Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

Magnetic fields and tokamak plasmas
Alan Wootton
R T
=
∫
S φ
TRdS φ
∫ TdS φ
S φ
12.19
L pt
= 8π
µ 0
I p
2
2
B p1
V
∫ dV 12.20
2µ 0
β I
= 8π
µ 0
I p
2
pdS φ
S φ
∫ 12.21
µ I
= − 8π
µ 0
I p
2
B 2 2
⎛
⎜ φ
− B φe ⎞
∫
⎟ dS
⎝ 2µ 0
⎠
φ
12.22
S φ
Note that µ 0 L pt /(4π) is a total inductance, and the usual "inductance per unit length times 4π/µ 0 "
is given as
B p
2
4
L p
=
2
µ 0
R ch
I p V
∫ dV 12.23
2µ 0
Now we can write Equation 12.16 as
B ⊥
= µ I 0 p
⎡
β I
+ µ I
+ ∂
8πR ch ⎣
⎢
∂R ch
⎤
( R ch
L p )
⎦
⎥
12.24
This is a generalization of the external vertical field needed to maintain a plasma, allowing for
non circularity.
m = 0, B 2 = 0
Now Equation 12.14 gives an integral expression along the minor radius:
V
∫
( )
∫
PdV + R − R T ch
2R T
( )
V
TdV
B 2 τ1
= 0.5
− B 2
⎡
n1
n •e ρ
− B B ⎤
τ 1 n1
∫ τ •e ρ
ρdS
⎢
⎣ 2µ 0
µ ⎥ n
− 0.5∫ j φ [ B z2 ( R − R ch )− B R2
z]dV
0 ⎦
V
S n
12.25
97