Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

Magnetic fields and tokamak plasmas
Alan Wootton
Figure 17.7. The (m,n) space, with the limits imposed by q 0 = 1
and q a = 3.2. The darker shaded area represents the space
considered in the analytic model described in the text.
The measurements of the fluctuating fields are usually restricted to root means square (rms)
fields, b rrms and b θrms , at r > a, the limiter radius. We would like to know how this is related to
the field components at the resonant surface, so that we can calculate associated island widths.
Typically we want to determine b rmn from a measured b θrms outside r mn . There is no unique
transformation from b θrms to b rmn , so a model must be invoked.
We proceed by assuming the fluctuating self generated field b θrms measured with magnetic
probes at r > a is proportional to the required b rmn at r mn ≈ a. Evidence for such a model comes
from correlation measurements between Langmuir probes and magnetic probes; the measured
b θrms at r > a is apparently determined by plasma current fluctuations at r ≈ a, i.e. at the plasma
edge. Outside the fluctuating current filaments (r > r mn ) the magnetic fields can be approximated
by
b r
=
⎛
r
∑( b rmn
) ⎜
m ,n r
r= r mn
⎝ mn
⎞
⎟
⎠
−(m +1)
cos(mθ + nφ +δ mn
) 17.29
with φ the toroidal angle, θ the poloidal angle, and δ mn a random phase. This ignores toroidal
effects, which introduce terms proportional to (m±1), and is strictly valid only for small n. If a
conducting wall is present at r = b then the expression is modified by the factor [1-(r/b) 2m ]. For
stationary ergodic turbulence the time and spatial averages are the same, so that the root mean
square is found as a spatial average. Outside the singular surface we measure
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