Mean Magnetic Field Generation in Sheared Rotators

A generalized mean magnetic field induction equation for differentialrotators is derived, including a compressibility, and the anisotropy induced onthe turbulent quantities from the mean magnetic field itself and a meanvelocity shear. Derivations of the mean field equations often do not emphasizethat there must be anisotropy and inhomogeneity in the turbulence for meanfield growth. The anisotropy from shear is the source of a term involving theproduct of the mean velocity gradient and the cross-helicity correlation of theisotropic parts of the fluctuating velocity and magnetic field,$\lb{\bfv}\cdot{\bfb}\rb^{(0)}$. The full mean field equations are derived tolinear order in mean fields, but it is also shown that the cross-helicity termsurvives to all orders in the velocity shear. This cross-helicity term canobviate the need for a pre-existing seed mean magnetic field for mean fieldgrowth: though a fluctuating seed field is necessary for a non-vanishingcross-helicity, the term can produce linear (in time) mean field growth of thetoroidal field from zero mean field. After one vertical diffusion time, thecross-helicity term becomes sub-dominant and dynamo exponentialamplification/sustenance of the mean field can subsequently ensue. Thecross-helicity term should produce odd symmetry in the mean magnetic field, incontrast to the usually favored even modes of the dynamo amplification insheared discs. This may be important for the observed mean field geometries ofspiral galaxies. The strength of the mean seed field provided by the cross-helicity depends linearly on the magnitude of the cross-helicity.Comment: 15 pages, LaTeX, matches version accepted to ApJ, minor revision