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 Configurations .................................. 15
2.1.1 The z-Pinch ..................................... 18
2.1.2 The Screw Pinch ................................. 18
2.2 Toroidal Configurations ................................ 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 A 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 δW ............................................ 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 δW in Tokamak Ordering .................. 55
4.2 External Kinks in a Tokamak with β = 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 β-
Limit ...................................................... 103
7.1 Tokamak Operational Scenarios ......................... 103
7.2 External Kink Modes in a Tokamak with Finite β ........ 105
7.3 The Effect 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 Effect of Tearing Modes on Kinetic Profiles ........... 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 .............................. 175
11.1.1 Phenomenology .................................. 175
11.1.2 Sawtooth Period and Onset Criterion ............ 176
11.1.3 Models for the Sawtooth Crash .................. 181
11.2 The Fishbone Instability .............................. 184
12 Tearing Modes in Finite β-Tokamaks ......................... 189
12.1 The Modified 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
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