Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

Magnetic fields and tokamak plasmas
Alan Wootton
21. TOKAMAK POSITION CONTROL
The axisymmetric instability
We first calculate the growth rate of an axisymmetric instability in a tokamak. We only consider
vertical motion, because it is easier than the calculation for horizontal motion. The driving force
is written in terms of the decay index n = -(R/B z )(dB z /dR). We consider a tokamak surrounded
by a conducting vacuum vessel. There is assumed to be a transverse (poloidal) insulating gap in
this vessel, so that only dipole currents can flow. The equation determining the vacuum vessel
current I s is
d
dt
( L s
I s
)+ d dt
( M sp
I p )+ Ω s
I s= 0
21.1
Here I p is the plasma current, L s is the vessel inductance, Ω s the vessel resistance, M sp the
mutual inductance between plasma and vessel. We introduce the vessel time constant τ s = L s /Ω s.
We can approximate the vessel as a circular shell, so that
L s
= µ 0π 2 R s
4
τ s
= µ 0σδ s
r s
2
21.2
21.3
where R s , r s , δ s are the vessel major radius, minor radius, thickness, and σ is the conductivity.
Approximating the plasma as a filament initially centered within the vessel (R = R 0 = R s , z = z s =
0), the mutual inductance and its spatial derivative are given by
M sp
= µ 0πR s
z
2r s
21.4
∂M sp
∂z
= µ 0πR s
2r s
21.5
From equations 22.1, 22.2 and 2.5 we can derive the relationship between the dipole current in
the vessel and the plasma displacement z:
dI s
dt + I s
τ s
= −2 I p
πr s
dz
dt
21.6
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