Gravity’s role in taming the Tayler instability in red giant cores

The stability of toroidal magnetic fields in radiative stellar interiors remains a key open problem in astrophysics. We investigate the Tayler instability of purely toroidal fields, B_ϕ, in a nonrotating stellar region with stable thermal stratification in spherical geometry using global linear perturbation analysis and 3D direct numerical simulations. Both approaches are based on an equilibrium whereby the Lorentz force is balanced by a gradient of the fluid pressure, and account for gravity and thermal diffusion. The simulations include finite resistivity and viscosity, and cover the full range from stable to highly supercritical regimes for the first time. The linear analysis, global in radius, reveals two classes of unstable nonaxisymmetric modes with azimuthal wavenumber m=1. High-latitude modes grow at Alfvénic rates with short radial length scales, consistent with solutions from the fully local, Wentzel–Kramers–Brillouin (WKB) approximation. Low-latitude modes, invisible to WKB analyses, exhibit larger radial scales and reduced growth rates due to the stabilizing buoyancy. The simulations strongly support these findings and yield threshold toroidal field strengths for both the instability onset and the transition between global and WKB unstable regimes. These thresholds correspond to the roots of two algebraic equations of the form B_ϕ^ - a_1 A _1 B_ϕ^ -a_0 A _0=0, where mathcal A _0 and mathcal A _1 are functions of the fundamental fluid properties, and a_0 and a_1 are coefficients determined from the numerical simulations. When combined with stellar evolution models of low-mass stars, our results suggest that the outer radiative cores of red giants are generally unstable, while deeper degenerate regions require toroidal fields above $10-100$ kG for instability. These findings provide new constraints for asteroseismic magnetic field detection and angular momentum transport in red giant cores, and establish a general framework for identifying instability conditions in other stars with radiative interiors.