Mean magnetic field generation in sheared rotators

A generalized mean magnetic held induction equation for differential rotato rs is derived, including a compressibility, and the anisotropy induced on t he turbulent quantities from the mean magnetic field itself and a mean velo city shear. Derivations of the mean held equations often do not emphasize t hat there must be anisotropy and inhomogeneity in the turbulence for mean f ield growth. The anisotropy from shear is the source of a term involving th e product of the mean velocity gradient and the cross-helicity correlation of the isotropic parts of the fluctuating velocity and magnetic field, < v . b >((0)). The full mean held equations are derived to linear order in mea n fields, but it is also shown that the cross-helicity term survives to all orders in the velocity shear. This cross-helicity term can obviate the nee d for a preexisting seed mean magnetic field for mean field growth: though a fluctuating seed field is necessary for a nonvanishing cross-helicity, th e term can produce linear (in time) mean held growth of the toroidal held f rom zero mean field. After one vertical diffusion time, the cross-helicity term becomes subdominant and dynamo exponential amplification/sustenance of the mean field can subsequently ensue. The cross-helicity term should prod uce odd symmetry in the mean magnetic field, in contrast to the usually fav ored even modes of the dynamo amplification in sheared disks. This may be i mportant for the observed mean field geometries of spiral galaxies. The str ength of the mean seed field provided by the cross-helicity depends linearl y on the magnitude of the cross-helicity.