Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

Magnetic fields and tokamak plasmas
Alan Wootton
where Λ = β I
+ l i
2 −1. In the local coordinate system (ρ, ω, φ) the necessary B z is then derived
from a flux (ignoring the constant of integration)
ψ ext
= µ I 0 pρcos( ω ) ⎡ ⎛
ln 8R g ⎞
4π ⎣ ⎢ ⎝ a ⎠ + Λ − 0. 5 ⎤
⎦ ⎥
6.1.3
ρ
ρ c
ω
ωc
∆
current filament center R
c
geometric center R
g
Figure 6.1.1. The geometry used in relating the geometric and current filament centers
Next we come to the flux ψ p produced by the plasma. Outside the plasma, where there is no
current and no pressure, the fields and fluxes we are looking for must be able to be constructed
from those due to circular filaments. This will not be true inside the plasma. For a first
approximation we will model the flux ψ p as being due to a single circular filament with current
I p . If we position this filament at a position R c then in a coordinate system (ρ c , ω c , φ) based on
the filament we have shown that the flux is well represented by
ψ p
≈ µ 0 I p R c
2π
⎡ ⎛ ⎛
ln⎜
8R c
⎞ ⎞ ⎛
⎜ ⎟ − 2 ⎟ − ⎜
⎢
⎣ ⎝ ⎝ ρ c
⎠ ⎠ ⎝
ρ c
R c
( )
⎞
⎟ cos ω c
⎠ 2
⎛ ⎛
ln⎜
8R c
⎞ ⎞ ⎤
⎜ ⎟ −1 ⎟
⎝ ⎝ ρ c
⎠ ⎠
⎥
⎦
6.1.4
However, we have to decide where to place this filament, that is, what to choose for R c , and how
ρ, ρ c , ω and ω c are related. Because we have derived ψ ext for a circular cross section (in the
calculation of B z we used inductances for circular current path), we must place the filament so
that, in the coordinate system (ρ, ω, φ) the surface ψ total = ψ ext + ψ p = constant is a circle of
radius at ρ = a.
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