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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 BriefHistory 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 Ampère'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 B 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:Alfvén'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 Kinematies 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:Alfvén's Theorem 104

4.3.1 Alfvén'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-Bénard convection 132

5.4.2 The governing equations 133

5.4.3 An energy analysis of the Rayleigh-Bénard 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 bv 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 Alfvén Waves and Magnetostrophic Waves 160

6.1.1 Alfvén 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 influenee 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 R？le 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 modem 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 fumace 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 405

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 B and Appendix 2 423

Subject Index 427