Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

Magnetic fields and tokamak plasmas
Alan Wootton
INTRODUCTION........................................................................................................................... 5
1. SOME CONCEPTS AND DEFINITIONS................................................................................ 7
Maxwell's equations.................................................................................................................... 7
Pick-up or Induction Coils .......................................................................................................... 7
Integration ................................................................................................................................... 8
Vector potential......................................................................................................................... 12
Mutual inductance..................................................................................................................... 16
Self inductance.......................................................................................................................... 17
Poloidal flux.............................................................................................................................. 18
Field lines and flux surfaces ..................................................................................................... 21
An example ............................................................................................................................... 23
Circuit equations ....................................................................................................................... 28
2. SOME NON STANDARD MEASUREMENT TECHNIQUES............................................. 30
Hall Probe ................................................................................................................................. 30
Faraday Effect ........................................................................................................................... 31
The Compass............................................................................................................................. 31
Flux gates.................................................................................................................................. 32
3. GENERAL FIELD CHARACTERIZATION.......................................................................... 34
Fourier components .................................................................................................................. 34
Field components on a rectangle............................................................................................... 36
4. PLASMA CURRENT.............................................................................................................. 39
Rogowski coil ........................................................................................................................... 39
5. LOOP VOLTS, VOLTS per TURN, SURFACE VOLTAGE.............................................. 40
Introduction............................................................................................................................... 40
The single volts per turn loop ................................................................................................... 40
Poynting’s theorem ................................................................................................................... 41
Uses of the Volts per turn measurement ................................................................................... 44
6. TOKAMAK EQUILIBRIA....................................................................................................... 45
6.0. AN INTUITIVE DERIVATION OF TOKAMAK EQUILIBRIUM ................................ 45
Introduction........................................................................................................................... 45
Energy associated with plasma pressure ............................................................................... 46
Energy associated with toroidal fields .................................................................................. 47
Energy associated with poloidal fields.................................................................................. 48
Total forces ........................................................................................................................... 49
6.1. THE FLUX OUTSIDE A CIRCULAR TOKAMAK ........................................................ 50
6. CIRCULAR EQUILIBRIUM .............................................................................................. 53
Derivation of the Grad Shafranov equation .......................................................................... 53
Solving the Grad Shafranov equation ................................................................................... 57
The poloidal field at the plasma edge ................................................................................... 59
Simple current distributions.................................................................................................. 60
The surface displacements: the Shafranov Shift................................................................... 64
Matching vacuum and plasma solutions ............................................................................... 65
More complicated configurations. ........................................................................................ 69
7. Position and βI + li/2 for the circular equilibrium ................................................................... 70
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