Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

Magnetic fields and tokamak plasmas
Alan Wootton
2 µ ∆Φ = πa p2
∆B φ
= πa 0
p
2πR I = πa 2 µ 0
p
2πR 2πRj s
2 µ
= πa 0
p
2πR 2πR p ⊥
p
= πa 2 p
µ ⊥
0
B φ
B φ
Using the definition of β I , discussed much more a little later, we then have
∆Φ = µ 2 0
I 2 p
β I
8πB φ
Macroscopic picture
Let us consider a toroidal device with no toroidal current plasma current, i.e. a stellarator, in
which the necessary rotational transform is produced only by external conductors. Starting with
the radial pressure balance, with p ⊥ = 0 at the plasma edge, and approximating the torus by a long
cylinder, then
dp ⊥
dr = j θ B φ
14.5
integrating over the minor radius (r = 0 to a p ) gives
and
a p
p ⊥
= −B φ ∫ j θ
( r' )dr'
14.6
r
p ⊥
= 1
πa p
2
= − B φ
πa p
2
a p
a p
∫
0
2πp ⊥
rdr = − B φ
πa p
2
a p
∫ 2πrdr∫
j θ
( r' ) dr'
∫ π r S( r)
2 j θ
( r)dr = −B φ ∫ j θ
( r)dr
0
a p
0
0
πa p
2
a p
r
14.7
Then j se = ∫ 0
ap
S(r)/(πap 2 )j θ (r)dr is the effective surface current density at the plasma edge as a
consequence of the finite plasma pressure.
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