Revised Tayler Instability (TSF)

• Revised Tayler Instability (TSF) is an updated framework that refines the classic m=1 kink instability of toroidal magnetic fields, crucial for stellar angular momentum transport and dynamo action.
• It extends traditional criteria using branch-dependent growth laws, incorporating inertial, viscous, and thermal diffusion effects within asteroseismic and MHD calibration.
• Its application in stellar modeling demonstrates improved core rotation rate predictions through nonlinear saturation physics and advanced instability mapping.

The revised Tayler instability ("TSF") encompasses an updated theoretical, computational, and phenomenological framework for the m=1 kink instability of toroidal magnetic fields, with direct implications for stellar evolution, angular momentum transport, and dynamo action. Modern variants refine the original Spruit–Tayler dynamo prescription via generalized instability criteria, branch-dependent growth laws, and saturation models rooted in asteroseismic and magnetohydrodynamic (MHD) constraints.

1. Fundamental Instability Mechanism and Generalized Criteria

The Tayler instability arises in differentially rotating, stably stratified stellar interiors when a toroidal magnetic field $B_\phi$ reaches a threshold to destabilize non-axisymmetric (primarily $m=1$) helical perturbations. In the ideal non-rotating, dissipationless limit, the instability occurs whenever the field profile satisfies

$$\dfrac{d}{dr}\big(r\,B_\phi^2\big) < 0 \tag{Classic TI}$$

(Zou et al., 2019). This criterion generalizes in the presence of rotation, viscosity, and resistivity. The extended Hain–Lüst equation yields the revised TSF criterion for cylindrical flows:

$$Rb > \dfrac{m^2}{4\alpha^2} - 1 \tag{TSF Instability Threshold}$$

where $Rb = r\mu'/2\mu$ is the toroidal-field gradient parameter and $\alpha$ encodes axial wave geometry (Zou et al., 2019). The threshold is robust against finite $Pm$ (magnetic Prandtl number) and differential rotation; it admits both short-wavelength TI and long-wavelength branches operative at low $k$.

For mixed poloidal–toroidal fields, non-axisymmetry alters the criterion, demanding the positivity of the quadratic form in the displacement field ($\xi$) via energy-principle conditions (Augustson et al., 2016). In spherical geometry, the axisymmetric limit recovers the classic condition, while additional constraints emerge for more general field topologies.

2. Linear Growth Rates, Suppression Mechanisms, and Branch Structure

In strongly stratified, rotating stellar interiors, the TSF formalism distinguishes multiple instability branches based on the dominant microphysical diffusion process (Skoutnev et al., 29 Apr 2024, Skoutnev et al., 13 Nov 2024):

• Inertial-wave branch: Active for $Pm \ll 1$, maximal growth at $k_\eta = \sqrt{2\Omega/\eta}$, $\sigma_\text{max} \sim \omega_A^2/4\Omega$, onset $\,\omega_A/(2\Omega) > Pm^{1/2}\,$.
• Viscous magnetostrophic branch: For $Pm \gg 1$, peak at $k_\nu = \sqrt{2\Omega/\nu}$, same $\sigma_\text{max}$, onset $\,\omega_A/(2\Omega) > Pm^{-1/2}\,$.
• Thermal/compositional magnetostrophic branch: Where thermal or compositional diffusion dominates, $k_{\kappa_{T,\mu}}$ defined by stratification and diffusivity; growth rate $\sim 0.21\,\omega_A^2/\Omega$.

The revised TSF instability map ("toggle-switch") in 1D stellar evolution codes applies analytic inequalities at each mass shell to switch angular momentum transport "on" if any branch is unstable (Skoutnev et al., 13 Nov 2024).

Stable stratification ($N^2$ via Brunt–Väisälä frequency) suppresses TI at large $\omega_{BV}/\omega_A$ ($\delta$). Robust anelastic MHD simulations locate a critical $\delta_\mathrm{crit} \sim 50$, above which growth rates fall to stellar timescales, demarcating the "magnetic desert" in Ap/Bp stars (Guerrero et al., 2019, Goldstein et al., 2018).

3. Nonlinear Saturation, Dynamo Loops, and Angular Momentum Transport

Magnetic energy extracted by the instability saturates by balancing amplification and the damping timescale set by diffusive, viscous, or turbulence-induced dissipation. In the TSF prescription [Fuller et al. 2019] and subsequent asteroseismic calibrations, this is parameterized via:

$$\nu_{\text{TSF}} = \alpha^3\,r^2\,\Omega\,\left(\frac{\Omega}{N_{\text{eff}}}\right)^2 \tag{TSF Diffusion Coefficient}$$

where $N_{\text{eff}}^2 = (\eta/K) N_T^2 + N_\mu^2$ combines thermal and compositional stratification (Si et al., 24 Dec 2025, Eggenberger et al., 2023). The minimum shear required for TI is

$$q_{\min} = \alpha^{-3}\left(\frac{N_{\text{eff}}}{\Omega}\right)^{5/2}\left(\frac{\eta}{r^2\Omega}\right)^{3/4}$$

Saturation is enforced when $q \sim q_{\min}$; viscous torque and associated mixing are active only when this threshold is exceeded.

Non-linear MHD simulations confirm the dominance of $m=1$ modes, secondary shear-induced saturation, and weak axisymmetric dynamo regeneration (Ji et al., 2022). The measured angular momentum transport efficiency is steeply sensitive to $\omega_A/\Omega$, scaling as $$\nu_T \propto r^2\Omega\,q\,\left(\frac{\omega_A}{\Omega}\right)^5$$ under turbulent quenching (Ji et al., 2022).

4. Application to Stellar Models: Calibration, Limitations, and Observational Constraints

The revised TSF prescription produces low core rotation rates in post-main-sequence stars, aligning more closely with observation than hydrodynamical or uncalibrated Tayler–Spruit models (Eggenberger et al., 2020, Si et al., 24 Dec 2025, Eggenberger et al., 2023). Its efficacy is controlled by the dimensionless coefficient $\alpha$, which encapsulates the uncertain saturation amplitudes.

Asteroseismic calibration in low-mass giants constrains $\alpha \sim 1$–6, whereas models of massive WNE stars require $\alpha \sim 0.01$ for optimal slowing of the core and envelope rotation (Si et al., 24 Dec 2025). However, a single value cannot reconcile transport at all evolutionary stages: matching subgiant and red giant interiors demands inconsistent $\alpha$ (Eggenberger et al., 2020). Compositional stratification (via $N_\mu$) further suppresses instability in layers around the hydrogen-burning shell, tightly restricting the region of effective angular momentum redistribution (Skoutnev et al., 13 Nov 2024).

Comparison of revised TSF, original Spruit, and Fuller prescriptions:

Prescription	Scaling of $\nu_T$	Calibration	Outcome
Original Spruit	$r^2\Omega\,q^2(\Omega/N_{\text{eff}})^4$	$n=1, C_T=1$	Too weak
Fuller (2019)	$r^2\Omega\,(\Omega/N_{\text{eff}})^2$	$n=3, C_T=1$	Over-braking (subgiants)
Calibrated TSF	$r^2\Omega\,q^2(\Omega/N_{\text{eff}})^4$	$n=1, C_T \gg 1$	"Best fit"

(Eggenberger et al., 2023)

5. Extensions: Mixed Field Topology, Liquid Metal Batteries, and Laboratory Analogs

The instability threshold and formalism generalize to arbitrary axisymmetric and non-axisymmetric mixed poloidal–toroidal fields via quadratic form energy-principle analysis (Augustson et al., 2016). Relaxing axisymmetry or introducing poloidal components sharpens the instability criteria and enables dynamo loops via the emergence of a growing $m=0$ axisymmetric mode—potentially closing the Tayler–Spruit dynamo (Ibañez-Mejia et al., 2015).

In laboratory settings, such as liquid metal batteries, the Tayler kink sets a critical current $I_{\text{crit}}$ and growth rate formulas for cylindrical geometries. Practical suppression strategies involve geometric modifications, return-path currents, and axial field stabilization (Stefani et al., 2010). These design principles directly apply the TSF theory to technological contexts.

6. Open Problems, Limitations, and Future Directions

Recent MHD simulations and theoretical analyses expose several unresolved aspects:

• Non-universality of transport scaling: Branch-dependent thresholds and diffusive parameters imply no universal dynamo law; time-dependent regimes and composition stratification further restrict transport regions (Skoutnev et al., 29 Apr 2024, Skoutnev et al., 13 Nov 2024).
• Nonlinear saturation physics: Secondary instabilities, flow feedback, and possible axisymmetric mode coupling remain insufficiently constrained, especially concerning oscillatory saturation and Hopf bifurcations (Bonanno et al., 2016).
• Parameter calibration: Empirical $\alpha$ values remain model-dependent; full validation requires systematic asteroseismic and spectroscopic comparison.
• Compositional mixing and feedback: Most implementations switch off mixing due to TI, whereas true magnetic fluctuations may couple momentum and composition, requiring self-consistent 3D MHD treatment (Si et al., 24 Dec 2025).
• Anelastic vs. compressible MHD: The anTI criterion modifies stability domains, with sound-wave filtering leading to less efficient transport than predicted by fully compressible analyses (Goldstein et al., 2018).

A plausible implication is that future unified models must merge TSF prescriptions with internal gravity wave transport, secular shear, and feedbacks involving composition and thermal gradients. Direct high-resolution global MHD simulations remain essential to refine saturation laws, instability mapping, and dynamo closure.

7. Summary Table: Core Revised TSF Equations and Criteria

Concept	Expression	Reference
Instability threshold	$Rb > m^2/4\alpha^2 - 1$	(Zou et al., 2019)
Canonical growth rate	$\gamma_{\text{TI}} \sim \omega_A^2 / 4\Omega$	(Skoutnev et al., 13 Nov 2024)
TSF viscosity	$$\nu_{\text{TSF}} = \alpha^3 r^2 \Omega (\Omega / N_{\text{eff}})^2$$	(Si et al., 24 Dec 2025)
Minimum shear	$$q_{\min} = \alpha^{-3}(N_{\text{eff}}/\Omega)^{5/2}(\eta/(r^2 \Omega))^{3/4}$$	(Si et al., 24 Dec 2025)
Anelastic instability criterion	$$q(\alpha-q/\gamma) + [2/\beta](\alpha-1) + [(\alpha-2)(q-\alpha\gamma)]/(\beta\gamma) < 0$$	(Goldstein et al., 2018)

The revised Tayler instability framework ("TSF") synthesizes advanced linear and nonlinear MHD theory with observational calibration, branch-dependent transport rules, and code-level implementation algorithms. It replaces earlier monolithic prescriptions, robustly quantifies magnetic angular-momentum transport in stellar interiors, and establishes rigorous conditions for instability onset and saturation across astrophysical and laboratory regimes.