Constructing stellarators with quasisymmetry to high order

Matt Landreman; Wrick Sengupta

REVIEW 4 major objections 4 minor 1 cited by

Constructing stellarators with quasisymmetry to high order

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper claims that a second-order near-axis construction with a corrected finite-radius boundary builds quasisymmetric stellarator equilibria whose symmetry-breaking modes follow the ideal $1/A^3$ scaling, reaching amplitudes below…

desk verdict Genuinely new O((r/R)^2) near-axis construction with strong numerical verification; the record-low quasisymmetry-breaking claim needs a resolution-convergence check before it can be trusted, but the central scaling result holds. read the letter →

arxiv 1908.10253 v2 pith:PN3L6NAD submitted 2019-08-27 physics.plasm-ph

Matt Landreman , Wrick Sengupta This is my paper · ORCID

classification physics.plasm-ph PACS 52.55.Hc52.30.Cv

keywords quasisymmetrystellaratornear-axisexpansionaspect-ratioscalingmagnetohydrodynamicequilibriumBoozercoordinatesoptimization

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

Quasisymmetric stellarators confine plasma using a magnetic field whose strength has a continuous symmetry, even though the field itself does not, avoiding the large currents that make tokamaks unstable. This paper aims to show that such configurations can be built directly, without iterating a three-dimensional equilibrium code, by solving the near-axis expansion through second order in $r/R$ and applying a corrected boundary construction. If the construction works, it replaces an expensive optimization search with an analytic procedure that runs in milliseconds and provides the first numerical confirmation that symmetry-breaking errors scale as the cube of the inverse aspect ratio. The reported symmetry-breaking amplitudes, below $2\times 10^{-7}$ T in a strongly nonaxisymmetric configuration, would mean quasisymmetry can be made essentially exact at high aspect ratio.

What carries the argument

The central object is the near-axis expansion about the magnetic axis: the position vector is written in the Frenet-Serret frame as $\mathbf{r} = \mathbf{r}_0(\phi) + X(r,\vartheta,\phi)\mathbf{n} + Y(r,\vartheta,\phi)\mathbf{b} + Z(r,\vartheta,\phi)\mathbf{t}$, with $X$, $Y$, $Z$, and the field strength $B$ expanded in powers of $r/R$ and the equilibrium and quasisymmetry conditions imposed order by order in Boozer coordinates $(\theta,\phi)$. The load-bearing mechanism is the finite-radius boundary correction of Section 3: substituting a finite minor radius $a$ into the truncated second-order expansion gives the leading field strength a spurious $a^2$ contribution, and including $X_3$, $Y_3$ proportional to $X_1$, $Y_1$ with the coefficient $\lambda = -Q B_0/(2 s_G \bar B)$ cancels it, so the true equilibrium inside the constructed boundary matches the target field through $O((r/R)^2)$.

What would settle it

For an axis shape not used in the paper, construct the boundary at several aspect ratios and compute the full three-dimensional equilibrium inside it; if the total quasisymmetry-breaking measure does not fall as the cube of the inverse aspect ratio, or if the individual $m=0$ and $m>0$ measures deviate from the predicted $1/A^2$, $1/A^3$, and $1/A^4$ scalings, the central scaling claim is wrong. A more elementary check is whether $[B_{m=0}(\phi,r=a)-B(\phi,r=0)]/a^2$ stops converging to the predicted $B_{20}(\phi)$ as $A$ grows.

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Extended reading notes

Core claim

The central claim is that a finite-aspect-ratio quasisymmetric stellarator can be generated from the second-order near-axis expansion by substituting a finite minor radius $a$ and adding a specific third-order correction: $X_3$ and $Y_3$ are set to $\lambda$ times $X_1$ and $Y_1$ with $\lambda = -Q B_0/(2 s_G \bar B)$, which removes the spurious $a^2$ term in the leading field strength. With this corrected boundary, independent three-dimensional MHD equilibrium computations reproduce the intended Boozer-coordinate field strength through $O((r/R)^2)$, and the symmetry-breaking measures fall with aspect ratio as $1/A^3$ for $m>0$ modes, $1/A^4$ for on-axis mirror modes, and eventually $1/A^2$ where the residual toroidal variation of $B_{20}$ dominates. This is the first numerical demonstration of the predicted ideal scaling: quasisymmetry can be realized to arbitrary precision at sufficiently high aspect ratio, and the paper reports symmetry-breaking amplitudes below $2\times 10^{-7}$ T in a strongly nonaxisymmetric configuration.

Load-bearing premise

The load-bearing premise is that the truncated near-axis expansion, evaluated at a finite minor radius and corrected by the required third-order shape term, produces a boundary whose true plasma equilibrium realizes the intended field strength to the claimed order; the paper verifies this for several examples but does not prove it for arbitrary inputs.

Editorial extensions

If this is right

Any axis shape with nonvanishing curvature can, in principle, be made quasisymmetric to arbitrary precision by choosing a sufficiently large aspect ratio.
Because the construction takes milliseconds per configuration, it enables rapid scans over families of stellarators and provides initial conditions for conventional optimization codes.
Second-order terms introduce triangularity and Shafranov shift, so generated shapes resemble previously optimized devices without fitting to them.
A vacuum configuration with $\iota > 0.4$ can have symmetry-breaking mode amplitudes below $2\times 10^{-7}$ T, and at a 5 T on-axis field the largest errors are about the size of the Earth's magnetic field.
The construction also works with nonzero pressure and current, producing a tokamak-stellarator hybrid with finite $\beta$.

Reading between the lines

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

One extension the authors leave implicit is that the same finite-radius correction logic should apply to omnigenous targets, since Section 3 is formulated for an arbitrary desired field strength, not only quasisymmetry.
If the $1/A^3$ scaling is universal, the near-axis construction becomes a generator for mapping the solution space of stellarators rather than finding isolated examples, allowing systematic exploration of input parameters.
Deliberately allowing $B_0$ to vary toroidally and canceling it with $B_{20}$ at one radius would move quasisymmetry off-axis, which the paper notes as a possible advantage for fast-particle confinement but does not test.
The extreme sensitivity of the quasi-helical example to rounding of the axis-shape coefficients suggests that precision quasisymmetry at moderate aspect ratio requires fine-tuning; quantifying that sensitivity would help predict where the construction is practical.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The thing to know: this paper actually delivers what it promises. Landreman and Sengupta solve the O((r/R)^2) Garren-Boozer equations that were derived but never solved, and they verify the result with VMEC/BOOZ_XFORM across a range of configurations. The 1/A^3 scaling of quasisymmetry-breaking is demonstrated numerically for the first time, and the evidence is convincing. This is a real step for fast stellarator design. What is new and good: the streamlined derivation in Appendix A is a genuine contribution, not just housekeeping. The finite-minor-radius boundary correction in Section 3, retaining a specially chosen piece of the (X3,Y3) shape, is clever and apparently necessary; without it the achieved field strength misses at O((r/R)^2). The numerical checks are thorough: B20 extracted from VMEC converges to the constructed function as A increases, in multiple settings including finite pressure, finite current, and non-stellarator-symmetric cases. The quasi-helical example is especially demanding, and it also tracks the predicted scaling. The method is orders of magnitude faster than conventional optimization, and the paper is honest about the limitations of the construction at finite aspect ratio. Soft spots, in proportion. The stress-test note about resolution convergence is a fair hit. The 'smallest ever reported' claim (<2e-7 T at A=320) rests on VMEC and BOOZ_XFORM resolving mode amplitudes around 1e-7 relative to a 1 T field. No resolution scan is shown for the A=320 QA point, and the authors themselves note convergence difficulties for the QH case at high A. I would not call this fatal, since the 1/A^3 scaling is visible over a wide range even if the last point is noisy, but the record claim should be softened or backed with a convergence study. The unproven 'generally linearly independent' assertion for the 6x6 system in Appendix B is a smaller issue: it is a gap in the proof of generality, not in the empirical demonstrations. Also, exact constancy of B20 is not proved; the authors say so themselves, which is fine, but the claim that it can be made arbitrarily small is heuristic. No code or data is released, which is a minor reproducibility complaint for a paper that would benefit enormously from it. Who this is for: plasma physicists working on stellarator design, near-axis expansions, or quasisymmetry theory. If that is your area, read it. If not, the scaling result is still a nice piece of applied mathematics. Recommendation: send it to serious peer review. The central argument holds up; the paper deserves referee time. The referees should push for either a resolution-convergence check on the A=320 point or a more modest wording of the record claim, and a clean statement of the linear-independence assumption in Appendix B.

Referee Report

4 major / 4 minor

Summary. The paper extends the Garren–Boozer near-axis expansion for quasisymmetric stellarators from O(r/R) to O((r/R)^2), deriving a streamlined set of equations (Appendix A) and a finite-minor-radius boundary construction that includes a carefully chosen subset of O((r/R)^3) terms (Section 3, Appendix B). The construction is solved numerically in milliseconds, and the resulting boundary shapes are fed into VMEC/BOOZ_XFORM to verify that the computed equilibria have the desired Boozer field-strength spectrum. The paper reports the first numerical demonstrations of the predicted 1/A^3 scaling of quasisymmetry-breaking modes, examples of quasi-axisymmetric, quasi-helically symmetric, tokamak-stellarator hybrid, and non-stellarator-symmetric configurations, and a claim of the smallest-ever symmetry-breaking amplitudes (<2e-7 T) in a strongly nonaxisymmetric vacuum equilibrium.

Significance. If the results are taken at face value, this is a substantial advance: it provides a fast, direct, analytically grounded method for generating high-precision quasisymmetric stellarator geometries, gives the first numerical evidence for the Garren–Boozer 1/A^3 scaling, and demonstrates quasisymmetry quality far beyond what conventional optimization has reported. The manuscript's strengths include a detailed and largely self-contained analytic derivation in Appendix A, a nontrivial finite-a correction in Section 3 with supporting analysis in Appendix B, and VMEC/BOOZ_XFORM checks spanning quasi-axisymmetry, quasi-helical symmetry, finite pressure/current, and broken stellarator symmetry. The method's speed and analytic output are genuinely useful for stellarator design and for initializing conventional optimization.

major comments (4)

[§5.2, Fig. 8; Eq. (5.4)] The abstract's central claims—the first numerical demonstration of the 1/A^3 scaling and the 'smallest ever reported' symmetry-breaking amplitudes (<2e-7 T)—rest on VMEC/BOOZ_XFORM resolving relative amplitudes of order 10^-7 in a 1 T field at A=320, but no resolution-convergence study is reported. The authors note in §5.4 that converged VMEC values are difficult to obtain for very small symmetry-breaking modes at the highest aspect ratios, so it is not established that the A=320 quasi-axisymmetric point is above the numerical noise floor. Please provide convergence tests with respect to VMEC radial/poloidal resolution and BOOZ_XFORM grids, state the estimated numerical uncertainty of each point in Fig. 8, and qualify the record/scaling claims accordingly.
[§4.1, §5.2, Fig. 4] The construction is described as achieving quasisymmetry 'fully' through O((r/R)^2), yet the paper states in §4.1 that exact constancy of B20 is not proved and that only minimization of its toroidal variation is performed. Residual φ-dependence of B20 contributes a symmetry-breaking term scaling as 1/A^2 (seen for Config 2 in Fig. 4 at large A), so the asymptotic scaling is not the claimed 1/A^3 unless B20 is exactly constant. Either prove existence/construct a solution with constant B20, or reframe the Garren–Boozer scaling claim as a finite-aspect-ratio result over the demonstrated range.
[Appendix B, paragraph following (B27)–(B28)] The proof that the finite-a boundary correction yields the desired field strength through O((r/R)^2) depends on the assertion that the six homogeneous linear equations for {ξ0, ξs, ξc, γ0, γs, γc} are 'generally linearly independent.' No proof or explicit condition is given, and a singular or near-singular system would invalidate the uniqueness step. Please supply a proof or a numerical verification of the determinant at the operating points used in Section 5, or state the conditions under which the system is nonsingular.
[Abstract; §5.2; §6] The 'smallest ever reported' superlative is not supported by a systematic quantitative comparison with previously published quasisymmetric configurations. Please cite the previous best published symmetry-breaking amplitudes using the same measure (e.g., Eq. (5.4)) and give the comparison criterion, or soften the claim to 'smaller than the values we have traced in the literature.'

minor comments (4)

[§5.4] The quasi-helically symmetric configuration is reported to be very sensitive to rounding of the axis coefficients from four to three significant digits; please quantify this sensitivity (for example, the resulting change in X3/Y3 and in the boundary shape) and discuss its implications for reproducibility of the numerical examples.
[§5.2] The section title 'Quasi-axisymmetry fully through O((r/R)^2)' conflicts with the text, which notes a 'small remaining toroidal variation' of B20; the terminology should be adjusted to reflect that the configuration is optimized rather than exactly quasisymmetric to that order.
[Fig. 8] The scaling plot would be much more informative with estimated error bars or shaded uncertainty bands derived from the requested convergence studies; currently the reader cannot distinguish genuine scaling from numerical floor effects.
[Eq. (5.2)] Please define the symbol N used in the mode-selection condition n ≠ N m explicitly at the point of first use in Section 5, since N denotes the quasisymmetry helicity earlier in the paper.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the O((r/R)^2) construction is solved analytically and independently checked with VMEC/BOOZ_XFORM; the 1/A^3 scaling is an emergent numerical verification, not a fitted input.

full rationale

The paper's central derivation is self-contained rather than circular. The near-axis equations through O((r/R)^2), including the quasisymmetry reduction and the finite-minor-radius correction, are derived explicitly in Appendices A and B; the lambda-scaled (X3,Y3) correction is chosen from an analytic calculation to cancel the spurious a^2 B20^(0) term, not by fitting VMEC output. The VMEC and BOOZ_XFORM calculations in Section 5 are genuinely independent checks: the construction supplies a boundary shape, and the full equilibrium code then produces the Boozer spectrum, with figures 3, 7, 11, 15, and 19 showing convergence to the predicted B20(phi) as A increases. The reported 1/A^3 scaling is the measured slope of symmetry-breaking measures computed from VMEC equilibria at a sequence of aspect ratios; it is not an input parameter of the near-axis model, even though it is expected from the order at which quasisymmetry is enforced. The optimization of input parameters targets internal expansion coefficients (X2, Y2, X3, Y3) and the toroidal variation of B20, not the VMEC symmetry-breaking amplitudes, so the fitted-input-called-prediction pattern does not apply. The paper does rely on self-citations to Landreman, Sengupta and Plunk for the O(r/R) parameterization, uniqueness, and numerical solution method, but these are foundational prior methods rather than load-bearing assertions of the new result; the O((r/R)^2) equations are re-derived in this paper. The acknowledged difficulty in obtaining converged VMEC values for the quasi-helical case at the highest aspect ratio is a numerical convergence concern, not evidence of circularity. Overall, the derivation chain is not equivalent to its inputs by construction.

Assumptions & free parameters 7 free parameters · 6 assumptions · 0 invented entities

The central claim rests on the near-axis expansion assumptions and a set of input parameters that are chosen or optimized. No new physical entities are introduced. The free parameters are the standard inputs of the near-axis construction and are not fitted to external data.

free parameters (7)

bar_eta (B1c/B0) = 0.640 m^-1 (Sec. 5.1), 0.632 (5.2), 0.95 (5.3), 1.569 (5.4), 2.5 (5.5)
Input constant controlling flux surface elongation; optimized in some examples.
I2 = 0 T/m in vacuum examples; 0.9 T/m (Sec. 5.3); 1.6 T/m (Sec. 5.5)
Leading on-axis toroidal current coefficient; set to zero for vacuum fields.
sigma(0) = 0 (stellarator-symmetric examples); 0.3 (Sec. 5.5)
Initial angle of elongation; controls orientation of elliptical surfaces.
p2 = 0 (vacuum); -6e5 Pa/m^2 (Sec. 5.3); -5e6 Pa/m^2 (Sec. 5.5)
Pressure profile coefficient at this order; sets pressure gradient.
B2c = -0.00322, -0.158, -0.7, 0.1348, 1 T/m^2 (examples)
cos(2*theta) mode of B2; controls triangularity of flux surfaces.
B2s = 0 (symmetric cases); 3 T/m^2 (Sec. 5.5)
sin(2*theta) mode of B2; controls stellarator-asymmetric triangularity.
Axis shape Fourier coefficients = See equations (5.1), (5.3), (5.5)-(5.7)
Coefficients of R0(phi) and z0(phi) are specified and optimized to minimize higher-order terms and B20 variation.

assumptions (6)

domain assumption Good nested flux surfaces exist in the region of interest near the axis.
Section 2, first paragraph: 'Throughout the analysis, we assume that good nested flux surfaces exist in the region of interest near the axis.'
domain assumption Profile functions G, I, p, iota are analytic in flux, so their expansions contain only even powers of r.
Section 2, after Eq. (2.11).
domain assumption Expansion coefficients are analytic in r, leading to the specific Fourier forms in Eq. (2.12).
Section 2, 'From analyticity considerations near the axis...'
domain assumption Boozer coordinates are well-defined with periodic angles.
Used throughout Section 2 to express the magnetic field and position vector.
domain assumption iota_N0 is not an integer, to avoid low-order magnetic islands.
Appendix A.3: 'To avoid large magnetic islands near the axis, we assume iota_N0 is not an integer.'
domain assumption The expansion parameter ordering a ~ r << R is valid for the finite-radius construction.
Section 3 and Appendix B use this ordering to analyze the effect of substituting a finite minor radius.

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Cite this review

Pith. "Pith review of Constructing stellarators with quasisymmetry to high order." pith.science (2026). https://pith.science/paper/PN3L6NAD

@misc{pith2026190810253,
  author       = {Pith},
  title        = {Pith review of: Constructing stellarators with quasisymmetry to high order},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PN3L6NAD}},
  note         = {Machine review of arXiv:1908.10253}
}

abstract

A method is given to rapidly compute quasisymmetric stellarator magnetic fields for plasma confinement, without the need to call a three-dimensional magnetohydrodynamic equilibrium code inside an optimization iteration. The method is based on direct solution of the equations of magnetohydrodynamic equilibrium and quasisymmetry using Garren and Boozer's expansion about the magnetic axis (Phys Fluids B 3, 2805 (1991)), and it is several orders of magnitude faster than the conventional optimization approach. The work here extends the method of Landreman, Sengupta and Plunk (J Plasma Phys 85, 905850103 (2019)), which was limited to flux surfaces with elliptical cross-section, to higher order in the aspect ratio expansion. As a result, configurations can be generated with strong shaping that achieve quasisymmetry to high accuracy. Using this construction, we give the first numerical demonstrations of Garren and Boozer's ideal scaling of quasisymmetry-breaking with the cube of inverse aspect ratio. We also demonstrate a strongly nonaxisymmetric configuration (vacuum $\iota > 0.4$) in which symmetry-breaking mode amplitudes throughout a finite volume are $< 2\times 10^{-7}$, the smallest ever reported. To generate boundary shapes of finite-minor-radius configurations, a careful analysis is given of the effect of substituting a finite minor radius into the near-axis expansion. The approach here can provide analytic insight into the space of possible quasisymmetric stellarator configurations, and it can be used to generate good initial conditions for conventional stellarator optimization.

Figures

Figures reproduced from arXiv: 1908.10253 by the authors.

**Figure 1.** The partially-quasi-axisymmetric example of section 5.1, for aspect ratio A = 10, Avmec = 9.75. The 3D surface shape in (a), shown from three angles, and the cross-sections in (b), are generated by the construction. In (a), magnetic field lines are shown as black lines, and color indicates the field strength computed by VMEC. 0.0 0.2 0.4 0.6 0.8 1.0 r/a = s 0.00 0.01 0.02 0.03 0.04 0.05 0.06 Fourier harmonics Bm, n … view at source ↗

**Figure 2.** The spectrum of B for the partially-quasi-axisymmetric example of section 5.1, computed by running the VMEC and BOOZ_XFORM codes inside the constructed boundary surface for aspect ratio A = 10, Avmec = 9.75 [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗

**Figure 3.** As the aspect ratio A increases, the B20(ϕ) component of the field strength of the numerical VMEC configurations converges to the function predicted by the Garren-Boozer construction. Data here are for the partially quasi-axisymmetric configuration of section 5.1. 10 1 10 2 Aspect ratio A = R00/a 10 9 10 8 10 7 10 6 10 5 10 4 10 3 10 2 10 1 S y m m etry-bre a kin g S Config 1, S r = a m > 0 Config 1, S r = a m = 0 C… view at source ↗

Figures from the paper (19 more)

**Figure 4.** Figure 4: The measures of quasisymmetry-breaking (5.2), computed by running the VMEC and BOOZ_XFORM codes inside the constructed boundary surfaces, scale as the expected power of aspect ratio. Data here are for the partially quasi-axisymmetric and optimized quasi-axisymmetric ex…

**Figure 5.** Figure 5: The quasi-axisymmetric example of section 5.2, for aspect ratio A = 10, Avmec = 9.71. The 3D surface shape in (a), shown from three angles, and the cross-sections in (b), are generated by the construction. In (a), magnetic field lines are shown as black lines, and colo…

**Figure 6.** Figure 6: The spectrum of B for the quasi-axisymmetric example of section 5.2, computed by running the VMEC and BOOZ_XFORM codes inside the constructed boundary surface for aspect ratio A = 10, Avmec = 9.71 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Boozer toroidal angle 0.0 0.1 0.2 0.3 0.4 0.…

**Figure 7.** Figure 7: As the aspect ratio A increases, the B20(ϕ) component of the field strength of the numerical VMEC configurations converges to the function predicted by the Garren-Boozer construction. Data here are for the quasi-axisymmetric configuration of section 5.2. is significant…

**Figure 8.** Figure 8: A numerical demonstration of the prediction by Garren & Boozer (1991a) that deviations from quasisymmetry can be made to scale as 1/A3 . Here, the deviations are measured by (5.4) for the configurations of sections 5.2 and 5.4. includes all quasisymmetry-breaking modes…

**Figure 9.** Figure 9: figure 9. Figure 10 shows the Boozer spectrum of the finite- [PITH_FULL_IMAGE:figures/full_fig_p017_9.png]

**Figure 9.** Figure 9: The tokamak-stellarator hybrid example of section 5.3, for aspect ratio A = 5, Avmec = 4.87. The 3D surface shape in (a), shown from three angles, and the cross-sections in (b), are generated by the construction. In (a), magnetic field lines are shown as black lines, a…

**Figure 10.** Figure 10: The spectrum of B for the tokamak-stellarator hybrid example of section 5.3, computed by running the VMEC and BOOZ_XFORM codes inside the constructed boundary surface for aspect ratio A = 5, Avmec = 4.87. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Boozer toroidal angle 0.00 0.25 0.5…

**Figure 11.** Figure 11: As the aspect ratio A increases, the B20(ϕ) component of the field strength of the numerical VMEC configurations converges to the function predicted by the Garren-Boozer construction. Data here are for the tokamak-stellarator hybrid configuration of section 5.3. large…

**Figure 12.** Figure 12: The measures of quasisymmetry-breaking (5.2), computed by running the VMEC and BOOZ_XFORM codes inside the constructed boundary surfaces, scale as the expected power of aspect ratio. Data here are for the tokamak-stellarator hybrid example of section 5.3. quasi-helica…

**Figure 13.** Figure 13: The quasi-helically symmetric example of section 5.4, for aspect ratio A = 8, Avmec = 7.14. The 3D surface shape in (a), shown from three angles, and the cross-sections in (b), are generated by the construction. In (a), magnetic field lines are shown as black lines, a…

**Figure 14.** Figure 14: The spectrum of B for the quasi-helically symmetric example of section 5.4, computed by running the VMEC and BOOZ_XFORM codes inside the constructed boundary surface for aspect ratio A = 8, Avmec = 7.14 [PITH_FULL_IMAGE:figures/full_fig_p021_14.png]

**Figure 15.** Figure 15: As the aspect ratio A increases, the B20(ϕ) component of the field strength of the numerical VMEC configurations converges to the function predicted by the Garren-Boozer construction. Data here are for the quasi-helically symmetric configuration of section 5.4. 10 1 1…

**Figure 16.** Figure 16: The measures of quasisymmetry-breaking (5.2), computed by running the VMEC and BOOZ_XFORM codes inside the constructed boundary surfaces, scale as the expected power of aspect ratio. Data here are for the quasi-helically symmetric example of section 5.4. nϕ) and sin(m…

**Figure 17.** Figure 17: The non-stellarator-symmetric quasi-helically-symmetric example of section 5.5, for aspect ratio A = 40, Avmec = 28.5. The 3D surface shape in (a), shown from three angles, and the cross-sections in (b), are generated by the construction. In (a), magnetic field lines …

**Figure 18.** Figure 18: The spectrum of B (including both ∝ cos(mθ − nϕ) and ∝ sin(mθ − nϕ) modes) for the non-stellarator-symmetric quasi-helically-symmetric example of section 5.5, computed by running the VMEC and BOOZ_XFORM codes inside the constructed boundary surface for aspect ratio A …

**Figure 19.** Figure 19: As the aspect ratio A increases, the B20(ϕ) component of the field strength of the numerical VMEC configurations converges to the function predicted by the Garren-Boozer construction. Data here are for the non-stellarator-symmetric quasi-helically-symmetric configurat…

**Figure 20.** Figure 20: The measures of quasisymmetry-breaking (5.2), computed by running the VMEC and BOOZ_XFORM codes inside the constructed boundary surfaces, scale as the expected power of aspect ratio. Data here are for the non-stellarator-symmetric quasi-helically-symmetric example of …

**Figure 21.** Figure 21: Contours of B(θ, ζ) at the boundaries of the configurations of sections 5.2 and 5.4, scaled to a mean field of 5 Tesla, at the aspect ratio for which the largest symmetry-breaking Fourier modes have amplitude 0.5 Gauss, the magnitude of Earth’s magnetic field. Departu…

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Extending near-axis equilibria in DESC
physics.plasm-ph 2025-06 conditional novelty 7.0 of 10

The paper derives linear Fourier-Zernike constraints that make global DESC equilibria match a prescribed near-axis expansion through second order in the flux-surface radius.

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