Torus Instability in Plasma & Astrophysics

• Torus instability is a magnetohydrodynamic phenomenon where the upward hoop force overcomes the restraining external field, causing flux rope eruptions.
• It is defined by a critical decay index that determines the onset of instability in toroidal structures, with applications in solar, astrophysical, and laboratory contexts.
• Laboratory experiments and theoretical models validate its role in predicting eruptive events and advance our understanding of complex plasma dynamics.

Torus instability is a critical ideal magnetohydrodynamic (MHD) phenomenon characterizing the loss of equilibrium of current-carrying toroidal structures under an external confining field, with important consequences in astrophysics, plasma physics, and fluid dynamics. In high-energy astrophysical systems, it governs the transition from stability to eruptivity in black hole–torus configurations, solar and stellar coronae, toroidal liquid droplets, and laboratory plasmas. Its onset, characterized by a critical decay in the restraining field with height, signifies the threshold at which a flux rope or toroidal structure can erupt or collapse. The physical, mathematical, and observational signatures of this instability exhibit both universality and system-specific subtleties that are central to understanding energetic processes such as coronal mass ejections, relativistic jet launching, and liquid filament dynamics.

1. Fundamental Physics and Stability Criteria

Torus instability emerges when the upward “hoop force” generated by azimuthal currents in a toroidal structure overcomes the downward Lorentz (strapping) force from the external poloidal field. The core mathematical criterion is the decay index $n$, which quantifies how rapidly the external field $B$ decreases with height or radius: $n = - \frac{d \ln |B_{ext,p}|}{d \ln h}$ A flux rope becomes unstable if $n$ exceeds a critical value $n_{crit}$, leading to loss of equilibrium and potentially rapid eruption or collapse (Demoulin et al., 2010, Alt et al., 2020). For idealized geometry:

• Circular (toroidal) current channel: $n_{crit} \approx 1.5$
• Straight (line) current channel: $n_{crit} \approx 1.0$
• For realistic, thick and deformable channels relevant to the solar corona: $n_{crit} \sim 1.1-1.3$ (Demoulin et al., 2010)

An essential insight is that both the “loss-of-equilibrium” (no neighboring stable state exists under a flux-conserving constraint) and linear MHD stability analysis yield the same instability threshold in terms of $n$ (Demoulin et al., 2010), confirming that the geometrical and evolutionary descriptions are congruent.

2. Astrophysical Manifestations: Solar and Stellar Contexts

In solar and stellar coronae, torus instability underpins the eruption of magnetic flux ropes, leading to coronal mass ejections (CMEs). The decay index of the external poloidal field above the active region sets a “critical height” $h_{crit}$, with instability and eruption occurring when the flux rope apex reaches the altitude where $n(h_{crit}) \approx n_{crit}$. Crucial findings from large-scale magnetogram studies over several solar cycles (James et al., 4 Sep 2024, James et al., 2022, Wang et al., 2017, Zuccarello et al., 2014, Sun et al., 2021) include:

• $h_{crit}$ is empirically proportional to the horizontal separation $d$ of opposite magnetic polarities: $h_{crit} \approx 0.5d$
• $h_{crit}$ also obeys a power law with total unsigned magnetic flux $\Phi$: $h_{crit} \propto \Phi^{0.31}$
• During solar maximum, active regions with greater flux and larger $d$ exhibit higher $h_{crit}$ values.
• CMEs are more frequent at solar maximum despite higher $h_{crit}$, likely due to increased flux emergence, magnetic complexity, and energetic injection.

Table: Empirical Relationships in Solar Active Regions

Parameter	Relationship to $h_{crit}$	Range/Typical Value
Polarity separation $d$	$h_{crit} \approx 0.5 d + 7.35$ Mm	$d \sim$ 30–100 Mm
Unsigned flux $\Phi$	$h_{crit} \propto \Phi^{0.31}$	$\Phi \sim 10^{20-22}$ Mx

Decrease in $h_{crit}$ (steeper decay of $B_{ext}$) lowers the threshold for eruption, favoring CME production (James et al., 4 Sep 2024, James et al., 2022, Wang et al., 2017). Multipolar field geometry or “saddle” decay index profiles can result in zones where instability is suppressed, causing confined flares (Wang et al., 2017).

In the stellar context, the “torus-stable zone” (TSZ) above starspots can confine erupting material high in the corona, inhibiting successful CME launch. The TSZ’s extent is set by the interplay between rapid decay from local spots and gradual decay from global dipole fields. Certain configurations lead to multiple TSZs, providing a plausible mechanism for “failed eruptions” (Sun et al., 2021).

3. Laboratory Plasma and Liquid System Analogues

Laboratory experiments on arched, line-tied magnetic flux ropes have reproduced torus instability under controlled conditions, validating theoretical predictions relevant to solar plasmas. These studies reveal:

• The critical decay index for line-tied, partial torus ropes is systematically lower than for a full torus ($n_{crit} \sim 0.9$, range 0.7–1.2), reflecting the influence of geometry and anchoring (Alt et al., 2020).
• “Failed torus” events exhibit initial instability but halt via current redistribution (edge “hollowing”) that boosts internal toroidal field and tension, re-stabilizing the system without full eruption (Alt et al., 2023).
• Energy measurements indicate that failed torus events can be largely explained by ideal MHD energy conservation, with non-ideal effects localized near footpoints.

In toroidal liquid drops (toroidal droplets), a shrinking instability unique to the topology arises—distinct from the classical Rayleigh-Plateau instability of cylinders. Driven by curvature-induced Laplace pressure asymmetry and volume conservation, the torus shrinks by reducing the inner “hole,” while the cross-section swells (Yao et al., 2010).

4. Extensions: Magnetic Topology, Pre-eruptive Processes, and Dynamical Evolution

Torus instability is modulated by the topology and internal properties of the toroidal structure:

• In the solar context, flux ropes may become eruptive only after pre-eruption processes (magnetic reconnection, flux cancellation, tether-cutting) raise the filament axis into the torus-unstable altitude or reduce the overlying field (Zuccarello et al., 2014, Prasad et al., 2023).
• The coupling of tether-cutting reconnection with rapid change in twist/current enhances instability independently of the background decay profile (Inoue et al., 4 Jul 2025).
• Eruptivity is best predicted using combined proxies: the degree of current neutralization (|DC/RC|), which controls the self-hoop force, and the normalized critical height. The dimensionless parameter $S = |DC/RC|/\mathrm{nch}$ (with nch the normalized $h_{crit}$) significantly improves discrimination between eruptive and non-eruptive active regions (Muhamad et al., 27 Mar 2025).

Table: Composite Predictors for Solar Eruptivity

Proxy	Physical Role	Discrimination Power
Decay index ($h_{crit}$)	Restraining field	Modest
Electric current neutralization ($|DC/RC|$)	Driving force	Modest
$S = |DC/RC|/\mathrm{nch}$	Composite predictor	High

Small-scale flux emergence, even at much smaller scale than the background field, can accelerate an eruption by raising the internal current and twist of a pre-existing flux rope, thereby enhancing the growth rate of torus instability at the nonlinear stage, with negligible changes to the large-scale decay index (Inoue et al., 4 Jul 2025).

5. Torus Instability in Relativistic and Exotic Contexts

In general relativistic black hole–torus systems, torus instability connects with several phenomena:

• The “runaway instability” in fluid tori arises when mass transfer across a “cusp” in the potential leads to accelerating accretion. However, full general relativistic simulations demonstrate that self-gravity alone does not induce runaway instability; axisymmetric oscillations around equilibrium are observed, and the onset of instability depends sensitively on the torus’s angular momentum profile and, when included, magnetic fields (Montero et al., 2010, Hamersky et al., 2013).
• In the presence of a toroidal magnetic field, sub-equipartition (strong) magnetization can tip a marginally stable configuration into instability by boosting internal pressure, though the oscillation frequencies remain tied to the radial epicyclic mode (Hamersky et al., 2013).
• Non-axisymmetric instabilities (such as the Papaloizou-Pringle instability) also play major roles in black hole–torus systems, with different dynamical signatures and emission properties (notably, strong gravitational wave emission from the $m=1$ mode deformations) (Kiuchi et al., 2011, Donmez, 2013, Donmez, 2017).

In soft-matter systems, toroidal nematics can be destabilized not only by elevated surface-like elasticity (saddle-splay constant $K_{24}$) but also by low twist-to-bend ratio ($K_2/K_3$), as occurs in chromonic liquid crystals. Instabilities emerge from the interplay of elastic anisotropy and boundary conditions (Pedrini et al., 2018).

6. Mathematical Formulations Across Domains

The central mathematical objects governing torus instability include:

• Decay Index (for external/strapping field in MHD):

$n = -\frac{d\ln B_{ext,p}}{d\ln h}$

with threshold values $n_{crit}$ as above.

• Pressure and Free Energy in Fluid Systems:

$$\Delta p(r, \alpha) = 0 , \qquad p-p_0 = \sigma H , \qquad dF/dR_1 = 2\pi^2 \sigma R_2$$

• Free Energy for Toroidal Nematics:

$$F[n] = \frac{1}{2}\int_{\mathcal{B}}\left\{ K_1 (\nabla \cdot n)^2 + K_2 (n \cdot \nabla \times n)^2 + K_3 |n \times \nabla \times n|^2 \right\} dV + K_{24} \int_{\partial \mathcal{B}}\{ \cdots \} dA$$

• Composite Eruptivity Parameter (Solar Flares):

$S = \frac{|DC/RC|}{nch}$

where $nch = h_{crit} / \langle h_{crit} \rangle$.

7. Implications and Future Directions

The torus instability sets a universal geometric and energetic threshold for eruption in diverse physical systems, but realized dynamics and observations depend on additional factors:

• The geometry (e.g., degree of line-tying or curvature, anchoring), flux rope thickness, and nature of current closure.
• Evolution of internal structure: current profile, twist, and the efficacy of reconnection processes in modifying them.
• Variability in $n_{crit}$ with geometry, propagation environment, and boundary conditions mandates that eruption forecasting in solar or stellar systems factor in specific region diagnostics, not just decay index estimates (Alt et al., 2020, Alt et al., 2023).
• The growing recognition that combined, multi-proxy approaches—merging field topology, current buildup, and environmental decay properties—are required for robust prediction.

Further research directions include quantifying $n_{crit}$ for more complex geometries and plasma conditions, modeling multi-scale coupling during eruption triggering (especially owing to small-scale flux emergence (Inoue et al., 4 Jul 2025)), and establishing standardized observational diagnostics, particularly in the context of operational space weather forecasting. Continued synergy between laboratory experiments and high-resolution observations is expected to refine the predictive understanding of torus instability and its role in dynamic astrophysical and laboratory plasmas.