Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

Magnetic fields and tokamak plasmas
Alan Wootton
G 0
( R, R' ) = µ 0
kπ
⎡ ⎛
RR' E( k2)− ⎜ 1 − k2 ⎞
⎣
⎢ ⎝ 2 ⎠ K k2
( )
⎤
⎦
⎥
8.16
where
k 2 =
4RR'
( R + R' ) 2 + ( z − z' ) 2
( )
8.17
Ideal MHD
Here we want to note only one important equation, which is a generalization of the “virial”
equation. We use Equations 6.1, 1.1 and 1.2. Allow the total equilibrium field to be split up into
two parts,
B = B 1
+ B 2
8.18
with ∇xB 2 = 0. We can choose the partitioning of B in a number of ways. Multiplying jxB =∇p
by an arbitrary vector Q, we can obtain
∫
V
⎡ ⎛
⎜
⎣
⎢ ⎝
p + B 2
1
⎞
⎟ ∇• Q − B 1
• ∇Q • B 1
2µ 0
⎠
µ 0
⎤
dV
⎦
⎥
⎡ ⎛
= p + B 2
⎜ 1
⎞
⎟
(
( Q • n) − B 1
• Q)• ( B 1
• n)
⎤
∫
dS
⎣
⎢ ⎝ 2µ 0
⎠
µ 0 ⎦
⎥ n
−
∫
Q • ( j × B 2 )dV
V
S n
8.19
with n the normal to the surface S n . We have made use of the vector identity
⎡
Q[ ∇ × B • B]= ∇ • ( Q • B)B − B2
⎣
⎢ 2 Q ⎤
⎦
⎥ + B2
2 ∇ • Q − B( B • ∇)Q
We shall use equation 8.19 later to derive important integral relationships.
Boundary conditions
Last in this section we turn to boundary conditions. Suppose we have our three regions, as in
Figure 8.2. Region 1 (S φplasma ) contains all the plasma current. Region 2 (S φvacuum ) contains no
source, but contains a contour l on which we make measurements. Region 3 (S φcoils ) is outside
region 2, extends to infinity, and contains all external currents. To find the plasma boundary we
have to know ψ(R,z) in region 2 between the contour on which parameters are measured, and the
plasma boundary itself. In principle we can do this knowing either
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