Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

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Magnetic fields and tokamak plasmas Alan Wootton i.e. the magnetic surfaces are described by circles with ρ = constant. Far away from the loop we have shown that the field looks like that due to a dipole with moment M: M = πR 2 0 I φ . 1.51 Then the equation for the magnetic surfaces becomes MR 2 ψ = ( R 2 + z 2 ) 3 2 = constant 1.52 Now we add a uniform field B z0 in the z direction. This has a vector potential given by A φ0 = B z 0 R 2 1.53 The magnetic surfaces are now given by R( A φ + A φ 0 )= constant 1.54 Finally we add a filament up the z axis, which produces a field ∝ 1/R in the φ direction (a ‘toroidal’ field. The vector potential due to this filament is A z = − µ I 0 z 2π ln( R) 1.55 Because this has only a z component, it does not affect the result (Equation 1.54). The results are shown in Figure 1.14 and Figure 1.15 for a vertical field (the form used is relevant to that required to maintain a circular tokamak in equilibrium) B z = ±µ I 0 φ ⎛ ⎛ ln 8R 0 ⎞ 4πR ⎝ 0 a ⎠ + Λ − 1 ⎞ ⎝ 2⎠ 1.56 with Λ = 2, R 0 = 1 m, and a = 0.2 m. In the figures the exact form for A φ from the circular current filament (written in terms of elliptic integrals) is actually used. The positive current I φ produces a field downwards (in the -z direction) at the outer equator (z = 0, R > 1 m). Adding a positive vertical field B z cancels this field at some point, producing a point where B z (z=0) = 0. This is called an X point; the flux surface through this point is called the separatrix. Note that, with the negative uniform vertical field applied, there is no inner X point. However, if a vertical 24
Magnetic fields and tokamak plasmas Alan Wootton field ∝ 1/R was applied the more negative B z at the inner equator would cancel the positive B z from the filament, and an X point would appear z (m) z (m) 0.1 0.2 0.61 0.6 0.3 0.4 0.5 0.61 0.3 0.2 0.15 0.1 0 -0.1 -0.2 major radius R (m) major radius R (m) Figure 1.14. Contours of constant flux ψ, Figure 1.15. Contours of constant flux ψ, with positive B z . with negative B z . x point P r1 r2 r • + 1 dipole current r4 • − dipole current 4 θ •+ "plasma" r3 d dipole current • 2 − y + • dipole current 3 Figure 1.16. The geometry used for calculating the flux surfaces of a straight filament in a quadrupole field. z is into the plane of the figure. 25
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Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

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