Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

Magnetic fields and tokamak plasmas
Alan Wootton
resonant, so that we cannot immediately calculate the magnetic island width using Equation 17.1.
Other problems to cope with include how to determine the poloidal (m) mode number from a set
of coils measuring b θ if they are not on a circular contour, so that a simple Fourier analysis is not
possible, and toroidal effects.
If the field perturbation is b, then outside the circular plasma at some radius r > a p we have
∇ •b = ∂b r
∂r + im r b θ
+ in R b φ
= 0 17.2
( ∇ × b)• e r
= im r b − in φ
R b 17.3
θ
Form the second equation we have
b θ
b φ
= mR
nr
17.4
which for low m, n is >>1, i.e. b θ >> b φ . Therefore with Equation 17.2 we determine that the
most important contributions to measure are b r and b θ , not b φ . If we are measuring just inside a
conducting wall (e.g. the vessel) then b r ≈ 0, so that only b θ should be monitored (but note the
conducting wall also affects the value of b θ : it can also affect the instability which produces b θ
itself).
plasma
θ
• coil at r
• filament at r = a
,θ = θ 0
θ
η
r
ξ
Figure 17.1a Figure 17.1b
A common representation of the fluctuations is as current filaments aligned along the field lines.
Initially let us consider the fields produced in a straight cylinder by a current filament (current I)
aligned along the cylinder, located at poloidal angle θ = θ 0 , at r = a (see Figure 17.1). Then.
b θ
= µ 0
I
2π
( r − a cos( θ − θ 0
)
a 2 + r 2 − 2ar cos θ − θ 0
( ( )
17.5
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