Magnetohydrodynamic instabilities in stellar radiative regions. I. Linear study of shear-driven instabilities

TL;DR

This work addresses how shear-driven magnetohydrodynamic instabilities operate in stellar radiative zones, with a focus on the magnetorotational instability ($MRI$) and the magnetised GSF instability ($MGSF$). It combines a local linear stability analysis that includes stratification, diffusion, and magnetic tension with a global Taylor–Couette study to validate the local predictions and to map instability domains and growth rates under realistic stellar conditions. The authors recover the standard $MRI$ criteria, derive a new criterion and growth rate for $MGSF$ in strongly sheared, magnetised flows, and provide practical growth‑time formulae that can be implemented in 1D stellar evolution codes. They further apply the framework to subgiants and young red giants, showing that shear‑driven instabilities can grow rapidly for fields below $\sim100\,\mathrm{kG}$, while strong axial fields confined to the hydrogen‑burning shell can suppress instabilities unless the shear is sufficiently remote. These results offer concrete criteria and timescales to incorporate into models of angular momentum transport in stellar evolution.

Abstract

This paper is the first in a series investigating magnetohydrodynamic instabilities that may contribute to angular-momentum transport and magnetic-field evolution in stellar radiative zones. We focus on shear-driven instabilities, specifically the Goldreich-Schubert-Fricke (GSF) instability and the magnetorotational instability (MRI), which are expected to play key roles in the internal dynamics of radiative regions. We carried out a local linear stability analysis using a numerical approach that extends beyond classical limiting cases and includes stabilizing effects such as stratification and magnetic tension, allowing the exploration of realistic flow regimes. These results were validated through a global mode analysis in a Taylor-Couette configuration. We recovered the standard MRI and azimuthal MRI stability criteria and quantified the effects of stratification, magnetic tension, and diffusion on their growth. In strongly sheared regimes, we derived a new criterion for the magnetised GSF (MGSF) instability and clarified the transition from SMRI to MGSF as stratification and magnetic effects narrow the unstable domain. We also provided approximate growth-time formulae that identify the dominant instability under given stellar conditions and can be implemented in 1D stellar evolution codes. Global Taylor-Couette calculations validate the local WKB analysis. Applied to subgiants and young red giants, our results show that shear-driven instabilities can grow rapidly for magnetic fields below 100 kG. Strong axial fields (100 kG) confined to the hydrogen-burning shell suppress instabilities unless the shear is sufficiently distant. These results support incorporating our criteria and growth estimates into stellar evolution models to assess the efficiency of shear-driven transport.

Figures (14)

• Figure 1: Sketch of the system under study. In the inertial frame, the differentially rotating flow is confined between two shells, subjected to a radial thermal gradient and immersed in a magnetic field with both axial and azimuthal components.
• Figure 2: Top panel: Growth rate, $\sigma_r$, normalised by $\Omega$ in the $(\operatorname{{P m}}, \Lambda_Z)$ plane of the axisymmetric SMRI, for a regime typical of DNS. See details in Table \ref{['table']}, with $\operatorname{{R o}} = -0.05, \operatorname{{(N / \Omega)^2}} = 1, \Lambda_\phi = 0$. The three dashed lines correspond respectively to the three criteria equation \ref{['MRI_crit']}. The blank region corresponds to the stable parameter domain. The dotted lines correspond to $\Lambda_{Z,L}, \Lambda_{Z,U}$, from the Eq. \ref{['bandpass']}, when $\Lambda_{Z,U} > \Lambda_{Z,L}$. Bottom panel: Growth rate, $\sigma_r$, normalised by $\Omega$ in the $(\operatorname{{P m}}, \Lambda_Z)$ for a regime of radiative stellar region (see Table \ref{['table']}, with $\operatorname{{R o}} = -0.05, \operatorname{{(N / \Omega)^2}} = 10, \Lambda_\phi = 0$).
• Figure 3: Growth time, $\tau_r/\tau_\Omega (=\Omega/\sigma_r),$ as a function of $\Lambda$ for different magnetic-field configurations, parametrised by $\Lambda_Z = \delta^2 \Lambda$, $\Lambda_\phi = \beta^2 \Lambda$. We consider regimes representative of stellar radiative zones (Table \ref{['table']}): a weakly stratified, low-shear case $(\vert \operatorname{{R o}} \vert = 0.5, \operatorname{{(N / \Omega)^2}} = 1)$ relevant to the MRI, and a strongly stratified, high-shear case $(\vert \operatorname{{R o}} \vert = 5, \operatorname{{(N / \Omega)^2}} = 5\times10^3)$ relevant to the GSF and MGSF instability (Sect. \ref{['sec:GSF']}). The dotted lines show the analytical growth rate estimates (Eqs. \ref{['gr_model']}, \ref{['grAMRI2']}, and \ref{['estimMGSF']}).
• Figure 4: Top panel: Growth rate $\sigma_r / \Omega$ in the $(\vert \operatorname{{R o}} \vert, \operatorname{{(N / \Omega)^2}})$ of the SMRI, GSF and SH instabilities, for a regime typical of radiative regions (see Table \ref{['table']}, $\Lambda_Z = 100, \Lambda_\phi = 0$). The three dashed lines correspond to instability thresholds, red for SMRI (Eq. \ref{['MRI_crit']}), blue for GSF, green for SH. The dotted lines separate the dominant type of instability inside the unstable domain from the mode properties. Bottom Panel: Stability map in the $(k,\alpha)$ plane for a stellar regime with $\lvert \operatorname{{R o}} \rvert = 3$ and $\Lambda_Z = 100$. Solid colored curves show the boundaries of the unstable domains for several values of the stratification parameter $\operatorname{{(N / \Omega)^2}}$. The blue region corresponds to the unstable range for $\operatorname{{(N / \Omega)^2}} = 10$, and the star markers indicate, for each $\operatorname{{(N / \Omega)^2}}$, the most unstable mode (corresponding to the regimes indicated by the stars in the top panel). The black region denotes modes excluded by our assumptions. The densely dashed horizontal line indicates the upper limit of the GSF-unstable domain set by magnetic stresses. The dashed lines show the modified stability boundaries due to the combined action of magnetic stresses and stratification, shown here for $\operatorname{{(N / \Omega)^2}} = 5\times 10^{3}$ and $5\times 10^{5}$. The loosely dashed line marks the limit imposed by viscous diffusion. Labels indicate the domains of SMRI$_\parallel$, SMRI$_\perp$, and MGSF modes.
• Figure 5: Ratio of the growth rate $\sigma_r$ to the analytical formula $\sigma_{\mathrm{estim}} \equiv \max(\sigma_{\mathrm{SMRI}}, \sigma_{\mathrm{MGSF}})$, which combines the predictions for SMRI (Eq. \ref{['gr_model']}) and MGSF (Eq. \ref{['estimMGSF']}). The flow parameters correspond to those of Fig. \ref{['sSMRI']} (top panel), with $\Lambda_Z = 10^5$. The white dotted lines delimit the different instability domains (see Appendix \ref{['ApMGSF']} and Fig. \ref{['kcri']} for details). The black dashed line indicate the sufficient MGSF stability criterion (Eq. \ref{['MGSFcrit']}). The grey dotted line correspond to the combination of the necessary criteria (Eq. \ref{['MGSFnece']}). See Appendix \ref{['mgsf_criteria']} for details.