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With application to inner stellar radiative zones, a linear theory is used toanalyze the instability of a dipole-parity toroidal background field, in thepresence of density stratification, differential rotation, and realisticallysmall Prandtl numbers. The physical parameters are the normalized latitudinalshear $a$ and the normalized field amplitude $b$. Only the solutions for thewavelengths with the maximal growth rates are considered. If these scales arecombined to the radial values of velocity, one finds that the (very small)radial velocity only depends slightly on $a$ and $b$, so that it can be used asthe free parameter of the eigenvalue system.  The resulting instability-generated tensors of magnetic diffusivity and eddyviscosity are highly anisotropic. The eddy diffusivity in latitudinal directionexceeds the eddy diffusivity in radial direction by orders of magnitude. Itslatitudinal profile shows a strong concentration toward the poles which is alsotrue for the effective viscosity which has been calculated via the angularmomentum transport of the instability pattern. The resulting effective magneticPrandtl number reaches values of $O(10^2)$, so that the differential rotationdecays much faster than the toroidal background field, which is {the} necessarycondition to explain the observed slow rotation of the early red-giant andsub-giant cores by means of magnetic instabilities.

The effective magnetic Prandtl number in magnetized and differentially rotating stellar radiation zones