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Extending near-axis equilibria in DESC

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Global stellarator equilibria can keep their near-axis design

desk verdict Solid new bridge between near-axis expansions and global DESC equilibria; abstract's 'guarantee' overstates what the method actually enforces. read the letter →

arxiv 2506.05170 v1 pith:IOIDJRSA submitted 2025-06-05 physics.plasm-ph

Dario Panici , Eduardo Rodriguez , Rory Conlin , Daniel Dudt , Egemen Kolemen This is my paper
classification physics.plasm-ph
keywords near-axisexpansionstellaratorglobalMHDequilibriumFourier-ZernikeDESCquasisymmetrymagneticwelllinearconstraints
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper aims to make the near-axis expansion (NAE) of stellarator fields actionable for global equilibrium construction: instead of feeding a near-axis surface far from the axis into a fixed-boundary solver, it constrains the global solve of the DESC code directly at the axis, so the global equilibrium provably reproduces the prescribed near-axis asymptotic behavior. The central technical step is a set of linear constraints on DESC's Fourier-Zernike coefficients (Eqs. (3.8) and (3.13)) that encode the magnetic axis, the elliptical first-order cross-sections, and the second-order triangular shaping of a given NAE solution, while the solver remains free to find the equilibrium far from the axis. If correct, this gives a practical bridge between fast near-axis design and high-fidelity global equilibria, preserving curated near-axis properties at low aspect ratios where the standard fixed-boundary approach loses them; the benchmarks show that on-axis rotational transform, field strength, quasisymmetry-error scaling, and magnetic well all match the NAE far better than the conventional construction.

What carries the argument

The load-bearing object is the linear map between the NAE's Taylor-Fourier coefficients and DESC's Fourier-Zernike coefficients, Eq. (2.13). It converts the requirement 'this flux surface has shape f at order l' into a weighted sum of Zernike modes, with weights that grow like $k^l$ for high radial order $k$; convergence is controlled by a finite truncation. On top of that map, the paper derives a geometric transform (Eqs. (3.6)-(3.7) and (3.12)) that rewrites the axis-frame elliptical and triangular surface shapes in cylindrical coordinates, accounting for the inclination of the axis. These two pieces combine into the concrete linear constraints (3.8) and (3.13) that the solver enforces while minimizing the magnetohydrostatic force residual.

What would settle it

Compute a second-order constrained equilibrium for a configuration with a strongly shaped, high-torsion axis (for example a quasi-isodynamic case), then Fourier-analyze the $R$ and $Z$ cross-sections at small fixed $\rho$ and compare the second-order coefficients to the NAE-predicted lab-frame values; a systematic deviation as torsion grows would indicate that the leading-order angle identification is the cause.

Watch

Extended reading notes

Core claim

The central claim is that a global ideal-MHD equilibrium in DESC can be forced to have exactly the near-axis behavior of a chosen NAE solution by imposing a handful of linear constraints on the solver's spectral coefficients. The near-axis Taylor-Fourier description of flux surfaces and the Zernike representation used by DESC are connected by an exact linear relation (2.13), and the near-axis surfaces, which are naturally written in the Frenet-Serret frame of the axis and in Boozer coordinates, are mapped to the cylindrical lab frame through the geometric relations (3.6)-(3.7) at first order and (3.12a)-(3.12b) at second order. The resulting constraints fix the axis shape, the elliptical cross-sections, and the second-order shaping (including the Shafranov shift and triangularity as seen in the lab frame) as linear combinations of Fourier-Zernike modes. With these constraints imposed through a feasible-direction method, DESC returns equilibria whose on-axis rotational transform, on-axis |B|, stream function, and magnetic well agree with the NAE to orders of magnitude better than equilibria built from a finite-radius near-axis boundary, and the quasisymmetry error scales as $O(\rho^2)$ (first-order constraints) or $O(\rho^3)$ (second-order constraints).

Load-bearing premise

The derivation assumes the poloidal angle used by DESC can be identified with the near-axis Boozer angle through the simple stream-function relation $\lambda = -\iota\nu$, which is only guaranteed to leading order; if that angle identification fails at higher order, the geometric constraints will not enforce the intended near-axis behavior.

Editorial extensions

If this is right

  • Near-axis-optimized properties—rotational transform, on-axis field strength, quasisymmetry error, and magnetic well—survive in a global equilibrium at low aspect ratio, where finite-radius fixed-boundary construction degrades them.
  • The NAE-constrained equilibrium provides a much better initial condition for conventional stellarator optimization than a fixed-boundary solve from a large-radius near-axis surface.
  • Because the constraints are linear, they can be added to the same constrained-optimization machinery already used for fixed-boundary solves, with modest extra cost.
  • Second-order constraints transfer magnetohydrodynamic-stability-linked features such as the magnetic well from the near-axis design into the global solution.
  • The method generalizes beyond vacuum fields, since the constraint derivation itself does not require a vacuum or quasisymmetry assumption.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same linear-constraint approach could be adapted to other spectral equilibrium codes or boundary-based solvers, as long as a radial-poloidal basis with the right near-axis regularity is available.
  • The angle-agnostic formulation sketched in Appendix B suggests a testable improvement: letting DESC use a generalized poloidal angle should reduce the toroidal-mode burden for quasi-isodynamic configurations with straight axis sections.
  • One could use the freedom left by the near-axis constraints to scan off-axis properties systematically, generating families of global equilibria that share the same core design but differ in boundary shaping, coils, or stability.
  • Soft or partial enforcement of the constraints, rather than exact imposition, may be useful during optimization to trade near-axis fidelity against other objectives.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper presents a method for constructing global ideal-MHD equilibria in the DESC code that match a prescribed near-axis expansion (NAE) to zeroth, first, or second order in the distance from the magnetic axis. The authors derive linear constraints on DESC's Fourier-Zernike coefficients by (i) relating the NAE Taylor-Fourier basis to the Zernike basis, and (ii) transforming the NAE Frenet-Serret description into the cylindrical-coordinate description used by DESC. They implement the constraints using pyQSC and pyQIC inputs, solve constrained equilibria, and then relax the constraints and re-solve as fixed-boundary equilibria as a final check. Benchmarks for quasi-axisymmetric, quasi-helically symmetric, and quasi-isodynamic configurations show that on-axis quantities such as B0, iota0, and V'' match the NAE values much better than conventional surface-built equilibria, and that the quasisymmetry error scales as expected with the order of the imposed expansion.

Significance. If the results hold, this is a useful methodological advance: it gives a systematic, transparent route from near-axis optimized designs to global equilibria, avoiding the common practice of using the NAE at a finite radius where it is least valid. The theoretical derivations in Sections 3.2-3.3 and Appendices D-E are carefully presented, and the paper ships data and plotting scripts on Princeton Data Commons, which is commendable. The numerical benchmarks show the expected O(rho^2) and O(rho^3) scaling of the quasisymmetry error and large improvements in on-axis rotational transform and magnetic well compared with the fixed-boundary approach. The main caveats are that the abstract's 'guarantees' is stronger than what the implemented workflow actually enforces, and the identification of DESC's poloidal angle with the Boozer angle is not directly verified.

major comments (3)
  1. [Section 4, paragraph before §4.1; Tables 1-2] The abstract's claim that the construction 'guarantees the correct asymptotic behaviour' is not supported by the implemented workflow, because the near-axis constraints are relaxed after the constrained solve and the equilibrium is re-solved as a fixed-boundary problem. The verification numbers in Tables 1 and 2 therefore describe a solution that is no longer subject to the constraints; for example, Table 1 reports Delta_lambda0 = 5.35e-02 for the QI case and Delta_lambda0 = 2.54e-02 for the QH case. Please either present the constrained solution itself as the deliverable, quantify the drift introduced by the relaxation step, or replace 'guarantees' with language that accurately describes the constrained solve and the subsequent verification step.
  2. [§2.2, Eq. (2.9); §3.2.1, Eq. (3.10)] The identification of DESC's computational poloidal angle theta with the Boozer angle theta_B through lambda = -iota*nu is an assumption, not an enforced condition. DESC solves for lambda as part of the equilibrium, and at first order only R and Z are constrained; if the computed lambda deviates from -iota*nu at O(rho), then the imposed R/Z constraints no longer correspond to the intended NAE behavior in Boozer coordinates. The nonzero values of Delta_lambda0 in Table 1 show that this deviation is not negligible, especially for QH and QI. Please add a diagnostic that compares the first-order stream function lambda_1 with Eq. (3.10), or impose a constraint on lambda_1 as well, before claiming that the asymptotic behavior is guaranteed.
  3. [§4.1 and §5] The verification compares quantities such as B0, iota0, V'', and the fB scaling against the same NAE data that were used to construct the constraints. This is a legitimate consistency check of the implementation, but it cannot independently validate that the global equilibrium has the intended near-axis behavior in Boozer coordinates. A more independent test, such as a direct Boozer-coordinate Fourier analysis of the DESC solution or a comparison against a separately generated NAE solution, would substantially strengthen the paper's central claim.
minor comments (6)
  1. [Section 1] The phrase 'Their benefits stride from the freedom' should be 'Their benefits stem from the freedom'.
  2. [Figure 4 caption] The word 'meaingful' should be 'meaningful'.
  3. [Section 5.2] The word 'equilibirum' should be 'equilibrium'.
  4. [Eq. (2.13) and Appendix C.3] Equation (2.13) is numbered identically in the main text and in Appendix C.3; renumber one of them or reference the equation only once.
  5. [Eq. (3.16b)] The symbols tau_tilde, kappa'_Z, and kappa_R are used in the equation but defined only in the surrounding prose; a brief definition immediately before the equation would improve readability.
  6. [References] Several references contain LaTeX artifacts such as 'tex.ids=' and stray 'publisher:' fields (e.g., Boozer 1983, Anderson et al. 1995, Landreman 2022b); these should be cleaned up before publication.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the near-axis inputs are external, and the verification diagnostics come from an independent force-balance solve.

full rationale

The derivation chain is self-contained and not circular. The near-axis expansion (NAE) data from pyQSC and pyQIC enter as external inputs into the linear constraints (3.2), (3.8), and (3.13), which map prescribed Taylor-Fourier near-axis coefficients onto DESC Fourier-Zernike degrees of freedom. These constraints fix the magnetic axis, first-order elliptical shaping, and second-order triangular shaping, but they do not prescribe the equilibrium diagnostics used for verification. The global equilibrium is obtained by minimizing the force residual (4.1) under these linear equality constraints, so quantities such as B0, iota0, lambda0, V'', and the fB scaling are outputs of an independent force-balance solve, not fitted parameters relabeled as predictions. Section 4.1 explicitly states that the comparison is made on derived quantities rather than the directly constrained flux-surface shapes: 'To make the comparison as impartial as possible, we do not compare field quantities that correspond to the shape of flux surfaces which we are directly enforcing by the constraint, but rather, derived ones.' The reported nonzero deviations, e.g., Delta-lambda0 = 5.35e-02 for the QI case in Table 1, confirm that the comparisons are not forced by construction. The paper's main limitation, namely that the identification theta = theta_B through lambda = -iota nu (Eq. 2.9) is assumed for the constraints rather than explicitly imposed on lambda, is an acknowledged correctness and scope caveat (Appendix B, Section 6), not a circular step. Self-citations (Panici et al. 2023; Dudt et al. 2023; Rodriguez et al. 2025) provide background and solver context but are not the load-bearing evidence for the central construction, which rests on the external NAE framework and the DESC force-balance solve. Therefore no claimed result reduces to its own input by construction.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

No physical constants or fitted parameters are added: the near-axis input is provided by external NAE codes (pyQSC and pyQIC), and the numerical truncations are implementation choices. The main ad hoc assumption is the angle identification lambda = -iota*nu in Eq. (2.9), which the authors acknowledge can be relaxed in Appendix B. No new physical entities are introduced.

free parameters (1)
  • Fourier-Zernike spectral truncation (L, M, N) = L=9-15, M=9-10, N=24-25 depending on configuration
    The mapping in Eq. (2.13) is an infinite sum, and truncation is required for implementation. Finite toroidal resolution is shown in Appendix F to discard part of the NAE Fourier content, especially for quasi-isodynamic configurations and at second order.
assumptions (6)
  • domain assumption The magnetic axis is regular enough for the Frenet-Serret frame to be used (no, or only a few isolated, flattening points).
    Section 2.1 limits the treatment to regular axes with only isolated zero-curvature points, which excludes axes with extended straight sections that are difficult to represent in cylindrical coordinates.
  • domain assumption The truncated NAE (to first or second order) is a quantitatively valid approximation of the desired global equilibrium near the axis.
    The method transfers NAE coefficients to global constraints; if the NAE itself is inaccurate for a configuration, the constrained equilibrium inherits that inaccuracy.
  • ad hoc to paper DESC's poloidal coordinate theta can be identified with the Boozer angle at leading order through lambda = -iota*nu, Eq. (2.9).
    This identification is what makes the simple constraints exact; the paper notes it is restrictive and Appendix B outlines a generalized angle treatment that is not implemented in the main text.
  • ad hoc to paper The Zernike expansion can be truncated without the omitted high-order modes spoiling the low-order NAE coefficients in Eq. (2.13).
    Eq. (2.13) involves an infinite sum with polynomial weights; implementation truncates it. Appendix F studies toroidal truncation effects but not the full radial truncation behavior.
  • ad hoc to paper Relaxing the NAE constraints and re-solving as a fixed-boundary equilibrium preserves the near-axis behavior.
    Section 4.1 performs this final relaxation as a sanity check; preservation is verified in benchmarks but is not guaranteed by the constraint construction itself.
  • standard math The Fourier-Zernike representation and the inverse-coordinate Clebsch form (2.1)-(2.5) faithfully represent smooth equilibria.
    The mapping in Eq. (2.13) relies on standard Zernike and Jacobi polynomial expansions with the assumed parity constraints; this is a standard basis representation in DESC.

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Cite this review

Pith. "Pith review of Extending near-axis equilibria in DESC." pith.science (2026). https://pith.science/paper/IOIDJRSA

@misc{pith2026250605170,
  author       = {Pith},
  title        = {Pith review of: Extending near-axis equilibria in DESC},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IOIDJRSA}},
  note         = {Machine review of arXiv:2506.05170}
}
read the original abstract

The near-axis description of optimised stellarator fields has proven to be a powerful tool both for design and understanding of this magnetic confinement concept. The description consists of an asymptotic model of the equilibrium in the distance from its centermost axis, and is thus only approximate. Any practical application therefore requires the eventual construction of a global equilibrium. This paper presents a novel way of constructing global equilibria using the \texttt{DESC} code that guarantees the correct asymptotic behaviour imposed by a given near-axis construction. The theoretical underpinnings of this construction are carefully presented, and benchmarking examples provided. This opens the door to an efficient coupling of the near-axis framework and that of global equilibria for future optimisation efforts.

Figures

Figures reproduced from arXiv: 2506.05170 by the authors.

Figure 1
Figure 1. Diagram illustrating the key element for the geometric transformation at first order. Schematic diagram showing the position of a point in the surface (at radial distance R and angle ϕ0 + δϕ), in reference to other quantities. These include the position along the magnetic axis (radial position R0 and angle ϕ0), and the ρx1 from the axis to the point, projected onto the R, ϕc plane (superindex π). 3.2. First order: e… view at source ↗
Figure 2
Figure 2. Definition of the slant angle α. Diagram showing the definition of the angle α measuring the inclination of the magnetic axis at the origin (ϕ = 0) with the ‘lab’ cylindrical coordinate system. The symbols have their usual meaning. 2019, Eq. (A20)), λ (0) = −ι0 Z ϕ 0  B0 G0 dℓ dϕ − 1  dϕ ′ , (3.9) where ℓ is the length along the magnetic axis, B0 the magnetic field magnitude, G0 the poloidal Boozer current and ι0 … view at source ↗
Figure 3
Figure 3. Geometric deformation of cross-sections from the near-axis to the lab frame. Example of the change in the cross-sections due to the geometric effects of going from the frame of the axis (broken lines) to the lab-frame (solid line). These correspond to the cross sections of the "precise QH" stellarator configuration from Landreman & Paul (2022) at one of its stellarator symmetric points. The left corresponds to the c… view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Verification of first order NAE-constrained equilibria. The figure shows a comparison of DESC 1st-order NAE constrained equilibrium solutions (in red) against DESC near-axis fixed-boundary solutions (in green) and the ideal near-axis field (broken line). From left to r…
Figure 5
Figure 5. Figure 5: Verification of second order NAE-constrained equilibria. The figure shows a comparison of DESC 2nd-order NAE constrained equilibrium solutions (in red) against DESC fixed-boundary solutions (in green) and the ideal near-axis field (broken line). The plots correspond to…
Figure 6
Figure 6. Figure 6: Diagram illustrating the key element for the geometric transformation at second order. Schematic diagram showing the position of a point in the surface (at radial distance R and angle ϕ0 + δϕ), in reference to other quantities. These include the position along the magn…
Figure 7
Figure 7. Figure 7: (Left) Fourier coefficients as a function of toroidal mode number of the [PITH_FULL_IMAGE:figures/full_fig_p028_7.png]

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Cited by 2 Pith papers

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Reference graph

Works this paper leans on

61 extracted references · 52 canonical work pages · cited by 2 Pith papers

  1. [1]

    , " * write output.state after.block = add.period write newline

    ENTRY address author booktitle chapter edition editor howpublished institution journal key month note number organization pages publisher school series title type volume year eprint label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sentence ...

  2. [2]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...

  3. [3]

    Anderson, F. Simon B. , Almagri, Abdulgader F. , Anderson, David T. , Matthews, Peter G. , Talmadge, Joseph N. & Shohet, J. Leon 1995 The Helically Symmetric Experiment , ( HSX ) Goals , Design and Status . Fusion Technology 27 (3T), 273--277 , publisher: Taylor & Francis \_eprint: https://doi.org/10.13182/FST95-A11947086

    work page doi:10.13182/fst95-a11947086 1995

  4. [4]

    CRC press

    Animov, Yu 2001 Differential geometry and topology of curves\/ . CRC press

    work page 2001

  5. [5]

    Berlin, Heidelberg: Springer Berlin Heidelberg

    Bauer, Frances , Betancourt, Octavio & Garabedian, Paul 1978 A Computational Method in Plasma Physics \/ . Berlin, Heidelberg: Springer Berlin Heidelberg

    work page 1978

  6. [6]

    Bernardin, M. P. , Moses, R. W. & Tataronis, J. A. 1986 Isodynamical (omnigenous) equilibrium in symmetrically confined plasma configurations . The Physics of Fluids 29 (8), 2605--2611

    work page 1986

  7. [7]

    1981 Plasma equilibrium with rational magnetic surfaces

    Boozer, Allen H. 1981 Plasma equilibrium with rational magnetic surfaces . The Physics of Fluids 24 (11), 1999--2003 , publisher: American Institute of Physics

    work page 1981

  8. [8]

    5 , tex.ids= boozer\_transport\_1983-1

    Boozer, Allen H 1983 Transport and isomorphic equilibria p. 5 , tex.ids= boozer\_transport\_1983-1

    work page 1983

Show all 61 references
  1. [9]

    Nuclear Fusion 61 (9), 096024

    Boozer, Allen H 2021 Stellarators as a fast path to fusion . Nuclear Fusion 61 (9), 096024

    2021

  2. [10]

    & Plunk, G

    Camacho Mata, K. & Plunk, G. G. 2023 Helicity of the magnetic axes of quasi-isodynamic stellarators . Journal of Plasma Physics 89 (6), 905890609

    2023

  3. [11]

    arXiv preprint arXiv:1311.5857

    Carroll, Daniel , K \"o se, Emek & Sterling, Ivan 2013 Improving frenet's frame using bishop's frame . arXiv preprint arXiv:1311.5857

    2013 arXiv

  4. [12]

    Cary, J. R. & Shasharina, S. G. 1997 Omnigenity and quasihelicity in helical plasma confinement systems . Physics of Plasmas 4 (9), 3323--3333 , arXiv:arXiv: https://pubs.aip.org/aip/pop/article-pdf/4/9/3323/12664528/3323\_1\_online.pdf

    1997

  5. [13]

    , Panici, Dario & Kolemen, Egemen 2024 Stellarator Optimization with Constraints

    Conlin, Rory , Kim, Patrick , Dudt, Daniel W. , Panici, Dario & Kolemen, Egemen 2024 Stellarator Optimization with Constraints . ArXiv:2403.11033 [physics]

    2024 arXiv

  6. [14]

    Dewar, R. L. & Hudson, S. R. 1998 Stellarator symmetry . Physica D: Nonlinear Phenomena 112 (1), 275--280 , tex.ids= dewar\_stellarator\_1998-1

    1998

  7. [15]

    D'haeseleer, W. D. , Hitchon, W. N. G. , Callen, J. D. & Shohet, J. L. 2012 Flux coordinates and magnetic field structure: a guide to a fundamental tool of plasma theory\/ . Springer Science & Business Media

    2012

  8. [16]

    Dudt, D. W. , Conlin, R. , Panici, D. & Kolemen, E. 2023 The DESC stellarator code suite Part 3: Quasi -symmetry optimization . Journal of Plasma Physics 89 (2), 955890201 , publisher: Cambridge University Press

    2023

  9. [17]

    , Goodman, Alan G

    Dudt, Daniel W. , Goodman, Alan G. , Conlin, Rory , Panici, Dario & Kolemen, Egemen 2024 Magnetic fields with general omnigenity . Journal of Plasma Physics 90 (1), 905900120

    2024

  10. [18]

    Dudt, D. W. , W., Conlin , D., Panici , Unalmis, K. , Kim, P. & Kolemen, E. 2022 Desc. https://github.com/PlasmaControl/DESC

    2022

  11. [19]

    Journal de math \'e matiques pures et appliqu \'e es 17 , 437--447

    Frenet, F 1852 Sur les courbes \`a double courbure . Journal de math \'e matiques pures et appliqu \'e es 17 , 437--447

  12. [20]

    Garren, D. A. & Boozer, A. H. 1991 a\/ Existence of quasihelically symmetric stellarators . Physics of Fluids B: Plasma Physics 3 (10), 2822--2834 , publisher: American Institute of Physics

    1991

  13. [21]

    Garren, D. A. & Boozer, A. H. 1991 b\/ Magnetic field strength of toroidal plasma equilibria . Physics of Fluids B: Plasma Physics 3 (10), 2805--2821 , publisher: American Institute of Physics

    1991

  14. [22]

    Garren, D. A. & Boozer, A. H. 1991 c\/ Magnetic field strength of toroidal plasma equilibria . Physics of Fluids B: Plasma Physics 3 (10), 2805--2821

    1991

  15. [23]

    arXiv preprint arXiv:2409.04826

    Giuliani, Andrew , Rodr \' guez, Eduardo & Spivak, Marina 2024 A comprehensive exploration of quasisymmetric stellarators and their coil sets . arXiv preprint arXiv:2409.04826

    2024 arXiv

  16. [24]

    In Theory of Fusion Plasmas-Proceedings of the Joint Varenna-Laussane International Workshop\/ , pp

    Gori, S , Lotz, W & N \"u hrenberg, J 1997 Quasi-isodynamic stellarators. In Theory of Fusion Plasmas-Proceedings of the Joint Varenna-Laussane International Workshop\/ , pp. 335--342

    1997

  17. [25]

    Greene, J M 1998 A BRIEF REVIEW OF MAGNETIC WELLS p. 15

    1998

  18. [26]

    & McNamara, Brendan 1975 Three‐dimensional equilibrium of the anisotropic, finite‐pressure guiding‐center plasma: Theory of the magnetic plasma

    Hall, Laurence S. & McNamara, Brendan 1975 Three‐dimensional equilibrium of the anisotropic, finite‐pressure guiding‐center plasma: Theory of the magnetic plasma . The Physics of Fluids 18 (5), 552--565

    1975

  19. [27]

    In Joint Varenna-Lausanne International Workshop on Theory of Fusion Plasmas\/

    Hindenlang, F , Plunk, G & Maj, O 2024 A generalized frenet frame for computing mhd equilibria. In Joint Varenna-Lausanne International Workshop on Theory of Fusion Plasmas\/

    2024

  20. [28]

    Plasma Physics and Controlled Fusion 67 (4), 045002 , publisher: IOP Publishing

    Hindenlang, Florian , Plunk, Gabriel G & Maj, Omar 2025 Computing MHD equilibria of stellarators with a flexible coordinate frame . Plasma Physics and Controlled Fusion 67 (4), 045002 , publisher: IOP Publishing

    2025

  21. [29]

    Hirshman, S. P. & Whitson, J. C. 1983 Steepest‐descent moment method for three‐dimensional magnetohydrodynamic equilibria . The Physics of Fluids 26 (12), 3553--3568 , publisher: American Institute of Physics

    1983

  22. [30]

    , Agostinho, B

    Jorge, R. , Agostinho, B. & Rodr\' i guez, E. 2023 rogeriojorge/pyqic: Version 0.2 of pyqic

    2023

  23. [31]

    Ku, L. P. & Boozer, A. H. 2010 New classes of quasi-helically symmetric stellarators . Nuclear Fusion 51 (1), 013004

    2010

  24. [32]

    Journal of Computational Physics 73 (1), 107--124

    Kuo-Petravic, G & Boozer, Allen H 1987 Numerical determination of the magnetic field line hamiltonian . Journal of Computational Physics 73 (1), 107--124

    1987

  25. [33]

    Plasma Physics and Controlled Fusion 61 (7), 075001 , tex.ids= landreman\_optimized\_2019-1 arXiv: 1902.01672 [physics] publisher: IOP Publishing

    Landreman, Matt 2019 Optimized quasisymmetric stellarators are consistent with the Garren - Boozer construction . Plasma Physics and Controlled Fusion 61 (7), 075001 , tex.ids= landreman\_optimized\_2019-1 arXiv: 1902.01672 [physics] publisher: IOP Publishing

    2019 arXiv

  26. [34]

    Journal of Plasma Physics 87 (1), 905870112 , tex.ids= landreman\_figures\_2021

    Landreman, Matt 2021 Figures of merit for stellarators near the magnetic axis . Journal of Plasma Physics 87 (1), 905870112 , tex.ids= landreman\_figures\_2021

    2021

  27. [35]

    ArXiv:2209.11849 [physics]

    Landreman, Matt 2022 a\/ Mapping the space of quasisymmetric stellarators using optimized near-axis expansion. ArXiv:2209.11849 [physics]

    2022 arXiv

  28. [36]

    2022 b\/ pyqsc

    Landreman, M. 2022 b\/ pyqsc. https://github.com/landreman/pyQSC

    2022

  29. [37]

    2012 Omnigenity as generalized quasisymmetry

    Landreman, Matt & Catto, Peter J. 2012 Omnigenity as generalized quasisymmetry . Physics of Plasmas 19 (5), 056103

    2012

  30. [38]

    Journal of Plasma Physics 86 (5), 905860510

    Landreman, Matt & Jorge, Rogerio 2020 Magnetic well and Mercier stability of stellarators near the magnetic axis . Journal of Plasma Physics 86 (5), 905860510

    2020

  31. [39]

    Physical Review Letters 128 (3), 035001

    Landreman, Matt & Paul, Elizabeth 2022 Magnetic Fields with Precise Quasisymmetry for Plasma Confinement . Physical Review Letters 128 (3), 035001

    2022

  32. [40]

    Landreman, Matt & Sengupta, Wrick 2018 Direct construction of optimized stellarator shapes. Part 1. Theory in cylindrical coordinates . Journal of Plasma Physics 84 (6), 905840616

    2018

  33. [41]

    Journal of Plasma Physics 85 (6), 815850601 , arXiv:1908.10253 [physics]

    Landreman, Matt & Sengupta, Wrick 2019 Constructing stellarators with quasisymmetry to high order . Journal of Plasma Physics 85 (6), 815850601 , arXiv:1908.10253 [physics]

    2019 arXiv

  34. [42]

    2019 Direct construction of optimized stellarator shapes

    Landreman, Matt , Sengupta, Wrick & Plunk, Gabriel G. 2019 Direct construction of optimized stellarator shapes. Part 2. Numerical quasisymmetric solutions . Journal of Plasma Physics 85 (1), publisher: Cambridge University Press

    2019

  35. [43]

    u hrenberg, J 1976 Equilibrium and stability of a three-dimensional toroidal mhd configuration near its magnetic axis . Zeitschrift f \

    Lortz, D & N \"u hrenberg, J 1976 Equilibrium and stability of a three-dimensional toroidal mhd configuration near its magnetic axis . Zeitschrift f \"u r Naturforschung A 31 (11), 1277--1288

    1976

  36. [44]

    Nuclear Fusion 4 (3), 213--226

    Mercier, Claude 1964 Equilibrium and stability of a toroidal magnetohydrodynamic system in the neighbourhood of a magnetic axis . Nuclear Fusion 4 (3), 213--226

    1964

  37. [45]

    Springer

    Nocedal, Jorge & Wright, Stephen J 1999 Numerical optimization\/ . Springer

    1999

  38. [46]

    & Zille, R

    Nührenberg, J. & Zille, R. 1988 Quasi-helically symmetric toroidal stellarators . Physics Letters A 129 (2), 113--117 , number: 2

    1988

  39. [47]

    , Conlin, R

    Panici, D. , Conlin, R. , Dudt, D.W. , Unalmis, K. & Kolemen, E. 2023 The DESC stellarator code suite. Part 1. Quick and accurate equilibria computations . Journal of Plasma Physics 89 (3), 955890303

    2023

  40. [48]

    Panici, Dario & Rodriguez Urretavizcaya, Eduardo 2025 Inputs, Outputs and Plotting Scripts for paper Extending near-axis equilibria in DESC

    2025

  41. [49]

    , Landreman, Matt & Helander, Per 2019 Direct construction of optimized stellarator shapes

    Plunk, Gabriel G. , Landreman, Matt & Helander, Per 2019 Direct construction of optimized stellarator shapes. Part 3. Omnigenity near the magnetic axis . Journal of Plasma Physics 85 (6), publisher: Cambridge University Press

    2019

  42. [50]

    Journal of Plasma Physics 90 (1), 945900102

    Rodr \' guez, Eduardo 2024 Magnetohydrodynami stability and the effects of shaping: a near-axis view for tokamaks and quasisymmetric stellarators--erratum . Journal of Plasma Physics 90 (1), 945900102

    2024

  43. [51]

    Physics of Plasmas 27 (6)

    Rodriguez, Eduardo , Helander, Per & Bhattacharjee, Amitava 2020 Necessary and sufficient conditions for quasisymmetry . Physics of Plasmas 27 (6)

    2020

  44. [52]

    ArXiv:2204.10234 [physics]

    Rodriguez, Eduardo , Sengupta, Wrick & Bhattacharjee, Amitava 2023 Constructing the space of quasisymmetric stellarators. ArXiv:2204.10234 [physics]

    2023 arXiv

  45. [53]

    Journal of Plasma Physics 89 (2), 905890211 , publisher: Cambridge University Press

    Rodríguez, Eduardo 2023 Magnetohydrodynamic stability and the effects of shaping: a near-axis view for tokamaks and quasisymmetric stellarators . Journal of Plasma Physics 89 (2), 905890211 , publisher: Cambridge University Press

    2023

  46. [54]

    , Paul, E

    Rodríguez, E. , Paul, E. J. & Bhattacharjee, A. 2022 a\/ Measures of quasisymmetry for stellarators . Journal of Plasma Physics 88 (1), 905880109 , publisher: Cambridge University Press

    2022

  47. [55]

    , Plunk, G.G

    Rodríguez, E. , Plunk, G.G. & Jorge, R. 2025 Near-axis description of stellarator-symmetric quasi-isodynamic stellarators to second order . Journal of Plasma Physics 91 (2), E44

    2025

  48. [56]

    Rodríguez, E , Plunk, G G & Jorge, R 2024 Near-axis description of stellarator-symmetric quasi-isodynamic stellarators to second order

    2024

  49. [57]

    Plasma Physics and Controlled Fusion 64 (10), 105006 , tex.ids= rodriguez\_phases\_2022 arXiv: 2202.01194

    Rodríguez, E , Sengupta, W & Bhattacharjee, A 2022 b\/ Phases and phase-transitions in quasisymmetric configuration space . Plasma Physics and Controlled Fusion 64 (10), 105006 , tex.ids= rodriguez\_phases\_2022 arXiv: 2202.01194

    2022 arXiv

  50. [58]

    Solov'ev, L. S. & Shafranov, V. D. 1970 Reviews of Plasma Physics 5 \/ . New York - London: Consultants Bureau

    1970

  51. [59]

    Stein, Elias M & Shakarchi, Rami 2011 Fourier analysis: an introduction\/ , , vol. 1 . Princeton University Press

    2011

  52. [60]

    Wesson, John & Campbell, David J 2011 Tokamaks\/ , , vol. 149 . Oxford university press

    2011

  53. [61]

    Zarnstorff, M. C. , Berry, L. A. , Brooks, A. , Fredrickson, E. , Fu, G.-Y. , Hirshman, S. , Hudson, S. , Ku, L.-P. , Lazarus, E. , Mikkelsen, D. , Monticello, D. , Neilson, G. H. , Pomphrey, N. , Reiman, A. , Spong, D. , Strickler, D. , Boozer, A. , Cooper, W. A. , Goldston, ...

    2001

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