Stretch, twist, fold : the fast dynamo

Cover ......Page 1
Lecture Notes in Physics: Monographs 37 ......Page 2
Stretch, Twist, Fold: The Fast Dynamo ......Page 4
Preface ......Page 6
Contents ......Page 10
1. The Fast Dynamo Problem ......Page 14
1.1 Magnetic Fields in Astrophysics ......Page 15
1.2 Kinematics of the Magnetic Field ......Page 18
1.3 Slow and Fast Dynamos ......Page 32
1.4 Stretch-Twist-Fold: the STF Picture ......Page 37
1.5 The Nature of the Perfectly Conducting Limit ......Page 40
1.6 Measures of Stretching ......Page 43
2.1 Computing a Fast Dynamo ......Page 44
2.2 Numerical Evidence for a Fast Dynamo in Otani's MW+ Flow ......Page 46
2.3 Eigenfunctions and the Stretch-Fold-Shear Mechanism ......Page 48
2.4 Magnetic Field Evolution with e = 0 ......Page 55
2.5 Discussion of the Flux Conjecture ......Page 60
2.6 Steady Three-Dimensional Flows ......Page 64
3. Fast Dynamos in Maps ......Page 74
3.1 Pulsed Beltrami Waves ......Page 75
3.2 The SFS Map ......Page 82
3.3 Generalized Baker's Maps ......Page 89
3.4 The Slide or SFS1 Map ......Page 94
3.5 Cat Maps ......Page 97
4.1 Measures of Dynamo Action ......Page 102
4.2 Anti-dynamo Theorems ......Page 112
4.3 Non-existence of Smooth Eigenfunctions ......Page 119
5. Magnetic Structure in Steady Integrable Flows ......Page 126
5.1 The Roberts Cell ......Page 129
5.2 Basic Mechanisms ......Page 131
5.3 Kinematic Prandtl-Batchelor Theory ......Page 137
5.4 Helical Dynamos ......Page 142
5.5 Analysis of the Roberts Cell and Extensions ......Page 146
5.6 Axisymmetric Boundary Layers ......Page 159
6.1 Field Growth and Topological Entropy ......Page 162
6.2 Oseledets' Conjecture ......Page 166
6.3 Fredholm Determinants, Periodic Orbits and Zeta Functions ......Page 168
6.4 Dynamo Action in the Cat Flow ......Page 178
6.5 Anosov Flows ......Page 181
6.6 An Approximate Green's Function ......Page 185
6.7 Noisy Trajectories and Diffusion ......Page 190
6.8 Shadowing and the Flux Conjecture ......Page 194
7.1 Intermittency ......Page 198
7.2 The Cancellation Exponent ......Page 203
7.3 Steady Flows ......Page 210
8. Nearly Integrable Flows ......Page 226
8.1 The Onset of Chaos ......Page 227
8.2 Dynamo Action in the Chaotic Web ......Page 234
8.3 A Model for Perfect Dynamo Action ......Page 244
8.4 Analysis ......Page 250
9. Spectra and Eigenfunctions ......Page 258
9.1 Preliminaries ......Page 259
9.2 The Spectrum of L_epsilon ......Page 263
9.3 Adjoints and Normality ......Page 269
9.4 A Model Problem ......Page 273
9.5 Baker's Maps Revisited ......Page 275
9.6 The SFS1 and SFS Maps ......Page 286
9.7 Pulsed Waves ......Page 297
10. Strongly Chaotic Systems ......Page 302
10.1 Diffeomorphisms of T^2 ......Page 303
10.2 Pulsed Beltrami Waves ......Page 308
10.3 SFS in the Limit of Strong Chaos ......Page 316
10.4 Cat Maps and Pseudo-Anosov Maps with Shear ......Page 319
11. Random Fast Dynamos ......Page 332
11.1 Random Renewing Flows ......Page 333
11.2 Diffusionless Evolution of Field ......Page 335
11.3 Stretching in Random Flows ......Page 339
11.4 Diffusion and Dynamo Action ......Page 343
11.5 Numerical Simulations ......Page 350
12. Dynamics ......Page 356
12.1 Formulation ......Page 358
12.2 The Post-kinematic Approach ......Page 363
12.3 Numerical Studies ......Page 373
12.4 MHD Turbulence and Dynamo Saturation ......Page 384
12.5 Astrophysical Implications ......Page 387
12.6 The Prospects for Modelling ......Page 392
References ......Page 394
Index ......Page 410