Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

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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 2
Magnetic fields and tokamak plasmas Alan Wootton An ‘exact’ circular equilibrium................................................................................................. 70 Extension of position measurement to non circular shapes ...................................................... 73 Extension of βI + li/2 measurement to non circular shapes...................................................... 74 Non-circular contours. .............................................................................................................. 75 8. SOME FUNDAMENTAL RELATIONS ................................................................................ 78 Geometry................................................................................................................................... 78 Field representation................................................................................................................... 79 Identities.................................................................................................................................... 80 Ideal MHD ................................................................................................................................ 82 Boundary conditions ................................................................................................................. 82 9. MOMENTS OF THE TOROIDAL CURRENT DENSITY.................................................... 84 10. PLASMA POSITION ............................................................................................................ 87 Position by multipole moments ................................................................................................ 87 Application to the large aspect ratio circular tokamak.............................................................. 89 11. PLASMA SHAPE.................................................................................................................. 92 12. MOMENTS OF PLASMA PRESSURE ............................................................................... 94 The Virial Equation................................................................................................................... 94 m = -1, B2 = Bext ..................................................................................................................... 96 m = 0, B2 = 0 ............................................................................................................................ 97 m = -1, B2 = 0........................................................................................................................... 98 13. βI + li/2 ................................................................................................................................ 100 Solution................................................................................................................................... 100 Separation of βI and li............................................................................................................. 101 Comments on the definition of poloidal beta.......................................................................... 102 14. DIAMAGNETISM .............................................................................................................. 103 Comments ............................................................................................................................... 103 Microscopic picture for a square profile plasma in a cylinder................................................ 103 Macroscopic picture................................................................................................................ 104 Paramagnetic and diamagnetic flux ........................................................................................ 105 Toroidal, non circular geometry.............................................................................................. 106 The meaning of βI................................................................................................................... 107 Measurements ......................................................................................................................... 109 15. FULL EQUILIBRIUM RECONSTRUCTION.................................................................... 113 16. FAST SURFACE RECONSTRUCTION............................................................................ 115 17. FLUCTUATING FIELDS ................................................................................................... 118 (MIRNOV OSCILLATIONS and TURBULENCE) .................................................................. 118 Mirnov Oscillations ................................................................................................................ 118 Analysis techniques................................................................................................................. 124 Turbulence .............................................................................................................................. 127 18. INTERNAL PLASMA MEASUREMENTS....................................................................... 133 Equilibrium ............................................................................................................................. 134 Mirnov Oscillations ................................................................................................................ 138 19 . THE CONDUCTING VACUUM VESSEL....................................................................... 140 Skin depths.............................................................................................................................. 140 Application to a diamagnetic loop .......................................................................................... 142 3
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Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

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