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Charged particle dynamics near an X-point of a non-symmetric magnetic field with closed field lines

Part of: Focus on Fusion

Published online by Cambridge University Press:  15 April 2020

Elena Elbarmi ,
Wrick Sengupta Open the ORCID record for Wrick Sengupta [Opens in a new window]  and
Harold Weitzner
Elena Elbarmi
Affiliation:
Courant Institute of Mathematical Sciences, New York University, NY10012, USA Friends Seminary, 22 E 16th St, New York, NY 10003, USA
Wrick Sengupta*
Affiliation:
Courant Institute of Mathematical Sciences, New York University, NY10012, USA
Harold Weitzner
Affiliation:
Courant Institute of Mathematical Sciences, New York University, NY10012, USA
*
†Email address for correspondence: [email protected]
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Abstract

Understanding particle drifts in a non-symmetric magnetic field is of primary interest in designing optimized stellarators in order to minimize the neoclassical radial loss of particles. Quasisymmetry and omnigeneity, two distinct properties proposed to ensure radial localization of collisionless trapped particles in stellarators, have been explored almost exclusively for magnetic fields with nested flux surfaces. In this work, we examine radial particle confinement when all field lines are closed. We then study charged particle dynamics in the special case of a non-symmetric vacuum magnetic field with closed field lines obtained recently by Weitzner & Sengupta (Phys. Plasmas, vol. 27, 2020, 022509). These magnetic fields can be used to construct magnetohydrodynamic equilibria for low pressure. Expanding in the amplitude of the non-symmetric fields, we explicitly evaluate the omnigeneity and quasisymmetry constraints. We show that the magnetic field is omnigeneous in the sense that the drift surfaces coincide with the pressure surfaces. However, it is not quasisymmetric according to the standard definitions.

Keywords

plasma confinementplasma dynamics
Type
Research Article
Information
Journal of Plasma Physics , Volume 86 , Issue 2 , April 2020 , 905860209
Copyright
© Cambridge University Press 2020

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References

Akerson, A. R., Bader, A., Hegna, C. C., Schmitz, O., Stephey, L. A., Anderson, D. T., Anderson, F. S. B. & Likin, K. M. 2016 Three-dimensional scrape off layer transport in the helically symmetric experiment hsx. Plasma Phys. Control. Fusion 58 (8), 084002.CrossRefGoogle Scholar
Andreeva, T.2002 Vacuum magnetic configurations of wendelstein 7-x. Tech. Rep. Max-Planck-Institut fuer Plasmaphysik.Google Scholar
Bader, A., Drevlak, M., Anderson, D. T., Faber, B. J., Hegna, C. C., Likin, K. M., Schmitt, J. C. & Talmadge, J. N. 2019 Stellarator equilibria with reactor relevant energetic particle losses. J. Plasma Phys. 85 (5), 905850508.CrossRefGoogle Scholar
Bauer, F., Betancourt, O. & Garabedian, P. 2012 A Computational Method in Plasma Physics. Springer Science & Business Media.Google Scholar
BenDaniel, D. J. 1965 Nonexistence of isotropic $\oint \text{d}l/B$ equilibria. Phys. Fluids 8 (8), 1567–1568.CrossRefGoogle Scholar
Boozer, A. H. 1995 Quasi-helical symmetry in stellarators. Plasma Phys. Control. Fusion 37 (11A), A103.CrossRefGoogle Scholar
Brakel, R., Anton, M., Baldzuhn, J., Burhenn, R., Erckmann, V., Fiedler, S., Geiger, J., Hartfuss, H. J., Heinrich, O., Hirsch, M. et al. 1997 Confinement in W7-AS and the role of radial electric field and magnetic shear. Plasma Phys. Control. Fusion 39 (12B), B273.CrossRefGoogle Scholar
Brakel, R.& the W7-AS Team 2002 Electron energy transport in the presence of rational surfaces in the Wendelstein 7-AS stellarator. Nucl. Fusion 42 (7), 903.CrossRefGoogle Scholar
Brizard, A. J.2011 Action-angle coordinates for the pendulum problem. Preprint, arXiv:1108.4970.Google Scholar
Burby, J. W., Kallinikos, N. & MacKay, R. S.2019 Some mathematics for quasi-symmetry. Preprint, arXiv:1912.06468.Google Scholar
Cary, J. R. 1982 Vacuum magnetic fields with dense flux surfaces. Phys. Rev. Lett. 49 (4), 276.CrossRefGoogle Scholar
Cary, J. R., Hedrick, C. L. & Tolliver, J. S. 1988 Orbits in asymmetric toroidal magnetic fields. Phys. fluids 31 (6), 1586–1600.CrossRefGoogle Scholar
Cary, J. R. & Littlejohn, R. G. 1983 Noncanonical hamiltonian mechanics and its application to magnetic field line flow. Ann. Phys. 151 (1), 1–34.CrossRefGoogle Scholar
Cary, J. R. & Shasharina, S. G. 1997a Helical plasma confinement devices with good confinement properties. Phys. Rev. Lett. 78 (4), 674.CrossRefGoogle Scholar
Cary, J. R. & Shasharina, S. G. 1997b Omnigenity and quasihelicity in helical plasma confinement systems. Phys. Plasmas 4 (9), 3323–3333.CrossRefGoogle Scholar
Feng, Y., Sardei, F., Grigull, P., McCormick, K., Kisslinger, J. & Reiter, D. 2006 Physics of island divertors as highlighted by the example of w7-as. Nucl. Fusion 46 (8), 807.CrossRefGoogle Scholar
Freidberg, J. P. 1982 Ideal magnetohydrodynamic theory of magnetic fusion systems. Rev. Mod. Phys. 54 (3), 801.CrossRefGoogle Scholar
Gardner, C. S. 1959 Adiabatic invariants of periodic classical systems. Phys. Rev. E 115 (4), 791.CrossRefGoogle Scholar
Garren, D. A. & Boozer, A. H. 1991 Existence of quasihelically symmetric stellarators. Phys. Fluids B: Plasma Phys. 3 (10), 2822–2834.CrossRefGoogle Scholar
Gori, S., Lotz, W. & Nührenberg, J. 1996 Quasi-isodynamic stellarators. Theory of Fusion Plasmas. In Proc. Joint Varenna-Lausanne Int. Workshop, pp. 335–342.Google Scholar
Grad, H. 1967 Toroidal containment of a plasma. Phys. Fluids 10 (1), 137–154.CrossRefGoogle Scholar
Grad, H. 1971 Plasma containment in closed line systems. In Plasma Physics and Controlled Nuclear Fusion Research 1971. Vol. III. Proceedings of the Fourth International Conference on Plasma Physics and Controlled Nuclear Fusion Research.Google Scholar
Hall, L. S. & McNamara, B. 1975 Three-dimensional equilibrium of the anisotropic, finite-pressure guiding-center plasma: theory of the magnetic plasma. Phys. Fluids 18 (5), 552–565.CrossRefGoogle Scholar
Hastie, R. J., Taylor, J. B. & Haas, F. A. 1967 Adiabatic invariants and the equilibrium of magnetically trapped particles. Ann. Phys. 41 (2), 302–338.CrossRefGoogle Scholar
Hazeltine, R. D. & Meiss, J. D. 2003 Plasma Confinement. Courier Corporation.Google Scholar
Helander, P. 2014 Theory of plasma confinement in non-axisymmetric magnetic fields. Rep. Prog. Phys. 77 (8), 087001.CrossRefGoogle ScholarPubMed
Helander, P., Beidler, C. D., Bird, T. M., Drevlak, M., Feng, Y., Hatzky, R., Jenko, F., Kleiber, R., Proll, J. H. E., Turkin, Y. et al. 2012 Stellarator and tokamak plasmas: a comparison. Plasma Phys. Control. Fusion 54 (12), 124009.CrossRefGoogle Scholar
Henneberg, S. A., Drevlak, M. & Helander, P. 2019 Improving fast-particle confinement in quasi-axisymmetric stellarator optimization. Plasma Phys. Control. Fusion 62 (1), 014023.Google Scholar
Hirsch, M., Baldzuhn, J., Beidler, C., Brakel, R., Burhenn, R., Dinklage, A., Ehmler, H., Endler, M., Erckmann, V., Feng, Y. et al. 2008 Major results from the stellarator Wendelstein 7-as. Plasma Phys. Control. Fusion 50 (5), 053001.CrossRefGoogle Scholar
Hudson, S. R. & Kraus, B. F. 2017 Three-dimensional magnetohydrodynamic equilibria with continuous magnetic fields. J. Plasma Phys. 83 (4), 715830403.CrossRefGoogle Scholar
Imbert-Gerard, L.-M., Paul, E. & Wright, A.2019 An introduction to symmetries in stellarators. Preprint, arXiv:1908.05360.Google Scholar
Kim, E., McFadden, G. & Cerfon, A. 2019 Elimination of mhd current sheets by modifications to the plasma wall in a fixed boundary model. Plasma Phys. Control. Fusion 62, 044002.Google Scholar
Klinger, T., Andreeva, T., Bozhenkov, S., Brandt, C., Burhenn, R., Buttenschön, B., Fuchert, G., Geiger, B., Grulke, O., Laqua, H. P. et al. 2019 Overview of first Wendelstein 7-x high-performance operation. Nucl. Fusion 59 (11), 112004.CrossRefGoogle Scholar
Kruskal, M. D. & Kulsrud, R. M. 1958 Equilibrium of a magnetically confined plasma in a toroid. Phys. Fluids 1 (4), 265–274.CrossRefGoogle Scholar
Landreman, M.2019 Quasisymmetry: A hidden symmetry of magnetic fields. Simons Collaboration on Hidden Symmetries and Fusion Energy, https://terpconnect.umd.edu/∼mattland/assets/notes/Introduction_to_quasisymmetry.pdf.Google Scholar
Landreman, M. & Catto, P. J. 2012 Omnigenity as generalized quasisymmetry a. Phys. Plasmas 19 (5), 056103.CrossRefGoogle Scholar
Landreman, M. & Sengupta, W. 2018 Direct construction of optimized stellarator shapes. Part 1. Theory in cylindrical coordinates. J. Plasma Phys. 84 (6), 905840616.CrossRefGoogle Scholar
Landreman, M. & Sengupta, W. 2019 Constructing stellarators with quasisymmetry to high order. J. Plasma Phys. 85 (6), 815850601.CrossRefGoogle Scholar
Landreman, M., Sengupta, W. & Plunk, G. G. 2019 Direct construction of optimized stellarator shapes. Part 2. Numerical quasisymmetric solutions. J. Plasma Phys. 85 (1), 905850103.CrossRefGoogle Scholar
Lortz, D. 1970 Über die existenz toroidaler magnetohydrostatischer gleichgewichte ohne rotationstransformation. Z. Angew. Math. Phys. 21 (2), 196–211.CrossRefGoogle Scholar
Mynick, H. E. 2006 Transport optimization in stellarators. Phys. plasmas 13 (5), 058102.CrossRefGoogle Scholar
Newcomb, W. A. 1959 Magnetic differential equations. Phys. Fluids 2 (4), 362–365.CrossRefGoogle Scholar
Nührenberg, J. 2010 Development of quasi-isodynamic stellarators. Plasma Phys. Control. Fusion 52 (12), 124003.CrossRefGoogle Scholar
Nührenberg, J., Sindoni, E., Lotz, W., Troyon, F., Gori, S. & Vaclavik, J. 1994 Quasi-axisymmetric tokamaks. In Proceedings of the Joint Varenna-Lausanne International Workshop on Theory of Fusion Plasmas, pp. 3–12.Google Scholar
Nührenberg, J. & Zille, R. 1988 Quasi-helically symmetric toroidal stellarators. Phys. Lett. A 129 (2), 113–117.CrossRefGoogle Scholar
Okamura, S., Matsuoka, K., Nishimura, S., Isobe, M., Nomura, I., Suzuki, C., Shimizu, A., Murakami, S., Nakajima, N., Yokoyama, M. et al. 2001 Physics and engineering design of the low aspect ratio quasi-axisymmetric stellarator chs-qa. Nucl. Fusion 41 (12), 1865.CrossRefGoogle Scholar
Plunk, G. G. & Helander, P. 2018 Quasi-axisymmetric magnetic fields: weakly non-axisymmetric case in a vacuum. J. Plasma Phys. 84 (2), 905840205.CrossRefGoogle Scholar
Plunk, G. G., Landreman, M. & Helander, P. 2019 Direct construction of optimized stellarator shapes. Part 3. Omnigenity near the magnetic axis. J. Plasma Phys. 85 (6), 905850602.CrossRefGoogle Scholar
Sengupta, W. & Weitzner, H. 2018 Radial confinement of deeply trapped particles in a non-symmetric magnetohydrodynamic equilibrium. Phys. Plasmas 25 (2), 022506.CrossRefGoogle Scholar
Sengupta, W. & Weitzner, H. 2019 Low-shear three-dimensional equilibria and vacuum magnetic fields with flux surfaces. J. Plasma Phys. 85 (2), 905850209.CrossRefGoogle Scholar
Spies, G. O. 1974 Nonlinear magnetohydrodynamic stability and plasma containment with closed field lines. Phys. Fluids 17 (6), 1188–1197.CrossRefGoogle Scholar
Taylor, J. B. 1963 Some stable plasma equilibria in combined mirror-cusp fields. Phys. Fluids 6 (11), 1529–1536.CrossRefGoogle Scholar
Taylor, J. B. & Hastie, R. J. 1968 Stability of general plasma equilibria-i formal theory. Plasma Phys. 10 (5), 479.CrossRefGoogle Scholar
Weitzner, H. 2014 Ideal magnetohydrodynamic equilibrium in a non-symmetric topological torus. Phys. Plasmas 21 (2), 022515.CrossRefGoogle Scholar
Weitzner, H. 2016 Expansions of non-symmetric toroidal magnetohydrodynamic equilibria. Phys. Plasmas 23 (6), 062512.CrossRefGoogle Scholar
Weitzner, H. & Sengupta, W. 2020 Exact non-symmetric closed line vacuum magnetic fields in a topological torus. Phys. Plasmas 27 (2), 022509.CrossRefGoogle Scholar
Wobig, H. 1987 Magnetic surfaces and localized perturbations in the Wendelstein VII-A stellarator. Z. Naturforsch. 42 (10), 1054–1066.CrossRefGoogle Scholar