On the Constraints and Observational Manifestations of Failed Solar Eruptions in Toroidal Magnetic Cage

Abstract: Observations show that many solar eruptions remain confined within strong overlying magnetic fields, forming a so-called magnetic cage. While confinement by poloidal overlying fields has been widely investigated, the role of strong external toroidal fields remains unclear. Using three-dimensional magnetohydrodynamic simulations, we study confined eruptions in a toroidal magnetic cage, focusing on the interplay between the Lorentz force and magnetic reconnection, and their observational signatures. We further employ a guiding-center test-particle approach to synthesize hard X-ray emission for comparison between thermal and nonthermal responses. We find that overlying toroidal fields play a crucial role in confinement by generating strong return currents that produce a significant downward Lorentz force, suppressing flux rope ascent. At the same time, they induce large-angle rotation of the flux rope, leading to reconnection with overlying fields and eventual break-up. Synthetic EUV emission reveals multi-ribbon flares with highly sheared, globally cowboy-hat-like loop structures. Hard X-ray diagnostics show that thermal and nonthermal emissions are not co-spatial, with return currents acting as an efficient accelerator of energetic electrons. These results demonstrate that toroidal-field-induced forces govern both the confinement and rotation of erupting flux ropes, providing an explanation for failed eruptions even under torus-unstable conditions. These results suggest that the morphology and shearing angle of flare loops are the useful diagnostics for distinguishing confined from eruptive events.

• The paper demonstrates that strong overlying toroidal fields generate return currents whose downward Lorentz force can exceed poloidal strapping by roughly an order of magnitude, reversing a rope launched at 150 km/s into a failed eruption.
• The simulations show that toroidal fields—not kink instability—dominate large-angle flux-rope rotation, while external slipping reconnection progressively breaks up the rope and explains confinement even when the torus criterion is met.
• The study identifies testable signatures of confined flares, including strongly sheared “cowboy-hat” loops, multiple ribbons, transient EUV CME-like structure, and hard X-ray sources preferentially located in return-current regions.

Motivation and scope

Confined flares—eruptions in which the eruptive core fails to escape the corona and produce a CME—are frequently observed, yet the physical conditions that distinguish them from eruptive events remain incompletely understood. Existing MHD models of failed eruptions have concentrated on strong overlying poloidal fields (strapping force) or on configurations containing null points. The scenario of strong overlying toroidal fields aligned with the flux-rope axis—the "toroidal magnetic cage"—has received little attention, partly because an erupting rope intuitively escapes laterally from such fields. Guo et al. (2603.21885) address this gap with 3D thermodynamic MHD simulations using MPI-AMRVAC, combining force analysis, reconnection diagnostics, EUV forward modelling, and test-particle computation of hard X-ray (HXR) sources.

The paper makes three claims that carry weight for eruption forecasting: (1) toroidal overlying fields generate return currents whose downward Lorentz force suppresses eruptions even when the rope is torus-unstable; (2) these same fields, not kink instability, drive large-angle flux-rope rotation; and (3) flare-loop morphology ("cowboy-hat" vs. saddle-like) and shearing degree can discriminate confined from eruptive flares observationally.

Numerical setup

The initial configuration follows the Titov–Démoulin modified (TDm) approach: a force-free toroidal flux rope ($R=30$ Mm, $a=12$ Mm) embedded via regularised Biot–Savart laws in potential fields produced by four sub-photospheric magnetic charges—two defining the external poloidal field $B_{\rm p}$ and two the external toroidal field $B_{\rm T}$ (model FR-PT). A control model FR-P retains only poloidal charges. Magneto-frictional relaxation establishes a near-force-free initial state; the rope sits near the torus-instability threshold ($n_p \approx 1.5$). The decay index computed from the total horizontal field exhibits a saddle-like profile, while $n_t$ and $n_p$ increase monotonically.

The simulations solve the full MHD equations with Spitzer thermal conduction, uniform resistivity, line-tied bottom boundary, constrained transport on staggered grids ($\langle|f_i|\rangle \sim 10^{-16}$), HLL fluxes, fifth-order WENO reconstruction, and four-level AMR on a $300^3$ Mm domain. Gravitational acceleration is scaled by $k=1.8$ to keep plasma $a=12$0 throughout, ensuring magnetically dominated dynamics—an assumption that shapes all subsequent conclusions.

Global evolution of the confined eruption

Triggered by torus instability, the rope accelerates to 150 km s$a=12$1 at $a=12$2, then decelerates to $a=12$3 km s$a=12$4 by $a=12$5 ($a=12$6 s), i.e., it reverses motion—a clear failed eruption. During ascent the rope rotates clockwise by roughly 90°, consistent with the observed sense of rotation for sinistral filaments. Twist number $a=12$7 decreases as twist converts to writhe and as reconnection with overlying fields proceeds; a shell of opposite-sign twist develops around the rope, co-spatial with the return current (RC).

Flare ribbons form along the PIL, separate at $a=12$8 km s$a=12$9, decelerate, and halt near $B_{\rm p}$0 Mm—demonstrating that ribbon separation occurs in confined flares too. Circular QSLs drift, expand, and merge with hooked outer QSLs; late-stage dot-like heated kernels appear at ribbon endpoints near the toroidal polarities.

Confinement mechanism: Lorentz-force decomposition

Decomposing $B_{\rm p}$1 allows separation of the vertical Lorentz force into self-, poloidal-, and toroidal-induced components. Three results stand out:

• Self-force: does not account for the net downward force.
• Poloidal strapping force: initially downward but weakens as the rope rises and expands, roughly balancing the upward magnetic-pressure gradient after $B_{\rm p}$2.
• Toroidal tension force: exceeds the poloidal contribution by roughly an order of magnitude because the environment is strongly sheared and $B_{\rm p}$3-dominated. It acts oppositely on upper and lower rope portions, and a concentric RC shell around the leading dome produces strong downward force from $B_{\rm p}$4 onward.

Field-line tracing shows that return currents anchored near toroidal polarities yield downward Lorentz forces, whereas those near poloidal polarities yield upward forces. This asymmetry provides a physical explanation for why confined flares preferentially occur in electric-current-neutralised active regions: the modeled |DC/RC| ratio drops rapidly from 8.9 to 3.5 within $B_{\rm p}$5. The authors note candidly that their TDm-based initial state yields |DC/RC| values larger than typically observed, so absolute neutralisation levels should not be compared directly with observations.

The control experiment confirms causality: without toroidal polarities (FR-P), the rope escapes to 300 Mm within $B_{\rm p}$6, reaching ~250 km s$B_{\rm p}$7, with |DC/RC| ≈ 30 versus ≈ 9 in FR-PT. A parameter survey varying $B_{\rm p}$8 further shows that rotation angle increases monotonically with toroidal-field strength and that sufficiently strong $B_{\rm p}$9 flips the system into the failed-eruption regime.

Rotation driven by external toroidal fields

Lateral Lorentz-force decomposition at $B_{\rm T}$0 Mm shows that the torque rotating the rope's upper section is dominated by $B_{\rm T}$1; the self-force drives counter-rotation, and the poloidal contribution is negligible. An isolated-rope run (no external fields) exhibits only weak rotation. Two implications follow directly. First, large-angle filament rotation cannot be taken as evidence of kink instability; kink may trigger initiation but does not govern rotation during eruption. Second, since the same toroidal field both drives rotation and supplies the downward tension force, the frequent association between rotating filaments and failed eruptions—and the fact that such filaments often halt where $B_{\rm T}$2—is self-consistently explained: the restraining force is toroidal-field-induced tension, not poloidal strapping, so the conventional decay-index criterion based on $B_{\rm T}$3 misdiagnoses confinement.

External reconnection destroying the flux rope

Rotation thins the QSLs between rope and overlying field, intensifying currents there. Traced field lines show slipping reconnection: one converts from a flux-rope line to a low flare loop (apex dropping from 150 Mm to 20 Mm within $B_{\rm T}$4, with sharp temperature increase); another transfers a footpoint to the outer toroidal polarity $B_{\rm T}$5. Both evolutions correspond to $B_{\rm T}$6–$B_{\rm T}$7 reconnection geometry, demonstrating progressive break-up of the rising rope. This identifies reconnection-induced disruption as a second confinement channel operating alongside the downward Lorentz force.

Thermal forward modelling: multi-ribbon flares and cowboy-hat loops

Synthesised 94 Å images reproduce a three-part CME-like structure (front, cavity, core) even though the eruption remains confined, plus conjugate dimmings inside ribbon hooks and late dot-like kernels at toroidal polarity borders—observational signatures of external reconnection. Side views reveal two groups of highly sheared arcades reconnecting into high-lying loops whose ensemble forms a transient "cowboy-hat" morphology (central loops arching above both ends). Comparison with a well-studied observed confined flare shows matching strongly sheared low-lying loops and multiple ribbons, supporting the model's observational fidelity.

Crucially, the paper contrasts this with the eruptive case: successful eruptions produce saddle-like loops (ends higher than centre) and weakly sheared loops nearly perpendicular to the PIL, whereas confined flares produce cowboy-hat loops with strong shear. Because the cowboy-hat shape relaxes within a few Alfvén times, detection requires observations close to flare peak in hot channels (94/131 Å). These morphological and shear diagnostics offer a practical proxy for CME productivity when coronagraphic classification is ambiguous (e.g., disk-center events). One caveat: "shear" here refers to coronal flare-loop geometry, not photospheric field shear, which behaves oppositely in statistical studies.

Non-thermal response: hard X-ray sources and return-current acceleration

Test-particle electrons ($B_{\rm T}$8, Maxwellian at 1 MK) propagated through the MHD fields via guiding-centre approximation, with HXR synthesis through a thick-target bremsstrahlung model, yield two notable findings:

• Spectral softening: the electron spectral index steepens from −4.39 at $B_{\rm T}$9 to −5.95 at $n_p \approx 1.5$0, tracking the decline of footpoint current.
• Spatial non-coincidence: >50 keV electrons deposit preferentially in return-current regions, while heated EUV ribbons coincide with direct-current regions. In the late phase, most energetic electrons land on the weakly heated RC ribbon.

The interpretation is that in a failed eruption the sub-rope current sheet develops inefficiently, while QSLs between rope and ambient field host extended, intense return-current channels that trap and accelerate electrons over long path lengths. This predicts asymmetric, displaced HXR sources relative to thermal ribbons—consistent with reported HXR asymmetries. The authors acknowledge that energetic-electron feedback on the MHD solution is neglected, so all heating is purely thermal (Joule, compressional, conductive); the thermal/non-thermal dichotomy is therefore a consequence of the one-way coupling assumption.

Limitations and open questions

Several limitations are stated explicitly. The idealised TDm configuration cannot capture how the return current forms temporally—whether via flux emergence or photospheric flows—so the claim that toroidal fields induce return currents requires validation in data-driven or radiative-MHD simulations. The conclusion that return currents produce substantial downward Lorentz force likewise awaits testing in realistic active-region models. The |DC/RC| values exceed observed ranges owing to the TDm construction. Finally, the proposed shear-angle diagnostic lacks a calibrated threshold separating confined from eruptive flares; determining that threshold is identified as the key objective of future parameter surveys.

Conclusion

This work establishes the toroidal magnetic cage as a physically distinct confinement channel: overlying toroidal fields simultaneously (i) induce return currents whose downward Lorentz force dominates the poloidal strapping force by about an order of magnitude, (ii) drive large-angle flux-rope rotation independently of kink instability, and (iii) promote external reconnection that destroys the rope. Together these effects explain why torus-unstable, strongly rotating filaments often fail and why current-neutralised active regions favour confined flares. The associated observational fingerprints—multi-ribbon structure, transient cowboy-hat flare loops, strong coronal-loop shear, and HXR sources displaced onto return-current ribbons—provide testable diagnostics for distinguishing confined from eruptive flares, pending quantitative calibration against real active regions.