Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

Magnetic fields and tokamak plasmas
Alan Wootton
contour l
vacuum region 2
plasma
region 1
τ
n
coil region 3
Figure 8.2. The boundaries between the plasma (S φplasma ) region 1, vacuum
(S φvacuum ) region 2 and coil (S φcoil ) region 3.
Suppose S φ can be split up into three regions, S φplasma , S φvacuum and S φcoils , as shown in Figure
8.2. Assume µ = µ 0 in the plasma and vacuum region. The exterior region (the complement of
S φ in the right half plane) is called S φext . Then if this external region has only linear magnetic
material, we can apply the last equation on the region S φ +S φext . Choosing the Greens function
G 0 so that G 0 (R,R') = 0 as |R| goes to infinity, and as R goes to 0, we have
ψ ( R' )= − ∫ G 0
j φ
dS φ
8.13
S φ + S φext
i.e. G 0 (R,R') equals the flux at R' caused by a negative current at position R. Therefore we define
ψ int
ψ ext
( R' )= − ∫ G 0
j φ
dS φ
8.14
S φ
1 ⎛
( R' ) = ∫ ψ ∂G 0
µR ∂n − G ∂ψ ⎞
0
dl 8.15
⎝
∂n ⎠
l
and ψ = ψ ext + ψ int from Equation 8.12. We understand ψ ext as the part of the flux caused by
currents in the exterior region, and ψ int is that part of the flux associated due to currents in S φ .
ψ int is homogeneous in the exterior region, and ψ ext is homogeneous in the interior region.
An analytic expression for G 0 if µ = µ 0 everywhere is:
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