This graduate level textbook develops the theory of magnetically confined plasma, with the aim of bringing the reader to the level of current research in the field of thermonuclear fusion. It begins with the basic concepts of magnetic field description, plasma equilibria and stability, and goes on to derive the equations for guiding center particle motion in an equilibrium field. Topics include linear and nonlinear ideal and resistive modes and particle transport. It is of use to workers in the field of fusion both for its wide-ranging account of tokamak physics and as a kind of handbook or formulary.This edition has been extended in a number of ways. The material on mode-particle interactions has been reformulated and much new information added, including methodology for Monte Carlo implementation of mode destabilization. These results give explicit means of carrying out mode destabilization analysis, in particular for the dangerous fishbone mode. A new chapter on cyclotron motion in toroidal geometry has been added, with comparisons of the analysis of resonances using guiding center results. A new chapter on the use of lithium lined walls has been added, a promising means of lowering the complexity and cost of full scale fusion reactors. A section on nonlocal transport has been added, including an analysis of Levy flight simulations of ion transport in the reversed field pinch in Padova, RFX.

Preface to the Third Edition vii

Preface to the Second Edition ix

Preface to the First Edition xiii

Acknowledgement xv

List of Figures xxiii

Chapter 1 Toroidal Configuration 1

1.1 INTRODUCTION 1

1.2 GENERAL COORDINATES 6

1.3 BASIS VECTORS, METRIC TENSOR 7

1.4 VECTOR OPERATORS 9

1.5 MAGNETIC FIELD REPRESENTATION 10

1.6 MAGNETIC SURFACES 11

1.7 MAGNETIC SURFACE DESTRUCTION 13

1.8 THE STANDARD MAP 22

1.9 Problems 29

1.10 References 31

Chapter 2 Equilibrium 33

2.1 INTRODUCTION 33

2.2 THE VIRIAL THEOREM 34

2.3 FIELD LINE CURVATURE 35

2.4 GENERAL 3-D EQUILIBRIA 36

      2.4.1 Hamada coordinates 40

      2.4.2 Boozer coordinates 41

      2.4.3 Cylindrical coordinates 42

      2.4.4 Zakharov coordinates 42

2.5 STELLARATORS 43

2.6 AXISYMMETRIC EQUILIBRIA 46

2.7 TOKAMAK ORDERING 48

2.8 THE SHAFRANOV EQUILIBRIUM 49

2.9 CYLINDRICAL TOKAMAK EQUILIBRIA 55

2.10 HIGH BETA EQUILIBRIUM 58

2.11 FLUX CONSERVING EQUILIBRIA 60

2.12 EQUILIBRIUM SCALING 61

2.13 EQUILIBRIUM TYPES 62

2.14 STEPPED PRESSSURE EQUILIBRIA 63

2.15 Problems 69

2.16 References 70

Chapter 3 Guiding Center Motion 73

3.1 INTRODUCTION 73

3.2 LAGRANGIAN AND HAMILTONIAN FORMULATION 77

3.3 ORBIT TYPES IN AXISYMMETRIC EQUILIBRIA 82

3.4 INTEGRAL INVARIANTS 93

3.5 TOROIDAL PRECESSION 96

3.6 LARGE ASPECT RATIO 98

3.7 DIAMAGNETIC CURRENT 102

3.8 CONFINEMENT OF FUSION ALPHA PARTICLES 104

3.9 MAGNETIC PERTURBATIONS 104

      3.9.1 Toroidal field ripple 104

      3.9.2 Flute modes 105

      3.9.3 Ideal MHD modes 110

      3.9.4 Kinetic Poincaré plots 111

3.10 SCATTERING AND ENERGY DIFFUSION 115

3.11 Problems 118

3.12 References 119

Chapter 4 Linear Ideal Modes 121

4.1 INTRODUCTION 121

4.2 PLASMA KINETIC AND POTENTIAL ENERGY 126

4.3 SELF ADJOINTNESS OF THE POTENTIAL ENERGY 127

4.4 THE ENERGY PRINCIPLE 128

4.5 CONVENIENT FORM FOR δW 129

4.6 CYLINDRICAL GEOMETRY ENERGY PRINCIPLE 133

4.7 MHD INSTABILITIES IN LOW β TOKAMAKS 137

4.8 KINK MODE 139

4.9 THE M =1 EXTERNAL KINK 144

4.10 THE INTERNAL KINK MODE 145

4.11 BALLOONING INSTABILITIES 148

4.12 MAGNETIC WELL 154

4.13 BALLOONING,SIMPLE EQUILIBRIA 155

4.14 MERCIER,SUYDAM CRITERIA 158

4.15 BALLOONING EQUATION MODIFICATION 162

4.16 TAE MODES 163

4.17 AXISYMMETRIC MODES 164

4.18 NUMERICAL MHD SPECTRUM 167

4.19 SHAPE AND ASPECT RATIO 168

4.20 Problems 172

4.21 References 173

Chapter 5 Linear Resistive Modes 177

5.1 INTRODUCTION 177

5.2 THE TEARING MODE 182

      5.2.1 The tearing mode, m ≠ 1 191

      5.2.2 The m = 1 tearing mode 194

5.3 THE SKIN CURRENT PROFILE 201

5.4 TOROIDAL AND SHAPING EFFECTS 202

5.5 THE RESISTIVE SURFACE KINK MODE 204

5.6 OPTIMIZED PROFILES 206

5.7 THE RIPPLING MODE 208

5.8 THE RESISTIVE INTERCHANGE MODE 209

5.9 RESISTIVE BALLOONING 209

5.10 DIAMAGNETIC ROTATION 211

5.11 Problems 215

5.12 References 217

Chapter 6 Mode-Particle Interaction 221

6.1 INTRODUCTION 221

6.2 IDEAL DISPLACEMENT~→ξ, α, δ~→B, AND POTENTIAL 226

6.3 RESONANCE 233

6.4 THE FIBONACCI SEQUENCE 236

6.5 LANDAU PHASE MIXING 238

6.6 PHASE VECTOR ROTATION 241

6.7 MODE INDUCED AVALANCHE 246

6.8 MODE-PARTICLE ENERGY TRANSFER 251

6.9 MODE EVOLUTION 253

      6.9.1 Frequency determined by equilibrium 253

      6.9.2 Frequency determined by particle distribution 257

      6.9.3 Calculation of δW_n 276

      6.9.4 Monte Carlo evaluation 278

6.10 TAE MODE DRIVE AND SATURATION 280

6.11 TRAPPED PARTICLE RESONANCE 284

6.12 FISHBONE INDUCED LOSS 287

6.13 FISHBONE DESTABILIZATION 289

6.14 THE FISHBONE CYCLE 299

6.15 RESISTIVE KINETIC INTERNAL KINK 302

6.16 STABILIZATION OF THE SAWTOOTH 303

6.17 BALLOONING DESTABILIZATION 308

6.18 ALPHA PARTICLE EFFECTS 309

6.19 Problems 311

6.20 References 312

Chapter 7 Cyclotron Motion 317

7.1 7.1 INTRODUCTION 317

7.2 SUB CYCLOTRON HEATING 317

7.3 STOCHASTIC DOMAIN 323

7.4 CYCLOTRON MOTION IN A TOROIDAL SYSTEM 325

7.5 CYCLOTRON AND GUIDING CENTER ANALYSES OF RESONANCE 328

7.6 PROFILE FLATTENING 331

7.7 References 333

Chapter 8 Nonlinear Behavior 335

8.1 INTRODUCTION 335

8.2 THE REDUCED EQUATIONS 339

8.3 NONLINEAR EXTERNAL KINK 342

8.4 VACUUM BUBBLES 345

8.5 NONLINEAR INTERNAL KINK 349

8.6 COMPLETE RESISTIVE RECONNECTION 352

8.7 NONLINEAR TEARING MODE ANALYSIS 352

8.8 SAWTOOTH OSCILLATIONS 360

8.9 DISRUPTIONS 365

8.10 EMPIRICAL LIMITS 370

8.11 THE GREENWALD DENSITY LIMIT 372

8.12 STABILIZATION OF TEARING MODES 374

8.13 Problems 377

8.14 References 378

Chapter 9 Transport 383

9.1 INTRODUCTION 383

9.2 THE DRIFT KINETIC EQUATION 389

9.3 CROSS FIELD DIFFUSION 392

9.4 BOOTSTRAP CURRENT 396

9.5 NEOCLASSICAL TEARING 401

9.6 WARE PINCH 401

9.7 MAGNETIC FIELD RIPPLE TRANSPORT 402

9.8 DIFFUSION IN A STOCHASTIC FIELD 415

9.9 ISLAND INDUCED DIFFUSION 418

9.10 ANOMALOUS TRANSPORT 421

9.11 CONFINEMENT SCALING 426

9.12 NONLOCAL TRANSPORT 428

      9.12.1 Determination of the Lévy flight distribution 432

      9.12.2 Two fluid Montroll equation 433

9.13 Burn Control 435

9.14 Problems 438

9.15 References 441

Chapter 10 The Lithium Wall Fusion Concept 447

10.1 THE IDEA OF MAGNETIC FUSION 447

10.2 CONFINEMENT REGIME CONTROLLED BY THERMAL CONDUCTION 449

10.3 DIFFUSION BASED CONFINEMENT REGIME 450

10.4 IMPLEMENTATION 452

10.5 ALPHA PARTICLE CONFINEMENT 454

10.6 DIVERTOR PLATES 455

10.7 References 456

Chapter 11 Phase Integral Methods 457

11.1 INTRODUCTION 457

11.2 CONNECTION FORMULAE 460

11.3 CAUSALITY 467

11.4 BOUND STATES-INSTABILITIES 468

11.5 OVERDENSE BARRIER-SCATTERING 472

11.6 UNDERDENSE BARRIER-SCATTERING 476

11.7 EIGENVALUE PROBLEMS 479

11.8 Problems 484

11.9 References 487

Index 489

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