Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

Magnetic fields and tokamak plasmas
Alan Wootton
j z
= 1 R
∂f
∂R
6.5
Now Amperes law µ 0 j = ∇ x B gives
j R
= − 1 µ 0
∂B φ
∂z
j z
= 1
µ 0
R
∂( RB φ )
∂R
6.6
6.7
so that, comparing Equation 6.4 with Equation 6.6, and Equation 6.5 with Equation 6.7, we have
µ 0
f = RB φ
6.8
That is, the function f includes the total current in the windings producing the toroidal field.
Since from Equation 6.1. j . ∇p = 0, using Equations 6.4 and 6.5 for j gives (∂f/∂R)(∂p/∂z) -
(∂f/∂z)(∂p/∂R) = 0, or
∇f × ∇p = 0 6.9
Since p is a function of ψ, i.e. p = p(ψ), we must have f = f(ψ) as well.
Now we want to derive the basic equilibrium equation in terms of ψ. Write Equation 6.1 as
j p
× e φ
B φ
+ j φ
e φ
× B p
= ∇p 6.10
where subscript p means poloidal and e φ is a unit vector in the φ direction. Now Equations 6.2.
and 6.3 can be written as
B p
= 1 (
R ∇ψ × e φ) 6.11
and Equations 6.4 and 6.5 can be written as
j p
= 1 (
R ∇f × e φ) 6.12
Substituting 6.11. and 6.12 into 6.10, (remember e φ . ∇ψ = e φ . ∇f = 0) gives
− B φ
R ∇f + j φ
∇ψ = ∇p 6.13
R
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