Moffatt drift driven large scale dynamo due to $\alpha$ fluctuations with nonzero correlation times

We present a theory of large-scale dynamo action in a turbulent flow that has stochastic, zero-mean fluctuations of the $\alpha$ parameter. Particularly interesting is the possibility of the growth of the mean magnetic field due to Moffatt drift, which is expected to be finite in a statistically anisotropic turbulence. We extend the Kraichnan-Moffatt model to explore effects of finite memory of $\alpha$ fluctuations, in a spirit similar to that of Sridhar & Singh (2014), hereafter SS14. Using the first-order smoothing approximation, we derive a linear integro-differential equation governing the dynamics of the large-scale magnetic field, which is non-perturbative in the $\alpha$-correlation time $\tau_{\alpha}$. We recover earlier results in the exactly solvable white-noise (WN) limit where the Moffatt drift does not contribute to the dynamo growth/decay. To study finite memory effects, we reduce the integro-differential equation to a partial differential equation by assuming that the $\tau_{\alpha}$ be small but nonzero and the large-scale magnetic field is slowly varying. We derive the dispersion relation and provide explicit expression for the growth rate as a function of four independent parameters. When $\tau_{\alpha}\neq 0$, we find that: (i) in the absence of the Moffatt drift, but with finite Kraichnan diffusivity, only strong $\alpha$-fluctuations can enable a mean-field dynamo; (ii) in the general case when also the Moffatt drift is nonzero, both, weak or strong $\alpha$ fluctuations, can lead to a large-scale dynamo; and (iii) there always exists a wavenumber ($k$) cutoff at some large $k$ beyond which the growth rate turns negative. Thus we show that a finite Moffatt drift can always facilitate large-scale dynamo action if sufficiently strong, even in case of weak $\alpha$ fluctuations, and the maximum growth occurs at intermediate wavenumbers.