  A generalized mean magnetic field induction equation for differential
rotators is derived, including a compressibility, and the anisotropy induced on
the turbulent quantities from the mean magnetic field itself and a mean
velocity shear. Derivations of the mean field equations often do not emphasize
that there must be anisotropy and inhomogeneity in the turbulence for mean
field growth. The anisotropy from shear is the source of a term involving the
product of the mean velocity gradient and the cross-helicity correlation of the
isotropic parts of the fluctuating velocity and magnetic field,
$\lb{\bfv}\cdot{\bfb}\rb^{(0)}$. The full mean field equations are derived to
linear order in mean 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 need for a pre-existing seed mean magnetic field for mean field
growth: though a fluctuating seed field is necessary for a non-vanishing
cross-helicity, the term can produce linear (in time) mean field growth of the
toroidal field from zero mean field. After one vertical diffusion time, the
cross-helicity term becomes sub-dominant and dynamo exponential
amplification/sustenance of the mean field can subsequently ensue. The
cross-helicity term should produce odd symmetry in the mean magnetic field, in
contrast to the usually favored even modes of the dynamo amplification in
sheared discs. This may be important for the observed mean field geometries of
spiral galaxies. The strength of the mean seed field provided by the cross-
helicity depends linearly on the magnitude of the cross-helicity.
