Control-matrix approach to stellarator design and control Page: 3 of 35
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the CM is simply the matrix Czj a 0Pz/&Zj of first derivatives at any point Zo in
Z space, and so can give topographical information only locally. One may provide
more information by expanding to second and higher orders, and most globally,
one may investigate the full nonlinear variation over ranges where a power-series
expansion is not practical. But even to compute the derivatives needed for Czj
correctly, one needs to know the characteristic scales on which the the Pi vary
in Z, so more global knowledge is important even for the local problem. And as
already indicated, the more global information may also be important in locating
genuinely different stellarator optima. Thus, this paper addresses both the more
local and more global sides of this topographical exploration.
In Sec. II we describe the mathematical basics of the CM method, and discuss
the means we use to reduce the dimensionality of Z space. Implementing the CM
procedure requires exploring the scales of variation of the Pi in Z in the vicinity
of a design point Zo, for which we choose "C10", a candidate NCSX configura-
tion. This is done in Sec. III. Our topographical study here finds that within an
appreciable domain (variations in the Z of order 1 cm) about C10, the P may
be well approximated by simple quadratic expressions, and in addition, we are
able to reduce the dimensionality of the Z space we need consider from an initial
N2 = 78 to 8. As a result, in Sec. IV we apply the machinery of the local CM
analysis to a greatly reduced parameter space, and within that space can compute
quantities of interest using analytically tractable quadratic expressions for the Pi.
We then provide the 'proof of principle' of the CM method, demonstrating that
the CM mathematics correctly produces perturbations ' with which we can in-
dependently vary the P2, and 'nullspace' perturbations v2 which produce different
configurations, but with unchanged values of the Pi. We discuss some of the fea-3
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Mynick, H.E. & Pomphrey, N. Control-matrix approach to stellarator design and control, report, February 9, 2000; Princeton, New Jersey. (https://digital.library.unt.edu/ark:/67531/metadc702053/m1/3/: accessed April 25, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.