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

Abstract

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 $B_φ$ 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 Alfvénic 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 $B_φ^{3/4} - a_1 \mathcal{A}_1 B_φ^{1/4} - a_0 \mathcal{A}_0 = 0$, 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.

Figures (13)

• Figure 1: Radial profiles of the real parts of selected unstable eigenmodes $B_{\theta}^\prime$ in the absence of thermal diffusion ($\epsilon=0$). Panels (a)-(c) show the high-latitude eigenmodes ($\theta=5^\circ$) with the nearly adiabatic growth rate of $\Gamma=0.95$ for different values of the stratification parameter $\delta$ (see legend insets). The inset in (c) provides a zoom on the eigenmode between $r=0.10$ and $0.12$. Panel (d) displays the most unstable eigenmodes at the equator ($\theta=90^\circ$); no unstable modes are found for $\delta\geq 25$. The horizontal axis is logarithmic in (a)-(c) and linear in (d).
• Figure 2: Radial wavenumber $k_r$ of unstable eigenmodes, evaluated from the real part of $B_\theta^\prime$, as a function of $\delta$ for (a) no thermal diffusion ($\epsilon=0$) and (b) finite thermal diffusion ($\epsilon=10^{-2}$). Triangles connected by solid lines indicate eigenmodes at colatitude $\theta=5^\circ$ with growth rates $\Gamma=0.8$ (blue), 0.9 (purple), and 0.95 (red). Selected eigenmodes of the $\Gamma=0.95$ tracks in (a) and (b) are displayed in Fig. \ref{['f:lin_noeps']}a-c and Fig. \ref{['f:lin_eps']}a-c, respectively. The dotted lines denote the minimum unstable wavenumber predicted by the local linear theory (Eqs. \ref{['e:kr_WKB_eps0']} and \ref{['e:kr_WKB']}). Open gray points connected by dashed lines correspond to the equatorial eigenmodes (Fig. \ref{['f:lin_noeps']}d and dashed lines in Fig. \ref{['f:lin_eps']}d).
• Figure 3: Same as Fig. \ref{['f:lin_noeps']} but for $\epsilon=10^{-2}$. In panel (d), dashed and solid lines correspond to the localized (type I) and distributed (type II) equatorial eigenmodes, respectively (see text for details). Type I eigenmodes are the most unstable (see Fig. \ref{['f:lin_grwr_delta']} for their growth rate values). The horizontal axis is logarithmic in all panels.
• Figure 4: Dimensionless growth rates of the type I (most unstable) equatorial eigenmodes, $\Gamma_\text{max}$, as a function of the stratification parameter $\delta$ for $\epsilon=10^{-2}$. The dashed line indicates the scaling $\Gamma_\text{max}\propto \delta^{-2}$ predicted by Bonanno12a.
• Figure 5: Temporal evolution of the volume-averaged toroidal magnetic energy (black lines) and poloidal kinetic energy (red lines). Results are shown for the unstratified run ($\delta_0=0$) and two stratified runs ($\delta_0=300$ and $900$). The Lundquist number is $\text{Lu}_0=917$. Both the flow and magnetic field remain purely axisymmetric throughout the evolution.