Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

Magnetic fields and tokamak plasmas
Alan Wootton
Suppose we place the filament at a position R 0 = R c + ∆. From Figure 6.1.1, and expanding in
the small parameter ∆/r, we then derive that
ρ 2 c
= ρ 2 + ∆ 2 − 2∆ρ cos( 180 − ω ) 6.1.5
ρ 2 = ρ 2 c
+ ∆ 2 − 2∆ρ c
cos( ω c
) 6.1.6
from which we have
and
⎛
ρ c
≈ ρ⎜
1 + ∆
⎝ ρ cos( ω ) ⎞
⎟ 6.1.7
⎠
cos ω c
( ) ≈ cos ω
⎛
( ) 1− ∆
⎝ r cos( ω)
⎞
⎠ + ∆
r
6.1.8
Substituting for ρ c and ω c in terms of ρ and ω, and using R 0 = R c + ∆, we obtain an expression
for the total flux ψ total (ρ, ω, φ) in the coordinate system based on the geometric center. Keeping
terms of order ∆/r and cos(ω) only (i.e. neglecting elliptic distortions to any surface) we find
ψ total
= µ 0 I p R g
2π
⎡ ⎛
ln⎜
8R g ⎞ ⎤
⎟ − 2
⎣
⎢ ⎝ ρ ⎠ ⎦
⎥ + µ I pρ cos ( ω ) ⎡
0 ⎛
ln ρ ⎞
4π ⎝ a⎠ + Λ + 1 2 − 2 ∆R g
⎤
⎣ ⎢
ρ 2
⎦ ⎥ 6.1.9
To ensure a circular outer contour (at r = a) we set the cos(ω) term to zero, that is we set
∆
a =
a ⎛
Λ + 1 ⎞
2R ⎝
g
2⎠
6.1.10
Substituting for ∆ form Equation 6.1.10 into Equation 6.1.9 gives the final expression for the flux
outside a circular tokamak:
ψ total
= µ 0 I p R g
2π
⎡ ⎛
ln⎜
8R g ⎞ ⎤
⎟ − 2
⎣
⎢ ⎝ ρ ⎠ ⎦
⎥ + µ I pρ cos ( ω )
0 ⎛
ln ρ ⎞
4π ⎝ a⎠ + ⎛
Λ + 1 ⎞
⎝ 2⎠ 1− a 2
⎡
⎛ ⎛
⎜ ⎞ ⎞ ⎤
⎟
⎢
⎜ ⎟
⎝ ⎝ ρ⎠
⎥
⎣
⎠ ⎦
6.1.11
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