Gravity's role in taming the Tayler instability in red giant cores
The stability of toroidal magnetic fields in radiative stellar interiors is a key open problem in astrophysics. We investigate the Tayler instability of purely toroidal fields $B_\phi$ in a nonrotating, thermally stably stratified stellar region using global linear perturbation analysis and 3D direct numerical simulations in spherical geometry. Both approaches assume a magnetohydrostatic equilibrium where the Lorentz force is balanced by a pressure gradient, and include gravity and thermal diffusion. The simulations incorporate finite resistivity and viscosity and span the full range from stable to highly supercritical regimes for the first time. The global linear analysis reveals two classes of unstable nonaxisymmetric $m=1$ modes. High-latitude modes grow at Alfv\'enic rates with short radial scales, consistent with local WKB solutions. Low-latitude modes, missed by local analyses, show larger radial scales and reduced growth rates due to the stabilizing buoyancy. Simulations support these findings and yield field strength thresholds for both instability onset and the transition between global and WKB regimes. These thresholds correspond to the roots of two algebraic equations of the form $B_\phi^{3/4} - a_1 \mathcal{A}_1 B_\phi^{1/4} - a_0 \mathcal{A}_0 = 0$, where $\mathcal{A}_0$, $\mathcal{A}_1$ depend on the fluid properties, and $a_0$, $a_1$ are simulation-derived coefficients. Combining our results with stellar evolution models of low-mass stars, we find that outer radiative cores of red giants are generally unstable, while deeper degenerate regions require toroidal fields above $10-100$ kG for instability. Our findings may help to constrain asteroseismic magnetic field detection and angular momentum transport in red giant cores, and provide a framework for identifying instability conditions in other stars with radiative interiors.