Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

Magnetic fields and tokamak plasmas
Alan Wootton
s 1
= 1
s 2
= R l
( λ 1
+ µ 1 )
a l
13.4
i.e., substituting 13.4 into 13.1:
β I
+ L p
2 = 1+ R l
( λ
a
1
+ µ 1
) 13.5
l
Let us apply this equation to the displaced analytic equilibrium discussed in sections 6 and 9.
The Rogowski coil with cosinusoidal varying winding density, and the saddle coil with sinusoidal
varying width, tell us λ 1 and µ 1 . First there is a mess with right and left handed coordinate
systems to unravel. Doing this then µ ⇒ -µ in Equation 13.5. Now Equation. 10.16 already tells
us that
λ 1
− µ 1
= a l
R l
⎡ ⎛
ln a ⎞
⎜
l
⎟ + β I
+ l i
⎝ a p ⎠ 2 −1
⎤
⎢
⎥
⎣
⎦
13.6
Therefore comparing Equations 13.5 and 13.6 we must have (allowing µ ⇒ -µ in Equation 13.5)
that
l i
2 + ln ⎛ a ⎞
⎜
l
⎟ = L p
⎝ ⎠ 2
a p
13.7
This is exactly what we expect, because the total inductance to a radius a l is given by:
⎛
L total
= 2πR µ 0 ⎞
⎝ 4π ⎠ L ⎡ ⎛
p
= µ 0
R ln a ⎞
⎜
l
⎟ + l ⎤
i
⎢
⎣ ⎝ a p ⎠ 2 ⎥
⎦
13.8
Separation of β I and l i
If we have β I + l i /2 from the poloidal field measurements just described, and β I from diamagnetic
measurements (see later) then obviously we can separate β I and l i
If no diamagnetic
measurement is available, two possibilities exist. For non circular plasmas, there is a third
integral relationship, which I have not derived, which gives in terms of a measurable line integral
the parameter ∫ V
(2p+B z 2 /µ 0 )dV. When V is the plasma volume, this is related to 2β I + L p . If the
volume averages and are different, as is the case for non circular discharges, then
this measurement allows the separation.
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