a1 Department of Atmospheric and Oceanic Sciences, University of California, Los Angeles, CA 90095-1565, USA
A quasi-linear theory is presented for how randomly forced, barotropic velocity fluctuations cause an exponentially growing, large-scale (mean) magnetic dynamo in the presence of a uniform parallel shear flow. It is a ‘kinematic’ theory for the growth of the mean magnetic energy from a small initial seed, neglecting the saturation effects of the Lorentz force. The quasi-linear approximation is most broadly justifiable by its correspondence with computational solutions of nonlinear magnetohydrodynamics, and it is rigorously derived in the limit of small magnetic Reynolds number, . Dynamo action occurs even without mean helicity in the forcing or flow, but random helicity variance is then essential. In a sufficiently large domain and with a small seed wavenumber in the direction perpendicular to the mean shearing plane, a positive exponential growth rate can occur for arbitrary values of , viscous Reynolds number , and random-force correlation time and orientation angle in the shearing plane. The value of is independent of the domain size. The shear dynamo is ‘fast’, with finite in the limit of . Averaged over random realizations of the forcing history, the ensemble-mean magnetic field grows more slowly, if at all, compared to the r.m.s. field (magnetic energy). In the limit of small and , the dynamo behaviour is related to the well-known alpha–omega ansatz when the force is slowly varying () and to the ‘incoherent’ alpha–omega ansatz when the force is more rapidly fluctuating.
(Received September 03 2011)
(Reviewed January 05 2012)
(Accepted February 27 2012)