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3D evolution of magnetic fields in a differentially rotating stellar radiative zone

The question of the origin and evolution of magnetic fields in stars possessing a radiative envelope, like the A-type stars, is still regarded as a challenge for stellar physics. Those zones are likely to be differentially rotating, which suggests that strong interactions between differential rotation and magnetic fields could be at play. We numerically compute the joint evolution of the magnetic and velocity fields in a 3D spherical shell starting from an initial profile for the poloidal magnetic field and differential rotation. The poloidal magnetic field is initially wound-up by the differential rotation to produce a toroidal field which becomes unstable. In the particular setup studied here where the differential rotation is dominant, the magneto-rotational instability is triggered. The growth rate of the instability depends mainly on the initial rotation rate, while the background state typically oscillates over a poloidal Alfvén time. We thus find that the axisymmetric magnetic configuration is strongly modified by the instability only if the ratio between the poloidal Alfvén frequency and the rotation rate is sufficiently small. An enhanced transport of angular momentum is found in the most unstable cases: the typical time to flatten the rotation profile is then much faster than the diffusion time scale. We conclude that the magneto-rotational instability is always favored (over the Tayler instability) in unstratified spherical shells when an initial poloidal field is sheared by a sufficiently strong cylindrical differential rotation. A possible application to the magnetic desert observed among A stars is given. We argue that the dichotomy between stars exhibiting strong axisymmetric fields (Ap stars) and those harboring a sub-Gauss magnetism could be linked to the threshold for the instability.