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

Generation of large-scale magnetic fields due to fluctuating $\alpha$ in shearing systems