Dynamics of Quantised Vortices in Superfluids

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