Instability of magnetized and differentially rotating stellar radiation zones with high magnetic Mach number

With applications to inner solar-type radiative zones, a linear theory is used to analyse the instability of a toroidal background field of dipolar parity, in the presence of density stratification, differential rotation and realistically small Prandtl numbers. The physical parameters are the Alfvén frequency Ω_A, the global rotation rate Ω and the buoyancy frequency N with Ω_A < Ω < N. Only the solutions for the wavelengths with the maximal growth rates are considered. If these scales are combined to estimate radial velocities, one finds that it hardly depends on the latitudinal shear and the magnetic Mach number. In the formulation of Schatzman the radial mixing of chemicals can be estimated as Re* = O(100) which indeed is necessary to dissipate the lithium in the solar tachocline with a time-scale of 1 Gyr. The calculated growth rates indicate a destabilization of the system for growing latitudinal shear except for small Mach numbers and antisolar shear. The ratio ɛ of the magnetic and the kinetic energy of the instability pattern only slightly depends on the shear but a strong dependence on the magnetic Mach number exists with ɛ ∝ Mm². The effective magnetic Prandtl number reaches values O(10³) so that for the stars with high magnetic Mach number the differential rotation decays much faster than the toroidal background field.