Marginal Stability Diagrams for Infinite-n Ballooning Modes in Quasi-symmetric Stellarators Page: 3 of 11
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Marginal stability diagrams for infinite-n ballooning modes in
quasi-symmetric stellarators
S.R.Hudson', C.C.Hegna2, R.Torasso3 & A.Ware4
1) Princeton Plasma Physics Laboratory, PO Box 451, Princeton NJ 08543, USA.
2) Department of Physics, University of Wisconsin-Madison, Madison WI 53706, USA.
3) Courant Institute of Mathematical Sciences, New York University, NY 10012, USA.
4) Department of Physics and Astronomy, The University of Montana, MT 59812, USA.
Abstract
By perturbing the pressure and rotational-transform profiles at a selected sur-
face in a given equilibrium, and by inducing a coordinate variation such that the
perturbed state is in equilibrium, a family of magnetohydrodynamic equilibria local
to the surface and parameterized by the pressure gradient and shear is constructed
for arbitrary stellarator geometry. The geometry of the surface is not changed. The
perturbed equilibria are analyzed for infinite-n ballooning stability and marginal
stability diagrams are constructed that are analogous to the (s, a) diagrams con-
structed for axi-symmetric configurations.
The method describes how pressure and rotational-transform gradients influence
the local shear, which in turn influences the ballooning stability. Stability diagrams
for the quasi-axially-symmetric NCSX, a quasi-poloidally-symmetric configuration
and the quasi-helically-symmetric HSX are presented. Regions of second-stability
are observed in both NCSX and the quasi-poloidal configuration, whereas no second
stable region is observed for the quasi-helically symmetric device.
To explain the different regions of stability, the curvature and local shear of the
quasi-poloidal configuration are analyzed. The results are seemingly consistent with
the simple explanation: ballooning instability results when the local shear is small
in regions of bad curvature. Examples will be given that show that the structure,
and stability, of the ballooning mode is determined by the structure of the potential
function arising in the Schrbdinger form of the ballooning equation.
1. Introduction
By employing the standard WKB-like formulation, the leading order stability of high-
k ideal modes in stellarator plasmas is governed by an ordinary differential equation,
the ballooning equation [1]. The ballooning equation shows that it is the interaction of
pressure gradients, curvatures and the local shear that determines ballooning stability.
The local pressure gradient at a given surface directly affects the ballooning stability. A
cursory investigation of ballooning stability suggests that, as the magnitude of pressure
gradient is increased, ballooning modes could only be destabilized. This however is not
the case. The local shear also influences ballooning stability, and the local shear is related
to the pressure gradient through the Pfirsh-Schluter currents. As the pressure gradient is
increased at a given surface with fixed geometry, the local shear must change to preserve
the magnetohydrodynamic (MHD) equilibrium condition. Depending on the geometry
of the surface, it may be the case that the local shear will be altered in such a manner
to stabilize infinite-n ballooning modes. This effect has been studied in axisymmetric
equilibria [2] and may be called 'second-stability'.1
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Hudson, S. R.; Hegna, C. C.; Torasso, R. & Ware, A. Marginal Stability Diagrams for Infinite-n Ballooning Modes in Quasi-symmetric Stellarators, report, December 5, 2003; Princeton, New Jersey. (https://digital.library.unt.edu/ark:/67531/metadc735377/m1/3/: accessed April 24, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.