Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

Magnetic fields and tokamak plasmas
Alan Wootton
11. PLASMA SHAPE
Using higher order moments we can obtain information on the plasma shape. Y 2 determines
ellipticity and Y 3 determines triangularity. Using equations 9.7 and 10.2, we obtain:
where
⎛
Y 2
= 1 − 2∆ ⎞
R
⎜ ⎟ s τ n
3
+ s 4
τ
⎝ ⎠ s 0
R l
( )
⎛
+ ∆ 2 z
− 1 − ∆ ⎞
R 2
⎜ ⎟ ∆ R
⎝ ⎠
R l
11.1
⎡ ⎛
s τ 3
= ξ 2 1 + ξ ⎞ ⎛
⎜ ⎟ −η 2
1+ 2ξ ⎞ ⎤
∫ ⎢
⎜ ⎟ ⎥ B
⎣ ⎝ R l ⎠ ⎝ R
τ
dl
l
l ⎠ ⎦
11.2
⎡ ⎛
s n 4
= 2ξη 1 + 3ξ ⎞ ⎤
∫ ⎢ ⎜ − η2
⎟ ⎥ B
⎣ ⎝ 2R l
3R l
ξ
n
dl
l
⎠ ⎦
11.3
s τ 0
= ∫ B τ
dl = µ 0
I p
11.4
l
That is, with the Rogowski coil measuring I p (i.e. s 0 τ ) and either modified Rogowski and saddle
coils, or single point measurements of B n and B τ suitably combined, we can construct Y 2 . If we
want to use modified Rogowski and saddle coils, then to obtain I p , ∆ R and Y 2 takes a total of 5
coils. For a circular contour, and ignoring toroidal effects, Equation 11.1 is written as
Y 2
= −∆ 2 R
+ ∆ 2 z
+ a 2
l
(
2 λ + µ 2 2) 11.5
That is, neglecting toroidal effects we need only λ 2 and µ 2 , in addition to ∆ R and I p .
To interpret the moments it is necessary to assume a plasma current distribution; because the
moment is an integral of the current density over the surface S φ there is no unique solution for the
boundary shape. As an example, consider a uniform current density and a surface described by an
2
⎛ x
ellipse with minor and major half width and half height a and b, so that
⎞
⎝ a⎠
+ z 2
⎛ ⎞
=1. Then
⎝ b⎠
for k = b −1, k