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

TL;DR

This paper develops a theory for large-scale magnetic field generation in turbulent flows with stochastic alpha fluctuations, highlighting how Moffatt drift and finite memory effects influence dynamo growth and cutoff scales.

Contribution

It extends the Kraichnan-Moffatt model to include finite correlation times of alpha fluctuations, revealing new conditions for dynamo action driven by Moffatt drift.

Findings

Finite Moffatt drift can enable dynamo action with weak alpha fluctuations.

A cutoff wavenumber exists beyond which dynamo growth ceases.

Finite memory effects modify the growth rate and scale of the dynamo.

Abstract

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.