An integral equation-based numerical solver for Taylor states in toroidal geometries
Abstract
Here we present an algorithm for the numerical calculation of Taylor states in toroidal and toroidal-shell geometries using an analytical framework developed for the solution to the time-harmonic Maxwell equations. Taylor states are a special case of what are known as Beltrami fields, or linear force-free fields. The scheme of this work relies on the generalized Debye source representation of Maxwell fields and an integral representation of Beltrami fields which immediately yields a well-conditioned second-kind integral equation. This integral equation has a unique solution whenever the Beltrami parameter λ is not a member of a discrete, countable set of resonances which physically correspond to spontaneous symmetry breaking. Several numerical examples relevant to magnetohydrodynamic equilibria calculations are provided. Lastly, our approach easily generalizes to arbitrary geometries, both bounded and unbounded, and of varying genus.
- Publication Date:
- Research Org.:
- New York Univ. (NYU), NY (United States)
- Sponsoring Org.:
- USDOE Office of Science (SC), Fusion Energy Sciences (FES); US Air Force Office of Scientific Research (AFOSR); US Department of the Navy, Office of Naval Research (ONR)
- OSTI Identifier:
- 1538427
- Alternate Identifier(s):
- OSTI ID: 1548839
- Grant/Contract Number:
- FG02-86ER53223; SC0012398; FA9550-10-1-0180; N00014-17-1-2059; N00014-17-1-2451
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Journal of Computational Physics
- Additional Journal Information:
- Journal Volume: 359; Journal Issue: C; Journal ID: ISSN 0021-9991
- Publisher:
- Elsevier
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 42 ENGINEERING; Beltrami field; generalized debye sources; plasma physics; force-free fields; Taylor states; magnetohydrodynamics
Citation Formats
O'Neil, Michael, and Cerfon, Antoine J. An integral equation-based numerical solver for Taylor states in toroidal geometries. United States: N. p., 2018.
Web. doi:10.1016/j.jcp.2018.01.004.
O'Neil, Michael, & Cerfon, Antoine J. An integral equation-based numerical solver for Taylor states in toroidal geometries. United States. https://doi.org/10.1016/j.jcp.2018.01.004
O'Neil, Michael, and Cerfon, Antoine J. Tue .
"An integral equation-based numerical solver for Taylor states in toroidal geometries". United States. https://doi.org/10.1016/j.jcp.2018.01.004. https://www.osti.gov/servlets/purl/1538427.
@article{osti_1538427,
title = {An integral equation-based numerical solver for Taylor states in toroidal geometries},
author = {O'Neil, Michael and Cerfon, Antoine J.},
abstractNote = {Here we present an algorithm for the numerical calculation of Taylor states in toroidal and toroidal-shell geometries using an analytical framework developed for the solution to the time-harmonic Maxwell equations. Taylor states are a special case of what are known as Beltrami fields, or linear force-free fields. The scheme of this work relies on the generalized Debye source representation of Maxwell fields and an integral representation of Beltrami fields which immediately yields a well-conditioned second-kind integral equation. This integral equation has a unique solution whenever the Beltrami parameter λ is not a member of a discrete, countable set of resonances which physically correspond to spontaneous symmetry breaking. Several numerical examples relevant to magnetohydrodynamic equilibria calculations are provided. Lastly, our approach easily generalizes to arbitrary geometries, both bounded and unbounded, and of varying genus.},
doi = {10.1016/j.jcp.2018.01.004},
journal = {Journal of Computational Physics},
number = C,
volume = 359,
place = {United States},
year = {Tue Jan 09 00:00:00 EST 2018},
month = {Tue Jan 09 00:00:00 EST 2018}
}
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Works referenced in this record:
On an exterior boundary-value problem for the time-harmonic maxwell equations with boundary conditions for the normal components of the electric and magnetic field
journal, January 1986
- Kress, R.; Leis, R.
- Mathematical Methods in the Applied Sciences, Vol. 8, Issue 1
A high-order Nyström discretization scheme for boundary integral equations defined on rotationally symmetric surfaces
journal, June 2012
- Young, P.; Hao, S.; Martinsson, P. G.
- Journal of Computational Physics, Vol. 231, Issue 11
Reconstruction of current profile parameters and plasma shapes in tokamaks
journal, November 1985
- Lao, L. L.; St. John, H.; Stambaugh, R. D.
- Nuclear Fusion, Vol. 25, Issue 11
The Beltrami spectrum for incompressible fluid flows
journal, September 1988
- Constantin, Peter; Majda, Andrew
- Communications in Mathematical Physics, Vol. 115, Issue 3
High-order accurate methods for Nyström discretization of integral equations on smooth curves in the plane
journal, June 2013
- Hao, S.; Barnett, A. H.; Martinsson, P. G.
- Advances in Computational Mathematics, Vol. 40, Issue 1
Computing Beltrami Fields
journal, January 2009
- Amari, Tahar; Boulbe, Cédric; Boulmezaoud, Tahar Zamène
- SIAM Journal on Scientific Computing, Vol. 31, Issue 5
Existence of three-dimensional toroidal MHD equilibria with nonconstant pressure
journal, July 1996
- Bruno, Oscar P.; Laurence, Peter
- Communications on Pure and Applied Mathematics, Vol. 49, Issue 7
Determination of normalized electric eigenfields in microwave cavities with sharp edges
journal, January 2016
- Helsing, Johan; Karlsson, Anders
- Journal of Computational Physics, Vol. 304
Anomalous diffusion of field lines and charged particles in Arnold-Beltrami-Childress force-free magnetic fields
journal, July 2014
- Ram, Abhay K.; Dasgupta, Brahmananda; Krishnamurthy, V.
- Physics of Plasmas, Vol. 21, Issue 7
“One size fits all” analytic solutions to the Grad–Shafranov equation
journal, March 2010
- Cerfon, Antoine J.; Freidberg, Jeffrey P.
- Physics of Plasmas, Vol. 17, Issue 3
Determination of Normalized Magnetic Eigenfields in Microwave Cavities
journal, May 2015
- Helsing, Johan; Karlsson, Anders
- IEEE Transactions on Microwave Theory and Techniques, Vol. 63, Issue 5
Exact Fourier expansion in cylindrical coordinates for the three-dimensional Helmholtz Green function
journal, November 2009
- Conway, John T.; Cohl, Howard S.
- Zeitschrift für angewandte Mathematik und Physik, Vol. 61, Issue 3
Relaxation and magnetic reconnection in plasmas
journal, July 1986
- Taylor, J. B.
- Reviews of Modern Physics, Vol. 58, Issue 3
Tokamak Startup Using Point-Source dc Helicity Injection
journal, June 2009
- Battaglia, D. J.; Bongard, M. W.; Fonck, R. J.
- Physical Review Letters, Vol. 102, Issue 22
Computing elliptic integrals by duplication
journal, March 1979
- Carlson, B. C.
- Numerische Mathematik, Vol. 33, Issue 1
Exact axisymmetric Taylor states for shaped plasmas
journal, June 2014
- Cerfon, Antoine J.; O'Neil, Michael
- Physics of Plasmas, Vol. 21, Issue 6
Scaling studies of spheromak formation and equilibrium
journal, April 1998
- Geddes, C. G. R.; Kornack, T. W.; Brown, M. R.
- Physics of Plasmas, Vol. 5, Issue 4
Robust and Efficient Solution of the Drum Problem via Nyström Approximation of the Fredholm Determinant
journal, January 2015
- Zhao, Lin; Barnett, Alex
- SIAM Journal on Numerical Analysis, Vol. 53, Issue 4
Hybrid Gauss-Trapezoidal Quadrature Rules
journal, January 1999
- Alpert, Bradley K.
- SIAM Journal on Scientific Computing, Vol. 20, Issue 5
On a selfadjoint realization of curl in exterior domains
journal, October 1998
- Picard, Rainer
- Mathematische Zeitschrift, Vol. 229, Issue 2
An explicit kernel-split panel-based Nyström scheme for integral equations on axially symmetric surfaces
journal, September 2014
- Helsing, Johan; Karlsson, Anders
- Journal of Computational Physics, Vol. 272
A fast, high-order solver for the Grad–Shafranov equation
journal, June 2013
- Pataki, Andras; Cerfon, Antoine J.; Freidberg, Jeffrey P.
- Journal of Computational Physics, Vol. 243
Universal quadratures for boundary integral equations on two-dimensional domains with corners
journal, November 2010
- Bremer, James; Rokhlin, Vladimir; Sammis, Ian
- Journal of Computational Physics, Vol. 229, Issue 22
On Force-Free Magnetic Fields
journal, January 1956
- Chandrasekhar, S.
- Proceedings of the National Academy of Sciences, Vol. 42, Issue 1
A Compact Cylindrical Green’s Function Expansion for the Solution of Potential Problems
journal, December 1999
- Cohl, Howard S.; Tohline, Joel E.
- The Astrophysical Journal, Vol. 527, Issue 1
Corner singularities for elliptic problems: Integral equations, graded meshes, quadrature, and compressed inverse preconditioning
journal, October 2008
- Helsing, Johan; Ojala, Rikard
- Journal of Computational Physics, Vol. 227, Issue 20
On the Nyström discretization of integral equations on planar curves with corners
journal, January 2012
- Bremer, James
- Applied and Computational Harmonic Analysis, Vol. 32, Issue 1
Debye Sources and the Numerical Solution of the Time Harmonic Maxwell Equations II
journal, August 2012
- Epstein, Charles L.; Greengard, Leslie; O'Neil, Michael
- Communications on Pure and Applied Mathematics, Vol. 66, Issue 5
Debye Sources, Beltrami Fields, and a Complex Structure on Maxwell Fields
journal, January 2015
- Epstein, Charles L.; Greengard, Leslie; O'Neil, Michael
- Communications on Pure and Applied Mathematics, Vol. 68, Issue 12
Self-adjoint curl operators
journal, February 2011
- Hiptmair, Ralf; Kotiuga, Peter Robert; Tordeux, Sébastien
- Annali di Matematica Pura ed Applicata, Vol. 191, Issue 3
Chandrasekhar-Kendall modes and Taylor relaxation in an axisymmetric torus
journal, October 2005
- Tang, X. Z.; Boozer, A. H.
- Physics of Plasmas, Vol. 12, Issue 10
Computation of multi-region relaxed magnetohydrodynamic equilibria
journal, November 2012
- Hudson, S. R.; Dewar, R. L.; Dennis, G.
- Physics of Plasmas, Vol. 19, Issue 11
On constant-alpha force-free fields in a torus
journal, January 1986
- Kress, R.
- Journal of Engineering Mathematics, Vol. 20, Issue 4
Computing Zeros on a Real Interval through Chebyshev Expansion and Polynomial Rootfinding
journal, January 2002
- Boyd, John P.
- SIAM Journal on Numerical Analysis, Vol. 40, Issue 5
Woltjer-Taylor State without Taylor’s Conjecture: Plasma Relaxation at all Wavelengths
journal, December 2012
- Qin, Hong; Liu, Wandong; Li, Hong
- Physical Review Letters, Vol. 109, Issue 23
Magnetic Field Confinement in the Solar Corona. I. Force‐free Magnetic Fields
journal, May 2004
- Flyer, N.; Fornberg, B.; Thomas, S.
- The Astrophysical Journal, Vol. 606, Issue 2
The Taylor relaxed state in tokamaks and the sawtooth collapse
journal, January 1989
- Hastie, R. J.
- Nuclear Fusion, Vol. 29, Issue 1
A Theorem on Force-Free Magnetic Fields
journal, June 1958
- Woltjer, L.
- Proceedings of the National Academy of Sciences, Vol. 44, Issue 6
Noncircular, finite aspect ratio, local equilibrium model
journal, April 1998
- Miller, R. L.; Chu, M. S.; Greene, J. M.
- Physics of Plasmas, Vol. 5, Issue 4
The role of edge current-driven modes in ELM activity
journal, September 2006
- Gimblett, C. G.; Hastie, R. J.; Helander, P.
- Plasma Physics and Controlled Fusion, Vol. 48, Issue 10
Non-axisymmetric, multi-region relaxed magnetohydrodynamic equilibrium solutions
journal, December 2011
- Hudson, S. R.; Dewar, R. L.; Hole, M. J.
- Plasma Physics and Controlled Fusion, Vol. 54, Issue 1
On an integral equation in electromagnetic diffraction theory
journal, June 1966
- Werner, Peter
- Journal of Mathematical Analysis and Applications, Vol. 14, Issue 3
Hydromagnetic equilibria and force-free fields
journal, September 1958
- Grad, Harold; Rubin, Hanan
- Journal of Nuclear Energy (1954), Vol. 7, Issue 3-4
Approximation of linear force-free fields in bounded 3-D domains
journal, January 2000
- Boulmezaoud, T. Z.; amari, T.
- Mathematical and Computer Modelling, Vol. 31, Issue 2-3
On Force-Free Magnetic Fields
journal, January 1956
- Chandrasekhar, S.
- Proceedings of the National Academy of Sciences, Vol. 42, Issue 1
A Theorem on Force-Free Magnetic Fields
journal, June 1958
- Woltjer, L.
- Proceedings of the National Academy of Sciences, Vol. 44, Issue 6
Observation of a Helical Self-Organized State in a Compact Toroidal Plasma
journal, November 2009
- Cothran, C. D.; Brown, M. R.; Gray, T.
- Physical Review Letters, Vol. 103, Issue 21
Minimally Constrained Model of Self-Organized Helical States in Reversed-Field Pinches
journal, July 2013
- Dennis, G. R.; Hudson, S. R.; Terranova, D.
- Physical Review Letters, Vol. 111, Issue 5
Determination of normalized electric eigenfields in microwave cavities with sharp edges
preprint, January 2015
- Helsing, Johan; Karlsson, Anders
- arXiv
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