Preface IX
1 The MHD Equations 1
1.1 Derivation of the MHD Equations 1
1.1.1 Multispecies MHD Equations 1
1.1.2 One-Fluid Model of Magnetohydrodynamics 4
1.1.3 Validity of the One-Fluid Model of Magnetohydrodynamics 6
1.2 Consequences of the MHD Equations 8
1.2.1 Magnetic Flux Conservation 8
1.2.2 MHD Equilibrium 10
1.2.3 Magnetohydrodynamic Waves 11
1.2.3.1 Compressional Alfven Waves 12
1.2.3.2 Shear Alfven Waves 13
2 MHD Equilibria in Fusion Plasmas 15
2.1 Linear Con?gurations 15
2.1.1 The z-Pinch 15
2.1.2 The Screw Pinch 18
2.2 Toroidal Con?gurations 22
2.2.1 The Tokamak 23
2.2.1.1 The Grad–Shafranov Equation 23
2.2.1.2 Circular Cross Section 27
2.2.1.3 Arbitrary Cross Section 32
2.2.1.4 The Straight Field Line Angle 34
2.2.2 The Stellarator 37
3 Linear Ideal MHD Stability Analysis 43
3.1 Linear MHD Stability as an Initial Value Problem 44
3.2 The Energy Principle of Ideal MHD 47
3.3 Forms of ?W 48
3.4 The Ideal MHD Energy Principle for the Tokamak 51
4 Current Driven Ideal MHD Modes in a Tokamak 55
4.1 Expression for ?W in Tokamak Ordering 55
4.2 External Kinks in a Tokamak with ?= 0 56
4.2.1 Modes with m = 1 56
4.2.2 Modes with m ≥ 2 58
4.3 Internal Kink Modes 61
4.4 n = 0 Modes: The Vertical Displacement Event (VDE) 63
5 Pressure Driven Modes in a Tokamak 69
5.1 Localized Interchange Modes in the Screw Pinch 69
5.2 Localized Pressure Driven Modes in the Tokamak 72
5.2.1 Interchange Modes in a Tokamak 73
5.2.2 Ballooning Modes 76
6 Combined Pressure and Current Driven Modes: Edge Localized Modes 83
6.1 ELM Phenomenology 84
6.2 Linear Stability of the Pedestal 86
6.3 Non-linear Evolution 90
6.3.1 Non-linear Cycles 90
6.3.2 Magnitude of the ELM Crash 92
6.3.3 Timescale of the ELM Crash 94
6.4 ELM Control 94
6.4.1 Small ELM Regimes 95
6.4.2 Active ELM Control 97
7 Combined Pressure and Current Driven Modes: The Ideal ?-Limit 103
7.1 Tokamak Operational Scenarios 103
7.2 External Kink Modes in a Tokamak with Finite ?105
7.3 The E?ect of a Conducting Wall on External Kink Modes 107
7.3.1 Ideally Conducting Wall 107
7.3.2 Resistive Wall 110
7.4 The Resistive Wall Mode (RWM) 112
7.5 The Troyon Limit 118
8 Resistive MHD Stability 123
8.1 Stability of Current Sheets 124
8.2 Reconnection in the Presence of a Guide Field 127
8.3 Magnetic Islands in Tokamaks 134
8.4 The Rutherford Equation 137
9 Current Driven (‘classical’) Tearing Modes in Tokamaks 141
9.1 E?ect of Tearing Modes on Kinetic Pro?les 141
9.2 Nonlinear Saturation 144
9.3 Tearing Mode Rotation and Locking 146
9.3.1 Rotation of Tearing Modes in Tokamaks 146
9.3.2 Locking of Pre-existing Magnetic Islands 148
9.3.3 Ab-initio Locked Modes 152
10 Disruptions 159
10.1 Phenomenology of Disruptions 159
10.1.1 The Density Limit 161
10.2 Consequences of Disruptions 165
10.2.1 Thermal Loads 165
10.2.2 Mechanical Loads 166
10.2.3 Runaway Generation 168
10.3 Disruption Avoidance and Mitigation 171
11 M=1 Modes beyond Ideal MHD: Sawteeth and Fishbones 175
11.1 The Sawtooth Instability 17
11.1.1 Phenomenology 175
11.1.2 Sawtooth Period and Onset Criterion 17
11.1.3 Models for the Sawtooth Crash 181
11.2 The Fishbone Instability 184
12 Tearing Modes in Finite ?-Tokamaks 189
12.1 The Modi?ed Rutherford Equation 189
12.2 The Neoclassical Tearing Mode (NTM) 190
12.3 Onset Criteria for NTMs 194
12.4 Frequently Interrupted Regime (FIR) NTMs 197
13 Control of Resistive MHD Instabilities by External Current Drive 201
13.1 Basic Properties of Localized Electron Cyclotron Current Drive (ECCD) 202
13.2 Criteria for Control of Resistive Instabilities 203
13.2.1 Control by Changing the Equilibrium Current Density 203
13.2.2 Control by Generating Helical Currents 206
13.3 Sawtooth Control 208
13.4 Tearing Mode Control 211
References 215
Index 221
