Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

Magnetic fields and tokamak plasmas
Alan Wootton
where q est,i is the value expected for q i when Equation 15.1 holds. q est
0 ,i is associated with ψ o ,
and the known Q ij is associated with χ j . Then the usual least squares approach will determine the
coefficients c j to minimize the function
M
( q i
− q est ,i ) 2
∑ 16.3
i =1
σ i
2
With σ i the standard error of the i th measurement. This procedure may not be stable, in which
case some numerical damping is added
One technique which avoids the iterations necessary to match the measured and predicted field
components is as follows. From the measured fields construct the multipole moments Y m using
the techniques outlined in sections 9 and 10. Then m toroidal filaments with current (I p /m) can
be positioned to give the same moments Y m as those measured. Because we have analytic forms
for the Y m produced by discrete current filaments (the integral Y m = ∫j φ f m dS φ only takes a finite
value at the filament location) we can derive analytic expressions for the filament positions in
terms of the measured Y m 's, thus avoiding the need for the iterative procedure. Just as discussed
above, we then use toroidal filaments with known currents for external windings, the m filaments
for the plasma, and plot the flux contours immediately. Solutions up to the plasma boundary are
as exact as our set of moments allows. I am not sure how unique the solution is, or to what extent
I should consider taking more than m filaments. We have still to ask if our solution is unique:
that is, do the m moments uniquely specify the fields on the contour l
An example of such a procedure is shown in Figure 16.1a. Figure 16.1b shows a full
equilibrium reconstruction with j φ (r) iterated until a good fit between measured and computed
moments was obtained. Clearly the 3 filament approximation, with the filaments chosen to give
the measured moments Y 1 ,Y 2 and Y 3 , gives a good description of the outer surface.
In principle we should be able to extend the "moments with filaments" method of finding the
plasma shape to the use of an analytic representation for the current density. Indeed, we did this
in section 11 to find a relationship between the second moment Y 2 and ellipticity. However,
there we made an arbitrary choice that the current density be flat. In fact we should specify that j φ
satisfy the Grad Shafranov equation: this problem is considerably more complicated. However, if
solved, we should be able to obtain analytic relationships between the measured moments and the
plasma shape. We would still have to parameterize the form for j φ : this would be restricted by,
for example, a knowledge of the ratio of q on axis to q at the edge.
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