Generation of large-scale magnetic fields due to fluctuating $\alpha$ in
  shearing systems
  We explore the growth of large-scale magnetic fields in a shear flow, due to
helicity fluctuations with a finite correlation time, through a study of the
Kraichnan-Moffatt model of zero-mean stochastic fluctuations of the $\alpha$
parameter of dynamo theory. We derive a linear integro-differential equation
for the evolution of large-scale magnetic field, using the first-order
smoothing approximation and the Galilean invariance of the $\alpha$-statistics.
This enables construction of a model that is non-perturbative in the shearing
rate $S$ and the $\alpha$-correlation time $\tau_\alpha$. After a brief review
of the salient features of the exactly solvable white-noise limit, we consider
the case of small but non-zero $\tau_\alpha$. When the large-scale magnetic
field varies slowly, the evolution is governed by a partial differential
equation. We present modal solutions and conditions for the exponential growth
rate of the large-scale magnetic field, whose drivers are the Kraichnan
diffusivity, Moffatt drift, Shear and a non-zero correlation time. Of
particular interest is dynamo action when the $\alpha$-fluctuations are weak;
i.e. when the Kraichnan diffusivity is positive. We show that in the absence of
Moffatt drift, shear does not give rise to growing solutions. But shear and
Moffatt drift acting together can drive large scale dynamo action with growth
rate $\gamma \propto |S|$.
