Parametric Study of the Torus Instability Threshold

This paper presents a parametric numerical study of the threshold of the torus instability, which is a key process in solar eruptions. The threshold, given by the critical decay index of the external poloidal (strapping) magnetic field, is insufficiently known and scatters in the range ncr ≈ 1–2 about its canonical value of ncr = 3/2. The study employs the force-free Titov-Démoulin equilibrium of a line-tied partial toroidal current channel and flux rope as the initial condition in zero-beta ideal MHD simulations.

The parameter range covered includes: footpoint distances Df = 1.1–1.63, minor radii of the current channel a = 0.55–0.8 (both normalized to the rope's apex height), and ratios of external toroidal (guide/shear) to poloidal (strapping) field Bet/Bep = 0–1.73. The study finds that the approximate analytical TD equilibrium expands in major radius during relaxation to a numerical equilibrium, with the expansion depending strongly on the aspect ratio. For the thinnest ropes (a = 0.55), the expansion remains moderate, while for the thickest ropes (a = 0.8), the apex height nearly doubles, and the current density redistributes significantly below the magnetic axis.

Key results, summarized at the effective critical height (weighted by the Lorentz force), are:

1. For zero guide field (Bet = 0), the critical decay index lies in the range n(h̃cr) = 1.1–1.3 for the smallest a = 0.55–0.65 and smallest Df = 1.1, consistent with analytical and many numerical studies. For a > 0.65, the critical decay index increases, reaching n(h̃cr) = 1.45 for a = 0.8, likely due to violation of the large aspect ratio approximation.

2. A guide field (Bet ≠ 0) strongly raises the threshold. The threshold n(h̃cr) rises above 1.5 for Bet/Bep ≳ 0.6 even for the thinnest ropes (a = 0.55) and for Bet/Bep ≳ 0.3 for thick ropes (a ≳ 0.7). Thresholds n(h̃cr) > 2 are found for Bet/Bep ≳ 1. This is likely the main reason for observational and numerical estimates of ncr > 3/2.

3. Line-tying is found to act stabilizing on the torus instability of moderately flat flux ropes (h ≲ Df). It stabilizes the TD flux rope by itself for Df > 1.63 if Bet = 0 and above even smaller Df in the presence of a guide field.

4. Most numerically relaxed marginally stable flux rope equilibria at Df ≤ 1.5 or a ≤ 0.65 are of pure O (BPS) type for Bet = 0. O-X-type (separator or HFT) equilibria form at larger Df, a, or Bet/Bep. A secondary current channel develops beneath the separator or HFT during relaxation, which raises the threshold.

5. The potential field based on the Green function yields a very good approximation of the instability threshold, provided the poloidal component (horizontal component perpendicular to the flux rope axis) is used. The total horizontal component yields a good approximation only for weak guide field, Bet/Bep ≲ 0.6.

6. The assumption of self-similar expansion, used in analytical estimates, is supported for weakly unstable equilibria.

The paper concludes that the strong stabilizing effect of the external toroidal field, the stabilizing effect of line-tying for flat, thick flux ropes, the good approximation provided by the poloidal component of the potential field, and the initially mostly nearly self-similar expansion of weakly unstable flux ropes are generally valid results. A further parametric study using an equilibrium with an exact image current below the photosphere is required to substantiate the critical decay index values and to infer their parametric dependence more completely.

Improvements to AI Systems:

1. Physics-Constrained Solar Eruption Prediction Models

• Integrate the parametric findings (e.g., critical decay index dependence on guide field ratio, footpoint distance, and rope thickness) into machine learning models that predict solar eruption onset.

• Improved AI can now classify active regions by computing effective critical height and decay index from magnetogram data, flagging regions with Bet/Bep > 0.6 as high-risk even if ncr < 1.5.

2. Surrogate Models for MHD Simulations

• Train neural networks to approximate the relaxation dynamics and threshold shifts (e.g., expansion factor, current redistribution) as functions of (Df, a, Bet/Bep), replacing costly 3D MHD runs.

• Improved AI can rapidly explore parameter space, generating stability maps for solar flux ropes in real-time, enabling ensemble forecasting.

3. Correction of Analytical Approximations

• Use the paper’s quantitative results to calibrate AI-based emulators of the Titov-Démoulin equilibrium, correcting for aspect-ratio-induced deviations (e.g., ncr rising to 1.45 for a=0.8).

• Improved AI can adjust decay-index estimates from potential field extrapolations, using the poloidal component (not total horizontal) for accurate thresholds, especially for guide fields <0.6.

4. Autonomous Classification of Magnetic Topology

• Develop a classifier that distinguishes O-type (BPS) vs. O-X-type (separator/HFT) equilibria based on (Df, a, Bet/Bep) thresholds identified in the paper.

• Improved AI can automatically detect secondary current channels in simulated or observed magnetograms, flagging them as threshold-raising features that delay eruptions.

5. Uncertainty Quantification in Eruption Timing

• Incorporate the stabilizing effects of line-tying (for flat ropes with h ≲ Df) and guide field into probabilistic eruption models.

• Improved AI can output confidence intervals for eruption likelihood, explicitly accounting for cases where ncr > 2 (e.g., Bet/Bep ≥ 1), reducing false alarms in space weather warnings.

6. Adaptive Mesh Refinement (AMR) Control in MHD Codes

• Use the paper’s finding that self-similar expansion holds for weakly unstable equilibria to design AI-driven AMR strategies that focus resolution along the expanding flux rope axis.

• Improved AI can dynamically allocate computational resources to resolve the secondary current channel beneath separators, capturing threshold shifts without full-grid refinement.

7. Transfer Learning for Stellar and Laboratory Plasmas

• Apply the parametric dependencies (e.g., guide field ratio, line-tying) to AI models of coronal mass ejections on other stars or tokamak disruptions.

• Improved AI can generalize stability criteria across different magnetic geometries, using the paper’s scaling laws as priors for new plasma regimes.

The torus instability of an arched current channel has been suggested to initiate and drive major solar and stellar eruptions. Its threshold, given by the critical decay index of the equilibrium external poloidal field (the so-called strapping field) at the position of the current channel, is insufficiently known. Here, we carry out a parametric numerical study of the threshold, employing the force-free Titov-D'emoulin equilibrium of a line-tied partial toroidal current channel and flux rope. This addresses the scatter of the threshold about its canonical value, n cr=3/2. Values scattering in the range n cr about,1--2 are typically found in numerical and observational studies of flux rope eruptions on the Sun. For zero external toroidal (guide, or shear) field and approximately semicircular geometry (corresponding to minimal photospheric line-tying), we find the threshold to lie in the theoretically expected range of about,1--1.5. An external toroidal field introduces a strong stabilizing effect on the instability, raising the threshold up to about,2.5, which can explain observational and numerical results above the canonical value. Line-tying is found to act stabilizing as well. We also consider the approximate threshold based on the potential field and find a very good agreement with the exact numerical value, provided the horizontal component perpendicular to the flux rope axis is used to approximate the external poloidal field.