Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

Magnetic fields and tokamak plasmas
Alan Wootton
Field representation
We shall try and be as general as possible. First we quickly re-derive the equilibrium equation.
For the case of axisymmetry, we discussed in section 6 the description of the fields using the
form
where
Ψ
2π = µ R I ⎛ ⎛
0 l p
ln 8R ⎞ ⎞
l
⎜ ⎜ ⎟ − 2⎟ − µ ⎡
0ρI p
1− a 2
⎛ ⎞
p
⎜
2π ⎝ ⎝ ρ ⎠ ⎠ 4π ⎝ ρ 2
⎠
⎟ ⎛
Λ + 1 ⎞ ⎛
⎜
⎝ 2
⎟ + ln
ρ ⎞
⎜ ⎟ − 2R ∆ ⎤
l g
⎢
⎥
⎣ ⎢
⎠ ⎝ a p ⎠ ρ 2
⎦ ⎥ cos( ω)
8.3
F = RB φ
= µf 8.4
and ψ= RA φ
From ∇xH = j, we can also write
j = − 1 ⎛
µ L* ψ∇φ + ∇⎜ F ⎞
× ∇φ 8.5
⎝ µ ⎠
where the operator L * is defined for axisymmetric scalar fields as
⎛
L * ψ = µR 2 ∇ • ⎜ ∇ψ ⎞
⎝ µR 2 ⎠
8.6
The operator L * satisfies the equation
L * ψ = −µRj φ
8.7.
For uniform permeability µ = µ 0 we have L * reducing to the operator ∆ * :
⎛
∆ * ψ = R 2 ∇ • ∇ψ ⎞
⎝ R 2 ⎠ = R ∂ ⎛ 1 ∂ψ ⎞
∂R ⎝ R ∂R ⎠ + ∂2 ψ
∂z ∆ 8.8
2
That is, ∆ * is the operator on the LHS of the Grad Shafranov equation. We also introduce the
operator L:
Lψ = 1 µ ∇ • ( µ∇ψ)
8.9
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