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

TL;DR

This paper investigates how fluctuating helicity in shear flows can generate large-scale magnetic fields, deriving a model that accounts for finite correlation times and identifying conditions for dynamo growth.

Contribution

It develops a non-perturbative model for large-scale magnetic field growth considering finite alpha-correlation time and shear, extending previous white-noise dynamo theories.

Findings

Shear and Moffatt drift together can drive dynamo action.

Growth rate of magnetic fields is proportional to shear magnitude.

Shear alone does not produce growth without Moffatt drift.

Abstract

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