Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

Magnetic fields and tokamak plasmas
Alan Wootton
Therefore if we can measure the difference in toroidal field with and without (when B φi = B φe )
plasma present, we can measure the average plasma pressure. This is discussed later in dealing
with “diamagnetism”.
The condition F R = 0 specifies the vertical field:
B z
= − aB θa
2R
= − aB θa
2R
⎡
ln ⎛ 8R
⎣
⎢ ⎝ a
⎡
ln ⎛ 8R
⎣ ⎝ a
⎞
⎠ − 3 2 + l i
⎞
⎠ − 3 2 + l i
2 + 2µ 0
p
2
2 + β ⎤
I
⎦
B θa
⎤
⎦
⎥
- 6.0.26
This was the field we used in section 1, Field lines and flux surfaces, to plot out the flux surfaces
which result from a combined circular filament and a vertical field.
6.1. THE FLUX OUTSIDE A CIRCULAR TOKAMAK
Later we will use the expression for the flux outside a circular tokamak. It can be considered to
come from two sources, that from the external maintaining fields ψ ext and that from the plasma
itself, ψ p . In the previous section we derived an expression for the vertical field B z necessary to
maintain a circular equilibrium (Equation 6.0.26). While the major radial term appearing as
⎛
ln 8R ⎞
in Equation 6.0.26 clearly refers to the geometric center R
⎝ a ⎠
g (it comes from the
inductance of a plasma with radius a with a geometric center R g ), it is not obvious to what radius
the term outside the square brackets refers. It could be either R g , or the coordinate itself, so that
B z ∝ 1/R. In the former case, in a right-handed cylindrical coordinate system (R,φ,z), the flux
would be derived from ψ ∝ R 2 , or in our local coordinate system (ρ,ω,φ) based on R g (See
Figure 1.7)
ψ = k 0
+ k 1
cos( ω) + k 2
cos( 2ω ) 6.1.1
with k i a constant. The constant is unimportant, but the cos(2ω) term means that such an external
field would introduce ellipticity, and we have specifically considered a circular plasma.
Therefore we must take B z ∝ 1/R, and in the coordinate system (R,φ,z) this is derived from a flux
ψ ext
= µ 0 I p R
4π
⎡ ⎛
ln 8R g ⎞
⎣ ⎢ ⎝ a ⎠ + Λ − 0.5 ⎤
⎦ ⎥
6.1.2
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