Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

Magnetic fields and tokamak plasmas
Alan Wootton
17.33
This expression can be further simplified for ρ > 1 by noting m 2 >> 1 (so all terms involving m 2
disappear) and m 1 > 1 (so the term involving q a
(m1+1) disappears) to give
b rms
b rmn
=
1
( ) q a
ln ρ
m1 +1
2ρ
( )
17.34
This is the required result, giving us b rmn at the plasma edge in terms of the measured b rms
outside the plasma. Because the terms involving m 2 have been ignored in deriving Equation
17.34 it predicts (b rms /|b rmn |) ρ⇒1
⇒ ∞. The more complete expression, remains finite, and
⎛
⎜
⎝
b rms
b rmn
⎞
⎟
⎠
r= a
≈
( m 2
− m 1 )
17.35
2q a
We conclude that the measured rms field outside the plasma (ρ > 1) is determined only by the
lowest m number (m 1 ); the higher m numbers fall off too rapidly to contribute. Indeed, if
Equation 17.34 is taken to represent the results, then a fit of the form b rms ∝ r -γ gives the result
γ = m 1
+ 1+
1
2 ln ρ ( )
17.36
We can also measure the frequency of the fluctuations. We assume that the observed frequency
is described by an expression of the form
f = ω 2π = m ⎛
κ T e
∂n e
2πr mn
B φ
e n e
∂r − E ⎞
⎜
r⎟ 17.37
⎝
⎠
This is consistent with experimental results, with κ a factor of order 1. We also know
experimentally that the term involving E r dominates the term involving κ, and that we can
approximate the expression by
f ≈ −cmE r
2πr mn
B φ
17.38
with χ a factor ≈ 1.5. E r is approximately constant for ρ < 1. We now relate r mn to q = m/n by
assuming a specific q profile. For the simple q profile used in deriving the analytic results the
transformation from r mn to m and n is given by Equation 17.31. A more realistic q profile gives
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