  The mixing of a passive scalar like lithium, beryllium or temperature
fluctuations due to the magnetic Tayler instability of a rotating axial pinch
is considered. Our study is carried out within a Taylor-Couette setup for two
rotation laws: quasi-Kepler and solid-body rotation. The minimum magnetic
Prandtl number used is 0.05 while the molecular Schmidt number Sc of the fluid
varies between 0.1 and 2. An effective diffusivity coefficient for the mixing
is numerically measured by the decay process of a global concentration peak
located between the cylinder walls. We find that only models with Sc>0.1 do
provide finite eddy diffusivity values. We also find that for quasi-Kepler
rotation at a magnetic Mach number Mm~2 the flow transits from the
slow-rotation regime to the fast-rotation regime. For fixed Reynolds number the
relation between the normalized eddy diffusivity and the Schmidt number of the
fluid is always linear so that also a linear relation between the
instability-induced diffusivity and the molecular viscosity results just in the
sense proposed by Schatzman (1977). The numerical value of the coefficient in
this relation will reach a maximum at Mm~2 and will decrease for Mm>>1 implying
that only toroidal magnetic fields of order kG can exist in the solar
tachocline.
