Self-Exciting Fluid Dynamos

Contents......Page 7
Preface......Page 17
PART I BASIC THEORY AND OBSERVATIONS......Page 20
1.1 What is dynamo theory?......Page 22
1.2.1 The geodynamo......Page 23
1.2.2 The solar dynamo......Page 27
1.3 The homopolar disc dynamo......Page 29
1.4 Axisymmetric and non-axisymmetric systems......Page 31
2.1.1 Solenoidality......Page 39
2.1.3 Lines of force (‘B-lines’)......Page 40
2.1.4 Helicity and flux tube linkage......Page 41
2.2 Chirality......Page 44
2.2.1 The rattleback: a prototype of dynamic chirality......Page 45
2.2.2 Mean response provoked by chiral excitation......Page 47
2.3.1 Spherical polar coordinates......Page 49
2.3.2 Toroidal/poloidal decomposition......Page 51
2.3.4 Two-dimensional fields......Page 53
2.4.1 Amp` ere’s law......Page 55
2.4.2 Multipole expansion of the magnetic field......Page 56
2.4.3 Axisymmetric fields......Page 57
2.5 Force-free fields......Page 58
2.5.1 Force-free fields in spherical geometry......Page 60
2.6 Lagrangian variables and magnetic field evolution......Page 62
2.6.1 Change of flux through a moving circuit......Page 63
2.6.3 Galilean invariance of the pre-Maxwell equations......Page 64
2.6.4 Ohm’s law in a moving conductor......Page 65
2.7 Kinematically possible velocity fields......Page 66
2.8 Free decay modes......Page 67
2.8.1 Toroidal decay modes......Page 68
2.8.2 Poloidal decay modes......Page 69
2.8.3 Behaviour of the dipole moment......Page 70
2.9 Fields exhibiting Lagrangian chaos......Page 72
2.10.1 Twist surgery......Page 73
2.10.2 Helicity of a knotted flux tube......Page 75
3.1 Alfven’s theorem and related results......Page 78
3.1.1 Conservation of magnetic helicity......Page 79
3.2 The analogy with vorticity......Page 81
3.4 Maintenance of a flux rope by uniform irrotational strain......Page 83
3.5 A stretched flux tube with helicity......Page 85
3.6 An example of accelerated ohmic diffusion......Page 86
3.7 Equation for vector potential and flux-function under particular symmetries......Page 87
3.7.2 Axisymmetric case......Page 88
3.8 Shearing of a space-periodic magnetic field......Page 89
3.9 Oscillating shear flow......Page 92
3.9.1 The case of steady rotation of the shearing direction......Page 94
3.10 Field distortion by differential rotation......Page 95
3.11 Effect of plane differential rotation on an initially uniform field: flux expulsion......Page 96
3.11.1 The initial phase......Page 97
3.11.2 The ultimate steady state......Page 98
3.11.3 Flow distortion by the flow due to a line vortex......Page 100
3.11.4 The intermediate phase......Page 101
3.11.6 Flux expulsion by Gaussian angular velocity distribution......Page 103
3.12 Flux expulsion for general flows with closed streamlines......Page 105
3.13 Expulsion of poloidal field by meridional circulation......Page 107
3.14 Generation of toroidal field by differential rotation......Page 108
3.14.2 The ultimate steady state......Page 109
3.15 Topological pumping of magnetic flux......Page 112
4.1 Planetary magnetic fields in general......Page 118
4.2 Satellite magnetic fields......Page 123
4.3 Spherical harmonic analysis of the Earth’s field......Page 125
4.4 Variation of the dipole field over long time-scales......Page 132
4.5 Parameters and physical state of the lower mantle and core......Page 135
4.6 The need for a dynamo theory for the Earth......Page 136
4.7 The core–mantle boundary and interactions......Page 137
4.8 Precession of the Earth’s angular velocity......Page 138
5.1 The solar magnetic field......Page 140
5.2.1 Surface observations......Page 141
5.2.2 Helioseismology......Page 143
5.3 Sunspots and the solar cycle......Page 145
5.4 The general poloidal magnetic field of the Sun......Page 150
5.5 Magnetic stars......Page 151
5.6 Magnetic interaction between stars and planets......Page 153
5.7 Galactic magnetic fields......Page 155
5.8 Neutron stars......Page 159
PART II FOUNDATIONS OF DYNAMO THEORY......Page 162
6.1 Formal statement of the kinematic dynamo problem......Page 164
6.2 Rate-of-strain criterion......Page 165
6.3 Rate of change of dipole moment......Page 167
6.4 The impossibility of axisymmetric dynamo action......Page 168
6.4.1 Ultimate decay of the toroidal field......Page 169
6.5 Cowling’s neutral point argument......Page 170
6.7 The impossibility of dynamo action with purely toroidal motion......Page 172
6.9 Rotor dynamos......Page 175
6.9.1 The 3-sphere dynamo......Page 177
6.9.2 The 2-sphere dynamo......Page 180
6.9.3 Numerical treatment of the Herzenberg configuration......Page 182
6.9.4 The rotor dynamo of Lowes and Wilkinson......Page 183
6.10 Dynamo action associated with a pair of ring vortices......Page 184
6.11 Dynamo action with purely meridional circulation......Page 188
6.12 The Ponomarenko dynamo......Page 190
6.14 The Bullard–Gellman formalism......Page 195
6.15 The stasis dynamo......Page 202
7.1 Turbulence and random waves......Page 204
7.2 The linear relation between E and B0......Page 207
7.3 The α-effect......Page 208
7.4 Effects associated with the coefficient βi jk......Page 212
7.5 First-order smoothing......Page 214
7.6 Spectrum tensor of a stationary random vector field......Page 215
7.7 Determination of αi j for a helical wave motion......Page 219
7.8 Determination of αi j for a random u-field under first-order smoothing......Page 221
7.9 Determination of βi jk under first-order smoothing......Page 224
7.10.1 Evaluation of αi j......Page 225
7.10.2 Evaluation of βi jk......Page 227
7.11 Effect of helicity fluctuations on effective turbulent diffusivity......Page 228
7.12 Renormalisation approach to the zero-diffusivity limit......Page 231
8.1 Introduction......Page 235
8.2 Lagrangian transformation of the induction equation when η = 0......Page 238
8.3 Effective variables in a Cartesian geometry......Page 240
8.4 Lagrangian transformation including weak diffusion effects......Page 241
8.5 Dynamo equations for nearly rectilinear flow......Page 242
8.6 Corresponding results for nearly axisymmetric flows......Page 244
8.7 A limitation of the pseudo-Lagrangian approach......Page 246
8.8 Matching conditions and the external field......Page 247
8.9 Related developments......Page 249
9.1 Dynamo models of α2- and αω-type......Page 250
9.1.1 Axisymmetric systems......Page 251
9.2 Free modes of the α2-dynamo......Page 252
9.2.2 Influence of higher-order contributions to E......Page 254
9.3 Free modes when αi j is anisotropic......Page 255
9.3.1 Space-periodic velocity fields......Page 256
9.3.2 The α2-dynamo in a spherical geometry......Page 257
9.3.3 The α2-dynamo with antisymmetric α......Page 260
9.4 Free modes of the αω-dynamo......Page 263
9.5 Concentrated generation and shear......Page 266
9.5.1 Symmetric U(z) and antisymmetric α(z)......Page 268
9.6 A model of the galactic dynamo......Page 270
9.6.1 Dipole modes......Page 273
9.6.2 Quadrupole modes......Page 274
9.6.3 Oscillatory dipole and quadrupole modes......Page 276
9.7 Generation of poloidal fields by the α-effect......Page 277
9.8 The αω-dynamo with periods of stasis......Page 279
9.9 Numerical investigations of the αω-dynamo......Page 280
9.10 More realistic modelling of the solar dynamo......Page 288
9.11 The Karlsruhe experiment as an α2-dynamo......Page 291
9.12 The VKS experiment as an αω-dynamo......Page 292
9.12.1 Field reversals in the VKS experiment......Page 294
9.13 Dynamo action associated with the Taylor–Green vortex......Page 295
10.1.1 Writhe and twist generated by the STF cycle......Page 298
10.1.2 Existence of a velocity field in R3 that generates the STF cycle......Page 300
10.1.3 Tube reconnection and helicity cascade......Page 301
10.2 Fast and slow dynamos......Page 302
10.3 Non-existence of smooth fast dynamos......Page 303
10.4 The homopolar disc dynamo revisited......Page 304
10.5 The Ponomarenko dynamo in the limit η → 0......Page 306
10.6 Fast dynamo with smooth space-periodic flows......Page 307
10.6.1 The symmetric case A = B = C = 1......Page 308
10.6.2 The Galloway–Proctor fast dynamo......Page 310
10.7 Large-scale or small-scale fast dynamo?......Page 312
10.8 Non-filamentary fast dynamo......Page 313
PART III DYNAMIC ASPECTS OF DYNAMO ACTION......Page 316
11.1 Dynamic characteristics of the segmented disc dynamo......Page 318
11.2 Disc dynamo driven by thermal convection......Page 321
11.2.1 The Welander loop......Page 322
11.2.2 Coupling of Welander loop and Bullard disc......Page 324
11.3 The Rikitake dynamo......Page 325
11.4 Symmetry-mode coupling......Page 327
11.5 Reversals induced by turbulent fluctuations......Page 329
11.5.1 Dipole-quadrupole model......Page 330
12.1 The momentum equation and some elementary consequences......Page 334
12.1.1 Alfven waves......Page 335
12.1.2 Alfven wave invariants and cross-helicity......Page 336
12.2 Lehnert waves......Page 338
12.2.1 Dispersion relation and up-down symmetry breaking......Page 339
12.2.2 Inertial and magnetostrophic wave limits......Page 341
12.3 Generation of a fossil field by decaying Lehnert waves......Page 342
12.4.1 A simple model based on weak forcing......Page 343
12.5 Magnetic equilibration due to α-quenching......Page 346
12.5.1 The case of steady forcing......Page 347
12.5.2 The case of unsteady forcing with ω2 ≫ ηνk4......Page 348
12.5.3 Cattaneo–Hughes saturation......Page 350
12.6 Quenching of the α-effect in a field of forced Lehnert waves......Page 352
12.7 Equilibration due to α-quenching in the Lehnert wave field......Page 355
12.7.1 Energies at resonance......Page 357
12.8 Forcing from the boundary......Page 358
12.9 Helicity generation due to interaction of buoyancy and Coriolis forces......Page 361
12.10 Excitation of magnetostrophic waves by unstable stratification......Page 362
12.11 Instability due to magnetic buoyancy......Page 367
12.11.1 The Gilman model......Page 369
12.12 Helicity generation due to flow over a bumpy surface......Page 372
13.1 Models for convection in the core of the Earth......Page 375
13.2 Onset of thermal convection in a rotating spherical shell......Page 376
13.2.1 The Roberts–Busse localised asymptotic theory for small E......Page 379
13.2.2 The Soward–Jones global theory for the onset of spherical convection......Page 380
13.2.3 Localised mode of instability in a spherical shell......Page 382
13.2.4 Dynamic equilibration......Page 384
13.3 Onset of dynamo action: bifurcation diagrams and numerical models......Page 385
13.3.1 Numerical models......Page 388
13.3.2 Model equations for super- and subcritical bifurcations......Page 390
13.3.3 Three regimes, WD, FM and SD; numerical detection......Page 391
13.3.4 The SD regime......Page 392
13.3.5 The WD / SD dichotomy......Page 394
13.4 The Childress–Soward convection-driven dynamo......Page 395
13.4.1 Mixed asymptotic and numerical models......Page 399
13.5 Busse’s model of the geodynamo......Page 400
13.6.1 Necessary condition for a steady solution U(x)......Page 403
13.6.2 Sufficiency of the Taylor constraint for the existence of a steady U(x)......Page 404
13.6.3 The arbitrary geostrophic flow (s)......Page 406
13.6.5 Torsional oscillations when the Taylor constraint is violated......Page 407
13.6.6 Effect of mantle conductivity......Page 408
13.7 Scaling laws......Page 409
14.1.1 Low-order models......Page 415
14.1.2 Mean-field models......Page 417
14.1.3 Direct numerical simulations......Page 419
14.2 From planets to stars......Page 421
14.3 Extracting dynamo mechanisms......Page 422
14.4 Dipole breakdown and bistability......Page 424
14.5 Kinematically unstable saturated dynamos......Page 425
14.6 The galactic dynamo......Page 427
14.7 Accretion discs and the magnetorotational instability (MRI)......Page 428
14.7.1 Rayleigh stability criterion......Page 429
14.7.2 Magnetorotational instability......Page 430
14.7.3 Shearing-box analysis......Page 431
14.7.4 Dynamo action associated with the magnetorota- tional instability......Page 433
14.7.5 Experimental realisation of the magnetorotational instability......Page 434
15.1 Effects of helicity on homogeneous turbulence......Page 436
15.1.1 Energy cascade in non-helical turbulence......Page 437
15.1.2 Intermittency......Page 438
15.1.3 Effect of helicity on energy cascade......Page 441
15.2 Influence of magnetic helicity conservation in energy transfer processes......Page 445
15.3 Modification of inertial range due to large-scale magnetic field......Page 451
15.4 Non-helical turbulent dynamo action......Page 452
15.5 Dynamo action incorporating mean flow effects......Page 454
15.6 Chiral and magnetostrophic turbulence......Page 457
16.1 Lower bound on magnetic energy......Page 460
16.3 Relaxation to a minimum energy state......Page 462
16.3.1 Alternative ‘Darcy’ relaxation procedure......Page 464
16.4 Two-dimensional relaxation......Page 465
16.5 The relaxation of knotted flux tubes......Page 467
16.6 Properties of relaxed state......Page 470
16.8 Structure of magnetostatic fields......Page 472
16.9 Stability of magnetostatic equilibria......Page 473
16.9.1 The two-dimensional situation......Page 475
16.10 Analogous Euler flows......Page 476
16.11 Cross-helicity and relaxation to steady MHD flows......Page 477
16.11.1 Structure of steady states......Page 478
16.11.2 The isomagnetovortical foliation......Page 479
16.11.3 Relaxation to steady MHD states......Page 480
17.1 Relaxation in a pressureless plasma......Page 482
17.2 Numerical relaxation......Page 484
17.3 The pinch effect......Page 486
17.4 Current collapse in an unbounded fluid......Page 487
17.4.1 Similarity solution when η = 0......Page 489
17.5 The Taylor conjecture......Page 490
17.6 Relaxation of a helical field......Page 494
17.7 Effect of plasma turbulence......Page 496
17.8 Erupting flux in the solar corona......Page 498
17.9 Conclusion......Page 500
Appendix Orthogonal Curvilinear Coordinates......Page 501
References......Page 504
Author index......Page 530
Subject index......Page 534