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Part A:The Fundamentals of MHD1

Introduction:The Aims of Part A1

1 A Qualitative Overview of MHD3

1.1 What is MHD?3

1.2 A BriefHistory of MHD6

1.3 From Electrodynamics to MHD:A Simple Experiment8

1.3.1 Some important parameters in electrodynamics and MHD8

1.3.2 A brief reminder of the laws of electrodynamics9

1.3.3 A familiar high-school experiment11

1.3.4 A summary of the key results for MHD18

1.4 Some Simple Applications of MHD18

2 The Governing Equations of Electrodynamics27

2.1 The Electric Field and the Lorentz Force27

2.2 Ohm's Law and the Volumetric Lorentz Force29

2.3 Ampère's Law31

2.4 Faraday's Law in Differential Form32

2.5 The Reduced Form of Maxwell's Equations for MHD34

2.6 A Transport Equation for B37

2.7 On the Remarkable Nature of Faraday and of Faraday's Law37

2.7.1 An historical footnote37

2.7.2 An important kinematic equation40

2.7.3 The full significance of Faraday's law42

2.7.4 Faraday's law in ideal conductors:Alfvén's theorem44

3 The Governing Equations of Fluid Mechanics47

Part 1:Fluid Flow in the Absence of Lorentz Forces47

3.1 Elementary Concepts47

3.1.1 Different categories of fluid flow47

3.1.2 The Navier-Stokes equation59

3.2 Vorticity,Angular Momentum and the Biot-Savart Law61

3.3 Advection and Diffusion of Vorticity64

3.3.1 The vorticity equation64

3.3.2 Advection and diffusion of vorticity:temperature as a prototype66

3.3.3 Vortex line stretching70

3.4 Kelvin's Theorem，Helmholtz's Laws and Helicity71

3.4.1 Kelvin's Theorem and Helmholtz's Laws71

3.4.2 Helicity74

3.5 The Prandtl-Batchelor Theorem77

3.6 Boundary Layers,Reynolds Stresses and Turbulence Models81

3.6.1 Boundary layers81

3.6.2 Reynolds stresses and turbulence models83

3.7 Ekman Pumping in Rotating Flows90

Part 2:Incorporating the Lorentz Force95

3.8 The Full Equations of MHD and Key Dimensionless Groups95

3.9 Maxwell Stresses97

4 Kinematies of MHD:Advection and Diffusion of a Magnetic Field102

4.1 The Analogy to Vorticity102

4.2 Diffusion of a Magnetic Field103

4.3 Advection in Ideal Conductors:Alfvén's Theorem104

4.3.1 Alfvén's theorem104

4.3.2 An aside:sunspots106

4.4 Magnetic Helicity108

4.5 Advection plus Diffusion109

4.5.1 Field sweeping109

4.5.2 Flux expulsion110

4.5.3 Azimuthal field generation by differential rotation114

4.5.4 Magnetic reconnection115

5 Dynamics at Low Magnetic Reynolds Numbers117

5.1 The Low-Rm Approximation in MHD118

Part 1:Suppression of Motion119

5.2 Magnetic Damping119

5.2.1 The destruction of mechanical energy via Joule dissipation120

5.2.2 The damping of a two-dimensional jet121

5.2.3 Damping of a vortex122

5.3 A Glimpse at MHD Turbulence128

5.4 Natural Convection in the Presence of a Magnetic Field132

5.4.1 Rayleigh-Bénard convection132

5.4.2 The governing equations133

5.4.3 An energy analysis of the Rayleigh-Bénard instability134

5.4.4 Natural convection in other configurations137

Part 2:Generation of Motion139

5.5 Rotating Fields and Swirling Motions139

5.5.1 Stirring of a long column of metal139

5.5.2 Swirling flow induced between two parallel plates142

5.6 Motion Driven bv Current Injection145

5.6.1 A model problem145

5.6.2 A useful energy equation146

5.6.3 Estimates of the induced velocity148

5.6.4 A paradox149

Part 3:Boundary Layers151

5.7 Hartmann Boundary Layers151

5.7.1 The Hartmann Layer151

5.7.2 Hartmann flow between two planes152

5.8 Examples of Hartmann and Related Flows154

5.8.1 Flow-meters and MHD generators154

5.8.2 Pumps,propulsion and projectiles155

5.9 Conclusion157

6 Dynamics at Moderate to High Magnetic Reynolds'Number159

6.1 Alfvén Waves and Magnetostrophic Waves160

6.1.1 Alfvén waves160

6.1.2 Magnetostrophic waves163

6.2 Elements of Geo-Dynamo Theory166

6.2.1 Why do we need a dynamo theory for the earth?166

6.2.2 A large magnetic Reynolds number is needed171

6.2.3 An axisymmetric dynamo is not possible174

6.2.4 The influenee of small-scale turbulence:the α-effect177

6.2.5 Some elementary dynamical considerations185

6.2.6 Competing kinematic theories for the geo-dynamo197

6.3 A Qualitative Discussion of Solar MHD199

6.3.1 The structure of the sun200

6.3.2 Is there a solar dynamo?201

6.3.3 Sunspots and the solar cycle201

6.3.4 The location of the solar dynamo203

6.3.5 Solar flares203

6.4 Energy-Based Stability Theorems for Ideal MHD206

6.4.1 The need for stability theorems in ideal MHD:plasma containment207

6.4.2 The energy method for magnetostatic equilibria208

6.4.3 An alternative method for magnetostatic equilibrium213

6.4.4 Proof that the energy method provides both necessary and sufficient conditions for stability215

6.4.5 The stability of non-static equilibria216

6.5 Conclusion220

7 MHD Turbulence at Low and High Magnetic Reynolds Number222

7.1 A Survey of Conventional Turbulence223

7.1.1 A historical interlude223

7.1.2 A note on tensor notation227

7.1.3 The structure of turbulent flows:the Kolmogorov picture of turbulence229

7.1.4 Velocity correlation functions and the Karman-Howarth equation235

7.1.5 Decaying turbulence:Kolmogorov's law,Loitsyansky's integral,Landau's angular momentum and Batchelor's pressure forces240

7.1.6 On the difficulties of direct numerical simulations247

7.2 MHD Turbulence249

7.2.1 The growth of anisotropy at low and high Rm249

7.2.2 Decay laws at low Rm252

7.2.3 The spontaneous growth of a magnetic field at high Rm256

7.3 Two-Dimensional Turbulence260

7.3.1 Batchelor's self-similar spectrum and the inverse energy cascade260

7.3.2 Coherent vortices263

7.3.3 The governing equations of two-dimensional turbulence264

7.3.4 Variational principles for predicting the final state in confined domains267

Part B:Applications in Engineering and Metallurgy273

Introduction:An Overview of Metallurgical Applications273

8 Magnetic Stirring Using Rotating Fields285

8.1 Casting,Stirring and Metallurgy285

8.2 Early Models of Stirring289

8.3 The Dominance of Ekman Pumping in the Stirring of Confined Liquids294

8.4 The Stirring of Steel298

9 Magnetic Damping Using Static Fields301

9.1 Metallurgical Applications301

9.2 Conservation of Momentum,Destruction of Energy and the Growth of Anisotropy304

9.3 Magnetic Damping of Submerged Jets308

9.4 Magnetic Damping of Vortices312

9.4.1 General considerations312

9.4.2 Damping of transverse vortices314

9.4.3 Damping of parallel vortices317

9.4.4 Implications for low-Rm turbulence323

9.5 Damping of Natural Convection324

9.5.1 Natural convection in an aluminium ingot324

9.5.2 Magnetic damping in an aluminium ingot329

10 Axisymmetric Flows Driven by the Injection of Current332

10.1 The VAR Process and a Model Problem332

10.1.1 The VAR process332

10.1.2 Integral constraints on the flow336

10.2 The Work Done by the Lorentz Force338

10.3 Structure and Scaling of the Flow340

10.3.1 Differences between confined and unconfined flows340

10.3.2 Shercliff's self-similar solution for unconfined flows342

10.3.3 Confined flows344

10.4 The Influence of Buoyancy346

10.5 Stability of the Flow and the Apparent Growth of Swirl348

10.5.1 An extraordinary experiment348

10.5.2 There is no spontaneous growth of swirl!350

10.6 Flaws in the Traditional Explanation for the Emergence of Swirl351

10.7 The R？le of Ekman Pumping in Establishing the Dominance of Swirl353

10.7.1 A glimpse at the mechanisms353

10.7.2 A formal analysis356

10.7.3 Some numerical experiments358

11 MHD Instabilities in Reduction Cells363

11.1 Interfacial Waves in Aluminium Reduction Cells363

11.1.1 Early attempts to produce aluminium by electrolysis363

11.1.2 The instability of modem reduction cells364

11.2 A Simple Mechanical Analogue for the Instability368

11.3 Simplifying Assumptions372

11.4 A Shallow-Water Wave Equation and Key Dimensionless Groups374

11.4.1 A shallow-water wave equation374

11.4.2 Key dimensionless groups378

11.5 Travelling Wave and Standing Wave Instabilities379

11.5.1 Travelling waves379

11.5.2 Standing waves in circular domains380

11.5.3 Standing waves in rectangular domains381

11.6 Implications for Reduction Cell Design385

12 High-Frequency Fields:Magnetic Levitation and Induction Heating387

12.1 The Skin Effect388

12.2 Magnetic Pressure,Induction Heating and High-Frequency Stirring390

12.3 Applications in the Casting of Steel,Aluminium and Super-Alloys394

12.3.1 The induction fumace394

12.3.2 The cold crucible397

12.3.3 Levitation melting398

12.3.4 Processes which rely on magnetic repulsion EM valves and EM casters403

Appendices405

1 Vector Identities and Theorems405

2 Stability Criteria for Ideal MHD Based on the Hamiltonian407

3 Physical Properties of Liquid Metals417

4 MHD Turbulence at Low Rm418

Bibliography422

Suggested Books on Fluid Mechanics422

Suggested Books on Electromagnetism422

Suggested Books on MHD423

Journal References for Part B and Appendix 2423

Subject Index427