A vorticity magnetic field dynamo instability

We generalize the mean field magnetic dynamo to include local evolution of the mean vorticity in addition to the mean magnetic field. The coupled equations exhibit a general mean field dynamo instability that enables the transfer of turbulent energy to the magnetic field and vorticity on larger scales. The growth of vorticity and magnetic field both require helical turbulence, and the helicity can be supplied by an underlying global rotation. The dynamo coefficients are derived, including the back-reaction from the mean magnetic field to lowest order. We find that a mean vorticity field can actually seed exponential growth of a mean magnetic field from only a small-scale seed magnetic field, without the need for a seed mean magnetic field. The equations decouple when the fluctuating velocity and magnetic field cross-correlations vanish, resulting in the separate mean field magnetic and mean field vorticity dynamo equations.