Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

Magnetic fields and tokamak plasmas
Alan Wootton
which reduces to the Laplacian ∆ for uniform permeability. In a current free region we can use
the representation H = ∇g, and ∇ . B = 0 is then equivalent to Lg = 0.
Identities
Now we turn to some identities. Green's first identity for L* is:
1
∫ ψL * ΘdS
µR
φ
=
S φ
Green's second identity (Green's theorem) is:
S φ
∫
∫
l
1
( ψL * Θ −ΘL * ψ)dS φ
=
µR
1
ψ ∂Θ
µR ∂n dl − 1
∫ ∇ψ • ∇ΘdS
µR
φ
8.10
S φ
∫
l
1 ⎛
ψ ∂Θ
µR ∂n − Θ ∂ψ ⎞
dl 8.11
⎝ ∂n ⎠
Both of these are derived by applying the divergence theorem to appropriate expression on V. In
particular see Smythe, static and dynamic electricity, page 53 eqn. 3.06(2) for a derivation of
Green’s theorem, which is, for scalars A, B and E,
[ A∇ •( E∇B)− B∇ • ( E∇A ) ⎡
∫ ] dV = E A ∂B
∂n − B ∂A ⎤
∫ dS
⎣ ∂n ⎦
n
. 8.11b
V
S n
Now let the function G(R,R') satisfy the equation L * G = µR'δ(R-R') in S φ , where G is considered
a function of R at fixed R'. No boundary conditions mean that G is specified to within a constant.
Then we obtain Green's third identity:
ψ ( R' )= − ∫ Gj φ
dS φ
+
S φ
∫
l
1 ⎛
ψ ∂G
µR ∂n − G ∂ψ ⎞
dl 8.12
⎝ ∂n ⎠
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