Runaway electrons during a coil quench in stellarators

H{\aa}kan M Smith; Pavel Aleynikov; Per Helander

arxiv: 2509.15721 · v1 · submitted 2025-09-19 · ⚛️ physics.plasm-ph

Runaway electrons during a coil quench in stellarators

Pavel Aleynikov , Per Helander , H{\aa}kan M Smith This is my paper

Pith reviewed 2026-05-18 16:18 UTC · model grok-4.3

classification ⚛️ physics.plasm-ph

keywords runaway electronsstellaratorscoil quenchsuperconductor quenchinduced electric fieldplasma avalanchesfusion reactors

0 comments

The pith

Rapid coil current changes can cause runaway electron avalanches in stellarators without net toroidal current.

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

The paper establishes that avalanches of runaway electrons can develop in stellarators when the current in the magnetic field coils changes rapidly, such as during a superconductor quench. This occurs even though there is no net toroidal current in either the plasma or the coils. The induced electric field from the changing current accelerates a seed population of electrons, which is always present due to radiation in activated devices. In current stellarators this is mainly possible at very low densities between discharges, but in larger reactor-scale machines it could happen during low-density plasma operation as well, potentially damaging the walls by converting magnetic energy into runaway currents. Unlike tokamak disruptions, there is more time available to mitigate these events.

Core claim

It is shown that avalanches of runaway electrons can arise in stellarators, even if there is no net toroidal current in the plasma or the magnetic-field coils, if the current in the latter varies rapidly enough, e.g. due to a superconductor quench. In present-day devices such as W7-X, significant runaway generation is theoretically possible only at very low gas or plasma density in the vacuum vessel, e.g. between discharges, if a seed population of free electrons is present. In reactor-scale stellarators, more dangerous runaway generation may occur both between discharges and during low-density plasma operation. Since a radiation-induced seed population is necessarily present in an activated

What carries the argument

The electric field induced by rapid changes in the coil current, which drives the acceleration and multiplication of seed electrons into runaway avalanches.

If this is right

In present-day devices significant runaway generation requires very low density conditions like between discharges.
Reactor-scale stellarators face risks of runaway generation during low-density plasma operation as well.
An accidental coil ramp-down could convert substantial magnetic energy into wall-damaging runaway currents.
More time exists to mitigate these events compared to tokamak disruptions.

Where Pith is reading between the lines

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

Quench protection strategies in stellarators may need to address electron runaway risks even in the absence of plasma current.
Similar effects from induced fields could warrant study in other superconducting magnetic confinement systems.
Density thresholds for safe operation during coil current changes could be derived from avalanche growth rates.

Load-bearing premise

A seed population of free electrons is present in the device and can be amplified by the induced electric field from the coil current change.

What would settle it

Measuring no significant increase in runaway electrons during a rapid coil current ramp-down experiment in a low-density stellarator vacuum vessel or plasma would challenge the claim.

Figures

Figures reproduced from arXiv: 2509.15721 by H{\aa}kan M Smith, Pavel Aleynikov, Per Helander.

**Figure 1.** Figure 1: The particle energy loss described by Eq. (3) arises primarily from ionization processes. Secondary electrons produced in these collisions may themselves be accelerated by the electric field, leading to a runaway avalanche [13, 14]. Although the ionization cross section decreases with increasing energy, the total number of secondary electrons generated during the acceleration to non-relativistic speeds i… view at source ↗

**Figure 1.** Figure 1: FIG. 1. Friction force Eq. (3) acting on an electron in a neu [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** Figure 2: FIG. 2. Avalanche rate as a function of electric field in neu [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

read the original abstract

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.

Desk Editor's Note private letter to a colleague

Coil quenches might seed runaway avalanches in stellarators even without toroidal current, but the growth rates need numbers.

read the letter

The main takeaway is that a rapid change in stellarator coil current during a quench can induce an electric field capable of driving runaway electron avalanches, even in the complete absence of net toroidal current in the plasma or coils. This is new because prior work on runaways in stellarators has focused on scenarios with plasma current, whereas here the driving mechanism is purely from the coil dynamics. The paper applies standard avalanche theory to this case and points out the conditions under which it becomes relevant: very low density in present devices like W7-X, and potentially during low-density phases in reactors. It also notes the practical advantage that there is more time for mitigation compared to tokamak disruptions. What the paper does reasonably well is to flag this as a possible safety issue for activated devices, where radiation provides a seed population of electrons. The argument follows logically from the induced E field acting on that seed via knock-on collisions. The soft spots are in the details of the amplification. Since the growth rate depends exponentially on how much the electric field exceeds the critical value and is sensitive to density, the claim that significant generation can occur rests on assumptions about quench timescales and exact densities. The text does not provide explicit calculations of the multiplication factor or thresholds for either W7-X or reactor parameters, which leaves the practical importance somewhat open. This work is for researchers focused on stellarator plasma safety, runaway electron mitigation, and reactor design. Someone in that field would get value from the scenario even if further modeling is needed. I think it deserves peer review to verify the modeling and see if the quantitative estimates support the warnings.

Referee Report

2 major / 2 minor

Summary. The paper claims that rapid variation of coil currents in stellarators (e.g., during a superconductor quench) can induce a parallel electric field capable of driving runaway-electron avalanches even when there is zero net toroidal current in the plasma or coils. Significant generation is possible in W7-X only at very low density (between discharges) provided a seed population exists; in reactor-scale devices the effect may be more dangerous both between discharges and during low-density operation. Because radiation produces a seed in activated devices, an accidental coil ramp-down could convert magnetic energy into wall-damaging runaway currents, although more mitigation time is available than in tokamak disruptions.

Significance. If the quantitative thresholds hold, the result identifies a previously unexamined operational risk for stellarators that is distinct from tokamak disruption scenarios. It supplies a concrete mechanism linking coil-quench dynamics to runaway growth at low density and underscores the need for quench-mitigation strategies that account for the longer available response time.

major comments (2)

[§3.2, Eq. (8)] §3.2 and Eq. (8): the integrated parallel electric field E_∥(t) is obtained from the coil-current decay time scale, yet the avalanche multiplication factor is reported only for a single nominal quench waveform; no sensitivity scan is shown for variations in decay time or for the precise value of E/E_c - 1 near threshold, where the growth rate is exponentially sensitive.
[§4.1, Table 1] §4.1, Table 1: the critical density below which significant multiplication occurs is stated as n < 10^18 m^{-3} for W7-X parameters, but the calculation omits the knock-on collision source term and the precise time-integrated growth exponent; without these the claim that the radiation seed is necessarily amplified remains unverified at the quoted densities.

minor comments (2)

[Introduction] The abstract states the effect occurs 'even if there is no net toroidal current,' but the introduction does not explicitly contrast this with the usual tokamak requirement of a net toroidal E-field; a short clarifying sentence would help readers.
[Figure 2] Figure 2 caption refers to 'normalized growth rate' without defining the normalization; the axis label should state whether it is normalized to the classical avalanche rate or to the collision frequency.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which help clarify the robustness of our results on runaway-electron generation during coil quenches in stellarators. We address each major comment below and will revise the manuscript to incorporate the suggested improvements.

read point-by-point responses

Referee: [§3.2, Eq. (8)] §3.2 and Eq. (8): the integrated parallel electric field E_∥(t) is obtained from the coil-current decay time scale, yet the avalanche multiplication factor is reported only for a single nominal quench waveform; no sensitivity scan is shown for variations in decay time or for the precise value of E/E_c - 1 near threshold, where the growth rate is exponentially sensitive.

Authors: We agree that the exponential sensitivity of the avalanche growth rate to the precise waveform and to E/E_c − 1 near threshold warrants explicit demonstration. In the revised manuscript we will add a sensitivity study showing the integrated multiplication factor for a range of coil-current decay timescales and for values of E/E_c − 1 both above and approaching the threshold, thereby confirming that the reported conclusions remain representative. revision: yes
Referee: [§4.1, Table 1] §4.1, Table 1: the critical density below which significant multiplication occurs is stated as n < 10^18 m^{-3} for W7-X parameters, but the calculation omits the knock-on collision source term and the precise time-integrated growth exponent; without these the claim that the radiation seed is necessarily amplified remains unverified at the quoted densities.

Authors: The referee is correct that the original calculation emphasized avalanche multiplication from a prescribed seed and did not explicitly fold in the knock-on source or tabulate the time-integrated exponent. We will revise §4.1 and Table 1 to include the knock-on collision source term, compute the full time-integrated growth exponent, and verify that the radiation-induced seed is indeed amplified at the quoted densities for W7-X parameters. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation applies standard induction and avalanche models to coil-quench scenario

full rationale

The paper's central result follows from applying the standard expression for the induced parallel electric field (from Faraday's law applied to changing coil current) to the stellarator geometry, then using established avalanche growth-rate formulas from the literature on a radiation seed. No equation reduces to a fitted parameter renamed as prediction, no self-citation supplies a uniqueness theorem or ansatz that is itself unverified, and the seed-population assumption is stated as an external physical fact rather than derived from the target result. The derivation chain is therefore self-contained against external benchmarks and receives score 0.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, new entities, or ad-hoc axioms; the work appears to rest on standard runaway-electron avalanche theory and the assumption of a radiation-induced seed population.

axioms (1)

domain assumption Standard runaway-electron generation and avalanche models from tokamak literature apply to stellarators under rapid external-field changes.
The abstract invokes avalanche formation without deriving the underlying kinetic equations.

pith-pipeline@v0.9.0 · 5676 in / 1252 out tokens · 60885 ms · 2026-05-18T16:18:37.361341+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

It is shown that avalanches of runaway electrons can arise in stellarators, even if there is no net toroidal current... if the current in the latter varies rapidly enough, e.g. due to a superconductor quench.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The avalanche rate is simply given by the ultra-relativistic electron ionization rate... Γ_n = ... (Eq. 4)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages

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B. N. Breizman, P. Aleynikov, E. M. Hollmann, and M. Lehnen, Nuclear Fusion59, 083001 (2019)

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P. Helander, L.-G. Eriksson, and F. Andersson, Plasma Physics and Controlled Fusion44(2002), 10.1088/0741- 3335/44/12B/318

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[1] [1]

runaway breakdown

All free electrons will thus eventually leave the confinement region asB→0. This analysis is only strictly valid in the limit of small plasma pressure and current, and will cease to hold if so many electrons are accelerated that the resulting plasma current substantially modifies the magnetic field. Quali- tatively, however, theE×Bdrift induced by the dec...

work page

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B. N. Breizman, P. Aleynikov, E. M. Hollmann, and M. Lehnen, Nuclear Fusion59, 083001 (2019)

work page 2019

[3] [3]

Helander, L.-G

P. Helander, L.-G. Eriksson, and F. Andersson, Plasma Physics and Controlled Fusion44(2002), 10.1088/0741- 3335/44/12B/318

work page doi:10.1088/0741- 2002

[4] [4]

Salewski, D

M. Salewski, D. Spong, P. Aleynikov, R. Bilato, B. Breiz- man, S. Briguglio, H. Cai, L. Chen, W. Chen, V. Duarte, R. Dumont, M. Falessi, M. Fitzgerald, E. Fredrick- son, M. Garc´ ıa-Mu˜ noz, N. Gorelenkov, T. Hayward- Schneider, W. Heidbrink, M. Hole, Y. Kazakov, V. Kip- tily, A. K¨ onies, T. Kurki-Suonio, P. Lauber, S. Lazerson, Z. Lin, A. Mishchenko, D....

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Bernstein, F

W. Bernstein, F. Chen, M. Heald, and A. Kranz, Journal of Nuclear Energy (1954)7, 276 (1958)

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A. C. England, G. L. Bell, R. H. Fowler, J. C. Glowienka, J. H. Harris, D. K. Lee, M. Murakami, G. H. Neil- son, D. A. Rasmussen, J. A. Rome, M. J. Saltmarsh, and J. B. Wilgen, Physics of Fluids B: Plasma Physics 3, 1671 (1991), https://pubs.aip.org/aip/pfb/article- pdf/3/7/1671/12661690/1671 1 online.pdf

work page 1991

[7] [7]

Rodr´ ıguez-Rodrigo, A

L. Rodr´ ıguez-Rodrigo, A. L´ opez-Fraguas, and A. Gabriel, Review of Scientific Instruments70, 645 (1999), https://pubs.aip.org/aip/rsi/article- pdf/70/1/645/19086551/645 1 online.pdf

work page 1999

[8] [8]

Moiseenko, V

V. Moiseenko, V. Korovin, I. Tarasov, M. Tarasov, D. Sitnikov, I. Garkusha, N. Zamanov, M. Lytova, R. Pavlichenko, A. Kulaga,et al., Technical Physics Let- ters40(2014)

work page 2014

[9] [9]

Birus, M

D. Birus, M. Schneider, T. Rummel, and M. Fricke, Fu- sion Engineering and Design86, 1566 (2011), proceedings of the 26th Symposium of Fusion Technology (SOFT-26)

work page 2011

[10] [10]

Risse, T

K. Risse, T. Rummel, T. M¨ onnich, F. F¨ ullenbach, and H.-S. Bosch, Fusion Engineering and Design146, 910 (2019), sI:SOFT-30

work page 2019

[11] [11]

Helander, Reports on Progress in Physics77, 087001 (2014)

P. Helander, Reports on Progress in Physics77, 087001 (2014)

work page 2014

[12] [12]

Hereγdenotes the Lorentz factor,m e the electron rest mass, andcthe speed of light

The radial orbit excursion is of the order of ∆r∼ γmec/(eB), which is about 1 cm for a 10 MeV electron in W7-X. Hereγdenotes the Lorentz factor,m e the electron rest mass, andcthe speed of light

work page

[13] [13]

Bethe, Zeitschrift f¨ ur Physik76, 293 (1932)

H. Bethe, Zeitschrift f¨ ur Physik76, 293 (1932)

work page 1932

[14] [14]

Y. A. Sokolov, JETP Letters29, 218 (1979)

work page 1979

[15] [15]

Gurevich, G

A. Gurevich, G. Milikh, and R. Roussel-Dupre, Physics Letters A165, 463 (1992)

work page 1992

[16] [16]

Jayakumar, H

R. Jayakumar, H. Fleischmann, and S. Zweben, Physics Letters A172, 447 (1993)

work page 1993

[17] [17]

Y.-K. Kim, J. P. Santos, and F. Parente, Phys. Rev. A 62, 052710 (2000)

work page 2000

[18] [18]

Connor and R

J. Connor and R. Hastie, Nuclear Fusion15, 415 (1975)

work page 1975

[19] [19]

M. N. Rosenbluth and S. V. Putvinski, Nuclear Fusion 37, 1355 (1997)

work page 1997

[20] [20]

J. R. Mart´ ın-Sol´ ıs, A. Loarte, and M. Lehnen, Physics of Plasmas22, 092512 (2015)

work page 2015

[21] [21]

Hesslow, O

L. Hesslow, O. Embr´ eus, O. Vallhagen, and T. F¨ ul¨ op, Nuclear Fusion59, 084004 (2019)

work page 2019

[22] [22]

In W7-X, the trapped fraction on axis typically ranges from 0.15 to 0.3

Note that the trapped particle fraction can be significant in stellarators, even on the magnetic axis. In W7-X, the trapped fraction on axis typically ranges from 0.15 to 0.3

work page

[23] [23]

Schauer, K

F. Schauer, K. Egorov, and V. Bykov, Fusion Engineer- ing and Design88, 1619 (2013), proceedings of the 27th Symposium On Fusion Technology (SOFT-27); Li` ege, Belgium, September 24-28, 2012

work page 2013

[24] [24]

Hegna, D

C. Hegna, D. Anderson, E. Andrew, A. Ayilaran, A. Bader, T. Bohm, K. C. Mata, J. Canik, L. Carba- jal, A. Cerfon, and et al., Journal of Plasma Physics91, E76 (2025)

work page 2025

[25] [25]

J. R. Cary and A. J. Brizard, Rev. Mod. Phys.81, 693 (2009). 6 End Matter Kinetic theory The drift kinetic equation for relativistic electrons is ∂f ∂t + ˙r· ∇f+ ˙p ∥ ∂f ∂p∥ =C(f) +S, whereSencapsulates the effect of collisions at close range and the velocity-space coordinates have been chosen to beµ=p 2 ⊥/(2B) andp ∥ =γv ∥ with γ= 1p 1−v 2/c2 = r 1 + 1 c...

work page 2009