Mathematical Aspects of Natural Dynamos (The Fluid Mechanics of Astrophysics and Geophysics)

Although the origin of Earth's and other celestial bodies' magnetic fields remains unknown, we do know that the motion of electrically conducting fluids generates and maintains these fields, forming the basis of magnetohydrodynamics (MHD) and, to a larger extent, dynamo theory. Answering the need for a comprehensive, interdisciplinary introduction to this area, Mathematical Aspects of Natural Dynamos provides a foundation in dynamo theory before moving on to modeling aspects of natural dynamos. Bringing together eminent international contributors, the book first introduces governing equations, outlines the kinematic dynamo theory, covers nonlinear effects, including amplitude saturation and polarity reversals, and discusses fluid dynamics. After establishing this base, the book describes the Earth's magnetic field and the current understanding of its characteristics. Subsequent chapters examine other planets in our solar system and the magnetic field of stars, including the sun. The book also addresses dynamo action on the large scale of galaxies, presents modeling experiments of natural dynamos, and speculates about future research directions. After reading this well-illustrated, thorough, and unified exploration, you will be well prepared to embark on your own journey through this fascinating area of research. Mathematical Aspects of Natural Dynamos......Page 1 CONTENTS......Page 6 CONTRIBUTORS......Page 14 LIST OF TABLES......Page 16 LIST OF NOTATIONS......Page 17 PREFACE......Page 19 ERRATUM......Page 20 PART I: FOUNDATIONS OF DYNAMO THEORY......Page 21 CHAPTER 1: INTRODUCTION TO SELF-EXCITED DYNAMO ACTION......Page 22 THE INDUCTION EQUATION......Page 23 INFLUENCE ON MATTER......Page 26 1.1.2. THERMODYNAMIC EQUATIONS......Page 27 1.1.3. NAVIER-STOKES EQUATION......Page 28 THE ANELASTIC APPROXIMATION......Page 29 THE BOUSSINESQ APPROXIMATION......Page 33 1.1.4. BOUNDARY CONDITIONS......Page 36 1.2.1. DISC DYNAMO......Page 37 1.2.3. BASIC MECHANISMS OF DYNAMO ACTION......Page 39 1.2.4. FAST AND SLOW DYNAMOS......Page 41 1.3.2. NON-NORMALITY OF THE INDUCTION EQUATION......Page 43 1.3.3. FLOW VELOCITY BOUNDS......Page 44 1.3.4. GEOMETRICAL CONSTRAINTS......Page 45 1.4.1. TWO SIMPLE EXAMPLES......Page 50 1.4.2. PULSED FLOWS......Page 51 1.5.1. THE TWO–SCALE CONCEPT AND PARKER’S MODEL......Page 53 1.5.2. MEAN FIELD ELECTRODYNAMICS......Page 54 1.5.3. MEAN FIELD MODELS......Page 57 1.6. LARGE MAGNETIC REYNOLDS NUMBERS......Page 60 1.6.1. SLOW DYNAMOS IN FLOWS......Page 61 1.6.2. THE STRETCH–TWIST–FOLD PICTURE......Page 66 1.6.3. FAST DYNAMOS IN SMOOTH FLOWS......Page 68 1.6.4. FAST DYNAMOS IN MAPPINGS......Page 69 1.6.5. THE STRETCH–FOLD–SHEAR MODEL......Page 73 2.1. GENERAL CONSIDERATIONS......Page 77 2.2. SATURATION OF A DYNAMO GENERATED BY A PERIODIC FLOW......Page 80 2.2.1. SCALE SEPARATION......Page 81 2.2.2. THE G.O. ROBERTS DYNAMO......Page 82 2.2.3. SATURATION OF DYNAMOS DRIVEN BY THE alpha–EFFECT......Page 83 2.3.1. A PONOMARENKO TYPE DYNAMO AS A TRACTABLE PROBLEM WITHOUT SCALE SEPARATION......Page 85 2.3.2. STRUCTURE OF THE PERTURBATION ANALYSIS......Page 86 2.3.3. THE LAMINAR SCALING......Page 87 2.4.1. DIMENSIONAL ARGUMENTS......Page 88 2.4.2. HIGH Re DYNAMOS CLOSE TO THE BIFURCATION THRESHOLD......Page 89 2.5.1. WEAK AND STRONG FIELD REGIMES OF THE GEODYNAMO......Page 90 2.5.2. FURTHER COMMENTS ON WEAK AND STRONG FIELD REGIMES......Page 91 2.5.3. SCALINGS OF MAGNETIC ENERGY USING DIMENSIONAL CONSIDERATIONS......Page 92 2.6. SCALING LAWS IN THE LIMIT OF LARGE Rm AND Re......Page 93 2.6.1. EFFECT OF TURBULENCE ON THE DYNAMO THRESHOLD......Page 94 2.6.2. BATCHELOR’S PREDICTIONS FOR TURBULENT DYNAMO THRESHOLD AND SATURATION......Page 95 2.6.3. A KOLMOGOROV TYPE SCALING IN THE LIMIT Re.........Page 96 2.7. NONLINEAR EFFECTS IN MEAN FIELD DYNAMO THEORY......Page 97 THE INCORPORATION OF PLAUSIBLE NONLINEARITIES......Page 99 NONLINEAR MODELS......Page 101 THE ROBUSTNESS OF NONLINEAR MODELS......Page 105 2.7.2. MHD TURBULENCE THEORIES......Page 106 2.7.3. DIRECT NUMERICAL SIMULATIONS......Page 110 2.8. PHYSICALLY-REALISTIC FARADAY-DISC SELF-EXCITED DYNAMOS......Page 115 2.8.1. HISTORICAL SURVEY......Page 116 2.8.2. CHARACTERISTICS OF SELF-EXCITED DYNAMOS......Page 118 2.8.3. GOVERNING EQUATIONS IN DIMENSIONAL FORM......Page 119 2.8.4. ENERGETICS AND EQUILIBRIUM SOLUTIONS......Page 121 2.8.5. DIMENSIONLESS EQUATIONS......Page 123 2.8.6. GENERIC SOLUTIONS......Page 125 VARIATIONS ON A THEME......Page 126 OTHER EXTENSIONS......Page 128 2.8.8. SOME NUMERICAL INTEGRATIONS......Page 129 CHAPTER 3: DYNAMICS OF ROTATING FLUIDS......Page 136 3.1. BOUNDARY AND SHEAR LAYERS IN ROTATING FLOWS......Page 137 3.1.1. EKMAN LAYERS......Page 138 3.1.2. SIDEWALL E1/3–LAYERS......Page 141 3.1.3. SIDEWALL E1/4–LAYERS......Page 144 3.1.4. DIFFERENTIALLY ROTATING SPHERES: THE PROUDMAN–STEWARTSON PROBLEM......Page 148 3.2. BOUNDARY AND SHEAR LAYERS IN ROTATING MHD FLOWS......Page 153 3.2.1. THE HARTMANN LAYER......Page 154 3.2.2. DIFFERENTIALLY ROTATING SPHERES......Page 156 3.2.3. THE EKMAN–HARTMANN LAYER......Page 159 3.2.4. ROTATING MHD FREE SHEAR LAYERS.........Page 162 3.3.1. INERTIAL WAVES......Page 168 REFLECTION AT A PLANE BOUNDARY......Page 170 BOUNDARY LAYERS......Page 172 LOW FREQUENCY MODES......Page 174 3.3.2. ALFVEN WAVES......Page 176 3.3.3. MHD WAVES IN ROTATING FLUIDS......Page 177 LEHNERT WAVES.........Page 179 TORSIONAL OSCILLATIONS WITH DISSIPATION......Page 180 3.3.4. STRATIFIED ROTATING MHD WAVES......Page 182 3.4.1. PHYSICAL MOTIVATIONS......Page 185 3.4.2. CONVECTION IN THE ROTATING CYLINDRICAL ANNULUS......Page 186 3.4.3. MATHEMATICAL FORMULATION OF THE PROBLEM OF CONVECTION IN ROTATING SPHERICAL SHELLS......Page 194 3.4.4. THE ONSET OF CONVECTION IN ROTATING SPHERICAL SHELLS......Page 196 3.4.5. ONSET OF INERTIAL CONVECTION AT LOW PRANDTL NUMBERS......Page 200 3.4.6. EVOLUTION OF CONVECTION COLUMNS AT MODERATE PRANDTL NUMBERS......Page 201 3.4.7. FINITE AMPLITUDE CONVECTION AT HIGHER PRANDTL NUMBERS......Page 207 3.4.8. FINITE AMPLITUDE INERTIAL CONVECTION......Page 210 3.4.9. PENETRATIVE AND COMPOSITIONAL CONVECTION......Page 212 3.4.10. CONCLUDING REMARKS ON CONVECTION......Page 214 PART II: NATURAL DYNAMOS AND MODELS......Page 216 4.1.1. A BRIEF HISTORY......Page 217 4.1.2. STRUCTURE OF THE EARTH......Page 218 4.1.3. THE GEOMAGNETIC FIELD......Page 220 4.1.4. ENERGY SOURCES......Page 223 4.2. GOVERNING EQUATIONS AND PARAMETERS......Page 225 4.3.1. TAYLOR’S CONSTRAINT......Page 228 4.3.2. THE “ARBITRARY” GEOSTROPHIC FLOW uG(s)......Page 230 4.3.3. EKMAN STATES, TAYLOR STATES AND MODEL-Z: DETERMINATION OF THE GEOSTROPHIC FLOW uG......Page 231 4.3.4. THE ROLE OF INERTIA......Page 233 4.4. PARAMETER CONSTRAINTS......Page 235 4.4.1. THE EKMAN NUMBER......Page 236 4.4.2. THE MAGNETIC REYNOLDS NUMBER......Page 237 4.4.4. THE RAYLEIGH NUMBER......Page 238 4.5. NUMERICAL MODELS......Page 239 4.5.2. 2.5D MODELS......Page 240 4.5.3. 3D MODELS......Page 241 4.6. TURBULENCE IN THE EARTH’S CORE: THE ENDS JUSTIFY THE MEANS?......Page 245 4.7.1. THE ENERGY SOURCE......Page 247 4.7.2. ORDERS OF MAGNITUDE......Page 248 4.7.3. BASIC EQUATIONS AND THEIR AVERAGES......Page 250 4.7.4. QUALITATIVE DESCRIPTIONS OF TURBULENCE......Page 251 4.8.1. A THREE-STEP PROGRAM......Page 255 4.8.2. LINEARISED MODES OF A SIMPLE MODEL......Page 256 4.8.3. THE MOST EASILY EXCITED MODE......Page 258 4.8.4. MORE COMPLICATED AND LESS COMPLICATED MODELS......Page 260 4.8.5. FINITE AMPLITUDES......Page 262 4.8.6. AN ALTERNATIVE APPLICATION: DNS......Page 264 4.9.1. FILTERING......Page 265 4.9.2. SIMILARITY AND DYNAMICAL SIMILARITY......Page 266 4.9.3. RELATED METHODS......Page 268 4.10.1. THE GEODYNAMO......Page 269 4.10.2. A CRITICAL SUMMARY OF TURBULENCE......Page 270 5.1. OBSERVATIONS OF PLANETARY MAGNETIC FIELDS......Page 272 5.2. SOME OUTSTANDING PROBLEMS IN PLANETARY DYNAMO THEORY......Page 278 5.3. CONDITIONS NEEDED FOR DYNAMO ACTION IN PLANETS......Page 280 5.4. ENERGY SOURCES FOR PLANETARY DYNAMOS......Page 282 Terrestrial planets......Page 284 5.5.1. GIANT PLANETS......Page 287 5.6. DYNAMICS OF PLANETARY INTERIORS......Page 288 5.6.1. TYPICAL VELOCITY AND FIELD ESTIMATES......Page 291 5.7. NUMERICAL DYNAMO MODELS FOR THE PLANETS......Page 293 6.1. STELLAR MAGNETIC ACTIVITY......Page 296 6.2. LINEAR alphaomega–DYNAMOS FOR THE SOLAR CYCLE......Page 299 6.2.1. DYNAMO WAVES......Page 300 6.2.3. THE omega–EFFECT......Page 302 6.2.4. THE alpha–EFFECT......Page 303 6.2.5. MAGNETIC PUMPING......Page 305 6.3. NONLINEAR QUENCHING MECHANISMS......Page 306 6.4. INTERFACE DYNAMOS......Page 308 6.4.1. SPHERICAL INTERFACE MODELS......Page 309 6.4.2. ZONAL SHEAR FLOWS......Page 310 6.5. MODULATION OF CYCLIC ACTIVITY......Page 312 6.5.1. DETERMINISTIC MODULATION......Page 314 6.5.2. LOW-ORDER MODELS......Page 319 6.5.3. ON–OFF AND IN–OUT INTERMITTENCY......Page 322 6.6. RAPIDLY ROTATING STARS......Page 323 6.7. THE FUTURE......Page 326 7.1. INTRODUCTION......Page 327 7.2.1. TURBULENCE AND MULTI-PHASE STRUCTURE......Page 330 7.2.2. GALACTIC ROTATION......Page 333 7.3. MAGNETIC FIELDS OBSERVED IN GALAXIES......Page 335 7.4. THE ORIGIN OF GALACTIC MAGNETIC FIELDS......Page 338 7.4.1. MEAN-FIELD MODELS OF THE GALACTIC DYNAMO......Page 339 THE LOCAL SOLUTION......Page 340 THE GLOBAL SOLUTION......Page 342 NON-AXISYMMETRIC, NONLINEAR AND NUMERICAL SOLUTIONS......Page 345 DYNAMO CONTROL PARAMETERS IN SPIRAL GALAXIES......Page 347 7.4.2. THE FLUCTUATION DYNAMO AND SMALL-SCALE MAGNETIC FIELDS......Page 349 7.4.3. MAGNETIC HELICITY BALANCE IN THE GALACTIC DISC......Page 352 7.5.1. MAGNETIC PITCH ANGLE......Page 356 7.5.2. THE EVEN (QUADRUPOLE) SYMMETRY OF MAGNETIC FIELD IN THE MILKY WAY......Page 358 7.5.3. THE AZIMUTHAL STRUCTURE......Page 360 7.5.4. A COMPOSITE MAGNETIC STRUCTURE IN M51 AND MAGNETIC REVERSALS IN THE MILKY WAY......Page 362 7.5.5. THE RADIAL MAGNETIC STRUCTURE IN M31......Page 364 7.5.6. STRENGTH OF THE REGULAR MAGNETIC FIELD......Page 366 7.6. ELLIPTICAL GALAXIES......Page 367 7.6.2. THE FLUCTUATION DYNAMO IN ELLIPTICAL GALAXIES......Page 368 7.7. ACCRETION DISCS......Page 369 7.8. CONCLUSIONS......Page 372 8.1. INTRODUCTION......Page 374 8.2. DESCRIPTION OF THE EXPERIMENTS......Page 377 8.2.2. A DYNAMO WITH TWO SOLID ROTATING CYLINDERS......Page 378 8.2.3. THE α–BOX EXPERIMENT......Page 379 8.2.4. A PRECESSING EXPERIMENT IN LIQUID SODIUM......Page 381 8.2.5. THE FIRST PONOMARENKO TYPE EXPERIMENT......Page 382 8.2.6. THE VORTICES OF GALLIUM......Page 383 8.2.7. THE RIGA DYNAMO......Page 385 8.2.8. THE KARLSRUHE DYNAMO......Page 387 8.2.9. THE COLLEGE PARK EXPERIMENTS......Page 389 8.2.10. VON KARMAN SODIUM EXPERIMENTS......Page 390 8.2.11. DERVICHE TOURNEUR SODIUM PROJECT......Page 391 8.2.13. THE PERM PROJECT......Page 392 LIQUID SODIUM AND ITS PROPERTIES......Page 393 EXTRACTION OF POWER AND SEALING......Page 394 MEASUREMENTS......Page 395 8.3.1. THE ω–EFFECT......Page 397 8.3.2. MAGNETIC FIELD EXPULSION......Page 398 8.3.3. THE α–EFFECT......Page 399 8.3.4. QUENCHING EFFECTS......Page 402 8.3.5. THE EXPERIMENTAL APPROACH TO A KINEMATIC DYNAMO......Page 403 THE RIGA DYNAMO......Page 407 THE KARLSRUHE DYNAMO......Page 408 8.3.7. THE EFFECT OF TURBULENCE......Page 410 8.3.8. SPECTRA......Page 412 8.3.9. THE β–EFFECT AND TURBULENT VISCOSITY......Page 415 8.3.10. SATURATION OF THE DYNAMO......Page 417 8.4. CONCLUSIONS......Page 418 CHAPTER 9: PROSPECTS......Page 421 APPENDIX A: VECTORS AND COORDINATES......Page 424 CYLINDRICAL POLAR COORDINATES......Page 425 SPHERICAL POLAR COORDINATES......Page 426 GREEN’S FORMULAE......Page 427 APPENDIX B: POLOIDAL–TOROIDAL DECOMPOSITION......Page 428 APPENDIX C: TAYLOR’S CONSTRAINT......Page 430 THE SUFFICIENCY OF TAYLOR’S CONSTRAINT (C.3)......Page 431 PLANE PARALLEL BOUNDARIES......Page 434 APPENDIX D: UNITS......Page 436 APPENDIX E: ABBREVIATIONS......Page 438 REFERENCES......Page 439 COLOR PLATES......Page 481