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Magnetohydrodynamic Stability of Tokamaks

Magnetohydrodynamic Stability of Tokamaks

Hartmut Zohm

ISBN: 978-3-527-41232-7

Feb 2015

240 pages

In Stock

$155.00

Description

This book bridges the gap between general plasma physics lectures and the real world problems in MHD stability. In order to support the understanding of concepts and their implication, it refers to real world problems such as toroidal mode coupling or nonlinear evolution in a conceptual and phenomenological approach. Detailed mathematical treatment will involve classical linear stability analysis and an outline of more recent concepts such as the ballooning formalism.

The book is based on lectures that the author has given to Master and PhD students in Fusion Plasma Physics. Due its strong link to experimental results in MHD instabilities, the book is also of use to senior researchers in the field, i.e. experimental physicists and engineers in fusion reactor science.

The volume is organized in three parts. It starts with an introduction to the MHD equations, a section on toroidal equilibrium (tokamak and stellarator), and on linear stability analysis. Starting from there, the ideal MHD stability of the tokamak configuration will be treated in the second part which is subdivided into current driven and pressure driven MHD. This includes many examples with reference to experimental results for important MHD instabilities such as kinks and their transformation to RWMs, infernal modes, peeling modes, ballooning modes and their relation to ELMs. Finally the coverage is completed by a chapter on resistive stability explaining reconnection and island formation. Again, examples from recent tokamak MHD such as sawteeth, CTMs, NTMs and their relation to disruptions are extensively discussed.

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 Alfvén Waves 12

1.2.3.2 Shear Alfvén Waves 13

2 MHD Equilibria in Fusion Plasmas 15

2.1 Linear Configurations 15

2.1.1 The z-Pinch 15

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

2.2.1.4 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 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 𝛽105

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 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 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

“The book is an excellent introduction and composition of modern results that may help not only the newcomers but also the specialists { congratulations to the author.”  (Zentralblatt MATH, 1 September 2015)