Three-dimensional evolution of magnetic fields in a differentially rotating stellar radiative zone

Context. 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 in such regions. Aims. We would like to analyse in detail the evolution of magnetic fields in a differentially rotating stellar radiative zone and the possible presence of magnetic instabilities. Methods. 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. In order to characterise the nature of the magnetic instabilities that may be expected, we use the predictions of a local linear analysis. Results. The poloidal magnetic field is initially wound up by the differential rotation to produce a toroidal field which becomes unstable. We find that different types of instabilities may occur, depending on the balance between the shear strength and the magnetic field intensity. In the particular setup studied here where the differential rotation is dominant, the instability is not of the Tayler-type but is the magneto-rotational instability. The growth rate of the instability depends mainly on the initial rotation rate, while the background state typically oscillates over a poloidal Alfven time. We thus find that the axisymmetric magnetic configuration is strongly modified by the instability only if the ratio between the poloidal Alfven 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 decay time associated with the phase-mixing mechanism, which also occurs in the stable cases. When the instability saturates before reaching a significant amplitude, the magnetic configuration relaxes into a stable axisymmetric equilibrium formed by several twisted tori. Conclusions. We conclude that the magneto-rotational instability is always favoured (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 harbouring a sub-Gauss magnetism could be linked to the threshold for the instability.