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Some mathematics for quasi-symmetry
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Burby, J. W., Kallinikos, N. and MacKay, Robert S. (2020) Some mathematics for quasi-symmetry. Journal of Mathematical Physics, 61 (9). 093503. doi:10.1063/1.5142487 ISSN 0022-2488.
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Official URL: https://doi.org/10.1063/1.5142487
Abstract
The concept of quasi-symmetry was introduced in (Booozer, 1983) and then distilled into a design principle for stellarators by N¨uhrenberg & Zille (1988). In its strongest sense it means integrability of first-order guiding-centre motion. An excellent survey of the subject was provided by Helander (2014), assuming magnetohydrostatic (MHS) fields, that is, magnetohydrodynamic equilibrium with isotropic pressure and no mean flow.
A fundamental step was made by Burby & Qin (2013), who stated necessary and sufficient local conditions for integrability of guiding-centre motion in terms of a continuous symmetry of three differential forms derived from the magnetic field and made clear that quasi-symmetry can be separated from the issue of whether the magnetic field is MHS or not.
Perturbative calculations of Garren & Boozer (1991), however, make it look very likely that the only possibility for exact quasi-symmetry for MHS fields with bounded magnetic surfaces is axisymmetry. Our paper gives first steps to deciding whether or not this is true.
In this paper we prove many consequences of quasi-symmetry and thereby restrictions on possible quasi-symmetric fields. In the case of a quasi-symmetric MHS field we derive a generalisation of the axisymmetric Grad-Shafranov equation.
Burby & Qin (2013) built in an assumption that a quasi-symmetry must be a circleaction. Here we relax this requirement, though prove that under some mild conditions it is actually a circle-action.
We write many equations using differential forms. For those unfamiliar with differential forms, (Arnol’d, 1978, chap. 7) is a classic and there is a tutorial (MacKay, 2019) specifically for plasma physicists.
Throughout the paper we will assume enough smoothness that the equations we write make sense, at least in a weak sense.
Item Type: | Journal Article | ||||||||||||
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Subjects: | Q Science > QA Mathematics | ||||||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||||||
Library of Congress Subject Headings (LCSH): | Quasisymmetric groups, Hamiltonian systems, Stellarators | ||||||||||||
Journal or Publication Title: | Journal of Mathematical Physics | ||||||||||||
Publisher: | American Institute of Physics | ||||||||||||
ISSN: | 0022-2488 | ||||||||||||
Official Date: | September 2020 | ||||||||||||
Dates: |
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Volume: | 61 | ||||||||||||
Number: | 9 | ||||||||||||
Article Number: | 093503 | ||||||||||||
DOI: | 10.1063/1.5142487 | ||||||||||||
Status: | Peer Reviewed | ||||||||||||
Publication Status: | Published | ||||||||||||
Access rights to Published version: | Open Access (Creative Commons) | ||||||||||||
Date of first compliant deposit: | 8 September 2020 | ||||||||||||
Date of first compliant Open Access: | 9 September 2020 | ||||||||||||
RIOXX Funder/Project Grant: |
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