Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

Magnetic fields and tokamak plasmas
Alan Wootton
ε 1
= dN 11
dt
= L 11
dI 1
dt
1.33
Poloidal flux
Suppose that a system consists only of toroidally wound loops producing only poloidal fields. In
a cylindrical coordinate system R,φ,z shown in Figure 1.7 (φ is also the ‘toroidal' angle in a quasi
cylindrical coordinate system) nothing depends on the angle φ. Then
M 12
= 2πR 1
A 2
I 2
1.34
and A has only a toroidal component A φ . In this case the fields are given by (B =∇xA):
B R
= − ∂A φ
∂z
B φ
= 0
B z
= 1 R
∂( RA φ )
∂R
These poloidal fields are also expressed in terms of the transverse (poloidal) flux function
Ψ: Ψ(R,z) = constant defines the form of the equilibrium magnetic surfaces, proved later:
1.35
B = 1 (
2πR ∇Ψ × e φ ) 1.36
with e φ a unit vector in the toroidal (φ) direction, so that
B z
= 1 ∂Ψ
2πR ∂R
B R
= − 1 ∂Ψ
2πR ∂z
1.37
But we know that we can write, in terms of the vector potential for our toroidally symmetric
system (∂/∂φ = 0),
B R
= − ∂A φ
∂z
B z
= 1 R
( )
∂R
∂ RA φ
1.38
That is, the poloidal flux can be written as (with subscripts implied but not given):
18