Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

Magnetic fields and tokamak plasmas
Alan Wootton
9. MOMENTS OF THE TOROIDAL CURRENT DENSITY
Let Θ be an arbitrary function satisfying the homogeneous equilibrium equation L * Θ = 0 in S φ ,
and let ψ be the poloidal flux function which satisfies L * ψ = -µRj φ . Apply Green's second
identity for the operator L * (Equation 8.11) to the pair (Θ,ψ):we obtain a fundamental integral
equation:
∫ Θj φ
dS φ
=
S φ
1 ⎛
µR ψ ∂Θ
∂n − Θ ∂ψ ⎞
∫ dl 9.1
⎝ ∂n ⎠
l
The moments of the current density (i.e. the integral on the LHS) are expressed in terms of ψ and
∂ψ/∂n on the boundary. From L * Θ = 0 we have ∫ l
R -1 µ -1 (∂Θ/∂n)dl = 0 so there is no dependence
upon the choice of the arbitrary constant in ψ. We introduce together with Θ a conjugate function
ξ according to the equation
⎛
∇⎜
ξ ⎞
⎝ µR⎠ =
∇Θ × ∇φ
µ
9.2
which admits a solution L*Θ = 0. In cylindrical geometry, Equation 9.2 is
∂ ⎛
⎜ ξ ⎞
∂R ⎝ µR⎠ = − 1
µR
∂ ⎛
⎜ ξ ⎞
∂z ⎝ µR⎠ = − 1
µR
∂Θ
∂z
∂Θ
∂R
9.3
9.4
The function ξ satisfies L(R -1 µ -1 ξ) = 0, (remember the operator L = µ −1 ∇ . (µ∇ψ), which reduces
to the Laplacian if µ is uniform). Therefore the definition of ξ implies through Equations 9.3 and
9.4 that −∂(R -1 µ -1 ξ)/∂τ = R -1 µ -1 ∂Θ/∂n, where ∂/∂τ is the partial derivative along l (clockwise on
the outer boundary). Now by partial integration we can eliminate ψ from Equation 9.1 and write
in terms of ∂ψ/∂τ:
∫ Θj φ
dS φ
=
S φ
1 ⎛
µR ξ ∂ψ
∂τ − Θ ∂ψ ⎞
∫ dl = 1 ⎝ ∂n ⎠ ∫ (
µ ξB + ΘB n τ
) dl 9.5
l
l
i.e. the "moments" over the current density can be measured as line integrals of the normal (B n =
(1/R)∂ψ/∂τ) and tangential fields (B τ = -(1/R)∂ψ/∂n) along the contour l.
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