  The purpose of this paper is to improve the modeling of the mixing of
chemical elements that occurs in stellar radiation zones. In addition to the
classical rotational mixing considered in our previous paper, which results of
the combined action of the thermally-driven meridional circulation and of the
turbulence generated by the shear of differential rotation, we include here the
effect of an axisymmetric magnetic field in a self-consistent way. We treat the
advection of the field by the meridional circulation, its Ohmic diffusion, and
the production of its toroidal component through the shear of differential
rotation. The Lorentz force is assumed not to exceed the centrifugal force; it
acts on the baroclinic balance and therefore on the meridional flow, and it has
a strong impact on the transport of angular momentum. All variables and
governing equations are expanded in spherical or spherical vectorial functions,
to arbitrary order: this yields a system of partial differential equations in
time and in the radial coordinate, which is ready to be implemented in a
stellar structure code.
