Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

Magnetic fields and tokamak plasmas
β I
= 2µ p ( 0)
0 0
2
( 1+ γ )B θ 0
( 1)
Alan Wootton
6.42
Now we can use the normalization x = r/a in equation 6.26 to obtain an expression for the surface
displacement:
d∆ 1
dx = − εa
2
⎡
β I
( γ +1)x 2 1− x 2
⎢
⎣
2x
( 1 −( 1− x 2
) 1+ α
) 2
x
( ) − ( 1 − ( 1 − x 2
) 1+α
) 2 dx
∫
( ) γ − β I
1 − ( 1 − x 2
) γ +1
0
x
⎤
⎥
⎦
6.43
Where ε = a/R g . The Shafranov Shift is defined as the distance between the magnetic and
geometric axis: ∆ s = ∆ 1 (1). Using a power series expansion for ∆ 1 (x) up to terms x 6 a general
expression can be derived:
∆ s
= εa
2
⎡
1 + 2α 9 + α 2
⎢ 72
+
⎢ 4
⎢ β I
γ ( γ +1)
⎧
2 + 11α
4( 1 + α) 2 9 + 5α 2
⎨
18 + γ − 1
⎣ ⎢ ⎩
6
( )( γ − 6)
⎤
⎥
⎥
4α γ −1
− ( ) ⎫ ⎥
⎬
9 ⎭ ⎦ ⎥
6.44
For the simple case of a flat current profile (α = 0, l i = 0.5) and a parabolic pressure profile (γ =
1) we obtain
∆ s
= εa ⎛
2 β + l i ⎞
⎝
I
2⎠
6.45
Matching vacuum and plasma solutions
Returning to the field outside the plasma, we must match the vacuum field to the solution for
B θ (a). The vacuum field is given by the solution to (∇xB) φ = 0. Expressing B R and B z in terms
of ψ (Equations 6.2 and 6.3), (∇xB) φ has the form of the LHS of Equation 6.18. This has a
solution of the form (for r