Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

Magnetic fields and tokamak plasmas
Alan Wootton
Substituting this form for ψ 1 (= -∆ 1 (r)cos(θ)∂ψ 0 /∂r) into the first order equation, Equation 6.21,
gives:
d
dr
⎛
⎝
2 d∆
rB 1
θ0
dr
⎞
⎠ =
r ⎛
2µ 0 r dp 0
R ⎝ g dr
2
− B θ0
⎞
⎠
6.23
where B θ 0
= 1 R g
dψ 0
dr
has been used.
The solution of this equation gives the horizontal shift ∆ 1 of a flux surface away from the
geometric axis, as illustrated in Figure 6.4. This, together with the solution of Equation. 6.22,
gives us what we want. If we wanted to consider non circular surfaces, we could write
∞
r = r 0
+ ∑ ∆ n
( r)cos( nθ)
for each surface. Then ψ(r) is a constant (by definition), so that
n=1
∞
∑
( )
ψ = ψ 0
− ∆ n
( r)cos nθ
n =1
dψ 0
dr .
Figure 6.4. Displaced circular flux surfaces.
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