Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-27T12:31:19.214Z Has data issue: false hasContentIssue false
  • English
  • Français

Near-axis expansion of stellarator equilibrium at arbitrary order in the distance to the axis

Part of: Focus on Fusion

Published online by Cambridge University Press:  28 January 2020

R. Jorge Open the ORCID record for R. Jorge [Opens in a new window] ,
W. Sengupta Open the ORCID record for W. Sengupta [Opens in a new window]  and
M. Landreman Open the ORCID record for M. Landreman [Opens in a new window]
R. Jorge*
Affiliation:
Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, MD 20742, USA
W. Sengupta
Affiliation:
Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA
M. Landreman
Affiliation:
Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, MD 20742, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

A direct construction of equilibrium magnetic fields with toroidal topology at arbitrary order in the distance from the magnetic axis is carried out, yielding an analytical framework able to explore the landscape of possible magnetic flux surfaces in the vicinity of the axis. This framework can provide meaningful analytical insight into the character of high-aspect-ratio stellarator shapes, such as the dependence of the rotational transform and the plasma beta limit on geometrical properties of the resulting flux surfaces. The approach developed here is based on an asymptotic expansion on the inverse aspect ratio of the ideal magnetohydrodynamics equation. The analysis is simplified by using an orthogonal coordinate system relative to the Frenet–Serret frame at the magnetic axis. The magnetic field vector, the toroidal magnetic flux, the current density, the field line label and the rotational transform are derived at arbitrary order in the expansion parameter. Moreover, a comparison with a near-axis expansion formalism employing an inverse coordinate method based on Boozer coordinates (the so-called Garren–Boozer construction) is made, where both methods are shown to agree at lowest order. Finally, as a practical example, a numerical solution using a W7-X equilibrium is presented, and a comparison between the lowest-order solution and the W7-X magnetic field is performed.

Keywords

fusion plasmaplasma confinement
Type
Research Article
Information
Journal of Plasma Physics , Volume 86 , Issue 1 , February 2020 , 905860106
Copyright
© Cambridge University Press 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bernardin, M. P., Moses, R. W. & Tataronis, J. A. 1986 Isodynamical (omnigenous) equilibrium in symmetrically confined plasma configurations. Phys. Fluids 29 (8), 2605.CrossRefGoogle Scholar
Bernardin, M. P. & Tataronis, J. A. 1985 Hamiltonian approach to the existence of magnetic surfaces. J. Math. Phys. 26 (9), 2370.CrossRefGoogle Scholar
Boozer, A. H. 1981 Plasma equilibrium with rational magnetic surfaces. Phys. Fluids 24 (11), 1999.CrossRefGoogle Scholar
Boozer, A. H. 2015 Stellarator design. J. Plasma Phys. 81 (6), 515810606.CrossRefGoogle Scholar
Chu, M. S., Guo, W., Liu, W., Ren, Q., Shaing, K. C. & Zhu, P. 2019 A three-dimensional magnetohydrodynamic equilibrium in an axial coordinate with a constant curvature. Nucl. Fusion 59 (8), 086004.CrossRefGoogle Scholar
D’haeseleer, W. D., Hitchon, W. N. G., Callen, J. D. & Shohet, J. L. 1991 Flux Coordinates and Magnetic Field Structure: A Guide to a Fundamental Tool of Plasma Theory. Springer.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
Garren, D. A. & Boozer, A. H. 1991a Existence of quasihelically symmetric stellarators. Phys. Fluids B 3 (10), 2822.CrossRefGoogle Scholar
Garren, D. A. & Boozer, A. H. 1991b Magnetic field strength of toroidal plasma equilibria. Phys. Fluids B 3 (10), 2805.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
Gradshteyn, I. S. & Ryzhik, M. 2007 Table of Integrals, Series, and Products. Elsevier.Google Scholar
Grieger, G., Lotz, W., Merkel, P., Nührenberg, J., Sapper, J., Strumberger, E., Wobig, H., Burhenn, R., Erckmann, V., Gasparino, U. et al. 1992 Physics optimization of stellarators. Phys. Fluids B 4 (7), 2081.CrossRefGoogle Scholar
Helander, P. 2014 Theory of plasma confinement in non-axisymmetric magnetic fields. Rep. Prog. Phys. 77 (8), 087001.CrossRefGoogle Scholar
Hirshman, S. P. & Whitson, J. C. 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
Kruskal, M. D. & Kulsrud, R. M. 1958 Equilibrium of a magnetically confined plasma in a toroid. Phys. Fluids 1 (4), 265.CrossRefGoogle Scholar
Kuo-Petravic, G. & Boozer, A. H. 1987 Numerical determination of the magnetic field line hamiltonian. J. Comput. Phys. 73 (1), 107.CrossRefGoogle Scholar
Landreman, M. 2019 Optimized quasisymmetric stellarators are consistent with the GarrenBoozer construction. Plasma Phys. Control. Fusion 61 (7), 075001.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. & 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. & Nuhrenberg, J. 1976 Equilibrium and stability of a three-dimensional toroidal MHD configuration near its magnetic axis. Z. Naturforsch. 31 (11), 1277.Google Scholar
Lortz, D. & Nuhrenberg, J. 1977 Equilibrium and stability of the $l=2$ stellarator without longitudinal current. Nucl. Fusion 17 (1), 125.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.CrossRefGoogle Scholar
Mynick, H. E. 2006 Transport optimization in stellarators. Phys. Plasmas 13 (5), 058102.CrossRefGoogle Scholar
Mynick, H. E., Pomphrey, N. & Xanthopoulos, P. 2010 Optimizing stellarators for turbulent transport. Phys. Rev. Lett. 105 (9), 095004.CrossRefGoogle Scholar
Olver, F. W. J., Lozier, D. W., Boisvert, R. F. & Clark, C. W. 2010 NIST Handbook of Mathematical Functions. Cambridge University Press.Google Scholar
Salat, A. 1995 Nonexistence of magnetohydrodynamic equilibria with poloidally closed field lines in the case of violated axisymmetry. Phys. Plasmas 2 (5), 1652.CrossRefGoogle Scholar
Solov’ev, L. S. & Shafranov, V. D. 1970 Reviews of Plasma Physics 5. Consultants Bureau.Google Scholar
Spivak, M. 1999 A Comprehensive Introduction to Differential Geometry: Volume 2. Publish or Perish Inc.Google Scholar
Spong, D. A., Hirshman, S. P., Berry, L. A., Lyon, J. F., Fowler, R. H., Strickler, D. J., Cole, M. J., Nelson, B. N., Williamson, D. E., Ware, A. S. et al. 2001 Physics issues of compact drift optimized stellarators. Nucl. Fusion 41 (6), 711.CrossRefGoogle Scholar
Weitzner, H. 2016 Expansions of non-symmetric toroidal magnetohydrodynamic equilibria. Phys. Plasmas 23 (6), 062512.CrossRefGoogle Scholar