  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, $\eta \rightarrow \infty$. 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 ($t_f \rightarrow \infty$) and to the
"incoherent" alpha--omega {\it ansatz} when the forcing is purely fluctuating.
