Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

Magnetic fields and tokamak plasmas
Alan Wootton
Energy associated with toroidal fields W 1
B
The energy associated with poloidal currents is written as
W B 1
= L I 2
1 1
2 + L I 2
1e 1e
2
+ M 1
I 1
I 1e
6.0.5
Here I 1 is the poloidal current in the plasma, and I 1e is the poloidal current in the toroidal field
coil (subscript e for external). I 1 is that poloidal current flowing in the plasma edge which
produces a toroidal field equal to the difference between the internal toroidal field B φi and the
external toroidal field B φe . By definition we have
L 1
I 1
2
( ) 2 V
B
2 = φi
− B φe
6.0.6
2µ 0
( B φi
− B φe )B φe
V
M 1
I 1
I 1e
=
6.0.7
µ 0
Now the circuits I 1 and I 1e are perfectly coupled, so that L 1 = M 1. The field B 1 = µ 0 I 1 /(2πR), and
so
B 1
2
M 1
= L 1
= 1 ⎛
2
I 1
∫ dV = µ 0
R − R 2 − a 2
µ ⎝
0
V
( ) 1 2
⎞
⎠ ≈ µ a 2
0
2R
6.0.8
for skin currents. To get the forces we will need only the functional dependencies, namely
∂L 1
∂a = 2 L 1
a
∂M 1
∂a = 2M 1
a
∂L 1
∂R = − L 1
R
∂M 1
∂R = − M 1
R
6.0.9
6.0.10
The forces will be computed at constant current. For example, the part of the force due to
∂/∂R(L 1 I 1 2 /2) is then written as (I 1 2 /2)∂/∂R(L 1 ) = -(I 1 2 /2)L/R = -(L 1 I 1 2 /2)(1/R). Using Equation
6.0.6 this becomes (I 1 2 /2)∂/∂R(L 1 ) = V/(2Rµ 0 ). Doing this for each component in
Equation 6.0.5 gives
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