Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

Magnetic fields and tokamak plasmas
Alan Wootton
Uses of the Volts per turn measurement
We can deduce an average value of the plasma conductivity, , by writing
2 2
2πR p
I p
πa 2 p
σ = j φ
∫ dV = I p
ε − ∂ 2
⎛ L i
I
⎜ p ⎞
⎟ 5.18
V
σ ||
∂t ⎝ 2 ⎠
From this we can define a conductivity temperature T σ . The conductivity deduced by Spitzer for
Coulomb collisions is given by (there are corrections for the fact that, in a torus, trapped particles
cannot carry current and so σ must be reduced)
σ = 1.9 ×10 4 T e
3 2
( )
Z eff
ln Λ s
5.19
Then T σ is defined as that temperature which gives a Spitzer conductivity (with Z eff = 1) equal to
the average conductivity , with an approximate value taken for ln(Λ s ). We can also derive an
average “skin time”, from the formula for the penetration of a field into a conductor of uniform
conductivity :
τ skin
= πµ 0σa p
2
16
5.20
The definition of energy confinement time for an ohmic heated plasma with major radius R p ,
cross sectional area S φ .(here we assume a circular minor radius a p, so S φ = πa p 2 ), total energy
content W = 3πR p ∫ S φ pdS φ, ohmic input power P oh = I p 2 Ω p , can we written in terms of the
poloidal beta value β I = 8π/(µ 0 I p 2 )∫ S φ pdS φ (discussed later) as
τ E
= W = 3µ 0β I
R p
= 3µ 0β I
a 2 p
σ
τ 5.21
P oh
8Ω p
16
Combining equations 5.20 and 5.21 shows that
τ E
τ skin
≈ β I
5.22
Therefore for ohmic heated plasmas, where typically β I ≈ 0.3, the currents penetrate
approximately 3 times slower than the energy escapes from the plasma.
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