The Elemental Shear Dynamo

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 shear flow, $\vec{U} = S x \vec{e}_y$. 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 magneto-hydrodynamics, and it is rigorously derived in the limit of large resistivity, $\eta \to \mathrm{\infty }$. 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 small wavenumber $k_z$ in the direction perpendicular to the mean shearing plane, a positive exponential growth rate $\gamma$ can occur for arbitrary values of $\eta$, the viscosity $\nu$, and the random-forcing correlation time $t_f$ and phase angle ${\theta }_f$ in the shearing plane. The value of $\gamma$ is independent of the domain size. The shear dynamo is “fast”, with finite $\gamma >0$ in the limit of $\eta \rightarrow 0$. Averaged over the random forcing ensemble, the mean magnetic field grows more slowly, if at all, compared to the r.m.s. field (or magnetic energy). In the limit of small Reynolds numbers ($\eta, \ \nu \rightarrow \infty$), the dynamo behavior is related to the well-known alpha–omega {\it ansatz} when the forcing is steady ($t_f\to \mathrm{\infty }$) and to the “incoherent” alpha–omega {\it ansatz} when the forcing is purely fluctuating.