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On quasisymmetric plasma equilibria sustained by small force

Published online by Cambridge University Press:  11 February 2021

Peter Constantin ,
Theodore D. Drivas  and
Daniel Ginsberg Open the ORCID record for Daniel Ginsberg [Opens in a new window]
Peter Constantin
Affiliation:
Department of Mathematics, Princeton University, Princeton, NJ08544, USA
Theodore D. Drivas
Affiliation:
Department of Mathematics, Princeton University, Princeton, NJ08544, USA
Daniel Ginsberg*
Affiliation:
Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ08544, USA
*
†Email address for correspondence: dg42@math.princeton.edu
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Abstract

We construct smooth, non-symmetric plasma equilibria which possess closed, nested flux surfaces and solve the magnetohydrostatic (steady three-dimensional incompressible Euler) equations with a small force. The solutions are also ‘nearly’ quasisymmetric. The primary idea is, given a desired quasisymmetry direction $\xi$, to change the smooth structure on space so that the vector field $\xi$ is Killing for the new metric and construct $\xi$–symmetric solutions of the magnetohydrostatic equations on that background by solving a generalized Grad–Shafranov equation. If $\xi$ is close to a symmetry of Euclidean space, then these are solutions on flat space up to a small forcing.

Keywords

fusion plasma
Type
Research Article
Information
Journal of Plasma Physics , Volume 87 , Issue 1 , February 2021 , 905870111
Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Arnold, V. I. & Khesin, B. A. 1999 Topological Methods in Hydrodynamics, vol. 125. Springer Science & Business Media.Google Scholar
Bernardin, M. P., Moses, R. W. & Tataronis, J. A. 1986 Isodynamical (omnigenous) equilibrium in symmetrically confined plasma configurations. Phys. Fluids 29 (8), 2605–2611.CrossRefGoogle Scholar
Burby, J. W., Kallinikos, N. & MacKay, R. S. 2020 Some mathematics for quasi-symmetry. J. Math. Phys. 61 (9), 093503.CrossRefGoogle Scholar
Burby, J. W., Kallinikos, N. & MacKay, R. S. 2020 Generalized Grad–Shafranov equation for non-axisymmetric MHD equilibria. Phys. Plasmas 27 (10), 102504.CrossRefGoogle Scholar
Bruno, O. P. & Laurence, P. 1996 Existence of three-dimensional toroidal MHD equilibria with nonconstant pressure. Commun. Pure Appl. Maths 49 (7), 717–764.3.0.CO;2-C>CrossRefGoogle Scholar
Constantin, P., Drivas, T. D. & Ginsberg, D. 2020 Flexibility and rigidity in steady fluid motion. arXiv:2007.09103.Google Scholar
Fecko, M. 2006 Differential Geometry and Lie Groups for Physicists. Cambridge University Press.CrossRefGoogle Scholar
Freidberg, J. 2014 Ideal MHD. Cambridge University Press.CrossRefGoogle Scholar
Garabedian, P. R. 2006 Three-dimensional equilibria in axially symmetric tokamaks. Proc. Natl Acad. Sci. USA 103 (51), 19232–19236.CrossRefGoogle ScholarPubMed
Garren, D. A. & Boozer, A. H. 1991 Existence of quasihelically symmetric stellarators. Phys. Fluids B 3 (10), 2822–2834.CrossRefGoogle Scholar
Grad, H. 1967 Toroidal containment of a plasma. Phys. Fluids 10 (1), 137–154.CrossRefGoogle Scholar
Grad, H. 1985 Theory and applications of the nonexistence of simple toroidal plasma equilibrium. Intl J. Fusion Energy 3 (2), 33–46.Google Scholar
Grad, H. & Rubin, H. 1958 Hydromagnetic equilibria and force-free fields. J. Nucl. Energy 7 (3–4), 284–285.Google Scholar
Hudson, S. R., Dewar, R. L., Dennis, G., Hole, M. J., McGann, M., Von Nessi, G. & Lazerson, S. 2012 Computation of multi-region relaxed magnetohydrodynamic equilibria. Phys. Plasmas 19 (11), 112502.CrossRefGoogle Scholar
Hudson, S. R., Dewar, R. L., Hole, M. J. & McGann, M. 2011 Non-axisymmetric, multi-region relaxed magnetohydrodynamic equilibrium solutions. Plasma Phys. Control. Fusion 54 (1), 014005.CrossRefGoogle Scholar
Jorge, R., Sengupta, W. & Landreman, M. 2019 Near-axis expansion of stellarator equilibrium at arbitrary order in the distance to the axis. arXiv:1911.02659.CrossRefGoogle Scholar
Kaiser, R. & Salat, A. 1997 New classes of three-dimensional ideal-MHD equilibria. J. Plasma Phys. 57, 425–448.CrossRefGoogle Scholar
Landreman, M. 2019 Quasisymmetry: a hidden symmetry of magnetic fields.Google Scholar
Lee, J. M. 2013 Smooth manifolds. In Introduction to Smooth Manifolds. Springer.CrossRefGoogle Scholar
Lichtenfelz, L., Misiolek, G. & Preston, S. C. 2019 Axisymmetric diffeomorphisms and ideal fluids on Riemannian 3-manifolds. arXiv:1911.10302.CrossRefGoogle Scholar
Lortz, D. 1970 Existence of toroidal magnetohydrostatic equilibrium without rotational transformation. Z. Angew. Math. Phys. 21, 196–211.CrossRefGoogle Scholar
Plunk, G. G. & Helander, P. 2018 Quasi-axisymmetric magnetic fields: weakly non-axisymmetric case in a vacuum. J. Plasma Phys. 84 (2).CrossRefGoogle Scholar
Rodriguez, E., Helander, P. & Bhattacharjee, A. 2020 Necessary and sufficient conditions for quasisymmetry. Phys. Plasmas 27 (6), 062501.CrossRefGoogle Scholar
Salat, A. & Kaiser, R. 1995 Three-dimensional closed field line magnetohydrodynamic equilibria without symmetries. Phys. Plasmas 2 (10), 3777–3781.CrossRefGoogle Scholar
Shafranov, V. D. 1966 Plasma equilibrium in a magnetic field. Rev. Plasma Phys. 2, 103.Google Scholar
Taylor, M. E. 1996 Partial Differential Equations III: Nonlinear Theory. Applied Mathematical Sciences, 117.Google Scholar
Weitzner, H. 2014 Ideal magnetohydrodynamic equilibrium in a non-symmetric topological torus. Phys. Plasmas 21 (2), 022515.CrossRefGoogle Scholar