Marginal Stability Boundaries for Infinite-n Ballooning Modes in a Quasi-axisymmetric Stellarator Page: 4 of 35
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quasi-poloidal equilibrium [5]. The existence of a second stable region in stellarators suggests the possibility of
high ) operation and is an enticing prospect.
A feature related to second stability, which we call 'self-stabilization' after Ref. [6] has been observed in various
stellarator experiments [7, 8, 9, 10]. In these results, geometrical deformations associated with the Shafranov shift
result in configurations which are stable with respect to Mercier modes as pressure is increased. In this article, we
use the term second stability to refer to the stabilization of ballooning modes due to pressure induced variations
in the local shear, with no geometrical deformations.
Hegna & Nakajima [11] extended the theory of Greene & Chance to three-dimensional configurations. In this
work, we study the ideal ballooning stability of a family of quasi-axisymmetric stellarator equilibria and present
marginal stability diagrams. We find that for this configuration, the existence of second-stability is observed on
some magnetic surfaces. While the conventional explanation of the appearance of second-stability (instability
ensues when regions of unfavorable curvature overlap regions of small local shear; second-stability occurs through
pressure modulations of the local shear) seems plausible, the non-axisymmetry of stellarators results in a more
complicated structure of both the curvature and the local shear and the mechanism of second-stability is less easily
demonstrated.
The outline of this paper is as follows. In Sec(II), the theory of the profile variation method is described. The
numerical implementation is discussed in Sec(III), and in Sec(IV) benchmarking results for an axisymmetric case
are presented. The axi-symmetric case is useful for developing intuition about the role of curvature and local
shear. Sec(V) will present marginal stability diagrams and analysis of a quasi-axisymmetric stellarator. Some
similar features of the marginal stability diagrams are observed. The existence of unstable regions for either sign
of averaged magnetic shear and the existence of second-stability is observed. Sec(VI) will discuss the effect of field
line variation on the stability diagrams. Some discussion of the physical mechanism of ballooning stability, and
the characteristics leading to second-stability will be mentioned is given in Sec(VII); however, a full quantitative
description of the onset of second-stability is an involved topic and requires further detailed analysis which is left
to future work.
II Theory
This work is essentially a numerical implementation of the theory presented by Hegna & Nakajima [11], who
generalized the work of Greene & Chance [3]. By applying variations to, in this case, the pressure gradient and
the average magnetic shear at a selected surface of a given equilibrium, and requiring that the coordinate response
to the variations be such that the resultant state is also an equilibrium, a two-dimensional family of perturbed
equilibria are constructed. Using this method, the effect on ballooning stability of changes in the pressure gradient
and changes in the shear may be separately studied, with the surface geometry held fixed. The following shall
outline the principles of the method and the key assumptions. For additional details the reader should consult
Ref. [11].
The analysis proceeds using Boozer coordinates [12] with (p, 0, () being the radial (toroidal flux), poloidal and
toroidal coordinates. The magnetic field is written in contravariant and covariant form as
B V@ x V + t(v)VC x V , (1)
B = (, 0, ()Va + I( p)V + G( )v(, (2)
where t is the rotational-transform, G is the poloidal current exterior to p, I is the toroidal current interior to p,
and ) is related to the Pfirsch-Schluter current.
The magnetic field is defined implicitly through the coordinate transform x(%, 0, () from Boozer coordinates to
Cartesian coordinates. The metric elements of the transformation gij e2 - ej are defined by the basis vectors
eg =OtJx, ee =Jex, and e = ax. The magnetic field may be written in terms of the basis vectors as
B= (tee + e4)/7g. (3)2
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Hudson, S. R. & Hegna, C. C. Marginal Stability Boundaries for Infinite-n Ballooning Modes in a Quasi-axisymmetric Stellarator, report, September 15, 2003; Princeton, New Jersey. (https://digital.library.unt.edu/ark:/67531/metadc740432/m1/4/: accessed April 19, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.