · v n ) = 0. ρn ρn

(6.47)

We have formulated the two-fluid theory on a purely phenomenological basis. Meanwhile, at low temperatures one can derive it from the Landau theory of the Bose superfluid. The theory suggests that thermodynamics of the Bose superfluid at finite but not too high temperatures follows from thermodynamics of the Bose gas of weakly interacting elementary excitations (quasiparticles). Any quasiparticle has a well defined energy ε0 (p), which is a function of a well defined quasiparticle momentum p. If the fluid moves with average velocity v 0 , Galilean invariance requires that the energy of the quasiparticle in the laboratory system is ε = ε0 (p) + p · v 0 . In the presence of quasiparticles the total mass current is equal to  1 (6.48) j = ρv 0 + 3 f (p)p d3 p, h where f (p) is the distribution function of quasiparticles in the momentum space. In thermal equilibrium, the distribution function is the Planck distribution f (p) = f0 (ε, v n ) with the drift velocity v n of quasiparticles: f0 (ε, v n ) =

1 exp

ε(p)−p·v n T

−1

=

1 exp

ε0 (p)+p·(v 0 −v n ) T

−1

.

(6.49)

6.2 Longitudinal modes: first and second sound

175

Linearising Eq. (6.49) with respect to the relative velocity v 0 − v n , one obtains from Eq. (6.48) that j = ρv 0 + ρn (v n − v 0 ).

(6.50)

This expression is equivalent to the two-fluid expression j = ρv s + ρn (v n − v s ) = ρs v s + ρn v n assuming that ρ = ρs + ρn and v 0 = v s , and the normal density is given by the usual two-fluid hydrodynamics expression (Khalatnikov, 2000): ρn = −

4π 3h3



∂f0 (ε0 ) 4 p dp, ∂ε0

(6.51)

where f0 (ε0 ) = f0 (ε, 0) is the Planck distribution of the quasiparticle gas at rest (v n = 0). In the same manner one can derive the two-fluid momentum flux tensor given by Eq. (6.17) but without the stress tensor σij related with vortices. For very low temperatures the main contribution to the normal mass comes from phonons (quanta of sound waves). The analysis of the sound wave momentum in Section 1.6 demonstrated a difference between the average velocity v 0 and the centre-of-mass velocity j /ρ. In two-fluid hydrodynamics the velocity v 0 becomes the superfluid velocity v s , while the centre-of-mass velocity is given by Eq. (6.46). The superfluid velocity vs κ of the two-fluid theory corresponds to the velocity 2π ∇θ , defined in the Gross–Pitaevskii theory via the gradient of the order parameter phase θ (Section 1.15).

6.2 Longitudinal modes: first and second sound When studying longitudinal modes we neglect vortex lattice elasticity. This means that there is no difference between the average and local superfluid velocities v s and v sl . We also ignore all dissipation processes except for mutual friction. The relevant linearised equations of motions are ∂ρ  + ρ(∇ · v) = 0, ∂t ∂S  + S(∇ · v n ) = 0, ∂t    B B ∂w ρs S∇T  + + 2 × 1 − w − sˆ × w = 0, ∂t ρn ρ 2 2 ∂v ∇P  + + 2 × v = 0, ∂t ρ

(6.52) (6.53) (6.54) (6.55)

where w = v n − v s is the relative velocity of the counterflow of the normal and the superfluid components. Introducing the entropy s = S/ρ per unit mass instead of the entropy S per unit volume, the continuity equation for entropy follows from Eqs. (6.52)

176

Vortex dynamics in two-fluid hydrodynamics

and (6.53): ρs s ∂s + ∇ · w = 0. ∂t ρ

(6.56)

In helium fluids the coefficient of thermal expansion is very small. This justifies a simple approach, which assumes that the mass density depends only on pressure P , while the entropy density s depends only on temperature T (Landau and Lifshitz, 1987; Khalatnikov, 2000). Then the continuity equations (6.52) and (6.56) can be written as 1 ∂P  + ρ(∇ · v) = 0, cs2 ∂t

(6.57)

ρs s cv ∂T  + ∇ · w = 0, T ∂t ρ

(6.58)

√ where cs = ∂P /∂ρ is the velocity of the usual sound, which is called the first sound in the two-fluid theory, and cv = T ∂s/∂T is the heat capacity. In this approximation the system of equations consists of two decoupled pairs of equations. The first pair, Eqs. (6.55) and (6.57), for the variables v and P  describes the first sound, which does not differ from the sound in a perfect one-component fluid with continuous vorticity analysed in Section 1.14. Vortex induced mutual friction does not play an essential role because all three components of the fluid move approximately with the same velocity: v L ≈ v s ≈ v n . The second pair of equations (6.54) and (6.58) for the variables w and T  describes the second sound with the second-sound velocity # ρs s 2 T c2 = . (6.59) ρn cv The second sound is a temperature wave without any centre-of-mass motion (v = 0), but there is a counterflow of the superfluid and the normal components with the relative velocity w = v n − v s . The counterflow leads to an essential effect of mutual friction. One can exclude the temperature variation and the axial velocity component wz from Eqs. (6.54) and (6.58), similarly to what was done with the density variation and vz in Section 1.14 for the hydrodynamical modes in a one-component compressible perfect fluid. Considering a plane wave ∝ eiK·R−iωt one obtains two equations for the two in-plane velocity components wk and wt parallel and normal to the in-plane component k of the wave vector K(k, p) [cf. Eq. (1.171)]:   ω2 − c22 K 2 − B wk − (2 − B  )wt = 0, − iω ω2 − c22 p2 (6.60) −(iω − B)wt + (2 − B  )wk = 0. Treating the mutual friction effect as a weak perturbation, one obtains a weakly damping second sound wave propagating in the xy plane (p = 0) with frequency

6.3 Hydrodynamical equations for incompressible superfluid

177

iB . (6.61) 2 A small imaginary part of the frequency leads to attenuation of the second sound. The mutual friction parameter B was determined from experimental investigations of attenuation of the second sound in pioneering work by Hall and Vinen (1956a), from which modern studies of vortex dynamics in superfluids started. ω = c2 k −

6.3 Hydrodynamical equations for a completely incompressible superfluid We shall call a superfluid completely incompressible when it is incompressible in the mechanical and thermal sense, that is, when the mass density and the entropy density are constants. In a completely incompressible fluid the superfluid and the normal densities do not vary in space and time.2 Formally this model corresponds to the limit of infinite velocities of the first and second sound. In this limit both v s and v n are divergence-free (∇ · v s = ∇ · v n = 0). Equations (6.42) and (6.47) for the velocities v s and v can be transformed into equations for the velocities v s and v n , which in a completely incompressible fluid are given by ∂v s + [2 × v L ]⊥ = 0, ∂t ∂v n ρs + [2 × v n ]⊥ + [2 × (v sl − v L )]⊥ + ν∇ × (∇ × v n ) = 0. ∂t ρn

(6.62) (6.63)

As in Section 1.6, the subscript ⊥ indicates that only the transverse part of the corresponding vector field is retained. Mutual friction and the first viscosity are the only possible dissipation processes in the incompressible fluid. All other dissipation processes involve compression of the superfluid or normal fluid, or both. Now let us perform the Fourier transformation of Eqs. (6.62), (6.63) and (6.45) using the earlier notation for the wave vector K and its components p and k on the z axis and in the xy plane. We present the resulting equations as two components in the xy plane along and normal to the wave vector k. The components will be denoted by the subscripts k and t, as before: p2 vLt = 0, K2 −iωvst + 2vLk = 0,

−iωvsk − 2

p2 ρs p2 v − 2 (vslt − vLt ) = 0, nt ρn K2 K2 ρs −(iω − νK 2 )vnt + 2vnk + 2(vslk − vLk ) = 0, ρn

(6.64)

−(iω − νK 2 )vnk − 2

2 This case was addressed in the problem to Section 140 of Landau and Lifshitz (1987).

(6.65)

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Vortex dynamics in two-fluid hydrodynamics

vslk − vsk vslt

νs p2 + =− iω

νs p2 − − vst = iω

cT2 k 2 2

cT2 k 2 2

vLt , (6.66)

vLk .

The z components vsz and vnz of the superfluid and normal velocities are determined from the incompressibility conditions. Vortex displacements have been excluded with the help of the relation v L = −iωu. Equations (6.64)–(6.66), together with Eq. (6.33) for the vortex velocity, constitute a closed system of equations governing oscillations in the completely incompressible fluid. Exclusion of the components of v s from Eqs. (6.64) and (6.66) yields two equations:   2 k2 c 1 2 + νs p2 p2 vslt ≈ (6.67) vslk = − 2 2 + νs p2 + T vLt , vLk . iω 2 iω K The solution of Eqs. (6.33), (6.65) and (6.67) yields all oscillation modes and their dispersion laws in the completely incompressible fluid. But the general dispersion law looks rather intricate, and we shall consider the most interesting cases separately.

6.4 Axial modes For axial modes propagating along the z axis (k = 0, K = p) the relation between the local and the average superfluid velocities differs from the similar relation (3.96) for the one-component perfect fluid by the presence of v sl and v s instead of v l and v: v sl = v s + νs zˆ ×

∂ 2u . ∂z2

(6.68)

The modes are circularly polarised and involve motion only in the xy plane, so it is convenient to use the j -complex representation for vectors in the xy plane introduced in Section 2.9. Then Eqs. (6.33) (6.64), (6.65) and (6.67) take a more compact form: ˜ v˜sl + j α˜ v˜n , v˜L = (1 − j α)

(6.69)

−iωv˜s + 2j v˜L = 0,

(6.70)

−(iω − νp2 )v˜n + 2j v˜n + v˜sl =

ρs 2j (v˜sl − v˜L ) = 0, ρn

2 + νs p2 j v˜L . iω

(6.71) (6.72)

Solving Eqs. (6.70)–(6.72), one can express all velocities through v˜s : v˜L =

iω v˜s , 2j

v˜sl =

2 + νs p2 v˜s , 2

v˜n = −

ρs 2 + νs p2 + j iω v˜s . ρn 2 + j (iω − νp2 )

(6.73)

6.4 Axial modes

179

Substitution of these velocities in Eq. (6.69) yields an equation that has a solution when the following dispersion equation holds:   ˜ (iω − νp2 ) iω − (2 + νs p2 ) (j + α)   − 2 iω (j + α˜ s ) − j (2 + νs p2 ) (j + α˜ s + α) (6.74) ˜ = 0, where α˜ s =

B˜ ρs ˜ − α˜ = B. 2 2ρ

Equation (6.74) is a quadratic equation for the frequency ω with two solutions:  i + 2 (2j + α˜ + α˜ s ) + νs p2 (j + α) ω=− ˜ + νp2 2  $) *2 2 2 2 ± 2 (α˜ − α˜ s ) + νs p (j + α) + 8(2 + νs p )α˜ α˜ s . ˜ − νp

(6.75)

(6.76)

Sometimes the underlined term in Eq. (6.76) can be neglected. This is always true if the wave number p is large enough. Then the viscous modes and the Kelvin modes are well separated. One sign before the square root yields the viscous modes with the dispersion relation     ρs ρs  2 2 (6.77) B − i  B + νp , ω = −i2 (j + α˜ s ) − iνp = ±2 1 − 2ρ ρ while the other yields the Kelvin modes3 : ˜ = (2 + νs p2 )[±(1 − α  ) − iα]. ω = −i(2 + νs p2 ) (j + α)

(6.78)

The explicit dispersion equations follow from j -complex expressions after the substitution j = ±i, each sign corresponding to one of two possible circular polarisations. The quality factor of Kelvin modes is (1 − α  )/α, and they are undamped only if 1 − α  α. In the long wavelength limit p → 0 one can solve Eq. (6.74) with respect to ω expanding in p2 . This yields two pairs of modes, each with two possible circular polarisations j = ±i. The first pair has oscillation frequencies     ρn ρn ρs ρs (6.79) ω = −ij 2 + νs p2 − i νp2 = ± 2 + νs p2 − i νp2 . ρ ρ ρ ρ These modes do not involve relative motion of the superfluid and normal part of the fluid; therefore mutual friction does not affect them, and they are undamped in the limit p → 0. However, mutual friction does affect the second pair of oscillation modes, since they involve counterflows. The frequencies are given by

3 For a single vortex ( = 0) this dispersion relation was derived by Barenghi et al. (1985) (see also Donnelly, 1991, Section 6.6).

180

Vortex dynamics in two-fluid hydrodynamics

   ρn B˜ ρs 2 ω = −ij 2 + νs p 1−j − i νp2 ρ 2 ρ      B B ρs ρn 1− ∓i − i νp2 . = ± 2 + νs p2 ρ 2 2 ρ

(6.80)

In the long wavelength limit the axial modes cannot be labelled as viscous or Kelvin because the vortex line tension parameter νs and the viscosity ν enter the expressions for the frequencies of the two pairs of modes on equal ground. The modes are classified by the character of fluid motion: either oscillations of two fluid components moving together (centre-of-mass modes), or oscillations of counterflows of two components affected by mutual friction (counterflow modes). The value of the wave number p, at which crossover from viscous and Kelvin modes to centre-of-mass and counterflow modes takes place, decreases with decreasing mutual friction. In the low frequency limit ω → 0 the dispersion relation (6.74) becomes   ˜ = 0. (6.81) ˜ + 2j (j + α˜ s + α) (2 + νs p2 ) νp2 (j + α) All axial modes are evanescent (p2 are complex) and penetrate only into boundary layers of finite width. The condition that the second factor in Eq. (6.81) vanishes determines a complex p for the viscous mode. The penetration depth of viscous modes differs from √ the width of the Ekman layer δE = ν/2 by a factor depending on the mutual friction parameters and deduced from Eq. (6.77). The Kelvin modes [zero first factor in Eq. (6.81)] √ penetrate into the superfluid Ekman layer of width E = kt−1 = νs /2, which does not differ from that width in a one-component perfect fluid [see Eq. (3.102) in Section 3.7]. This is because according to Eq. (6.73) in the low-frequency limit the normal and the vortex velocities become very small in Kelvin modes.

6.5 In-plane modes A simple example of an in-plane mode (p = 0, K = k) is the viscous mode, involving motion of the normal fluid along the z axis. Rotation has no effect on such a mode, and its spectrum is the same as in the fluid at rest and is given by Eq. (1.90): ω = −iνk 2 . Let us turn to in-plane modes involving motion only in the xy plane. The velocities v s and v n have no k components because of the incompressibility conditions. We can exclude from the equation of motion for vortices, Eq. (6.33), all velocities excepting v L by using Eqs. (6.64)–(6.66). This yields equations for the components of the velocity v L :     2 2 cT k 2αs 2α 2αs  − = 0, + vLt 1 − α − vLk 1 − iω iω − νk 2 iω − νk 2 2iω   (6.82)   cT2 k 2 αs cT2 k 2 α 2(1 − α  ) 2αs −vLk + vLt 1 − − + = 0. iω 2iω iω − νk 2 iω(iω − νk 2 )

6.5 In-plane modes

181

The dispersion relation for this system of linear equations is:   2 cT2 k 2 2  2 2 (iω − νk ) 1 − α − 2 [(1 − α ) + α ] iω ω     cT2 k 2 cT2 k 2   (ααs + α αs ) = 0. − 2 αs 1 − 2 − 2iω ω

(6.83)

If mutual friction parameters vanish, the dispersion relation reduces to two uncoupled modes: the viscous mode ω = −iνk 2 in the normal component and the Tkachenko mode ω = cT k in the superfluid component. But as in the case of axial modes, mutual friction leads to mixing of these modes. Let us neglect for a while the crystalline order in the vortex lattice (cT = 0). Then the dispersion equation becomes: k2 =

iω iω − B . ν iω − 2α

(6.84)

This is the dispersion law of the viscous mode modified by mutual friction [see Eq. (4.38) of the review by Andronikashvili et al. (1961)]. When the mutual friction is strong (Bρn /ρ /ω) one obtains that ρn 2 (6.85) νk . iω = ρ This is a centre-of-mass viscous mode in which the superfluid and the normal fluid oscillate together, coupled by mutual friction. However, one cannot ignore the crystalline order of the vortex lattice in the low frequency and the long wavelength limits. Expansion of the dispersion equation (6.83) in small k yields:   2αs − (1 − α  )2 − α 2 − ααs − α  αs iω(iω − B) − cT2 k 2 iω − νk 2 (iω − 2α) = 0.

(6.86)

One solution of this equation gives a gapless Tkachenko mode ω = ct k + iωI ,

(6.87)

where a small imaginary part of the frequency is     cT2 B 2 B2 ρn 2 1− ωI = − k ν + + 2ρ B 2 4    2 B B2 κ ρn 2 1− + =− k ν+ , (6.88) 2ρ 8π B 2 4 √ ρs /ρcT differs from the zero-temperature and the Tkachenko wave velocity ct = √ Tkachenko wave velocity cT = (κ/8π )1/2 by the factor ρs /ρ. This factor arises

182

Vortex dynamics in two-fluid hydrodynamics

because at very small k the mutual friction effectively couples the normal and the superfluid components, making them oscillate together, while the elastic shear modulus C66 = ρs cT2 is proportional to the superfluid density. The mode with the spectrum (6.87) is a centreof-mass mode without essential counterflow. According to Eq. (6.88) the imaginary part ωI of the frequency grows with decreasing mutual friction parameter B. But the equation √ itself is valid until k  ρs /ρB/cT . When this inequality becomes invalid, a crossover to the regime of decoupled oscillations of the superfluid and normal component occurs (the regime of weak mutual friction), where damping of the Tkachenko wave remains weak. Another solution of the dispersion equation (6.86) corresponds to an overdamped counterflow mode with the spectrum:    B 2 B2 ρs ρn cT2 k 2 1− + (6.89) − i νk 2 . ω ≈ −iB + i ρ B 2 4 ρ At zero frequency the dispersion equation (6.83) yields the evanescent mode with νk 2 = −

B 2α ρs ρs =− . ρ (1 − α  )2 + α 2 ρn (1 − α  )2 + α 2

(6.90)

The ratio of the vortex velocity to the normal velocity in the in-plane viscous mode (both have only t components) is small when ω → 0 and is given by βν =

vLt ρn ν iω ρn 8π ν . = iω = vnt ρs cT2  ρs κ

(6.91)

Equation (6.90) cannot be obtained from Eq. (6.84), which ignores the crystalline order. So the crystalline order is also essential for the zero-frequency limit. Equation (6.90) demonstrates that the in-plane viscous mode in the low-frequency limit √ has a finite penetration depth of the order of the Ekman layer width δE = ν/2 [see Eq. (1.144)], in contrast to the in-plane mode with p = 0 in the classical rotating viscous fluid. The latter is gapless according to Eq. (1.142). Note that although the shear rigidity ∝ cT2 does not appear in Eq. (6.90), the equation itself arises from the terms ∝ cT2 k 2 /ω2 in the dispersion equation (6.83) and therefore is a consequence of the crystalline order. The drastic effect of crystalline order on the low frequency behaviour of the in-plane viscous mode can be used for experimental confirmation of the crystalline order (see Section 7.4). Our analysis was based on the elasticity theory of the vortex lattice which is valid as long √ as k is smaller than the inverse intervortex distance 1/rv ∼ /κ.

6.6 Slow modes in a completely incompressible superfluid Now we shall consider soft oscillation modes with wave vector K directed at small angles to the xy plane (p  k). As in the perfect fluid, these will be called slow modes. We shall solve equations for the centre-of-mass velocity v = ρρs v s + ρρn v n and the relative velocity w = v n − v s . This will better illustrate the classification of modes as weakly damped centre-of-mass modes and strongly damped counterflow modes. The equations for v and w

6.6 Slow modes in incompressible superfluid

are readily derived from Eqs. (6.33), (6.64) and (6.65):     p2 ρs ρn 2 ρs −iωvk − 2 2 vt + (vslt − vst ) + νk vk + wk = 0, ρ ρ ρ K     ρs ρn ρs −iωvt + 2 vk + (vslk − vsl ) + νk 2 vt + wt = 0, ρ ρ ρ

183

(6.92)

  p2 p2 p2 − iω − 2 B wk − (2 − B  ) 2 wt − B 2 (vslk − vsl ) K K K   2 p ρs  2 +(2 − B ) 2 (vslt − vst ) + νk vk + wk = 0, ρ K − (iω − B) wt + (2 − B  )wk − B(vslt − vst )   ρs −(2 − B  )(vslk − vsl ) + νk 2 vt + wt = 0. ρ

(6.93)

Substitution of v s = v − (ρn /ρ)w and v n = v + (ρn /ρ)w into Eq. (6.33) yields v L = v + (1 − α  )(δv s − w) − α zˆ × (δv s − w).

(6.94)

In Eq. (6.66) for the components of δv s = v sl − v s one can neglect the vortex line tension ∝ νs . Then cT2 k 2 c2 k 2 vLt , δvst = − T vLk . (6.95) 2iω 2iω In the long wavelength limit the equations for v and for w are uncoupled and describe a centre-of-mass mode and a counterflow mode respectively. We shall start from the centreof-mass slow mode. The transverse velocities of vortices and the fluid, vLt ≈ vt , are larger than the longitudinal velocities, since vLk ∼ vk ∝ ω/ . Correspondingly, according to Eq. (6.95), the longitudinal component of the difference δvsl between the local and the average superfluid velocities is larger than its transverse component δvst , and the latter can be neglected. Also neglecting viscosity and coupling with relative motion, we may write Eq. (6.92) in a much simpler form: δvsl = −

p2 −iωvk − 2 2 vt = 0, K   2 2 ρs cT k − iω + vt + 2vk = 0. ρ iω

(6.96)

This system of the equations yields the dispersion law ω2 = 42

p2 p2 ρs + ct2 k 2 = 42 2 + cT2 k 2 , 2 ρ K K

(6.97)

which differs from the dispersion law of the slow mode in the perfect fluid by the temperature-dependent factor ρs /ρ in the Tkachenko contribution. The physical origin

184

Vortex dynamics in two-fluid hydrodynamics

of this factor has already been explained in Section 6.5 for a pure Tkachenko wave at p = 0. The small imaginary correction to the frequency Eq. (6.97) can be found using perturbation theory that implies calculation of the small relative velocity w from Eq. (6.93). As a result we have       B 2 B2 42 p2 ρs cT4 k 4 ρn 2 1− + νK 1 + 2 2 + . (6.98) Imω = − 2ρ ρ Bω2 2 4 ω K When p = 0, Eqs. (6.97) and (6.98) coincide with Eqs. (6.87) and (6.88) for the pure Tkachenko wave. We shall restrict our analysis of the counterflow slow mode to the case of continuous vorticity, neglecting the elastic shear modulus (cT = 0). Also neglecting coupling with centre-of-mass motion, the system of equations (6.93)) becomes   p2 p2 ρs − iω − νk 2 − 2 B wk − (2 − B  ) 2 wt = 0, ρ K K   (6.99) ρs − iω − νk 2 − B wt + (2 − B  )wk = 0. ρ This yields the dispersion relation [cf. Eq. (71) of Chandler and Baym (1986)]     2  ρs 2 B 2 ρs 2 p2 2p iω − νk − B iω − νk − 2 B + 4 2 1 − = 0. ρ ρ 2 K K

(6.100)

If mutual friction vanishes (B = B  = 0), Eq. (6.100) yields the dispersion law of the classical inertial wave. At small p/K there is an overdamped counterflow slow mode with frequency ω = −i

ρs p2 (2 − B  )2 + B 2 − i νk 2 . B ρ K2

(6.101)

6.7 Vortex dynamics in the clamped regime In the clamped regime the normal part of the fluid moves as a solid body together with the solid surfaces confining the superfluid. This is possible, for example, when the viscous penetration depth is much larger than the width of the fluid layer. The fluid layer, nevertheless, may be larger than other relevant hydrodynamics scales. Such a regime of motion was realised for superfluid 4 He in porous media (Shapiro and Rudnick, 1965). The clamped regime takes place in most cases in superfluid 3 He because of its high viscosity. The clamped regime has been assumed to exist in neutron stars, where the charged normal part of the fluid is clamped to the solid outer crust of the star by a large magnetic field (Baym et al., 1969). In the clamped regime one may delete the Navier–Stokes equation from the system of hydrodynamical equations and assume v n = 0 in the remaining equations. Then the number

6.7 Vortex dynamics in the clamped regime

185

of degrees of freedom reduces to that for the perfect fluid. The difference from the perfect fluid is that superfluid motion involves a smaller mass density and is affected by mutual friction. The equation (6.33) of vortex motion becomes v L = (1 − α  )v sl − α sˆ × v sl .

(6.102)

The difference between the local superfluid velocity v sl and the average superfluid velocity v s is determined by Eq. (6.66). The continuity equations for mass and entropy become ∂ρ 1 ∂P + ∇(ρs v s ) = 2 + ∇(ρs v s ) = 0, ∂t cs ∂t

(6.103)

∂S ∂s ∂ρ cv ρ ∂T s ∂P =ρ +s = + 2 = 0. ∂t ∂t ∂t T ∂t cs ∂t

(6.104)

In general there is no conservation law for the superfluid momentum with density j s = ρs v s . In an incompressible fluid the balance equation of the superfluid momentum can be derived directly from the Euler equation (6.10). The mutual friction force is an external force, which transfers the momentum to the superfluid component. Taking into account the relation (6.30) connecting the mutual friction force f f r with the velocity difference v L − v sl , the balance equation of the superfluid momentum is ∂jsi (s) + ∇j ij = −[ω˜ × (v L − v sl )]i , ∂t

(6.105)

where the momentum flux tensor

  ω˜ i ω˜ j (s) ˜ ij − ij = Ps δij + ρs vsi vsj + ρs νs ωδ ω˜

(6.106)

differs from the momentum flux tensor (3.62) in the HVBK theory for a one-component fluid by the superfluid mass density ρs and velocity v s instead of ρ and v, and by the partial superfluid pressure Ps instead of the global pressure P . The Gibbs–Duhem equation for the partial pressure Ps is [cf. Eq. (3.60)] ˜ dPs = ρs dμs − ρs νs d ω.

(6.107)

The only longitudinal mode in the clamped regime is the fourth sound, which involves coupled oscillations of the pressure and temperature connected by the continuity equation (6.104) for entropy. Neglecting the effect of rotation in the linearised Euler equation and using the Gibbs–Duhem relation and the entropy continuity equation (6.104) for excluding the temperature variation, one obtains (Landau and Lifshitz, 1987) ∂v s ∇P ∂v s ∇P ∂v s + ∇μ = + − s∇T = + = 0, ∂t ∂t ρ ∂t ρs c42

(6.108)

where c42 =

ρs 2 ρs T s 2 ρn ρs c + = cs2 + c22 . ρ s ρcv ρ ρ

(6.109)

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Vortex dynamics in two-fluid hydrodynamics

A solution of Eqs. (6.108) and the mass continuity equation (6.103) is a fourth-sound plane wave ∝ eiK·R−iωt with the fourth-sound velocity c4 = ω/K given by Eq. (6.109).

6.8 Oscillations of an incompressible fluid in the clamped regime The dispersion equation of the Kelvin waves does not change in the clamped regime from that given by Eq. (6.78) because the Kelvin wave does not excite motion of the normal component and is not affected by the normal viscosity. We focus here on the slow mode in an incompressible fluid. The mode is described by Eqs. (6.64), (6.66) and (6.102), neglecting line tension contributions νs p2 . In terms of longitudinal and transverse components of velocities the equations are: −iωvsk − 2

p2 vLt = 0, K2

(6.110)

−iωvst + 2vLk = 0, vslk − vsk = − vslt

cT2 k 2 vLt , iω2

c2 k 2 − vst = − T vLk , iω2

vLk = (1 − α  )vslk + αvslt , vLt = (1 − α  )vslt − αvslk .

(6.111)

(6.112)

Only the longitudinal component vslk − vsk of the difference between the local and the average superfluid velocities v sl and v s is of importance, while vslt − vst ≈ 0. After exclusion of the superfluid velocities v sl and v s one obtains the equations for the two components of the vortex velocity v L :   cT2 k 2 p2  (iω − 2α)vLk + (1 − α ) 2 2 + vLt = 0, 2 K (6.113)    cT2 k 2 p2  −(1 − α )(iω − 2α)vLk + iω − α 2 2 + vLt = 0. 2 K Excluding the small longitudinal component vLk and neglecting some insignificant terms proportional to small p2 or k 2 , one obtains the equation for the transverse component vLt :     p2 ω2 + iωα2 − 42 2 + cT2 k 2 [(1 − α  )2 + α 2 ] vLt = 0. (6.114) K This is a quadratic equation for the frequency with two solutions: #   p2 ω = −iα ± −2 α 2 + [(1 − α  )2 + α 2 ] 42 2 + cT2 k 2 . K

(6.115)

6.8 Oscillations of an incompressible fluid

187

At p = 0 this formula agrees with those obtained by Stauffer (1967) and Volovik and Dotsenko (1980) for the Tkachenko wave in the clamped regime. At weak mutual friction (small mutual friction parameters α and α  ) the frequency is # p2 (6.116) ω = ±(1 − α  ) 42 2 + cT2 k 2 − iα. K In the long wavelength limit (small p/k and k) there are two overdamped modes with imaginary frequencies: ω1 = −i2α = −i ω2 = −i

(1 − α  )2 + α 2 2α

ρn B, ρ

  p2 42 2 + cT2 k 2 . K

(6.117) (6.118)

If mutual friction is not too weak, i.e., α is not very small compared to unity, the first mode with the frequency ω1 is not a slow mode because it is scaled by the frequency  and does not depend on elastic deformations of vortices. The local superfluid velocity v sl does not differ from the average superfluid velocity v s , and the incompressibility condition requires that the longitudinal component vslk = vsk of the superfluid velocity vanishes. The mode with the frequency ω1 is elliptically polarised with the ratio of two vortex velocity components following from the equations of motion (6.113) in the clamped regime: vLk α iω = = . vLt 2 (1 − α  ) 1 − α

(6.119)

The mode describes relaxation of an array of straight vortices to a new equilibrium state after spin-up or spin-down of the container. In general it is a non-linear process, which is described by the continuity equation (3.11) for vorticity, which for straight vortices parallel to the z axis can be written as ∂ ω˜ z + ∇ · (ω˜ z v L ) = 0. ∂t

(6.120)

We assume that during the relaxation process the vorticity ω˜ z , does not vary in space (solid body rotation), and therefore in the rotating coordinate frame v s = (ω˜ z /2 − )[ˆz × r], while according to Eq. (6.102) ∇ · v L = (1 − α  )∇ · v s + α zˆ · [∇ × v s ] = α(ω˜ z − 2).

(6.121)

Then Eq. (6.120) transforms to ∂ ω˜ z + α ω˜ z (ω˜ z − 2) = 0. ∂t For small deviation of vorticity ω˜ z from the equilibrium value 2, this equation describes relaxation governed by mutual friction and by the imaginary frequency ω1 . If  = 0, the

188

Vortex dynamics in two-fluid hydrodynamics

equation describes decay of vorticity after rotation of the container has stopped: ω˜ z = κnv ∝

1 . α(t − t0 )

(6.122)

Later (Section 14.2) we shall see that the same law describes the decay of the vortex line density in the chaotic vortex tangle in superfluid turbulence. The second frequency ω2 corresponds to the slow mode governed by vortex bending and the Tkachenko shear rigidity ∝ cT2 . For the slow mode of small frequency ω  2α, one can neglect the term ω2 in Eq. (6.114). After performing the inverse Fourier transformation one obtains the partial differential equation describing the slow mode in the configurational space:   2 (1 − α  )2 + α 2 ∂v L 2 ∂ vL 2 4 (6.123) − 4 − cT ⊥ v L = 0. ⊥ ∂t 2α ∂z2 As in the case of the one-component perfect fluid, in the clamped regime in the extreme long wavelength limit, fluid compressibility becomes important. While in the onecomponent perfect fluid this occurs at k ∼ /cs , the corresponding scale in the clamped regime is k ∼ /c4 , where c4 is the velocity of the fourth sound.

6.9 Phenomenological theory close to the critical temperature The first phenomenological theory of superfluidity and superconductivity dealing with the order parameter wave function was the Ginzburg–Landau theory for superconductors (Ginzburg and Landau, 1950). Its mathematical form is similar to the Gross–Pitaevskii theory (Gross, 1961; Pitaevskii, 1961) which appeared later, but it was derived using different physical reasoning and is valid for another area of physical parameters, namely, close to the critical temperature at which the transition to the superfluid (superconducting) state takes place. Both theories use an expansion in the order parameter wave function and its gradients, but in the Gross–Pitaevskii theory the expansion is justified by weakness of particle interaction, while the Ginzburg–Landau theory uses smallness of the wave function amplitude close to the critical point. So, to study dynamics close to the critical temperature one should use the Ginzburg– Landau approach. Within this approach Ginzburg and Pitaevskii (1958) suggested the theory for a stationary state of a superfluid close to the critical temperature, which was called the Ginzburg–Pitaevskii theory. The time-dependent theory was developed by Pitaevskii (1958b). Before turning to the Ginzburg–Pitaevskii theory itself, let us extend the two-fluid hydrodynamics, which considered the superfluid density ρs as a defined function of other thermodynamic variables. Now we consider it as an independent variable.4 Close to the critical

4 A detailed hydrodynamical theory treating ρ as an independent thermodynamic variable was developed by Hills and Roberts s

(1978).

6.9 Phenomenological theory near critical temperature

189

temperature, relaxation of the order parameter becomes very slow. Since the superfluid density ρs is related to the order parameter, it cannot relax instantaneously to its equilibrium value when the thermodynamic state is altered. Instead of Eq. (6.4), the Gibbs relation is given by dE = μdρ ˜ + μ˜ s dρs + T dS + v n · dj 0 + j · dv s , where two chemical potentials were introduced:   ∂E  ∂E  μ˜ = , μ˜ s = . ∂ρ ρs ∂ρs ρ

(6.124)

(6.125)

In the standard two-fluid theory the superfluid chemical potential μ˜ s must vanish, while μ˜ transforms to the usual chemical potential μ. In contrast to Eq. (6.4), we consider a vortexfree superfluid, and the contribution to the energy, which was connected with vortex lattice deformations ∇j ui , is now absent. So the energy density in an arbitrary inertial coordinate frame is given by E(ρ, ρs , S, j 0 , v s ) = E0 (ρ, ρs , S) +

j02 ρv 2 + j 0 · vs + s , 2ρn 2

(6.126)

where E0 (ρ, ρs , S) is the energy density at rest and j 0 = ρn (v n − v s ) is the mass current in the coordinate frame moving with the superfluid velocity v s . The chemical potentials μ˜ and μ˜ s are connected with the chemical potentials μ˜ 0 = ∂E0 /∂ρ and μ˜ s0 = ∂E0 /∂ρs of the resting fluid by the relations μ˜ = μ˜ 0 −

(v s − v n )2 v2 + s, 2 2

μ˜ s = μ˜ s0 +

(v n − v s )2 . 2

(6.127)

The pressure is defined as P = −E + μρ ˜ + μ˜ s ρs + T S + v n · j 0 ,

(6.128)

while the Gibbs–Duhem relation is dP = ρd μ˜ + ρs d μ˜ s + SdT + j 0 · dv n − j · dv s = ρd μ˜ 0 + ρs d μ˜ s0 + SdT . (6.129) Deriving the dynamical equations in the new theory, one needs an additional continuity equation for the superfluid component of the superfluid: ∂ρs + ∇ · (ρs v s ) = −λμ˜ s , (6.130) ∂t where the right-hand side takes into account possible transformation of the superfluid component to the normal component and vice versa. This is a dissipative process, which is described in the linear response theory by a phenomenological constant λ. Together with a similar continuity equation for the normal component, ∂ρn + ∇ · (ρn v n ) = λμ˜ s , ∂t this yields the mass continuity equation (6.8) for the whole fluid.

(6.131)

190

Vortex dynamics in two-fluid hydrodynamics

In the momentum conservation law (6.11) the momentum flux tensor is given by Eq. (6.17) without the stress tensor σij and with the pressure given by Eq. (6.128). The Euler equation in the extended theory becomes ∂v s (6.132) + ∇(μ˜ + μ˜ s ) = ∇{ζ3 ∇ · [ρs (v s − v n )]}. ∂t The right-hand side is the second-viscosity term, which is crucial for vortex dynamics close to the critical point. The presence of the sum μ˜ + μ˜ s of two chemical potentials in the gradient term on the left-hand side is required by the momentum conservation law. Indeed, using the Gibbs relation (6.124) and the equations of motion, one can check that the energy conservation law (6.13) holds with the energy flux Q = μj ˜ + μ˜ s ρs v s + [ST + (j 0 · v n )]v n − ζ3 ρs (v s − v n )∇ · [ρs (v s − v n )],

(6.133)

and the dissipation function λμ˜ 2s ζ3 {∇ · [ρs (v s − v n )]}2 + . (6.134) 2 2 Instead of determination of the chemical potentials at fixed entropy and current j 0 in the superfluid coordinate frame, it is convenient to use the thermodynamic potential different from the energy, namely, the free energy determined by variables v n and T . The free energy is obtained after the Legendre transformation: R=

F (ρ, ρs , T , v n , v s ) = E − T S − j 0 · v n ρn (v s − v n )2 ρvs2 − . (6.135) 2 2 The relations (6.127) between chemical potentials in two coordinate frames remain intact, although the derivatives, which determine the chemical potential, are taken at fixed T and v n (instead of fixed S and j 0 ). A key assumption of the Ginzburg–Landau and the Ginzburg–Pitaevskii theories is that the superfluid density and the superfluid velocities are connected with the modulus and the phase of the order parameter wave function ψ and that the free energy can be expanded in small ρs = m|ψ|2 near the critical temperature:   2  i h¯ ρv 2 B m F = F0 (ρ, 0, T ) − n + A(T )|ψ|2 + |ψ|4 +  − ∇ − v n ψ  , (6.136) 2 2 2 m = F0 (ρ, ρs , T ) +

where the parameters A and B are A=m

 ∂F  , ∂ρs ρ

B = m2

 ∂ 2 F  . ∂ρs2 ρ

(6.137)

Neglecting gradients of the wave function modulus, i.e., neglecting gradients of ρs = m|ψ|2 , Eq. (6.136) is an expansion of the phenomenological free energy density (6.135) in ρs . While the parameter B depends weakly on temperature and can be treated as a constant,

6.9 Phenomenological theory near critical temperature

191

the parameter A = a(T − Tc ) is a linear function of temperature and becomes negative below the critical temperature Tc . This is a second-order transition to the superfluid state with finite value of the order parameter ψ. The minimisation of the free energy (6.136) with respect to the wave function ψ yields the stationary Ginzburg–Pitaevskii equation:  2 i h¯ 1 − ∇ − v n ψ + μ˜ s0 ψ = 0, (6.138) 2 m where μ˜ s0 = A(T ) + B(T )|ψ|2 .

(6.139)

The time-dependent theory (Pitaevskii, 1958b) deals with the effective non-linear Schr¨odinger equation:   ∂ψ vn2 = −i hv ψ i h¯ ¯ n · ∇ψ + m μ˜ 0 − ∂t 2    2 i h¯ 1 − ∇ − v n ψ + μ˜ s0 ψ , + (1 − i)m (6.140) 2 m where  is a dimensionless dissipation parameter. The Madelung transformation reduces the complex equation for ψ = aeiθ to two real equations for the superfluid density ρs = h¯ ∇θ . Taking into ma 2 and the phase θ , which determines the superfluid velocity v s = m account that close to the critical point  1, these equations are:   (v s − v n )2 2m h¯ 2 ∇ 2 a ∂ρs + ∇ · (ρs v s ) = − + ρs μ˜ s0 − , (6.141) ∂t 2 h¯ 2m2 a h h¯ ∂θ (v n − v s )2 v2 ¯ ∇ · [ρs (v s − v n )]. + μ˜ 0 − + s = m ∂t 2 2 2mρs

(6.142)

Neglecting gradients of the order parameter modulus a, these two equations agree with the phenomenological equations (6.130) and (6.132) derived from the conservation laws, provided that the dissipative parameters in these equations are ζ3 =

h ¯ , 2mρs

λ=

2m ρs . h¯

(6.143)

Taking the gradient of Eq. (6.142) one obtains the Euler equation for two-fluid hydrodynamics. But Eq. (6.141) is absent in two-fluid hydrodynamics. Equations (6.141) and (6.142) should be solved together with the other equations of hydrodynamics. A solution of the stationary Ginzburg–Pitaevskii theory [Eq. (6.138)] for a single vortex in the coordinate frame, where the normal component is at rest, is mathematically similar to the vortex solution of the Gross–Pitaevskii theory discussed in Section 2.1. However, now |ψ|2 is not the total particle density ρ/m but the superfluid particle density ρs /m. The entropy and the total mass density S and ρ vary in space but do not vanish at the vortex

192

Vortex dynamics in two-fluid hydrodynamics

axis. The entropy variation as a function of the distance r from the vortex axis can be found from the thermodynamic relation: ∂A(T ) aρs (r) ∂F = S0 − |ψ0 |2 = S0 − , (6.144) ∂T ∂T m where S0 is the entropy in the normal state of the fluid just above the critical temperature. One can find the pressure variation from the Gibbs–Duhem relation (6.129) together with the conditions that the chemical potential μ˜ 0 and the temperature T are constant. The pressure variation leads to an insignificant density variation inversely proportional to the sound velocity squared. Equation (6.140) for the wave function ψ was also used for the Bose–Einstein condensate of cold atoms interacting with thermal excitations and was called the dissipative Gross–Pitaevskii equation (Tsubota et al., 2013). In the theory of superconductivity a more general form of the time-dependent Ginzburg–Landau equation was used (Kopnin, 2001). In the coordinate frame moving with normal velocity v n (in superconductors v n becomes a velocity of a crystal) this equation is5   h¯ 2 2 ∂ψ (i R − ) h¯ (6.145) − mμ˜ 0 ψ = − ∇ ψ + mμ˜ s0 ψ. ∂t 2m S(r) = −

One obtains Eq. (6.140) with v n = 0 from this equation if R

+i

=

1 . 1 − i

(6.146)

Sometimes when dealing with the phenomenological equation (6.145) the chemical potential μ˜ 0 of the total fluid was ignored (Pismen, 1999; Tsubota et al., 2013). This is justified if μ˜ 0 is constant. Then μ˜ 0 can be removed by adding a linear function of time to the phase of the wave function ψ. The transformation is simply a shift of the zero energy level. But as we shall see later (Section 8.7), in a moving vortex μ˜ 0 varies in space and cannot be ignored in the calculation of mutual friction parameters. Taking into account variation of μ0 eliminates a logarithmic divergence in the dissipative mutual friction parameter. The Ginzburg–Pitaevskii theory is a mean-field theory of the second-order superfluid phase transition. It was suggested at the time when superfluid 4 He was the only superfluid available in laboratories. The theory provided a useful illustration of processes close to the critical temperature, but strictly speaking in superfluid 4 He there was no temperature window where the theory could be quantitatively accurate. Large critical fluctuations near the critical temperature Tc invalidated the mean-field theory down to temperatures where the order parameter wave function is not small enough to justify the Ginzburg–Landau expansion in the order parameter magnitude. Like the Gross–Pitaevskii theory at zero temperature, in weakly interacting BEC of cold atoms the Ginzburg–Pitaevskii theory 5 Following the analogy with superconductors one should take into account that for charged superfluids the chemical potential

must be replaced by the electrochemical potential. Moreover, in metals the chemical potential does not vary in space because of the condition of quasineutrality. This reduces the electrochemical potential to the scalar electric potential, which replaces the chemical potential μ˜ s0 (Kopnin, 2001).

6.9 Phenomenological theory near critical temperature

193

became a quantitatively reliable theory, since weakness of interaction confines critical fluctuations to a very narrow temperature interval near Tc . There were attempts to modify the Ginzburg–Pitaevskii theory in order to extend it to temperatures where critical fluctuations invalidate the mean-field approach. Mamaladze (1967) proposed renormalising parameters of the theory, making them non-analytical functions of Tc − T fitted to experimental data and scaling laws. The resultant theory, called the phenomenological  theory, was further developed and detailed in the reviews by Ginzburg and Sobyanin (1976, 1982). It is difficult (if possible) to justify rigorously this rather heuristic approach by microscopic analysis. Nevertheless, it is an effective model providing a good physical picture of dynamical processes close to the critical temperature, despite its possible quantitative inaccuracy. We shall use this theory further in the calculation of mutual friction parameters close to the critical temperature (Section 8.7).

7 Boundary problems in two-fluid hydrodynamics

7.1 Boundary conditions on a horizontal solid surface At finite temperatures the number of oscillation modes of a fluid increases, so the number of boundary conditions should increase also. On a horizontal solid surface confining a perfect fluid along the rotation axis (the z axis), we had the Bekarevich–Khalatnikov condition (5.19) imposed on the vortex velocity vL and the condition (5.40) that the mass current normal to the solid surface vanishes. In the two-fluid theory, the latter condition is imposed on the centre-of-mass velocity, ρs ρn (7.1) vz = vsz + vnz = 0, ρ ρ and new conditions should be added: the no-slip condition for the normal velocity v n and the thermal boundary condition connecting the normal heat flux and the variation of temperature on the surface. The no-slip condition is that the component of v n in the horizontal plane coincides with the velocity v B of the solid surface: v B = v n − zˆ (ˆz · v n ).

(7.2)

As a thermal condition one may assume that there is no heat flux across the solid surface. This means that wz = vnz − vsz = 0.

(7.3)

Together with Eq. (7.1) this means that neither v s nor v n has a z component normal to the surface. An alternative to the adiabatic condition, as one can call Eq. (7.3), is the isothermal condition: the temperature of the fluid near the solid surface is kept constant. But later on we shall use only the adiabatic thermal condition.

7.2 Pile-of-disks oscillations and effective boundary condition Section 5.4 analysed the motion of a one-component perfect fluid between two oscillating horizontal solid surfaces separated by a distance 2L. Now we turn to this analysis in 194

7.2 Pile-of-disks oscillations

195

the framework of two-fluid hydrodynamics. Surfaces of oscillating disks generate a fluid velocity field, which is a superposition of all six oscillation modes possible in two-fluid hydrodynamics: two Kelvin modes, two axial viscous modes, and two inertial waves. Each of these corresponds to a solution of the dispersion equation for p2 at given k and ω. The general analysis of such a system of equations is quite cumbersome (see Sonin, 1987, and references therein), and we restrict ourselves to discussion of simpler particular cases. When the frequency is large (ω ), there are Hall resonances with frequencies obtained from Eq. (6.78) with the wave number p satisfying the condition pL = π(2n − 1)/2 for Hall resonance:     π 2n − 1 2 (7.4) (1 − α  − iα). ωn = 2 + νs 2 L Without mutual friction (α = α  = 0) this coincides with Eq. (5.39) derived for a onecomponent perfect fluid. For an analysis of slow motion at ω  , one should reconsider derivation of the effective boundary condition (5.52) for the slow mode in the framework of the two-fluid theory. We look for a boundary condition imposed on the bulk centre-of-mass slow mode without essential counterflow, which is weakly affected by mutual friction. At small ω   one can ignore viscous modes and the counterflow slow mode everywhere including the superfluid Ekman layers near horizontal solid surfaces. This returns us to the three-mode superposition (the slow mode and the two Kelvin modes) in the expression (5.44) for the vortex velocity field in a one-component perfect fluid. A difference between the onecomponent and the two-fluid theories appears in the expression (5.46) for the longitudinal in-plane component of the fluid velocity, which is the centre-of-mass velocity in the twofluid theory: 

ρs Rej vk (z, r, t) = − ρ



  2j 2 p2 ipz ik·r−iωt −kE z e v˜K e + vS e . iω iω K 2

(7.5)

The factor ρs /ρ before the contribution from the Kelvin modes appears because the Kelvin wave does not generate motion of the normal part of the fluid, and therefore in the Kelvin wave the centre-of-mass velocity is v = ρρs v s . The same factor appears in the condition of the absence of the centre-of-mass velocity normal to a solid surface, which instead of Eq. (5.47) becomes   2j 2 p ρs k v˜K + vS = 0. Rej (7.6) ρ ikE iω iω k The same factor ρs /ρ must be added to the expression Eq. (5.50), and the parameter A in the effective boundary condition (5.52) in the two-fluid theory becomes A=

(b + kE )2 + b2 ρ kE . ρs b(b + kE ) + b2

(7.7)

196

Boundary problems in two-fluid hydrodynamics

The frequency of the inertial wave resonance for strong pinning in the two-fluid theory, instead of Eq. (5.42), is #    ρs 1 ρs νs 1/4 = (2)3/4 . (7.8) ωL = 2 ρ kE L ρ L2 As discussed in Section 5.4, the inertial wave resonance was observed in the pile-ofdisks experiment of Andereck and Glaberson (1982). The observed temperature dependence of the resonance frequency is shown in Fig. 7.1, reproduced from Fig. 19 of Andereck and Glaberson (1982). The experimental points are compared with the theoretical tempera√ ture dependence ρs /ρ following from Eq. (7.8) (the solid line). The proportionality factor

Frequency (rad/sec)

13

12

11 1.0

1.2

1.4 Temperature (K)

1.6

Figure 7.1 Dependence of the resonance frequency on the temperature in the pile-of-disks experiment. Black circles show data of Andereck and Glaberson (1982) obtained for an angular velocity  = 10.1 rad/sec and a distance of 0.051 cm between disks. The dashed line shows the dependence ∝ (ρs /ρ)1/4 following from the density-of-states peak theory (see the text). The dependence was scaled to match the lowest temperature data. The solid line shows the temperature dependence ∝ (ρs /ρ)1/2 derived for inertial-wave resonance [Eq. (7.8)]. The line was added to the figure from Andereck and Glaberson (1982).

7.3 Oscillations in the clamped regime

197

of the theoretical curve was chosen to match the low temperature experimental points. The theoretical and experimental dependence agree quite well, despite the fact that the condition ω  2 was not well satisfied in the experiment. Andereck and Glaberson (1982) interpreted their experiments differently, relating observed resonances with the frequency of the density-of-state peaks given by Eq. (5.122). In the two-fluid theory one should replace the zero-temperature Tkachenko wave velocity √ cT in Eq. (5.122) by the temperature-dependent Tkachenko wave velocity ct = ρs /ρcT . Then the resonance frequency is proportional to (ρs /ρ)1/4 . The temperature dependence following from the density-of-state peak theory is shown in Fig. 7.1 by the dashed line. The theory of inertial wave resonance explains the experimental temperature dependence at least as well as the density-of-state peak theory.

7.3 Oscillations in the clamped regime: damped slow mode In the clamped regime the normal fluid in the space between two oscillating solid surfaces oscillates together with the solid surfaces as a single solid. Therefore, the number of possible modes (degrees of freedom) drops to that in a one-component perfect fluid: two Kelvin modes and the slow mode (see Section 6.7). The normal fluid clamped by the oscillating disks exerts a driving force on vortices via mutual friction. We do not present here the general analysis of the pile-of-disks oscillations in the clamped regime (for that see Sonin, 1987), restricting ourselves instead to some particular cases. Since in Hall resonances the normal part of the fluid does not participate in oscillation, the expression (7.4) for frequencies of Hall resonances remains valid in the clamped regime. In the pile-of-disks experiment in the clamped regime, the superfluid component is dragged by oscillating disks even in the absence of surface vortex pinning (ideally flat surfaces of disks). The necessary hydrodynamical equations are Eqs. (6.33) and (6.64), where it is assumed that v n = v B and v sl = v s . The latter condition means that vortices oscillate without deformations. We are thus considering a case within the continuous vorticity model, and p → 0 and k → 0, but at the same time p/k → 0 also. The longitudinal component vsk of the superfluid velocity vanishes because of the incompressibility condition. Eventually Eqs. (6.33) and (6.64) take a quite simple form: −iωvst + 2vLk = 0, vsk = 0,

(7.9)

vLk = −α(vB − vst ),

(7.10)

vLt = vst + α  (vB − vst ).

(7.11)

The solution of these equations is vst =

2α vB , 2α − iω

vLk =

iωα vB , 2α − iω

vLt =

2α − iωα  vB . 2α − iω

(7.12)

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As was discussed in Section 5.4, in the pile-of-disks geometry the direction of k is identical to the radial direction. These longitudinal (subscript k) and transverse (subscript t) components of velocities correspond to radial and azimuthal components respectively. The slow mode at ω   in the clamped regime is described by the partial differential equation (6.123). To describe of slow columnar motion in the layer −L < z < L one should average this equation over the layer using the effective boundary condition Eq. (5.52), as was done in Section 5.5 for a one-component perfect fluid. After averaging one obtains the 2+1 partial differential equation for the vortex velocity v L :  (1 − α  )2 + α 2 (1 − α  )2 + α 2  2 ∂v L 2 + ωL v L − cT2 ⊥ v L = ωL vB , (7.13) ∂t 2α 2α √ where the frequency ωL = 2/ AL is given by Eq. (5.59) derived for a one-component perfect fluid. The parameter A in the effective boundary condition Eq. (5.52) is also determined by Eq. (5.50) obtained for a one-component perfect fluid, and not by Eq. (7.7), which differs from Eq. (5.50) by the factor ρ/ρs . This is because Eq. (7.7) referred to the case when the normal component oscillated together with the superfluid component, while in the clamped regime only the superfluid component oscillates. Neglecting the Tkachenko shear rigidity ∝ cT2 , Eq. (7.13) describes slow relaxation governed by vortex pinning via the frequency ωL : ω = −i

2 (1 − α  )2 + α 2 ωL . α 2

(7.14)

This mode was detected experimentally by observing relaxation of 3 He-B after spin-down (reduction of the rotation speed) with the NMR method (Krusius et al., 1993; Sonin et al., 1993). Two stages of the relaxation process were observed: (i) a relatively fast relaxation on a time scale of a few seconds, and (ii) an exponentially relaxing mode with a time constant of a few minutes. The fast relaxation, called the fast mode, was related to the frequency ω1 given by Eq. (6.117). Measurement of the fast mode allowed extraction of the dissipative mutual friction parameter. The slow mode was related to weak vortex pinning when the Tkachenko shear rigidity can be ignored and Eq. (7.14) for the slow mode reduces to (1 − α  )2 + α 2 νs b . (7.15) α L If isolated non-interacting vortices are pinned at the solid surface, the surface pinning parameter b does not depend on the angular velocity  proportional to the vortex density √ nv = 2/κ. Then according to Eq. (7.15), the relaxation time τs = 1/|ω| of the slow mode depends only weakly on  via a logarithmic factor ln  in the expression (2.50) for νs . However, experimentally it was found (Krusius et al., 1993) that τs does depend on  and this dependence was opposite to that from the weak logarithmic dependence of νs . This was explained by collective pinning which was discussed in Section 5.3. Although a direct Tkachenko shear rigidity contribution to the slow mode frequency was negligible in the experiment, indirectly it affected the observed frequency because collective pinning is impossible without shear rigidity of the vortex array. ω ≈ −i

7.4 Boundary condition on a vertical solid surface

199

7.4 Boundary condition on a vertical solid surface In Sections 4.3 and 5.8 we used the effective boundary condition on the lateral wall [Eqs (4.21) and (5.108)], implying that there is a force sticking vortices to the wall, which is parallel to them. Such a force is provided by mutual friction between the vortices and the normal fluid and sticking of the normal fluid to the wall. Now we are going to derive this boundary condition. Suppose that low frequency axisymmetric oscillations are excited in a superfluid put in a cylindrical container of radius R. The assumption that vortices do not interact directly with the wall means that the component σϕr of the stress tensor of the vortex lattice vanishes at the wall, and the boundary condition (4.18) must be satisfied. Another boundary condition is that the normal component sticks to a solid surface (no-slip condition) moving with the velocity v B : v n (R) = v B . The low frequency oscillation modes in the superfluid are exhausted by slow modes, which involve motion of both parts of the fluid together with vortices with the same velocity. Slow modes cannot satisfy both boundary conditions simultaneously, so we resort to the concept of the boundary layer again, this time supposing that there is a layer in which the moving wall generates an evanescent viscous mode and that the width of the boundary layer is the penetration depth of the mode. If the container radius is much larger than the penetration depth, the viscous mode in the boundary layer can be approximated by a plane wave with wave vector k ν normal to the wall. Then the azimuthal velocity components in the boundary layer are vnϕ = vϕ (r) + vν eikν (r−R) ,

vLϕ = vϕ (r) + βν vν eikν (r−R) .

(7.16)

Here vϕ is the azimuthal component of the velocity in the slow wave that propagates in the bulk, while the terms ∝ eikν (r−R) are related to the viscous mode in the boundary layer. The wave number kν and the ratio vLϕ /vnϕ = vLt /vnt in the viscous mode are given by Eqs. (6.90) and (6.91) respectively. We consider harmonic oscillations with frequency ω, but the time-dependent factor e−iωt is omitted. Substitution of Eq. (7.16) into the boundary conditions (4.18) and the no-slip condition v n (R) = v B (vLϕ = −iωuϕ ) yields ∂vϕ (R) vϕ (R) − + ikν βν vν = 0, ∂r R vϕ (R) + vν = vB .

(7.17) (7.18)

Excluding vν from these equations, one obtains the effective boundary condition (5.108), with the parameter αb equal to αb = −

1 . ikν βν

(7.19)

Substitution of kν and βν given by Eqs. (6.90) and (6.91) for the in-plane viscous mode in the low frequency limit yields

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Boundary problems in two-fluid hydrodynamics

# κ αb = − 8π iω

2 ρs (1 − α  )2 + α 2 . ν ρn α

(7.20)

Using this expression one can evaluate the role of the term ∝ αb in the boundary condition (5.108). One can ignore this term if kαb  1, where k is the in-plane wave vector of the slow mode in the bulk. Then the fluid sticks to the oscillating solid surface: vϕ = vB . Since k < ω/ct , this requires an inequality, # κ ρ (1 − α  )2 + α 2 . (7.21) |αb k|  16π ν ρn α Our derivation of the effective boundary condition relies on the elasticity theory of the √ vortex lattice and therefore is valid only for kν rv  1, where rv ∼ κ/ is the intervortex distance. When studying the oscillation of a freely suspended rotating container with a superfluid in Section 5.8 with the help of the effective boundary condition (5.108), we neglected the contribution of boundary layers to the net angular momentum. To estimate the accuracy of such an approximation we consider a two-dimensional problem of an array of straight vortices when all parameters vary only in the xy plane. The variation of the total fluid angular momentum is determined by the total flux of the angular momentum through the fluid boundary: dM = −2π R 2 ϕr , (7.22) dt where ϕr is the component of the net momentum flux tensor, which determines the force applied by the superfluid to the wall of the container. Since the stress tensor σϕr vanishes at the fluid boundary, ϕr contains only the viscosity tensor given by Eq. (1.81) in Cartesian coordinates. In cylindrical coordinates, neglecting the second viscosity ζ , the off-diagonal component, ϕr of the net momentum flux tensor is   ∂vnϕ (R) vnϕ (R) − ϕr = τϕr = −ρn ν ∂r R   ∂vϕ (R) vϕ (R) − + ikν vν = −ρn ν ∂r R    ∂vϕ (R) vϕ (R) 1 − 1− = −ρn ν ∂r R βν     ∂vϕ (R) vϕ (R) 1 ρs cT2 = ρn ν − −1 . (7.23) ∂r R iω ρn ν For low frequencies when ω  ρs cT2 /ρn ν, the off-diagonal component ϕr becomes equal to the stress tensor component σϕr determined at distances r close to R but still outside the boundary layer where the evanescent viscous mode is present. The fact that the momentum flux at the very border of the fluid does not differ from that outside the

7.5 Oscillations of a cylinder in a rotating superfluid

201

boundary layer means that the angular momentum in the boundary layer does not affect the total momentum balance, which yields the dispersion equation (5.115) in the limit of a very long cylinder (ωL = 0).

7.5 Oscillations of a cylinder immersed in a rotating superfluid Let us turn now to oscillations of a cylinder immersed in a rotating superfluid. The boundary condition for the fluid around the cylinder is Eq. (5.108), but αb ∝ 1/kν has different sign because the choice of the sign of kν should provide attenuation of the viscous mode deep within the fluid. Another sign must also be chosen in Eq. (7.22) connecting the time variation of the fluid angular momentum with the momentum flux component ϕr since the momentum flux now brings the momentum to the fluid and not from the fluid. Then the balance of angular momenta provides the following dispersion equation for eigenfrequencies of the oscillator: ω2 − ω02 − iω

2π R 3 ϕr = 0. Ic vB

(7.24)

Now one should approximate ϕr not by the viscous tensor, as in Eq. (7.23), but by the stress tensor σϕr because we analyse the fluid bulk outside the boundary layer using the effective boundary condition. Here ω0 is the eigenfrequency of the torsional oscillator without the superfluid, and Ic is the moment of inertia of the container per unit length. When ω0 = 0, Eq. (7.24) determines the eigenfrequencies of a freely suspended cylinder. The oscillating cylinder irradiates a cylindric Tkachenko wave with the spectrum ω = √ ct k, where ct = ρs /ρcT is the temperature-dependent Tkachenko velocity. Assuming the boundary condition (5.108) with αb = 0 and approximating the cylindric Tkachenko wave by a plane wave vLϕ = vB exp[ik(r − R) − iωt], near the surface of the cylinder the off-diagonal momentum flux component is ∂uφ kvLϕ = ρct2 = ρct vB . ∂r ω Using this expression in Eq. (7.24) one obtains ϕr = σϕr = −ρs cT2

ω2 = ω02 − iωct

2πρR 3 . Ic

(7.25)

(7.26)

The frequency has an imaginary part associated with energy losses √ due to emission of the Tkachenko wave. The damping δ ∼ Imω/ω is proportional to /ω when ω  . In the past the theory of oscillations of a cylinder in a rotating superfluid did not take into account the shear rigidity of the vortex lattice and assumed that the oscillating cylinder only generated a viscous oscillation mode in the surrounding fluid (Andronikashvili et al., 1961). Then according to Eq. (7.23), ϕr = −ikν ρn νvν with vν = vB . Substituting for kν the value of k given by Eq. (6.84) yields  2π R 3 iω − B 2 2 . (7.27) ω = ω0 − ρn ω iων Ic iω − 2α

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Boundary problems in two-fluid hydrodynamics

At slow rotation   ω this formula predicts the linear dependence of the damping δ on  [see Eq. (7.3.1) in the review by Andronikashvili and Mamaladze (1967)]. This prediction was confirmed by the experiments of Tsakadze and Chkheidze (1960). The shear rigidity effect is not expected to be important at   ω and cannot be calculated within the continuum theory because the wave number of the Tkachenko wave is much larger than the inverse intervortex distance in this case. But for fast rotation  ω, the correct formula for eigenfrequencies is Eq. (7.26), allowing for the emission of the Tkachenko √ wave. Equation (7.27) in this case predicts the dependence δ ∝ 1/ ω, instead of δ ∝ √ /ω following from Eq. (7.26). An experimental study of the oscillations of a long cylinder in a rotating superfluid in the regime of fast rotation could provide information on vortex lattice shear rigidity, as Tkachenko (1974) hoped for.

7.6 Single vortex line terminating at a lateral wall In Section 2.12 we considered precession of a vortex line terminating at the lateral wall of a container filled with a one-component perfect fluid. Now we shall address the same problem in the two-fluid theory, taking into account mutual friction (Sonin, 2012b). We shall consider the clamped regime, when the normal component of the fluid corotates with the container as a solid body. The vortex line goes along the container axis (the z axis), but deviates from it starting from some height and eventually terminates at the lateral wall. In analogy with the vortex front of the vortex bundle, which will be addressed in the next section, we shall call the end segment of the vortex line connecting the axis and the lateral wall the vortex front. Stationary propagation of the vortex front supposes that it moves as a solid body with constant vertical velocity vf , at the same time rotating around the z axis with angular velocity f different from . Thus the vortex velocity v L inside the vortex front has only two components in the rotating coordinate frame: the z component vf and the azimuthal component (f − )r. Mutual friction produces an external force on the vortex, F f r = ρs κ{d[ˆs × [ˆs × (v L − v n )]] − d  [ˆs × (v L − v n )]},

(7.28)

which is balanced by the Magnus force: −ρs κ[ˆs × (v L − v sl )] = F f r .

(7.29)

Equation (7.28) differs from the right-hand side of Eq. (6.38) by the normalisation of a vortex line per unit length instead of per unit volume as in Eq. (6.38). In the clamped regime the normal component rotates as a solid body (v n =  × r) and is at rest in the rotating coordinate frame. For a single vortex in the local induction approximation, the average superfluid velocity v s vanishes, and the local superfluid velocity v sl is determined only by the line tension force: d sˆ (7.30) [ˆs × v sl ] = −νs . dl

7.6 Single vortex line terminating at a lateral wall

203

Without mutual friction when v L = v sl this equation is identical to Eq. (2.49) derived for a one-component perfect fluid. The magnitude of the curvature vector d sˆ /dl is equal to the inverse curvature radius given by Eq. (2.51). Two functions z(r) and ϕ(r) determine the shape of the vortex line in cylindrical coordinates r, ϕ, z. Using Eq. (7.30) and the expression for the curvature vector in the cylindrical coordinate frame, the two vector equations (7.28) and (7.29) give two equations for the axial and the azimuthal components of forces on the vortex:   dz/dr d sr f r + νs dr [1 + r 2 (dϕ/dr)2 + (dz/dr)2 ]1/2 = (1 − sz2 )dvf + (sr d  − sz sϕ d)(f − )r, (7.31)   r 2 (dϕ/dr) νs d sr −vf + r dr [1 + r 2 (dϕ/dr)2 + (dz/dr)2 ]1/2 = (1 − sϕ2 )d(f − )r − (sr d  + sz sϕ d)vf .

(7.32)

The third radial component of the force balance equation (7.28) is not an independent equation, being a consequence of Eqs. (7.31) and (7.32). Integrating Eqs. (7.31) and (7.32) over " the whole vortex line [keeping in mind that the line length element is dl = dr/sr = 1 + r 2 (dϕ/dr)2 + (dz/dr)2 dr] and taking into account the boundary conditions dz/dr = ∞ at r = 0 and dz/dr = dϕ/dr = 0 at r = R one obtains f

R2 R2 − νs = dz vf R + (d  − dzϕ )(f − ) , 2 2

(7.33)

R2 R2 R3 = dϕ (f − ) − (d  + dzϕ )vf , (7.34) 2 3 2 where three new mutual friction parameters related to the dissipative component ∝ d of the mutual friction force were introduced:    d R 1 − sz2 3d R 1 − sϕ2 2 2d R sz sϕ dz = dr, dϕ = 3 r dr, dzϕ = 2 r dr. R 0 sr sr R 0 R 0 sr (7.35) −vf

Equation (7.33) is the balance of axial forces on the vortex, while Eq. (7.34) is the balance of moments around the z axis. This becomes evident if one rewrites them as   R2 , (7.36) e − f mz = −ρs κ dz vf R + (d  − dzϕ )(f − ) 2   R2 R3  vf mz = −ρs κ dϕ (f − ) − (d + dzϕ )vf , (7.37) 3 2 where e = ρs κνs is the energy and mz = ρs κR 2 /2 is the angular momentum around the z axis per unit length of straight vortex line far below the termination point respectively.

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Boundary problems in two-fluid hydrodynamics

The left-hand side of Eq. (7.36) is a mechanical force on the vortex front, which is balanced by the axial friction force on the right-hand side. Equation (7.37) presents the balance between the moment transferred to the fluid because of the vortex front propagation (the left-hand side) and the mutual friction torque (the right-hand side). None of the terms in the balance equations, except for the dissipative terms ∝ d on the right-hand side, depend on the shape of the vortex. In particular, the terms ∝ vf and f originate from the Magnus force term ρs κ[ˆs × v L ] in Eq. (7.28). The equations of linear and angular momenta balance lead to the balance of the total energy, which determines the energy dissipation: dE = vf (e − f mz ) + (f − )vf mz = vf (e − mz ) dt   3 2 2R 2 − dzϕ (f − )vf R . = −ρs κ dz vf R + dϕ (f − ) 3

(7.38)

Solving Eqs. (7.33) and (7.34) one obtains vf =

2dϕ R( − c1 ) , 2 ] 4dz dϕ + 3[(1 − d  )2 − dzϕ

3(1 − d  − dzϕ )( − c1 ) f −  = − , 2 ] 4dz dϕ + 3[(1 − d  )2 − dzϕ

(7.39)

where c1 = 2νs /R 2 is the first critical angular velocity [see Eq. (2.13)], at which the energies per unit length of the straight vortex line far below the termination point and of the vortex-free region far above this point are equal in the rotating coordinate frame. One cannot use these expressions to determine the propagation and the rotation velocities vf and f directly because the mutual friction parameters dz , dϕ , and dzϕ depend on the vortex shape determined from the differential equations (7.31) and (7.32). However, assuming that the friction force is weak, either because d and d  are small or because the difference  − c1 is small, one can determine the values dz , dϕ , and dzϕ using Eq. (2.120), which describes the shape of the vortex line in the state of equilibrium solid body rotation together with the container. Then √ $ 1 r 2R 2 − r 2 R2 − r 2 sr = " = , s = 1 − sr2 = , sϕ = 0, z 2 R R2 1 + (dz/dr)2 (7.40) and Eq. (7.35) yields √ 2−1 2 dz = d ρ 2 − ρ 2 dρ = d = 0.609d, 3 0  1 √ ρ dρ = 3( 2 − 1)d = 1.24d, dzϕ = 0. dϕ = 3d " 0 2 − ρ2 

1

"

(7.41)

7.6 Single vortex line terminating at a lateral wall

205

Using these values in Eq. (7.39) one obtains vf =

2.48dR( − c1 ) , 3.02d 2 + 3(1 − d  )2

f −  = −

3(1 − d  )( − c1 ) , 3.02d 2 + 3(1 − d  )2

(7.42)

the dissipation rate being dϕ ( − c1 )2 R 3 dE 1.24d( − c1 )2 R 3 = −ρ κ . = −ρs κ s dt 4dz dϕ + 3(1 − d  )2 3.02d 2 + 3(1 − d  )2

(7.43)

The relations between the mutual friction parameters d and d  on one side and the parameters α and α  on the other side are given by Eq. (6.37). One can see that in general the dissipation rate and the vortex front velocity vf depend on both α and α  , and not only on α, as was supposed earlier (Eltsov et al., 2009). However, in the limit of the weak friction force one can neglect the tiny difference between the factors 3 and 3.02 in the denominators of Eqs. (7.42) and (7.43). Then according to Eq. (6.37) the dissipation rate and the vortex front depend only on α. In particular, vf ≈ 0.83αR( − c1 ).

(7.44)

The single-vortex front dynamics was simulated numerically on the basis of the Biot–Savart law by Karim¨aki et al. (2012). The results are in qualitative agreement with those presented here and obtained in the local induction approximation. In particular, expansion of the expression (11) of Karim¨aki et al. (2012) for the front velocity (denoted vLz there) with respect to  − c1 gives vf ≈ 0.741αR( − c1 ), which does not differ essentially from Eq. (7.44). A more quantitative comparison requires more numerical data for lower angular velocities  ∼ c1 while the available data of Karim¨aki et al. (2012) focus on high angular velocities  c1 . It is useful to generalise the analysis for a helical vortex ending at a lateral wall. This will help in understanding the more complicated case of the vortex bundle addressed in the next section. As in Section 2.10, the helical vortex line spirals around a cylinder of radius a with pitch 2π/p. Now the equations of motion (7.31) and (7.32) must be integrated not from 0 to R but from a to R. This yields more general equations replacing Eqs. (7.33) and (7.34): f −vf

νs R2 − a2 R2 −" = dz vf R + (d  − dzϕ )(f − ) . 2 2 1 + p2 a 2

(7.45)

R2 − a2 R2 R3 νs pa 2 = dϕ (f − ) −" − (d  + dzϕ )vf . 2 3 2 1 + p2 a 2

(7.46)

Here the mutual friction parameters dz , dϕ , and dzϕ are given by Eq. (7.35) with the lower limit in the integral being a instead of 0. After multiplication of these equations by the superfluid density ρs , terms proportional to f and vf in the left-hand sides of these equations contain the angular momentum mz = ρs (R 2 − a 2 )/2 per unit length of the

206

Boundary problems in two-fluid hydrodynamics

helical vortex. Keeping in mind that the angle between the vortex line and the z axis is γ = arctan(pa), the terms proportional to νs in the equations are the axial component of the line tension force ρs νs cos γ [Eq. (7.45)] and the moment −ρs νs a sin γ of this force around the z axis [Eq. (7.46)]. The moment of the line tension force is nothing other than the angular momentum flux  R Gzz r dr = −ρs νs a sin γ (7.47) Jm = 2πρ 0

along the container axis introduced in Section 5.6. On the other hand, the axial component of the line tension force is related to the component zz of the linear momentum flux tensor and can be written as "

ρs νs cos γ = e + pJm ,

(7.48)

where e = ρs νs 1 + p2 a 2 is the energy per unit length of the helical vortex, and the angular momentum flux statisfies the canonical relation Jm = −∂e/∂∇z ϕ, where the gradient of the azimuthal angle ϕ along the vortex line is equal to the pitch p. Eventually one obtains the equations of balance   R2  , (7.49) e − f mz + ∇z ϕJm = −ρs κ dz vf R + (d − dzϕ )(f − ) 2   R2 R3 vf mz − Jm = −ρs κ dϕ (f − ) − (d  + dzϕ )vf , (7.50) 3 2 which reduce to Eqs. (7.36) and (7.37) in the limit a → 0 when Jm = 0. The left-hand side of Eq. (7.49) is a driving mechanical force on the vortex front balanced by the z component of the friction force on the right-hand side of the equation. Equation (7.50) tells us that the angular momentum brought by the angular momentum flux Jm is compensated for by the growth of the total angular momentum (the term mz vf ) due to front propagation and by the friction torque (the right-hand side of the equation). Note that a similar term Pz vf is absent in the balance of the linear momentum since we consider the case when the fluid does not move along the z axis and Pz = 0. It is important that in the wake of the vortex front propagating with velocity vf , a helical vortex remains uniform and rotates with the container angular velocity  only if the linear velocity vf and the angular velocity f of the propagating vortex front are related with the kinematic relation f −  = vf ∇z ϕ.

(7.51)

Bearing in mind this relation, one can find that at front propagation the balance of the energy in the rotating coordinate frame is given by Eq. (7.38) derived for an axial vortex. Only two of the derived balance equations for the linear and the angular momenta and for the energy are independent. Any one of them follows directly from the other two. It is more convenient to choose the balance of the energy and the angular momentum [Eqs. (7.38) and (7.50)].

7.7 Vortex bundle terminating at a wall

207

Equations (7.38) and (7.50) show that in principle the vortex front can propagate with finite stationary velocity vf even in the absence of friction when the right-hand sides of the equations vanish. Then the left-hand sides must also vanish. The quantity vf (e − mz ) on the left-hand side of Eq. (7.38) has a dimensionality of force like the mechanical force in the balance of the linear momentum (7.49) but differs from the former in general. One can call it the effective force. Absence of effective force at finite vf requires that the energy e−mz per unit length in the rotating coordinate frame vanishes. Since this energy vanishes in the vortex-free region well above the vortex front, the condition e − mz = 0 means that the vortex propagation does not affect the energy balance. Referring now to the angular momentum balance (7.50), one can see that frictionless propagation of the vortex front is possible if the angular momentum, which is necessary for propagation of the vortex front, is supplied by the angular momentum flux along the helical vortex. Then the vortex front propagates with the velocity vf =

Jm . mz

(7.52)

Frictionless propagation of the vortex front is possible if the helical vortex is pinned at the bottom of the container.

7.7 Vortex bundle terminating at a wall: propagation of the vortex front Let us now consider a vortex bundle terminating at the lateral wall. The part of the bundle diverging to the wall is a vortex front separating the vortex-filled and the vortex-free parts of the container (Fig. 7.2). Motion of the front along the container axis is a transient process of vorticity penetration into a container in the spin-up experiments (Eltsov et al., 2009). There are stationary states of the bundle when the front does not move along the z axis and the bundle and the front rotate as a solid body with the angular velocity determined from the thermodynamic analysis (see Section 5.6). Here we address the case of a moving front (Sonin, 2012b). The motion of the vortex front leads to a change of the energy and the angular momentum and is accompanied by mutual friction with the normal component moving rigidly with the walls and possibly by friction of vortex ends at rough surfaces of walls. We look for the state with the vortex front moving with constant velocity vf , when the angular momentum necessary for this motion is supplied from vortex pinning at the bottom and is transferred to the front by the flux of the angular momentum Jm . The flux is related to the twist Q = ∇z ϕ of the vortex bundle stem (Section 5.6). A twisted vortex bundle is a collection of helical vortices with the same pitch 2π/p = 2π/Q. A uniformly twisted bundle stem in the wake of the front moving with velocity vf is possible if the kinematic relation (7.51) is satisfied. For a full solution of the problem one should solve the equation (6.38) of vortex motion. Now the equation of motion also includes the average superfluid velocity v s , which incorporates the interaction between vortices in the bundle. An analytical solution of the problem is not known and we refer to the balance equations for the energy and the linear and the

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Boundary problems in two-fluid hydrodynamics

(a)

(b)

Figure 7.2 Untwisted and twisted vortex bundle. (a) Untwisted vortex bundle. Any vortex line is in an axial plane, i.e., is not twisted. Such a structure corresponds to the equilibrium state in the coordinate frame rotating with angular velocity  (Sonin and Nemirovskii, 2011). (b) Twisted vortex bundle. The twisted stem of the bundle is at rest in the coordinate frame rotating with angular velocity . The vortex front propagates with the velocity vf along the z axis and rotates around the z axis with relative velocity  = f −  with respect to the container, the normal fluid, and the bundle stem rotating together. Figure from Sonin (2012b).

angular momenta. Generalising the three balance equations, Eqs. (7.38), (7.49) and (7.50), derived for a single helical vortex one obtains (Sonin, 2012b): (e − mz )vf = −(f − )Tf r − vf Ff r ,

(7.53)

mz vf − Jm = −Tf r ,

(7.54)

e − f mz + ∇z ϕJm = −Ff r .

(7.55)

Here Ff r is the z component of the friction force and Tf r is friction torque around the z axis, which will be specified later on. None of the terms on the left-hand sides of the equations, i.e., those not connected with mutual friction, depend on the velocity and vorticity distribution inside the vortex front, but depend only on the distribution inside the

7.7 Vortex bundle terminating at a wall

209

bundle stem far below the front. The front velocity vf is absent in the equation of the linear momentum balance (7.55) because elongation of the bundle due to front propagation does not change the linear momentum since the latter is exactly zero above and below the front. The balance equations (7.53)) and (7.54) for the energy and the angular momentum seem evident.1 But Eq. (7.55) presenting the linear momentum balance is less evident. The simplest argument in its favour follows from the fact that the three balance equations (7.53)– (7.55) are not independent, and it is easy to check that Eq. (7.55) follows from Eqs. (7.53) and (7.54). If one believes the first two, one must accept the third. One can also derive it directly from the equations of hydrodynamics. For derivation of the balance of the linear momentum one should consider the momentum fluxes in the vortex-free region above and in the vortex bundle stem below the vortex front. The latter was found in Section 5.6 apart from an undefined constant pressure P0 on the bundle axis. The calculation was done there for a one-component perfect fluid. Repeating this in the two-fluid theory for the clamped regime one should use the linear (s) momentum balance (6.105) with the momentum flux tensor ij given by Eq. (6.106). The latter contains the partial superfluid pressure Ps instead of the global pressure P . This yields a partial pressure distribution across the bundle cross-section similar to that given by Eq. (5.82) but with the total density ρ and the fluid velocity v replaced by the superfluid density ρs and the superfluid velocity v s : ρs v02 ρs vs2 + ρs rvsϕ − ρs νs ω˜ + + ρs νs ω(0) ˜ + P0 . (7.56) 2 2 Above the vortex front the superfluid part of the fluid does not move, and the superfluid (s) momentum flux ij and the partial pressure Ps vanish. One can find P0 from the momentum conservation law without any knowledge of the velocity and the vorticity field in the vortex front. The vortex front moves with velocity vf along the z axis, and the balance of momentum must take into account the time derivative dvsz /dt = −vf ∇z vsz , but does not contain the contribution of the normal component, which does not move at the bundle axis. Then the partial pressure gradient along the z axis is determined from the relation   v2 (7.57) ∇z Ps = −ρs ∇z νs ω˜ + s − vf vsz . 2 Ps = −

Integration over the whole axis z yields the value of the partial pressure P0 at the axis z far below the vortex front:     vs (0)2 vs (0)2 mz − vf v0 = −ρs νs ω˜ + + (f − ) , (7.58) P0 = −ρs νs ω˜ + 2 2 π R2 where Eqs. (5.71) and (7.51) were used. Eventually integration of the momentum flux over the cross-section of the bundle yields the left-hand side of the linear momentum balance equation (7.55).

1 The importance of the angular momentum balance was discussed by Hosio et al. (2013).

210

Boundary problems in two-fluid hydrodynamics

Next let us calculate the total friction force and the total friction torque, integrating the mutual friction force per unit volume given by Eq. (6.36) over the whole fluid:  ∞  R +   , dz ω˜ d vf (1 − sz2 ) − sz sϕ (f − )r + d  sr (f − )r r dr Ff r = 2πρs κ −∞ ∞

0





  ωd ˜ vf (1 − sz2 ) − sz sϕ (f − )r −∞ 0   ∂vsϕ −d (f − )r r dr = dvf mF + d  (f − )mz , ∂z  ∞  R +   , = 2πρs κ dz ω˜ d (f − )r(1 − sϕ2 ) − sz sϕ vf − d  sr vf r 2 dr = 2πρs κ

Tf r



R

dz

−∞

0

= d(f − )mT − d  vf mz . Here

 mF = 2πρs κ mT = 2πρs κ

(7.59) 

∞

−∞  ∞ −∞

R

dz 

0 R

dz 0

ω(1 ˜ − sz2 − sz sϕ Qr)r dr,   sz sϕ r 3 dr, ω˜ 1 − sϕ2 − Qr

(7.60)

are effective moments determining dissipation proportional to the parameter d. Only the vortex front region contributes to the bulk integrals, since there is no vorticity in the vortexfree region above the front, while below the front the relative velocity v L − v n has no component normal to the vortex lines and integrands also vanish. The dissipation rate depends only on the dissipative mutual friction parameter d: (f − )Tf r + vf Ff r = d[vf2 mF + (f − )2 mT ] = dvf2 [mF + (∇z ϕ)2 mT ]. (7.61) The explicit expressions for the friction force and the friction torque allow the relations determining the front velocity vf and the twist Q = ∇z ϕ to be derived from the balance equations : 1 − α mT Q m F + Q2 m T 1 − d = =− + Jm , d α mz (mz  − e)mz

vf =

mz  − e . d(mF + mT Q2 )

(7.62)

As a rather rough estimation of the moments mF and mT , one can approximate the vortex line in the front by straight line segments normal to the axis (Sonin, 2012b). Then sr = 1 and sz = sϕ = 0, and ω˜ = −∂vϕ /∂z. After integration over z, Eq. (7.60) reduces to  R  R mF = 2π vsϕ (r)r dr, mT = 2π vsϕ (r)r 3 dr. (7.63) 0

0

An essential assumption of the analysis was pinning of the vortex bundle at the bottom, which provided the constant angular momentum flux towards the vortex front. Although pinning at the lower end of the bundle is quite possible, especially if the ‘bottom’ is in fact an interface separating the B phase from the A phase with the periodic vortex structure, one

7.7 Vortex bundle terminating at a wall

211

can also address the case without pinning. Then coupling between the vortex bundle and the container can be provided only by mutual friction. This would complicate the analysis since below the vortex front the angular momentum flux proportional to the twist will not be uniform along the container axis. At very low temperatures mutual friction coupling of the vortex bundle and the container also becomes ineffective, and the vortex bundle rotates at a smaller angular velocity than the container (Hosio et al., 2011). In the limit of strong dissipative friction parameter d compared to 1 − d  , vf and Q are expected to be small. Neglecting the line tension effects in the relations given in Section 5.6 and expanding them in Q, Eq. (7.62) reduces to 56 1 − α → − QR, α 45

vf →

3R . 8d

(7.64)

Note that for our choice of the rotation sense the twist Q is negative. One can compare this result with experimental measurements of the twist Q and the vortex front velocity vf at temperatures higher than about 0.5Tc . Figures 15 and 17 of Eltsov et al. (2009) show that Q grows and vf drops with decreasing temperature. This agrees qualitatively with Eq. (7.64) since both d and (1 − α  )/α = (1 − d  )/d grow with decreasing temperature according to measurements of Bevan et al. (1997b) for 3 He-B at T > 0.6 Tc (see their Figs. 5, 6, 8, and 9). The results of the experiment by Eltsov et al. (2009) at high temperatures were described by the expression vf ≈ αR. It is worthwhile noting that at α 1 − α  there is no difference between α and 1/d, so the quoted result differs from Eq. (7.64) only by the factor 3/8. This may be explained by inaccuracy of our assumption of the shape of the bundle inside the vortex front. Nowadays there is great interest in the question: what is the dynamics of the vortex front in the T = 0 limit? In this limit mutual friction must vanish, and steady propagation of the vortex front is possible only if there is no force accelerating the vortex front. As in the case of the helical vortex considered at the end of Section 7.6, this requires that lefthand sides of the balance equations (7.53)–(7.55) vanish. Only two of these three equations are independent, and it is enough to require zero effective force e − mz in Eq. (7.53) and zero torque mz vf − Jm in Eq. (7.54). The zero torque condition simply determines the vortex front velocity vf = Jm /mz , and only the zero effective force condition is not trivial. Since the energy e = e − mz in the rotating coordinate frame per unit length also vanishes in the vortex-free region, this condition means that the energies of the states without vortices and with twisted vortex bundle are equal. According to Eq. (5.77) in the continuous vorticity approach, i.e., neglecting the line tension contributions, e is always negative independently of how strong the twist Q is. But quantum line tension terms linear and quadratic in νs make the condition e = 0 possible. In an untwisted vortex bundle the condition e = 0 corresponds to the angular velocity  = 8νs /R 2 (Sonin and Nemirovskii, 2011). This angular velocity is of the same order as the critical angular velocity c1 [see Eq. (2.13)] at which the first vortex appears in the rotating container. The numerical difference between the two critical angular velocities arises because the velocity  = 8νs /R 2 was determined from macroscopic hydrodynamics, which assumes

212

Boundary problems in two-fluid hydrodynamics

the presence of many vortices in the container. In reality, at this value of  the number of vortices is of the order of the large logarithm ln(R/rc ), which is not so large, but still can be larger than unity. A growing twist Q shifts the system closer to the state where e = e − mz vanishes. For large numbers of vortices the state is possible only for very large QR. Using the expressions of Section 5.6, in this limit Eq. (7.62) yields: Q=−

R , 2νs

vf = −

3 . 4Q ln(|Q|R)

(7.65)

The angular momentum, which is necessary for propagation of the vortex front, is supplied by the angular momentum flux. Thus in the laminar regime, which our analysis addressed, steady front motion without mutual friction is possible only for very high twists or rather small numbers of vortices. Physical and numerical experiments provided evidence that at low temperatures the laminar regime transforms to the turbulent regime, while at extrapolation to T = 0 the vortex front velocity remains finite (Eltsov et al., 2010). Later in this book (Section 14.1) we shall show that a comparatively weak twist of the vortex bundle leads to the Glaberson–Johnson– Ostermeier instability, which can be considered a precursor of this transition. It is natural to expect that a turbulent vortex bundle has higher energy than a bundle in the laminar regime. This would allow the condition e − mz = 0 to be reached at larger vortex numbers and weaker twists and would make our scenario of steady vortex front propagation without mutual friction more plausible. However, one should remember a more trivial explanation of finite vortex front velocity in the T = 0 limit: surface friction of vortex ends moving along a rough wall. The surface friction does not disappear at T = 0.

8 Mutual friction

8.1 Mutual friction and macroscopic hydrodynamics Since the concept of mutual friction was first formulated by Hall and Vinen (1956a) it has remained one of the most important and probably most intricate problems of the hydrodynamics of rotating superfluids. Calculations of mutual friction parameters inevitably refer to ‘microscopic’ hydrodynamics and even to the really microscopic kinetic theory based on the Boltzmann equation. The present chapter considers these calculations, assuming that vortices are well separated and their cores do not overlap even close to the critical temperature where the core radius becomes very large. Thus we remain fully within the VLL state explained in Section 3.1. Then the mutual friction problem is reduced to that of a single straight vortex. If the vortex moves with the velocity v L relative to the normal component moving with the velocity v nl , one expects a force on the vortex proportional to the relative velocity v L − v nl similar to the force on a body moving through a viscous fluid. In an isotropic fluid where rotational symmetry is broken only by the presence of a vortex parallel to the axis zˆ , a general linear relation between the force and the relative velocity is F f r = −D[ˆz × [ˆz × (v nl − v L )]] + Dz zˆ {ˆz · (v nl − v L )} + D  [ˆz × (v nl − v L )].

(8.1)

The transverse component ∝ D  of the force normal to the relative velocity v nl − v L is possible because of broken time-inversion invariance. Here v nl is the local normal velocity in the area close to the vortex line, which in general is different from the average normal velocity v n in macroscopic hydrodynamics. If the vortex is ideally straight, the fluid is translationally invariant along the z axis and a friction force along the z axis is impossible, i.e., the parameter Dz must vanish. However, experimental investigations of the second sound attenuation revealed a small friction force along the rotation axis (Donnelly, 1991, Section 3.6). One can explain this by interaction of bulk quasiparticles with thermally excited Kelvin waves, or kelvons in the quantum mechanical approach on the vortex line (Sonin, 1975, Appendix II). Both the experiment and the theory show that Dz is small and will be ignored later on. Assuming that all velocities are in the plane normal to the vortex line (the axis zˆ ) the linear relation between the force and the relative velocity is F f r = D(v nl − v L ) + D  [ˆz × (v nl − v L )].

(8.2) 213

214

Mutual friction

The difference between v nl and v n is similar to the difference between the velocity v 0 close to the point, where a local force is applied to a fluid, and the velocity v very far from this point in a classical viscous fluid. Therefore the analysis of this effect for a classical viscous fluid in Section 1.9 is fully relevant for a normal component of the superfluid, after replacing the total mass density ρ by the normal mass density ρn , the fluid velocity v by the normal velocity v n , and the fluid velocity v 0 by the local normal velocity v nl . Then Eq. (1.115) transforms to v n − v nl =

ln(rm /rl ) Ffr. 4πρn ν

(8.3)

Hall and Vinen (1956b) called this effect viscous drag. As in a classical viscous fluid (Sections 1.8 and 1.9), the upper cut-off rm in the logarithm argument should be the smaller of two lengths: the Oseen length rO = ν/|v nl − v n | or the viscous penetration depth √ δ = ν/ω for time-dependent motion with frequency ω. In rotating superfluids two other √ scales should be added to this list: the Ekman layer width ∼ ν/  and the intervortex √ distance ∼ κ/. The lower cut-off rl is usually chosen to be of the order of the mean free path lqp of quasiparticles (Hall and Vinen, 1956a). Such a choice of the lower cut-off assumes that the quasiparticle flux on the vortex is entirely determined by the equilibrium Planck distribution at the border between the ballistic and the two-fluid hydrodynamics region at r ∼ lqp .1 With the help of Eq. (8.3) one can rewrite the expression (8.2) for the friction force in terms of the velocities v n and v L : F f r = −D(v L − v n ) − D [ˆz × (v L − v n )],

(8.4)

where the parameters D and D are connected with D and D  in Eq. (8.2) by a j -complex relation 1 ln(rm /rl ) 1 + = . (8.5)  D + jD 4πρn ν D + j D The friction force on the vortex must be balanced by the Magnus force: ρs [κ × (v sl − v L )] = F f r .

(8.6)

After excluding the local normal velocity v nl and the force F f r , Eqs. (8.2), (8.3) and (8.6) yield the expression (6.33) for the vortex velocity v L with α and α  given by the j -complex formula   1 1 ln(rm /rl ) 1 −1  + α − jα = − . (8.7) κρs 4πρn ν D + j D jρs κ This connects the mutual friction parameters α and α  in macroscopic hydrodynamics with the mutual friction parameters in single-vortex dynamics. 1 However, scattered quasiparticles colliding with other quasiparticles immediately after scattering can return back and modify

the quasiparticle flux on the vortex. As a result of this, the lower cut-off could be a combination of the mean free path lqp and some effective cross-section of the vortex. This logarithmically weak effect was discussed briefly by Sonin (1975).

8.2 Semiclassical scattering (geometric optics)

215

For steady motion of a single vortex, the upper cut-off rm in Eq. (8.3) is the Oseen length, which diverges in the linear theory for low velocities. In the limit of very large logarithm, Eq. (8.7) yields that α=

4πρn ν , ρs κ ln(rm /rl )

α  = 0.

(8.8)

In this limit the velocity v L is equal to the local normal velocity v nl , as in the case of a cylinder moving in a classical viscous fluid (Section 1.8). This model of vortex motion was used by Matheiu and Simon (1980) to interpret experimental data on mutual friction parameters at intermediate temperatures (Section 8.8). Later in this chapter we shall see that according to the scattering theory of noninteracting quasiparticles by a vortex, the vortex moves approximately with the local centre-of-mass velocity v L = ρρs v sl + ρρn v nl . In this approximation (Sonin, 1975, 1976) D  = −κρn and D ≈ 0, and   ρn κρs ln(rm /rl ) −1 α − jα = . j+ ρ 4πρν 

(8.9)

Here, as well as in the previous case, the entire dissipation is associated with viscous losses in the normal component. The calculations of the force follow along different lines, depending on the ratio between various space scales. The theory of Hall and Vinen (1956b) was based on the assumption that the mean free path of quasiparticles is much larger than the size of the area responsible for scattering. Therefore the mutual friction force emerged from scattering of noninteracting quasiparticles by a vortex. This assumption is accurate at low temperatures when quasiparticle interaction is weak. When the temperature rises and approaches the critical temperature, the radius of the vortex core becomes larger than the mean free path of quasiparticles, and one expects that some sort of phenomenological theory similar to the time-dependent Ginzburg–Landau theory may be valid. This was realised in the theory of mutual friction near the critical temperature discussed in Section 8.7.

8.2 Semiclassical scattering of quasiparticles (geometric optics) Quasiparticles are scattered by the velocity field outside the core and by the inhomogeneous field of the order parameter inside the core. Only the first contribution can be calculated more or less rigorously, and we focus on it. The semiclassical scattering theory for rotons was suggested by Lifshitz and Pitaevskii (1957) long ago. The semiclassical equations of quasiparticle motion are the Hamilton equations for the quasiparticle position vector R and the quasiparticle momentum p: ∂ε dR = , dt ∂p

dp ∂ε =− . dt ∂R

(8.10)

216

Mutual friction

Here ε(p, r) = ε0 (p) + p · v v

(8.11)

is the energy of the quasiparticle in the fluid velocity field around the rectilinear vortex, vv =

[κ × r] , 2π r 2

(8.12)

ε0 is the quasiparticle energy in the resting fluid, and r is a position vector in the plane normal to the vortex line (the projection of R on that plane). This approach assumes that any quasiparticle has well defined position and momentum and, correspondingly, a well defined classical trajectory. In wave mechanics this approach is called geometric optics. The vortex velocity field produces a force −∇(p · v v ) on a quasiparticle. The force is weak and the quasiparticle trajectory is nearly rectilinear. Suppose that the trajectory is in the xy plane and parallel to the y axis (Fig. 8.1a) and its impact parameter (the distance between the vortex line and the trajectory) is b = x. Then Eq. (8.10) gives dy = vG , dt

dp = −∇(p · v v ). dt

(8.13)

Here v G = ∂ε0 (p)/∂p is the quasiparticle group velocity in the resting fluid, which in our case is approximately parallel to the y axis . Excluding time from these equations, one has a differential equation determining the variation in quasiparticle momentum along the trajectory: 1 dp = − ∇(p · v v ). dy vG

(8.14)

In the Hamilton–Jacoby theory the momentum is connected to the classical action: p = ∂S/∂r. The momentum component px normal to the trajectory varies from 0 at y = −∞ (before scattering) to the value px =

∂δS(b) ∂b

(8.15)

at y = ∞ (after scattering). Here δS(b) is the total variation of the classical action along the trajectory with the impact parameter b produced by the velocity field around the vortex. It is the difference between the action variation with and without the vortex:  ∞ δS(b) = [p(y) − p]dy, (8.16) −∞

where p is the magnitude of the momentum p = p(−∞) at y = −∞ (before scattering). A small scattering angle is φ(b) ≈ −

1 ∂δS(b) px =− . p p ∂b

(8.17)

8.2 Semiclassical scattering (geometric optics)

217

(a)

–4

–2

2

4

(b)

1.0 0.5

–4

2

–2

4

–0.5 –1.0 –1.5 (c)

2

1

–4

–2

2

4

–1

Figure 8.1 Semiclassical scattering of rotons. (a) Rotons at trajectories with impact parameters b = x < 0 or b > b∗ move past the vortex. Rotons at trajectories with 0 < b < b∗ are reflected by the vortex (Andreev reflection). The shaded area (Andreev shadow) is classically forbidden for rotons. (b) Action variation δS(b) along the trajectory as a function of the impact parameter b (dimensionless variables). (c) Scattering angle φ(b) as a function of the impact parameter b (dimensionless variables). Figure from Sonin (2013b).

218

Mutual friction

Dependence of the scattering angle on the impact parameter determines the differential cross-section for a quasiparticle scattered by the vortex σ (φ) =

db . dφ

(8.18)

The scattered quasiparticle transfers momentum to the vortex, i.e., produces a force on the vortex. The momentum transferred to the vortex has a y component p(1 − cos φ) along the initial momentum p of the quasiparticle and an x component p sin φ normal to p. If there is a uniform flow of quasiparticles with quasiparticle density nqp and momentum density (mass current) j qp = nqp p, the parallel and the normal components of the force on the unit length of the vortex are F = j qp vG σ , F⊥ = j qp vG σ⊥ .

(8.19)

The dissipative longitudinal force F related to the longitudinal momentum transfer is determined by the transport cross-section π σ =

∞ σ (φ)(1 − cos φ)dφ =

−π

(1 − cos φ)db,

(8.20)

−∞

while the non-dissipative transverse force F⊥ is determined by another effective crosssection, which can be called the transverse cross-section: π σ⊥ =

∞ σ (φ) sin φdφ =

−π

sin φ db.

(8.21)

−∞

In our case the scattering angle φ is small, and the two effective cross-section are approximately given by ∞ σ ≈ −∞

φ2 db, 2

(8.22)

and ∞ σ⊥ ≈ −∞

1 dbφ(b) = − p

∞ −∞

δS(−∞) − δS(+∞) ∂δS(b) db = . ∂b p

(8.23)

Thus the transverse cross-section depends only on the action variations along trajectories with very large impact parameters on both sides of the vortex line. At large impact parameters one can neglect variation of the group velocity vG along the trajectory, and integration of Eq. (8.14) yields p(y) = p −

p v v (b, y), vG

(8.24)

8.2 Semiclassical scattering (geometric optics)

where p = p(−∞) is the momentum at y = −∞. Then according to Eq. (8.16)  ∞  ∞ p b pκ κp δS(b) = − vvy dy = − dy = −signb , vG −∞ 2π vG −∞ b2 + y 2 2vG

219

(8.25)

and the expression (8.23) yields the transverse cross-section σ⊥ =

κ . vG

(8.26)

We have obtained a universal expression for the transverse cross-section, which looks valid for any quasiparticle spectrum. The cross-section is proportional to the total variation of the classical action around the vortex line, which is proportional to variation of the quantum mechanical phase in quantum theory. This points out the connection of the transverse force with the geometric phase, or the Aharonov–Bohm effect (Sonin, 1975), which will be discussed in Section 8.5. We shall see that Eq. (8.26) yields a correct transverse crosssection for phonons even though the semiclassical theory is not valid for phonons: there are no well defined classical trajectories for phonons except for large impact parameters b, at which the scattering angle φ is negligible. In the case of phonons the group velocity vG is the sound velocity cs . Equation (8.23) contains a double integral. The action variation δS(b) is an integral over the trajectory defined by Eq. (8.25), and the second integration is over impact parameters b. This double integral is improper: its value depends on which integration is done first. The correct procedure is to integrate along the trajectory first, and over the impact parameters afterwards. In order to check this one should introduce finite limits in the double integral of Eq. (8.23) which means that the integration is restricted by some area around the vortex line. The integral depends on the shape of this area. For example, a circular border of the area yields σ⊥ which is less than that of Eq. (8.26) by a factor 2. But the analysis of the collisionless Boltzmann equation for quasiparticles (Sonin, 1975) showed that other terms, which are also originate from the slow decrease of the velocity field, contribute to the momentum balance. Taking into account all of these terms, we always arrive at the expression for the transverse force via the transverse cross-section given by Eq. (8.26) independently of the shape of the integration area. The order of integrations in Eq. (8.23) assumes that the proper integration area is rectangular in shape, with a much longer side along the quasiparticle trajectory. For such a shape all additional shape-dependent contributions to the transverse force vanish exactly. The transverse cross-section (8.26) does not depend on details of the dependence of the scattering angle on the impact parameter, because it is determined by an integral over a derivative of the action. Nevertheless, it is necessary to find this dependence in order to calculate the transport cross-section. Now we shall do this for rotons with the energy spectrum ε0 (p) =  +

(p − p0 )2 . 2μ

(8.27)

220

Mutual friction

According to the energy conservation law following from the Hamilton equations (8.10) one has: +

(p − p0 )2 [p(y) − p0 ]2 + p(y) · v v (y) =  + , 2μ 2μ

(8.28)

where the right-hand side is the energy far from the vortex line. Then variation of the roton group velocity along the trajectory with the impact parameter b is given by # $ p(y) − p0 bb∗ 1 2 (p − p0 ) − 2μpvvy = vG 1 − 2 . vG (y) = = μ μ b + y2

(8.29)

Here the characteristic scattering length b∗ =

p κμp κ = 2 π vG p − p0 π(p − p0 )

(8.30)

is introduced and vG = vG (−∞) = (p − p0 )/μ is the roton group velocity far from the vortex line. In the classical scattering theory the point y = 0 on the trajectory is a turning point. At y < 0 the quasiparticle approaches the scattering centre (vortex line in our case) while at y > 0 the quasiparticle moves away from the vortex line. Equation (8.29) shows that for impact parameters 0 < b < b∗ the quasiparticle cannot reach the common turning point since at y = −y ∗ , where y∗ =

"

b∗ b − b2 ,

(8.31)

the group velocity changes sign, and the quasiparticle starts to move back to y = −∞ without an essential change in its momentum. At the point y = −y ∗ the momentum reaches its value p = p0 at the minimum of the roton spectrum, and the transition between two branches of the roton spectrum with p > p0 (positive branch, parallel momentum and group velocity) and with p < p0 (negative branch, antiparallel momentum and group velocity) occurs. This is Andreev reflection, which is well known in the theory of superconductivity (Andreev, 1964). We shall encounter this interesting phenomenon again later in the book when considering vortex core bound states and scattering of BSC quasiparticles in Fermi superfluids (Chapter 9). Due to Andreev reflection, a shadow region is formed near the vortex line which is not available for roton classical trajectories. This shadow (Andreev shadow) region is shown in Fig. 8.1a. Let us find the variation of the classical action along trajectories with impact parameters b > b∗ or b < 0 when there is no Andreev reflection and the incident roton stays an the same branch after the collision. Taking into account variation of p(y) − p along the trajectory given by Eq. (8.29), for the incident momentum p > p0 the expression (8.16) for the action variation yields

8.2 Semiclassical scattering (geometric optics)

∞ # δS(b) = (p − p0 ) −∞

221

 b∗ b 1− 2 − 1 dy b + y2 

= 2sign(b)(p − p0 )

b−b

∗

!

 K

b∗ b



 − bE

b∗ b

 ,

(8.32)

where K(k) and E(k) are complete elliptic integrals of the first and second orders [Eq. (2.56)]. In the limits b → ±∞, Eq. (8.32) reduces to Eq. (8.25). In the interval 0 < b < b∗ a trajectory ends at the Andreev reflection point with the coordinate y = −y ∗ . The incident roton with momentum p = p0 + (p − p0 ) > p0 returns after Andreev reflection to y = −∞ on the other branch with the same energy but a slightly different momentum p− = p0 −(p −p0 ) < p0 . The variation of the action along the whole path is −y δS(b) =

∗

−∞ a −∞ p(y) dy + p− (y) dy − p dy − p− dy −y ∗

−∞

⎡ ⎢ = 2(p − p0 ) ⎣

−∞

∗ −y #

−∞

1−

b∗ b b2 + y 2

a



⎤

⎥ − 1 dy − y ∗ − a ⎦ .

(8.33)

Here a is an undefined constant, which does not depend on b and therefore has no effect on the scattering angle φ. Choosing a = 0 one eliminates any discontinuity of S(b) at b = 0 and b = b∗ .2 One can also reduce Eq. (8.33) to the expression        ! b∗ b∗ ∗ δS(b) = 2(p − p0 ) b − b F γ , − bE γ , , (8.34) b b in terms of incomplete elliptic integrals of the first and second order F (γ , k) and E(γ , k) √ [Eq. (2.118)] with γ = arcsin b/b∗ . In Figs. 8.1b and 8.1c the action δS(b) and the scattering angle φ(b) = ∂δS(b)/p∂b are plotted as functions of the impact parameter b (in dimensionless variables). The angle φ has singularities at b = 0 and b = b∗ . The strongest of these is at negative b approaching to 0: # # b∗ p − p0 b∗ ln . (8.35) φ≈ p0 |b| |b|

2 If one chooses non-zero a , discontinuity of S(b) at b = 0 and b = b∗ leads to two δ -function contributions of opposite signs in

the angle φ , which cancel each other in the transverse cross-section σ⊥ .

222

Mutual friction

The singularity is integrable in the transverse cross-section σ⊥ , but provides the main contribution to the transport cross-section: (p − p0 )2 σ ≈ 2p02

b∗ b0

 # 2  3 p0 b∗ b∗ κμ ln d|b| ≈ , ln |b| |b| 3πp0 p − p0

(8.36)

where the lower cut-off b0 ∼ b∗ (p − p0 )2 /p02 of the logarithmic divergence was chosen from the condition that the angle φ becomes of the order of 1. The transverse cross-section (8.26) is less than the impact parameter b∗ by a small parameter |p − p0 |/p0 and the transport cross-section (8.36) is less than the impact parameter b∗ by a small parameter (p − p0 )2 /p02 (ignoring the logarithmic factor). The impact parameter b∗ determines the width of the Andreev shadow. If the quasiparticle momentum is not in the xy plane, the transverse and the transport cross-sections (8.26) and (8.36) contain additional trigonometric factors:  3 p0 κ κμ ln , σ ≈ , (8.37) σ⊥ = vG sin ϑ p − p0 3πp0 sin3 ϑ where ϑ is the angle between the quasiparticle momentum and the z axis. The total mutual friction force on the vortex should be determined by averaging the single-quasipartice forces in Eq. (8.19) over the Planck distribution (6.49) for quasiparticles with drift velocity v n . For quasiparticles moving at an arbitrary angle to the vortex line, the group velocity vG in Eq. (8.26) is the component of the group velocity in the plane normal to the vortex line. The Planck distribution should be linearised with respect to the relative velocity v s − v n = (v s − v L ) + (v L − v n ). But the velocity v s − v L enters the energy ε(p) = ε0 (p) + p · (v s − v L ) of the quasiparticle in the reference frame connected with the vortex. According to the principle of detailed balance, the quasiparticle distribution function, which depends only on the energy, cannot produce a force. Thus only the part of the distribution function which is linear in the relative drift velocity v n − v L contributes to the force. Finally the parameters D and D  are given by the expressions  1 ∂f0 (ε0 ) 2 D=− 3 p⊥ σ vG⊥ d3 p, (8.38) ∂ε0 2h  1 ∂f0 (ε0 ) 2  D = 3 p⊥ σ⊥ vG⊥ d3 p, (8.39) ∂ε0 2h where p⊥ = p sin ϑ and vG⊥ = vG sin ϑ are magnitudes of projections of the momentum p and the group velocity v G in the xy plane. Using the transverse and the transport crosssections given by Eq. (8.37), one obtains (Lifshitz and Pitaevskii, 1957): √   p0 3 κρn μT ln √ D=√ , (8.40) μT 2π 3 p0 D  = −κρn .

(8.41)

8.3 Scattering of phonons by a vortex

223

√ The parameter μT /p0 is small and the longitudinal component ∝ D of the mutual friction force F f r [see Eq. (8.2)] is small compared to the transverse component ∝ D  . As we shall see, this is also true for low temperatures when the normal component consists mostly of phonons. Like any force acting upon the vortex, the force F f r should be balanced by the Magnus force in accordance with Eq. (8.6). Neglecting the longitudinal force ∝ D, Eq. (8.6) yields that the vortex moves with the velocity vL =

ρs ρn v sl + v nl . ρ ρ

(8.42)

This is a generalisation of Helmholtz’s theorem for two-fluid hydrodynamics: the vortex moves with the local centre-of-mass velocity of the fluid. A rather simple and universal expression D  = −κρn for the amplitude of the transverse force (valid both for rotons and for phonons) tempts us to think about its universal topological origin, since κ in this expression is a topological charge. However, in Section 9.2 we shall see that the expression is not universal. For quasiparticles in a BCS superconductor with energy much exceeding the gap, an additional small factor appears in this expression.

8.3 Scattering of phonons by a vortex The study of phonon scattering by a vortex line began with the works by Pitaevskii (1958a) and Fetter (1964). A phonon is a quantum of a sound wave, and one can investigate phonon scattering within the framework of the classical hydrodynamics of a one-component perfect fluid presented in Chapter 1. We consider a sound wave propagating in the plane xy normal to a vortex line (the axis z). In the linearised hydrodynamical equations of Section 1.6 the fluid velocity v 0 should be replaced by the velocity v v (r) around the vortex line:

∂v (1) ∂t

∂ρ(1) + ρ0 ∇ · v (1) = −v v · ∇ρ(1) , ∂t * ) c2 + s ∇ρ(1) = − (v v · ∇)v (1) + (v (1) · ∇)v v . ρ0

(8.43) (8.44)

Using the vector identity (1.32) for the velocity v = v v + v (1) , Eq. (8.44) can be rewritten as c2 ∂v (1) + s ∇ρ(1) = −∇(v v · v (1) ) + [v (1) × κ]δ(r). ∂t ρ0

(8.45)

The perturbation from the vortex (the right-hand side) contains a two-dimensional δ-function of the position vector r. The vortex line is not at rest when the sound wave propagates past the vortex. One can eliminate the singularity by introducing a timedependent vortex velocity v v (r, t) as a zero-order approximation for the velocity field (Sonin, 1975). Then r in the expression for v v (r, t) must be replaced by r − v L t and ∂v v /∂t = −(v L · ∇)v v = −∇(v L · v v ) + [v L × κ]δ(r). Since there is no external force on the fluid, the vortex moves with the velocity in the sound wave: v L = v (1) (0, t).

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Mutual friction

Now the fluid acceleration in Eq. (1.186) must be presented as ∂v/∂t = ∂v v /∂t + ∂v (1) /∂t. As a result, Eq. (8.45) is replaced by: ∂v (1) c2 + s ∇ρ(1) = ∇[v v · v (1) (r)] − ∇[v v · v (1) (0)]. ∂t ρ0 Equation (8.46) yields: ρ(1)

ρ0 κ =− 2 cs 2π



 ∂θ + v v · [∇θ (r) − ∇θ (0)] , ∂t

(8.46)

(8.47)

where θ is the order parameter phase, which determines the scalar potential for the velocity κ ∇θ . v (1) = 2π Substitution of ρ(1) in Eq. (8.43) yields the linear sound equation for the phase θ : ∂ ∂ 2θ − cs2 ∇ 2 θ = −v v (r) · ∇ [2θ (r) − θ (0)] , 2 ∂t ∂t

(8.48)

or taking into account the time dependence e−iωt , −(k 2 + ∇ 2 )θ =

ik v v (r) · ∇ [2θ (r) − θ (0)] . cs

(8.49)

In perturbation theory one can use the zero-order approximation for the phase θ (r) ≈ θ0 eik·r−iωt on the right-hand side: (k 2 + ∇ 2 )θ = θ0 k

k · v v (r) ik·r (2e − 1). cs

(8.50)

A parameter of the perturbation theory is κk/cs . It is small in the long-wavelength limit, when the wavelength 2π/k essentially exceeds the vortex core radius rc ∼ κ/cs . A solution of Eq. (8.50) in the Born approximation is    ik (1) θ = θ0 e−iωt eik·r − (8.51) d2 r 1 H0 (k|r − r 1 |)k · v v (r 1 )(2eik·r 1 − 1) . 4cs Here H0(1) (z) is the zero-order Hankel function of the first kind, and 4i H0(1) (k|r − r 1 |) is the Green function for the two-dimensional wave equation, which satisfies the equation −(k 2 + ∇ 2 )θ (r) = δ(r − r 1 ).

(8.52)

According to standard scattering theory, at large distances from a scattering centre the wave function is a superposition of the incident plane wave ∝ eik·r and the scattered wave ∝ eikr :   f (φ) ikr −iωt ik·r θ = θ0 e + √ e e . (8.53) r The scattering amplitude f (φ) determines the scattering differential cross-section σ (φ) = |f (φ)|2 . It is a function of the scattering angle φ between the wave vector k of the incident wave and the wave vector k  = kr/r after scattering (see Fig. 8.2).

225

sc at w a t ere ve d

8.3 Scattering of phonons by a vortex

incident wave

Figure 8.2 Scattering of a sound wave by a vortex. Figure from Sonin (2002).

The asymptotic expression (8.53) follows from Eq. (8.51) and the asymptotic expansion of the Hankel function at large values of the argument:  2 i(z−π/4) (1) e . (8.54) lim H (z) = z→∞ 0 πz For perturbation confined to the vicinity of the vortex line r1  r, (r 1 · r) , (8.55) r and integration in Eq. (8.51) yields (Pitaevskii, 1958a)     κ k 1 −i π k −i π sin φ cos φ 1 1  4 f (φ) = − e [κ × k ] · k − 2 = e 4 , (8.56) 2 2π cs 2cs 2π 1 − cos φ q 2k |r − r 1 | ≈ r −

where q = k − k  is the momentum transferred by the scattered phonon to the vortex, and q 2 = 2k 2 (1 − cos φ). The scattering amplitude obtained by Fetter (1964) differs from Eq. (8.56) by the absence of the factor (1 − q 2 /2k 2 ) = cos φ. The factor arises from motion of the vortex line dragged by the incident sound wave [the δ-function term on the right-hand side of Eq. (8.45)]. Fetter ignored this motion so the result of Fetter is valid if the vortex line is kept at rest by strong pinning. Since Fetter used the method of partial waves, we shall return to this difference when discussing the partial-wave approach in Section 8.5. Due to a very slow decrease in the velocity vv ∝ 1/r far from the vortex, the scattering amplitude is divergent at small scattering angles φ → 0:  k κ −i π 1 (8.57) e 4 , lim f (φ) = φ→0 2π cs φ while the differential cross-section σ (φ) = |f (φ)|2 diverges as 1/φ 2 . The divergence is integrable in the integral for the transport cross-section σ , Eq. (8.20), which yields σ of

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Mutual friction

the order of κ 2 k/cs2 . Using this value in Eq. (8.38), one obtains the value of the mutual friction parameter D: D ∼ κρn

T , mcs2

(8.58)

where ρn ∼ T 4 /h¯ 3 cs5 is the mass density of the normal component consisting of phonons (Khalatnikov, 2000). The differential cross-section σ (φ) is quadratic in the circulation κ and even in φ. Therefore in the Born approximation the transverse cross-section σ⊥ vanishes. But contrary to the transport cross-section, the integrand in Eq. (8.21) for the transverse cross-section has a pole at φ = 0. The principal value of the integral vanishes, but there is no justification for the choice of the principal value, and the contribution of the small angles requires a special analysis. This analysis (see the next section) demonstrates the existence of the transverse force, which was revealed by Iordanskii (1964, 1965a) who studied phonon scattering by the method of partial wave expansion.

8.4 Iordanskii force √ At small scattering angles φ  1/ kr the asymptotic expansion Eq. (8.53) is invalid, and one cannot use the differential cross-section or the scattering amplitude for description of the small-angle scattering (Iordanskii, 1964, 1965a; Sonin, 1975; Berry et al., 1980; Shelankov, 1998). Meanwhile, the small-angle behaviour is crucial for the transverse force, as demonstrated below. The accurate calculation of the integral in Eq. (8.51) for small angles (see Sonin, 1975, 1997, for more details) gives3 that at φ  1     kr iκk θ = θ0 exp(−iωt + ik · r) 1 +  φ . (8.59) 2cs 2i Using an asymptotic expression for the error integral at |z| → ∞, 2 (z) = √ π

z 0

e−t dt −→ 2

1 z 2 − √ e−z , |z| πz

√ one obtains for angles 1 φ 1/ kr:      iκk φ iκ k 1 ikr+iπ/4 −iωt ik·r 1+ − e e . θ = θ0 e 2cs |φ| cs 2π r φ

(8.60)

(8.61)

The second term in square brackets coincides with scattering waves at small angles φ  1 with the amplitude given by Eq. (8.57). But now one can see that the standard scattering

3 Olariu and Popescu (1985) presented a more general expression, which does not use perturbation theory with the small

parameter κk/cs .

8.4 Iordanskii force

227

theory misses an important non-analytical correction to the incident plane wave, which changes sign when the scattering angle φ crosses zero. Its physical meaning is discussed below. Let us consider the momentum balance in the area where scattering occurs. The force on the vortex  phfrom scattering is determined by the sound wave contribution to the momentum flux ij dSj through the surface surrounding this area. Replacing the fluid velocity v 0 by the velocity v v induced by the vortex in Eq. (1.78) one obtains: ph

ij = P(2) + ρ(1) v(1)i vvj + ρ(1) v(1)j vvi + ρ0 v(1)i v(1)j .

(8.62)

The term ∝ vvj does not contribute to the momentum flux through a cylindrical surface around the vortex, since the velocity v v is tangent to this surface. The term vvi ρ(1) v(1)j , in which the mass current ρ(1) v(1)j is given by Eq. (1.76) for the plane wave in the absence of the vortex, gives a contribution to the momentum flux which exactly cancels the contribution of the term ρ0 (v(1) )i (v(1) )j outside the interference region (Fig. 8.3), where the velocity is given by v (1)

  ik κ κ ∇θ = θ0 ik − v v eik·r−iωt . = 2π 2π cs ph

Finally only the interference region contributes to the force Fi a scattered phonon on the vortex.

=



(8.63) dSj ij produced by

~Ö r

Figure 8.3 Aharonov–Bohm interference. The phase (action) variations along trajectories above and √ below the vortex are different. The shaded area of the width λr is the interference region. Here δS± = δS(±∞), and δ± are the scattering phase shifts δl of partial waves (see Section 8.5) at l → ±∞. Figure from Sonin (2002).

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Mutual friction

In the interference region the velocity components in the cylindrical system of coordinates are κk κ ∂θ ≈ θ0 e−iωt+ik·r , v(1)r = 2π ∂r 2π  $  (8.64) kr 2 k ∂ φ 2i iκ κ ∂θ v(1)φ = = θ0 e−iωt+ik·r , 2π r ∂φ 4π cs r ∂φ and the transverse x component of the force on the vortex is   ρ0 θ02 κ 3 k 2 ph = κj ph , F⊥ = dSj φj = ρ0 v(1)φ v(1)r rdφ = 8π 2 cs

(8.65)

or in vector form: F ph = [j ph × κ].

(8.66)

Here j ph is the mass current in the sound wave given by Eq. (1.76). Thus the interference region, which corresponds to an infinitesimally small angle interval, yields a finite contribution to the transverse force, which one could not obtain from the standard scattering theory using the differential cross-section. In fact the details of the solution in the interference region are not essential: only a jump of the phase across the interference region is of importance. The transverse force (8.65) on the vortex produced by a sound wave coincides with that given in Eq. (8.19) and obtained from geometric optics, bearing in mind that in geometric optics according to Eq. (8.26) σ⊥ vG = κ. This is despite the fact that, strictly speaking, geometric optics is not valid for describing phonon scattering. Correspondingly, the amplitude of the transverse force D  = −ρn κ obtained in Section 8.2 from geometric optics is also confirmed. As in the case of rotons, D  significantly exceeds the amplitude of the longitudinal force D given by Eq. (8.58) since T  mcs2 . In order to elucidate further the physics behind our derivation of the Iordanskii force, let us consider the semiclassical solution of the sound equation (8.49), which is given by Eq. (8.61) after deleting the second term in square brackets:   iκk ϕ , (8.67) θ = θ0 e−iωt+ik·r+iδS/h¯ ≈ θ0 e−iωt+ik·r 1 + 2π cs where the angle ϕ = ±π − φ is an azimuthal angle for the position vector r measured from r v v · dl = the direction opposite to the wave vector k (see Fig. 8.2) and δS = −(hk/c ¯ s) hθ ¯ κk/2π cs is the variation of the action due to interaction with the circular velocity from the vortex along semiclassical trajectories. In Eq. (8.67) the phase θ is multi-valued, and one must choose a cut for θ in the direction k, where ϕ = ±π . The phase jump on the line behind the vortex is a manifestation of the Aharonov–Bohm effect: the sound wave after its interaction with the vortex has different phases on the left and on the right of the vortex line (above and below the vortex line in Fig. 8.3). This results in interference at √ a distance r from the vortex line in the region of the width dint ∼ r/k (Sonin, 1975).

8.4 Iordanskii force (a)

229 (b)

3.0

– – Figure 8.4 Aharonov–Bohm interference on scattering of a wave by a vortex. The wave fronts shown (lines of constant phase) demonstrate the presence of a wave front dislocation. (a) Numerical simulation for a plane wave incident from x = +∞ by Berry (2010). (b) Experimental observation of a wave front dislocation on scattering of a surface wave by a vortex in water (Berry et al., 1980).

√ The interference region corresponds to very small scattering angles ∼ dint /r = 1/ kr. Equation (8.67) satisfies the sound equation (8.49) in the first order of the parameter κk/cs except in the interference area, where it must be replaced by Eq. (8.59). Although the Aharonov–Bohm effect was discovered in quantum mechanics, it is an effect of classical wave mechanics. The only presence of quantum mechanics was our choice of quantised circulation. The Aharonov–Bohm effect at scattering by a classical vortex in water was observed by Roux et al. (1997) for a sound wave (acoustic Aharonov– Bohm effect) and by Berry et al. (1980) for a surface wave. Aharonov–Bohm interference results in the emergence of a dislocation of wave fronts (lines of constant phase). Figure 8.4 shows the wave front dislocation calculated numerically by Berry (2010) and observed by Berry et al. (1980). The Aharonov–Bohm effect for scattering of light by optical vortices was also studied (Neshev et al., 2001). From the very beginning, the transverse force from quasiparticles scattered by a vortex was a subject of controversy and of lively debate. Iordanskii noticed that his force cannot be obtained from the differential cross-section, like the Lifshitz–Pitaevskii force for rotons. On this ground he suggested that his force and the Lifshitz–Pitaevskii force had different origins and for rotons they should be summed. Since the force calculated by Lifshitz and Pitaevskii and the Iordanskii force were equal in amplitude but opposite in sign, he concluded that the transverse force from rotons vanished. But later it was revealed (Sonin, 1975) that the transverse force from rotons was calculated in the original paper by Lifshitz and Pitaevskii (1957) with incorrect sign. After correction, the transverse force on the

230

Mutual friction

vortex had the same sign and value both for rotons (the Lifshitz–Pitaevskii force) and for phonons (the Iordanskii force). Connection of the transverse force from quasiparticles with the Aharonov–Bohm effect (Sonin, 1975, 1997, 2010) led to the conclusion that the Iordanskii force for phonons and the Lifshitz–Pitaevskii force for rotons have the same origin and they must not be added. Later on, debates on the transverse force (especially on its phonon version, i.e., the Iordanskii force) continued with a number of publications rejecting the existence of the transverse force from quasiparticles. In part, confusion about the Iordanskii force originated from attempts to define of the Iordanskii force a priori, before (or without) any dynamical analysis (see Sonin, 1987, for references to old debates on the transverse force). Meanwhile, any force is a transfer of momentum between two subsystems, and a more careful approach is first to derive rigorously some momentum balance equation and only a posteriori to label terms that enter this equation as such-and-such force. Ao and Thouless (1993) suggested that the Berry phase provides a universal exact value for the total transverse force on the vortex, which does not depend on the presence of quasiparticles or impurities (see also Thouless, 1998). This ruled out transverse forces on the vortex from quasiparticles and impurities. Arguments on this issue continued for a number for years (Hall and Hook 1998; Sonin 1998; Wexler et al. 1998a, 1998b), but eventually some consensus was reached that the original calculation of the Berry phase missed the contribution from the normal-fluid circulation (Thouless et al., 2001; Sonin, 2002), which led to the transverse force from quasiparticles. The Berry phase in vortex dynamics will be discussed in Section 8.6.

8.5 Partial-wave analysis and the Aharonov–Bohm effect The Aharonov–Bohm effect follows from the stationary Schr¨odinger equation for an electron in the presence of a magnetic flux  confined to a thin tube: e 2 1  −i h∇ (8.68) Eψ(r) = ¯ − A ψ(r). 2m c Here ψ is the electron wave function with energy E and A is the electromagnetic vector potential connected with the magnetic flux  by the relation [ˆz × r] . (8.69) 2π r 2 This vector potential does not produce a magnetic field outside the magnetic flux tube, but still affects properties of the electron, including electron scattering, even if the magnetic flux tube is very thin and the probability of finding the electron inside the tube is negligible. The Aharonov–Bohm effect is periodic with respect to the parameter γ = /1 with period 1. Here 1 = hc/e is the magnetic-flux quantum for one electron (two times larger than the magnetic flux quantum 0 = hc/2e for a Cooper pair). The analogy between phonon scattering by a vortex and the Aharonov–Bohm effect (Aharonov and Bohm, 1959) for electrons scattered by a magnetic flux tube becomes A=

8.5 Partial-wave analysis and the Aharonov–Bohm effect

evident if one rewrites the sound equation (8.49) in the presence of the vortex as 2  k k 2 θ − −i∇ + v v θ = 0. cs

231

(8.70)

The expression (8.12) for the velocity v v around the vortex line is similar to the expression (8.69) for the vector potential A. Equation (8.70) differs from Eq. (8.49) by the term of the second order in vv ∝ κ and by the absence of the contribution from the vortex line motion [the term ∝ θ (0) on the right-hand side of Eq. (8.49)]. For calculation of the transverse force, which is linear in κ, this difference is not important. But terms quadratic in κ and motion of the vortex line are important for calculation of the scattering amplitude, as will be shown below. The small parameter κk/2π cs in the phonon scattering theory corresponds to the parameter γ = /1 in the Aharonov–Bohm effect, which can be arbitrarily large, and one cannot use perturbation theory in general. However, Aharonov and Bohm (1959) found an exact solution of the Schr¨odinger equation for a very thin magnetic flux tube using the partial-wave expansion. From this solution one can derive the transverse force on the flux tube. Expansion of the wave function ψ in partial cylindrical waves in the polar system of coordinates (r, φ) is  ψl (r) exp(ilφ). (8.71) ψ= l

The partial-wave amplitudes should satisfy equations d 2 ψl (l − γ )2 1 dψl − + ψl + k 2 ψl = 0. 2 r dr dr r2

(8.72)

Here k is the wave number of the electron far from the flux tube with energy E = h¯ 2 k 2 /2m. A solution of this equation regular at r → 0 is the Bessel function J|l−γ | (kr) with asymptotic expression at large arguments:   π π 2 cos kr − |l − γ | − . (8.73) J|l−γ | (kr) ≈ π kr 2 4 On the other hand, the partial-wave expansion of the plane wave, which propagates in the absence of the scattering potential, is   eik·r = eikr cos φ = Jl (kr)eil(φ+π/2) = J|l| (kr)ei|l|π/2 eilφ (8.74) l

l

with the asymptotic expression at large r   π  i|l|π/2 ilφ 2  π ik·r e e = cos kr − |l| − e π kr 2 4 l   1  iπ(|l|+1/4)  −ikr e = e + eikr−iπ(|l|+1/2) eilφ . 2π kr l

(8.75)

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Mutual friction

In order to obtain the asymptotic expression ‘the incident plane wave + the scattered cylindrical wave’ for the Aharonov–Bohm solution, f (φ) ψ ∼ eik·r + √ eikr , r

(8.76)

similar to Eq. (8.53), one should determine the amplitudes ψl of partial waves in Eq. (8.71) from the condition that the incoming components ∝ e−ikr in the plane wave and in Eq. (8.71) coincide. This yields that  2  π cos[kr − (|l| + 1/2) + δl ]ei|l|π/2+iδl eilφ ψ= π kr 2 l   1  iπ(|l|+1/4)  −ikr e = e + eikr−iπ(|l|+1/2)+2iδl eilφ , (8.77) 2π kr l

where δl = (|l| − |l − γ |)

π 2

(8.78)

are the partial-wave phase shifts. Because of periodicity in γ it is enough to consider − 12 < γ < 12 . Then Eq. (8.78) yields δl = γ π/2 for l  0 and δl = −γ π/2 for l < 0 if γ > 0 and δl = γ π/2 for l > 0 and δl = −γ π/2 for l  0 if γ < 0. The scattering amplitude in Eq. (8.76) is given by  1  2iδl (e − 1)eilφ . (8.79) f (φ) = 2π ik l

Equation (8.21) for the transverse cross-section may be rewritten as an expansion in partial waves: π σ⊥ =

π σ (φ) sin φ dφ =

−π

|f (φ)|2 sin φ dφ −π

  1  i2δl−1  i2δl+1  i2(δl+1 −δl )  i2(δl−1 −δl ) = e − e + e − e . 2ik l

l

l

(8.80)

l

Shifting the number l by 2 in the first sum and by 1 in the fourth sum one obtains the expression for the transverse cross-section in the partial-wave method derived long ago by Cleary (1968): 1 sin(2δl − 2δl+1 ). (8.81) σ⊥ = k l

The transport cross-section is given by π σ =

|f (φ)|2 (1 − cos φ)dφ = −π

1 [1 − cos(2δl − 2δl−1 )] . k l

(8.82)

8.5 Partial-wave analysis and the Aharonov–Bohm effect

233

Using the phase shift values for the Aharonov–Bohm effect, Eq. (8.78), the transverse cross-section is 1 σ⊥ = − sin 2π γ . k

(8.83)

The shift of l in the first sum of Eq. (8.80) is not an innocent operation because of divergence of the first and second sums at l → ±∞. The derivation of Cleary’s formula (8.81) assumes that the first and the second divergent sums in Eq. (8.80) should cancel exactly after the shift of l. But if one does not shift l, the difference of the first and the second sums is finite. Moreover, this difference cancels the contribution of the third and the fourth sums, and σ⊥ vanishes in the first order with respect to the phase shift δl . Ambiguity in the calculation of the partial-wave sum for the transverse cross-section because of divergence at large l is dual to the small-scattering-angle problem in the configurational space. The way to avoid the ambiguity is similar to that in the configurational space: one should not use the concept of the scattering amplitude for calculation of the transverse force. Instead one can refer to the momentum balance. The momentum flux tensor for the electron Schr¨odinger equation (8.68) follows from the expression (1.184) for momentum flux tensor after neglecting the interaction term ∝ |ψ|4 and the gauge transformation introducing the vector potential A: ij =

+   1 e e ∗ ∗ Re −i h∇ ¯ i ψ − Ai ψ i h∇ ¯ j ψ − Aj ψ 2m c c  e  e  , ∗ −ψ −i h∇ ¯ i − Ai −i h∇ ¯ j − Aj ψ . c c

(8.84)

If the axis y is directed along the wave vector k of the incident wave, the transverse force is determined by the momentum flux tensor component xr , where r is the radial " coordinate in the cylindrical coordinate system r = x 2 + y 2 and φ = − arctan(x/y). Neglecting terms inversely proportional to r one obtains that xr = −

    ∗ ∂ψl  ∂ψl ∂ 2 ψl i(l−l  )φ h¯ 2   e , − ψl∗ Re sin φ 2m  ∂r ∂r ∂r 2 l

(8.85)

l

and the transverse force is  F⊥ = −

  ∗ 2 ∗ ∂ψl h¯ 2  ∂ψl+1 ∗ ∂ ψl − ψl+1 xr rdφ = π r Im . m ∂r ∂r ∂r 2

(8.86)

l

Inserting the partial-wave amplitudes given by Eq. (8.77) one obtains F⊥ = j vσ⊥ , where j = h¯ kn and v = h¯ k/m are the momentum density and the velocity in the incident plane wave, and σ⊥ is the transverse cross-section given by Eq. (8.81). Another derivation of the force on the Aharonov–Bohm flux tube using the wave-packet presentation was suggested by Shelankov (2002).

234

Mutual friction

Although we derived relations connecting the scattering amplitude and scattering crosssections with the scattering phase shifts for the particular case of Aharonov–Bohm scattering, they are universal for any axisymmetric potential scattering particles or quasiparticles. For quasiparticles scattered by the vortex, the scattering phase shifts δl are small, and according to Eq. (8.81) the transverse cross-section depends only on the phase shifts at l → ±∞: 2(δ−∞ − δ∞ ) , (8.87) k while dependence of δl on l is of no importance. Bearing in mind that in the classical limit the partial wave l corresponds to the semiclassical trajectory with impact parameter b = l/k = hl/p and the action variation along the trajectory with impact parameter b ¯ is connected with the phase shift by the relation δS(b) = 2hδ ¯ l (see Fig. 8.3), Eq. (8.87) derived from the partial-wave analysis reproduces the expression (8.23) with p = hk ¯ for the transverse cross-section obtained from geometric optics. One would expect that the linear in κ scattering amplitude (8.56) for phonon scattering obtained from perturbation theory in the configurational space should also follow from the partial-wave analysis of Aharonov–Bohm scattering assuming that γ = −κk/2π cs is small. But this expectation is not fully realised since two revisions of the analysis are needed. First, the partial wave l = 0 (S wave) requires special attention. A perturbation in Eq. (8.72) for the partial-wave amplitude l = 0 is quadratic in γ . Nevertheless, the perturbation yields the phase shift δ0 = ±γ π/2 which is linear in γ . Apparently, this is because of the singular character of the perturbation growing as 1/r 2 at r → 0. Anyway, in the perturbation theory of Section 8.3 the quadratic perturbations were totally ignored and correspondingly, for comparison with this theory, the l = 0 phase shift δ0 , which emerged from the quadratic perturbation, should be ignored also. Thus one must exclude the term l = 0 calculating the sum in Eq. (8.79) for small phase shifts:   1  ilφ k  κ −i π −i π4 δl e = −i e 4 signleilφ f (φ) = 2ie 2π k 2cs 2π l=0 l=0  sin φ k κ −i π . (8.88) = e 4 2cs 2π 1 − cos φ σ⊥ =

This agrees with the result of Fetter (1964), who also used the partial-wave expansion.4 But Eq. (8.88) still differs from Eq. (8.56) by the factor cos φ. As was already discussed in Section 8.3, the factor cos φ is absent if the vortex line is strongly pinned and does not participate in oscillation. Under this assumption, all partial waves must be regular at r → 0 being proportional to the Bessel functions Jl−γ (kr). But if the vortex line is free to move, the velocity field becomes singular at r → 0. One should remember that the vortex 4 In particular, Fetter analysed perturbations of the second order in κ in phonon scattering, which are different from those

following from the analogy with the Aharonov–Bohm effect. His conclusion was that they do not yield an essential contribution to δ0 , and the latter must really be ignored.

8.6 Transverse force and Berry phase

235

velocity v L = du/dt is a Lagrange variable for a point on a moving vortex line, while the fluid velocity v(r) is an Euler variable defined at a fixed point of space. Therefore at small r v(r) = v L − (u · ∇)v v (r) = v L − ∇[u · v v (r)] = v L + The phase, which determines the velocity v(r) =

κ 2π ∇θ (r)

1 ∇[v L · v v (r)]. iω

(8.89)

at small r, is

2π 2π (8.90) vL · r − [u · v v (r)], κ κ or in cylindrical coordinates with the azimuthal angle φ measured from the direction of v L , θ (r) =

θ (r, φ) ≈

2π 1 vL vL r cos φ − sin φ = θ1 eiφ + θ−1 e−iφ , κ iω r

(8.91)

where π 1 vL vL r ± . (8.92) κ 2ω r The partial cylindrical waves l = ±1, which provide singular terms ∝ 1/r at r → 0, are θ±1 =

θ±1 ∝ J1±γ (kr) + α±1 N1±γ (kr).

(8.93)

At small γ and r, J1±γ (kr) ≈ J1 (kr) ≈ kr/2 and N1±γ (kr) ≈ N1 (kr) ≈ 2/πkr. Equation (8.93) agrees with Eq. (8.92) if α± = ±κk/8cs . The asymptotic expression for the partial waves l = ±1 at large arguments is   π 2 π cos kr − |l − γ | − α± − . (8.94) θ±1 ∝ π kr 2 4 Thus α± are phase shifts produced by vortex motion, which must be added to the shifts δ±1 given by Eq. (8.78). This does not change the expression for the transverse force, but adds the lacking factor cos φ to the expression (8.88). After this, the scattering amplitude obtained from the partial wave expansion does not differ from that given by Eq. (8.56).

8.6 Transverse force and Berry phase in two-fluid hydrodynamics Up to now we analysed spatial scales much less than the mean free path lqp of quasiparticles (ballistic region). Quasiparticles interacted with the velocity field generated by a vortex, but their interaction was neglected. Now we shall see what is going on at scales much larger than lqp , where two-fluid hydrodynamics is valid. All relevant scales are shown in Fig 8.5. In accordance with Newton’s third law, the force on the vortex is accompanied by a force −F f r of opposite direction on the normal fluid. The force acts at distances much less than the quasiparticle mean free path lqp . At hydrodynamical scales the force on the normal fluid is a δ-function local force concentrated along the vortex line. In Section 1.9 we saw that a momentum, which is brought into a viscous fluid by a local force, is transported to large distances by a viscous momentum flux Ff r i = − τij dSj determined by the viscosity tensor τij . Viscous momentum transport also takes place in the normal component of the

236

Mutual friction

in

two-fluid-hydrodynamics region

er

t ia ls ub

us

g io

co

re

v is

ub

n

s re g io

n

ballistic region

Figure 8.5 Relevant scales for momentum transport produced by quasiparticle scattering: the scale of the interaction between the quasiparticle and the velocity field induced by the vortex (the wavelength λ in the case of phonons), the quasiparticle mean free path lqp , and the Oseen length rO . Figure from Sonin (2002).

superfluid (Section 8.1). This leads to a difference between the local normal velocity v nl close to the point where the force is applied and the average normal velocity v n given by Eq. (8.3). Momentum transport by a viscous momentum flux is relevant in the viscous subregion shown in Fig. 8.5 at distances from the vortex less than the Oseen length rO ∼ ν/|v n − v L |. At distances r > rO (inertial subregion in Fig. 8.5) non-linear inertial terms in the Navier–Stokes equation are more important than viscous terms everywhere except for the laminar wake behind the moving vortex (Section 1.9). The viscous momentum flux in the laminar wake is still able to transport the momentum related to the longitudinal component of the force ∝ D in Eq. (8.4). But transmission of the transverse component of the force ∝ D is impossible without the existence of circulation of the normal velocity as in a perfect fluid.5 Replacing in Eq. (1.123) the circulation K∞ by κn , v by v n , and the total density ρ by the normal density ρn and using Eq. (8.4), one obtains the relation connecting the normal circulation κn with the parameter of the transverse force D :  D (8.95) κn = dl · v n = − . ρn Neglecting the longitudinal component ∝ D in the ballistic region and using the relation D  = −κρn for the Iordanskii force in Eq. (8.5) one obtains that 5

Existence of normal circulation at large distances in the presence of the transverse force on the vortex was pointed out by Pitaevskii (unpublished).

8.6 Transverse force and Berry phase

κn =

1+



κ κ ln(rm /rl ) 4π ν

2 .

237

(8.96)

Now let us consider a connection between the transverse force and the Berry phase. We shall use the hydrodynamical description with the Lagrangian obtained by the Madelung transformation of Eq. (1.179): κ 2ρ 2 V 2 κρ ∂θ − ∇θ − ρ . (8.97) 2π ∂t 8π 2 The first term with the time derivative of the phase θ (Wess–Zumino term) is responsible for the Berry phase " = SB /h, ¯ which is the variation of the phase of the quantum mechanical wave function resulting from adiabatic transport of the vortex round a closed loop (Berry, 1984). In a two-dimensional system with the density ρ being the mass per unit area,  κρ (v L · ∇ L )θ (8.98) SB = dr dt 2π L=−

is the classical action variation around the loop and ∇ L θ is the gradient of the phase θ [r − u(t)] with respect to the vortex position vector u(t). The gradient ∇ L θ = −∇θ differs by sign from the gradient ∇θ with respect to r. The integral over the loop yields the circulation of the total current j = (κ/2π )ρ∇θ for points inside the loop, but vanishes for points outside. As a result, the Berry phase action is given by (Geller et al., 1998)  κ (dl · j ), (8.99) SB = −A 2π where A is the loop area. Contrary to Eq. (8.98), the loop integral in Eq. (8.99) is related to the variation of the position vector r, the vortex position vector u being fixed. Since the Berry phase is proportional to the current circulation, which determines the transverse force, there is a direct connection between the Berry phase and the amplitude of the transverse force on a vortex (Thouless et al., 1996). If the circulation of the normal velocity at large distances vanished [as was assumed by Ao and Thouless (1993); Thouless  et al. (1996); Geller et al. (1998)], the current circulation would be (dl · j ) = ρs κ, and the Berry phase (as well as the transverse force) would be proportional to ρs . On this ground Ao and Thouless (1993) ruled out any contribution to the transverse force from quasiparticles, including the Iordanskii force and the Kopnin–Kravtsov force in Fermi superfluids, which will be considered later in Section 9.7. However, on determining the Berry phase, one should choose a loop radius much larger than any relevant scale including the Oseen length rO . So one should consider the total current circulation in the inertial subregion in Fig. 8.5. We have seen that in this subregion the momentum flux connected with the transverse force is impossible without normal circulation κn , which is given by Eq. (8.95). Therefore the  total current circulation (dl · j ) = ρs κ + ρn κn contains the normal circulation. Then the Berry phase yields the same transverse force as that determined from the momentum balance. The presence of normal circulation in the Berry phase, which corrects their original conclusion, was later confirmed by the detailed analysis of Thouless et al. (2001).

238

Mutual friction

The normal circulation κn is not a topological charge and depends on details of interaction of quasiparticles with a vortex. So the Berry phase analysis itself cannot provide the value for normal circulation without analysis of processes at small distances from the vortex. Topology is not sufficient to determine of the transverse force on the vortex. The magnitude of the transverse force must be determined not from the Berry phase, but vice versa: calculation of the transverse force from the momentum balance is necessary for determination of the Berry phase. We shall return to the role of topology in vortex dynamics later, in Section 13.5, addressing the Magnus force and the Hall effect in lattice superfluids.

8.7 Mutual friction near the critical point Experiments by Matheiu et al. (1976) showed that when T approaches the critical temperature Tc , the mutual friction parameters B ∼ α and B  ∼ α  diverge as (Tc − T )−1/3 . Such critical behaviour was explained by Pitaevskii (1977) on the basis of the dynamic scaling hypothesis, supposing that relaxation of the order parameter modulus was the principle energy-dissipation mechanism. He found the same critical exponent as in experiment, but no conclusions concerning the magnitudes of B and B  were drawn. Approaching the critical point, the vortex core radius becomes very large, and some phenomenological theory like the time-dependent Ginzburg–Landau theory must be applicable. The equation of vortex motion in type II superconductors was derived from the time-dependent Ginzburg–Landau theory by Gor’kov and Kopnin (1975). In Section 6.9 we discussed the mean-field Ginzburg–Pitaevskii theory and its extension, which took into account critical fluctuations. Now we use it for derivation of the equation of vortex motion in neutral superfluid (Sonin, 1981) using a method similar to that of Gor’kov and Kopnin (1975). Let us start from the general phenomenological equation (6.140) where μ˜ s0 is given by Eq. (6.139). At steady vortex motion with constant velocity v L , the time derivatives are expressed through gradients of hydrodynamical variables: ∂/∂t = −v L · ∇. The parameter  diverges on approaching the critical point. At  1 the Ginzburg–Pitaevskii equation (6.140) becomes     2  2 1 i h¯ h¯ vn (v L − v n ) · ∇ψ + i − μ˜ 0 ψ =  ∇ + v n ψ + μ˜ s0 ψ . (8.100) m 2 2 m Performing the Madelung transformation and separating the real and the imaginary parts one obtains two equations:   (v s − v n )2 2m h¯ 2 ∇ 2 a (v L − v n ) · ∇ρs = + ρs μ˜ s0 − , (8.101) 2 h¯ 2m2 a h v2 h¯ ¯ (v L − v n ) · ∇θ − μ˜ 0 + n = − ∇ · [ρs (v s − v n )]. m 2 2mρs

(8.102)

8.7 Mutual friction near the critical point

239

Equations (6.141) and (6.142) for the superfluid density and the phase must be supplemented with the continuity equations for the mass and the entropy: ∂ρ + ∇ · (ρs v s + ρn v n ) ≈ −(v L − v n ) · ∇ρ + ∇ · [ρs (v s − v n )] + ρ∇ · v n = 0, ∂t (8.103) ∂S + ∇ · (Sv n ) ≈ −(v L − v n ) · ∇S + S∇ · v n = 0. (8.104) ∂t Excluding ∇ · v n from the continuity equations (8.103) and (8.104) one obtains the relation   ρ (v L − v n ) · ∇ρ − ∇S = ∇ · [ρs (v s − v n )]. (8.105) S Comparison of Eqs. (8.102) and (8.105) yields that   h¯  v2 ρ h¯ (v L − v n ) · ∇θ − μ˜ 0 + n = − (8.106) (v L − v n ) · ∇ρ − ∇S . m 2 2mρs S Now we return back to one complex Schr¨odinger equation for the wave function ψ = aeiθ , this time uniting two real equations (8.101) and (8.105):    2   ρ 1 i h¯ h¯ ∇S − ∇ρ ψ =  ∇ + v n ψ + μ˜ s0 ψ . (v L − v n ) · ∇ρs + i 2mρs S 2 m (8.107) In the new Schr¨odinger equation the left-hand side of Eq. (8.102) is replaced by the expression via gradients of ρ and S given by Eq. (8.106). At low vortex velocities one can linearise the right-hand side of Eq. (8.107) with respect to a small correction ψ  = ψ − ψ0 produced by vortex motion, where ψ0 = a0 (r)eiϕ is a solution of the equation for a resting single-quantum vortex when the left-hand side of the equation vanishes. The linearisation procedure must take into account that the chemical potential μ˜ 0 and the partial superfluid chemical potential μ˜ s0 also have corrections connected with vortex motion and proportional to the relative vortex velocity v L − v n . Bearing in mind the critical behaviour of the relaxation parameter  ∼ (Tc − T )−1/3 and of the core radius (coherence length) rc ∼ (Tc − T )−2/3 , and keeping only the leading divergent terms, one can neglect pressure and density gradients. The theory of mutual friction in this approximation was developed in the adiabatic regime when the entropy is conserved (Sonin, 1981). Neglecting pressure variation and other terms proportional to ρs , the Gibbs–Duhem relation (6.129) connects the chemical potential variation μ˜ 0 only with the temperature variation T  : S μ˜ 0 ≈ − T  . ρ

(8.108)

One can determine T  from Eq. (8.106): S  hρ ¯ T = −(v L − v n ) · v s + (v L − v n ) · ∇S. ρ 2mSρs

(8.109)

240

Mutual friction

Then the variation of the partial superfluid chemical potential μ˜ s0 is   hρ ∂ μ˜ s0  ∂S ρ ¯  T =− vs − ∇S · (v L − v n ). μ˜ s0 (T ) = ∂T ∂ρs S 2mSρs

(8.110)

Eventually linearisation of the non-linear Schr¨odinger equation (8.107) yields:    ∂S 2mρρs 2 ρ 2 ρ ∇S + i ∇S vv − (v L − v n ) · ∇ρs − hS ∂ρs S S2 ¯    2 ∂ μ ˜ 2mρs 1 i h¯ s0 ψ0 (ψ0∗ ψ  + ψ0 ψ ∗ ) . ∇ + v n ψ  + μ˜ s0 ψ  + = 2 m ∂ρs h¯ 

(8.111)

Equation (8.111) is a non-uniform linear differential equation for ψ  . Such an equation has a solution only if the non-uniform term [the left-hand side of Eq. (8.111)] is orthogonal to the solution of the corresponding uniform equation (i.e., with zero left-hand side). As a consequence of translational invariance, a proper solution of the uniform equation is a variation of the equilibrium wave function produced by an arbitrary translation t of the vortex line:   ∇a0 m + i v v ψ0 . (8.112) ψt = (t · ∇)ψ0 = t · a0 h¯ The solvability condition, which is also called the orthogonality condition or the condition of the absence of the secular terms, is obtained by multiplication of both sides of Eq. (8.111) by ψt∗ , integration over the whole space and taking the real part of the obtained equation (Gor’kov and Kopnin, 1975). The differential linear operator acting on ψ  on the right-hand side is self-conjugated. Since ψt is an eigenfunction of this operator with zero eigenvalue, integration by parts reduces the integral over the right-hand side to an integral over a distant surface. So the solvability condition is 

    ∂S mρ h¯ ρ 2 h¯ vv − ∇ρs − ∇S dr (t · ∇ρs )(v L − v n ) · 4ρs ∂ρs 2S 4ρs S 2   mρ (t · v v )[(v L − v n ) · ∇S] = Re (ψt∗ ∇ψ  − ψ  ∇ψt∗ ) · dS. + 2S

(8.113)

The value of ψ  at large distance r → ∞ is known: m [(v sl − v n ) · r]a0 (∞)eiϕ , h¯

(8.114)

  m m2 ∇ψ = i (v sl − v n ) − 2 [(v sl − v n ) · r]v v a0 (∞)eiϕ . h¯ h¯

(8.115)

m

ψ  = ψ0 (∞)ei h¯ (v sl −v n )·r − ψ0 (∞) ≈ i its gradient being 

Here v sl is a transport superfluid velocity at the location of the vortex and ψ0 (∞) = a0 (∞)eiϕ is the wave function far from the vortex. Taking into account that Eq. (8.113)

8.8 Comparison with experiments and other theories

241

must be satisfied for arbitrary translation t, and performing integration by parts on the right-hand side, one obtains the equation (6.38) of vortex motion with the mutual friction parameters       ∞ ∂ρs (r) 2 r dr ρs ∂S(r) 2 1 + d= , 4ρs 0 ρs (r) ∂r S ∂r (8.116) ρ[S − S(0)] C Tc − T . d − 1 = =− ρs S ρs S Tc In Eq. (8.116) S and ρs are values of S(r) and ρs (r) far from the vortex line at r → ∞, and C is the discontinuity of the specific heat at the critical temperature. The expression for the dissipative parameter d consists of two terms. The first term determined by the gradient of ρs originates from relaxation of the superfluid order parameter modulus described by the phenomenological equation (6.141). Just this process was considered by Pitaevskii (1977). The second term in d is determined by the gradient of the entropy S and is connected with relaxation of the order parameter phase (superfluid velocity) characterised by the second viscosity coefficient ζ3 [see Eq. (6.132)]. The parameter 1 − d  related with the transverse force does not require any knowledge of details of the vortex core structure like in the case of mutual friction originating from scattering of non-interacting quasiparticles at low temperatures. It is fully determined by the entropy difference in the normal and the superfluid states. This demonstrates again that the transverse force cannot be found from pure topology (cf. Section 8.6). For our derivation it is important that the chemical potential μ˜ 0 varies in space, and therefore should not be ignored [see Section 6.9 after Eq. (6.146)]. Vortex motion contributes both to the phase variation ∂θ/∂t = −(v L · ∇)θ and to the chemical potential μ˜ 0 . Each of these two contributions decays at large distances as 1/r. Keeping only the first of them, one obtains the logarithmically divergent mutual friction parameter (Pismen, 1999). But their sum according to Eq. (8.106) decays faster, as 1/r 2 , and there is no logarithmic divergence. This statement is rather general and is not restricted to our adiabatic case when chemical potential variation reduces to temperature variation. One can consider a hypothetical case when thermal conductivity is strong enough for realisation of the isothermal case. Then the chemical potential variation reduces to pressure variation, which also eliminates logarithmic divergence in mutual friction parameters.6

8.8 Comparison with experiments and other theories The experimental results concerning mutual friction in superfluid 4 He have been exhaustively reviewed by Barenghi et al. (1983). In Fig. 8.6 the experimental temperature dependence of the transverse force parameter D  is reproduced from Fig. 9.A of Barenghi et al.

6 In superconductors, vortex motion produces an electric field (Kopnin, 2001), and the electric scalar potential plays the same

role as the chemical potential in neutral superfluids.

242

Mutual friction 120

D (106 g cm–1sec–1)

100 80 60 40 20 0 –20 –40 1.2

1.4

1.6 1.8 Temperature (K)

2.0

2.2

Figure 8.6 Dependence of the mutual friction parameter D  on the temperature in superfluid 4 He. The solid line was obtained by fitting the experimental data. Figure from Barenghi et al. (1983) with the added dashed line showing the theoretical value D  = −ρn κ.

(1983).7 The theoretical values D  = −κρn obtained for low temperatures are shown by the dashed line. They are 2–3 times smaller than the experimental values at the lowest temperatures for which comparison is possible. Probably the agreement would be better at much lower temperatures. The disagreement between theory and experiment for the longitudinal force parameter is also considerable (Barenghi et al., 1983). This has been attributed to the contribution of the core to mutual friction. It is difficult to estimate this contribution rigorously because of the lack of a reliable theory to describe the vortex core in superfluid 4 He (excepting for a narrow critical region discussed in the previous section). Various speculative models describing the core contribution to mutual friction have been proposed (Barenghi et al., 1983). The observed longitudinal force is explained if the core absorbs all rotons falling on it and the collision diameter of the core coincides with its diameter. The concept of the absorbing core was suggested by Lifshitz and Pitaevskii (1957) and was investigated by Hillel and Vinen (1983), who found that the core radius, deduced from experiment on the basis of this concept, is in reasonable agreement with some models of core structure. According to the experiments of Matheiu and Simon (1980) the mutual friction parameters close to the critical temperature are described by the expressions     Tc − T 1/3 Tc − T 1/3  , d − 1 = −2 . (8.117) d = 2.8 Tc Tc

7 D  corresponds to D of Barenghi et al. (1983). t

8.8 Comparison with experiments and other theories

243

Meanwhile, a rough estimation using the theoretical expressions in Eq. (8.116) yielded (Sonin, 1981)     Tc − T 1/3 Tc − T 1/3 , d  − 1 ≈ −1.5 . (8.118) d≈ Tc Tc The estimation used the following values of parameters of the phenomenological theory (Ginzburg and Sobyanin, 1976, 1982):   Tc − T −1/3 7 −3 −1  = 0.3 , C = 0.76 × 10 erg cm deg , Tc (8.119) ρs = 0.21 g cm−3 deg2/3 (Tc − T )−2/3 , Theoretical and experimental values agree better for d  − 1 than for d. Apparently this is because, according to Eq. (8.116), in contrast to d calculation of d  − 1 does not require knowledge of the relaxation parameter  and the spatial distribution of S and ρs inside the core. The theory of mutual friction near the critical point has also been developed by Onuki (1983). He used the phenomenological theory, which is a generalisation of the model F of Hohenberg and Halperin (1977). This theory is equivalent to the  theory of Ginzburg and Sobyanin (1976, 1982) except that Onuki assumed the relaxation parameter  to be complex. In addition, Onuki took into account the thermal conductivity in the continuity equation (8.104) for entropy, which had been ignored by Sonin (1981). Without the thermal conductivity corrections and for real , the resultant formulas of Onuki agree with the expressions for d and d  − 1 in Eq. (8.116). Onuki calculated mutual friction parameters numerically and obtained d = 0.62−1 ,

d  − 1 = −0.58−1 .

(8.120)

After substitution of the value  from Eq. (8.119) this agrees better with the experiment than d and d  − 1 in Eq. (8.118). The corrections due to thermal conductivity are not large, according to Onuki, so the numerical difference between values of d obtained by Sonin (1981) and by Onuki (1983) probably results from a more careful numerical calculation of the distribution of S and ρs in the vortex core by Onuki. In all, the theory provides a satisfactory qualitative explanation of experimental data in the critical region, in particular, the negative sign of B  . For the intermediate temperature range 1.7–2.1 K, just where the theories discussed here are not satisfactory, Matheiu and Simon (1980) suggested a model of mutual friction, which led to the conclusion that the vortex moves with local normal velocity. Then the mutual friction parameters are given by Eq. (8.8). Arguing for motion of vortices together with the normal component, Mathieu and Simon relied on not quite rigorous conjectures, but their model was in rather good agreement with experiment.

9 Mutual friction and vortex mass in Fermi superfluids

9.1 Bardeen–Cooper–Schrieffer theory and Bogolyubov–de Gennes equations In Fermi superfluids, calculation of the forces on a vortex from quasiparticle scattering and of the vortex mass have some peculiarities, which are absent in Bose superfluids. The present chapter addresses them. The analysis must start from a microscopic description of quasiparticles in Fermi superfluids using the Bardeen–Cooper–Schrieffer (BCS) theory. We shall summarise the features of the BCS theory, but only those which are relevant for our goals. For a more thorough description of the BCS theory one can refer to numerous textbooks on superconductivity (for example, de Gennes, 1966; Saint-James et al., 1969; Tinkham, 1975; Kopnin, 2001). The wave function of quasiparticles in the BCS theory is a spinor with two components,  ψ(R, t) =

u(R, t) v(R, t)

 .

(9.1)

The wave function describes a quantum state, which is a superposition of a state with one particle (upper component u) and a state with one antiparticle, or hole (lower component v). So the number of particles (charge) is not a quantum number of the state. The spinor components are determined from the Bogolyubov–de Gennes equations (de Gennes, 1966):  h¯ 2  2 ∂u =− ∇ + kF2 u + eiθ v, ∂t 2m  2 h¯  2 ∂v i h¯ = ∇ + kF2 v + e−iθ u. ∂t 2m

i h¯

(9.2) (9.3)

Here kF is the Fermi wave number, and the order parameter, or gap, eiθ can vary in space and time. The equations correspond to the Hamiltonian with the density H= 244

h¯ 2 h¯ 2 (|∇u|2 − kF2 |u|2 ) − (|∇v|2 − kF2 |v|2 ) + eiθ u∗ v + e−iθ v ∗ u. 2m 2m

(9.4)

9.1 BCS theory and Bogolyubov–de Gennes equations

245

In a resting uniform superfluid the modulus  and the phase θ of the order parameter are constant, and solutions of the Bogolyubov–de Gennes equations are plane waves   u0 (9.5) eik·R−iε0 t/h¯ , v0 eiθ−iε0 /h¯ where 

u0 v0



⎛   1 ⎜ 2 1+ ⎜  =⎝  1 2 1−

ξ ε0 ξ ε0

 ⎞ ⎟  ⎟ ⎠.

The quasiparticle energy is given by the well known BCS expression " ε0 = ± ξ 2 + 2 .

(9.6)

(9.7)

Here ξ = (h¯ 2 /2m)(k 2 − kF2 ) ≈ hv ¯ F (k − kF ) is the quasiparticle energy in the normal Fermi liquid and vF = kF /m is the Fermi velocity. The two wave numbers # $ 2m k± = kF2 ± 2 ε02 − 2 (9.8) h¯ correspond to the particle-like (+) and the hole-like (−) branches of the spectrum. As in the case of the Schr¨odinger equation, there is a continuity equation for the probability |u|2 + |v|2 of finding the quasiparticle in some point in space: ∂(|u|2 + |v|2 ) = −∇ · g qp , dt

(9.9)

where i h¯ ∗ i h¯ ∗ (u ∇u − u∇u∗ ) + (v ∇v − v∇v ∗ ) (9.10) 2m 2m is the probability flux. Equation (9.9) is a manifestation of the conservation law for the number of quasiparticles. But the number of quasiparticles is not the same as the number of particles (charge). The Hamiltonian of Eq. (9.4) is not gauge invariant and there is no conservation law for the particle number. The Bogolyubov–de Gennes equations lead to the following balance equation for the particle density (|u|2 − |v|2 ): g qp = −

1 2i ∂(|u|2 − |v|2 ) + ∇ · j = (e−iθ v ∗ u − eiθ vu∗ ), dt m h¯

(9.11)

where i h¯ ∗ i h¯ (u ∇u − u∇u∗ ) − (v ∗ ∇v − v∇v ∗ ) (9.12) 2 2 is the mass current. Equation (9.11) contains a source on the right-hand side related to possible changes of the total particle number. Globally the number of particles is of course a conserved quantity. The source in the balance equation for the particle number corresponds to conversion of the superfluid part of the fluid to normal fluid and vice versa in j =−

246

Mutual friction and vortex mass in Fermi superfluids

inhomogeneous states. In order to restore the global conservation law one should solve the Bogolyubov–de Gennes equations together with the self-consistency equation for the order parameter proportional to the gap. This property of the Bogolyubov–de Gennes equations is well known in the theory of superconductivity (Aronov et al., 1981; Blonder et al., 1982). We do not use the self-consistency equation further in this book, assuming that the gap  is already known, and refer the reader to textbooks on superconductivity for derivation and discussion of this equation. Now let us consider a quasiparticle eigenstate in a moving superfluid. The superfluid κc ∇θ is determined by the order parameter phase gradient, where κc = velocity v s = 2π h/2m is the circulation quantum for the Cooper-pair condensate taking into account that the mass of the Cooper pair is 2m. Assuming that the absolute value of the gap  and the gradient of the phase do not vary in space, the solution of the Bogolyubov–de Gennes equations is     u u0 ei(k+∇θ1 )·R e−iεt/h¯ . (9.13) = v0 ei(k−∇θ2 )·R v Here we introduced separate phases θ1 and θ2 for the two spinor components. Their sum determines the order parameter phase θ = θ1 + θ2 . Evidently the quasiparticle wave function (9.13) is not an eigenfunction of the momentum operator since the wave vectors k + ∇θ1 and k − ∇θ2 of the two spinor components are different. The spinor (9.13) corresponds to the energy (neglecting terms of the second order in phase gradients) hκ ∂ε0 ∇θ1 − ∇θ2 ¯ c k · ∇θ + · 2π  ∂k 2  ξ κc (∇θ1 − ∇θ2 ) . = ε0 (k) + h¯ k · v s + ε0 2π

ε = ε0 (k) +

(9.14)

It looks as if the phase difference θ1 − θ2 will be of no importance since it can be removed by redefinition of the wave vector k. Choosing θ1 = θ2 one obtains the expression for the quasiparticle energy following from the Galilean invariance and well known from textbooks on superconductivity (de Gennes, 1966): ε = ε0 + hk ¯ · v s . For such a choice of phases one defines the wave vector k as an average of wave vectors of the two spinor components. But in Section 9.2 we shall see that in the analysis of quasiparticle scattering by a vortex, cyclic boundary conditions for spinor components at a closed path around the vortex require another choice: either θ1 = 0 or θ2 = 0. This means that the wave vector k is defined as a wave vector of either the particle or the hole component of the spinor wave function. For the choice θ1 = θ2 , Eq. (9.12) yields the following expression for the mass current in the plane-wave state: 2 2 j = hk ¯ + m(|u0 | − |v0 | )v s .

(9.15)

So the superfluid velocity contribution to the mass current is proportional to the charge |u0 |2 − |v0 |2 of the quasiparticle.

9.2 Friction from scattering of free BCS quasiparticles

247

9.2 Mutual friction from scattering of free Bardeen–Cooper–Schrieffer quasiparticles by a vortex The mutual friction force from scattering of continuum BCS quasiparticles by a vortex was calculated for pure type II superconductors by Gal’perin and Sonin (1976) and by Kopnin and Kravtsov (1976b). These calculations were based on the partial-wave expansion and on the Boltzmann equation, which can be derived from microscopic theory (Aronov et al., 1981; Kopnin, 2001). Since the BCS theory also describes superfluid 3 He and the effect of the magnetic field is insignificant for mutual friction in type II superconductors, the aforementioned calculations are also relevant for singular vortices in neutral superfluids, like superfluid 3 He or Fermi superfluids of cold atoms. When the energy of the quasiparticles is close to the energy gap of the superconductor (ξ  ), the BCS quasiparticle spectrum ε0 ≈  + vF2 h¯ 2 (k − kF )2 /2 [see Eq. (9.7)] is identical to the roton spectrum (8.27) with the roton minimum momentum p0 replaced by the Fermi momentum hk ¯ F /m ¯ F and the roton mass μ replaced by /vF2 , where vF = hk is the Fermi velocity. So one can extend geometric optics for rotons to BCS quasiparticles, replacing the circulation quantum κ = h/m by the circulation quantum κc = h/2m for the Cooper-pair condensate. The group velocity for the BCS quasiparticles close to the Fermi surface is vG = vF ξ/. This yields the width b∗ of the Andreev shadow and the transverse cross-section for BCS quasiparticles propagating in the xy plane: b∗ =

 kF kF κc = , π vG |k − kF | 2εF (k − kF )2

σ⊥ =

1 κc π , = vG 2εF |k − kF |

(9.16)

where εF = h¯ 2 kF2 /2m is the Fermi energy. In a similar manner one can obtain the transport cross-section for BCS quasiparticles, but for BCS quasiparticles one should modify the lower cut-off b0 in the logarithmically divergent integral determining the transport crosssection in Eq. (8.36). Now the cut-off impact parameter b0 must be of the order of the coherence length ξ0 ∼ h¯ vF /, which determines the vortex core radius in the BCS theory. At this impact parameter Eq. (8.35) yields the scattering angle (after replacing roton parameters by BCS parameters), which is rather small: # # b∗  kF k − kF b ∗ . (9.17) ln ∼ ln φ∼ kF b0 b0 εF εF (k − kF ) Then the transport cross-section for BCS quasiparticles is   3 3 kF kF κc  k − kF  1 ln ln σ ≈ = . 3π vG kF εF |k − kF | 6εF kF εF |k − kF |

(9.18)

Repeating the further derivation of the mutual friction parameters done in Section 8.2 for rotons one obtains √    3 T κc ρn ln , D  = −κc ρn . (9.19) D= T 16(2π )3/2 εF

248

Mutual friction and vortex mass in Fermi superfluids

Despite nearly 180% reflection of quasiparticles from the area of the Andreev shadow of the width b∗ shown in Fig. 8.1 for rotons, the effective cross-sections σ and σ⊥ for BCS quasiparticles scattered by vortices are much smaller than b∗ . This is because Andreev reflection does not change quasiparticle momenta essentially. However, 180% Andreev reflection of quasiparticles is important for description of zero-temperature superfluid turbulence when the Andreev shadow width b∗ becomes a relevant scale (Barenghi et al., 2009; Fisher and Pickett, 2009; Suramlishvili et al., 2012). If the quasiparticle energy is much larger than the superconducting gap, the group velocity vG approaches the Fermi velocity vF and geometric optics yields the transverse crosssection κc /vF . This result does not look reasonable, because the cross-section, although small, still does not vanish in the limit  → 0. Indeed, the partial-wave calculations (Gal’perin and Sonin, 1976; Kopnin and Kravtsov, 1976b) yielded that in the limit of small /ξ the transverse cross-section differed from the geometric optics result of Eq. (8.26) by a small factor 2 /2ξ 2 . This also followed from solution of the Bogolyubov–de Gennes equations in the Born approximation (Sonin, 1997) as shown below. We use the perturbation theory with respect to the gap  and the gradient of the orderparameter phase ∇θv , which determines the superfluid velocity v v = (κc /2π )∇θv induced by the vortex. In the zero-order approximation u = u0 eik·r and v = 0. In the first-order approximation the second Bogolyubov–de Gennes equation (9.3) yields  v = e

−iθv

 h¯ 2 k · ∇θv 1 + u0 eik·r . 2ξ(k) 4m ξ(k)2

(9.20)

The first term in brackets is a correction to the quasiparticle energy ∝ 2 , which does not contribute to scattering determined by the order parameter phase gradients. So we keep only the second term proportional to ∇θv . Inserting it in the first Bogolyubov–de Gennes equation (9.2) one obtains the equation for the first-order correction to the quasiparticle amplitude u: (∇ 2 + k 2 )u(1) = (k · ∇θv )

2 u0 eik·r . 2ξ 2

(9.21)

This equation is similar to equation (8.50) for the sound wave. Using this analogy with phonon scattering, one obtains at small scattering angles φ the differential cross-section of the order σ (φ) ∼ 4 /ξ 4 kF φ 2 . Calculating the transport cross-section σ with the help of Eq. (8.20) one must take into account that for trajectories outside the vortex core, i.e., with impact parameters more than the core radius rc ∼ ξ0 , the scattering angle φ does not exceed the very small value /εF [see Eq. (9.17)], which must replace π in the limits of the integral over φ in Eq. (8.20). Then the transport cross-section is of the order of σ ∼

5 . εF ξ 4 kF

(9.22)

9.2 Friction from scattering of free BCS quasiparticles

249

Also using the analogy with phonon scattering, one easily obtains the transverse crosssection, σ⊥ =

2 π 2 κ c = , 2ξ 2 kF 2ξ 2 vF

(9.23)

which is smaller by the factor 2 /2ξ 2 than the value obtained from geometric optics. The question arises where the geometric optics went wrong. A source of inaccuracy is ambiguity in the definition of the canonical momentum h¯ k, which is not an exact momentum of the quasiparticle (Sonin, 2013b). This was discussed at the end of Section 9.1 in connection with Eqs. (9.13) and (9.14). Up to now in geometric optics the wave vector k has been defined choosing θ1 = θ2 = θ/2 in Eqs. (9.13) and (9.14). However, this choice violates the cyclic boundary conditions at a closed path around the vortex line. Let us move the spinor given by Eq. (9.13) along this path. After closing the path the shift of the phase θ is 2π , but the shifts of the phases θ1 and θ2 are equal to π and the spinor components are not single-valued. The cyclic boundary conditions for the spinor components u and v are satisfied only if either θ1 or θ2 vanishes. According to Eq. (9.14) this modifies the expression for the quasiparticle energy in the vortex velocity field: ε = ε0 (k) + (hk ¯ ± mv G ) · v v .

(9.24)

Then the value of p in Eq. (8.25) must be replaced by h¯ k ± mvG . Choosing − for quasiparticles and + for quasiholes (this is dictated by a physically reasonable condition that the transverse cross-section vanishes far from the Fermi surface) and approximating k by the Fermi wave number kF , one obtains the action variation along the whole trajectory:     κc k κc h¯ k 1 1 π = −signb h¯ . (9.25) δS(b) = −signb − − 2 vG vF 2vG 2 Then Eq. (8.23) yields the transverse cross-section (Gal’perin and Sonin, 1976; Kopnin and Kravtsov, 1976b) ⎛ ⎞ σ⊥ =

ε0 κc κc κc ⎜ ⎟ − = − 1⎠ . ⎝$ vG vF vF 2 2 ε0 − 

(9.26)

In the limit ε0  this agrees with the expression (9.23) obtained from perturbation theory with respect to . A more rigorous partial-wave analysis in Section 9.4 confirms Eq. (9.26). This reveals a shortcoming of the naive geometric optics analysis: it ignored peculiarities of the Aharonov–Bohm effect for BCS quasiparticles and used an improper definition for the quasiparticle phase shift along the trajectory. The high energy correction to the transverse cross-section leads to a correction to the transverse mutual friction parameter D  given by Eq. (9.19): D  = −κc ρn + 2κc ρf0 (),

(9.27)

250

Mutual friction and vortex mass in Fermi superfluids

where f0 (ε) =

1 eε/T

(9.28)

+1

is the equilibrium Fermi distribution function of quasiparticles. Close to the critical temperature where   T , the normal density ρn differs from the total density ρ by terms quadratic in , and Eq. (9.27) yields D  ≈ −κc ρ

 . 2T

(9.29)

Using the transport differential cross-section (9.22) close to the critical temperature one can estimate the dissipative mutual friction parameter D as D ∼ κc ρ

2 . εF T

(9.30)

The values of D obtained both at low temperatures and close to the critical temperature are too low compared to those deduced from experimental observations in type II superconductors (Kopnin, 2001) or in superfluid 3 He (Bevan et al., 1997b). One possible reason is that our analysis did not take into account scattering inside the vortex core. Moreover, in Fermi superfluids it is not sufficient to consider only scattering of free (delocalised) quasiparticles by the vortex. An essential contribution to mutual friction comes from bound quasiparticles localised inside the vortex core. This is a topic of Section 9.7.

9.3 Semiclassical theory of partial waves versus geometric optics: accuracy Revision of the geometric optics analysis in the previous section was based on a correction π/2 [the second term in Eq. (9.26)] to the quantum mechanical phase δS(b)/h. ¯ A question arises whether the semiclassical analysis, which deals with large phase variations, is able to estimate such corrections accurately. Therefore before doing the partial-wave analysis of scattering by a vortex it is useful to evaluate the accuracy of the semiclassical theory of partial waves (Landau and Lifshitz, 1982a). This can be done for a trivial case without any real scattering: a plane wave propagating in a uniform state. The partial-wave expansion for a two-component spinor in the polar coordinates r, φ (we consider scattering in the xy plane) is u=

 l

ul eilφ ,

v=



vl eilφ .

(9.31)

l

Generalising the partial-wave expansion (8.75) of a plane wave for a single-component wave function, one comes to the partial-wave expansion for a plane-wave solution of the

9.3 Semiclassical theory versus geometric optics

Bogolyubov–de Gennes equations given by (9.5) and (9.6):     π 2 ul u0 = cos[kr − (|l| + 1/2)]ei|l|π/2 eilφ−iεt/h¯ vl v0 π kr 2    1 iπ(|l|+1/4)  −ikr u0 e = + eikr−iπ(|l|+1/2) eilφ−iεt/h¯ . e v0 2π kr

251

(9.32)

The difference between the phases of the outgoing wave ∼ eikr and the incoming wave ∼ e−ikr at large r is   1 0,l = 2kr − π |l| + . (9.33) 2 Let us calculate the same phase difference using the semiclassical approximation for partial √ √ waves. It is convenient to rescale the partial-wave amplitudes: ul = Ul / r, vl = Vl / r. The Bogolyubov–de Gennes equations for new amplitudes do not contain first derivatives:     h¯ 2 kF2 h¯ 2 d 2 Ul l 2 − 1/4 − Ul + Vl = ε + Vl , − 2m dr 2 2m r2 (9.34)     h¯ 2 kF2 h¯ 2 d 2 Vl l 2 − 1/4 − Vl + Ul = ε − Vl . 2m dr 2 2m r2 The semiclassical solution of these two coupled one-dimensional wave equations yields the following expression for the incoming and the outgoing partial waves: ⎛ #   ⎞ √ 

Ul Vl



⎜ 12 1 + ε ε− ⎜ 1 ⎜ #  ∼"  √ k± (r) ⎜ ⎝ 1 ε2 −2 2 1− ε 2

2

⎟ ⎟ ±i  r k (r)dr ± ⎟e , ⎟ ⎠

(9.35)

l 2 − 1/4 2m " 2 l 2 − 1/4 2 = k2 − ± ε −  . ± h¯ r2 r2

(9.36)

where k± (r)2 = kF2 −

Here k± are the values of k± (r) at r → ∞. In the semiclassical approach the phase difference 0,l is the phase accumulated after propagation of the cylindrical wave from large r towards the centre point r = 0 down to the turning point, where it is reflected and returns back to the same r. The turning point is at r equal to " l 2 − 1/4 . (9.37) rt = k±

252

Mutual friction and vortex mass in Fermi superfluids

At this point the outgoing wave must be fitted with the incoming (incident) wave. For a particle-like branch (upper sign) this yields:  0,l = 2

# r rt

# = 2r

2 k+

2 − k+

l 2 − 1/4 π dr − 2 2 r

" $ l 2 − 1/4 π l 2 − 1/4 2 − − 2 l − 1/4 arccos − . k+ r 2 r2

(9.38)

Here the phase shift −π/2 is accumulated at reflection in the turning point and follows from the boundary condition in the turning point, where the semiclassical approach becomes invalid (Landau and Lifshitz, 1982a). At large r, Eq. (9.38) differs from the exact value in Eq. (9.33) only by terms of the order 1/ l, which are insignificant at large orbital numbers l. Thus at large l the correction π/2 is within the accuracy of the semiclassical theory of partial waves. Motion of a quasiparticle along a trajectory with an impact parameter b corresponds to a superposition of partial waves (wave packet) with orbital numbers l close to√the value l = k+ b. We assume that the trajectory is parallel to the axis y, so x = b and y = r 2 − b2 . Then the phase variation along the trajectory from y = −y∞ to y = y∞ must correspond to the phase shift of the lth partial wave ψl eilφ . The azimuthal angle φ = arctan yx = arccos br varies along the whole trajectory by π for positive b and by −π for negative b. Eventually, according to the partial-wave analysis of the plane wave, the whole phase variation along the trajectory with a large impact parameter is tr = 0,l + π |l| = 2k+ r −

π . 2

(9.39)

Naive geometric optics predicts that the phase shift along the trajectory between points y = −y∞ ≈ −r and y = y∞ ≈ r must be 2k+ r. So geometric optics loses the phase shift − π2 . This is exactly the phase shift accumulated by the wave at the reflection point of the partial cylindrical wave at y = 0.

9.4 Semiclassical partial-wave theory for scattering of free Bardeen–Cooper–Schrieffer quasiparticles by a vortex Now let us turn to the case of real scattering of quasiparticles by a vortex. In the cylindrical coordinates in the presence of a vortex, the phase of the order parameter eiθ around the vortex is θ = φ, and the partial-wave expansion for the wave function is u=

 l

ul eilφ ,

v=

 l

vl ei(l−1)φ .

(9.40)

9.4 Partial-wave theory for scattering of quasiparticles

253

√ √ For the scaled amplitudes Ul = ul r and Vl = vl r the Bogolyubov–de Gennes equations for the lth partial wave are:     h¯ 2 kF2 h¯ 2 d 2 Ul l 2 − 1/4 − − Ul + Vl = ε + Ul , 2m dr 2 2m r2   (9.41)   h¯ 2 kF2 (l − 1)2 − 1/4 h¯ 2 d 2 Vl − Vl + Ul = ε − Vl . 2m dr 2 2m r2 The semiclassical solution of the these equations is ⎛ -   $ . . ε2 −2 ⎜ /1 1 + l ⎜ 2 εl   1 ⎜ Ul ⎜ ∼√ ⎜  $ Vl k± ⎜ .  2 −2 ⎜ . ε l ⎝ /1 1 − 2

εl

⎞ ⎟ ⎟ ⎟ r ⎟ i k± (r)dr , ⎟e ⎟ ⎟ ⎠

(9.42)

where εl = ε −

h¯ 2 l − 1/2 2m r 2

(9.43)

and k± (r)2 = kF2 −

(l

− 1/2)2 r2

2 . 2m . h¯ 2 l − 1/2 / ± − 2 . ε− h¯ 2m r 2

(9.44)

In the semiclassical theory, a quasiparticle with the wave number k+ will either stay in the same branch of the spectrum with the same absolute value of the wave number k+ after reflection, or Andreev reflection will take place and the quasiparticle will be reflected as a hole with the wave number k− . The usual reflection occurs at the turning point determined by the condition k+ (rr ) = 0. At the Andreev reflection point r = r ∗ the radical in Eq. (9.44) must vanish, and # l − 1/2 . (9.45) r ∗ = h¯ 2m(ε − ) Bearing in mind that l ≈ bkF and the correspondence between the roton and the " BCS quasi∗ particle parameters at ε −   , the radius r coincides with the distance y ∗2 + b2 = √ ∗ b b of the roton Andreev reflection point from the vortex line, where y ∗ and b∗ are given by Eqs. (8.31) and (9.16) respectively. The type of reflection depends on which turning point is reached first: reflection is usual if rr > r ∗ , while Andreev reflection takes place at rr < r ∗ .

254

Mutual friction and vortex mass in Fermi superfluids

At large orbital numbers l, which correspond to large impact parameters, only usual reflection is possible, and one can expand the inner radical in Eq. (9.44) with respect to (l − 1/2)/r 2 : √ (l − 1/2)2 ± (l − 1/2)ε/ ε2 − 2 2 2 k ≈ k± − r2 √ (l − 1/2 ± ε/2 ε2 − 2 )2 2 ≈ k± − . (9.46) r2 The total phase accumulated by a quasiparticle moving from very large r to the turning point and then back to large r is # √  r (l − 1/2 ± ε/2 ε2 − 2 )2 π 2 k± − dr − l = 2 2 2 r rt    π  1 ε − . (9.47) = 2k± r − π l − ± √ 2 2 ε2 − 2  2 Effects of scattering emerge because of differences between the phases l and the phases 0,l in the plane wave [see Eq. (9.33)]. The differences determine the scattering phase shifts δl equal to    1 π  ε l − 0,l  + |l| π = − l − + √ δl = 2 2 2 2 2 2 ε −  2   π ε = signl. (9.48) 1− √ 2 4 ε − 2 The variation of the classical action along the quasiparticle trajectory is connected with the quantum mechanical scattering phase shift by the relation δS(b) = 2hδ ¯ l , where b ≈ l/kF . This yields Eq. (9.25) for δS(b). In the case of the hole branch (the lower sign in the expressions above) one should subtract from the phase shift l in the vortex state the phase shift 0,l−1 = 2k− r − π(|l − 1| + 1/2) of the (l − 1)-partial wave function in the uniform state. It is worth mentioning that if one chooses the spinor with violated cyclic boundary conditions, as explained above Eq. (9.24), instead of Eq. (9.40) the partial-wave expansion would be   ul ei(l+1/2)φ , v= vl ei(l−1/2)φ . (9.49) u= l

l

Then the correction ±π/4 to the scattering phase shift δl would not appear. In the semiclassical approximation, at any scattering event there is only one outgoing wave after reflection, which belongs to the same branch as an incident wave at usual reflection, or to the other branch if Andreev reflection took place. In general there is some probability for the two types of reflections, and there are two outgoing waves at any scattering event (Cleary, 1968; Sonin, 2013b).

9.5 Bound Andreev states in a planar SNS junction

255

9.5 Bound Andreev states in a planar SNS junction One of the main features which makes vortex dynamics in Fermi superfluids different from that in Bose superfluids is quasiparticle bound states in a vortex core, which were revealed in the BCS theory long ago (Caroli et al., 1964). These states do not exist in Bose superfluids. The energies of the bound states were accurately calculated for realistic distribution of the order parameter inside the core (de Gennes, 1966). Their essential effect on vortex dynamics was discovered by Kopnin and Kravtsov (1976a). In order to analyse the role of the core states in vortex dynamics, Stone (1996) suggested a simplified approach based on geometric optics. He used the model of a normal vortex core, exploiting its analogy with the one-dimensional problem of Andreev bound states in the ballistic Superconductor–Normal metal–Superconductor (SNS) junction. We shall also investigate this useful analogy, although our subject is neutral superfluids and the electric charge is absent. In the past a number of authors have addressed the question whether and what Josephson current is possible through such a SNS junction in full absence of the order parameter in the normal layer (Kulik, 1969; Ishii, 1970; Bardeen and Johnson, 1972). They concluded that the Josephson current is possible due to phase coherence of the Andreev states, which are sensitive to the phase difference on the junction. We consider a normal layer of width L, which is perpendicular to the axis y. A fluid in superfluid regions y < 0 and y > L moves with superfluid velocity v s . Let us look for a state with the energy ε = ε0 + h¯ k · v s ≈ ε0 + h¯ k 0 · v s ,

(9.50)

where the wave vector k 0 (kx , kf , kz ) has a modulus equal to the Fermi wave number kF , $ 2 so that the component ky is equal to kf = kF − kx2 − kz2 . The energy of the state is inside the gap (|ε0 | < ), and the state is a propagating wave inside the normal layer, but is an evanescent wave inside superfluid regions. The wave function, which satisfies the Bogolyubov–de Gennes equations, is given by 

u v



 =

Aeim(v s ·R)/h¯ +imε0 y/h¯ kf 2 Be−im(v s ·R)/h¯ −imε0 y/h¯ kf 2

 eik 0 ·R

(9.51)

inside the normal layer 0 < y < L, 

u v



 =

u− eiθ− /2+im(v s ·R)/h¯ v− e−iθ− /2−im(v s ·R)/h¯

 e

ik 0 ·R+

m

√

2 −ε02

h¯ 2 kf

y

(9.52)

inside the superfluid region at y < 0, and 

u v



 =C

eiθ+ /2+im(v s ·R)/h¯

u+ v+ e−iθ+ /2−im(v s ·R)/h¯

 e

ik 0 ·R−

m

√

2 −ε02

h¯ 2 kf

y

(9.53)

256

Mutual friction and vortex mass in Fermi superfluids

inside the superfluid region at y > L. Here - ⎛ ⎞ $ . . 2 − ε02 ⎟ .1 ⎜ u± = v∓ = . ⎠. / 2 ⎝1 ± i ε0

(9.54)

The constants A and B are determined from the conditions of continuity of spinor components at the interface y = 0: B = v− e−iθ− /2 .

A = u− eiθ− /2 ,

(9.55)

The third constant C must be determined from the conditions of continuity of spinor components at the interface y = L. This is only possible for discrete values of the energy ε0 satisfying the following Bohr–Sommerfeld condition (Stone, 1996): 2mε0 L h¯ 2 kf

  1 ε0 = 2π s + + (θ+ − θ− ) − 2 arcsin . 2 

(9.56)

Here s is an integer. At small energy ε0   this yields the spectrum of the bound states:  ε0 =

2mL

2 + 2  h¯ kf

−1 

   1 2π s + + (θ+ − θ− ) . 2

(9.57)

Both components of the wave functions in these states have wave numbers ky close to the large positive value kf , while the directions of the group velocities of the particle (u) and (v) components are opposite. This means that at interfaces between normal and superfluid regions only Andreev reflection is possible, which does not change the quasiparticle momentum essentially. Therefore, the states can be called Andreev bound states. In this approximation the boundary conditions at the interfaces require continuity only of two spinor components. The approximation is accurate in the weak-coupling limit when the superfluid gap  is small compared to the Fermi energy εF = h¯ 2 kF2 /2m. A more general approach would consider a superposition including the wave function with negative ky close to −kf and requiring continuity of wave function gradients at the interfaces. Now let us find the contribution of bound states to the momentum of the fluid. Since in the bound states particle and hole components are equal in amplitude, the charge |u|2 − |v|2 in these states vanishes, and according to Eq. (9.15) the mass current normal to the layers in any bound state is about hk ¯ f . Let us consider the case of a normal layer with width L much larger than the coherence length ξ0 = h¯ 2 kF /m = hv ¯ F /. In this case there is a large number of bound states and the sum over them can be replaced by an integral. Then the total contribution of the bound states to the mass current is simply a product of h¯ kf and a difference of the number of states with the wave vectors in opposite directions. At T = 0

9.5 Bound Andreev states in a planar SNS junction

257

all states with ε = ε0 + h¯ k 0 · v s < 0 are filled. Then the total momentum in the bound states (per unit area of the SNS junction) normal to the layers is  Pbs = −

h¯ 2 kf2 vs dk  = −Lnmvs , δε 4π 2

(9.58)

k ) of a quasiparticle propagating from y = −∞ to y = ∞, the spinor in the normal layer 0 < y < L is given by the same expression as Eq. (9.51) for the bound state, whereas in superfluid regions the states are described by spinors √   m ε02 −2 iθ /2+im(v ·R)/ h ¯ + s ik ·R+i y 0 u0 e h¯ 2 kf (9.60) e t −iθ /2−im(v ·R)/ h ¯ + s v0 e for y > L, and ⎡



⎣

eiθ− /2+im(v s ·R)/h¯

u0 v0 e−iθ− /2−im(v s ·R)/h¯ 

+r



i

e

eiθ− /2+im(v s ·R)/h¯

v0 u0 e−iθ− /2−im(v s ·R)/h¯

m

√2

ε0 −2 y h¯ 2 kf

 e

−i

m

√2

ε0 −2 y h¯ 2 kf

⎤ ⎦ eik 0 ·R

(9.61)

for y < 0. Here t and r are amplitudes of transmission and reflection (|t|2 +|r|2 = 1) which are determined from the continuity of spinor components (Bardeen and Johnson, 1972) at y = 0 and y = L. As in the case of bound states, the analysis considers only the Andreev reflection, neglecting the probability of usual reflection, which changes the direction of the wave vector. The amplitudes of the spinor components in the normal layer [see Eq. (9.51)] are A = tu0 e B = tv0 e

$ iθ+ /2+im( ε02 −2 −ε0 )L/h¯ 2 kf

,

$ −iθ+ /2+im( ε02 −2 +ε0 )L/h¯ 2 kf

(9.62) .

The transmission probability is |t|2 =

ε02 − 2 ε02 − 2 cos2 [ε0 mL/h¯ 2 kf − (θ+ − θ− )/2]

.

(9.63)

The transmission probability differs from unity in the small energy interval of the order   εF , and the effect of reflection is not essential for the contribution of delocalised states to the supercurrent, which can be found by summation of Eq. (9.15) over the whole continuum of free bulk states. The whole particle density is accumulated in delocalised but not bound states. Neglecting reflection for the continuum states, the density and the current in the normal and the superfluid regions do not differ essentially. So the whole ensemble of delocalised quasiparticles is a fluid of nearly constant density n moving with the spatially uniform velocity vs . This points out the nearly ideal transparency of ballistic normal layers

9.6 Bound vortex core states in a normal core

259

for the supercurrent of delocalised quasiparticles. Note that scattering of continuum states by impurities is impossible since all continuum states are fully occupied. On the other hand, we saw that bound quasiparticles do not contribute to the total particle density but provide the particle current −nvs exactly opposite to the particle current of delocalised quasiparticles. So inside the normal layer the total particle current vanishes while in the superfluid area the particle current is nvs . According the particle conservation law in one-dimensional geometry, the current cannot vary in space. A natural conclusion from this is that superfluid transport (but not diffusive transport with dissipation!) with finite superfluid velocity vs in this geometry is impossible. But this does not rule out superfluid when discreteness of the Andreev transport with very low superfluid velocities vs ≤ h/mL ¯ bound states and the phase difference across the normal layer cannot be ignored. This returns us again to the problem of the Josephson effect in the SNS junction (Kulik, 1969; Ishii, 1970; Bardeen and Johnson, 1972).

9.6 Bound vortex core states in a normal core Now let us consider bound states in the normal core of a vortex. The model of the normal core assumes that the order parameter inside the vortex core of radius rc is totally absent, and at the core boundary the gap  jumps from zero to a finite constant value. On the other hand, in contrast to the empty core in Bose superfluids, the particle density inside the core is approximately the same as outside. Bound quasiparticles inside the core, similar to those in the SNS junction, form a normal fluid, which exists even at zero temperature (Volovik, 2003b). A reliable assumption is that a quasiparticle inside the core, where the superfluid order parameter vanishes, moves along an approximately straight trajectory back and forth, reversing its direction of motion via Andreev reflection at the boundary of the core as shown in Fig. 9.2. In the figure the trajectory with the impact parameter b is chosen to be parallel to the y axis. The quasiparticle is a particle for motion up and a hole for motion down. The linear momentum of the quasiparticles is about hk ¯ f , while the angular momentum is Lz = hk ¯ f b. The quantised angular momentum Lz = hl ¯ is determined by the integer quantum number l = bkf . For trajectories with impact parameters b much less than the core radius, the bound states are similar to those in the SNS junction with the width of the normal layer L equal to the core diameter 2rc . On the other hand the phase difference θ+ − θ− = θv + θs consists of the phase difference θv related to the phase of the κc ∇θv , vortex-induced velocity field v v = 2π θv = π − 2 arcsin

2l b ≈π− , rc kf rc

(9.64)

and the phase difference θs , produced by the superflow past the vortex. Further using the analogy with the SNS junction, for the chiral zero-crossing branch s = −1 the energy of the bound state in the normal core with radius rc exceeding the coherence length ξ0 is ε(l) = ε0 (l) + hk ¯ f vsc cos α,

(9.65)

260

Mutual friction and vortex mass in Fermi superfluids

Figure 9.2 Bound state in the normal core. The vertical solid arrowed line shows the trajectory of the particle and the vertical dashed arrowed line shows the trajectory of the hole after Andreev reflection of the quasiparticle at the core boundary. The picture is purely schematic, and in fact the analysis was done for the case when the impact parameter b is much less than the core radius rc but still much larger than the interatomic distance 1/kF . Figure from Sonin (2013b).

where h¯ 2 kf ε0 (l) = 2mrc

  l θs − . + kf rc 2

(9.66)

Here α is the angle between the trajectory (the axis y) and the near-core superfluid velocity v sc (see Fig. 9.2). If the vortex core is transparent for the superflow, the superfluid velocity does not vary in space and v sc = v s . But in general v sc is different from the superfluid velocity v s far from the vortex because of the backflow (see below). If the flow cannot penetrate the core at all (as in the case of a cylinder moving in a perfect fluid considered in Section 1.3), then v sc = 2v s . In reality the vortex core radius rc is of the same order as ξ0 = hv ¯ F /, and our analysis cannot provide accurate numerical factors, but this inaccuracy was already presumed in the very model of a normal core with sharp borders. Since the bound quasiparticle has an angular momentum Lz = h¯ l, the trajectory slowly rotates around the vortex axis with a frequency determined by a canonical relation ω0 = −

∂ε0 h¯ 2π κc = = 2 . ∂Lz 2mrc2 rc

(9.67)

Introducing the isotropic part of the spectrum ε00 = −ω0 Lz ,

(9.68)

9.6 Bound vortex core states in a normal core

261

the total energy (9.65) becomes ε(l) = ε00 + hk ¯ · (˜v s − v L ),

(9.69)

where the presence of the vortex velocity v L shows that this is an energy in the coordinate frame connected with the vortex, and v˜ s = v sc + v θ

(9.70)

is the effective superfluid velocity, which includes all effects of the superflow on the energy of the bound states. Here the velocity v θ is parallel to v sc and connected with the phase difference θs : θs =

4mrc vθ cos α. h¯

(9.71)

In our case rc ξ0 , v θ does not differ from v sc , but this is not true in general. Later we shall see that θs and v θ do not appear in final expressions for physical quantities of interest. Despite its simplicity, the model of the normal core gives a qualitatively correct energy spectrum, different from that in more accurate theories only by a numerical factor of order unity. In the model of the normal core, Andreev reflection for all bound states occurs at the core boundary, and the energies of bound states are easily determined analytically from the semiclassical Bohr–Sommerfeld condition. Now we consider a more realistic model of the core with linear growth of the order parameter in the core at small r. This model is not as accurate as the calculation, which takes into account the real variation of the order parameter at any r (Caroli et al., 1964; de Gennes, 1966). However, it allows a simple analytical solution, which shows how far one can apply the semiclassical approach to describe the motion of bound quasiparticles inside the core (Sonin, 2013b). As before, we consider a quasiparticle inside the core, which moves back and forth along an approximately straight trajectory parallel to the y axis, changing its direction of motion via Andreev reflection. The gap in the Bogolyubov–de Gennes equations (9.2) and (9.3) depends linearly on the distance r: (R) = r/rc where the constant  is the gap outside the core. For the sake of simplicity we assume that there is no component of the wave vector parallel to the z axis. Eliminating fast oscillations of the wave function, we make a transformation to new spinor amplitudes u˜ = ue−ikF y , v˜ = ve−ikF y . Neglecting second derivatives of u˜ and v, ˜ the Bogolyubov–de Gennes equations become −i hv ¯ F

d u(b, ˜ y) r iθ + e v(b, ˜ y) = εu(b, ˜ y), dy rc

d v(b, ˜ y) r −iθ + i hv e u(b, ˜ y) = εv(b, ˜ y). ¯ F dy rc

(9.72)

262

Mutual friction and vortex mass in Fermi superfluids

"

Here r = b2 + y 2 . In the absence of superfluid motion through the core, the phase θ coincides with the azimuthal angle φ = arctan(y/b). Then one can transform the Bogolyubov– de Gennes equations to: −i hv ¯ F

d u(b, ˜ y) (b + iy) + v(b, ˜ y) = εu(b, ˜ y), dy rc

d v(b, ˜ y) (b − iy) i hv u(b, ˜ y) = εv(b, ˜ y). + ¯ F dy rc

(9.73)

The normalised solution of the Bogolyubov–de Gennes equations is e−y /2rc ξ0 √ 2 π rc ξ0 2

u˜ = −v˜ =

(9.74)

with the energy ε equal to ε00 given by Eq. (9.68) with the angular velocity of slow trajectory precession  . (9.75) hk ¯ F rc The calculated energy spectrum differs insignificantly from the spectrum obtained in the original paper (Caroli et al., 1964) and in the book by de Gennes (1966) using the partialwave analysis and a more realistic variation of the gap  in the space. This agreement confirms a simple picture of the bound states assuming well defined trajectories of quasiparticle motion. However, it is necessary to stress that though the trajectory is well defined in the sense that the impact parameter is well defined, the motion along the trajectory cannot be described semiclassically. In particular, our solution shows that there are no well defined Andreev reflection points. Using the semiclassical approach and the Bohr–Sommerfeld condition for calculation of energy levels, one obtains a totally incorrect spectrum, which is not linear in the angular momentum. So the semiclassical theory of motion along the trajectory of the bound state is valid only for the model of the totally normal core and not for more realistic models with non-zero order parameter in the core. ω0 =

9.7 Mutual friction in a vortex core: Kopnin–Kravtsov force If there are impurities in superconductors or collisions of bound quasiparticles with free bulk quasiparticles in superfluid 3 He, core bound states give an essential contribution to the balance of forces on the vortex, producing an additional mutual friction called the Kopnin–Kravtsov force (Kopnin and Kravtsov, 1976a). For derivation of this force, Kopnin and Kravtsov (1976a) replaced the ensemble of discrete bound states by the continuum of these states characterised by the two Hamiltonian-conjugate quantities ‘angle α - moment Lz ’. They suggested the Boltzmann equation for the distribution function in the core state continuum:  ∂f ∂ε ∂f ∂ε ∂f ∂f  + . (9.76) − = ∂t ∂α ∂L ∂L ∂α ∂t  z

z

col

9.7 Friction in a vortex core: Kopnin–Kravtsov force

The collision term on the right-hand side in the relaxation-time approximation is  ∂f  f − fn (ε, v n ) =− , ∂t col τ

263

(9.77)

where τ is the relaxation time. It takes into account elastic collisions with impurities in superconductors (then v n is the velocity of the crystal lattice) or with bulk free quasiparticles in superfluids. Here 1

fn (ε, v n ) = e

ε−h¯ k·(v n −v L ) T

+1

1

= e

ε00 −h¯ k·(v n −˜v s ) T

+1

(9.78)

is the Fermi distribution function for bound states, which are at equilibrium with the normal component. We expand the distribution functions around the isotropic equilibrium distribution function f0 (ε00 ) equal to fn at v n = v˜ s = v L : f (p) = f0 (ε00 ) + f1 (ε, v n ). The equation for the first-order correction linear in the relative velocities is   1 ∂f1 ∂f0 ∂f0 h¯ ω0 k · [(˜v s − v L ) × zˆ ] − ω0 =− f1 − h¯ k · (v n − v˜ s ) , ∂ε ∂α τ ∂ε where the effective superfluid velocity v˜ s is given by Eq. (9.70). Its solution is   ∂f0 ω0 τ k · [(v n − v L ) × zˆ ] + k · (v n − v L ) h¯ k · (˜v s − v L ) − f1 = . ∂ε 1 + ω02 τ 2

(9.79)

(9.80)

(9.81)

The contribution of the bound states to the mutual-friction force is determined by the momentum transferred from bound states confined in the vortex core to normal quasiparticles or impurities via collisions:

Fc =

1 2

kF −kF

dkz 2π

π −π

dα 2π

max L z

dLz k Lmin z

 ∂f  . ∂t col

(9.82)

The factor 1/2 before the integral takes into account that the bound state is a superposition of a particle and a hole and that summation over all bound states counts any particle (hole) state twice. The limits of integration over the angular momentum Lz of bound states are values of Lz at the energy ε = ±, i.e., at the border between bound (localised) and delocalised states. Roughly the limits are close to the values ±L0 = ±/ω0 , but taking into account the phase shift θs , they are Lmax = L0 + hk ¯ f rc θs /2, z

Lmin = −L0 + hk ¯ f rc θs /2. z

(9.83)

264

Mutual friction and vortex mass in Fermi superfluids

The first-order correction f1 to the isotropic distribution function is confined to the close vicinity of the Fermi surface ε = 0, and when calculating the mutual friction force one can calculate the integral over Lz with infinite limits. At very low temperatures ∂f0 /∂ε = −δ(ε), and substitution of f1 from Eq. (9.81) into Eq. (9.82) yields F c = π h¯ n

ω0 τ (v n − v L ) − [(v n − v L ) × zˆ ] . 1 + ω02 τ 2

(9.84)

The force component transverse to v n − v L is the Kopnin–Kravtsov force. In the balance of forces on the vortex this force should be added to the Magnus and the Lorentz forces: mns [(v L − v s ) × κ c ] = F c ,

(9.85)

where v s is the superfluid velocity far from the vortex1 different from the effective superfluid velocity v˜ s near the core. The latter does not affect the Kopnin–Kravtsov force and the total balance of forces. At T = 0 (ns = n) Eqs. (9.84) and (9.85) yield the equation of vortex motion     ω0 τ v L ω0 τ v n − [v n × zˆ ] = mnκc [ˆz × v s ] + , (9.86) mκc nM [ˆz × v L ] + n 1 + ω02 τ 2 1 + ω02 τ 2 where the force transverse to the vortex velocity v L , which may be considered as the effective Magnus force on the vortex, is proportional to the density nM =

ω02 τ 2 1 + ω02 τ 2

n.

(9.87)

In the limit ω0 τ → 0 the Kopnin–Kravtsov force fully compensates for the Magnus force, and the total transverse force on the vortex vanishes.

9.8 Vortex mass in Fermi superfluids The two contributions to the vortex mass (from the backflow and the fluid compressibility), which were discussed in Section 2.5 for the Bose superfluid, are also relevant in principle for the Fermi superfluid. However, the compressibility mass becomes inessential in the weak-coupling limit despite a large logarithmic factor. The difference from the Bose superfluid is that while in the Bose superfluid the sound velocity goes down (compressibility goes up) in the weak-interaction limit, in the Fermi superfluid the sound velocity remains high, being always of the order of the Fermi velocity. As a result, the compressibility mass, apart from the logarithmic factor, is of the order of the small vortex mass ∼ ρ/kF2 ∼ mn−1/3 obtained by Suhl (1965) for superconductors using the time-dependent Ginzburg–Landau

1 When we say ‘far from the vortex’ we mean distances longer than the vortex core radius. If vortices are not straight or form a

vortex array, the superfluid velocity v s must be replaced by the local superfluid velocity v sl , which is defined at distances which are short compared with the vortex line curvature or the intervortex distance but still much larger than the vortex core radius.

9.8 Vortex mass in Fermi superfluids

265

theory. But the most important difference between the Bose and the Fermi superfluids comes from bound core states, which contribute not only to the mutual friction force but also to the vortex mass (Kopnin, 1978b, 2001). For analysis of the vortex mass in Fermi superfluids, one needs to know the contribution of bound states to the total momentum of the fluid if the superfluid flows past the vortex. As in the case of the SNS junction, every bound state has a momentum of magnitude about hk ¯ f directed along the bound-state trajectory. The total contribution of occupied bound states to the momentum component Pbs parallel to the near-core superfluid velocity v sc is

Pbs

1 = 2

kF −kF

dkz 2π

π −π

dα h¯ kf cos αN(α), 2π

(9.88)

$ where kf = kF2 − kz2 and N (α) is the density of occupied bound states in the angle interval dα. Taking into account that the energy interval between levels is hω ¯ 0 , this density is N (α) =

 − hk ¯ f vsc cos α . hω ¯ 0

(9.89)

Eventually in an arbitrary coordinate frame, where the velocity v sc must be replaced by the relative velocity v sc − v L , the total momentum of bound states is

P bs

1 = 2

kF −kF

dkz 2π

π −π

2

hk ¯ f dα cos2 α (v L − v sc ) = μK (v L − v sc ). 2π ω0

(9.90)

Here μK = π h¯ n/ω0 is the vortex mass calculated by Kopnin (1978b). Using Eq. (9.67) for ω0 and assuming that the momentum is uniformly distributed over the area π rc2 of the core, this momentum corresponds to the mass current j bs = P bs /πrc2 = −2mn(v sc − v L ) inside the core. The momentum in the core bound states depends only on the near-core superfluid velocity v sc outside the core and not on the phase difference θs . The reason is the same as in the case of the SNS junction: the phase difference θs , like the phase difference θ+ − θ− in the SNS junction (see Fig. 9.1), shifts energy levels but does not change their number since any crossing of the zero energy by a level is compensated for by an entry or an exit of a level at the bottom of the forbidden gap. The full vortex mass is not reduced to the Kopnin mass. The mass current j bs = P bs /πrc2 in the bound states exists only inside the core and must transform to the superfluid mass current outside the core. The latter current forms the backflow velocity field, which must be determined from continuity of the total mass current. As a result, the Kopnin mass will be renormalised by the backflow effect. In analogy with the analysis of the backflow in a perfect superfluid (Section 1.3), the near-core superfluid velocity at the core boundary is v sc = v s + v bf , where v bf is the

266

Mutual friction and vortex mass in Fermi superfluids

backflow velocity near the core boundary. Continuity of the mass current at the core boundary requires that h¯ n(v L − v s − v bf ) = mnv bf . ω0 rc2

j bs =

(9.91)

Note that the mass current in the continuum of delocalised states does not affect this condition because it has no discontinuity at the core boundary and contributes the same term mnv sc to the two sides of this equation. The equation yields that μK (v L − v s ), (9.92) v bf = μcore + μK where μcore = π mnrc2 is the core mass. The total momentum including the backflow momentum (Kelvin impulse) P K given by Eq. (1.47) is 2μcore μK (v L − v s ). μcore + μK

P bs + μcore v bf =

(9.93)

Thus the Kopnin mass μK is renormalised by the factor 2μcore /(μK + μcore ) equal to 4/3 for the value of ω0 given by Eq. (9.67). We have considered the vortex mass without taking into account possible interaction of core bound quasiparticles with impurities or delocalised quasiparticles. If this interaction is essential one should determine the total momentum in the vortex core bound states via the distribution function f (α, Lz ) found from the Boltzmann equation in Section 9.7: P bs

1 = 2

kF −kF

dkz 2π

π −π

max L z

dα 2π

dLz kf (α, Lz ).

(9.94)

Lmin z

Here Lmax and Lmin are given by Eq. (9.83). Dependence of the integral limits on the z z phase difference θs does not mean that the momentum P bs depends on θs . Indeed, the anisotropic part of the distribution function f1 obtained from the Boltzmann equation depends on the θs -dependent v˜s given by Eq. (9.70) because the Boltzmann equation only takes into account events near the Fermi surface. However, entries and exits of the bound states to and from the forbidden gap at the top and at the bottom of the gap are also and Lmin important. These events are accounted for with direction-dependent limits Lmax z z of the integral in Eq. (9.94). Eventually all θs -dependent terms cancel out. One can see this by changing variables in this integral and introducing the modified angular momentum Lz = Lz − h¯ kf rc θs /2. Then P bs

1 = 2

kF −kF

dkz 2π

π −π

dα 2π

L0 −L0

dLz k

  ∂f0 f1 − hk ¯ · (v θ − v L ) . ∂ε

(9.95)

The second term in brackets cancels the θs dependent term in f1 given by Eq. (9.81), and the total momentum of core bound states depends only on the near-core superfluid velocity v sc .

9.9 Spectral flow and vortex dynamics

267

In the limit of zero temperature, ∂f0 /∂ε = −δ(ε), where δ(ε) is the δ-function of the energy, and the momentum in the bound states is P bs

  π h¯ n ω0 τ [(v n − v L ) × zˆ ] + v n − v L = v L − v sc + . ω0 1 + ω02 τ 2

(9.96)

The expression reduces to Eq. (9.90) in the limit τ → ∞. The part of the momentum linear in v L determines the vortex mass, which is now a tensor: ω0 τ μˆ K = μK 1 + ω02 τ 2



ω0 τ 1

−1 ω0 τ

 .

(9.97)

Repeating the process of renormalisation of the Kopnin mass by the backflow effect, one obtains the same renormalisation factor 2μcore /(μK + μcore ) as obtained in the previous section without collisions. Kopnin and Vinokur (1998) called the off-diagonal part of the vortex mass tensor the transverse vortex mass. The transverse vortex mass does not lead to a conservative inertial force, which follows from some Hamiltonian. It determines a high frequency correction to the dissipative (longitudinal) mutual friction force. A term in the dissipative function related to this correction is not positively defined. Therefore, the transverse mass correction makes sense only if it is small compared to the usual friction force (Sonin, 2013b). In the case of frequent collisions (τ  1/ω0 ), the velocity v L drops out of the expression (9.96) for the momentum, and the Kopnin mass vanishes. This is because in this limit the effect of reflections from the walls of the core is fully suppressed by frequent collisions with impurities or quasiparticles. This means that the vortex mass vanishes in our approximation, which neglected effects of the order /εF . It is worth noting that a small ω0 τ does not necessarily invalidate the assumption that the mean free path lqp of quasiparticles is much longer than the core radius. Indeed, since τ = lqp /vF and ω0 ∼ h¯ /mrc2 , the condition ω0 τ  1 reduces to the condition lqp /rc  εF /. In the weak-coupling limit εF / is very large so even large lqp /rc can satisfy this condition.

9.9 Spectral flow and vortex dynamics The spectral flow concept is known in both mathematics (Booss-Bavnbek and Wojciechowski, 1993) and physics. According to its mathematical definition, the spectral flow is the number of eigenstates of an operator with eigenvalues passing zero value on tuning of some parameter, on which the operator (and correspondingly its eigenstates) depends. A physical example of the spectral flow is the flow of the Andreev bound states in the SNS junction (Stone, 1996), which we addressed in Section 9.5. According to Eq. (9.57) the energy of the Andreev state depends linearly on the superfluid phase difference θ+ − θ− between the superconductors forming the junction. When the phase difference varies monotonously in time (the ac Josephson effect), discrete energy levels cross the whole

268

Mutual friction and vortex mass in Fermi superfluids

superconducting gap passing zero energy. In this example the parameter governing the spectral flow is the phase difference and the operator corresponds to the Bogolyubov–de Gennes equations, which determine the Andreev bound states inside the gap. Volovik (1993, 2003b) suggested that the process of vortex motion is accompanied by a steady motion of core bound-state levels from the negative-energy continuum to the positive-energy continuum, i.e., by the spectral flow across the superconducting gap similar to that in the SNS junction. He argued that any crossing of the gap by a bound state leads to transfer of the momentum, which leads to the transverse Kopnin–Kravtsov force. So momentum transfer from the vortex moving with relative velocity v L − v n with respect to the normal component (or to impurities in superconductors) is realised not via jumps of particles between energy levels caused by collisions but via motion of energy levels themselves in the energy space. Volovik (2003b) also used the concept of spectral flow to explain of the correction to the Iordanskii force in Fermi superfluids at quasiparticle energies much exceeding the superconducting gap [the second term ∝ 1/vF in the transverse cross-section (9.26)]. The interpretation of the Kopnin–Kravtsov force in terms of the spectral flow was was widely accepted (Kopnin, 2001, 2002) and led to a far-reaching conclusion that the spectral flow in vortex dynamics models the cosmological baryogenesis in the early Universe (Bevan et al., 1997a). Evidence of the Kopnin–Kravtsov force in mutual friction measurements (Bevan et al. 1997a, 1997b) was interpreted as evidence of the spectral flow in the core of a moving vortex. Meanwhile, original theoretical papers, which predicted the Kopnin–Kravtsov force (Kopnin and Kravtsov, 1976a) and the correction to the Iordanskii force in Fermi superfluids (Gal’perin and Sonin, 1976; Kopnin and Kravtsov, 1976b), did not refer to the spectral flow. We explained these phenomena in this chapter also without using the spectral flow concept. The spectral flow cannot be used as an alternative explanation, simply because it is absent in the core of a moving vortex, as was already noticed by Stone (1996). In contrast to the SNS case, where the phase difference across the normal layer can vary monotonously, the phase difference θs across the normal core of the moving vortex can only oscillate without crossing the forbidden gap. This rules out steady spectral flow. The oscillating spectral flow is related to rotation of the bound state with angular velocity ω0 and dependence of the level position with respect to the gap on the α-dependent phase difference θs in Eq. (9.66) (Stone, 1996). Because of the great attention to the concept of spectral flow in the literature on vortex dynamics, it would be useful to reassess Volovik’s arguments in favour of spectral flow (Sonin, 2013b). Deriving the spectral flow, Volovik considered the angular momentum Lz = zˆ · [(u − (v L − v n )t) × p] around the axis z in the coordinate frame, which moves together with the thermal bath (normal component). Here u is the position vector with the origin on the symmetry axis of the moving vortex. Volovik’s angular momentum varies in time: dLz = −ˆz · [(v L − v n ) × p]. dt

(9.98)

9.9 Spectral flow and vortex dynamics

269

Since the energy of the bound state is proportional to the angular momentum, Volovik concluded that the energy levels move in the energy space and cross the zero energy level with a rate proportional to v L − v n . The problem with this argument is that the position of the bound state energy with respect to the gap depends on the angular momentum about the symmetry axis of the vortex in the coordinate frame moving together with the vortex. Then the angular momentum is conserved and provides a good quantum number, which determines the energy of the bound state. In any other coordinate frame with the reference axis, which does not coincide with the vortex axis, the angular momentum is not conserved and is

(a)

–

0

–

0

–

0

(b)

(c)

Figure 9.3 Effect of a shift of energy levels on the density of states n0 () at various ω0 τ . The density of states is shown by solid lines before the shift and by dashed lines after the shift. (a) ω0 τ → ∞. The density of states is a chain of sharped peaks. (b) ω0 τ  1. Very broad peaks strongly overlap and cause only weak oscillations of the density of states, which are still noticeable in principle. (c) ω0 τ = 0. The plot of the density of states is totally flat and its shift does not lead to any physical consequence. Figure from Sonin (2013b).

270

Mutual friction and vortex mass in Fermi superfluids

not a quantum number. Moreover, deriving Eq. (9.98), Volovik assumed that the momentum p of the bound state does not vary in time. Meanwhile, in a bound state the momentum p rotates with angular velocity ω0 and vanishes on average. As a result, dLz /dt vanishes also, and the angular momentum determined with reference to any axis does not differ on average from the angular momentum around the vortex symmetry axis. This is a direct consequence of the theorem of mechanics, which tells us that for a system with vanishing centre-of-mass velocity, the angular momentum does not depend on the choice of the reference axis. So the vortex motion with respect to the thermal bath does not lead to spectral flow. Volovik stressed that his derivation referred to the continuum limit ω0 τ → 0 when levels are strongly broadened and in fact cease to be discrete levels. Originally spectral flow concept was considered only for discrete levels. In the continuum limit the very concept of spectral flow becomes ambiguous. This is illustrated in Fig. 9.3, which shows the effect of the shift of the levels on the density of states n0 () for various ω0 τ . Without collisions (ω0 τ → ∞) the density of states is a chain of very narrow peaks and a shift of the levels with respect to the forbidden gap is a clear effect (Fig. 9.3a). For very small but still finite ω0 τ , the effect of a level shift on the density of states is much weaker but still noticeable (Fig. 9.3b). In the extreme case ω0 τ = 0, when oscillations of the density of states are totally undetectable, the level shift does not lead to any effect and cannot influence any physical process. Without taking into account tiny oscillations of the density of states it is impossible even to define it. In conclusion it is necessary to stress that the absence of spectral flow in a moving vortex puts in question not the Kopnin–Kravtsov force itself but the connection of the force with spectral flow. The claim that experiments on the mutual friction force confirm spectral flow (Bevan et al., 1997a; Volovik, 2003b) is not justified. Only the Kopnin–Kravtsov force was revealed.

10 Vortex dynamics and hydrodynamics of a chiral superfluid

10.1 Order parameter in the A phase of superfluid 3 He Up to now we considered isotropic superfluids, in which gauge invariance was broken but they remained invariant with respect to any three-dimensional rotation. In particular, in the Fermi superfluids the order parameter, or gap , was a scalar independent of the direction. This means that the wave function of Cooper pairs was in the s state with zero orbital angular momentum and spin. Superconductors with such symmetry of the order parameter are called s-wave superconductors. In superfluid 3 He the Cooper pair has a total spin and a total orbital moment equal to 1 (in unit h). ¯ Superconductors (charged superfluids), in which Cooper pairs have orbital momentum and spin equal to 1, are called spin-triplet or p-wave superconductors. In p-wave superfluids the order parameter is a 3 × 3 matrix with complex elements (18 parameters) in general (Vollhardt and W¨olfle, 1990). We focus our attention on the A phase of superfluid 3 He, for which the order parameter matrix is a direct product of two three-dimensional vectors, which correspond to wave functions with spin 1 in the spin space and with orbital moment 1 in the orbital space. The unit vector d in the spin space determines the axis along which the spin of the Cooper pair exactly vanishes, although the spin modulus is equal to 1. Spin components along any other axis also vanish but only on average. So this spin wave function has no spin polarisation, and the state is analogous to the spin state in antiferromagnets with d being an analogue of the antiferromagnetic vector. In the orbital space there are two √ orthogonal unit vectors m and n, which determine a complex unit vector (m + in)/ 2 and a unit vector l = m × n. The vector l is called the orbital vector. It delineates the axis along which the orbital moment of the Cooper pair is directed. Neutral and charged superfluids with such an order parameter are called chiral or px + ipy superfluids. So the condensate of Cooper pairs has a spontaneous angular momentum along l, which is called an intrinsic angular momentum. In charged superfluids (px + ipy -wave superconductors) the intrinsic angular momentum leads to spontaneous magnetisation. This is orbital ferromagnetism, since spontaneous magnetisation originates from the orbital moment, in contrast to the more common case of ferromagnetism related to spin. It is widely believed that an order parameter similar to that in the A phase of superfluid 3 He exists in some superconductors, in particular in Sr2 RuO4 (Mackenzie and Maeno, 2003). The spontaneous orbital moment breaks time-inversion symmetry. 271

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Vortex dynamics and hydrodynamics of a chiral superfluid

The spin degree of freedom is connected with the orbital degree of freedom by dipoledipole (spin-orbit) interaction. We shall consider only the dipole-locked regime, when dipole-dipole interaction orients l and d parallel or antiparallel to each other, i.e., the spin vector d ceases to be an independent variable. This approach is accurate as long as fluid parameters vary slowly at the scale of the order of the dipole length ∼ 10−3 cm (Brinkman and Cross, 1978). Then it is sufficient to describe the order parameter of the A phase by a complex vector: ψ0 ψ = √ (m + in), 2

(10.1)

where ψ0 is a real constant. The two unit vectors m and n together with the third vector l = m × n form a triad of three real orthogonal unit vectors. It is important for hydrodynamics that the gauge transformation of the px + ipy order parameter, m + in → (m + in)eiθ = (m cos θ − n sin θ ) + i(m sin θ + n cos θ ), is equivalent to rotation around the axis l by the angle φl = −θ and therefore is not an independent symmetry transformation. So the full point symmetry group of the order parameter is the group of three-dimensional rotations. The group is not Abelian and the angle of rotation around any axis, including the axis l, depends on the path along which the transformation is performed. In particular, if we deal with the phase θ = −φl , a result of two small consecutive variations δ1 and δ2 of θ depends on the order of their realisations: δ1 δ2 θ − δ1 δ2 θ = l · [δ1 l × δ2 l].

(10.2)

This means that the phase θ is not well defined globally, although its infinitesimal variations still make sense and the quantum mechanical definition of the superfluid velocity vs =

h¯ ∇θ , 2m

(10.3)

is valid. Here 2m is the mass of the Cooper pair of 3 He atoms each with mass m. Because of Eq. (10.2), variation of the superfluid velocity is determined not only by variation of the phase θ but also by variation of the orbital vector l. Namely, assuming δ1 → d and δ2 → ∇i in Eq. (10.2) one obtains dvsi =

h¯ (∇i dθ + [∇i l × l] · dl). 4m

(10.4)

Moreover, the superfluid velocity is not curl-free. Relating δ1 and δ2 with two gradients ∇1 and ∇2 along two different directions (x and y, or y and z, or z and x), Eq. (10.3) yields the Mermin–Ho relation (Mermin and Ho, 1976) between vorticity and spatial variation of l: ∇ × vs =

h¯ ikn li ∇lk × ∇ln . 4m

This relation has a dramatic impact on the hydrodynamics of chiral superfluids.

(10.5)

10.2 Gross–Pitaevskii theory for superfluids

273

10.2 Gross–Pitaevskii theory for p x + ipy -wave superfluids Let us consider the extension of the Gross–Pitaevskii theory to bosons with internal degrees of freedom (Sonin, 1984). The wave function is given by a three-component complex vector like that in Eq. (10.1). The Lagrangian of the theory is   ∂ψ ∗ i h¯ ∗ ∂ψ ψ · −ψ · − H (ψ, ψ ∗ ). (10.6) L= 2 ∂t ∂t The most general Hamiltonian allowed by symmetry is  h¯ 2  K1 ∇i ψj∗ ∇i ψj + K2 ∇i ψj∗ ∇j ψi + K3 ∇i ψi∗ ∇j ψj + V (ψ, ψ ∗ ), H = (10.7) 2M where M is the mass of a boson, K1 and K2 are constants and the non-linear interaction energy V (ψ, ψ ∗ ) depends on components of the wave function ψ but not on its gradients. For further analysis we do not need its explicit expression. The invariant ∝ K3 can be reduced to the invariant ∝ K2 after integration by parts. The Hamilton equations for canonically conjugate fields ψ and ψ ∗ are  δH (ψ, ψ ∗ ) ∂ψ h¯ 2  2 = K + V |ψ|2 ψ, (10.8) i h¯ = − ∇ ψ + (K + K )∇(∇ ψ 1 2 3 j j j ∂t δψ ∗ 2M and the equation which is complex conjugate to this. According to Noether’s theorem we can write down the conservation law following from the gauge invariance. This is the mass continuity equation (1.18), where the mass density is ρ = mψ ∗ · ψ, and the mass current is given by i h¯  K1 (ψj∗ ∇i ψj − ψj ∇i ψj∗ ) + K2 (ψj∗ ∇j ψi − ψj ∇j ψi∗ ) ji = − 2  + K3 (ψi∗ ∇j ψj − ψi ∇j ψj∗ ) . (10.9) On the other hand, Noether’s theorem connects translational invariance with the conservation law ∂gi + ∇j ij = 0, (10.10) ∂t where the current i h¯ gi = − (ψj∗ ∇i ψj − ψj ∇i ψj∗ ) (10.11) 2 is different from the mass current j in the mass continuity equation (1.18). The flux tensor in Eq. (10.10) is ij =

! h¯ 2 ) K1 ∇i ψk ∇j ψk∗ + ∇i ψk∗ ∇j ψ 2M   + K2 ∇i ψj ∇k ψk∗ + ∇i ψj∗ ∇k ψk !* + K3 ∇j ψi ∇k ψk∗ + ∇j ψi∗ ∇k ψk + δij P ,

(10.12)

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Vortex dynamics and hydrodynamics of a chiral superfluid

and the pressure is given by  V |ψ|4 h¯ 2  K1 ∇ 2 |ψ|2 + (K2 + K3 )∇i ∇j (ψi∗ ψj ) . (10.13) − 2 4M The Lagrangian and the Hamiltonian are invariant with respect to simultaneous rotation of the configurational space and the wave function vector ψ, and Noether’s theorem connects this invariance with the conservation law ∂(L0i + [R × g]i ) + ∇j Gij = 0, (10.14) ∂t where the angular momentum P =

∗ L0 = i h[ψ ¯ ×ψ ]

(10.15)

can be called the intrinsic angular momentum. The angular momentum flux tensor Gij = (i) (e) Gij + Gij consists of two parts: the flux of the intrinsic angular momentum L0 , (i)

Gij =

) h¯ 2 ikn K1 (ψk∗ ∇j ψn + ψk ∇j ψn∗ ) 2M  + K2 (ψk∗ ∇n ψj + ψk ∇n ψj∗ ) ,

(10.16)

and the flux of the extrinsic angular momentum [R × g], (e)

Gij = ikn kj Rn .

(10.17)

The two currents j and g coincide only in a simple particular case K1 = 1, K2 = K3 = 0. Only in this case is the generalised Gross–Pitaevskii theory Galilean invariant, and the intrinsic and the extrinsic angular momenta are totally decoupled, each of them satisfying its own conservation law: ∂[R × g]i ∂L0i + ∇j G(i) + ∇j G(e) (10.18) ij = 0, ij = 0. ∂t ∂t One can perform the generalised Madelung transformation, after which the vector wave function ψ is described by the mass density ρ = Mψ02 , the superfluid velocity v s = h¯ M ∇θ , and the orbital vector l. In the spirit of the hydrodynamical approach, we neglect dependence of the energy on density gradients (gradients of ψ0 ). Then the Hamiltonian (10.7) becomes a hydrodynamical Hamiltonian    K2 + K3 K2 + K3 ρ K1 + vs2 − (v s · l)2 H = 2 2 2 h¯ {v s · [∇ × (Cl)] − C0 (l · v s )(l · [∇ × l]) + 2M h¯ 2 ρ Vρ 2 2 , (10.19) + {K ∇ l · ∇ l + (K + K )[(l · ∇)l] } + 1 i i 2 3 2 4M 2 where C = ρK3 ,

C0 = ρ(K2 + K3 ).

(10.20)

10.2 Gross–Pitaevskii theory for superfluids

275

In hydrodynamical variables the generalised Gross–Pitaevskii theory yields the following canonical equations of motion, two of which, h¯ ∂θ ∂ρ + μ = 0, + ∇ · j = 0, M ∂t ∂t are the same as in a non-chiral fluid, while the third one,   δH ∂l = 0, L0 + (j · ∇)l + l × ∂t δl

(10.21)

(10.22)

is new. Equation (10.22) is the equation of orbital dynamics for chiral superfluids. Here is the magnitude of the intrinsic angular momentum, L0 = hρ/M ¯ μ=

∂H ∂ρ

is the chemical potential, and    ∂H K2 + K3 K2 + K3 vs − (v s · l)l + j l j= = ρ K1 + ∂v s 2 2

(10.23)

(10.24)

is the mass current, while h¯ (10.25) {[∇ × (Cl)] − l(l · [∇ × (C0 l)])} 2M is the current related to spatial variation of l (l-texture). The functional derivative in Eq. (10.22) was determined at fixed superfluid velocity v s : jl =

∂H ∂H δH = − ∇i . δl ∂l ∂∇i l

(10.26)

Meanwhile, variation of l also produces variation of v s . Bearing in mind Eq. (10.4) connecting the two variations, one can redefine the functional derivative with respect to l as ˜ ∂H h¯ h¯ δH δH δH + [∇i l × l] = + [(j · ∇)l × l]. = ˜δl δl ∂vsi 4m δl 4m Then Eq. (10.22) transforms to

  ˜ δH ∂l L0 + l × = 0. ˜ ∂t δl

(10.27)

(10.28)

In the absence of the normal component at zero temperature, the Gross–Pitaevskii theory must be Galilean invariant, and only the trivial choice of parameters (K1 = 1, K2 = K3 = 0) is allowed. Then the current does not contain any contribution from gradients of l. In analogy with the spin-related current in the Landau–Lifshitz theory of ferromagnetism (Landau and Lifshitz, 1984), one can introduce a solenoidal current related to the intrinsic angular momentum: jL =

1 ∇ × (L0 l). 2

(10.29)

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Vortex dynamics and hydrodynamics of a chiral superfluid

This current is divergence-free, and one can redefine the current as j˜ = j + j L with the textural current j l redefined as j˜ l = j l + j L =

h¯ {[∇ × {(C + L0 )l)}] − l(l · [∇ × (C0 l)])}. 2M

(10.30)

Except for the case of the Galilean invariant theory with zero K2 and K3 (zero C and C0 ), it is impossible in the total current to distinguish the contribution related to the intrinsic angular momentum (∝ L0 ) from the term ∝ C in the transport current related to the extrinsic orbital momentum. So in general there is no single-valued angular momentum which can be proclaimed unambiguously as an intrinsic angular momentum. This is a matter of convention and semantics. Following Cross (1977) we relate the term ‘intrinsic angular momentum’ to the magnitude L0 , which determines the orbital dynamics. The angular momentum, which is related with the current determined by gradients of l, was called the ‘textural angular momentum’ by Cross (1977). We shall return to a discussion of the angular momentum after the analysis of a more general hydrodynamical theory in the next section.

10.3 Hydrodynamics of a chiral superfluid with an arbitrary intrinsic angular moment Turning to two-fluid hydrodynamics of chiral superfluids, one can use the standard procedure for deriving hydrodynamical equations from thermodynamical relations and conservation laws (Landau and Lifshitz, 1987). This procedure has already been used in Chapter 6. Isotropic fluids and gases are the most symmetric condensed matters characterised by the minimal number of the hydrodynamical variables and correspondingly by the minimal number of hydrodynamical modes. A phase transition can lead to a ground state of the fluid with symmetry lower than the symmetry of the Hamiltonian (Lagrangian) describing the fluid. This is the phenomenon of broken symmetry. Since the Hamiltonian remains symmetric, a transformation corresponding to the broken symmetry changes the ground state, but is not able to change the energy of the ground state. So it provides a new ground state but with the same energy as the original one. Thus broken symmetry inevitably results in degeneracy of the ground state and one should introduce a new parameter labelling different ground states. If broken symmetry is continuous (the corresponding symmetry transformation is continuous) the parameter labelling different ground states is also continuous. According to the Goldstone theorem, degeneracy of the ordered phase leads to the existence of a soft gapless mode. The mode is gapless since the energy does not depend on the new hydrodynamical variable (degeneracy parameter) but only on its gradients. Since the frequency of the Goldstone mode is low for wavelengths which are long enough, it may be treated within the hydrodynamical theory, and the continuous degeneracy parameter can be considered as a new hydrodynamical variable on equal standing with standard hydrodynamical variables (Martin et al., 1972). At the transition to the A phase of superfluid 3 He, not only the gauge symmetry is broken but

10.3 Hydrodynamics of a chiral superfluid

277

also the rotational symmetry, since there is a preferable direction determined by the orbital vector l. In hydrodynamics of chiral superfluids the intrinsic angular momentum related to the orbital vector l is still being debated. The simplest original idea was (Anderson and Morel, 1961) that since any Cooper pair has an angular momentum hl, ¯ the density of the total intrinsic angular moment L0 is the Planck constant h¯ times the number density n/2 = ρ/2m of Cooper pairs, i.e., L0 = h¯ ρ/2m at T = 0 or h¯ ρs /2m at T > 0. This follows from phenomenological theories based on the non-linear Schr¨odinger equation discussed in the previous section. But the situation turned out to be more complicated. One source of confusion was semantics (Brinkman and Cross, 1978). As was discussed above, there is an intrinsic orbital momentum L0 , which determines orbital dynamics, and there is a textural angular momentum determined by gradients in l textures. We focus on the former. Some microscopic calculations within the BCS theory predicted that the intrinsic angular momentum L0 is reduced by a very small factor (Tc /εF )2 due to particle-hole symmetry (Volovik, 1975; Cross, 1975, 1977). Meanwhile, Khalatnikov and Lebedev (1977) developed a hydrodynamical theory based on the canonical Lagrange formalism, assuming the large intrinsic angular momentum equal to L0 = h¯ ρ/2m at T = 0. At the same time Hu and Saslov (1977) suggested the hydrodynamical theory for a vanishing intrinsic angular momentum magnitude L0 = 0. Since then a considerable number of papers on 3 He-A hydrodynamics have been published, but it has remained a matter of controversy (Hall and Hook, 1986). Bearing this controversy in mind, we shall present the hydrodynamical theory allowing an arbitrary intrinsic angular momentum L0 (Sonin, 1984). In contrast to the direction of the intrinsic angular momentum given by the unit vector l, its magnitude L0 is not a hydrodynamical variable, and eventually the theory must consider L0 as some function of density and temperature. But for a better insight into the physical picture it is useful for a while to include L0 in the list of independent variables. In the light of this, the Gibbs relation for variation of the energy density in an arbitrary inertial coordinate frame is dE = μdρ + T dS + v n · dj 0 + j · dv s + ωL dL0 +

∂E ∂E dl + d∇j l, ∂l ∂∇j l

(10.31)

where ωL = ∂E/∂L0 . The pressure is determined by the usual thermodynamic formula: P = −E + T S + μρ + j 0 · v n + ωL L0 .

(10.32)

Differentiation of Eq. (10.32) yields the Gibbs–Duhem relation: dP = ρdμ + SdT + j 0 · dv n + L0 dωL − j · dv s −

∂E ∂E dl − d∇j l. ∂l ∂∇j l

(10.33)

In addition to the continuity equations of two-fluid hydrodynamics for mass and entropy, ∂ρ + ∇ · j = 0, ∂t

∂S + ∇ · (Sv n ) = 0, ∂t

(10.34)

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Vortex dynamics and hydrodynamics of a chiral superfluid

there are a modified equation for the phase (Josephson equation),   h¯ ∂θ + ωL + μ = 0, 2m ∂t and the orbital dynamics equations     ∂l h¯ δE L0 = − L0 vnj + (jj − ρvnj ) ∇j l − l × , ∂t 2m δl   ∂L0 h¯ + ∇ · L0 v n + (j − ρv n ) = T . ∂t 2m

(10.35)

(10.36) (10.37)

We chose the boson mass M equal to the mass 2m of two particles forming a Cooper pair. Here T is the magnitude of the torque, which will be defined below. The torque T enters the balance equation for the intrinsic angular momentum, violating its conservation law:    hl ∂E ∂(L0 l) ¯ + ∇j L0 lvnj + (jj − ρvnj ) − l × ∂t 2m ∂∇j l     ∂E ∂E =− l× − ∇j l × + T l. (10.38) ∂l ∂∇j l ˜ /δl ˜ defined by Eq. (10.27), one can introduce the functional derivative In analogy with δH ˜ δl ˜ taking into account the dependence of v s on l: δE/ ˜ δE h¯ δE + [{(j − ρv n ) · ∇}l × l]. = ˜δl δl 2m Then Eq. (10.36) becomes  L0

   ˜ δE ∂l . + (v n · ∇)l = − l × ˜ ∂t δl

(10.39)

(10.40)

The Euler equation for the superfluid velocity v s is obtained by differentiation in space of the equation (10.35) for the phase. In doing this one must take into account non-commutativity of the differentiation operators applied to the phase [see Eq. (10.4)]:   h¯ ∂l h¯ ∂vsi + × ∇i l = −∇i μ − ∇i ωL . (10.41) ∂t 2m ∂t 2m In the momentum conservation law, ∂ji + ∇j ij = 0, ∂t

(10.42)

the momentum flux tensor is ∂E ∂E ∂E T + vsi + ∇i lk + εij k lk ∂j0j ∂vsj ∂∇j lk 2 ∂E T = P δij + j0i vnj + vsi jj + ∇i lk + εij k lk . ∂∇j lk 2

ij = P δij + j0i

(10.43)

10.3 Hydrodynamics of a chiral superfluid

279

The momentum flux tensor is not symmetric. Therefore there is no conservation law for the extrinsic angular momentum [R × j ]. Its balance equation is ∂[R × j ]i + ∇j {εikl Rk lj } ∂t       ∂E ∂E ∂E − vs × − ∇lk × − T li . = − j0 × ∂j 0 ∂v s ∂∇lk

(10.44)

One can consider the balance equation for the sum of the extrinsic and intrinsic angular momenta:    hl ∂{[R × j ]i + L0 li } ∂E ¯ i + ∇j εikl Rk lj + L0 li vnj + (jj − ρvnj ) − l × ∂t 2m ∂∇j l i     ∂E ∂E = − j0 × − vs × ∂j 0 i ∂v s i       ∂E ∂E ∂E − ∇lk × − l× − ∇j l × . (10.45) ∂∇lk i ∂l i ∂∇j l i The right-hand side of this equation is the total derivative of the energy with respect to the angle of rotation around the ith coordinate axis. The energy must be invariant under such rotations. So the right-hand side vanishes, and the conservation law for the total angular momentum holds although there is no conservation law for the intrinsic and extrinsic angular momenta separately since they are coupled. A crucial check for consistency of the system of hydrodynamical equations is that it must satisfy the energy conservation law. The time variation of the energy density can be determined with the help of the Gibbs relation (10.31) and the equations of motion for all hydrodynamical variables. This yields the energy balance equation   1 ∂E + ∇ · Q = T ωL − (l · [∇ × v n ]) , (10.46) ∂t 2 with the energy flux defined as

  h¯ (j − ρv n ) Q = μj + [ST + (j 0 · v n )]v n + ωL L0 v n + 2m ∂E ∂li T . + [l × v n ] − 2 ∂∇li ∂t

(10.47)

The energy conservation law is fulfilled in two cases. In the first case T = 0, and inclusion of the magnitude of the intrinsic angular momentum into the list of independent hydrodynamical variables makes sense. Despite T = 0, the intrinsic and extrinsic angular momenta are not yet decoupled and there is no conservation law for the intrinsic and extrinsic angular momenta separately: the right-hand side of Eq. (10.38) contains the torque related to rotations of the orbital vector l. It is natural to expect that in the zero-temperature limit without the normal component, the normal velocity v n must drop out of the equations. This condition can be satisfied only if the magnitude of the intrinsic angular momentum

280

Vortex dynamics and hydrodynamics of a chiral superfluid

is L0 = hρ/2m. This was assumed in the canonical formalism developed by Khalatnikov ¯ and Lebedev (1977), and we shall call the intrinsic angular momentum of such a magnitude the canonical intrinsic angular momentum. For the canonical intrinsic angular momentum Eq. (10.38) becomes     jj δE ∂(L0 l) + ∇j L0 l =− l× . (10.48) ∂t ρ δl Apart from the term related to the momentum transport with the centre-of-mass velocity j /ρ of the fluid, this equation is similar to the Landau–Lifshitz equation for the spontaneous magnetisation in ferromagnets. A similar equation describes precession of a mechanical rotator driven by the external torque. For us the second case is more interesting, when the energy conservation law holds because the expression in curled brackets on the right-hand side of Eq. (10.46) vanishes. The energy conservation law is satisfied independently of the value of T if ωL =

1 (l · [∇ × v n ]). 2

(10.49)

Because of this relation, the magnitude L0 of the intrinsic angular momentum ceases to be an independent variable and is determined from minimisation of the energy with respect to L0 . The minimisation of the energy in the rotating coordinate frame, where the normal component rotates as a solid body, yields Eq. (10.49). At not very high velocities and gradients one can assume that L0 (ρ, S) is a function of only mass and entropy densities ρ and S, i.e., dL0 =

∂L0 ∂L0 dρ + dS. ∂ρ ∂S

(10.50)

Then the equation (10.37) for L0 is not independent. It is compatible with the continuity equations in (10.34) only if the torque magnitude T is given by     ∂L0 ∂L0 ∂L0 h¯ − ∇ · (j − ρv n ) + L0 − ρ −S ∇ · vn. (10.51) T = 2m ∂ρ ∂ρ ∂S After the dependence of L0 (ρ, S) on ρ and S has been properly specified, Eq. (10.37) can be removed. Now we are ready to formulate the system of hydrodynamical equations for our case of interest when L0 is not an independent hydrodynamical variable. Because of the differential relation (10.50), the Gibbs relation (10.31) transforms to dE = μ dρ + T  dS + v n · dj 0 + j · dv s +

∂E ∂E dl + d∇j l, ∂l ∂∇j l

(10.52)

where the renormalised values of the chemical potential and the temperature were introduced: μ = μ + ωL

∂L0 , ∂ρ

T  = T + ωL

∂L0 . ∂S

(10.53)

10.3 Hydrodynamics of a chiral superfluid

281

We redefine the pressure as P  = −E + T  S + μ ρ + j 0 · v n .

(10.54)

This leads to a new Gibbs–Duhem relation: dP  = ρdμ + SdT  + j 0 · dv n − j · dv s −

∂E ∂E dl − d∇j l. ∂l ∂∇j l

In modified hydrodynamics the Josephson equation (10.35) becomes    h¯ ∂θ h¯ ∂L0 + ωL − + μ = 0, 2m ∂t 2m ∂ρ

(10.55)

(10.56)

while the Euler equation is

     h¯ ∂l ∂L0 ∂vsi h¯  + × ∇i l = −∇i μ − ωL − . ∂t 2m ∂t 2m ∂ρ

(10.57)

The expressions for the momentum and the energy flux change to    ∂L0 ∂L0 −S δij + j0i vnj + vsi jj ij = P  + ωL L0 − ρ ∂ρ ∂S + ∇i lk

∂E T + εij k lk , ∂∇j lk 2

T ∂E ∂li [l × v n ] − 2 ∂∇li ∂t      h¯ ρ ∂L0 ∂L0 h¯ − vn + − j . + ωL L0 − S ∂S 2m 2m ∂ρ

(10.58)

Q = μ j + [ST  + (j 0 · v n )]v n +

(10.59)

For further concretisation of the hydrodynamical theory, one needs an explicit expression for the energy, which is a generalisation of the two-fluid expression (6.18) taking into account the energy related to gradients of the orbital vector l: 1 Es = E0 (ρ) + (j0i − jli )(ρn−1 )ij (j0j − jlj ) + El , 2

(10.60)

where E0 is the energy of a uniform resting fluid, which depends only on the mass and entropy densities, El =

h¯ 2 ρ {K1 ∇i l · ∇i l + (K2 + K3 )[(l · ∇)l]2 } 4m2

(10.61)

is the energy density, which depends only on gradients of l, jl =

h¯ [C∇ × l − C0 l(l · ∇ × l)] 2m

(10.62)

is the contribution to the mass current coming from gradients of l, j0i = (ρn )ij (vnj − vsj ) + jli

(10.63)

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Vortex dynamics and hydrodynamics of a chiral superfluid

is the total muss current in the coordinate frame moving with the superfluid velocity v s , and (ρn−1 )ij are matrix elements of the matrix inverse to the normal density matrix (ρn )ij = (ρnl + ρnt )δij − ρnt li lj .

(10.64)

In general the parameters C and C0 are not necessarily connected with K2 and K3 by the relations (10.20). Using Eq. (10.63) one can write down the expression for the energy in the arbitrary inertial coordinate frame: 1 E = Es + j 0 · v s + ρvs2 2 1 1 = E0 (ρ) + vni (ρn )ij vnj + vsi (ρs )ij vsj + v s · j l + El , 2 2

(10.65)

where (ρs )ij = ρδij − (ρn )ij is the superfluid density matrix. One should remember that j 0 is in our set of hydrodynamical variables and must be kept constant when varying other variables. Now let us turn to the ground state angular momentum, which is expected in chiral superfluids even without rotation and therefore can be called the spontaneous angular momentum. It includes the contribution from the textural current j˜ l given by Eq. (10.30) and does not differentiate between the intrinsic and the extrinsic angular momenta, taking both into account. If the current is purely solenoidal (C0 = 0) there is a well defined angular momentum density L0 + C, which is similar to the spontaneous magnetic moment of spin in the Landau–Lifshitz theory of ferromagnetism (Landau and Lifshitz, 1984). Anderson and Morel (1961) called the appearance of the ground state angular momentum in 3 He-A ‘orbital ferromagnetism’. But the textural current j˜ also contains the non-local l term ∝ C0 , which cannot be reduced to a curl of some moment and was called anomalous. Some microscopic theories predicted a large C0 approximately equal to C at T = 0 [see Mermin and Muzikar (1980); Volovik (2003b), and references therein]. Together with the conception that L0 is very small, this leads to a conclusion that the ground state angular momentum is very small compared to hn/2. This conclusion was challenged by the latest ¯ analysis of the two-dimensional chiral superfluid (Tada et al., 2015), which supported the large value hn/2 of the ground state angular momentum. The debate on this issue continues ¯ (Volovik, 2014). A further complication of the problem is that the total spontaneous angular momentum of a chiral superfluid at equilibrium depends on edge currents through edge states and therefore depends on boundary conditions at the fluid border (Sauls, 2011; Huang et al., 2015). Because of this dependence the angular momentum has no universal value. We have already seen above that a self-consistent local Galilean hydrodynamics satisfying the momentum conservation law in the limit T → 0, ρn → 0 exists only if L0 → ρ h/2m and C0 → 0 in this limit [see a more detailed discussion with additional refer¯ ences by Sonin (1984)]. In order to reconcile the hydrodynamics with the microscopic theory predicting small L0 and large C0 , Volovik and Mineev (1981) suggested that the normal density remains finite even at T → 0 if l varies in space [see also Combescot

10.4 Gauge wheel

283

and Dombre (1986), and references therein]. However, no self-consistent detailed theory based on this idea was suggested. Further discussion of problems and various approaches in hydrodynamics of chiral superfluids can be found in the book by Kagan (2013). Experimental attempts to reveal the intrinsic angular momentum L0 will be discussed in Section 10.6 after discussion of the connection between orbital dynamics and mutual friction. The existence of the ground state angular momentum is also a manifestation of chirality. Walmsley and Golov (2012a) investigated torsional oscillations of a cavity filled by superfluid 3 He-A with a texture created by cooling down while continuously rotating. This must orient the total angular momentum of the fluid parallel to the angular velocity during cooling down. After cooling below the critical temperature the original rotation was stopped, but the cavity was rotated again, with the rotation sense the same or opposite to the sense of the original rotation, i.e., the same or opposite to the direction of the angular momentum. The response to the new rotations was asymmetric, which was evidence of the ground state angular momentum. The problem of the angular momentum also exists in chiral px + ipy -wave superconductors, where the angular momentum produces a magnetic moment (spontaneous magnetisation). Thus in chiral superconductors superconductivity coexists with orbital ferromagnetism. Spontaneous magnetisation must create stray magnetic fields outside samples. But sensitive scanning magnetometry did not reveal any sign of spontaneous magnetisation in the px + ipy -wave superconductor Sr2 RuO4 (Kirtley et al., 2007; Curran et al., 2014), although the resolution was high enough even for detection of the lowest value of spontaneous magnetisation among those predicted by various theories. Various explanations were discussed (domains, disorder) but the puzzle has not been resolved (see Lederer et al., 2014; Huang et al., 2014, and references therein). In chiral px + ipy -wave superconductors an expression similar to Eq. (10.30) connects the electric current with the orbital vector variation. If the anomalous current ∝ C0 does not vanish, the basic assumption of the Landau–Lifshitz theory of spin ferromagnetism that the current is a curl of a well defined magnetic moment (Landau and Lifshitz, 1984) does not hold, and the Landau–Lifshitz theory is not valid for orbital ferromagnetism. As a result, the Zeeman energy is not defined, and the theory must deal with currents but not with momenta (Braude and Sonin, 2006). 10.4 Gauge wheel An interesting consequence of hydrodynamics of chiral superfluids, which has been widely discussed and debated in the literature on superfluid 3 He, is the gauge wheel effect first pointed out by Liu and Cross (1979). The effect is connected with the term ωL = 12 (l · [∇ × v n ]) in the Josephson equation (10.56) for the phase, which can lead in principle to acceleration of the superfluid component. Ho and Mermin (1980) suggested a Gedanken experiment for observation of the effect. There are two containers, each in a state of equilibrium rotation around the same rotation axis (one container is above the other) but at different angular velocities 12 [∇ × v n ]. Connecting them by an axisymmetric tube and

284

Vortex dynamics and hydrodynamics of a chiral superfluid

assuming that the orbital vector l is the same at the ends of the tube, one has a gradient of ωL along the tube, which must accelerate a superfluid there. As already discussed in Section 6.1, terms in the Euler equation, which depend on the normal velocity and are responsible for the gauge wheel, depend on the definition of the chemical potential. The latter is not measured experimentally, and it is better to replace its gradient by gradients of the pressure and temperature using the Gibbs–Duhem relation. Repeating this procedure for a chiral superfluid and using the Gibbs–Duhem relation (10.55), the Euler equation (10.57) becomes   ∇i P  j h¯ ∂l S ∂vsi + × ∇i l = − + ∇i T  + 0 · ∇ i v n ∂t 2m ∂t ρ ρ ρ j 1 ∂E · ∇i v s − ∇i l ρ ρ ∂l    ∂L0 1 ∂E h¯ ∇i ∇j l − ∇i ωL − . − ρ ∂∇j l 2m ∂ρ −

(10.66)

In the limit T = 0 when the normal component vanishes, the only term on the right-hand side, which depends on the normal velocity, is the last term containing ωL . One can see that there is no gauge wheel without normal component for the canonical value of the intrinsic For any other value of L0 the gauge wheel remains at angular momentum L0 = hρ/2m. ¯ T = 0 (Sonin, 1984). In particular, this follows from the hydrodynamical theory of Hu and Saslov (1977), which assumed that L0 = 0. However, as discussed above, for any noncanonical intrinsic angular momentum without normal component, there is no consistent Galilean invariant hydrodynamical theory of chiral superfluids. The paradox of the gauge wheel effect without normal component is more evidence of this. 10.5 Vortices and macroscopic hydrodynamics of chiral superfluid A phase of 3 He Although the structure of the hydrodynamical equations for 3 He-A is not entirely clear at present, we may distract ourselves from this difficulty when formulating the ‘macroscopic’ hydrodynamics of the rotating chiral superfluid, which is a cruder theory evolving from the original ‘microscopic’ hydrodynamics after a coarse-graining procedure. However, the unresolved problem of the intrinsic angular momentum becomes vital again if one attempts to estimate some parameters in macroscopic hydrodynamics, for example, those for mutual friction (see Section 10.6). As a consequence of the Mermin–Ho relation (10.5) the solid body rotation of the superfluid as a whole does not require the creation of isolated singular vortex lines with vorticity concentrated at these lines. Energetically it is more beneficial to create continuous vortices without singular cores. In a sense we return to continuous vorticity in rotating classical fluids (Chapter 1), but not completely. Now continuous vorticity cannot appear without spatial variation of l (l-texture), and vorticity in solid body rotation is not constant but is a doubly periodic function in the plane normal to the rotation axis (the xy plane)

10.5 Vortices and macroscopic hydrodynamics

285

(Volovik and Kopnin, 1977). This costs some energy but the latter does not contain a large logarithm factor, in contrast to singular vorticity. A large number of various vortex structures have already been proposed. What vortex structure is in equilibrium depends on the angular velocity and the magnetic field (Sepp¨al¨a and Volovik, 1983; Ohmi, 1984; Maki and Zotos, 1985). The cell of the periodic vortex structure may be considered as an elementary superfluid vortex. Despite continuous distribution over the cell, circulation is always quantised as in a conventional superfluid. This follows directly from the Merlin–Ho relation (10.5) or more readily from the integral form of the relation derived by Ho (1978). The Ho circulation theorem is obtained by integration of the z component of the vorticity ∇ × v s , over the vortex structure cell in the xy plane. It yields the circulation around the elementary cell of the doubly periodic continuous vorticity structure:     h¯ ∂l ∂l × . (10.67) dx dy zˆ · [∇ × v s ] = dx dy l κv = 2m ∂x ∂y cell

cell

The direction of l at any point within the cell is presented by a point on the surface of a unit sphere. Then the l-texture yields a mapping of the vortex structure cell, which is topologically equivalent to a torus, on a spherical surface (Volovik, 1984). The integral in Eq. (10.67) is the area of mapping, which is equal to the surface area 4π of the unit sphere multiplied by an integer topological charge p. This charge indicates how many times the unit sphere is covered at mapping of the l-texture on the sphere. As a result, the circulation quantum of the elementary cell is κv = 2pκc .

(10.68)

So a non-singular continuous vortex is always characterised by an even number 2p of circulation quanta κc = h/2m for the Fermi superfluid. In contrast, the circulation of the singular vortex may be any integer number of quanta.1 Macroscopic hydrodynamics of a rotation superfluid is obtained by averaging the original hydrodynamical equations over the vortex cell (the procedure of coarse-graining). Thus its formal structure should not depend on details of the vortex structure within the vortex cell, in particular, on whether the vortex is singular or not. As usual, instead of the coarse-graining derivation one can deduce the macroscopic hydrodynamics by referring to thermodynamics, the conservation laws, and the symmetry of the order-parameter variables. In rotating 3 He-A, the degeneracy with respect to the direction of l is lifted, since l-texture is fixed by the vortex structure determined by minimisation of energy in the rotating coordinate frame. Then the vector l drops out from the list of independent hydrodynamical variables, and the order parameter is characterised by the same set of variables as in non-chiral superfluids: the superfluid velocity v s and the deformation tensor of the vortex structure determined by its displacements u. The velocity v L = ∂u/∂t 1 We do not consider here a more complicated case of a half-integer vortex, which is hybridised with a disclination in the field of

the spin vector d (Salomaa and Volovik, 1985).

286

Vortex dynamics and hydrodynamics of a chiral superfluid

determines time variation of l: ∂l/∂t = −vLi ∇i l. With help of the Mermin–Ho relation (10.5) one can rewrite the Euler equation (10.57) as    h¯ ∂L ∂vsi  + ω˜ × v L = −∇i μ − ωL − . ∂t 2m ∂ρ

(10.69)

This differs from the Euler equation for non-chiral superfluids only by the gauge wheel term containing ωL . Eventually the formal structure of the macroscopic hydrodynamics, as given in Section 6.1, remains valid for the chiral superfluid, but the equation of motion for vortices, Eq. (6.33), should be replaced by a more general equation allowing a lower symmetry of the superfluid, and the superfluid density becomes a tensor. The theory, developed for a superfluid with singular vortex lines (VLL state), was simplified by the assumption that the vortex lines perturb the superfluid only in their immediate vicinity, which made it possible to use the scalar superfluid densities while ignoring lowering of the symmetry due to the vortex lines (see Section 6.1). This assumption is not valid in a superfluid with continuous vorticity, in which the superfluid densities along the rotation axis (the z axis) and in the xy plane differ no matter what the symmetry of the vortex structure is. But the oscillation modes that we have considered involved mostly the motion in the xy plane or of the fluid as a whole, when the difference between ρs along the z axis and in the xy plane does not matter. In such cases the results of the theory developed for isotropic superfluids can be extended to an anisotropic superfluid with high symmetry of the vortex structure without any modification. In general, the analysis of the vortex oscillations in an anisotropic superfluid will require symmetry classification of the vortex structures (Section 3.2). 10.6 Mutual friction for continuous vortices in the A phase of 3 He The theory of mutual friction in 3 He-A for continuous vortices was developed by Kopnin (1978a) on the basis of the Boltzmann equation close to Tc . But the results of Kopnin can be obtained within the hydrodynamical theory without referring to the Boltzmann equation, and their validity is not restricted by the Ginzburg–Landau region near Tc . Here we shall give the hydrodynamical derivation of the equation of vortex motion, including the effect of orbital inertia (Sonin, 1986), which was not considered by Kopnin. Suppose that a continuous vortex moves through the superfluid. The latter is assumed to be incompressible in the mechanical and the thermal sense, i.e., the densities of mass and entropy do not vary. We consider the clamped regime, in which the normal fluid is not dragged by the vortex and v n is constant. Then we need only the equation of orbital dynamics and the condition of incompressibility. We add to the equation (10.40) of orbital dynamics the dissipative term proportional to the orbital viscosity χ (Cross, 1977):      ∂l ∂l δE + (v n · ∇)l − χ + (v n · ∇)l = . L0 l × ∂t ∂t δl

(10.70)

10.6 Mutual friction for continuous vortices

287

The functional derivative δE/δl takes into account the dependence of the superfluid velocity variation on variation of l, though the sign ‘tilde’ was omitted [see Eqs (10.26) and (10.27)]. The condition of incompressibility is 0=

∂E h¯ ∂E h¯ δE = −∇ =− ∇ ∇ · j sn , =− δθ ∂∇θ 2m ∂v s 2m

(10.71)

where j sn = j − ρv n

(10.72)

is the mass current in the coordinate frame moving with the normal velocity v n . The vortex moves with the constant velocity v L , so ∂l/∂t = −(v L · ∇)l. Linearisation of Eqs. (10.70) and (10.71) with respect to the small perturbations produced by vortex motion yields L0 × [(v s − v L ) · ∇] l − χ [(v s − v L ) · ∇] l = 0=

δ2E  δ2E  l + 2 δθ . δlδθ δθ

δ2E δl 2

l +

δ2E  θ, δlδθ

(10.73) (10.74)

The symbols δ 2 E/δl 2 , δ 2 E/δlδθ , and δ 2 E/δθδl denote here linear differential operators applied to the perturbations l  and θ  . Equations (10.73) and (10.74) are inhomogeneous linear equations for l  (r) and θ  (r). As in Section 8.7 addressing mutual friction in the Ginzburg–Pitaevskii theory, we derive the equation of vortex motion from the solvability condition. The equations have a solution only if the left-hand sides (the inhomogeneous terms) are orthogonal to any solution of the adjoint homogeneous equations. Because of translational invariance, the solution of the homogeneous equations is obtained by an arbitrary translation of the solution for the vortex at rest. The solution will be denoted by the superscript t: 2m (t · v s ), l t = (t · ∇)l, h¯ (10.75) h¯ h¯ t t ∇i θ t + l · [∇i l × l]. vsi = 2m 2m Here t is an arbitrary vector of translation. In order to obtain the solvability condition it is necessary to multiply Eq. (10.73) by l t and Eq. (10.74) by θ t , to integrate both equations over the whole plane, and to sum integrals. Because the linear differential operators are self-adjoint, integration by parts reduces the volume integrals of the right-hand sides of Eqs. (10.73) and (10.74) to surface integrals. Following Kopnin (1978a), we consider the axisymmetric vortex with l parallel to the z axis far from the vortex. Then l  and l t vanish at the surface far from the vortex, and the surface integrals contain only θ  and θ t . Eventually the solvability condition is  dr(t · ∇) {L0 × [(v s − v L ) · ∇] l − χ [(v s − v L ) · ∇] l}  h¯ =− dSi [θ t (r)jn (r) − θ  (r)jnt (r)]. (10.76) 2m θ t = (t · ∇)θ =

288

Vortex dynamics and hydrodynamics of a chiral superfluid

Here the differential vector dS is normal to the surface and directed outside. The currents far from the vortex are j sn = j tsn

∂ 2E  v = ρs⊥ (v sl − v n ), ∂v s ∂vsj sj

(10.77)

∂ 2E t h¯ = vsj = ρs⊥ ∇θ t , ∂v s ∂vsj 2m

where ρs⊥ is the superfluid density in the plane normal to l. The superfluid velocity v sl is a velocity far from the vortex, but still at a small distance compared to other hydrodynamical scales, the vortex line curvature radius, the distance from other vortices or from the wall, for example. Since the condition (10.76) must hold at any translation t, we obtain after integration: −

2mL0 κv zˆ × (v L − v n ) + χ γˆ (v L − v n ) = −ρs⊥ κv zˆ × (v sl − v n ). h¯

(10.78)

The integral in the orbital inertial term [the first term on the left-hand side of Eq. (10.78)] has been reduced to the integral for circulation in Eq. (10.67). The tensor γˆ has components  (10.79) γij = dr∇i l · ∇j l only in the xy plane, since ∇z l = 0. For the axisymmetric vortex it is reduced to a scalar γ δij (i, j  = z) and     2   ∞ dβ 2 1 dα 2 r dr + sin β 2 + γ =π (10.80) dr dr r 0 for l-texture given by lx = sin β(r) cos[α(r)+φ], ly = sin β(r) sin[α(r)+φ], lz = cos β(r) where r and φ are polar coordinates in the plane. Hall (1985) estimated γ = π 3 /2 or π 2 /3 for different models of the Anderson–Toulouse two-quantum continuous vortex (Anderson and Toulouse, 1977). Comparing Eq. (6.38) with Eq. (10.78) we obtain the mutual friction parameters in the chiral superfluid: d=

χγ , ρs⊥ κv

d = 1 −

2mL0 . ρs⊥ h¯

(10.81)

The expressions for the mutual friction parameters α and α  can be obtained from Eq. (10.81) with the help of Eq. (6.37): α=

1+



αK 2mL0 ρs⊥ h¯ αK

Here αK =

2 ,

α = 1 −

2mL0 2 ρs⊥ h¯ αK

1+



2mL0 ρs⊥ h¯ αK

  3.5 T −1/2 ρs⊥ κv ≈ 1− χγ γ Tc

2 .

(10.82)

(10.83)

10.6 Mutual friction for continuous vortices

289

is the value of α obtained by Kopnin (1978a) and Cross (1983) neglecting orbital inertia (L0 = 0). Equation (10.82) shows that the orbital inertia term ∝ L0 contributes to the active and reactive components of the mutual friction force. The quantity 2mL0 κv 1 − α 2mL0 αK = = hχ α ρs⊥ h¯ ¯ γ

(10.84)

is of the same order as the quality factor L0 /χ of an orbital wave described by the linearised equation (10.70) of orbital dynamics. Thus at a very small intrinsic angular momentum L0 , the orbital wave is strongly overdamped. On the other hand, according to Eq. (6.78) (1 − α  )/α is the quality factor of the Kelvin mode, which is also overdamped at small L0 . This is no surprise, since in a chiral superfluid the Kelvin wave itself is nothing other than an orbital wave propagating in the periodic vortex structure. When the temperature approaches the critical value Tc , Kopnin’s theory (L0 = 0) predicts that α ∝ (1 − T /Tc )−1/2 , while assuming L0 ∝ ρs , the critical behaviour is α ∝ (1 − T /Tc )1/2 . Thus orbital inertia changes the critical behaviour of mutual friction, and measurement of the mutual friction can give evidence of the existence of an intrinsic angular momentum, as was pointed out first by Hall (1985). Hall revealed a discrepancy between values of mutual friction parameters measured experimentally in 3 He-A and those obtained in the theory with L0 = 0. Although his explanation of this discrepancy was different from that given here, it was also related to the effect of the intrinsic angular momentum (see a comparison of the two interpretations by Sonin, 1987). Experimental investigation of orbital relaxation by Paulson et al. (1976) did not reveal any traces of orbital inertia. They claimed the absence of any tendency to oscillation in the process of orbital relaxation, which means that L0  χ . Detailed measurements of mutual friction parameters in 3 He-A by Bevan et al. (1997b) did not reveal any measurable deviation of α  from 1, i.e., no intrinsic angular momentum L0 was detected. In summary, one should repeat after Leggett (2006) that the problem of the intrinsic angular momentum (chiral anomaly) ‘is more than 30 years old and still has apparently not attained a universally agreed resolution’. The only correction needed now is ‘40 years’ instead of ‘30 years’.

11 Nucleation of vortices

11.1 Thermal nucleation of vortices in a uniform flow Thermal and quantum nucleation of vortices in superfluids attracted the attention of theorists long ago (Iordanskii, 1965b; Langer and Fisher, 1967; Muirihead et al., 1984). The quantum nucleation of vortices by superflow in small orifices (Davis et al., 1992; Ihas et al., 1992) and by moving ions (Hendry et al., 1988) has been reported. The process of vortex nucleation is crucial for onset of essential dissipation when superfluid velocities reach the critical velocity for penetration of vortices into a container. The original state is a metastable state with a persistent vortex-free superfluid flow. Vortex nucleation is necessary for transition to a state with a smaller superfluid velocity (and eventually to the stable equilibrium state with zero velocity) in the case of uniform flows in channels, or for transition to solid body rotation with an array of straight vortices parallel to the rotation axis in the case of rotating containers. In the process of vortex nucleation a small vortex loop appears, which grows in size. Eventually the vortex loop transforms to a straight vortex line in the case of rotation, or the vortex line crosses the channel crosssection decreasing the phase difference between ends of the channel by 2π (the phase slip). The latter process is illustrated in Fig. 11.1. Although vortex nucleation is a key process, which determines critical velocities, the problem of critical velocities does not reduce to the nucleation problem. The theory of critical velocities requires introduction of additional definitions and assumptions. One can find discussion of critical velocities with relevant references elsewhere (Donnelly, 1991; Varoquaux, 2015). Vortex nucleation is possible due to either thermal or quantum fluctuations in the fluid. This section addresses the Iordanskii–Langer–Fisher theory of thermal nucleation (Iordanskii, 1965b; Langer and Fisher, 1967). The rate of thermal nucleation of vortices is governed by the Arrhenius law ∝ e−Em /T . The energetic barrier Em is determined by a maximum of the energy of a vortex loop in the process of its growth. In order to find Em one should solve a static problem: it is necessary to find a vortex loop corresponding to an energy extremum (saddle point) at fixed mass current through the loop. When there is a superflow past the vortex line, the total velocity field is v s +v v , where v s is the velocity field line. Then the energy produced by the superflow and v v is the velocity induced by the vortex  of interaction between the vortex and the superflow is Ef = ρs (v v · v s ) dR. If the vortex

290

11.1 Thermal nucleation of vortices in a uniform flow

291

Figure 11.1 Cross-sections of the surfaces of constant order parameter phase θ showing schematically the successive stages of the vortex loop nucleation. Figure from Langer and Fisher (1967).

line is a closed loop, the integral determining Ef can be transformed by integration by parts to the integral over the surface SL attached to the vortex line. At this surface encircled by the closed vortex loop the phase θv , which determines the velocity v v = (κ/2π )∇θv induced by the vortex line, jumps by 2π, and

Ef =

ρs κ 2π



 (∇θv · v s )dR = ρs κ

(v s · dS). SL

(11.1)

292

Nucleation of vortices

Here the infinitesimal vector dS is normal to the surface SL and points in the same direction as the velocity v v inside the loop. Thus the interaction energy is simply the factor κ times the mass current through the loop. For a uniform superflow when v s does not vary in space, Eq. (11.1) yields Ef = v s · P , where P = ρs κS

(11.2) 

is the momentum of the vortex loop, and a projection of the vector S = dS on some axis is equal to the area of the projection of the vortex loop area on a plane normal to this axis. The whole energy of the vortex loop in the uniform superflow is E = E0 +Ef = E0 +v s ·P , as follows directly from Galilean invariance. If the superflow is not uniform, the energy of the vortex loop lying in a plane is E = εL − ρs κ v¯s S,

(11.3)

where ε is the vortex line tension, L and S are the vortex loop length and area respectively, and v¯s is the component of the superfluid velocity along the direction inverse to S averaged over the loop area. The stationary shape of the vortex loop in a uniform superflow is a circumference, which is a curve of constant radius of curvature r (vortex ring). Then L = 2π r and S = π r 2 . However, a two times smaller barrier, which allows more effective nucleation, is realised at nucleation of a half-ring near a plane border of a superfluid with length L = π r and area S = π r 2 /2. Taking into account an image of the half-ring in the wall, the velocity field induced by the half-ring is the same as that of the full ring far from the wall. The maximum of the half-ring energy is at the radius κ ε κ = ln2 . (11.4) rm = mns κvs 4π vs vs rc The energy barrier, which corresponds to the energy maximum at r = rm , is Em = E0 (rm ) − vs P (rm ) =

ρs κ 3 κ ln 32π vs vs rc

(11.5)

One may consider the energy barrier for thermal nucleation of a point vortex in a thin film. The energy of the vortex in a uniform flow is E = ε − ρs κvs r =

r ρs κ 2 ln − ρs κvs r, 4π rc

(11.6)

where r is the distance from the film border, and ρs is a two-dimensional superfluid mass density, in contrast to the three-dimensional ρs in Eq. (11.5). The energy extremum is at κ , (11.7) rm = 4π vs and the energy barrier is Em ≈

κ ρs κ 2 rm ρs κ 2 ln ln ≈ . 4π rc 4π vs rc

(11.8)

11.2 Thermal nucleation of vortices in a non-uniform superflow

293

A more detailed analysis of the nucleation process requires taking into account interaction of the vortex loop with thermal excitations (quasiparticles) and solving the Fokker– Planck equation resulting from this interaction (Iordanskii, 1965b). Eventually this yields the nucleation rate per unit volume: R = R0 e−Em /T ,

(11.9)

where R0 is interpreted as some attempt frequency. In the case of the thermal escape of a particle from a potential well, the meaning of the attempt frequency is more or less clear: it is a frequency of particle oscillations in the well, which determines how frequently the particle hits the well wall. In the case of the vortex loop it is a more tricky issue. Langer and Fisher (1967) suggested some atomic collision rate for estimation of R0 . Fortunately an accurate definition of R0 is not so important. For determination of the critical velocity at which nucleation becomes intensive enough, the value of the argument of the exponential function is much more essential than the value of the pre-exponential factor.

11.2 Thermal nucleation of vortices in a non-uniform superflow Experimental investigations have shown qualitative agreement with the Iordanskii–Langer– Fisher theory [see Brown and Hess (1982) and references therein]. However, usually the experimental critical velocities were lower than predicted. It was suggested that roughness of channel walls and inhomogeneity of superfluid velocity resulting from this were responsible for the lower critical velocities (Vinen, 1961b; Brown and Hess, 1982; Soininen and Kopnin, 1994). Namely, the nucleation rate is determined not by an average velocity, but mostly by values in areas where it is much higher than average. This urges us to consider nucleation in non-uniform superflows. An example of such a flow is the flow past a sharp wedge, shown in Fig. 11.2 (Sonin, 1995). The wedge has an angle β and is uniform along the z axis, the superflow being in the xy plane and directed along the x axis on average. So this is a two-dimensional problem, which has a simple solution in terms of analytical functions on the complex plane (Landau and Lifshitz, 1987). The phase field, which is a κ ∇θ , in polar coordinates (r, φ), is scalar potential for the potential velocity field v s = 2π    2π vs0 a  r k π −β θ =− , cos k φ + kκ a 2

(11.10)

where the azimuthal angle φ is measured from the axis x. The wedge sides correspond to 3π −β φ = β−π and the condition that there is no velocity component normal to 2 and φ = 2 the wedge sides is k=

π . 2π − β

(11.11)

294

Nucleation of vortices (a)

(b)

Figure 11.2 Vortex nucleation in a superflow past a wedge of angle β. (a) Superflow without the vortex. View along the wedge. (b) Vortex line at the wedge (solid line). View normal to the wedge. Figure from Sonin (1995).

For further analysis one needs the velocity field along the y axis (φ = π/2), where the velocity is directed parallel to the x axis and is a function of y:  1−k a . (11.12) vs (y) = vs0 y Here vs0 is the velocity at the distance y = a from the wedge edge. The case β = π (k = 1) corresponds to a uniform superflow near a plane wall, which was considered in the previous section. The case β = 0 (k = 1/2) describes an edge of an orifice in a thin partition dividing two fluid bulks, which will be discussed in more detail below. Now we present a qualitative scaling estimate for a general case (Brown and Hess, 1982). For this estimate we use Eq. (11.4) which connects the size rm of the loop with the superfluid velocity vs , but we assume that vs is the velocity vs (y) at the distance y ∼ rm . Then according to Eq. (11.12): 1/k   r 1−k k−1 ε κ κ m , or rm ∼ ln a k . (11.13) rm ∼ mns κvs0 a 4π vs0 vs0 rc Finally the activation barrier is  Em ∼ εrm ∼ ε

κ κ ln 4π vs0 vs0 rc

1/k a

k−1 k

.

(11.14)

Ignoring weak logarithmic vs0 -dependence of the line tension ε, the dependence of the 2 (orifice activation barrier on vs0 may vary from ∝ 1/vs0 (ideal plane wall) to ∝ 1/vs0 edge). Later we shall see that the effect of sharp wedges on quantum nucleation (Sections 11.3 and 11.5) is even stronger. The barrier for nucleation near the wedge differs by the

11.2 Thermal nucleation of vortices in a non-uniform superflow

295

factor (κ/vs0 a)(1−k)/k (k < 1) from the barrier for vortex nucleation in a uniform flow. This factor is small if the size rm of the extremal vortex loop is small compared to a. Since our scaling estimation is valid only if rm  a, within limits of this estimation a sharp wedge decreases the activation barrier for vortex nucleation as expected. We can go beyond a scaling estimation for a vortex loop near the edge of a small orifice of radius a0 in a thin partition dividing two fluid bulks (Sonin, 1995). This is the limit of a wedge with β = 0. The velocity field of the initial vortex-free superflow can be found analytically using oblate spheroidal coordinates (Morse and Feshbach, 1953, p. 1294). We need to know only the velocity in the partition plane where it is normal to the plane: vs0 a0 . (11.15) vs (r) = $ 2 a02 − r 2 Here r is the distance from the centre of the orifice and vs0 is the average superfluid velocity in the orifice plane. We shall consider a loop which is small in size compared to the orifice radius a0 . Then one may assume that the edge of the orifice is straight and coincides with Then the velocity the z axis, whereas y = a0 − r is the distance from the edge (Fig. 11.3b). √ near the orifice edge is given by Eq. (11.12) with k = 1/2 and a = a0 /2 2. According to Eq. (11.3) the total energy is: #  zc   zc " a0 dy(z) 2 1+ dz − ρs κvs0 y(z)dz. (11.16) E=ε dz 2 −zc

−zc

Here y(z) describes the shape of the vortex loop and ±zc are the coordinates of points where the vortex line enters the edge. Varying the energy with respect to y(z), one obtains the differential equation for the shape of the vortex line y(z):   dy/dz 1 d = 0, (11.17) − √ − " 2 dz 2 ym y(z) 1 + (dy/dz) where the length ym is given by 2 2   2 2 ε κ κ ym = = ln . mns κvs0 a0 4π vs0 vs0 rc a0 The first and the second integrations of Eq. (11.17) yield at z > 0:  y dz =− , dy ym − y # ym # " y ym − y  z(y) = dy = y(ym − y) + ym arcsin .  ym − y ym

(11.18)

(11.19)

y

Equation (11.19) shows that ym is the maximal distance of the vortex line from the orifice edge. The dimension of the loop along the orifice edge is 2zc = πym . The vortex loop is shown in Fig. 11.3b.

296

Nucleation of vortices (a)

2a0

Z

ZC y

(b)

ym Figure 11.3 Nucleation of the vortex loop in an orifice of radius a0 in the partition dividing two superfluid bulks. (a) The cross-section of the whole orifice. (b) The vortex loop nucleated (thermally or via quantum tunnelling) in the plane of the orifice near its edge. The loop is small compared to the orifice radius, and in the figure the circular orifice edge is shown as a straight border. Figure from Sonin (1995).

Next we can calculate the energy of the extremal loop using Eq. (11.16). This energy determines the height of the activation barrier for vortex nucleation:  2 κ 4 8ε κ Em = εym = ln . (11.20) 3 3a0 4π vs0 vs0 rc This agrees with the qualitative estimation (11.14) for k = 1/2 (β = 0).

11.3 Nucleation of a massless vortex via macroscopic quantum tunnelling: semiclassical theory Approaching zero temperature, thermal nucleation of vortices becomes more and more improbable, but vortex nucleation can occur via quantum tunnelling. Experimental evidence of the crossover from thermal to quantum nucleation could be saturation of the temperature dependence of the critical velocity, although sometimes one should filter out other less exciting explanations for saturation (Varoquaux, 2015). The semiclassical theory of quantum tunnelling of a particle through a barrier is presented in any textbook on quantum mechanics (see Landau and Lifshitz, 1982a, for example). However, the vortex is a macroscopic perturbation of a fluid, and its quantum tunnelling is a process of macroscopic quantum tunnelling, which changes states of a huge number of particles. The central assumption of the macroscopic quantum tunnelling

11.3 Nucleation of a massless vortex

297

concept is that this many-body process can be reduced to dynamics of a single macroscopic degree of freedom. This assumption is not self-evident and, as we shall see, not ideally exact, but there is a broad consensus (including the present author) that the assumption provides a reasonably simple but at the same time a reasonably reliable quantitative description of a very complicated process. Here we restrict ourselves to an elementary theory of macroscopic tunnelling, putting aside such interesting topics as the effect of environment (Caldeira and Leggett, 1983). The quantum tunnelling theory considers motion of a particle in a classically forbidden area under the barrier by transition to imaginary time or coordinate. One needs to calculate  the action S = L dt along the trajectory crossing the classically inaccessible underbarrier region. The exponent of the probability of tunnelling W ∼ e− is determined by the imaginary part of the action: = 2ImS/h. ¯ If the trajectory is found from the equation of motion, the action along the trajectory follows from the Hamilton–Jacobi theory:  Pi dxi , (11.21) S= i

L

where summation is over all pairs of conjugate variables (xi , Pi ) and integration is over the trajectory L determined from the equations of motion. For application of this procedure to the vortex, it is important that the dynamics of the vortex is essentially different from that of a particle governed by Newton’s second law. As already stressed many times in this book, usually one can ignore the inertia force (i.e., ignore the vortex mass) in the equation of vortex motion, assuming that any external force on the vortex is opposed not by the inertia force, but by the Magnus force. Calculation of the classical action, which determines the probability of quantum tunnelling, must be based on the equation of motion of a massless vortex. The Magnus force is responsible for the Hall effect in superconductors, and this type of tunnelling is sometimes called ‘Hall tunnelling’ (Blatter et al., 1994). Since we deal with neutral superfluids we shall call tunnelling of a massless vortex Magnus tunnelling. The first analysis of Magnus tunnelling for a superfluid vortex was done by Volovik (1972), who considered nucleation of a circular vortex half-loop near a plane boundary. The Lagrangian for the half-loop is L = xP ˙ x (r) − Ev (r) − V (x, r),

(11.22)

where the vortex momentum Px (r) and the vortex energy Ev (r) are given by Eqs. (11.2) and (11.3) with length L = π r and area S = π r 2 /2, and V (r, z) is the energy of interaction with the defect of the plane boundary. One should look for a trajectory at fixed total energy Ev (r) + V (x, r) = 0 since the initial state without the vortex has zero energy. The underbarrier region 0 < r < rf , where the energy Ev is positive, is classically inaccessible. Here rf =

2ε κ κ = ln mnκvs 2π vs vs rc

(11.23)

298

Nucleation of vortices

is the radius of the half-loop, which corresponds to zero energy Ev . We consider zero temperatures, replacing everywhere the superfluid density ns by the total density n. For an ideal plane boundary, i.e., a plane without defects [V (x, r) = 0], there is no trajectory which crosses the classically inaccessible underbarrier region. This is a consequence of translational invariance along the axis x. The x component Px (r) of the momentum is conserved, and the radius r remains constant along any trajectory, even on the complex plane, and there is no quantum process which would be able to change its value. So the presence of the term V (x, r) in the Lagrangian, which breaks translational invariance, is crucial. As a defect responsible for this term, Volovik (1972) considered a half-sphere bulge on the plane boundary, looking for half-loop trajectories in terms of spherical functions. Meanwhile, in order to calculate the tunnelling exponent, the shape of the defects is not so essential. It can affect only a pre-exponential factor, which is difficult to calculate anyway. The only requirement for interaction with a small-size defect localised in the point r = x = 0 is a singularity of the potential V (x, r) at x 2 + r 2 → 0 (Sonin and Horovitz, 1995; Sonin, 1995). In the Lagrangian (11.22) for a three-dimensional vortex half-loop one can choose the simplest singular δ-function potential V (x, r) = −gδ(r 2 + x 2 ). The trajectory must start from a microscopic loop of negligible radius r ≈ rc ≈ 0 at x = 0, where the defect is located. The trajectory goes into the complex x plane along the line x = −ir. This satisfies the condition r 2 + x 2 = 0, at which the vortex along the trajectory interacts with the defect. When the trajectory reaches the point r = rf , x = −irf it crosses the trajectory with constant r = rf along which interaction with the defect is absent. The path continues along this new trajectory until the point r = rf , x = 0. Here the path returns from the complex x plane to the real x axis. Eventually the path connects the values r = rc ≈ 0 and r = rf on both sides of the barrier. The tunnelling exponent for such a path is ⎧ ⎫ −ir ⎪ ⎪  f 0 ⎨ ⎬ 2 2ImS = Im Px (ix) dx + Px (rf ) dx = ⎪ ⎪ h¯ h¯ ⎩ ⎭ −irf

0

2 = h¯

rf 0

4π 2 3 κ 3n nrf = [Px (rf ) − Px (r)]dr = 3 6π vs3

  κ 3 ln . vs rc

(11.24)

The tunnelling exponent does not depend on the amplitude of the potential, but the latter can influence the pre-factor of the exponential function, which is not considered here. For a two-dimensional system describing point vortices in thin films the Lagrangian is L = xP ˙ x (y) − Ev (y) − V (x, y),

(11.25)

where Px (y) = κmny,

Ev (y) =

y mnκ 2 ln − vs Px (y) 4π rc

(11.26)

11.4 Quantum nucleation of a vortex with mass

299

are the momentum and the energy of a point vortex at distance y from the film edge. The barrier region where the energy Ev (y) is positive extends from y ≈ rc ≈ 0 to y = yf with κ κ yf = ln . (11.27) 4π vs vs rc As in the three-dimensional case, the shape of the defect responsible for tunnelling is not important, and it is possible to choose the simplest interaction with the defect: V (x, y) = −gδ(x 2 + y 2 ).1 The underbarrier trajectory starts at the point r ≈ 0, x = 0, goes along the trajectory x = −iy, and after reaching the point x = −iyf , y = yf goes to the point y = yf , x = 0 along the trajectory y = yf . Eventually the logarithm of tunnelling probability W ∼ e− is ⎫ ⎧ −iy ⎪ ⎪  f 0 ⎬ ⎨ 2ImS 2 = Px (ix) dx + Px (yf ) dx = Im ⎪ ⎪ h¯ h¯ ⎭ ⎩ 0

2 = h¯

−iyf

yf [Px (yf ) − Px (y)]dy = 0

2π nyf2

κ 2n = 8π vs2

  κ 2 . ln vs rc

(11.28)

In both the three-dimensional and two-dimensional cases, the probability logarithm is roughly equal to the number of particles in in the area occupied by the velocity field induced by the vortex after nucleation (nrf3 in the three-dimensional case and nyf2 in the two-dimensional case). This can be used for scaling estimation of in non-uniform flows. We shall consider quantum nucleation in the non-uniform flow past a wedge (Sonin, 1995). In a uniform flow the size rf of the nucleated vortex loop [Eq. (11.23)] differs from the size rm of the loop of maximal energy given by Eq. (11.4) only by a numerical factor. Assuming that this also holds in a non-uniform flow, one can estimate the probability logarithm for quantum tunnelling at the wedge using the value of rm given by Eq. (11.13):  3/k 3(k−1) κ κ 3 k ∼ nrm ∼ na ln . (11.29) 4π vs0 vs0 rc Thus the wedge has an even more pronounced effect on quantum nucleation than on thermal nucleation. For flow through the orifice in the thin partition (k = 1/2) the probability 6 instead of 1/v 3 for a uniform superflow. In the case of logarithm is proportional to 1/vs0 s 4.5 . the direct-angle wedge (k = 1/3) the probability logarithm is proportional to 1/vs0 11.4 Quantum nucleation of a vortex with mass at a thin film edge It is interesting to discuss how vortex mass can affect quantum vortex nucleation via Magnus tunnelling. Muirihead et al. (1984) addressed this issue for a two-dimensional 1 This is supported by the analysis of Magnus tunnelling of a two-dimensional vortex near a circular (Sonin and Horovitz, 1995)

and an elliptic (Fischer, 1998) bulge of the wall.

300

Nucleation of vortices

vortex near a film edge using the analogy of a vortex subject to the inertia, Magnus and Lorentz forces to a charge in crossed magnetic and electric fields (see also Donnelly, 1991, Section 8.5.4). They used the simplified version of the real potential for the vortex. Here we present a semiclassical theory for this case, taking into account the real potential. This allows us to consider the crossover from the case of a massive vortex to a massless vortex nucleated via Magnus tunnelling. For a two-dimensional vortex with mass, the Lagrangian (11.25) must be modified to μv y˙ 2 μv x˙ 2 ˙ − Ev (y), (11.30) + + hnM xy 2 2 where μv is the vortex mass and the energy Ev (y) is given by Eq. (11.26) as before. In ˙ responsible for the Magnus force we replaced the particle density n by the term hnM xy some effective density nM . In a Galilean invariant fluid at T = 0, the effective density nM cannot differ from n. But introducing nM one can tune the Magnus force in order to compare its effect with the effect of the inertia force. In the present analysis one need not retain the energy of interaction with a defect V (x, y). Now we have two degrees of freedom, in contrast to one degree of freedom for a massless vortex. Correspondingly we have two momenta canonically conjugate to coordinates x and y: L=

Px =

∂L = μv x˙ + hnM y, ∂ x˙

Py =

∂L = μv y. ˙ ∂ y˙

(11.31)

The energy (Hamiltonian) for the Lagrangian (11.30) is H=

Py2 ∂L μv x˙ 2 μv y˙ 2 ∂L (Px − hnM y)2 x˙ + y˙ − L = + + Ev (y) = + + Ev (y). ∂ x˙ ∂ y˙ 2 2 2μv 2μv (11.32)

The classical equations of motion are ˙ μv x¨ = −hnM y,

μv y¨ = hnM x˙ −

∂Ev (y) . ∂y

(11.33)

The equations have two integrals. The first integral is the momentum Px , which is conserved because of translational invariance along the axis x. The second integral is the energy. The trajectory must correspond to zero energy. At y = 0, where the trajectory must start, Eq. (11.32) shows that the zero energy condition yields the relation connecting ˙ at y = 0 where Ev (y) also vanishes: Px with the initial value of y(0) Px2 ˙ 2 μv y(0) = 0. + 2μv 2

(11.34)

Evidently one can satisfy this condition only at imaginary y(0), ˙ and we shall introduce the imaginary time t = −iτ looking for an underbarrier trajectory. The first integration of the equations of motion yields $ 1 Px − hnM y dy dx = = 2μv Ev (y) + (Px − hnM y)2 , . (11.35) dτ μv dτ iμv

11.5 Quantum nucleation of vortices: many-body approach

301

The relevant trajectory, which starts at y ≈ 0, ends at the point y = yf where Ev (y) = 0 and the classically accessible area begins. Then the tunnelling exponent is: ⎧ ⎫ ⎫ ⎧y x(y ⎪ ⎪  ⎬  f) yf ⎨ ⎬ 2 ⎨ f  dx 2 2ImS Px + Py dy = Im Px dx + Py dy = Im = ⎭ ⎪ ⎩ ⎪ dy h¯ h¯ ⎩ ⎭ h¯ 0

=

2 h¯

yf 0

0

0

2μv Ev (y) + hnM y(hnM y − Px ) dy. " 2μv Ev (y) + (hnM y − Px )2

(11.36)

The coordinate x is imaginary along the trajectory, but at the end of the underbarrier trajectory it must become real again. This imposes a condition on the value of the momentum Px : yf x(yf ) = 0

dx dy = i dy

yf " 0

hnM y − Px 2μv Ev (y) + (hnM y − Px )2

dy = 0.

(11.37)

For a massless vortex (μv = 0) at nM = n, Eq. (11.36) yields exactly the same probability exponent as Eq. (11.28), which was obtained using the complex coordinate but not complex time. In this limit the momentum Px is cancelled out in Eq. (11.36). In the opposite limit, when there is no Magnus force (nM = 0), the condition (11.37) requires that Px vanishes, and   yf # mnκ 2 y 2 2" ln − vs hny dy. 2μv Ev (y) = 2μv (11.38) = h¯ h¯ 4π rc 0

This is the standard expression for tunnelling of a particle of mass μv through the potential barrier described by the energy Ev (y). With logarithmic accuracy, i.e., replacing the logarithm in the integrand by a constant, one obtains # 4yf μv mnκ 2 yf ln . (11.39) = 3h¯ 2π rc While at Magnus tunnelling the probability logarithm is proportional to yf2 , i.e., the area occupied by the velocity field induced by the vortex, for a massive vortex not subject to the Magnus force it is proportional to yf . Ignoring logarithmic factors and estimating the vortex mass μv as the core mass π mnrc2 , the crossover from mass to Magnus tunnelling takes place at nM yf ∼ nrc . At low velocities, yf ∼ κ/vs is much larger than the core radius, and the vortex mass can influence quantum tunnelling only if the Magnus force is strongly suppressed, i.e., nM  n.

11.5 Quantum nucleation of vortices: many-body approach Up to now we considered the traditional approach to macroscopic quantum tunnelling based on the semiclassical theory for one or two macroscopic degrees of freedom. Meanwhile, an

302

Nucleation of vortices

attempt was made (Sonin, 1973) to address the problem within a more general many-body theoretical framework. The analysis was based on the Gross–Pitaevskii theory of a weakly non-ideal Bose gas (Section 1.15). In this theory the quantum mechanical N -particle wave function at T = 0 is a product of the condensate wave functions ψ(R) as given by Eq. (1.178). Quantum nucleation is a transition from an initial state with the condensate wave function √ ψ (R) = n exp[iθs (R)] to a final state with the condensate wave function ψv (R) = √0 n exp{i[θs (R) + θ (R)]}. Here n is the particle density, the phase θs determines the velocity v s = (κ/2π )∇θs of the initial superflow (without vortices), and the phase θ (R) is induced by the nucleated vortex loop. The energy difference between the final state and the initial state is [cf. Eq. (11.1)]   mnκ 1 mnκ 2 2 dR + (11.40) (R)] v s · ∇θ (R)dR. E= [∇θ 2 2π 4π 2 Suppose that the transition is due to interaction with a sum of single-particle potentials acting on the N particles of the fluid: Hint =

N 

V (R i ).

(11.41)

i=1

The spatially varying potential V (R i ) breaks translational invariance and can be connected with roughness of the channel walls. The transition probability is proportional to the square of the modulus of the matrix element for the transition between two states: 2    W ∝ | 0N |Hint |vN |2 = N −N+1 ψ0 |V (R)|ψv ( ψ0 |ψv )N −1  = N −2N +2 | ψ0 |V (R)|ψv |2 | ψ0 |ψv |2N −2 ≈ | ψ0 |V (R)|ψv |2 e− , where

(11.42)

    iθ(R) = −2Re{ln | ψ0 |ψv |} = 2Re n [1 − e ]dR = 2n [1 − cos θ (R)]dR (11.43)

is the probability logarithm, which does not depend on interaction with a defect, but is determined by overlapping of the order parameter wave functions. This means that the wave functions of the initial and the final states are not orthogonal. Non-orthogonality can be a source of inaccuracy of the derivation. However, this is inaccuracy of the preexponential factor and not of the exponent argument (Sonin, 1973). Various aspects of the relation between vortex tunnelling and the wave function overlap were discussed by Auerbach et al. (2006). Suppose that the final state after tunnelling is similar to that assumed in the semiclassical theory of Section 11.3: a vortex half-ring of radius rf [see Eq. (11.23)], which

11.5 Quantum nucleation of vortices: many-body approach

303

induces the velocity field corresponding to an extremum of the energy (11.40) and satisfying the incompressibility condition. The phase for such a field satisfies the Laplace equation θ = 0. Then according to Eq. (11.43) the probability logarithm is of the order of nrf3 , as in the semiclassical theory. However, this is not the most effective channel of tunnelling. In order to find this, one should determine a configuration for which the probability logarithm is minimal at the condition that the energies of the initial and the final states are the same, in other words, at the condition that the vortex loop with zero energy is nucleated. Thus one should find an extremum of the functional {θ (R)} = {θ (R)} + qE{θ (R)},

(11.44)

where q is the Lagrange multiplier. The Euler–Lagrange equation for the functional {θ (R)} is the sine-Gordon equation sin θ = 0. l2 The length l is connected to the Lagrange multiplier: θ −

(11.45)

8π 2 l 2 . (11.46) mκ 2 The relevant solution of the sine-Gordon equation is that for a plane 2π -domain wall (or soliton) which is restricted by the vortex line and is oriented normally to the velocity v s . When v s is parallel to the x axis, the phase distribution across the wall far from the vortex line is   x  . (11.47) θ = 4 tan−1 − exp − l At distances from the vortex line less than l, the phase distribution does not differ from that obtained from the Laplace equation. The stable phase distribution, which satisfies the Laplace equation θ = 0, and the phase distribution obtained from the sine-Gordon equation (11.45) are shown in Fig. 11.4. If the radius of curvature of the vortex line exceeds the domain-wall width l, one can easily calculate , E, and : q=

= 8nlS, mnκ 2 l mnκ 2 S + L ln − mnκvs S, 4π rc π2 l   π 2 l π 2 vs 2  = 16n lS + l L ln − l S . 8 rc 2κ

E=

(11.48) (11.49) (11.50)

Here S and L are the area and the length of the vortex loop. Note that now the upper cut-off in the line tension energy proportional to the length L is equal to l. One must find the extremum of the functional  at zero energy E. At the extremum the shape of the vortex line is circular, and we consider nucleation of a half-ring of radius r

304

Nucleation of vortices (a)

p/4

7p/4

0 p/2

3p/2 r 3p p 5p 4 4 0

p/4

(b)

7p/4 3p 2

p 2

r

3p p 5p 4 4 l Figure 11.4 Phase pattern around the nucleated vortex line. This is a cross-section normal to the vortex line (the filled circle). The arrow shows the superflow direction around the vortex line. Curved thin solid lines are lines of constant phase. Phase values are shown in radians. The dashed line shows a cut where the phase change is 2π . (a) The stable phase pattern, which corresponds to the minimal energy. The phase satisfies the Laplace equation θ = 0 for an incompressible fluid. (b) The phase pattern which corresponds to the maximal probability of many-body quantum nucleation. The phase satisfies the sine-Gordon equation (11.45) and θ = 0. In comparison with the stable case, the phase pattern is compressed in the transverse direction. Its transverse dimension is the width l of the 2π domain wall and l  r. The phase pattern is not stable and relaxes to the stable pattern by irradiation of phonons. Figure from Sonin (1973).

near a plane wall with S = π r 2 /2 and L = π r. The two conditions E = 0 and ∂/∂r = 0 at fixed l and vs yield that l=

3κ , π 2 vs

(11.51)

while the radius of the half-ring is connected with the domain-wall width by the relation r=

l πl ln . 4 rc

(11.52)

11.5 Quantum nucleation of vortices: many-body approach

Then according to Eq. (11.48) the probability logarithm is   κ 2 27  κ 3 2 ln = 4π nr l = n . v vs rc 4π 3

305

(11.53)

This is smaller by the factor (2π 2 /81) ln(κ/vs rc ) than the probability logarithm in Eq. (11.24) obtained from the semiclassical theory. In both cases is of the order of the number of particles in the volume perturbed by the vortex loop, but in the present theory the perturbed volume is oblate along the superflow direction, being restricted in this direction with the domain-wall width l instead of the loop radius r. In the final state after nucleation, the velocity is not divergence-free at distances from the vortex line larger than l. Therefore immediately after nucleation the fluid density starts to vary: the 2π -domain wall encircled by the vortex line becomes a source of phonons, which will propagate in the fluid and then relax somewhere. But the latter process is not impeded by an activation energy barrier and thus our estimation of the probability logarithm refers to the bottleneck of the nucleation. In summary, our calculation shows that the quantum tunnelling is more effective if a packet of phonons is created simultaneously with the vortex loop. This is an example of phonon-assisted macroscopic quantum tunnelling. If it were possible to detect phonon radiation in nucleation experiments, this would provide evidence of the presented scenario of nucleation. Our derivation was based on the Gross–Pitaevskii theory of Bose superfluids with a rather simple expression for the N -particle wave function of the coherent state. One would expect that the derived relation between the probability logarithm and the overlap of the order parameter wave functions of the initial and the final states is also valid in the BCS theory of Fermi superfluids, although it is more difficult to check this starting from the many-particle wave function of the BCS state. This problem was addressed by Auerbach et al. (2006). In a non-uniform superflow, the constant velocity vs in Eqs. (11.49) and (11.50) for E and  must be replaced by the average velocity v¯s over the area S of the nucleated vortex loop. Now the optimal shape of the loop is not circular and must be found by variation of the functional . We shall consider the superflow through the orifice when the initial superflow before nucleation is given by Eq. (11.15). The shape of the nucleated loop in the plane yz is determined by the function y(z) = a0 − ρ  a0 (see Fig. 11.3). The particle flow through the vortex loop is  nv¯s S = nvs0

a0 2

zc " y(z)dz, −zc

while the energy E and functional  are given by ⎡ ⎤ #  2 # zc y(z) mnκ 2 y(z) dy l ⎣ ⎦ dz, E= − ln + 1+ 4π rc r dz rm −zc

(11.54)

(11.55)

306

Nucleation of vortices

π l  = 16n l 2 ln 8 rc

⎡

zc

#

⎣ 2y(z) + r

−zc

 1+

dy dz

#

2 −

⎤ y(z) ⎦ dz, rm

(11.56)

where  rm =

κ l ln 4π vs0 rc

2

2 , a0

(11.57)

and the length r is connected with the wall thickness l by the relation Eq. (11.52) as before, but now it is not a loop radius since the loop is not circular. Varying the functional  with respect to y(z) one obtains the Euler–Lagrange equation for y(z):   1 d dy/dz 2 − √ − = 0. (11.58) " r 2 rm y dz 1 + (dy/dz)2 Integration of Eq. (11.58) at z > 0 yields: " √ dy 1 − ( y/rm − 2y/r)2 =− . √ dz y/rm − 2y/r

(11.59)

Dimensions of the nucleated vortex loop along the y and z axes are ym and zc (they are shown in Fig. 11.3b): #   "  ym y  /rm − 2y  /r r r r r −4− −8 , zc = dy  . ym = $ " 8 rm rm rm 1 − ( y  /rm − 2y  /r)2 0 (11.60) The ratio γ = rm /r = 0.117 is determined from the condition E = 0, which yields an equation tm 0

√ √ 4γ 2 − ( t − 12 t)( t − t) dt = 0, " √ 4γ 2 − ( t − t)2

(11.61)

1√

where tm = 12 − 2γ − 2 1 − 8γ . After numerical calculation of the integral in Eq. (11.60), one obtains the dimensions of the nucleated loop: ym = 2.58rm ,

zc = 6.21rm .

(11.62)

Now one can obtain the relation between the average velocity vs0 in the orifice and the domain-wall width l, using Eqs. (11.52) and (11.57): l=

κ2 κ2 ln . 2 2 r 2π 3 γ a0 vs0 2π 3 a0 γ vs0 c

(11.63)

11.5 Quantum nucleation of vortices: many-body approach

307

2 ) where The area of the loop is S = αr 4 /(16rm

tm α= 0

√ t ( t − t) dt = 3.68 × 10−2 . " √ 2 2 4γ − ( t − t)

(11.64)

Finally the value of the probability logarithm is 5 κ2 ln 2 r 2π 3 a0 γ vs0 c  5 6 κ2 −3 nκ = 2.13 × 10 . ln 6 a3 2 r 2π 3 a0 γ vs0 vs0 c 0

nκ 6 α α 2 nlr = = 8nSl = 6 a3 2γ 2 28 π 7 γ 5 vs0 0



(11.65)

6 , in contrast to 1/v 3 in uniform superflows, and The probability logarithm grows as 1/vs0 s0 3 is smaller by the factor (κ/vs0 a0 ) than the probability logarithm in uniform superflows. Thus non-uniformity of the velocity field essentially increases the probability of quantum nucleation at low velocities, bearing in mind also a rather small numerical factor in this expression. In addition, let us obtain a qualitative estimate of for the superflow past the wedge with the method used at the end of Section 11.3 for the semiclassical theory:  3   3 −1 k 3(k−1) κ k κ ln . (11.66) ∼ na k vs0 vs0 rc

As in the case of uniform flow, the many-body approach yields a probability logarithm which is smaller by the logarithmic factor ln(κ/vs0 rc ) than the probability logarithm (11.29) obtained in the semiclassical theory.

12 Berezinskii–Kosterlitz–Thouless theory and vortex dynamics in thin films

12.1 Statical theory The Berezinskii–Kosterlitz–Thouless (BKT) theory (Berezinskii, 1970, 1971; Kosterlitz and Thouless, 1973), which was developed to describe superfluid phenomena in twodimensional systems, had a huge impact not only on condensed matter physics but on the whole of modern physics. An impetus for the creation of this theory was the formal absence of Bose–Einstein condensation in two-dimensional quantum Bose liquids that put in doubt the possibility of superfluid phenomena there. Still some sort of transition was shown to be possible. Let us start from a heuristic picture of this transition. If there is a pair of vortices of opposite circulation (vortex + antivortex) the energy of the pair is twice the energy of one vortex, and depends logarithmically on the distance r between the vortex and the antivortex: ε = εc +

ρs κ 2 r ln , 2π rc

(12.1)

where εc ∼ ρs κ 2 /2π is of the order of the energy of two vortex cores. This is the energy of a bound state of the vortex pair with infinite energy of ionisation. But for realisation of the bound state, not only the energy but also the entropy associated with the vortex pair of size r is important. One can estimate the latter as S ≈ 2 ln N = 2 ln

r2 , rc2

(12.2)

where N is the number of possible positions of a vortex inside the area ∼ r 2 . The free energy therefore is F = ε −TS =

r r ρs κ 2 ln − 4T ln . 2π rc rc

(12.3)

The free energy changes sign at the critical temperature πρs h¯ 2 ρs κ 2 = . (12.4) 8π 2m2 A negative free energy at r → ∞ means that the vortex and the antivortex cease to be bound, i.e., dissociation of vortex pairs sets in, which leads to a loss of superfluidity. Tc =

308

12.1 Statical theory

309

However, our estimation of the two-dimensional critical temperature ignored an essential factor of the process: approaching the phase transition, the superfluid density ρs is affected by vortex pairs and must be properly renormalised. Kosterlitz and Thouless (1973) suggested an approximate but plausible procedure for such a renormalisation. One can consider the vortex pair as an elementary excitation with energy ε(r) and momentum p = −ρs [κ × r], where r = r 1 − r 2 is the separation vector for the vortex pair with the vortex at the position vector r 1 and the antivortex at the position vector r 2 . Then vortex pairs must contribute to the normal mass of the fluid on the same ground as other excitations (phonons and rotons), and one could use the same recipe to estimate this contribution. But there is an essential difference between usual excitations and vortex pairs. For usual excitations the spatial coordinates and the momenta are independent variables, and the number of their states in the elementary volume of the phase space is dr dp/ h2 . On the other hand, the number of degrees of freedom for vortices is two times less, and the momentum is not independent of the spatial coordinate. So the phase space for the pair of the vortex and the antivortex is determined by their two position vectors r 1 and r 2 , and the number of states in the elementary volume of this space is dr 1 dr 2 /rc4 . Using the separation vector r and the average position vector r 0 = (r 1 + r 2 )/2 as variables, the number of pair states per unit volume in the space of r is dr/rc4 (in contrast to dp/ h2 for usual quasiparticles). Then the mass current produced by vortex pairs is determined by the integral in the two-dimensional space of r:  1 (12.5) j p = 4 f (r)p(r) d2 r, r0 where f (r) is the distribution function of pairs in the r space. The current appears if the equilibrium gas of pairs moves with some drift velocity. The drift velocity is expected to be close to the normal velocity v n , which is a drift velocity for other excitations (phonons and rotons). Then the distribution function of pairs is f (r) = e−[ε(r)−p(r)·v n ]/T ,

(12.6)

where the energy in the presence of the transport superfluid velocity is ε(r) = ε0 (r) + p(r) · v s .

(12.7)

Expanding in low velocities v s and v n , the distribution function is f (r) = f0 +

∂f0 [p · (v s − v n )], ∂ε

(12.8)

and the mass current j p = ρp (v n − v s ) is determined by the proportionality coefficient ρp , which can be called the vortex pair normal mass density:  1 ∂f0 2 p d2 r. (12.9) ρp = − 4 ∂ε 2r0

310

Berezinskii–Kosterlitz–Thouless theory

Here f0 = e−ε0 /T is the equilibrium Boltzmann distribution function for vortex pairs. This expression is similar to Eq. (6.51) for the three-dimensional normal density but now integration is performed in the two-dimensional space of r. A crucial assumption of the BKT theory was that the energy of interaction between the vortex and the antivortex with the distance r between them is affected only by pairs of separation less than r. This leads to the concept of a length-dependent renormalised superfluid density ρ˜s (r) = ρs − ρp (r), which must be determined by the integral in Eq. (12.9) cut off at some separation r. The energy and the momentum in the integrand must be determined by the renormalised superfluid density ρ˜s (r):  p(r) = −

r rc

 r , ρ˜s (r  ) dr  κ × r

ε0 (r) = εc +

κ2 2π



r rc

ρ˜s (r  ) dr  . r

(12.10)

Equation (12.9) yields the following integral equation for the renormalised superfluid density ρ˜s (r): π κ 2 −εc /T ρs − ρ˜s (r) = e T



r



2

r



ρ˜s (r )dr



rc

rc



κ2 exp − 2π T



r

rc

ρ˜s (r  ) dr  r 



r  dr  . r04 (12.11)

By analogy with the Coulomb two-dimensional gas, Kosterlitz and Thouless (1973) introduced the dielectric constant (r) =

ρs . ρ˜s (r)

(12.12)

Then Eq. (12.11) transforms to the integral equation for the r-dependent (r): 1 πρs κ 2 −εc /T 1− = e (r) T



r



rc

r

dr  (r  )

rc

2



ρs κ 2 exp − 2π T



r rc

dr  (r  )r 



r  dr  . (12.13) r04

Differentiation of this equation with respect to r yields: πρs κ 2 −εc /T 1 d(r) = e 2 T (r) dr



r

rc

dr  (r  )

2

   r ρs κ 2 r dr  exp − . 2π T rc (r  )r  r04

(12.14)

In the analogy with the two-dimensional Coulomb gas, a vortex pair is compared with a pair of electric charges of opposite sign in a constant electric field. Using this analogy, Kosterlitz and Thouless (1973) obtained an equation for the dielectric constant  different from Eq. (12.13): πρs κ 2 −εc /T e (r) − 1 = T



r rc



ρs κ 2 exp − 2π T



r

rc

dr  (r  )r 



r 3 dr  . r04

(12.15)

12.1 Statical theory

311

So the analogy with the two-dimensional Coulomb gas is not absolutely exact. But at large scales one can write approximately that  r r dr  ≈ , (12.16) ) (r (r) rc and Eq. (12.13) becomes 1 πρs κ 2 −εc /T 1− = e (r) T



r

rc



ρs κ 2 exp − 2π T



r

rc

dr  (r  )r 



r 3 dr  . (r  )2 r04

(12.17)

After differentiation with respect to r, Eqs. (12.15) and (12.17) yield the same integrodifferential equation  3   r πρs κ 2 −εc /T d(r) ρs κ 2 r dr  = e exp − . (12.18) dr T 2π T rc (r  )r  r04 In order to understand where the analogy with the two-dimensional Coulomb gas stops, let us write down the energy of an electric dipole in a dielectric medium with the length dependent (r) in a constant electric field E:  r dr  2 − qE · r, (12.19) (r) = q   rc (r )r where q is the electric charge and rc is the size of charged particle. In the Coulomb gas analogy the electric field translates into superfluid hydrodynamics as [ˆz × v s ] (apart from a dimensional factor), the bare charge squared q 2 translates as the bare superfluid density ρs , while q 2 /(r) translates to the length-dependent superfluid density ρ˜s (r) = ρs /(r). In the Coulomb gas approach the expression (12.10) for the momentum and the energy of the vortex pair transforms to p = −ρs [κ × r],

ε0 = εc +

κ2 2π

r

ρ˜s (r  )dr  . r

(12.20)

rc

Thus the Coulomb gas approach uses the renormalised superfluid density ρ˜s in the energy ε0 of interaction between a vortex and an antivortex, but retains the bare superfluid density in the momentum p of the vortex pair, in contrast to the hydrodynamical approach [see Eq. (12.10)]. It is worth noting that if one applied the logic of the Coulomb gas approach to the vortex pair affected by usual quasiparticles like phonons and rotons, one would use the superfluid density ρs in the expression of the vortex pair energy but the total density ρ for the vortex pair momentum, in conflict with two-fluid hydrodynamics. From this position the hydrodynamical approach looks more consistent than the Coulomb gas approach. Actually the difference between the two approaches is not so essential for the small  − 1 usually assumed in the BKT theory. But it yields a different dependence of the vortex pair dissociation rate on the transport velocity vs (Section 12.3).

312

Berezinskii–Kosterlitz–Thouless theory

We introduce notation widely used in papers on the Kosterlitz–Thouless theory (Kosterlitz and Thouless, 1973; Kosterlitz, 1974): l = ln

r , rc

y(l)2 = y02 e

K(l) = κ2 4l− 2π T

l 0

ρ˜s (l)κ 2 , 4π 2 T

ρ˜s

(l  )dl 

,

K(0) =

ρs κ 2 , 4π 2 T

(12.21)

y0 = 4π e−εc /2T .

Then Eq. (12.18) reduces to the Kosterlitz recursion equations: −

π dK(l) = K(l)2 y 2 (l), dl 4

dy 2 (l) = y 2 (l)[4 − 2π K(l)]. dl

(12.22)

This system of equations has a singular point at 4 − 2π K(l) = y(l) = 0 at l → ∞. The point corresponds to the phase transition at K(∞) = 2/π, which occurs at the critical temperature given by Eq. (12.4) after replacing the bare superfluid density ρs by the renormalised value ρ˜s determined as ρ˜s (r) in the limit r → ∞. At the critical temperature the renormalised superfluid density ρ˜s drops discontinuously to zero by a universal amount determined by Eq. (12.4) with the bare superfluid density ρs replaced by the renormalised superfluid density ρ˜s (∞). At temperatures close to critical one can introduce a small x(l) = π [K(l) − Kc (∞)], where Kc (l) yields values of K(l) at the critical point and at l → ∞, Kc (∞) = 2/π. Then the recursion relations can be written as dx(l) = −y 2 , dl

dy 2 (l) = −2xy 2 . dl

(12.23)

This system of equations has an integral: x(l)2 − y(l)2 = C.

(12.24)

The singular point x = y = 0 of this system of equations, where the constant C vanishes, corresponds to the critical temperature. Assuming analytical dependence on temperature, it is natural to expect a linear dependence of C on relative temperature t = (T − Tc )/Tc : C = −4b2 t. Equation (12.24) describes hyperbolas in the plane xy. If the fugacity y0 is low, x(l) is small at any l and Kosterlitz’s recursion equations (12.23) have an analytical solution in the low temperature superfluid phase t < 0 (Kosterlitz, 1974; Ambegaokar et al., 1980):   x0 cosh(x∞ l) + x∞ sinh(x∞ l) −1 x0 , = x∞ x(l) = x∞ coth x∞ l + coth x∞ x0 sinh(x∞ l) + x∞ cosh(x∞ l)   y0 x∞ x0 = . y(l) = x∞ csch x∞ l + coth−1 x∞ x0 sinh(x∞ l) + x∞ cosh(x∞ l)

(12.25) (12.26)

12.1 Statical theory

313

√ √ Here x0 = x(0) and x∞ = x(∞) = C = 2b |t|. This allows us to find the temperature dependence of the renormalised superfluid density ρ˜s (∞) close to the critical temperature, which has a square-root cusp:   " x(∞) = ρ˜sc (1 + b |t|), (12.27) ρ˜s (∞) = ρ˜sc 1 + 2 where ρ˜sc is the superfluid density at the critical temperature. The cusp, which was revealed by Nelson and Kosterlitz (1977) with numerical calculations, plays an important role in vortex dynamics in superfluid films on rotating porous substrates (Section 12.5). When the analytical solution is valid for all scales down to r ∼ rc , the parameter b can be determined from the temperature dependence of the initial values x0 and y0 (Nemirovskii and Sonin, 2007). In general the solution is valid only at very long scales l 1. However, the assumption that the initial parameters are analytical functions of the temperature is sufficient for the existence of the square-root temperature dependence given by Eq. (12.27), although the parameter b in this equation cannot be determined analytically in general. One can define the scale, which satisfies the condition x∞ l = x∞ ln(r/rc ) ∼ 1 of the crossover from short-scale to long-scale behaviour, as the coherence length ξ− for the BKT phase transition (Ambegaokar et al., 1980): ξ− = rc e1/x∞ = rc e1/2b

√ |t|

.

(12.28)

At long scales r ξ− (x∞ l 1) the solution of the recursion equations is x(l) = x∞ (1 + 2e−2x∞ l ),

y(l) =

2x∞ y0 −x∞ l e , x0

(12.29)

while at short scales r  ξ− (x∞ l  1) 2 l x0 + x∞ y0 , y(l) = . (12.30) 1 + x0 l 1 + x0 l √ At the critical temperature when x∞ = C = 0, the parameters x(l) and y(l) are equal, and in original dimensional variables Eq. (12.30) corresponds to the following expressions for the renormalised density and the dielectric constant at scales l 1/x0 :     π π ρs 1− , c (r) = , (12.31) ρ˜sc (r) = ρ˜sc 1 + 2 ln(r/rc ) ρ˜sc 2 ln(r/rc )

x(l) =

where ρ˜sc = ρ˜sc (∞) is the final renormalised superfluid massed density at long scales. Above the critical temperature (t > 0) the analytical solution of the recursion equations is y0 xi x0 cos(xi l) − xi sin(xi l) , y(l) = , (12.32) x(l) = xi x0 sin(xi l) + xi cos(xi l) x0 sin(xi l) + xi cos(xi l) √ √ where xi = −C = 2b t. In contrast to the superfluid phase t < 0, the solution does not converge to any point at the trajectory in the xy plane in the long-scale limit l → ∞ but oscillates strongly between 0 and ∞ instead. This invalidates our solution, which is

314

Berezinskii–Kosterlitz–Thouless theory

accurate only at small x, and supports the view that above the critical temperature there is no renormalised superfluid density. Nevertheless, oscillations begin only at l > 1/xi , and Kosterlitz (1974) suggested defining the scale at which xi l ∼ 1 as the coherence length in the normal phase, √

ξ+ = rc e1/xi = rc e1/2b t .

(12.33)

This is a counterpart of the coherence length ξ− [Eq. (12.28)] in the superfluid phase t < 0.

12.2 Dynamical theory Ambegaokar et al. (1978, 1980) developed the dynamical theory (the Ambegaokar– Halperin–Nelson–Siggia theory) on the basis of the Fokker–Planck equation for vortex pairs in the space of the vectors r. Let us start from the equation of vortex motion (6.102) in the clamped regime in the coordinate frame moving with normal velocity v n : v L = (1 − α  )v s − α zˆ × v s .

(12.34)

Here we ignore a difference between the average superfluid velocity v s and the local superfluid velocity v sl . One can rewrite Eq. (12.34) in terms of the effective Lorentz force F = −∂ε/∂r = −ρs [κ × v s ] acting on the vortex: vL =

1 − α α F+ [κ × F ]. ρs κ ρs κ 2

(12.35)

This looks like the standard linear relation between a velocity and a force. If Eq. (12.35) is written for a vortex located at the position vector r 1 , the equation for the antivortex at r 2 differs by the opposite sign of the circulation κ, and the time derivative for the separation vector r = r 1 − r 2 does not depend on the transverse mutual friction force ∝ 1 − α  : dr dr 1 dr 2 2α = − = 2μF = F, dt dt dt ρs κ

(12.36)

where μ=

α ρs κ

(12.37)

is the vortex mobility in the space of the vector r. One can formulate the Fokker–Planck equation for the distribution function f (r), ∂f = −∇ · j , ∂t with the pair current j = −μ∇ε(r) − 2D∇f = −2D

(12.38) 

 f ∇ε(r) + ∇f . T

(12.39)

12.2 Dynamical theory

315

The energy ε is determined by Eqs. (12.7) and (12.10). The second term in this expression is the diffusion current, which originates from the noise force on the vortex in the Langevin equation (Ambegaokar and Teitel, 1979; Ambegaokar et al., 1980). It can also be obtained directly from the fluctuation-dissipation theorem, which requires that for the equilibrium Boltzmann distribution of pairs in the r space the total pair current vanishes. This requirement yields the relation between the mobility and the diffusion coefficient (the Einstein relation) D α = . ρs κ T

(12.40)

The factor 2 in the diffusion term of the Fokker–Planck equation appears because the diffusion coefficient D, as well as the vortex mobility [Eq. (12.37)], was defined for diffusion of a single vortex, which is two times less than the diffusion coefficient in the r space of vortex pairs. The force F = F 0 +F s consists of two parts. The first part F 0 = −∂ε0 /∂r is connected with the superfluid velocity induced by a partner of a vortex (antivortex). The second force F s = −ρs [κ × v s ] can be considered as the external Lorentz force corresponding to the superfluid velocity v s induced by the transport current and all other vortices except the two forming the pair. We consider the linear response of the superfluid to a small external time-dependent Lorentz force linearising the Fokker–Planck equation with respect to F s ∝ e−iωt . The solution of the linearised Fokker–Planck equation should have the following form: ∂f0 (12.41) [p · (v s − v n )]g(r, ω). f (r) = f0 (ε0 ) + ∂ε The dynamical effects are accumulated in the factor g(r, ω), which is equal to unity in the static limit ω → 0 [cf. Eq. (12.8)]. Now the vortex pair density ρp = ρs − ρ˜s depends on frequency, and correspondingly one can introduce the frequency-dependent dielectric constant connected with the length-dependent static dielectric constant (r) via the following formula: ρs =1+ ˜ (ω) = ρ˜s (ω)

∞ dr

d(r) g(r, ω). dr

(12.42)

0

The Fokker–Planck equation reduces to the second-order differential equation for g. The equation has no simple analytical solution and various approximations have been suggested (Ambegaokar and Teitel, 1979; Ambegaokar et al., 1980). We restrict ourselves to simple scaling arguments, which provide a good estimation of the linear response apart from a numerical factor. Since the Fokker–Planck equation depends analytically on ω, one can expect that for low frequency 1 − g ∝ iω and for high frequency g ∝ 1/ iω.$On the ˜ other hand, the new length scale, which appears in dynamics, is the distance rD ∼ 2D/ω at which the vortex pair diffuses during the oscillation period 2π/ω. The parameter D˜ may differ from the diffusion coefficient by a numerical factor. Ambegaokar et al. (1978)

316

Berezinskii–Kosterlitz–Thouless theory

chose D˜ = D, Ambegaokar and Teitel (1979) estimated that D˜ = 4D, while Ambegaokar et al. (1980) assumed D˜ = 7D. Apparently the crossover from the low frequency to the high frequency behaviour must occur at r ∼ rD . Altogether this leads to the following extrapolation formula (Ambegaokar et al., 1978; Ambegaokar and Teitel, 1979): g(ω, r) =

2D˜ 2D˜ − iωr 2

=

2 rD 2 − ir 2 rD

.

(12.43)

The derivative of the static dielectric constant d(r)/dr is proportional to the number of vortex pairs of separation r and is given by Eq. (12.18). It is a slowly varying function of r, and Ambegaokar et al. (1980) approximated the faster varying complex function g(ω, r) by 2 − ωr 2 ) + g(ω, r) ≈ θ (rD

iπr δ (r − rD ) . 4

(12.44)

This means that only pairs with separations r less than rD renormalise the real part of the superfluid density, and the renormalised superfluid density ρ˜s has an imaginary part related to dissipation of pairs with separations r close to rD . Using the analytical solution close to the critical temperature, which was discussed in the previous section, one obtains that K(0) dx(l) K(0) 4π Tc d =− = y(l)2 = (l)2 y(l)2 . 2 2 dl π K(l) dl π K(l) ρs κ 2

(12.45)

At x0 l x∞ l 1, y(l) is given by Eq. (12.29), and √ 16π Tc y02 d 64π Tc y02 2 2 −2x∞ 2 2 −4bl |t| = (l) x e = (l) b |t|e . ∞ dl ρs κ 2 x02 ρs κ 2 x02

(12.46)

If the imaginary part is small one obtains from Eq. (12.42) that  −4b√|t| 1 16π 2 Tc y02 2 1 rc = −i b |t| . ˜ (ω) (rD ) rD ρs κ 2 x02

(12.47)

An important application of the theory is experiments with torsional oscillators of piles of disks covered by superfluid films. We shall use the expressions for the period and the inverse quality factor shifts in Eq. (5.8) relating the dragged fluid density ρ  with the contribution of vortex pairs to the normal component, i.e., ρ  = ρp = ρs − ρ˜s . Then Eq. (5.8) yields:   1 ρs T ρs 1 1 − Re = , Q−1 = − (12.48) Im . T 2ρt ˜ 2ρt ˜ According to Eq. (12.47) the inverse quality factor grows with temperature, reaching its maximum near the critical temperature.

12.2 Dynamical theory

317

In the normal phase at T > Tc there are no bound pairs but mostly free vortices and antivortices. The theory of Ambegaokar et al. (1980) assumed that their velocities were determined by the same mobility as in the superfluid phase. Then they exploited again the analogy with the two-dimensional Coulomb gas for the contribution of free vortices to the linear response. As in statics, we prefer to analyse the linear response directly in terms of superfluid hydrodynamics. The Lorentz force on the vortex depends on the superfluid density, which strictly speaking is absent in the normal phase, at least in statics. However, one can introduce some sort of effective superfluid density, which will be denoted by ρ˜s and will be purely imaginary, as we shall see, and vanish at ω → 0. The longitudinal velocity of a vortex driven by the Lorentz force is v L = μF [see Eq. (12.35)] with the mobility μ determined by Eq. (12.37) and with the Lorentz force F = −ρ˜s [κ × (v s − v n )]. The velocity of an antivortex has the opposite direction because of inversion of the circulation vector κ. The counterflow of vortices creates the variation of the momentum of the superfluid dragged by the oscillator with rate dP v = nf ρ˜s [κ × v L ] = nf (ρ˜s κ)2 μ(v s − v n ), dt

(12.49)

where nf is the total density of vortices and antivortices. Although the bare superfluid density vanishes above the critical temperature, the effective superfluid density does not, and the momentum of the dragged part of the superfluid is P v = ρ˜s (v s − v n ). Comparing it with Eq. (12.49) one obtains the expression for the effective superfluid density for monochromatic oscillation with frequency ω: ρ˜s = −

iω iωT =− . μκ 2 nf Dκ 2 nf

(12.50)

This determines the contribution to the imaginary part of the inverse dielectric constant from free vortices: Im

ρ˜s iωT 1 = = . ˜ (ω) ρs Dρs κ 2 nf

(12.51)

Ambegaokar et al. (1980) suggested that the vortex density is determined by the coherence length in the normal phase ξ+ given by Eq. (12.33): nf ∼ 1/ξ+2 . Note that the bare superfluid velocity ρs cancels out in the expression for the inverse quality factor since, according to Eq. (12.40), the diffusion coefficient D is inversely proportional to ρs . The coherence length decreases with temperature. Losses grow on approaching the critical temperature from both sides of the transition, and dissipation has a peak close to the BKT transition. The dissipation peak at the critical temperature was observed experimentally (Bishop and Reppy, 1980).

318

Berezinskii–Kosterlitz–Thouless theory

12.3 Rate of pair dissociation Ambegaokar et al. (1980) also analysed the non-linear process of thermally activated dissociation of vortex-antivortex pairs in the BKT theory (see also Gillis et al., 1985; Bowley et al., 1992; Giorgini and Bowley, 1996, for references to later works) on the basis of the Iordanskii–Langer–Fisher theory addressed in Section 11.1. The pair dissociation is accompanied by overcoming the potential barrier. Solution of the Fokker–Planck equation for the distribution of vortex-antivortex pairs by Ambegaokar et al. (1980) yields the Arrhenius law for the dissociation rate: R=

2D −ε(rs )/T e . rs4

(12.52)

Here rs is the pair size r where the vortex pair energy ε(r) = ε0 (r) + p(r) · v s [see Eq. (12.7)] has an extremum (saddle point). For determination of the saddle point position, the difference between the hydrodynamical approach and the Coulomb gas approach discussed in Section 12.1 becomes important. In the hydrodynamical approach both the vortex pair energy ε0 (r) in the resting fluid and the vortex pair momentum p(r) are renormalised by vortex pairs with separation less than r [see Eq. (12.10)], and the energy ε(r) has an extremum (saddle point) at the separation rs =

κ , 2π vs

(12.53)

which does not depend on either the bare or the renormalised superfluid densities, ρs or ρ˜s (r). The value of rs is two times larger than the distance rm [see Eq. (11.7)] of a point vortex from a plane wall, which determines the energy barrier for thermal vortex nucleation. In contrast, in the Coulomb gas approach adopted by Ambegaokar et al. (1980), only the energy ε0 (r) is renormalised [see Eq. (12.20)], and the separation in the saddle point must satisfy the equation rs =

κ κK(ls ) = , 2π (rs )vs 2π K(0)vs

(12.54)

where ls = ln(rs /rc ). If the velocity vs is small the values of ls and rs are large, and approximately rs =

κ κK(∞) = . 2π (∞)vs 2π K(0)vs

(12.55)

However, in both approaches the energy at the saddle point is given by the same expression after excluding vs with help from the expression (12.54) or (12.55): ⎡ r ⎤ s ρs κ 2 ⎣ dr  1 ⎦ ε= − . (12.56) 2π (r  )r  (rs ) rc

12.4 Coreless vortices in superfluid films

319

Using this in Eq. (12.52) yields the expression for the dissociation rate in the Kosterlitz– Thouless variables obtained by Ambegaokar et al. (1980): R=

2D 2 y (ls )e2π K(ls ) . rs4

(12.57)

In all, the two approaches differ only by the value of rs given by either Eq. (12.53) or Eq. (12.55).

12.4 Coreless vortices in superfluid films on rotating porous substrates: from two-dimensional to three-dimensional vortex dynamics A superfluid 4 He film adsorbed in porous media was an actual topic of study in the physics of superfluidity (see Reppy, 1992, and references therein). Study of this system gives a unique possibility to investigate the crossover between two-dimensional and threedimensional vortex dynamics. On the one hand, torsional oscillator experiments reveal the dissipation peak near the temperature of the superfluid onset Tc , which is predicted by the dynamical theory of vortex-antivortex pairs for two-dimensional films discussed in Section 12.2. On the other hand, it was found that in films on porous substrates the superfluid density critical index was ∼ 2/3 (Reppy, 1992) as in the three-dimensional system and the sharp cusp of the specific heat (Murphy and Reppy, 1990) at Tc was similar to that near the critical temperature in the bulk 4 He. In torsional oscillator experiments an additional rotation-induced peak in dissipation (inverse quality factor) was observed as a function of temperature. This was shifted from the stationary (static) peak that was observed without rotation (Fukuda et al., 1998, 2005). The double-peak structure differs essentially from the case of plane films, where the only effect of rotation was broadening of the stationary peak (Bishop and Reppy, 1978, 1980; Adams and Glaberson, 1987). A porous substrate was usually modelled with a ‘jungle gym’ structure (Minoguchi and Nagaoka, 1988; Machta and Guyer, 1988, 1989; Gallet and Williams, 1989; Obata and Kubota, 2002): a three-dimensional cubic lattice of intersecting cylinders of diameter a with period c (Fig. 12.1). In the jungle gym structure a three-dimensional coreless or pore vortex is possible, which is just a flow around the vortex pores having non-zero circulation. The vortex ‘line’ in this case is not a line at all; this is a chain of the jungle gym structure cells with non-zero circulation around them. While continuous vortices in chiral superfluids, which also have no core, arise because of topology of the order parameter space (Section 10.5), coreless vortices in films on porous substrates are related to the topology of the configurational space, namely multiple connectivity of superfluid films in pores. The coreless vortex is depicted schematically in Fig. 12.1. Three-dimensional coreless vortices play a key role in hydrodynamics of superfluid films in porous media. For coreless vortices the usual type of vortex motion in a continuous fluid is ruled out simply because the coreless vortex has no continuous coordinate: its position is discrete and is determined by a cell with non-zero circulation around it. The only way for the vortex to move is to jump from cell to cell (Fukuda et al., 2005). These jumps are

320

Berezinskii–Kosterlitz–Thouless theory

l

a Figure 12.1 Vortex creep in the jungle gym structure: the coreless vortex (dashed line) crosses a cylinder between two cells via dissociation of the vortex-antivortex pair. Arrowed curved lines show the direction of circulation around the cells where the vortex line is located. Figure from Fukuda et al. (2005).

related to dissociation of the vortex-antivortex pair on one side of a rod separating different pores, with subsequent annihilation of the pair on the other side of the rod. The result of the jump is a shift of velocity circulation to a neighbouring pore. This type of vortex motion is called vortex creep. Any jump is in fact a process in time (however short) during which the vortex line inevitably crosses a rod covered by a superfluid film. So during the jump the ‘coreless’ vortex does have cores: at the place where it enters the rod and at the place where it exits from the rod (Fig. 12.1). These two cores together form a two-dimensional vortexantivortex pair, which should grow, dissociate, and eventually annihilate on the other side of the rod. For better illustration, a two-dimensional picture of this process is shown in Fig. 12.2. It shows two cells: upper (u) and lower (l). Before the process (Fig. 12.2a) there is circulation κ = h/m around the lower cell. Provided that there are no other coreless vortices nearby, the same circulation exists around any path inclusive of the cell (l) as shown for the path around the two cells. The transient process of the ‘jump’ from the cell (l) to the cell (u) is shown in Fig. 12.2b. The vortex-antivortex pair is present in the film between two pores. This makes circulation around any of the two cells undefined: it depends on whether the path goes outside or inside the pair. Only circulation around the two cells together remains equal to κ. Figure 12.2c shows the state after the process: the coreless vortex is now located in cell (u). Although this scenario is shown in the plane picture, it is directly applicable to the three-dimensional jungle gym structure with the two-dimensional vortex and antivortex moving around a cylindrical rod. During the process of displacement of the vortex from one cell to another, the vortex energy is higher than in the initial and the final states. Therefore the process is accompanied by overcoming the energy barrier. Without fluctuations, thermal or quantum, this is impossible, i.e., the vortex is pinned. This is an example of intrinsic pinning, which impedes motion of a vortex from one crystal cell to another in superconductors. In our case the energy barrier is the barrier for nucleation and dissociation of the vortex pair. Since we

12.4 Coreless vortices in superfluid films

(a)

(b)

321

(c)

Figure 12.2 Vortex creep. The superfluid moves with velocity v s along the axis x, and the vortex creeps along the axis y. (a) The state before the jump. There is circulation κ around the lower cell (l) as it is shown in the picture. The same circulation exists around any path inclusive of cell (l). The circulation around two cells (l) and (u) is shown. The coreless vortex is located at the cell (l). (b) The vortex on the way between cells (l) and (u). There is a vortex-antivortex pair in the film. Circulations around the paths inclusive only of cell (l) or (u) are not defined, while for the path around the two cells circulation is equal to κ as before. (c) The state after the jump. The coreless vortex is located at the cell (u). There is no longer circulation around the cell (l) but there is circulation around the cell (u), or around the two of them. Figure from Nemirovskii and Sonin (2007).

focus on phenomena close to the critical temperature, the main mechanism of the process is thermal dissociation of a vortex pair, which was considered in Section 12.3. Creation and dissociation of vortex-antivortex pairs is a stochastic process, which is determined by the average dissociation rate R(vs , T ) (number of dissociation events per second and per unit area of a film), which depends on temperature T and, most important for us, on average superfluid velocity vs in the film. In our case the total superfluid velocity field consists of two parts: v˜ s = v c + v s , where v c originates from a circular flow around the vortex line at rest and v s is a transport superfluid velocity with respect to a moving substrate. In Fig. 12.2 the velocity v s is directed along the axis x. So it is added to the circular velocity v c above the cell with circulation κ and is subtracted from v c below this cell. Therefore the jumps of the vortex line up and down are unbalanced and some average drift (creep) of the vortex line up the picture in Fig. 12.2 arises. The creep of coreless vortices can be described approximately in the terms usually used for three-dimensional vortices moving in a continuous medium. One can introduce an average vortex velocity determined as vL = Ac[R(vc + vs , T ) − R(vc − vs , T )] ≈ 2Ac

∂R vs , ∂vs

(12.58)

where c is the period of the jungle gym structure, A = π ac is the area of the rod separating neighbouring pores, and R(vs , T ) is the dissociation rate of vortex pairs in the superfluid film, which was considered in Section 12.3.

322

Berezinskii–Kosterlitz–Thouless theory

Equation (12.58) is a particular case of the general relation (12.34) connecting the vortex velocity v L with the superfluid velocity v s at the following values of the mutual friction parameters: α = 2π ac2

∂R , ∂vs

α  = 1.

(12.59)

The condition 1 − α  = 0 means that vortices move normally to the superfluid velocity and the ‘effective’ Magnus force, which is defined as the term ∝ [ˆz × v L ], is absent. The reason for this is clear. In our scenario the vortex can only creep normally to the average superfluid velocity directed along some of main axes of the jungle gym structure, i.e., axes of cylinders forming the structure. In Fig. 12.2 the creep direction is vertical. Though the vortex can randomly jump to neighbouring cells in horizontal directions (the direction of v s ), the probabilities of jumps to left and to right are equal, and there is no creep in the horizontal direction on average. Suppression of the transverse (effective Magnus) force is typical for superfluids with broken Galilean invariance. One example was discussed in Section 9.7: the transverse force in Fermi superfluids with impurities, when the Kopnin– Kravtsov force cancels the bare Magnus force. Another example is superfluids in lattices, which will be considered in Chapter 13. The energy of the vortex line per unit length (vortex line tension) can be estimated in the usual manner [see Eq. (2.6)] with the superfluid density ρs replaced by the effective three-dimensional density ρs3 and the lower cut-off in logarithm being the size of pore c (instead of the core radius rc ): ε=

ρs3 κ 2 rm ln . 4π c

(12.60)

Here, as before, rm is some external scale, for example, intervortex distance, radius of curvature of line, etc. Usually this quantity is several times larger than the period of the pore structure, and ln(rm /c) is not so large. The effective three-dimensional superfluid mass density ρs3 in the porous glass substrate of volume V is connected with the twodimensional superfluid density ρ˜s of the film by the relation ρs3 =

ρ˜s Atot . V

(12.61)

Here Atot is the total area of the film. For the ‘jungle gym’ structure with cell size c and rod diameter a, πa Atot ≈ 2. V c

(12.62)

Since any jump shifts only a small segment of the three-dimensional vortex of length ∼ c it should be accompanied by random local lengthening of the three-dimensional vortex line, which increases the barrier for vortex creep. The increase in the barrier can be estimated via the three-dimensional line tension as ρs3 κ 2 c, where ρs3 is the effective three-dimensional superfluid mass density. This means that the line tension barrier is smaller by a factor

12.5 Torsional oscillations in films

323

a/c ln 2π κvs rc than the pair-dissociation barrier, and in the following analysis the line tension contribution to the activation barrier will be ignored.

12.5 Torsional oscillations in films on rotating porous substrates: rotation dissipation peak In torsional oscillator experiments, Fukuda et al. (1998) revealed an additional rotationinduced peak in dissipation, different from the dissipation peak that was observed in superfluid films without rotation. A heuristic interpretation of the rotation-induced dissipation peak was suggested by Fukuda et al. (2005). Rotation can affect dissipation via rotation contribution to the velocity field. So it is a non-linear correction to the response. Instead of a derivation of such a correction from the theory, Fukuda et al. (2005) used the data on the non-linear response taken from independent experiments on large-amplitude torsional oscillations. This provided a qualitative explanation of the rotation peak, and even of some quantitative features of it, but could not pretend to be a full theory of the effect. A more detailed theory (Nemirovskii and Sonin, 2007) connected the parameters of the rotation peak with the parameters of the film and the substrate. The parameters of the three-dimensional ‘effective-medium’ description must be determined within the theory of two-dimensional films covering the multi-connected substrate. The crucial parameter to be determined is the derivative dR/dvs of the dissociation rate R of vortex-antivortex pairs in the two-dimensional film. In Section 12.3 we considered the thermally activated dissociation of vortex-antivortex pairs in superfluid films on plane substrates, while in our case films cover cylindrical surfaces. But as we shall see below, the relevant scale (size of the pair at the saddle point) is small compared to the substrate curvature, and one can use the BKT theory for plane films. Within the Coulomb gas approach using Eqs. (12.22) and (12.55) Nemirovskii and Sonin (2007) obtained: 8π 2 DK(0) 2 dR = y (ls ) exp[2π K(ls )] . dvs κrs3

(12.63)

The hydrodynamical approach yields a slightly different expression, but for further analysis this difference is not essential. Close to the critical temperature the coherence length ξ− is very long and even at rather low superfluid velocities vs the pair separation rs at the extremum is still shorter than ξ− . In this case, according to Eq. (12.30), the Kosterlitz– √ 2 ∝ t but not on x |t|. As Thouless parameters y(ls ) and x(ls ) can depend on x∞ ∞ ∝ a result of this, in both approaches the dissociation rate R and its derivative are analytic functions of temperature, and close to Tc the temperature dependence is described by the relation dRc dR = (1 + γ t), dvs dvs where Rc is the dissociation rate at the critical point.

(12.64)

324

Berezinskii–Kosterlitz–Thouless theory

Let us now apply the theory to oscillatory motion of the substrate superimposed on its steady rotation with angular velocity . The substrate oscillates as a solid body with linear velocity v B e−iωt in the rotational coordinate frame, where v B = [ × r], and  is the oscillating component of the substrate angular velocity. The superfluid oscillates in the clamped regime, and the normal velocity v n coincides with v B . Because of mutual friction, the superfluid component is dragged by the oscillating substrate, and according to Eq. (7.12) the superfluid component oscillates with the velocity vs =

2α vB . 2α − iω

(12.65)

Thus rotation increases the effective density ρ  of the fluid participating in the oscillation. The slow rotation with angular velocity   ω/α yields a purely imaginary contribution to ρ  equal to 2αρs3 / iω and according to Eq. (5.8) the rotation contribution to the inverse quality factor of the torsional oscillator is: Q−1 =

ρs3 2α . ρt ω

(12.66)

Using Eqs. (12.59), (12.61), and (12.62), one obtains that Q−1 =

4π 2 a 2 ρ˜s 2 ∂R . ρt ω ∂vs

(12.67)

Taking into account the temperature dependence of ρ˜s and ∂R/∂vs given by Eqs. (12.27) and (12.64) close to the critical temperature, the temperature dependence of the inverse quality factor is " (12.68) Q−1 = Q−1 (Tc )(1 + b |t| − γ |t|). The square-root cusp is more important at small t, and the inverse quality factor has a maximum at t = −b2 /4γ 2 close to the critical point. A more quantitative numerical analysis and comparison with experiments was done by Nemirovskii and Sonin (2007). Here we give some experimental parameters of Fukuda et al. (2005), which allow us to judge the relevance of the presented approach. The porous glass substrate can be modelled with the jungle gym structure with the diameter of rods a ≈ 1 μm and structure period c ≈ 2.5 μm. Then the circulation velocity around the pore is estimated as vs = κ/4l ≈ 1 cm/sec. The transition temperature was Tc ≈ 0.628 K. From comparison of theory and experiment, one can derive the pair size rs ≈ 2.98 × 10−5 cm at the saddle point. The inequality rs  a justifies our treatment of vortex-antivortex unbinding using the theory for plane films. The estimation of the creep velocity vL of the pore vortex using Eq. (12.58) yields vL ≈ 0.7 cm/sec. For the experimental frequency 477 Hz in the experiment, this corresponds to vortex displacements of about 1.5 × 10−3 cm during one oscillation period. The displacement is about 6 periods of the substrate structure. This looks sufficient for approximating sequences of jumps between neighbouring cells by quasi-continuous vortex motion.

12.5 Torsional oscillations in films 1

2

3

325

1

0.8 0.6 0.4 0.2 0 0.59

0.60

0.61

0.62

0.63

0.64

0.65

T (K) Figure 12.3 Comparison of the experimental and the theoretical inverse quality factor of torsional oscillations. The right peak (curve 1) is the static peak, which is present without rotation. The group of left peaks (curves 2) is rotation-induced peaks obtained experimentally for various  and scaled (divided) by . The fact that all these curves collapse well onto the same curve proves linearity on . All peaks are scaled so that heights of their apexes are equal to 1. The dashed line shows the theoretical inverse quality factor. Its height was fitted to the height of the experimental rotation peak, but the width and the position were calculated without any fitting. The theoretical curve ends with the critical point. The experimental superfluid density scaled by its zero-temperature value is also displayed (curve 3). Figure from Fukuda et al. (2005) with the dashed line added by Nemirovskii and Sonin (2007).

In Fig. 12.3 we plot the theoretical Q−1 (T ) (dashed line) together with the experimental data (Fukuda et al., 2005). The observed rotation-induced dissipation peak (the left peak) was scaled by the angular velocity  reducing it to the value measured at  = 6.28 rad/sec. The plots for different values of  collapse onto the same curve. This proves linear dependence of rotation induced dissipation on . For better comparison of the peak shape, the theoretical height of the peak was fitted to the experimental value. After this a reasonably good agreement in the peak shape and position was achieved without any additional fitting parameter.

13 Vortex dynamics in lattice superfluids

13.1 Magnus force in Josephson junction arrays The vortex dynamics in lattice models of superfluids has important features which are absent in continuous superfluids. The first feature is intrinsic pinning. As in the case of intrinsic pinning of vortices in superfluid films on porous substrates (Section 12.4), displacement of the vortex from one lattice cell to another is impeded by an energetic barrier. So the vortex is able to move only if the force on the vortex is strong enough. Since the force on the vortex is provided by a current, there is a critical current for vortex depinning above which steady vortex motion becomes possible. The second important feature is the suppressed or totally absent Magnus force. This force leads to motion of the vortex along the current and the Hall effect in superconductors. The best known lattice model of the superfluid is the Josephson Junction Array (JJA). We shall discuss only an ideal array, which is a regular lattice of superfluid islands (nodes) with weak Josephson links between neighbouring islands. Usually the theory of vortex motion in this model is studied in the continuous limit when the lattice is approximated by the continuous fluid with effective parameters derived from the parameters of the original lattice. These studies have not revealed any Magnus force normal to the vortex velocity (Eckern and Schmid, 1989). Moreover, there is experimental evidence of ballistic vortex motion in the JJA (van der Zant et al., 1992), which is possible only in the absence of the Magnus force. There has been much theoretical work trying to explain this important feature of the JJA. Although this book is about neutral superfluids, this section addresses a charged superfluid in order to have a common ground with the majority of papers on JJA dynamics. Let us consider a conductor in a magnetic field H . When its symmetry is not lower than three-fold (which includes triangular and square lattices), the Ohm law is: ρH [H × I ], (13.1) E = ρL I + H where E is the electrical field, ρL is the longitudinal resistance and ρH is the Hall resistance in the magnetic field H . Now let us consider the transformation in which the directions of the fields E and H and the current I are reversed: E → −E, 326

H → −H ,

I → −I .

(13.2)

13.1 Magnus force in Josephson junction arrays

327

The Ohm law Eq. (13.1) is only invariant with respect to this field-current inversion for a system without the Hall effect (ρH = 0). Our field-current inversion is in fact charge inversion. Symmetry with respect to it takes place in semiconductors with equal numbers of particles and holes and weak-coupling superconductors with particle-hole symmetry, which was shown to forbid the Hall effect in the Ginzburg–Landau theory (see Dorsey, 1992, and references therein). Next we consider the JJA with the energy  1  −1 Ql Cl,k Qk − EJ cos (φl − φk ) . (13.3) E= 2 l,k

Here Ql is the electric charge, EJ is the Josephson coupling energy, and φl (t) is the gauge invariant phase at the island specified by the integer index l. The double summation −1 are the elements of the inverse capacity expands only on the neighbouring nodes, and Cl,k matrix, which characterises interaction (mostly Coulomb) between the lth and the kth nodes −1 ). In an ideal uniform lattice there are and inside every island (the diagonal elements Cl,l constant values for all diagonal and all off-diagonal elements of the capacitance matrix. In the external magnetic field H = ∇ × A, the gauge invariant phase φl is not single valued. Only the difference between neighbouring nodes is well defined:  2π (k) A · dl. (13.4) φk − φl = θk − θl − 0 (l) Here 0 = hc/2e is the magnetic flux quantum for the Cooper-pair condensate, θl is the canonical phase at the lth node, and the integral over the electromagnetic vector potential A is taken along the path connecting the two neighbouring kth and lth nodes. Considering the energy (13.3) as a Hamiltonian for the pairs of canonical conjugate variables {Ql /2e, θl }, one can write the Hamilton equations supplemented with the dissipation terms, which are connected with the Ohmic conductance matrix σl,k : dθl = −χl = −μl − el , dt   dQl χ l − χk = IC . sin(φl − φk ) + σl,k dt e h¯

k

(13.5) (13.6)

k

Here IC = 2eEJ /h¯ is the critical Josephson current, l and μl are electric and chemical potentials of the lth island, and χl is the electrochemical potential determined from 9 inversion of the relation Ql = k Cl,k χk /e. The canonical phase is the phase of the condensate wave function at a given superconducting island. It is not single valued too, but its circulation along any closed path through the nodes of the JJA is always an integer number of 2π , while the circulation of the gauge invariant phase may be any number depending on the magnetic field. There is a fundamental difference between vortices in a lattice and vortices in a continuous superfluid. In the lattice there are no singular vortex lines. They only appear in the continuum limit. At best, one can define a lattice cell containing the vortex centre: it is

328

Vortex dynamics in lattice superfluids

the cell around which the circulation of the phase θ is equal to 2π . However, in the lattice the circulation around a closed path is also not well defined. Let us consider the closed path around the cell with the vortex centre. One can change the phase difference by −2π between any two neighbouring nodes on this path without any effect on observed physical parameters (currents, voltages and so on). Then the phase circulation around the chosen cell vanishes, but must appear along a path around a neighbouring cell. Thus one cannot locate unambiguously the position of the phase singularity. In order to avoid this ambiguity in the JJA model, a special rule has been formulated: the phase difference between two neighbouring nodes must not exceed π . When for some Josephson link the phase difference achieves the value π , one must redefine the phases; as a result, the vortex centre is put into another cell. This procedure is usual for numerical studies of the vortex motion in JJAs (Bobbert, 1992). The field-current inversion discussed in the beginning of the section reduces to the change of signs of the phases and the electromagnetic potentials in the equations of motion (13.5) and (13.6). It is evident that the equations are invariant with respect to this inversion. This forbids the Hall effect and the Magnus force in the model (Sonin, 1997). The presence of the external charge has no effect on our symmetry analysis. In order to take into account 9 the external charge, one should use the Gibbs potential G = E − V ex l Ql , where E is given by Eq. (13.3) and V ex is the electric potential which creates the external charge 9 Qex = V ex k Cl+k . Then introducing the charge deviation Ql = Ql − Qex , one returns back to the energy E with Ql instead of Ql . So the effect of the external charge reduces to a shift of the Fermi level. Thus symmetry rules out any Magnus force in the JJA, since the model of the JJA discussed up to now was derived assuming particle-hole symmetry of the Fermi superfluid inside islands. Meanwhile, this symmetry is not exact, and one of the consequences of its violation is dependence of the Josephson energy EJ on the number of island electrons. Taking into account this dependence, the continuous description of the lattice JJA model allows the existence of the Magnus force although it is usually rather weak. We shall discuss the vortex dynamics in the continuous approximation for the lattice model in the next section, and finish this section by introducing the continuous approximation for the JJA model. We replace values of charges and phase in discrete nodes by the continuous fields of these variables, assuming that they vary slowly from node to node. Then differences of variables at neighbouring nodes are approximated by their small gradients: θl − θk ≈ (r lk · ∇)θ (r)  1,

Nl − Nk =

Ql − Qk ≈ (r lk · ∇)N (r)  1. (13.7) e

Here Nl is the number of particles in the lth island and r lk are the position vectors connecting the neighbouring lth and the kth nodes. Assuming for the sake of simplicity the two-dimensional quadratic JJA, these position vectors have a modulus equal to the lattice cell size a and are directed along the x and the y axes. Eventually the equations of motion

13.2 Vortex dynamics in continuous approximation

329

become partial differential equations for continuous fields corresponding to the Hamiltonian (energy) per unit area (from now on the magnetic field is ignored): E=

α1 h¯ 2 n˜ α (n − n0 )2 + (∇n)2 + (∇θ )2 . 2 2 2m

(13.8)

Here n = N/a 2 is the average two-dimensional density of particles, a is the lattice constant, and n0 is the value of n in the ground state. The parameters α=

e2 a 2 , C

α1 =

e2 a 4 2C0

(13.9)

are determined by two capacitances C and C0 connected with the original capacitances in Eq. (13.3) by the relations 1 = (C −1 )l,l + 4(C −1 )l,k , C

1 = (C −1 )l,k . C0

(13.10)

The parameter n˜ has dimensionality of the density and is determined by the Josephson energy: h¯ 2 n˜ = EJ . m

(13.11)

13.2 Vortex dynamics in continuous approximation for a lattice superfluid The continuous approximation for lattice superfluids in general gives the phenomenological theory, which corresponds to the Lagrangian: ˜ )2 h¯ 2 n(∇θ − Ec (n), (13.12) 2m where Ec (n) is the energy of a resting fluid which depends only on n. For simplicity we consider the two-dimensional problem, where n is the particle number per unit area. The Hamiltonian (energy) for this Lagrangian, L = −hn ¯ θ˙ −

h¯ 2 n(∇θ ˜ )2 ∂L θ˙ − L = + Ec (n), (13.13) 2m ∂ θ˙ differs from the Hamiltonian (13.8) by the absence of the term dependent on ∇n, which is usually neglected in hydrodynamical theories, and by a more general form of the term, which depends only on the density n [the term ∝ α in Eq. (13.8)]. The chemical potential in the model is H =

d n˜ h¯ 2 (∇θ )2 , (13.14) dn 2m where μ0 = ∂Ec /m∂n is the chemical potential of the fluid at rest. According to the thermodynamic definition the pressure is  2  h¯ (∇θ )2 d n˜ − n˜ . (13.15) P = mnμ − E = mnμ0 − Ec + n dn 2m μ = μ0 +

330

Vortex dynamics in lattice superfluids

The Hamiltonian (13.13) recalls the hydrodynamical Hamiltonian, which follows, in particular, from the Gross–Pitaevskii theory. But there is an essential difference. The continuous approximation for the lattice model restores translational invariance, but not Galilean invariance. The latter is absent since the effective density n, ˜ which characterises the stiffness of the phase field, can be much less than the particle density n because of weak links between islands. So it is necessary to reassess vortex dynamics, especially the transverse force, in the light of this difference. Let us discuss the conservation laws, which follow from Noether’s theorem. The gauge invariance provides the conservation law for charge (particle number):   ∂L d ∂L = 0. (13.16) + ∇k dt ∂ θ˙ ∂∇k θ Since the canonical definition for the particle density is n = −∂L/h∂ ¯ θ˙ , this is the mass continuity equation (the first Hamilton equation) for the fluid: m

dn = −∇ · j , dt

(13.17)

where j =−

m ∂L = h¯ n∇θ ˜ = mnv ˜ s h¯ ∂∇θ

(13.18)

h¯ is the mass current, and the superfluid velocity v s = m ∇θ is defined as in the Galilean invariant superfluid. The mass current is at the same time the momentum density. The second Hamilton equation is the Josephson equation:

h¯ 2 d n˜ ∂H ∂θ =− = −mμ0 − (∇θ )2 . (13.19) ∂t ∂n 2m dn Linearising the Hamilton equations with respect to small variations of density and phase, one obtains the sound-like spectrum ω = c˜s k of the plane-wave collective mode ∝ eik·r−iωt . The sound velocity is given by h¯

∂μ0 n˜ (13.20) = cs2 , ∂n n and is normally much less than the sound velocity cs of the uniform fluid because of the small n˜ determined by the low Josephson energy. From translational invariance, Noether’s theorem gives an equation     ∂L ∂ ∂L (13.21) ∇l θ − Lδkl = 0. ∇k θ + ∇l ∂t ∂ θ˙ ∂∇k θ c˜s2 = n˜

This is the conservation law ∂gk ˜ kl = 0, + ∇l  ∂t

(13.22)

∂L ∇θ = hn∇θ = mnv s . ¯ ∂ θ˙

(13.23)

for the current g=−

13.2 Vortex dynamics in continuous approximation

331

Here h2 ˜ kl = ¯ n∇  ˜ k θ ∇l θ + P δkl . m

(13.24)

In the Galilean invariant system the current g, which appears in the Noether conservation law following from translational invariance, coincides with j . But in our case with broken Galilean invariance (n˜  = n), the currents g and j differ. The true mass current (true momentum density) is j and not g. This follows from the fact that the density n and the current j in the mass continuity equation (13.17) are averages of their relevant quantum mechanical operators nˆ = ψˆ † ψˆ and i h¯ ˆ jˆ = − (ψˆ † ∇ ψˆ − ∇ ψˆ † ψ), 2

(13.25)

where ψˆ and ψˆ † are the annihilation and the creation operators normalised to the particle density. The continuity equation (13.17) is universal and valid for any gauge invariant system, including lattice superfluids, independently of what forces are applied to the system or how particles interact. So Noether’s theorem for a translational invariant but not Galilean invariant system does not provide the conservation law for the true momentum. In the next section we shall see that for particles in a periodic potential the current g coincides with the quasimomentum density. Correspondingly the flux tensor (13.24) can be called the quasimomentum flux tensor. Although Noether’s theorem does not lead to the conservation law for the true momentum, the true momentum conservation law still holds approximately as can be checked using the Hamilton equations (13.17) and (13.19): ∂jk + ∇l kl = 0, ∂t

(13.26)

h¯ 2 d n˜ n∇ ˜ k θ ∇l θ + P˜ δkl , m dn

(13.27)

where the momentum flux tensor is kl =

and the partial pressure P˜ is determined from the relation d P˜ = ndμ ˜ 0 . These equations reduce to those in hydrodynamics of a Galilean invariant fluid when n˜ = n. When deriving the momentum conservation law we treated d n/dn ˜ as a constant. Variation of d n/dn ˜ is quadratic in phase gradients and enters the theory in combination with terms which are quadratic in phase gradients. So neglecting variation of d n/dn ˜ one neglects phase gradient terms higher than second order. The next step is the derivation of the forces on the vortex: the Magnus force proportional to the vortex velocity v L and the Lorentz force proportional to the superfluid mass current. Since both forces are linear in velocities of the vortex and the superfluid, we can consider the two forces independently. Deriving the Lorentz force one can assume that the vortex is at rest in the laboratory system connected with the lattice. Solving Eq. (13.19) for the

332

Vortex dynamics in lattice superfluids

time-independent phase θ , one obtains the quadratic in ∇θ correction to the chemical potential (Bernoulli law): h¯ 2 d n˜ (∇θ )2 . (13.28) 2m2 dn The Lorentz force must be derived from the quasimomentum balance. Because of Eq. (13.28) the quasimomentum flux tensor (13.24) becomes   h2 ˜ kl = ¯ n∇  ˜ k θ ∇l θ + P0 − n(∇θ ˜ )2 δkl , (13.29) m where P0 is a constant pressure in the absence of any velocity field. Similarly to hydrodynamics of a perfect fluid (Section 2.3), if the quasimomentum conservation law holds, the components of the Lorentz force are equal to the quasimomentum flux through a  ˜ kl dSl . The phase gradient cylindrical surface surrounding the vortex line: FLk = −  ∇θ = ∇θv + ∇θs consists of the gradient ∇θv = [ˆz × r]/r 2 induced by the vortex line and the gradient ∇θs = j /h¯ n˜ produced by the transport current. The force arises from the cross-terms ∇θv ·∇θs in the quasimomentum flux tensor. The integration yields the Lorentz force μ0 = −

˜ s × κ]. F L = [j × κ] = mn[v

(13.30)

This expression for the Lorentz force is well known and is not disputed (in contrast to the Magnus force). One can derive it from the definition of a force in mechanics: a force is a derivative of the energy with respect to displacement. In the present case it is a displacement of the vortex. The Lorentz force follows from the quasimomentum balance because it is a momentum exchange between the transport velocity field and the vortex. Any variation of the transport velocity must be accompanied by the momentum transfer to or from the lattice, as follows from the Bloch band theory for particles in a periodic potential (see the next section). It was suggested that the Magnus force must be derived from the balance of the true momentum (Sonin, 2013c). We shall consider this balance in the coordinate frame moving with the vortex where variables are time independent. In the coordinate frame moving with velocity w, the energy E  is connected with the energy E in the laboratory frame by the well known relation (we ignore terms quadratic in w): E  = E − j · w.

(13.31)

Then according to the Hamiltonian (13.13) written in the laboratory frame E =

h¯ 2 n˜ (∇θ  )2 + Ec (n), 2m

(13.32)

where the phase gradient in the moving frame is ∇θ  = ∇θ − mw/h. ¯ This does not differ from the Hamiltonian in the laboratory frame, and one can use the momentum flux tensor (13.27) replacing θ by θ  . The pressure variation is determined from the Bernoulli law as before. Deriving the Magnus force one may ignore the transport superfluid current in the

13.2 Vortex dynamics in continuous approximation

333

laboratory  frame, i.e., ∇θ = ∇θv , and replace w by v L . Then the Magnus force components FMi = kl dSl are given by the momentum flux through a cylinder around the vortex line (cf. the Lorentz force), and the Magnus force is FM =

d n˜ nm[v ˜ L × κ]. dn

(13.33)

Without external forces F L = F M , and the vortex moves with a velocity which differs ˜ from the superfluid velocity v s by the factor d n/dn. An argument for derivation of the Magnus force from the true momentum balance is that the vortex moves without any superfluid motion with respect to the lattice at infinity and therefore the motion is not accompanied by momentum transfer to the lattice (Sonin, 2013c). Vortex motion produces a backflow, which removes particles before the vortex with some value of the vortex-induced velocity v v and transports them behind the vortex, where v v has opposite sign. Adding or removing particles can affect the momentum density only if n˜ depends on the particle density n. This is why the Magnus force is determined from the balance of the true momentum and is proportional to d n/dn. ˜ Anyway, our approach yields correct values of the Magnus force at least in the two limits when the force is known exactly: the Galilean invariant fluid and the JJA with particle-hole symmetry when d n/dn ˜ = 0 and the Magnus force vanishes. None of these arguments are free from assumptions and they cannot be a substitute for a rigorous derivation of the Magnus force in a lattice superfluid, which is still lacking. The rigorous derivation requires going beyond the continuous approximation. As usual, one may define the vortex core radius rc as a distance from the vortex axis at which a correction to the effective density n, ˜ which determines the kinetic energy, becomes of the same order as n˜ itself. The correction n to the density n can be derived from the generalised Bernoulli law (13.28): n = −

h¯ 2 ∂n d n˜ h¯ 2 d n˜ (∇j θ )2 = − n˜ 2 2 (∇j θ )2 , ∂μ0 dn 2m dn 2m c˜s

(13.34)

where the sound velocity c˜s is given by Eq. (13.20). The distance rc , where ∇θ ∼ 1/rc , is ˜ ∼ n: ˜ determined by the condition n (d n/dn) rc ∼

d n˜ κ . dn c˜s

(13.35)

The core radius can be rather small because of the small factor d n/dn ˜ and it even vanishes in the standard JJA model where d n/dn ˜ = 0. But one should remember that rc cannot be smaller than the lattice constant a, which must be accepted as a core radius if Eq. (13.35) gives a value less than a. Similarly to the case of the uniform Galilean invariant superfluid, the suppression of the effective density n˜ inside the core leads to the backflow around a moving vortex, which determines the core vortex mass. The backflow velocity field is determined from the condition that the component of the velocity v L normal to the core boundary is equal

334

Vortex dynamics in lattice superfluids

to the normal component of the center-of-mass velocity of the fluid determined as vg =

n˜ j = vs . mn n

(13.36)

In the next section we shall see that this velocity is a group velocity of a quasiparticle in the Bloch band theory. Calculation of the energy of the backflow veloctiy field, similar to the calculation done in Section 1.3 for a moving cylinder, yields the core mass of the vortex in a superfluid with broken Galilean invariance: μcore = π rc2

mn2 . n˜

(13.37)

The first evaluation of the vortex mass in the lattice JJA model was undertaken by Simanek (1983). Using the continuous approximation he took into account the contribution of the term dependent on ∇n [see Eq. (13.8)], which originates from the Coulomb interaction between islands and is determined by the capacitance C0 defined by Eq. (13.10).

13.3 Vortex dynamics from Bloch band theory The meaning of our approach becomes more transparent if one applies it to particles in a periodic potential U (r) = U (r + a). Here a can be any translation vector of the periodic structure. The density of the single-particle energy is H =

h¯ 2 |∇ψ|2 + U (r)|ψ|2 . 2m

(13.38)

The eigenstates are described by Bloch wave functions: ψ(r, t) = un (r, k)eik·r−iE(k)t/h¯ ,

(13.39)

where un (r, k) is a periodic function. The quasimomentum h¯ k differs from the true momentum of the quantum state. The latter can be calculated by averaging the quantum mechanical expression (13.25) for the momentum operator over the lattice unit cell:   ∗ (13.40) p = −i h¯ ψ(r) ∇ψ(r) dr = h¯ k − i h¯ u(r)∗ ∇u(r) dr. Here, in contrast to Eq. (13.25), the wave function is normalised to the unit cell: |ψ|2 dr = 1. Calculating the band energy E(k) in the kp approximation for small k, i.e., at the band bottom (the energy minimum at k = 0) one obtains E(k) =

h¯ 2 k 2 , 2m∗

p = mv g ,

vg =

hk dE(k) ¯ = ∗, h¯ dk m

where v g is the group velocity and m∗ is the effective mass.

(13.41)

13.3 Vortex dynamics from Bloch band theory

335

Bose condensation of particles in a single Bloch state was analysed by Pitaevskii (2006). The wave vector k = ∇θ is a gradient of the phase θ . Then the true momentum density (mass current) j = np coincides with that given by Eq. (13.18) if n˜ = nm/m∗ . On the other hand, the current g = nhk ¯ is the quasimomentum density. So the Bloch band theory for bosons clearly connects the currents j and g derived in Section 13.3 from the phenomenological Lagrangian with the densities of the true momentum and the quasimomentum respectively (Sonin, 2013c). It is well known from solid state physics that an external force on a particle in the energy band determines time variation of the quasimomentum but not the true momentum: h¯

dv g dk = m∗ = f. dt dt

(13.42)

In the absence of Umklapp processes the total quasimomentum is also a conserved quantity, and the quasimomentum conservation law for the Bose condensate in a single Bloch state is given by Eq. (13.22). Only the part dp/dt of the whole momentum variation h¯ dk/dt brought to the system by the external force is transferred to particles. The rest is transferred to the lattice supporting the periodic potential. In the Bloch theory for particles in a periodic potential, the true momentum differs from the quasimomentum by the constant factor m/m∗ . Therefore the true momentum conservation law is exact and directly follows from the quasimomentum (Noether’s) conservation law after multiplying the latter by m/m∗ . But in the general case considered in the previous section, only the quasimomentum conservation law was exact. At the Galilean transformation (r = r  + wt, t = t  ) to the coordinate frame moving with the velocity w, the Hamiltonian and the Schr¨odinger equation retain their form, but the  2 wave function must transform as ψ = ψ  eimv·r /h¯ +imv t/2h¯ (see the problem to Section 17 of Landau and Lifshitz, 1982a). Correspondingly, in the moving coordinate frame the Bloch function (13.39) becomes 



ψ  (r  , t  ) = un (r  + wt  , k)eik ·r −iEf (k)h¯ ,

(13.43)

where the wave number k  and the energy Ef (k) are connected with those in the laboratory frame by the relations k  = k − mw, h¯ 2 (k − m∗ w)2 mw 2 (m − m∗ )w 2 Ef = E(k) − h¯ k · w + ≈ . + 2 2m∗ 2

(13.44)

In the moving frame the Schr¨odinger equation contains the time dependent periodic potential, and its solution is not an eigenstate of the quantum mechanical energy operator i h∂/∂t. Equation (13.43) is a solution of this equation following from the Floquet theorem, ¯ the energy Ef being the Floquet quasienergy. The true energy of the state is an average value (expectation value) of the energy operator independently of whether the state is an

336

Vortex dynamics in lattice superfluids

eigenstate of the energy operator, or not. It is different from the Floquet quasienergy and is given by 



E =

∂ψ ψ i h¯ dr = Ef + ∂t



∂u dr ∂r mw 2 = E(k) − hp ¯ ·w+ 2 m h¯ 2 (k − mw)2 mw 2  1 − ≈ . + 2m∗ 2 m∗

∗

u∗n i hw ¯ ·

(13.45)

The Galilean transformation demonstrates the difference between the quasimomentum and the quasienergy on one side, and the true momentum and the true energy on the other. If one ignores the difference and treats the quasiparticle as a real particle with mass m∗ , the ground ∗ − w). state in the moving frame is the state with zero particle current j  /m = n(hk/m ¯  But at the minimum of the true energy Eq. (13.45) the particle current j /m does not vanish. This is the effect of dragging of particles by a moving periodic potential, which is especially pronounced in the limit of large effective mass when the particles are totally trapped by the periodic potential and cannot move relatively to it. The effect was observed for a potential produced by a running acoustic wave, which drags electrons (acoustoelectric effect) (Ilisavskii et al., 2001) or excitons (Cerda-M`endez et al., 2010). The analysis of this section addressed only single-particle states and the ideal Bose– Einstein condensation in a single-particle state. But a real vortex with a well defined core is impossible without interaction. However, adding weak particle-particle interaction one can develop the Gross–Pitaevskii theory similar to that theory for uniform translational invariant fluids. This approach is valid as long as the interaction is not too strong and the vortex core radius essentially exceeds the lattice constant.

13.4 Vortex dynamics in the Bose–Hubbard model The JJA model can describe not only Fermi superfluids, but also Bose superfluids. In the latter case the fluid in islands consists of bosons, and the JJA model follows from the Bose– Hubbard model (Fisher et al., 1989), which is described by the Hamiltonian H = −J

 i,j

 U ˆ ˆ bˆi† bˆj + Ni (Ni − 1) − mμ Nˆ i . 2 i

(13.46)

i

Here μ is the chemical potential, the operators bˆi and bˆi† are the operators of annihilation and creation of a boson at the ith node (island), and Nˆ i = bˆi† bˆi is the particle number operator at the same node. The first sum is over neighbouring lattice nodes i and j . The periodic structure of potential wells, which leads to the Bose–Hubbard model, is realised for cold-atom BEC in experiments with optical lattices (Bloch et al., 2008).

13.4 Vortex dynamics in the Bose–Hubbard model

337

In the superfluid phase with large numbers of particles Ni per node, all operators can be replaced by the classical numbers in the spirit of the Bogolyubov theory: " " bˆi† → Ni e−iθi , (13.47) bˆi → Ni eiθi , where θi is the phase at the ith node. According to Eq. (13.46) and assuming that the particle numbers Ni and the phases θi vary slowly from node to node, the energy at the ith node is Ei = −zi J Ni +

U Ni (Ni − 1) − mμNi , 2

(13.48)

 (θj − θi )2 , 2

(13.49)

where zi =



ei(θj −θi ) ≈ z0 −

j

j

and z0 is the number of nearest neighbours in the lattice. For the transition to continuous description, we assume that θi = k · r i where r i is the position vector of the ith node. For a square lattice with z0 = 4 and lattice constant a zi = 4 − k 2 a 2 .

(13.50)

Since the wave vector k corresponds to the gradient operator ∇ in the configurational space, one obtains in the continuum limit for small k the Hamiltonian (13.13) with n=

N , a2

n˜ =

4mJ a 2 h2 ¯

Ec (n) =

n,

U a2 2 n − mμn − 4J n. 2

(13.51)

This is the tight-binding limit of the Bose condensate of particles in a Bloch state (see the previous section) with the effective mass m∗ =

h¯ 2 . 4J a 2

(13.52)

When the energy J of the internode hopping decreases, the phase transition from superfluid to Mott insulator must occur (Fisher et al., 1989). In the limit z0 J /U → 0 when the hopping term ∝ J can be ignored, the eigenstates are given by Fock states |N = |N

with fixed number N of particles at any node. At growing J the transition line can be found in the mean-field approximation (Ueda, 2010). One takes hopping into account by introducing a mean field equal to the average value of the annihilation operator (its complex conjugate is the averaged creation operator): bˆi = ψi = |ψ|eiθi ,

bˆi† = ψi∗ = |ψ|e−iθi .

(13.53)

It is assumed that only the phase and not the modulus of the order parameter ψ varies from node to node. In general |ψ|2 is not equal to N as Eq. (13.47) assumed and it must be

338

Vortex dynamics in lattice superfluids

determined from the condition of self-consistency (see below). Introducing the mean field, one reduces the problem to the single-node problem with the Hamiltonian Hs = −zi J (bˆi† ψ + ψ ∗ bˆi ) +

U ˆ ˆ Ni (Ni − 1) − mμNˆ i , 2

(13.54)

where zi is given by Eq. (13.49). The multi-node wave function is a product of the single-node wave functions. Calculating the energy of the original Hamiltonian (13.46) for this wave function and minimising it with respect to ψ one obtains the self-consistency equation, which determines ψ. Following this approach (Ueda, 2010) one obtains the phase diagram shown in Fig. 13.1. The Mott

Figure 13.1 The phase diagram of the Bose–Hubbard model. The Mott insulator phase occupies lobes corresponding to fixed integer numbers N of bosons. The shaded beaks of the superfluid phase, which penetrate between insulator lobes, are analysed in the text. The dashed line separates the region with the inverse Magnus force from the rest of the superfluid phase. The line is schematic since only its left horizontal part was really calculated. The region of the inverse Magnus force exists under any lobe but is shown only for the beak between the N = 1 and N = 2 lobes. Figure from Sonin (2013c).

13.4 Vortex dynamics in the Bose–Hubbard model

339

insulator phases with fixed numbers N of particles per node occupy the interiors of lobes at small z0 J /U . We address vortex dynamics close to the Mott transition at minimal values of J , i.e., at beaks of the superfluid phase between lobes, which are shaded in Fig. 13.1. Here the mean-field approximation is simplified by the fact that only two states with N and N + 1 particles1 interplay at the superfluid phase in the beak between the lobes N and N + 1. This is because at μ = N U/m these two states have the same energy, whereas all other states are separated by a gap of the order of the high energy U . For a beak between the lobes corresponding to Mott insulators with numbers of bosons per node N and N + 1, we look for a solution in the form of a superposition of two Fock states: |N = fN |N + fN +1 |N + 1 .

(13.55)

The wave function is an eigenfunction of the single-node Hamiltonian (13.54) if .$  2 . μ2 . 4 + mz 2 J 2 (N + 1)|ψ|2 ∓ μ2 . , fN ,N+1 = / $ 2 2 2 μ4 + mz 2 J 2 (N + 1)|ψ|2

(13.56)

where the upper sign corresponds to N and the lower one corresponds to N + 1. The energy of the ground state is #   1 m2 μ2 U N (N + 1) − mμ N + − + z2 J 2 (N + 1)|ψ|2 , (13.57) εN = − 2 2 4 where μ = μ − U N/m. The average number of particles per node is a function of μ : ˆ =N+1+ $ N

2 2 μ2 +

μ 4z2 2 J (N m2

.

(13.58)

+ 1)|ψ|2

The self-consistency equation follows either from the minimisation of the total energy with ˆ respect to ψ or from the condition that ψ is the average value of the operator b: zJ (N + 1) ˆ = $ ψ = b

ψ. 2 2 2 m 4μ + z2 J 2 (N + 1)|ψ|2

(13.59)

A non-trivial (i.e., non-zero) solution of this equation is |ψ|2 =

m2 μ2 N +1 − 2 2 . 4 4z J (N + 1)

(13.60)

The eigenvalue εN of the Hamiltonian (13.46) determines the Gibbs thermodynamic potential GN = zJ |ψ|2 + εN of the grand canonical ensemble per node. It is useful to go from the grand canonical ensemble with the Gibbs potential being a function of μ to the

1 We assume that numbers of particles at all nodes are equal and write N without the subscript i . i

340

Vortex dynamics in lattice superfluids

canonical ensemble where the energy density is a function of the average particle number Nˆ per node. Then the energy per node is    2 √ 1 U N ˆ = + U N Ne + zJ |ψ|2 − 2 N + 1 − Ne2 |ψ| , EN = GN + mμ N

2 4 (13.61) where Ne = Nˆ − N − 12 . The energy has a minimum at   1 − Ne2 . |ψ|2 = (N + 1) 4

(13.62)

As in any second-order phase transition, ψ vanishes at the phase transition lines, where Ne = ± 12 and the number of particles reaches N at the lower border and N + 1 at the upper border. But in contrast to the Landau–Lifshitz theory of second-order transitions, there is no analytic expansion in ψ near the critical temperature because of the term linear in |ψ|. One can repeat the transition to continuous description done for the superfluid phase far from the Mott transition [see Eqs. (13.49)–(13.51)]. This yields the Hamiltonian (13.13) with   1 2m (13.63) − n2e a 4 , n˜ = 2 J (N + 1) 4 h¯ and the energy density of the fluid at rest equal to   EN UN2 4J 1 2 4 Ec = 2 = − ne a . + U N ne − 2 (N + 1) 4 a 2a 2 a

(13.64)

Here ne = Ne /a 2 = n − (N + 12 )/a 2 is the effective density, while the true particle density is n = Nˆ /a 2 . The density dependent factor in the expression (13.33) for the Magnus force is   1 8m2 2 4 d n˜ 2 2 4 (13.65) n˜ = − 4 J a (N + 1) ne − ne a . dn 4 h¯ A remarkable feature of the Magnus force in beaks of the superfluid phase is that its sign can be inverse with respect to that dictated by the sign of velocity circulation around the vortex. This happens in the upper halves of the beaks as shown in Fig. 13.1. The regions of the inverse Magnus force neighbour any insulator lobe from below, where ne is positive. Since at the upper borders of the lobes ne is negative, the line ne = 0, where the Magnus force changes its sign, must end somewhere at the border of the lobe. In Fig. 13.1 it is shown by a dashed line for the beak between the N = 1 and N = 2 lobes. According to Eqs. (13.20) and (13.35) the sound velocity and the vortex core radius in beaks of the superfluid phase are  1 4J a ne a 2 − n2e a 4 , (N + 1) rc ∼ $ a. (13.66) c˜s = 4 h¯ 1 − n2 a 4 4

e

13.5 Magnus force, Hall effect and topology

341

Except for the close vicinity to the Mott transition where ne a 2 = ± 12 , this yields a core radius of the order of the lattice constant a or less. In the latter case one must estimate the core radius as equal to a.

13.5 Magnus force, Hall effect and topology The Magnus force leads to the Hall effect if particles have a charge q. The electric field is determined by the vortex velocity: E = 1c [B × v L ]. The value of the Hall conductivity q j is the charge current normal to the electric field, depends on σH = jq /E, where jq = m the amplitude of the Magnus force. According to the analysis within the Bloch band theory (Section 13.3), the Hall conductivity σH = (m/m∗ )2 qnc/B is less by the factor (m/m∗ )2 than the Hall conductivity qnc/B known for normal electrons in solids and derived from the same Bloch band theory as used by us. There is no conflict between these two results. In the normal electron fluid the magnetic force lines (counterparts of our vortex lines) move ∗ in the Bloch with the same velocity as charges, i.e., with the group velocity v g = hk/m ¯ band, in accordance with Helmholtz’s theorem of classic hydrodynamics. In fact, this is the only relevant velocity, since there is no coherent phase, which determines the superfluid h¯ ∇θ . In the superfluid case, Helmholtz’s theorem is not valid in general, velocity vs = m and the velocity v L is a velocity of a phase singularity, which is less than the velocity v g by the factor m∗ /m. The uniform magnetic field is crucial for dynamics of normal electrons, causing them to move along cycloid trajectories. In dynamics of superconducting vortices, the non-uniform magnetic field is localised in fluxons and is commonly neglected as being weak compared to the effect of the phase gradient around the phase singularity (Kopnin, 2001). In the light of this connection between the Magnus force and the Hall conductivity, let us compare our theory with that of Lindner et al. (2010) and Huber and Lindner (2011), who were the first to notice that in the Bose–Hubbard model for charged particles the Hall conductivity changes its sign together with the sign of ne .2 However, in their theory the Hall conductivity σH remains finite on approaching the line ne = 0. So the change in the sign of σH is accompanied by a jump of σH , whereas according to our analysis σH must be continuous at ne = 0 [see Eq. (13.65)]. Moreover, in the superfluid phase far from the superfluid–insulator transition, our theory predicts the Hall conductivity, which differs from that of Lindner et al. (2010) and Huber and Lindner (2011) by the factor (m/m∗ )2 proportional to J 2 [see Eq. (13.52)]. The factor can be very small in the tightbinding limit. This is the same factor which differentiates our Hall conductivity from that of the normal fluid (see the previous paragraph). The Hall conductivity of Lindner et al. (2010) and Huber and Lindner (2011) far from the superfluid–insulator transition would be obtained if one considered the effective mass as a true mass of particles in a superfluid and 2 Note that Huber and Lindner (2011) used the name ‘Magnus force’ for the force proportional to the superfluid velocity v but s not to the vortex velocity v L . This disagrees with the nomenclature usually used in the theory of superconductivity: the force ∝ v s is the Lorentz force and the force ∝ v L is the Magnus force.

342

Vortex dynamics in lattice superfluids

the quasimomentum as a true momentum. This means that in the theory of Lindner et al. (2010) and Huber and Lindner (2011), the broken Galilean invariance does not suppress the Magnus force. A possible source of disagreement is that the theory of these papers used topological arguments without directly addressing the momentum balance. In the past there have been other attempts to derive the Magnus force in lattice superfluids from topology. In particular, the topological origin of the Wess–Zumino term −hn ¯ θ˙ in the Lagrangian (13.12) has been widely discussed in the literature (Volovik, 2003b). The arguments were about whether the total fluid density n must be replaced by some other density. It is evident that adding any constant C to the density n in the Wess–Zumino term does not affect the Hamilton equations (13.17) and (13.19). This leads to formal redefinition of the current hn∇θ in the Lagrangian formalism, which becomes h(n+C)∇θ . However, in ¯ ¯ the conservation law (13.22) following from Noether’s theorem, the constant C appears in ∂L/∂ θ˙ (the first term) and in L (the last term), and the two corrections cancel one another. Moreover, as discussed above, the true current given by Eq. (13.18) is determined not by ∂L/∂ θ˙ but by ∂L/∂∇θ , which does not contain the constant C. So the choice of the constant C does not matter at all. But the role of the Wess–Zumino term changes after transition from the continuous model in terms of fields to the reduced description in terms of the vortex coordinates r L (xL , yL ). This leads to substitution of the phase field θv (r − r L ) for a vortex moving with the velocity v L = dr L /dt into the Wess–Zumino term and its integration over the whole space. Bearing in mind that θ˙v = −(v L · ∇)θv , the Wess–Zumino term becomes  (13.67) LW Z = h(n ¯ + C) (v L · ∇) θv (r, r L )dr = h(n ¯ + C)v L · [ˆz × r L ]. Varying the Lagrangian with respect to r L (t), one obtains the equation of vortex motion with the effective Magnus force ∝ (n + C): F M = 2π h(n ¯ + C)[v L × zˆ ].

(13.68)

So the constant C does matter for the value of the Magnus force. It was argued that the contribution ∝ C is of topological origin and can be found from the topological analysis (Volovik, 2003b). We think that there is no general principle which dictates the charge in the Wess–Zumino term. It is not the undefined Wess–Zumino term that determines what and whether the transverse force is, but vice versa; one must derive the transverse force from dynamical equations and only after this does one know what the Wess–Zumino term should be in the vortex Lagrangian. In the Galilean and translational invariant fluid one obtains from the momentum conservation law that the amplitude of the Magnus force is proportional to the total density. Then only the latter enters the Wess–Zumino term and C = 0. On the other hand, if the Magnus force vanishes, then the Wess–Zumino term vanishes also (C = −n). This is an additional illustration of the view of the relation between topology and the transverse force on the vortex presented in Sections 8.6 and 9.7 when discussing the Iordanskii and the Kopnin–Kravtsov forces: topology is not sufficient for determination of the transverse force on the vortex.

14 Elements of a theory of quantum turbulence

14.1 A tour to classical turbulence: scaling arguments, cascade and Kolmogorov spectrum The theory of classical turbulence starts from the Navier–Stokes equation (1.87) for an incompressible fluid. But it is a long way from this starting point to a full picture of turbulent flows. The linearised Navier–Stokes equation can be solved more or less straightforwardly for various geometries of laminar flows. If the velocity of the flow grows, the non-linear inertial term (v · ∇)v in the Navier–Stokes equation becomes relevant and eventually leads to instability of the laminar flow. The instability threshold is controlled by the Reynolds number (1.88). After the instability threshold is reached, the flow becomes strongly inhomogeneous in space and time in a chaotic manner. This is despite the fully deterministic character of the Navier–Stokes equation. The emergence of chaos from the deterministic description is a fundamental problem in physics and mathematics, but we mostly skip this transient process of turbulence evolution except for a few comments in Section 14.8. We are interested in a discussion of developed turbulence, which arises at rather high Reynolds numbers of the order of a few thousands or more. Usually it is assumed that in the state of developed turbulence the fluid is infinite and homogeneous in space and time, but only on average. At scales smaller than the size L of the fluid there are intensive temporal and spatial fluctuations of the velocity field. The natural description of such a chaotic field is in terms of probability distributions and random correlation functions. The velocity v averaged over a scale comparable with the fluid size L is not so important since it can be removed by the Galilean transformation. More important are fluctuations of velocity v  = v − v . Further we omit the ‘prime’, assuming that v = 0. The amplitude of velocity fluctuations depends on the scale at which it is considered. Following Landau and Lifshitz (1987, Chapter III) let us introduce the scale-dependent Reynolds number Rel = lv(l)/ν for the velocity fluctuation v(l) at the scale l. The scale-dependent Reynolds number characterises the effectiveness of the viscosity, which becomes important at small scales with Rel ∼ 1. Since we consider the case when the global Reynolds number Re = Lv(L)/ν is very large, at all scales between the global scale L and the ‘viscous’ scale ld , at which Rel ∼ 1, one may neglect viscosity. Bearing this in mind, the following qualitative picture of developed turbulence emerges. The turbulent flow must be supported

343

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Elements of a theory of quantum turbulence

by an input of energy, which is pumped at the global scale L by forces supporting the flow. This energy input must be transferred to small scales ∼ ld of the turbulence structure, where the energy can be absorbed by viscosity dissipation. So the energy must flow through the long interval of scales between the large L and much smaller ld practically without dissipation. This interval is called the inertial range and the process itself is called a ‘Richardson cascade’ (Richardson, 1922; Frisch, 1995). A crucial assumption of the further qualitative scaling analysis is locality: the energy flux through the scales l in the inertial range is fully determined by parameters for the same scales. In other words, this is really a cascade when the energy is transferred only to adjacent scales without jumps over a large interval of scales. The intensity of the energy flux does not depend on the viscosity parameter, even though the cascade scenario is definitely impossible without eventual dissipation via viscosity at the smallest scales (Landau and Lifshitz, 1987; Frisch, 1995). This is true for very large Reynolds numbers Re when the inertial range is long enough. Then the bottleneck for the dissipation process is not smallscale dissipation itself but the transfer of the energy from large to small scales through the very long inertial range (in analogy with the electric resistance of a long sample, which is determined mostly by bulk resistivity and not by the resistance of contacts). Although the viscosity ν does not affect the energy flux in the inertial range, it does affect the condition for the existence of the long inertial range: a high Reynolds number requires low values of the viscosity. After these assumptions have been made, scaling arguments are fully sufficient for derivation of the distribution of energy in the inertial range. Instead of the interval of scales one can introduce the interval in the space of wave numbers K, which are absolute values of three-dimensional wave vectors K and inversely proportional to relevant scales. Because of locality, the energy flux gε (K) at some point in the K space is determined only by the value of K and of the velocity fluctuation v(K) (which corresponds to the previous vl for the spatial scale l ∼ 1/K) at this point. The only expression for the energy flux which has correct dimensionality is −

dE = gε ∼ ρv(K)3 K. dt

(14.1)

 1/ l The total energy stored in the inertial range is determined by the integral E = 1/Ld e(K)dK over the inertial range, where e(K) is the energy density in the one-dimensional K space. According to Eq. (14.1), the fluctuating kinetic energy ∼ ρv(K)2 relaxes with the characteristic time 1/v(K)K, which is the turnover time necessary for fluid particles with the velocity v(K) to travel a distance l ∼ 1/K. The kinetic energy ∼ ρv(K)2 includes contributions from all scales larger than 1/K (Landau and Lifshitz, 1987), i.e., the estimation of the energy density e(K) is e(K) ∼ ρ

v(K)2 ρ d[v(K)2 ] ∼ρ ∼ dK K K



gε ρK

2/3

= (ρgε2 )1/3 K −5/3 .

(14.2)

14.2 Vinen’s theory of quantum vortex tangle

345

This is the famous Kolmogorov–Obukhov law −5/3 (frequently referred to in the literature as the Kolmogorov law), which was confirmed by physical and numerical experiments performed in classical fluids (Frisch, 1995). The law is valid down to the scale ld = ν/v(ld ) at which Rel ∼ 1 and viscous dissipation sets in. According to Eq. (14.1), this scale is  3 1/4 ρν ld ∼ . (14.3) gε Below we shall see that the Kolmogorov law may also be relevant for quantum turbulence. In classical turbulence, vorticity is absent on average, but there are vorticity fluctuations with the mean-square vorticity ω˜ 2 . Vorticity fluctuations at scale l ∼ 1/K are also determined from dimensional arguments:  2/3 gε K 4/3 . (14.4) ω˜ 2 ∼ K 2 v(K)2 ∼ ρ The picture of fully homogeneous and isotropic turbulence, which is presented above, is certainly an oversimplification. In real life, patterns of turbulent flows are not strictly homogeneous and there are alternating regions in space and time with stronger and weaker eddies. This phenomenon is called intermittency and requires a more elaborate analysis (Frisch, 1995). But the model of homogeneous and isotropic turbulence still catches remarkably well most of the essential qualitative features. The scaling theory also provides a rough picture of the turbulence decay when the energy input supporting turbulence is stopped. The whole energy 1/ ld 

e(K)dK ∼ ρ 1/3 (gε L)2/3

E∼

(14.5)

1/L

is concentrated at large scales of the inertial range. The equation describing turbulence decay is dgε dE ∼ ρ 1/3 L2/3 gε−1/3 = −gε . dt dt The solution of this equation is

(14.6)

gε ∼

ρL2 . t3

(14.7)

E∼

ρL2 . t2

(14.8)

Thus the energy decays as 1/t 2 :

14.2 Vinen’s theory of quantum vortex tangle In contrast to classical fluids, superfluids are described by the two-fluid theory, which assumes that a superfluid consists of a viscous normal component and an inviscid superfluid component. One might expect that turbulence would arise in the normal component

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Elements of a theory of quantum turbulence

similarly to in the classical fluid. But what about the superfluid component? Vorticity, which is an obligatory element of a turbulent flow, can appear in the superfluid component only in the form of quantised vortex lines, and their motion through the normal component is impeded by a mutual friction force. Quantised vorticity and mutual friction are the main features, which distinguish turbulence in the superfluid component from classical turbulence. The possibility of quantum turbulence in the form of a chaotic tangle of vortex lines was suggested by Feynman (1955). Vinen performed the first experimental (Vinen, 1957a, 1957b) and quantitative theoretical (Vinen, 1957c) analysis of quantum turbulence. The object of Vinen’s investigation was the turbulence which arises in the heat current and leads to the non-linear dependence of the mutual friction force on the relative velocity between the superfluid and the normal components (the Gorter–Mellink force). Vinen’s theory exploited dimensional arguments, which were so successful in the theory of classical turbulence. In the following we address only developed homogeneous turbulence, neglecting wall effects which were also discussed by Vinen and his followers (Baggaley and Laizet, 2013; Marakov et al., 2015). The turbulent vortex tangle can be characterised by a length per unit volume L, which, in contrast to a usual length, has dimensionality cm−2 and not cm. Similarly to the hypothesis of self-preservation in the theory of classical turbulence, the length L is the only parameter which determines the structure of the vortex tangle, the distance between lines, and their curvature radius estimated by the scale l0 ∼ L−1/2 . The goal is to obtain an equation describing evolution of the vortex tangle, i.e., evolution of the parameter L. This equation presents a balance between processes increasing and decreasing L. The source of the vortex length growth is mutual friction. In the presence of mutual friction the velocity of the vortex line is given by Eq. (6.33) or (6.34). Only the component of the vortex velocity v L , which lies in the plane of the curved vortex line, is able to change the vortex line length. Thus the line stretching is determined by the dissipative mutual friction parameter α. A simple example of line stretching is growth of the vortex ring of radius R lying in the plane normal to the counterflow velocity v n − v s : dR ∼ α|v n − v s |. (14.9) dt The radius R grows if the counterflow velocity exceeds the self-induced velocity κ/R which tries to shrink the ring. Then the local superfluid velocity v sl does not differ from the superfluid velocity v s induced by other vortices. Replacing R with the typical curvature radius L−1/2 of the vortex tangle, one obtains that the growth of L in time is determined by  dL  ∼ αV L3/2 |v n − v s |, (14.10) dt  gr

where the parameter αV may differ from the mutual friction parameter α by a geometric factor of the order unity. Decay of turbulence (shortening of L) is determined by mutual annihilation of vortex lines. As in the Kolmogorov theory of classical turbulence, the decay rate is estimated from purely dimensional arguments based on the only possible combination of relevant variables,

14.2 Vinen’s theory of quantum vortex tangle

347

namely, velocity v and the length scale l ∼ 1/K. Translating this to quantum turbulence, a length scale is l0 ∼ L−1/2 and a suitable velocity scale in the vortex tangle is the velocity κ/ l0 at distance l0 from a vortex line. Eventually this does not leave any choice other than the decay rate  dL  ∼ −χ κL2 , (14.11) dt d where χ is a dimensionless factor also determined by mutual friction. This dimensional analysis is also supported by the same example of the vortex ring. But now one should consider shrinking of the ring by the self-induced velocity in the absence of the counterflow. This yields [instead of Eq. (14.9)] dR/dt = −ακ/R. Eventually the following equation (Vinen’s equation) describes evolution of the vortex tangle:   dL  dL  dL = + ∼ αL3/2 |v n − v s | − χ κL2 . (14.12) dt dt gr dt d In the stationary state (dL/dt = 0) this yields the following dependence of L on the counterflow velocity v n − v s : L=

α2 |v n − v s |2 . χ 2κ 2

(14.13)

Since the mutual friction force per unit volume, either in the heat flow or in the second sound propagation, is proportional to the product of the length L and the counterflow velocity v n − v s , one immediately comes to the conclusion that in the turbulent regime the mutual friction force per unit volume is proportional to |v n − v s |3 . This is exactly a cube law for the Gorter–Mellink force, which was revealed in experiments on the heat flow and the second sound propagation. In contrast to the decay rate, Eq. (14.11), the growth rate given by Eq. (14.10) cannot be obtained only from dimensional arguments. One√cannot rule out an additional factor depending on a dimensionless parameter |v n − v s |/κ L. Assuming this dependence to be linear one obtains  dL  ∼ α  L|v n − v s |2 . (14.14) dt  gr

The argument in favour of this dependence is purely ‘aesthetic’: in contrast to Eq. (14.10), this expression contains analytic functions of L and v n −v s . Moreover, this growth rate also provides a correct cubic dependence of the Gorter–Mellink force on v n − v s . On the other hand, the agreement of Eq. (14.10) with the clear-cut case of a propagating ring cannot be discarded. One can find further discussion of various versions of the growth rate and comparisons with experiment by Nemirovskii and Fiszdon (1995), Nemirovskii (2013) and Khomenko et al. (2015). Vinen’s equation (14.12) cannot be used for the initial stage of the formation of the vortex tangle, when L is small and a complicated process of vortex nucleation is going on. But the equation is suitable for decaying turbulence, when the counterflow is stopped.

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Elements of a theory of quantum turbulence

In this case the growth term (the first term on the right-hand side of the equation) is absent and the time-dependent solution of the equation is L∝

1 . χ κ(t − t0 )

(14.15)

In a uniform ordered vortex lattice the vortex line length L is equal to the two-dimensional vortex density nv , and the time dependence (14.15) reproduces the time dependence (6.122) describing the decay of uniform ordered vorticity. The decay law (14.15) was confirmed by numerical and real experiments, but only at the first stage of tangle decay (Schwarz and Rozen, 1991). At long times the decay rate becomes much slower. The problem of slow decay requires a more complicated treatment (Nemirovskii and Fiszdon, 1995; Nemirovskii, 2013). At some stage of long time decay experiments revealed the time dependence of the classic turbulence decay (see the end of Section 14.3).

14.3 Classical versus quantum turbulence The original theory of Vinen addressed turbulence produced by the counterflow of the superfluid and the normal components. However, its importance went far beyond this particular problem. This theory has developed into an advanced field of modern physics (Donnelly, 1991, Chapter 7), in which research work has especially intensified in recent years. Nowadays different experimental methods are used for generation of turbulence: oscillating objects (grids, wires, or forks), impulsive spin-down of containers to rest, injections of vortex rings and charges, propagating vortex fronts and others (Walmsley and Golov, 2008; Eltsov et al., 2009; Fisher and Pickett, 2009; Vinen and Skrbek, 2014). Direct visualisation of quantum turbulence became possible, either with the help of tracer micrometre particles (Fonda et al., 2014; Van Sciver and Barenghi, 2009), or using metastable helium molecules (Guo et al., 2010; Vinen, 2014; Marakov et al., 2015). The latter method was especially effective for investigation of turbulence generated by a thermal counterflow when both components (superfluid and normal) become turbulent. The counterflow accompanied by mutual friction is a pure two-fluid effect, which has no analogue in classical hydrodynamics. In the presence of mutual friction there is no inertial range of scales, where the energy flows without dissipation. In counterflow turbulence, pumping of the energy and the energy drain (dissipation) occur at the same scales of the order of l0 . However, mutual friction between superfluid and normal components has no effect in two limiting cases, namely, when mutual friction is either very strong or very weak. In the former limit the two components flow together with the same velocity (this is called co-flow), while in the latter limit processes in the superfluid component simply are not affected by what is going in the normal component. The no-mutual-friction case is realised at very low temperatures when the normal component (and mutual friction, as a result of it) disappears. In both limits one returns back from the two-fluid to the one-fluid case. Estimations have shown that mutual friction in superfluid 4 He is strong enough for practically complete coupling between the motions of superfluid and normal components at

14.3 Classical versus quantum turbulence

349

temperatures above 1 K (Vinen and Niemela, 2002). The situation in 3 He is totally different. Normal viscosity in superfluid 3 He is a few orders higher, and turbulence in the normal fluid is hardly possible. But turbulence in the superfluid component is possible at low temperatures, when the normal density is low enough. When the temperature is growing, turbulence can be affected by the mutual friction force, which is able eventually to suppress turbulence. The role of mutual friction in superfluid turbulence was investigated by Volovik (2003a), L’vov et al. (2004), Vinen (2005), and Bou´e et al. (2012). We focus on the T = 0 limit, which is now being intensively investigated both for 4 He and 3 He (Davis et al., 2001; Hosio et al., 2011; Walmsley and Golov, 2012b). The superfluid component is the only component, and neither normal viscosity nor mutual friction are present. Vinen’s quantitative theory connecting the length L with the counterflow and mutual friction cannot be used, although the vortex tangle concept remains intact. Oscillating objects (for example, grids) pump the energy into flows at scales much larger than the intervortex distance l0 ∼ L−1/2 in the vortex tangle. Vinen (2000) suggested that at large scales exceeding l0 , quantisation of circulation is not so important and superfluid turbulence must behave similarly to turbulence in a conventional classical fluid. In purely superfluid turbulence the classical cascade described by the Kolmogorov law is interrupted not by viscosity at the scale ld but by quantum effects at the scale l0 . In analogy with the Reynolds number Re in classical hydrodynamics [see Eq. (1.88)], one can introduce the superfluid Reynolds number (Finne et al., 2003; Volovik, 2003a) Lv(L) , (14.16) κ where the circulation quantum κ replaces the kinematic viscosity ν in the classical Reynolds number. Here v(L) is the velocity at the largest scale L where the energy is pumped for supporting turbulence. A long Kolmogorov inertial range between the scale L, where the energy is pumped into turbulence, and the smallest scale l0 of the range is possible only for large superfluid Reynolds numbers. The suggestion that the classical Kolmogorov theory can describe quantum turbulence leads to a conclusion that there are fluctuations of vorticity at large scales, i.e., fluctuations of the random vortex tangle result in fluctuating polarisation of large-scale vortex bundles (Vinen, 2000). In the Kolmogorov theory, scaling estimation of the mean-square vorticity is given by Eq. (14.4). On the other hand, a dimensional estimation of vorticity in the vortex tangle is κL (Stalp et al., 1999; Vinen, 2000). This value is the maximum possible vorticity, which requires full polarisation of vortex lines all directed in the same direction. The case is realised for the vortex array of the equilibrium rotating superfluid, when the vortex line length per unit volume is L = 2/κ, and the dimensional estimation yields the correct value κL = 2 for the vorticity at solid body rotation. In a random vortex tangle one can expect a mean-square vorticity of order κL at scales l ∼ l0 where1 Res =

1 Vinen (2000) warned against estimation of the squared vorticity by averaging it over the whole space occupied by the vortex

tangle. Such an average diverges at short scales and must be cut-off by the vortex core radius. We look for vorticity at scales of the order of l0 and fluctuations at shorter scales must be excluded. This eliminates divergence and brings us to the estimation of Eq. (14.17).

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Elements of a theory of quantum turbulence

ω(l ˜ 0 )2 ∼ κ 2 L 2 ∼

κ2 . l04

(14.17)

Indeed, only one or a few vortex lines can pierce any volume with linear size of the order of the intervortex distance l0 . This leads to strong polarisation in any volume, although the polarisation direction varies randomly from one volume to another. Comparing this with the dimensional estimation Eq. (14.4) of the mean-square vorticity in the classical inertial range L  l  l0 , one finds that it coincides with Eq. (14.17) at K ∼ 1/ l0 , if gε = −

dE ρκ 3 ∼ 4 ∼ ρκ 3 L2 . dt l0

(14.18)

This agrees with the dissipation rate per unit mass gε /ρ ∼ νeff κ 2 L2 determined by Vinen (2010), where νeff was defined as an effective kinematic viscosity. In the zero-temperature limit νeff is the circulation quantum, and Eq. (14.18) yields the only expression of correct dimensionality for the energy flux (dissipation rate), which one can construct from the circulation quantum and the length scale. This value of gε ensures that when the scale l approaches to l0 the mean-square velocity fluctuation becomes of the order of κ 2 / l02 ∼ κ 2 L, i.e., the square of the velocity induced by a vortex at the distance l0 . In Section 14.5 we shall see that Eq. (14.18) follows also from the condition of smooth continuity of the energy spectrum at the crossover between the classical and the quantum inertial ranges, which takes place at l ∼ l0 . According to Eqs. (14.4) and (14.17) the mean-square vorticity in the classical inertial range depends on the scale l as  4/3 l0 2 2 ω(l) ˜

∼ ω(l ˜ 0)

. (14.19) l Thus for most of the classical inertial range the partial polarisation of vortex bundles is rather weak due to the small ratio l0 / l. Although weak polarisation of vortex lines at large scales is inevitable, with respect to application of the Kolmogorov theory to quantum superfluid there have been intensive discussions and debates on how this polarisation can emerge. One can find a detailed discussion of the issue with relevant references in the reviews by Vinen and Niemela (2002) and Nemirovskii (2013). Equations (14.5) and (14.18) yield the relation connecting the total energy E of the turbulence with the length L: E ∼ ρκ 2 L2/3 L4/3 .

(14.20)

Together with Eq. (14.8) this shows that in decaying turbulence the length L decreases as (Stalp et al., 1999; Walmsley et al., 2007) L∝

1 . (κt)3/2

(14.21)

Vinen (2010), who arrived at the time dependence ∼ 1/t 3/2 with a slightly different course of reasoning, discussed experimental evidence for this law. Recent experimental observations of the decay law t −3/2 (Babuin et al., 2015; Zmeev et al., 2015) have shown that it

14.4 Kelvin wave cascade in quantum inertial range

351

can take place at late stages of the turbulence decay even if the turbulence is generated by counterflow. The experiments on superfluid turbulence revealed the classical Kolmogorov spectrum in some interval of scales (Smith et al., 1993; Maurer and Tabeling, 1998; Stalp et al., 1999; Salort et al., 2012). The Kolmogorov spectrum was also obtained in numerical simulations (Nore et al., 1997). 14.4 Kelvin wave cascade in the quantum inertial range A question arises of what happens with the energy spectrum and the energy flux at scales l shorter than the intervortex spacing l0 ∼ L−1/2 in the vortex tangle where the Kolmogorov approach becomes invalid. In the low temperature limit dissipation at short scales l ≤ l0 is still very weak and the scales belong to the inertial range, but now it is the quantum inertial range. It was suggested that the cascade in the quantum range is connected with Kelvin waves along the vortex lines forming the vortex tangle (Svistunov, 1995; Vinen, 2000; Kivotides et al., 2001). The intervortex spacing l0 also describes the typical radius of curvature in the vortex tangle. Therefore one may expect that at scales much smaller than l0 it is possible to neglect vortex line curvature and consider Kelvin waves along a straight vortex line. Such an approach was used by Kozik and Svistunov (2004) who assumed that the energy flows from lower to higher Kelvin wave numbers without dissipation (pure Kelvin wave cascade) up to very high wave numbers of the order of the inverse core radius, at which the energy ultimately dissipates via phonon radiation. The analysis described only an asymptotic behaviour at K much larger than 1/ l0 . The theory of Kozik and Svistunov (2004) was based on the Boltzmann equation for the distribution function of Kelvin modes (Svistunov et al., 2015). It is important that the simple local induction approximation is not sufficient for the non-linear theory of Kelvin waves (Adachi et al., 2010), since in the local induction approximation there is an exact non-linear solution for oscillating vortex lines (Hasimoto, 1972). This leads to an infinite set of integrals of motion, which impose severe restrictions eliminating stochastic Kelvin wave interactions involved in the Boltzmann equation. So the non-local Biot–Savart law must be used for derivation of terms presenting non-linear wave interactions. This does not rule out using the local induction approximation in the expression for the total energy of a distorted vortex line [see Eqs. (2.66) and (2.67)]: E=

ρκ 2 L  2 p |u(p)|2 , 8π p

(14.22)

where  ∼ ln(l0 /rc ) and u(p) are the amplitudes of the Fourier expansion u(z) = 9 ipz of the z-dependent in-plane displacement of the vortex line from its original p u(p)e position at the z axis. An actual derivation of the Boltzmann equation for interacting Kelvin waves is a straightforward but rather tiresome procedure, especially because only multi-wave processes are allowed by the conservation laws. Fortunately, for the scaling analysis in

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Elements of a theory of quantum turbulence

the spirit of Kolmogorov, it is not necessary to have explicit expressions for vertices determining Kelvin wave interactions. It is enough to know the dimensional structure of various terms. Suppose that the main interaction involves n Kelvin waves. Introducing the intensity of the Kelvin mode m(p) = l0 |u(p)|2 , the Boltzmann equation for the variation of m(p) in time is (Sonin, 2012d)  V (p, p1 , p2 , . . . , pn−1 )m(p1 )m(p2 ), . . . , m(pn−1 ). (14.23) m(p) ˙ = p1 ,p2 ,...,pn−1

The vertex V (p, p1 , p2 , . . . , pn−1 ) describes the interaction between n Kelvin modes and contains δ-functions δ(p +p1 +p2 +· · ·+pn−1 ) and δ(ω+ω1 +ω2 +· · ·+ωn−1 ) providing the conservation laws of energy and momentum. The assumption of locality requires that all relevant p are of the same order, and a proper dimensionality is provided (keeping in mind that the dimensionality of m is cm3 ) if κp3n−4 m(p)n−1 . (14.24)  The large logarithm  appears from scaling δ(ω + ω1 + ω2 + · · · + ωn−1 ) by 1/ω with the Kelvin wave frequency ω = κp2 /4π linear in . In the local induction approximation, the logarithmic factor  is a constant multiplier in the expression (14.22) for the energy, which can be removed by rescaling of time. This is evident from Eq. (14.24). Taking the time derivative of the energy and using Eq. (14.24), one obtains the energy flux in the space of p:  ρκ 2 L p  2 ρκ 3 L0 dE = −gε = m(p)n−1 p3n−1 . dp p m(p ˙ ) ∼ (14.25) dt 4π 4π m(p) ˙ ∼

In the inertial range, where there is no dissipation, the energy flux gε does not depend on p. This yields   1  1  2 n−1 n−1 g l 3n−1 3n−1 gε ε 0 p− n−1 = p− n−1 , (14.26) m(p) = 3 3 ρκ L ρκ and according to Eq. (14.22) the energy density in the p space is given by   1 gε l02 n−1 − n+1 ρκ 2 e(p) ∼  p n−1 . ρκ 3 8π l02

(14.27)

The conservation laws of energy and momentum forbid interactions with n less than 6. On the other hand, for weak turbulence the interaction of 6 Kelvin waves is stronger than the interaction of larger numbers n of waves. For n = 6, Eq. (14.27) yields the energy spectrum  1/5 gε l02 ρκ 2  p−7/5 , (14.28) e(p) ∼ ρκ 3 8π l02 predicted by Kozik and Svistunov (2004) for weak turbulence (weakly nonlinear Kelvin waves).

14.5 Crossover from Kolmogorov to Kelvin wave cascade

353

Our scaling analysis was based on the perturbation theory used in the theory of weak wave turbulence (Nazarenko, 2011). For strong turbulence it is invalid. But the perturbation theory can provide some hint of what happens when turbulence is strong. The terms with growing number n of Kelvin waves become important. A reasonable conjecture of the spectrum of strong turbulence in the p space is Eq. (14.27) in the limit n → ∞ (Sonin, 2012d): e(p) ∼

ρκ 2  −1 p . 8π l02

(14.29)

Remarkably the spectrum does not depend on the energy flux gε at all. This spectrum was derived by Vinen et al. (2003) using different but also scaling arguments. It is important to stress a difference between wave numbers p of Kelvin waves forming a one-dimensional space and wave numbers K, which are magnitudes of three-dimensional wave vectors K forming the three-dimensional Fourier space of the three-dimensional velocity field (Kivotides et al., 2001). Correspondingly, in the quantum inertial range the spectrum e(p) is not the same as the spectrum e(K). Araki et al. (2002) studied numerically the evolution of quantum turbulence within the vortex filament model on the basis of the Biot–Savart law. They looked for the energy spectrum e(K) and found that at K of the order of the inverse line spacing l0 , the Kolmogorov law e(K) ∼ K −5/3 transforms into the law e(K) ∼ K −1 . This was also confirmed by analytical estimations (Nemirovskii et al., 2002). The law e(K) ∼ K −1 follows from the Fourier transformation of the r −1 velocity field for an isolated vortex.2 Thus the estimation of the spectrum e(K) including dimensional factors yields e(K) ∼

ρκ 2 −1 K , l02

(14.30)

which differs from the spectrum e(p) by the absent logarithmic factor . The large factor  appears in the energy density e(p) just because it is a result of integration over components of the wave vector K normal to the vortex line. The integral diverges at large K with a cut-off at K ∼ 1/rc (ultraviolet catastrophe). Since the energy density e(K) does not suppose such an integration, this factor cannot appear in e(K).

14.5 Crossover from Kolmogorov to Kelvin wave cascade There have been discussions in the literature about how the crossover from √ the Kolmogorov to the Kelvin wave cascade occurs at scales of the order of l0 ∼ 1/ L. The scale l0 is connected with the energy flux gε by Eq. (14.18). The latter follows from matching of the velocity and the vorticity in the classical inertial range with those expected in the vortex

2 The Fourier transformation of the in-plane velocity field ∼ 1/r is v(k) ∼ 1/k . We look for the energy density not in the two-

dimensional space of wave vectors k but in the one-dimensional space of wave vector magnitudes k . The energy density in this space is ρv(k)2 k ∼ 1/k . The wave vector component p along the vortex line is absent, and k = K .

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Elements of a theory of quantum turbulence

tangle at the scale l0 separating the classical and the quantum inertial ranges. One can check also the continuity of the energy density (energy spectrum) e(K) at the crossover. Indeed, the energy spectra in the Kolmogorov and the Kelvin wave cascades given by Eqs. (14.2) and (14.30) match one another at K ∼ 1/ l0 if gε is given by Eq. (14.18). It is also necessary to check how close to the crossover one can use the theory of the Kelvin wave cascade. To address this issue, let us estimate the total elongation L of the tangled vortex line in the Kelvin wave cascade (Vinen, 2000). For small displacement gradients ∂u/∂z, 1 L = 2

 

∂u ∂z

2 (14.31)

dz.

After Fourier transformation this integral becomes 

∞



gε l04 L ∼ L p m(p)dp ∼ L κ 3ρ 1/ l0 2



1 2n−1

.

(14.32)

Here Eq. (14.26) was used. In the Kelvin wave cascade the elongation L must not exceed the original length L of the tangle. Otherwise displacements induced by Kelvin modes are larger than the intervortex distance l0 . This signals a leading role of reconnections (see Section 14.8), which invalidate the conditions for the pure Kelvin wave cascade. Equation (14.32) together with Eq. (14.18) shows that the condition L ∼ L is reached at the scale l0 independently of the number n of interacting waves. Thus the Kelvin wave cascade theory can be used up to the scale l0 , and one has a coherent picture of the crossover governed by the single scale l0 . Independently of the energy pumped into turbulence, close to the crossover, the turbulence is always strong and one should not expect the weak turbulence spectrum e(p) ∼ p−7/5 there. The weak turbulence spectrum is possible only asymptotically very far from the crossover. In the past more complicated scenarios of the crossover were suggested. L’vov et al. (2007, 2008) argued that the crossover is impeded by mismatch of the energy distributions on the two sides of the crossover. This must lead to a bottleneck for the energy flux. However, the bottleneck arose because L’vov et al. (2007, 2008) tried to match the Kolmogorov cascade energy density e(K) given by Eq. (14.2) with the energy density e(p) in the Kelvin wave cascade in the one-dimensional p space (Sonin, 2012d). At the crossover wave number p ∼ 1/ l0 , the spectrum e(p) is larger than e(K) by the logarithmic factor . But as was discussed above, e(p) and e(K) are different physical quantities determined in different spaces, and there is no ground for the requirement that they match one another. The counterpart of the Kolmogorov energy in the Kelvin wave cascade is the kinetic energy e(K) related with the velocity induced by the vortex tangle, which is also determined in the three-dimensional K space. As demonstrated above, there is no mismatch between these two counterparts, and the bottleneck problem is non-existent.

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Another picture of the crossover was suggested by Kozik and Svistunov (2008), who believed that the Kolmogorov cascade becomes invalid at a scale lc ∼ l0 1/2 essentially longer than l0 because of the large logarithmic factor . So according to Kozik and Svistunov there is an intermediate range of scales much longer than l0 , where neither the Kolmogorov nor the Kelvin wave cascade is valid. The additional scale lc in this scenario appeared due to the large logarithmic factor . However, as was discussed in the end of Section 14.4,  results from divergent integration over large components of wave vectors K normal to the vortex line in the calculation of e(p) [but not e(K)]. There is no evident reason why very large K ∼ 1/rc can affect the crossover at much smaller K ∼ 1/ l0 . This would be in conflict with the spirit of the locality assumption, which excludes the effect of wave number K distant from those values at which the energy flux is investigated [see Sonin (2012d) for further details]. 14.6 Symmetry of Kelvin wave dynamics and Kelvin wave cascade For justification of the scaling analysis of the Kelvin wave cascade, one should check the condition of locality. This condition is satisfied if the multiple integrals over p, which arise in calculation of the energy flux, are convergent. In the scaling estimation of the energy flux the convergence was taken for granted. It is very tiresome to check convergence of the integral analytically. Instead Kozik and Svistunov (2004) did a numerical check, claiming that it gave a positive result. However, L’vov and Nazarenko (2010) challenged this conclusion, arguing that the relevant integrals diverge if one of p goes to zero, and the vertices in the Boltzmann equation essentially depend on this small p linearly. This means that the original idea of the cascade is invalid. L’vov and Nazarenko (2010) suggested a modified cascade scenario treating the long-wavelength mode as a static deformation of the originally straight vortex line. Static deformation breaks rotational invariance, lifting the restriction imposed by the momentum conservation law. This allows four-wave interaction, and for n = 4 Eq. (14.27) yields the L’vov–Nazarenko spectrum e(p) ∼ p−5/3 instead of the Kozik–Svistunov spectrum e(p) ∼ p−7/5 . The difference between the two cascade power-law exponents was not so large (powerlaw exponent −1.4 versus −1.67), especially keeping in mind difficulties in the accurate extraction of the exponent from numerical and physical experiments. But this is a case where the source of disagreement is much more important than the disagreement itself. The dispute is about the fundamentals of non-linear vortex dynamics: what is proper symmetry and how it manifests. Moreover, the absence of locality must lead to dramatic consequences for non-linear Kelvin wave dynamics. Kozik and Svistunov (2010a) argued that symmetry with respect to rotations of the vortex line around the axis normal to the line (tilt symmetry) rules out linear dependence of any physically meaningful quantity on very small p, which determines the tilt of a vortex line. So the correct asymptotic at small p must be not p but p2 , which corresponds to dependence on the vortex line curvature (instead of dependence on a tilt angle). Kozik and Svistunov (2010a) pointed out that the theory of L’vov and Nazarenko is

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in conflict with tilt symmetry because in the analysis of interaction of Kelvin waves they (Laurie et al., 2010) replaced the integral Biot–Savart law by an approximate non-linear differential equation violating tilt symmetry. This controversy led to lively discussion (Kozik and Svistunov, 2010b; Lebedev and L’vov, 2010; Lebedev et al., 2010; L’vov and Nazarenko, 2012; Sonin, 2012c). L’vov and Nazarenko (2012) accepted that their approximate Hamiltonian was not tilt invariant. But they argued that ‘a truncated Hamiltonian cannot and must not be tilt invariant’. Indeed, if one uses a weak-perturbation expansion of the original tilt invariant Hamiltonian in a series, any truncation of the series strictly speaking leads to violation of the tilt symmetry. This is because the expansion parameter is a displacement from a straight line, and one must choose a straight line (the axis z) with respect to which displacements are measured. The appearance of some special axis formally leads to broken tilt symmetry. Accepting the argument of L’vov and Nazarene per se one may conclude that the theory is not able to respect rotational symmetry of our space in principle. Putting aside the debatable logic of the conclusion that if the theory cannot it must not, the theory in fact is not so impotent. If one makes the truncation properly, tilt symmetry is broken only in terms beyond the accuracy of the truncation, which should be ignored anyway (Sonin, 2012d). Namely, in the theory of the Kelvin wave cascade, which requires terms up to the sixth order, a proper truncation violates tilt symmetry only in terms higher than the sixth order. This is enough to say that the Hamiltonian properly respects tilt symmetry. Meanwhile, the approximate Hamiltonian used by L’vov et al. violated this symmetry in essential terms of less than sixth order [see Kozik and Svistunov (2010a); Sonin (2012d) for more details]. Definitely the mechanism of L’vov et al. is not forbidden by symmetry of their Hamiltonian, but this is detached from the original problem of Kelvin wave dynamics in the real isotropic space. Pursuing the idea that a tilt of a vortex line in our fully isotropic space may affect interaction of Kelvin modes, Lebedev and L’vov (2010) pointed out a non-linear frequency shift of a Kelvin mode with a wave number p by another Kelvin mode with a wave number p  . The effect of the second mode reduces to a pure tilt of the vortex line if p  p and is equivalent to rotation of the coordinate frame around some axis normal to the unperturbed vortex line (the axis z). The rotation by the angle φ around the axis y is shown in Fig. 14.1. This is a transformation from one resting coordinate frame to another resting frame, and time is not touched by the transformation. Naturally the transformation cannot lead to any frequency shift of the Kelvin wave. But the period of the Kelvin mode along the new axis will be rescaled due to transformation. This is evident from the geometry in Fig. 14.1 [see more details in Sonin (2012d)]. Rescaling of the Kelvin wavelength does not produce any frequency shift and does not affect scaling arguments used at the derivation of the energy spectrum of the Kelvin wave cascade. It is much more difficult to reveal the Kelvin wave spectrum e(p) experimentally in the one-dimensional space of the tangled vortex line than it is to reveal the spectrum e(K) related with velocity fluctuations in the whole three-dimensional space, and there are no reports on progress in this problem. But a number of numerical simulations were

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Figure 14.1 Kelvin wave in the coordinate frame with the z axis along an unperturbed straight vortex line and in the rotated coordinate frame with the z˜ axis tilted to the original z axis by the angle φ. The spatial periods 2π/p and 2π/p˜ are different in the two frames. Figure from Sonin (2012d).

undertaken attempting to determine the exponent in the power-law Kelvin wave spectrum and to resolve the dispute between two views on the Kelvin wave cascade. Simulations were done for Kelvin wave turbulence excited on a single vortex line (see Baggaley and Laurie, 2014; H¨anninen and Baggaley, 2014, for references). Realisation and interpretation of these simulations is a serious challenge for the numerical community, as was stressed by H¨anninen and Hietala (2013). The results may depend on power pumped for support of turbulence, the interval where a power law was observed was not long, and the difference between the two competing exponents was not large. Anyway, Krstulovic (2012) and Baggaley and Laurie (2014) reported numerical simulations of the Kelvin wave cascade claiming a proof of the cascade exponent predicted by L’vov and Nazarenko. Krstulovic (2012) solved the Gross–Pitaevskii equation numerically, while calculations of

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Baggaley and Laurie (2014) were based on the Biot–Savart law in an incompressible fluid. In general, it is not usual to check symmetry laws by numerical simulations. Rather, symmetry laws are used as a check of the credibility of simulations. If a numerical result contradicts a symmetry law one should look for a reason why symmetry in numerics was broken. In numerical experiments with a single vortex line stretched between two parallel solid surfaces, tilt symmetry is broken by these surfaces. One may argue that probably one cannot [and following the logic of Lebedev and L’vov (2010) must not] reach tilt-symmetric conditions in numerical simulation for a single vortex. As in the case of the Hamiltonian truncation, this view is too pessimistic. The influence of symmetry-breaking boundary conditions on the Kelvin wave cascade is expected to weaken when one moves far away in the p space from the interval of wave numbers p, where the energy is pumped for turbulence support. Krstulovic (2012) and Baggaley and Laurie (2014) observed the power law of L’vov and Nazarenko for wave numbers which were not more than about one order of magnitude larger than wave numbers in the pumping interval. So the recipe for observation of the Kelvin wave cascade respecting tilt symmetry of our space is to increase the inertial range of p in numerical simulations. Unfortunately, according to H¨anninen and Hietala (2013) this would not be easy to realise. It should also be mentioned that in a chaotic vortex tangle tilt symmetry broken in single-vortex simulations can be restored by disorder.

14.7 Short-wavelength cut-off of Kelvin wave cascade: sound emission In classical fluids the inertial range of the Kolmogorov cascade ends when the viscosity becomes essential. In zero-temperature superfluid turbulence where dissipation is absent, the Kelvin wave cascade stops when essential sound emission by Kelvin waves starts (Vinen, 2000, 2001). Sound emission by moving vortices was a subject of investigations in classical hydrodynamics (Kambe, 1986). This is a particular case of a more general problem of sound emission by bodies oscillating in a fluid (Landau and Lifshitz, 1987, Section 74). If the sound wavelength exceeds the body size, the problem is solved by the method of matched asymptotic expansions, which Vinen (2001) used for analysing sound radiation by a rectilinear vortex moving in a circle. In Section 8.5 it was shown that an oscillating vortex line produces the velocity field containing a singular term proportional to the displacement u of the vortex line. Now we want to add to the expression (8.90) for the phase around the vortex line a non-linear contribution quadratic in displacements u: θ (r) = =

2π π 2π (v L · r) − [u · v v (r)] + (u · ∇)[u · v v (r)] κ κ κ 2π (u · [ˆz × r]) (r · [ˆz × u])(u · r) + . (v L · r) − κ r2 r4

(14.33)

This expression yields the phase induced by the Kelvin wave at distances r much shorter than the Kelvin wavelength 2π/p (but longer than the displacement amplitude u). Bearing

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in mind the expressions (2.77) for the components of the displacement vector u(ux , uy ) in the Kelvin wave, the singular term in Eq. (14.33) which is linear in u is a real part of ia ipz−iϕ−iωt e , (14.34) r where a is the amplitude of the Kelvin wave and ϕ is an azimuthal angle of the position vector r. On the other hand, the phase in an outgoing dipole cylindrical wave, which corresponds to the partial wave with angular momentum −1, is θ (1) =

θ = AH−1 (kr)e−iϕ−iωt = A[J−1 (kr) + iN−1 (kr)]eipz−iϕ−iωt , (1)

(14.35)

where k is the wave number in the xy plane. Expanding Eq. (14.35) in kr and matching singular terms ∝ 1/r in Eqs. (14.34) and (14.35) one obtains the value of the constant A = π ka/2. The asymptotic expression for θ at kr 1 is  π k ipz+ikr−iϕ− iπ −iωt 4 e θ ≈a . (14.36) 2r Using the expression (1.30) for the energy flux Q and integrating it over the surface of a cylinder surrounding the vortex line, one obtains the total flux of energy brought away by the emitted sound wave in the direction normal to the vortex line:   1 ∂θ 2 ∇r θ r dϕ = ρκ 2 ωk 2 a 2 . (14.37) Qd = ρμ0 vr r dϕ = ρκ ∂t 8 Vinen (2001) obtained this expression for sound radiation by a rotating rectilinear vortex when p = 0 and k = ω/cs . For the Kelvin wave with finite p the wave number k must be determined from the condition that the Kelvin wave and the emitted sound wave have the same frequency and the same wave vector component p along the z axis: $ 2 (14.38) ω = νs p = cs p2 + k 2 . Since for sound emission k must be real, emission is possible only for a Kelvin wavelength 2π/p shorter than νs /cs , which differs from the vortex core radius rc ∼ κ/cs only by a logarithmic factor. Kozik and Svistunov (2009) considered the quadruple sound radiation by Kelvin waves using their canonical formalism developed for studying vortex-phonon interaction. Within the method of matched asymptotic expansions the quadruple radiation is connected with the term quadratic in u in Eq. (14.33): θ (2) =

ia 2 i(p1 +p2 )z−2iϕ−i(ω1 +ω2 )t e . r2

(14.39)

(1)

The outgoing quadruple cylindrical wave θ ∝ H−2 (kr)eipz−2iϕ−iωt of angular momentum −2, where p = p1 +p2 and ω = ω1 +ω2 , must match Eq. (14.39). Repeating the procedure of matching similar to that for dipole radiation one obtains the flux of energy brought away by the emitted quadruple sound wave: Qq =

1 ρκ 2 ωk 4 a 4 . 32

(14.40)

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In quantum mechanical language, quadruple radiation is a process in which two kelvons annihilate, emitting one phonon. The process is allowed by the energy and momentum conservation laws if $ (14.41) νs (p12 + p22 ) = cs (p1 + p2 )2 + k 2 . Deep in the inertial range of the Kelvin wave cascade the wave numbers p cs /νs , and the condition can be satisfied only for two kelvons moving in opposite direction with p1 ≈ −p2 . The analysis addressed a system which is translationally invariant along the unperturbed vortex line. This assumption is hardly valid in a vortex tangle, where the momentum along the vortex line is not conserved. Then wave numbers p of the Kelvin wave and of the emitted sound wave must not coincide, and the condition (14.37), which restricts the possibility of dipole sound radiation, is not relevant. Taking this into account Vinen (2001) made heuristic estimates for possible sound emission in a vortex tangle. They do not affect the qualitative conclusion that the short-scale cut-off of the Kelvin wave cascade is the core radius rc .

14.8 Beyond the scaling theory of developed homogeneous superfluid turbulence Despite its effectiveness and simplicity (or just because of its simplicity), the scaling theory of homogeneous developed turbulence is phenomenological and certainly could not be the last word in the field. Starting from Schwarz (1977), much effort was invested in developing a more microscopic approach and in derivation of Vinen’s equation from principles of vortex dynamics. The theory must address stochastic dynamics of extended linear objects. Schwarz (1978) suggested a model describing evolution of distribution functions for parameters of the vortex tangle and derived Vinen’s equation from this model. Nemirovskii (2006, 2008) suggested a kinetic theory based on the Boltzmann-type rate equation, which described evolution (merging and splitting) of vortex loops. The starting point of the analysis was the original idea of Feynman (1955) of the cascade-like process of consequent breaking down of vortex loops, degenerating them into thermal excitations. These theories had to take into account complicated random processes of stretching, shortening and reconnection of vortex lines. The process of reconnection3 occurring at crossing of vortex lines (Fig. 14.2) is the most difficult for theoretical analysis, since the event takes place at length scales of the order of the vortex core radius and less. Neither the local induction approximation, nor the Biot–Savart law is sufficient. The vortex reconnection was simulated numerically by Koplik and Levine (1993) using the Gross– Pitaevskii theory (non-linear Schr¨odinger equation). Nazarenko and West (2003) found an analytical solution at very small scales. Reconnections were detected experimentally by

3 The process of vortex reconnection in superconductors is usually called flux-line cutting (Clem et al., 2011).

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Figure 14.2 Reconnection of two vortex lines crossing one another. States before (left) and after (right) the reconnection are shown.

direct visualisation of tiny solid particles (tracers) trapped and moving together with vortex lines (Paoletti et al., 2010). Because of the complexity of the problem, theoretical researchers widely used numerical simulations. Schwarz (1985, 1988) used the local induction approximation simulating turbulence generated by the thermal counterflow. But because the non-linear dynamics of the vortex line in the local induction approximation has an exact solution, it is impossible to reach a random vortex tangle within this approximation. Therefore Schwarz used an artificial procedure introducing randomness into the numerical procedure, in which reconnections played a key role. This procedure enabled Schwarz to reach a self-sustained random vortex tangle. One of his simulations of a vortex tangle is shown in Fig. 14.3a. A more rigorous approach is to exploit the Biot–Savart law in numerical simulations, as was done by Adachi et al. (2010) (see an example of their simulation in Fig. 14.3b). A number of groups now perform numerical simulations of various kinds of superfluid turbulence using the Biot–Savart law for singular vortex lines or the Gross–Pitaevskii theory (see Nemirovskii, 2013; Barenghi et al., 2014; H¨anninen and Baggaley, 2014; White et al., 2014; Kondaurova et al., 2014, for references). In real experiments it is impossible to deal with a fully homogeneous turbulence pattern. One cannot rule out that inhomogeneity affects the turbulence pattern significantly. The theory incorporating possible inhomogeneity of the turbulent vortex tangle was developed by Nemirovskii and Fiszdon (1995). They developed a macroscopic approach similar to that in the HVBH theory but applied it to a random vortex tangle instead of to a regular vortex array. This approach generalises Vinen’s equation by including the vortex line diffusion term. The role of diffusion in the decay of an inhomogeneous vortex tangle was analysed by Tsubota et al. (2003) numerically. They concluded that diffusion is slow with a diffusion coefficient about Dv ≈ 0.1κ. However, Nemirovskii (2010) obtained numerically a much larger diffusion coefficient Dv ≈ 2.2κ and suggested that vortex tangle diffusion can

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Figure 14.3 Numerical simulation of a vortex tangle in a real channel. (a) Simulation by Schwarz (1988). (b) Simulation by Adachi et al. (2010)

explain recent results of experiments and numerical simulations on the turbulence decay without using the decay scenarios proposed for homogeneous turbulence (see previous sections). This issue requires further experimental and numerical investigations. Up to now we assumed that turbulence was not only homogeneous but also isotropic. There were fluctuations of polarisation (vorticity) at finite scales, but without total average polarisation of the vortex tangle. But turbulence has also been observed and analysed in rotating superfluids when the fluid (and turbulence pattern) is not isotropic (see Walmsley et al., 2014, for a review). Investigations of superfluid turbulence were not limited to its developed stage. Numerous experimental and theoretical studies addressed transient processes on the way to fully developed turbulence starting from the laminar flow. The question of when the transition occurs from the laminar to the turbulent regime has no satisfactory answer in the theory. Although in classical hydrodynamics it is well established that the transition is governed by the Reynolds number, it is difficult to provide a quantitative explanation why the critical Reynolds number can reach a few thousands or a few tens of thousands. In the case of superfluid turbulence the problem is even more formidable. It starts from the questions how vortices first nucleate in the laminar flow (intrinsic nucleation), or how remnant vortices, which could remain from the prehistory of the laminar flow (spin-down from the state of rotation, for example), develop into a vortex tangle (extrinsic nucleation). This is a problem of critical velocities, which was mentioned in the very beginning of Chapter 11. Among the suspects for triggering the transition to the turbulent regime was the Glaberson–Johnson–Ostermeier instability, which was discussed in Section 3.10. The threshold for the Glaberson–Johnson–Ostermeier instability was evaluated for a twisted

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vortex bundle behind the propagating vortex front (Sonin, 2012b). Twisting of the vortex bundle leads to axial mass currents along vortex lines, which are a source of instability. To evaluate the threshold one should replace the uniform vorticity 2 in the expression (3.124) ˜ of the space-dependent vorticity, i.e., for the critical velocity vcr by the absolute value ω(r) √ √ ˜ s . This approach is justified since ω(r) ˜ varies slowly at the scale νs /2 at vcr = ων which instability sets in. The Glaberson–Johnson–Ostermeier critical velocity vcr should be compared with the superfluid velocity vl = (vφ − r)sφ + vz sz along the vortex lines in the rotating coordinate frame. For a large number of vortices when R 2 νs , one can use the calculations for continuous vorticity (Section 5.6). Then the twisted vortex bundle is stable only for rather weak twist satisfying the inequality Q2 R 2