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.