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arxiv: 2604.27317 · v1 · submitted 2026-04-30 · ⚛️ physics.plasm-ph

Recognition: unknown

Improved n=1 Empirical Error Field Penetration Threshold Scaling with Ohmic and L-Mode Conventional Tokamak Plasma Discharges

C.F.B. Zimmermann, C. Paz-Soldan, E.M. Bursch, EUROfusion Tokamak Exploitation Team, F. Mao, G. Szepesi, H. Wang, JET contributors, J.K. Park, L. Piron, M. Pharr, N.C. Logan, N. Wang, R.J. Buttery, S.M. Yang

Pith reviewed 2026-05-07 09:38 UTC · model grok-4.3

classification ⚛️ physics.plasm-ph
keywords error field penetrationn=1 modetokamak scalingOhmic dischargesL-modeJETJ-TEXTconventional tokamak
0
0 comments X

The pith

An updated n=1 error field penetration threshold scaling derived solely from Ohmic and L-mode conventional tokamak data improves fit quality and lowers uncertainty in projections to future devices.

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

The paper establishes an improved empirical scaling for the n=1 error field amplitude at which penetration occurs by restricting analysis to an expanded database of only Ohmic and L-mode discharges from conventional tokamaks, including added J-TEXT and new JET results. This choice yields a better statistical fit than prior scalings while reducing uncertainty when the relation is extrapolated to next-generation machines. A sympathetic reader cares because error field tolerances set by such scalings directly constrain the design of correction coils and vacuum vessel tolerances, which are major cost and schedule drivers in tokamak construction. By focusing on the regime expected to have the lowest threshold, the scaling supplies the conservative limit needed for safe engineering decisions.

Core claim

By compiling data exclusively from Ohmic and L-mode conventional tokamak plasmas and incorporating new J-TEXT and JET measurements, the authors obtain an n=1 error field penetration threshold scaling that demonstrates higher fit quality and reduced projection uncertainty relative to earlier empirical laws. Because H-mode plasmas exhibit greater resilience to error field penetration, the resulting relation is presented as the most dangerous regime for new device design.

What carries the argument

The n=1 error field penetration threshold scaling, an empirical multi-parameter fit that relates plasma current, density, toroidal field and other quantities to the critical error field strength required for mode penetration.

If this is right

  • Future conventional tokamaks can adopt tighter yet more reliable error field correction requirements based on the reduced projection uncertainty.
  • Device design teams gain a clearer basis for setting allowable error field amplitudes during the most vulnerable plasma phases.
  • The improved fit quality enables more accurate interpolation of thresholds across varying plasma conditions within the Ohmic and L-mode domain.
  • Engineering margins for error field coils and vessel alignment can be optimized with greater statistical confidence.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same database-expansion and regime-restriction approach could be applied to derive separate scalings for H-mode or for higher-n error fields.
  • Real-time plasma control systems might incorporate this threshold to adjust error field correction coil currents dynamically during Ohmic or L-mode phases.
  • The scaling could inform cost-benefit analyses for correction coil power supplies by quantifying how much margin is truly required.

Load-bearing premise

That Ohmic and L-mode discharges exhibit the lowest penetration thresholds and therefore supply the appropriate conservative bound for new tokamak engineering tolerances.

What would settle it

A new conventional tokamak experiment that measures a lower n=1 penetration threshold in an Ohmic or L-mode plasma than the scaling predicts, or that finds H-mode thresholds falling below the Ohmic/L-mode values.

Figures

Figures reproduced from arXiv: 2604.27317 by C.F.B. Zimmermann, C. Paz-Soldan, E.M. Bursch, EUROfusion Tokamak Exploitation Team, F. Mao, G. Szepesi, H. Wang, JET contributors, J.K. Park, L. Piron, M. Pharr, N.C. Logan, N. Wang, R.J. Buttery, S.M. Yang.

Figure 1
Figure 1. Figure 1: Parameter space plot demonstrating additions of J-TEXT and JET and absence of spherical tori. view at source ↗
Figure 2
Figure 2. Figure 2: R2 and condition number versus number of parameters added. Note that number of variables is truncated at 8 for visualization due to an exponential increase in condition number, at which the scaling suffers from colinearity. variables into the fit. However, the third highest correlated parameter is not major radius, which was selected third, but rather toroidal field, highlighting the importance of secondar… view at source ↗
Figure 3
Figure 3. Figure 3: Correlation matrix for a selection of key variables in the reported error field penetration threshold scaling equations. 3.3. Changes in Scaling Dependencies The best quality fit found with the outlined decisions and scaling technique has R2 = 0.63 and a condition number of 17.76: δ = 10−4.31±0.03  βn li 0.25±0.02 |Ip| −0.97±0.02R 1.88±0.04 0 n 0.77±0.02 e |BT | 0.20±0.03 (9) Instead of the outlined ordi… view at source ↗
Figure 4
Figure 4. Figure 4: Best full-device fit using WLS (equation view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of error field penetration threshold scalings between view at source ↗
Figure 7
Figure 7. Figure 7: NSTX predicted and actual δ using the new error field penetration threshold scaling. explicit BT and Ip dependence, and therefore the first to fully capture some amount of the q95 dependence. Future work could shed more light on this issue through adding in other spherical tokamaks such as MAST-U, or by dedicated modeling of spherical tokamak error field penetration. 6. Projecting the Error Field Threshold… view at source ↗
Figure 8
Figure 8. Figure 8: Monte-Carlo probability density function, view at source ↗
read the original abstract

This paper presents an updated n=1 error field penetration threshold scaling, which increases fit quality compared to previous error field scaling laws, is produced from an expanded database, and exhibits reduced uncertainty in projections to future conventional tokamaks. It improves confidence in tokamak engineering tolerances, which are a significant driver of cost and time constraints on device construction. We add J-TEXT data, new JET data, and create the scaling using only conventional tokamak Ohmic and L-mode experiments. Since H-mode plasmas are more resilient to error field penetration, this scaling predicts what is likely the most dangerous regime of error field penetration for new tokamak designs. These decisions improve confidence in the error field penetration threshold scaling and its application in the construction and design decisions of any future conventional tokamak or FPP.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript presents an updated empirical scaling law for the n=1 error field penetration threshold in conventional tokamaks. It is derived from an expanded database restricted to Ohmic and L-mode discharges (including new J-TEXT and JET data), claims higher fit quality and lower projection uncertainty than prior scalings, and argues that the restriction yields the most conservative (lowest) threshold because H-mode plasmas are more resilient to error field penetration.

Significance. If the scaling derivation is robust and the conservative justification holds, the result would strengthen engineering tolerances for error fields in future conventional tokamaks and FPPs, directly addressing a major cost and schedule driver. The expanded conventional-tokamak database is a clear strength for applicability.

major comments (3)
  1. [Abstract and scaling derivation section] Abstract and § on scaling derivation: the claims of 'increased fit quality' and 'reduced uncertainty' are stated without the explicit functional form of the scaling, the numerical fit coefficients, goodness-of-fit metrics (R², reduced χ², or residual analysis), data-selection criteria, or any cross-validation procedure. These omissions make the improvement over previous laws unverifiable and load-bearing for the central claim.
  2. [Abstract and database section] Abstract and database section: the assertion that 'H-mode plasmas are more resilient' and therefore the Ohmic/L-mode scaling 'predicts what is likely the most dangerous regime' is presented without any direct threshold comparison, cited H-mode data points, or quantitative demonstration that H-mode thresholds exceed the new fit across the relevant parameter space. This leaves the conservative-bound justification unsupported.
  3. [Projections section] Projections section: the reduced-uncertainty projections to future devices rest on the empirical fit to the chosen database; without explicit uncertainty propagation (including extrapolation range and covariance of coefficients), the claimed improvement in engineering confidence cannot be assessed.
minor comments (2)
  1. [Notation and equations] Notation for the scaling variables (e.g., definitions of normalized quantities) should be collected in a single table or equation block for clarity.
  2. [Figures] Figures showing the new scaling would benefit from overlaying the previous scaling laws for immediate visual comparison of fit quality.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive feedback and the recommendation for major revision. We agree that the manuscript requires additional explicit details to make the central claims verifiable. We will revise the paper to include the functional form and fit metrics for the scaling, supporting citations and discussion for the conservative justification, and explicit uncertainty propagation for the projections. Our point-by-point responses follow.

read point-by-point responses

  1. Referee: [Abstract and scaling derivation section] Abstract and § on scaling derivation: the claims of 'increased fit quality' and 'reduced uncertainty' are stated without the explicit functional form of the scaling, the numerical fit coefficients, goodness-of-fit metrics (R², reduced χ², or residual analysis), data-selection criteria, or any cross-validation procedure. These omissions make the improvement over previous laws unverifiable and load-bearing for the central claim.

    Authors: We agree that these elements are required for independent verification. The revised manuscript will explicitly state the functional form of the n=1 error field penetration threshold scaling, report the numerical coefficients with uncertainties, include goodness-of-fit metrics (R², reduced χ², and residual analysis), detail the data-selection criteria (Ohmic and L-mode conventional tokamak discharges from the expanded database including J-TEXT and JET), and describe any cross-validation performed. This will enable direct quantitative comparison to prior scalings and substantiate the claimed improvements in fit quality and projection uncertainty. revision: yes

  2. Referee: [Abstract and database section] Abstract and database section: the assertion that 'H-mode plasmas are more resilient' and therefore the Ohmic/L-mode scaling 'predicts what is likely the most dangerous regime' is presented without any direct threshold comparison, cited H-mode data points, or quantitative demonstration that H-mode thresholds exceed the new fit across the relevant parameter space. This leaves the conservative-bound justification unsupported.

    Authors: We acknowledge that a more quantitative demonstration would strengthen the conservative-bound argument. The database was deliberately restricted to Ohmic and L-mode discharges to focus on the most vulnerable regime, but the revised manuscript will add citations to published H-mode error field penetration threshold studies (from devices such as DIII-D and ASDEX Upgrade) that report higher thresholds than the corresponding L-mode values at similar parameters. We will also include a brief discussion or supplementary comparison illustrating that the new scaling lies below reported H-mode thresholds in the relevant parameter space, thereby supporting its use as a conservative engineering bound. revision: partial

  3. Referee: [Projections section] Projections section: the reduced-uncertainty projections to future devices rest on the empirical fit to the chosen database; without explicit uncertainty propagation (including extrapolation range and covariance of coefficients), the claimed improvement in engineering confidence cannot be assessed.

    Authors: We agree that explicit uncertainty propagation details are necessary to assess the claimed reduction in projection uncertainty. The revised manuscript will describe the uncertainty propagation procedure, including the treatment of coefficient covariances from the fit, the extrapolation ranges for future conventional tokamaks and FPPs, and the resulting uncertainty bands on the projected thresholds. This will allow readers to evaluate the improvement in engineering confidence relative to previous scalings. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical scaling update from expanded database

full rationale

The paper updates an empirical n=1 error field penetration threshold scaling by fitting coefficients to an expanded database of conventional tokamak Ohmic and L-mode discharges. The resulting scaling law and its projections to future devices are direct outputs of this fit, with no first-principles derivation claimed. No equations or steps reduce by construction to prior inputs, no self-citations are used for load-bearing uniqueness or ansatz justification, and the H-mode resilience statement is presented as background justification for database choice rather than a derived result. This matches standard empirical scaling work that remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on an empirical fit to experimental data. Free parameters are the coefficients of the scaling law. The key domain assumption is the relative resilience of H-mode versus Ohmic/L-mode plasmas.

free parameters (1)
  • scaling law coefficients
    Empirical scaling laws are defined by multiple numerical coefficients that are fitted to match observed penetration thresholds in the database.
axioms (1)
  • domain assumption H-mode plasmas are more resilient to error field penetration than Ohmic and L-mode plasmas
    This premise is invoked to justify that the Ohmic/L-mode scaling represents the most dangerous regime for new tokamak designs.

pith-pipeline@v0.9.0 · 5508 in / 1344 out tokens · 66911 ms · 2026-05-07T09:38:35.902300+00:00 · methodology

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Reference graph

Works this paper leans on

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