Periodic Korteweg–de Vries soliton potentials generate quasisymmetric magnetic field strength in a finite plasma-β equilibrium

Sengupta, W.; Madan, R.; Buller, S.; Nikulsin, N.; Paul, E. J.; Nies, R.; Kaptanoglu, A. A.; Hudson, S. R.; Bhattacharjee, A.

doi:10.1063/5.0289641

Periodic Korteweg–de Vries soliton potentials generate quasisymmetric magnetic field strength in a finite plasma-β equilibrium

Journal Article · Wed Oct 22 00:00:00 EDT 2025 · Physics of Plasmas

DOI:https://doi.org/10.1063/5.0289641· OSTI ID:2999846

^[1]; Madan, R. ^[2]; ^[1]; ^[3]; ^[4]; ^[2]; ^[5]; ^[6]; ^[1]

Princeton Univ., NJ (United States)
Princeton Univ., NJ (United States); Princeton Plasma Physics Laboratory (PPPL), Princeton, NJ (United States)
Princeton Univ., NJ (United States); Max Planck Institute for Plasma Physics, Greifswald (Germany)
Columbia Univ., New York, NY (United States)
New York Univ. (NYU), NY (United States)
Princeton Plasma Physics Laboratory (PPPL), Princeton, NJ (United States)

Quasisymmetry (QS) is a hidden symmetry of the magnetic field strength, B, that enables effective confinement of charged particles in a fully three-dimensional (3D) toroidal plasma equilibrium. Such equilibria are typically modeled by the ideal magnetohydrostatic (MHS) equations. The nonlinear, overdetermined nature of the QS MHS equations severely complicates our understanding of the interplay between 3D shaping, equilibrium properties such as pressure and rotational transform, and B. Progress has been made through expansions near the magnetic axis; however, a more comprehensive theory is desirable. Using a combination of analysis and regression on a large dataset of numerically optimized quasisymmetric stellarators, we demonstrate that there is a hidden lower dimensionality of B on a magnetic flux surface with connections to the theory of periodic solitons. We show that B on a flux surface is determined by three or at most four flux functions, each of which determines a critical value of the derivative of B along the field line. While being consistent with the near-axis models, our results are global and hold even on the last closed flux surface.

View Accepted Manuscript (DOE)

Research Organization:: Princeton Plasma Physics Laboratory (PPPL), Princeton, NJ (United States)

Sponsoring Organization:: USDOE

Grant/Contract Number:: SC0024548

OSTI ID:: 2999846

Journal Information:: Physics of Plasmas, Journal Name: Physics of Plasmas Journal Issue: 10 Vol. 32; ISSN 1070-664X; ISSN 1089-7674

Publisher:: American Institute of Physics (AIP)Copyright Statement

Country of Publication:: United States

Language:: English

References (58)

The periodic problem for the Korteweg?de vries equation Novikov, S. P. Functional Analysis and Its Applications, Vol. 8, Issue 3 https://doi.org/10.1007/BF01075697	journal	January 1975
Über die Existenz toroidaler magnetohydrostatischer Gleichgewichte ohne Rotationstransformation Lortz, Dietrich Zeitschrift für angewandte Mathematik und Physik ZAMP, Vol. 21, Issue 2 https://doi.org/10.1007/BF01590644	journal	March 1970
Some considerations on closed configurations of magnetohydrostatic equilibrium Palumbo, D. Il Nuovo Cimento B Series 10, Vol. 53, Issue 2 https://doi.org/10.1007/BF02710251	journal	February 1968
Changes in adiabatic invariants Keller, Joseph B.; Mu, Ye Annals of Physics, Vol. 205, Issue 1 https://doi.org/10.1016/0003-4916(91)90243-2	journal	January 1991
Single-stage gradient-based stellarator coil design: Optimization for near-axis quasi-symmetry Giuliani, Andrew; Wechsung, Florian; Cerfon, Antoine Journal of Computational Physics, Vol. 459 https://doi.org/10.1016/j.jcp.2022.111147	journal	June 2022
Nested toroidal flux surfaces in magnetohydrostatics. Generation via soliton theory Schief, W. K. Journal of Plasma Physics, Vol. 69, Issue 6 https://doi.org/10.1017/S0022377803002472	journal	November 2003
Quasi-axisymmetric magnetic fields: weakly non-axisymmetric case in a vacuum Plunk, G. G.; Helander, Per Journal of Plasma Physics, Vol. 84, Issue 2 https://doi.org/10.1017/S0022377818000259	journal	April 2018
Direct construction of optimized stellarator shapes. Part 1. Theory in cylindrical coordinates Landreman, Matt; Sengupta, Wrick Journal of Plasma Physics, Vol. 84, Issue 6 https://doi.org/10.1017/S0022377818001289	journal	December 2018
Direct construction of optimized stellarator shapes. Part 2. Numerical quasisymmetric solutions Landreman, Matt; Sengupta, Wrick; Plunk, Gabriel G. Journal of Plasma Physics, Vol. 85, Issue 1 https://doi.org/10.1017/S0022377818001344	journal	January 2019
Direct construction of optimized stellarator shapes. Part 3. Omnigenity near the magnetic axis Plunk, Gabriel G.; Landreman, Matt; Helander, Per Journal of Plasma Physics, Vol. 85, Issue 6 https://doi.org/10.1017/S002237781900062X	journal	November 2019
Constructing stellarators with quasisymmetry to high order Landreman, Matt; Sengupta, Wrick Journal of Plasma Physics, Vol. 85, Issue 6 https://doi.org/10.1017/S0022377819000783	journal	November 2019
Perturbing an axisymmetric magnetic equilibrium to obtain a quasi-axisymmetric stellarator Plunk, G. G. Journal of Plasma Physics, Vol. 86, Issue 4 https://doi.org/10.1017/S0022377820000902	journal	August 2020
Magnetic well and Mercier stability of stellarators near the magnetic axis Landreman, Matt; Jorge, Rogerio Journal of Plasma Physics, Vol. 86, Issue 5 https://doi.org/10.1017/S002237782000121X	journal	October 2020
On quasisymmetric plasma equilibria sustained by small force Constantin, Peter; Drivas, Theodore D.; Ginsberg, Daniel Journal of Plasma Physics, Vol. 87, Issue 1 https://doi.org/10.1017/S0022377820001610	journal	February 2021
Vacuum magnetic fields with exact quasisymmetry near a flux surface. Part 1. Solutions near an axisymmetric surface Sengupta, Wrick; Paul, Elizabeth J.; Weitzner, Harold Journal of Plasma Physics, Vol. 87, Issue 2 https://doi.org/10.1017/S0022377821000039	journal	March 2021
Direct construction of stellarator-symmetric quasi-isodynamic magnetic configurations Camacho Mata, Katia; Plunk, Gabriel G.; Jorge, Rogerio Journal of Plasma Physics, Vol. 88, Issue 5 https://doi.org/10.1017/S0022377822000812	journal	October 2022
A single-field-period quasi-isodynamic stellarator Jorge, R.; Plunk, G. G.; Drevlak, M. Journal of Plasma Physics, Vol. 88, Issue 5 https://doi.org/10.1017/S0022377822000873	journal	September 2022
Mapping the space of quasisymmetric stellarators using optimized near-axis expansion Landreman, Matt Journal of Plasma Physics, Vol. 88, Issue 6 https://doi.org/10.1017/S0022377822001258	journal	December 2022
Magnetohydrodynamic stability and the effects of shaping: a near-axis view for tokamaks and quasisymmetric stellarators Rodríguez, Eduardo Journal of Plasma Physics, Vol. 89, Issue 2 https://doi.org/10.1017/S0022377823000211	journal	April 2023
Constructing precisely quasi-isodynamic magnetic fields Goodman, A. G.; Camacho Mata, K.; Henneberg, S. A. Journal of Plasma Physics, Vol. 89, Issue 5 https://doi.org/10.1017/S002237782300065X	journal	September 2023
Asymptotic quasisymmetric high-beta three-dimensional magnetohydrodynamic equilibria near axisymmetry Sengupta, Wrick; Nikulsin, Nikita; Gaur, Rahul Journal of Plasma Physics, Vol. 90, Issue 2 https://doi.org/10.1017/S0022377824000357	journal	April 2024
Energetic particle tracing in optimized quasi-symmetric stellarator equilibria Figueiredo, P. A.; Jorge, R.; Ferreira, J. Journal of Plasma Physics, Vol. 90, Issue 2 https://doi.org/10.1017/S0022377824000400	journal	April 2024
Direct stellarator coil design using global optimization: application to a comprehensive exploration of quasi-axisymmetric devices Giuliani, Andrew Journal of Plasma Physics, Vol. 90, Issue 3 https://doi.org/10.1017/S0022377824000412	journal	May 2024
An asymptotic Grad–Shafranov equation for quasisymmetric stellarators Nikulsin, Nikita; Sengupta, Wrick; Jorge, Rogerio Journal of Plasma Physics, Vol. 90, Issue 6 https://doi.org/10.1017/S0022377824000916	journal	December 2024
A family of quasi-axisymmetric stellarators with varied rotational transform Buller, S.; Landreman, M.; Kappel, J. Journal of Plasma Physics, Vol. 91, Issue 1 https://doi.org/10.1017/S0022377824001351	journal	January 2025
Collisionless zonal-flow dynamics in quasisymmetric stellarators Zhu, Hongxuan; Lin, Z.; Bhattacharjee, A. Journal of Plasma Physics, Vol. 91, Issue 1 https://doi.org/10.1017/S0022377824001739	journal	January 2025
An extension of Palumbo’s method of solution for the Grad–Shafranov equation Hernandes, J. A.; Clemente, R. A. Physics of Plasmas, Vol. 16, Issue 8 https://doi.org/10.1063/1.3208041	journal	August 2009
“One size fits all” analytic solutions to the Grad–Shafranov equation Cerfon, Antoine J.; Freidberg, Jeffrey P. Physics of Plasmas, Vol. 17, Issue 3 https://doi.org/10.1063/1.3328818	journal	March 2010
Omnigenity as generalized quasisymmetry Landreman, Matt; Catto, Peter J. Physics of Plasmas, Vol. 19, Issue 5 https://doi.org/10.1063/1.3693187	journal	May 2012
Some mathematics for quasi-symmetry Burby, J. W.; Kallinikos, N.; MacKay, R. S. Journal of Mathematical Physics, Vol. 61, Issue 9 https://doi.org/10.1063/1.5142487	journal	September 2020
Magnetic field strength of toroidal plasma equilibria Garren, D. A.; Boozer, A. H. Physics of Fluids B: Plasma Physics, Vol. 3, Issue 10 https://doi.org/10.1063/1.859915	journal	October 1991
Existence of quasihelically symmetric stellarators Garren, D. A.; Boozer, A. H. Physics of Fluids B: Plasma Physics, Vol. 3, Issue 10 https://doi.org/10.1063/1.859916	journal	October 1991
Three-dimensional equilibrium of the anisotropic, finite-pressure guiding-center plasma: Theory of the magnetic plasma Hall, Laurence S.; McNamara, Brendan Physics of Fluids, Vol. 18, Issue 5 https://doi.org/10.1063/1.861189	journal	January 1975
Guiding center drift equations Boozer, Allen H. Physics of Fluids, Vol. 23, Issue 5 https://doi.org/10.1063/1.863080	journal	January 1980
Plasma equilibrium with rational magnetic surfaces Boozer, Allen H. Physics of Fluids, Vol. 24, Issue 11 https://doi.org/10.1063/1.863297	journal	January 1981
Transport and isomorphic equilibria Boozer, Allen H. Physics of Fluids, Vol. 26, Issue 2 https://doi.org/10.1063/1.864166	journal	January 1983
Degenerate toroidal magnetohydrodynamic equilibria and minimum B Bishop, C. M.; Taylor, J. B. The Physics of Fluids, Vol. 29, Issue 4 https://doi.org/10.1063/1.865913	journal	April 1986
Omnigenity and quasihelicity in helical plasma confinement systems Cary, John R.; Shasharina, Svetlana G. Physics of Plasmas, Vol. 4, Issue 9 https://doi.org/10.1063/1.872473	journal	September 1997
Necessary and sufficient conditions for quasisymmetry Rodríguez, E.; Helander, P.; Bhattacharjee, A. Physics of Plasmas, Vol. 27, Issue 6 https://doi.org/10.1063/5.0008551	journal	June 2020
Solving the problem of overdetermination of quasisymmetric equilibrium solutions by near-axis expansions. I. Generalized force balance Rodríguez, E.; Bhattacharjee, A. Physics of Plasmas, Vol. 28, Issue 1 https://doi.org/10.1063/5.0027574	journal	January 2021
Islands and current singularities in quasisymmetric toroidal plasmas Rodríguez, E.; Bhattacharjee, A. Physics of Plasmas, Vol. 28, Issue 9 https://doi.org/10.1063/5.0057186	journal	September 2021
Generalized Boozer coordinates: A natural coordinate system for quasisymmetry Rodríguez, E.; Sengupta, W.; Bhattacharjee, A. Physics of Plasmas, Vol. 28, Issue 9 https://doi.org/10.1063/5.0060115	journal	September 2021
Weakly quasisymmetric near-axis solutions to all orders Rodríguez, E.; Sengupta, W.; Bhattacharjee, A. Physics of Plasmas, Vol. 29, Issue 1 https://doi.org/10.1063/5.0076583	journal	January 2022
Optimization of quasi-symmetric stellarators with self-consistent bootstrap current and energetic particle confinement Landreman, M.; Buller, S.; Drevlak, M. Physics of Plasmas, Vol. 29, Issue 8 https://doi.org/10.1063/5.0098166	journal	August 2022
A Grad-Shafranov model for compact quasisymmetric stellarators Nikulsin, N.; Sengupta, W.; Buller, S. Physics of Plasmas, Vol. 32, Issue 4 https://doi.org/10.1063/5.0258308	journal	April 2025
Meromorphic solutions of algebraic differential equations Eremenko, A. E. Russian Mathematical Surveys, Vol. 37, Issue 4 https://doi.org/10.1070/RM1982v037n04ABEH003967	journal	August 1982
Theory of plasma confinement in non-axisymmetric magnetic fields Helander, Per Reports on Progress in Physics, Vol. 77, Issue 8 https://doi.org/10.1088/0034-4885/77/8/087001	journal	July 2014
Fake Airy functions and the asymptotics of reflectionlessness Berry, M. V.; Howls, C. J. Journal of Physics A: Mathematical and General, Vol. 23, Issue 6 https://doi.org/10.1088/0305-4470/23/6/002	journal	March 1990
Physics of the compact advanced stellarator NCSX Zarnstorff, M. C.; Berry, L. A.; Brooks, A. Plasma Physics and Controlled Fusion, Vol. 43, Issue 12A https://doi.org/10.1088/0741-3335/43/12A/318	journal	November 2001
Stellarator and tokamak plasmas: a comparison Helander, P.; Beidler, C. D.; Bird, T. M. Plasma Physics and Controlled Fusion, Vol. 54, Issue 12 https://doi.org/10.1088/0741-3335/54/12/124009	journal	November 2012
The use of near-axis magnetic fields for stellarator turbulence simulations Jorge, R.; Landreman, M. Plasma Physics and Controlled Fusion, Vol. 63, Issue 1 https://doi.org/10.1088/1361-6587/abc862	journal	November 2020
Phases and phase-transitions in quasisymmetric configuration space Rodríguez, E.; Sengupta, W.; Bhattacharjee, A. Plasma Physics and Controlled Fusion, Vol. 64, Issue 10 https://doi.org/10.1088/1361-6587/ac89af	journal	August 2022
Constructing the space of quasisymmetric stellarators through near-axis expansion Rodríguez, E.; Sengupta, W.; Bhattacharjee, A. Plasma Physics and Controlled Fusion, Vol. 65, Issue 9 https://doi.org/10.1088/1361-6587/ace739	journal	July 2023
Properties of a new quasi-axisymmetric configuration Henneberg, S. A.; Drevlak, M.; Nührenberg, C. Nuclear Fusion, Vol. 59, Issue 2 https://doi.org/10.1088/1741-4326/aaf604	journal	January 2019
Construction of quasisymmetric stellarators using a direct coordinate approach Jorge, R.; Sengupta, W.; Landreman, M. Nuclear Fusion, Vol. 60, Issue 7 https://doi.org/10.1088/1741-4326/ab90ca	journal	June 2020
Critically Balanced Ion Temperature Gradient Turbulence in Fusion Plasmas Barnes, M.; Parra, F. I.; Schekochihin, A. A. Physical Review Letters, Vol. 107, Issue 11 https://doi.org/10.1103/PhysRevLett.107.115003	journal	September 2011
Magnetic Fields with Precise Quasisymmetry for Plasma Confinement Landreman, Matt; Paul, Elizabeth Physical Review Letters, Vol. 128, Issue 3 https://doi.org/10.1103/PhysRevLett.128.035001	journal	January 2022
Asymptotics of reflectionless potentials Gjaja, I.; Bhattacharjee, A. Physical Review Letters, Vol. 68, Issue 16 https://doi.org/10.1103/PhysRevLett.68.2413	journal	April 1992

Similar Records

Asymptotic quasisymmetric high-beta three-dimensional magnetohydrodynamic equilibria near axisymmetry

Journal Article · Thu Apr 04 20:00:00 EDT 2024 · Journal of Plasma Physics · OSTI ID:2335376

Characterization of admissible quasisymmetries

Journal Article · Sun Feb 09 19:00:00 EST 2025 · Journal of Plasma Physics · OSTI ID:2513530

Related Subjects

Differential geometry
Magnetic fields
Plasma confinement
Quasisymmetric
Stellarators
Tokamaks

Periodic Korteweg–de Vries soliton potentials generate quasisymmetric magnetic field strength in a finite plasma-β equilibrium

Citation Formats

References (58)

Similar Records

Related Subjects