Large-scale dynamo action due to $\alpha$ fluctuations in a linear shear
  flow
  We present a model of large-scale dynamo action in a shear flow that has
stochastic, zero-mean fluctuations of the $\alpha$ parameter. This is based on
a minimal extension of the Kraichnan-Moffatt model, to include a background
linear shear and Galilean-invariant $\alpha$-statistics. Using the first order
smoothing approximation we derive a linear integro-differential equation for
the large-scale magnetic field, which is non perturbative in the shearing rate
$S\,$, and the $\alpha$-correlation time $\tau_\alpha\,$. The white-noise case,
$\tau_\alpha = 0\,$, is solved exactly, and it is concluded that the necessary
condition for dynamo action is identical to the Kraichnan-Moffatt model without
shear; this is because white-noise does not allow for memory effects, whereas
shear needs time to act. To explore memory effects we reduce the
integro-differential equation to a partial differential equation, valid for
slowly varying fields when $\tau_\alpha$ is small but non zero. Seeking
exponential modal solutions, we solve the modal dispersion relation and obtain
an explicit expression for the growth rate as a function of the six independent
parameters of the problem. A non zero $\tau_\alpha$ gives rise to new physical
scales, and dynamo action is completely different from the white-noise case;
e.g. even weak $\alpha$ fluctuations can give rise to a dynamo. We argue that,
at any wavenumber, both Moffatt drift and Shear always contribute to increasing
the growth rate. Two examples are presented: (a) a Moffatt drift dynamo in the
absence of shear; (b) a Shear dynamo in the absence of Moffatt drift.
