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The on-axis magnetic well and Mercier's criterion for arbitrary stellarator geometries

Part of: Focus on Fusion

Published online by Cambridge University Press:  04 May 2021

P. Kim Open the ORCID record for P. Kim [Opens in a new window] ,
R. Jorge Open the ORCID record for R. Jorge [Opens in a new window]  and
W. Dorland Open the ORCID record for W. Dorland [Opens in a new window]
P. Kim*
Affiliation:
Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, MD20742, USA
R. Jorge
Affiliation:
Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, MD20742, USA
W. Dorland
Affiliation:
Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, MD20742, USA Department of Physics, University of Maryland, College Park, MD20742, USA
*
†Email address for correspondence: pkim1236@terpmail.umd.edu
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Abstract

A simplified analytical form of the on-axis magnetic well and Mercier's criterion for interchange instabilities for arbitrary three-dimensional magnetic field geometries is derived. For this purpose, a near-axis expansion based on a direct coordinate approach is used by expressing the toroidal magnetic flux in terms of powers of the radial distance to the magnetic axis. For the first time, the magnetic well and Mercier's criterion are then written as a one-dimensional integral with respect to the axis arclength. When compared with the original work of Mercier, the derivation here is presented using modern notation and in a more streamlined manner that highlights essential steps. Finally, these expressions are verified numerically using several quasisymmetric and non-quasisymmetric stellarator configurations including Wendelstein 7-X.

Keywords

fusion plasmaplasma confinementplasma instabilities
Type
Research Article
Information
Journal of Plasma Physics , Volume 87 , Issue 2 , April 2021 , 905870231
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Ahlfors, L. 1979 Complex Analysis, 3rd edn. McGraw-Hill Education.Google Scholar
Anderson, F.S.B., Almagri, A.F., Anderson, D.T., Matthews, P.G., Talmadge, J.N. & Shohet, J.L. 1995 The helically symmetric experiment, (HSX) goals, design and status. Fusion Technol. 27 (14), 273–277.CrossRefGoogle Scholar
Beidler, C., Grieger, G., Herrnegger, F., Harmeyer, E., Kisslinger, J., Lotz, W., Maassberg, H., Merkel, P., Nuehrenberg, J., Rau, F., et al. 1990 Physics and engineering design for Wendelstein VII-X. Fusion Technol. 17 (1), 148–168.CrossRefGoogle Scholar
Boozer, A.H. 2015 Stellarator design. J. Plasma Phys. 81 (6), 515810606.CrossRefGoogle Scholar
Drevlak, M., Beidler, C.D., Geiger, J., Helander, P. & Turkin, Y. 2019 Optimisation of stellarator equilibria with ROSE. Nucl. Fusion 59 (1), 016010.CrossRefGoogle Scholar
Freidberg, J.P. 2007 Plasma Physics and Fusion Energy. Cambridge University Press.CrossRefGoogle Scholar
Freidberg, J.P. 2014 Ideal MHD. Cambridge University Press.CrossRefGoogle Scholar
Garren, D.A. & Boozer, A.H. 1991 Existence of quasihelically symmetric stellarators. Phys. Fluids B 3 (10), 2822–2834.CrossRefGoogle Scholar
Geiger, J., Beidler, C.D., Feng, Y., Maaßberg, H., Marushchenko, N.B. & Turkin, Y. 2015 Physics in the magnetic configuration space of W7-X. Plasma Phys. Control. Fusion 57 (1), 014004.CrossRefGoogle Scholar
Glasser, A.H., Greene, J.M. & Johnson, J.L. 1975 Resistive instabilities in general toroidal plasma configurations. Phys. Fluids 18 (7), 875–888.CrossRefGoogle Scholar
Helander, P. 2014 Theory of plasma confinement in non-axisymmetric magnetic fields. Rep. Prog. Phys. 77 (8), 087001.CrossRefGoogle ScholarPubMed
Hirshman, S.P. 1983 Steepest-descent moment method for three-dimensional magnetohydrodynamic equilibria. Phys. Fluids 26 (12), 3553.CrossRefGoogle Scholar
Jorge, R. 2019 SENAC: Stellarator Near-axis Equilibrium Code. Dataset on Zenodo. DOI:10.5281/zenodo. 3575116.CrossRefGoogle Scholar
Jorge, R., Sengupta, W. & Landreman, M. 2020 a Construction of quasisymmetric stellarators using a direct coordinate approach. Nucl. Fusion 60 (7), 076021.CrossRefGoogle Scholar
Jorge, R., Sengupta, W. & Landreman, M. 2020 b Near-axis expansion of stellarator equilibrium at arbitrary order in the distance to the axis. J. Plasma Phys. 86 (1), 905860106.CrossRefGoogle Scholar
Kuo-Petravic, G. & Boozer, A.H. 1987 Numerical determination of the magnetic field line Hamiltonian. J. Comput. Phys. 73 (1), 107–124.CrossRefGoogle Scholar
Landreman, M. 2019 Optimized quasisymmetric stellarators are consistent with the Garren–Boozer construction. Plasma Phys. Control. Fusion 61 (7), 075001.CrossRefGoogle Scholar
Landreman, M. & Jorge, R. 2020 Magnetic well and Mercier stability of stellarators near the magnetic axis. J. Plasma Phys. 86 (5), 905860510.CrossRefGoogle Scholar
Landreman, M. & Sengupta, W. 2019 Constructing stellarators with quasisymmetry to high order. J. Plasma Phys. 85 (6), 815850601.CrossRefGoogle Scholar
Lortz, D. & Nührenberg, J. 1977 Equilibrium and stability of the $\ell =2$ stellarator without longitudinal current. Nucl. Fusion 17 (1), 125–133.CrossRefGoogle Scholar
Mercier, C. 1964 Equilibrium and stability of a toroidal magnetohydrodynamic system in the neighbourhood of a magnetic axis. Nucl. Fusion 4 (3), 213–226.CrossRefGoogle Scholar
Mercier, C. & Luc, H. 1974 Lectures in Plasma Physics. The Magnetohydrodynamic Approach to the Problem of Plasma Confinement in Closed Magnetic Configurations. Commission of the European Communities, Directorate General Scientific and Technical Information and Information Management, 1974.Google Scholar
Solov'ev, L.S. & Shafranov, V.D. 1970 Plasma confinement in closed magnetic systems. In Reviews of Plasma Physics (ed. L. A. Leontovich), pp. 1–247. Springer.CrossRefGoogle Scholar
Spivak, M. 1999 A Comprehensive Introduction to Differential Geometry, vol. 2. Publish or Perish Inc.Google Scholar