Preface page ................................................. xvii
Part A: The Fundamentals of MHD ................................. 1
Introduction: The Aims of Part A ................................ 1
1. A Qualitative Overview of MHD ................................ 3
1.1. What is MHD? ............................................ 3
1.2. A Brief History of MHD .................................. 6
1.3. From Electrodynamics to MHD: A Simple Experiment ........ 8
1.3.1. Some important parameters in electrodynamics
and MHD .......................................... 8
1.3.2. A brief reminder of the laws of
electrodynamics .................................. 9
1.3.3. A familiar high-school experiment ............... 11
1.3.4. A summary of the key results for MHD ............ 18
1.4. Some Simple Applications of MHD ........................ 18
2. The Governing Equations of Electrodynamics .................. 27
2.1. The Electric Field and the Lorentz Force ............... 27
2.2. Ohm's Law and the Volumetric Lorentz Force ............. 29
2.3. Ampere's Law ........................................... 31
2.4. Faraday's Law in Differential Form ..................... 32
2.5. The Reduced Form of Maxwell's Equations for MHD ........ 34
2.6. A Transport Equation for В ............................. 37
2.7. On the Remarkable Nature of Faraday and of
Faraday's Law .......................................... 37
2.7.1. An historical footnote .......................... 37
2.7.2. An important kinematic equation ................. 40
2.7.3. The full significance of Faraday's law .......... 42
2.7.4. Faraday's law in ideal conductors: Alfven's
theorem ......................................... 44
3. The Governing Equations of Fluid Mechanics .................. 47
Part 1: Fluid Flow in the Absence of Lorentz Forces ......... 47
3.1. Elementary Concepts .................................... 47
3.1.1. Different categories of fluid flow .............. 47
3.1.2. The Navier-Stokes equation ...................... 59
3.2. Vorticity, Angular Momentum and the Biot-Savart Law ... 61
3.3. Advection and Diffusion of Vorticity ................... 64
3.3.1. The vorticity equation .......................... 64
3.3.2. Advection and diffusion of vorticity:
temperature as a prototype ...................... 66
3.3.3. Vortex line stretching .......................... 70
3.4. Kelvin's Theorem, Helmholtz's Laws and Helicity ........ 71
3.4.1. Kelvin's Theorem and Helmholtz's Laws ........... 71
3.4.2. Helicity ........................................ 74
3.5. The Prandtl-Batchelor Theorem .......................... 77
3.6. Boundary Layers, Reynolds Stresses and Turbulence
Models ................................................. 81
3.6.1. Boundary layers ................................. 81
3.6.2. Reynolds stresses and turbulence models ......... 83
3.7. Ekman Pumping in Rotating Flows ........................ 90
Part 2: Incorporating the Lorentz Force ..................... 95
3.8. The Full Equations of MHD and Key Dimensionless
Groups ................................................. 95
3.9. Maxwell Stresses ....................................... 97
4. Kinematics of MHD: Advection and Diffusion of a Magnetic
Field ...................................................... 102
4.1. The Analogy to Vorticity .............................. 102
4.2. Diffusion of a Magnetic Field ......................... 103
4.3. Advection in Ideal Conductors: Alfven's Theorem ....... 104
4.3.1. Alfven's theorem ............................... 104
4.3.2. An aside: sunspots ............................. 106
4.4. Magnetic Helicity ..................................... 108
4.5. Advection plus Diffusion .............................. 109
4.5.1. Field sweeping ................................. 109
4.5.2. Flux expulsion ................................. 110
4.5.3. Azimuthal field generation by differential
rotation ....................................... 114
4.5.4. Magnetic reconnection .......................... 115
5. Dynamics at Low Magnetic Reynolds Numbers .................. 117
5.1. The Low-Rm Approximation in MHD ....................... 118
Part 1: Suppression of Motion .............................. 119
5.2. Magnetic Damping ...................................... 119
5.2.1. The destruction of mechanical energy via
Joule dissipation .............................. 120
5.2.2. The damping of a two-dimensional jet ........... 121
5.2.3. Damping of a vortex ............................ 122
5.3. A Glimpse at MHD Turbulence ........................... 128
5.4. Natural Convection in the Presence of a Magnetic
Field ................................................. 132
5.4.1. Rayleigh-Benard convection ..................... 132
5.4.2. The governing equations ........................ 133
5.4.3. An energy analysis of the Rayleigh-Benard
instability .................................... 134
5.4.4. Natural convection in other configurations ..... 137
Part 2: Generation of Motion ............................... 139
5.5. Rotating Fields and Swirling Motions .................. 139
5.5.1. Stirring of a long column of metal ............. 139
5.5.2. Swirling flow induced between two parallel
plates ......................................... 142
5.6. Motion Driven by Current Injection .................... 145
5.6.1. A model problem ................................ 145
5.6.2. A useful energy equation ....................... 146
5.6.3. Estimates of the induced velocity .............. 148
5.6.4. A paradox ...................................... 149
Part 3: Boundary Layers .................................... 151
5.7. Hartmann Boundary Layers .............................. 151
5.7.1. The Hartmann Layer ............................. 151
5.7.2. Hartmann flow between two planes ............... 152
5.8. Examples of Hartmann and Related Flows ................ 154
5.8.1. Flow-meters and MHD generators ................. 154
5.8.2. Pumps, propulsion and projectiles .............. 155
5.9. Conclusion ............................................ 157
6. Dynamics at Moderate to High Magnetic Reynolds' Number ..... 159
6.1. Alfven Waves and Magnetostrophic Waves ................ 160
6.1.1. Alfven waves ................................... 160
6.1.2. Magnetostrophic waves .......................... 163
6.2. Elements of Geo-Dynamo Theory ......................... 166
6.2.1. Why do we need a dynamo theory for the
earth? ......................................... 166
6.2.2. A large magnetic Reynolds number is needed ..... 171
6.2.3. An axisymmetric dynamo is not possible ......... 174
6.2.4. The influence of small-scale turbulence:
the α-effect ................................... 177
6.2.5. Some elementary dynamical considerations ....... 185
6.2.6. Competing kinematic theories for the geo-
dynamo ......................................... 197
6.3. A Qualitative Discussion of Solar MHD ................. 199
6.3.1. The structure of the sun ....................... 200
6.3.2. Is there a solar dynamo? ....................... 201
6.3.3. Sunspots and the solar cycle ................... 201
6.3.4. The location of the solar dynamo ............... 203
6.3.5. Solar flares ................................... 203
6.4. Energy-Based Stability Theorems for Ideal MHD ......... 206
6.4.1. The need for stability theorems in ideal
MHD: plasma containment ........................ 207
6.4.2. The energy method for magnetostatic
equilibria ..................................... 208
6.4.3. An alternative method for magnetostatic
equilibrium .................................... 213
6.4.4. Proof that the energy method provides both
necessary and sufficient conditions for
stability ...................................... 215
6.4.5. The stability of non-static equilibria ......... 216
6.5. Conclusion ............................................ 220
7. MHD Turbulence at Low and High Magnetic Reynolds Number .... 222
7.1. A Survey of Conventional Turbulence ................... 223
7.1.1. A historical interlude ......................... 223
7.1.2. A note on tensor notation ...................... 227
7.1.3. The structure of turbulent flows: the
Kolmogorov picture of turbulence ............... 229
7.1.4. Velocity correlation functions and the
Karman-Howarth equation ........................ 235
7.1.5. Decaying turbulence: Kolmogorov's law,
Loitsyansky's integral, Landau's angular
momentum and Batchelor's pressure forces ....... 240
7.1.6. On the difficulties of direct numerical
simulations .................................... 247
7.2. MHD Turbulence ........................................ 249
7.2.1. The growth of anisotropy at low and high Rm ... 249
7.2.2. Decay laws at low Rm ........................... 252
7.2.3. The spontaneous growth of a magnetic field
at high Rm ..................................... 256
7.3. Two-Dimensional Turbulence ............................ 260
7.3.1. Batchelor's self-similar spectrum and the
inverse energy cascade ......................... 260
7.3.2. Coherent vortices .............................. 263
7.3.3. The governing equations of two-dimensional
turbulence ..................................... 264
7.3.4. Variational principles for predicting the
final state in confined domains ................ 267
Part B: Applications in Engineering and Metallurgy ............ 273
Introduction: An Overview of Metallurgical Applications ....... 273
8. Magnetic Stirring Using Rotating Fields .................... 285
8.1. Casting, Stirring and Metallurgy ...................... 285
8.2. Early Models of Stirring .............................. 289
8.3. The Dominance of Ekman Pumping in the Stirring
of Confined Liquids ................................... 294
8.4. The Stirring of Steel ................................. 298
9. Magnetic Damping Using Static Fields ....................... 301
9.1. Metallurgical Applications ............................ 301
9.2. Conservation of Momentum, Destruction of Energy
and the Growth of Anisotropy .......................... 304
9.3. Magnetic Damping of Submerged Jets .................... 308
9.4. Magnetic Damping of Vortices .......................... 312
9.4.1. General considerations ......................... 312
9.4.2. Damping of transverse vortices ................. 314
9.4.3. Damping of parallel vortices ................... 317
9.4.4. Implications for low-Rm turbulence ............. 323
9.5. Damping of Natural Convection ......................... 324
9.5.1. Natural convection in an aluminium ingot ....... 324
9.5.2. Magnetic damping in an aluminium ingot ......... 329
10.Axisymmetric Flows Driven by the Injection of
Current .................................................... 332
10.1.The VAR Process and a Model Problem ................... 332
10.1.1.The VAR process ................................ 332
10.1.2.Integral constraints on the flow ............... 336
10.2.The Work Done by the Lorentz Force .................... 338
10.3.Structure and Scaling of the Flow ..................... 340
10.3.1.Differences between confined and unconfined
flows .......................................... 340
10.3.2.Shercliff s self-similar solution for
unconfined flows ............................... 342
10.3.3.Confined flows ................................. 344
10.4.The Influence of Buoyancy ............................. 346
10.5.Stability of the Flow and the Apparent Growth of
Swirl ................................................. 348
10.5.1.An extraordinary experiment .................... 348
10.5.2.There is no spontaneous growth of swirl! ....... 350
10.6.Flaws in the Traditional Explanation for the
Emergence of Swirl .................................... 351
10.7.The Role of Ekman Pumping in Establishing the
Dominance of Swirl .................................... 353
10.7.1.A glimpse at the mechanisms .................... 353
10.7.2.A formal analysis .............................. 356
10.7.3.Some numerical experiments ..................... 358
11.MHD Instabilities in Reduction Cells ....................... 363
11.1.Interfacial Waves in Aluminium Reduction Cells ........ 363
11.1.1.Early attempts to produce aluminium by
electrolysis ................................... 363
11.1.2.The instability of modern reduction cells ...... 364
11.2.A Simple Mechanical Analogue for the Instability ...... 368
11.3.Simplifying Assumptions ............................... 372
11.4.A Shallow-Water Wave Equation and Key
Dimensionless Groups .................................. 374
11.4.1.A shallow-water wave equation .................. 374
11.4.2.Key dimensionless groups ....................... 378
11.5.Travelling Wave and Standing Wave Instabilities ....... 379
11.5.1.Travelling waves ............................... 379
11.5.2.Standing waves in circular domains ............. 380
11.5.3.Standing waves in rectangular domains .......... 381
11.6.Implications for Reduction Cell Design ................ 385
12.High-Frequency Fields: Magnetic Levitation and Induction
Heating .................................................... 387
12.1.The Skin Effect ....................................... 388
12.2.Magnetic Pressure, Induction Heating and High-
Frequency Stirring .................................... 390
12.3.Applications in the Casting of Steel, Aluminium
and Super-Alloys ...................................... 394
12.3.1.The induction furnace .......................... 394
12.3.2.The cold crucible .............................. 397
12.3.3.Levitation melting ............................. 398
12.3.4.Processes which rely on magnetic repulsion
EM valves and EM casters ....................... 403
Appendices
1.Vector Identities and Theorems .............................. 405
2.Stability Criteria for Ideal MHD Based on the
Hamiltonian ................................................. 407
3.Physical Properties of Liquid Metals ........................ 417
4.MHD Turbulence at Low Rm .................................... 418
Bibliography .................................................. 422
Suggested Books on Fluid Mechanics ............................ 422
Suggested Books on Electromagnetism ........................... 422
Suggested Books on MHD ........................................ 423
Journal References for Part В and Appendix 2 .................. 423
Subject Index ................................................. 427
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