The shear dynamo problem for small magnetic Reynolds numbers

TL;DR

This paper investigates the possibility of large-scale dynamo action driven by turbulence in a shear flow at low magnetic Reynolds numbers, providing a nonperturbative analysis and deriving an integro-differential equation for magnetic field evolution.

Contribution

It offers a nonperturbative approach to the shear dynamo problem at low Rm, explicitly calculating the mean electromotive force and deriving a comprehensive evolution equation.

Findings

D terms do not support shear-current dynamo effect at lowest order in Rm.

Normal modes are shearing waves with kernels expressed via the velocity spectrum tensor.

The analysis is valid for arbitrary fluid Reynolds number but low magnetic Reynolds number.

Abstract

We study large-scale dynamo action due to turbulence in the presence of a linear shear flow, in the low conductivity limit. Our treatment is nonperturbative in the shear strength and makes systematic use of both the shearing coordinate transformation and the Galilean invariance of the linear shear flow. The velocity fluctuations are assumed to have low magnetic Reynolds number (Rm) but could have arbitrary fluid Reynolds number. The magnetic fluctuations are determined to lowest order in Rm by explicit calculation of the resistive Green's function for the linear shear flow. The mean electromotive force is calculated and an integro-differential equation is derived for the time evolution of the mean magnetic field. In this equation, velocity fluctuations contribute to two different kinds of terms, the C and D terms, in which first and second spatial derivatives of the mean magnetic field, respectively, appear inside the spacetime integrals. The contribution of the D terms is such that the time evolution of the cross-shear components of the mean field do not depend on any other components excepting themselves. Therefore, to lowest order in Rm but to all orders in the shear strength, the D terms cannot give rise to a shear-current assisted dynamo effect. Casting the integro-differential equation in Fourier space, we show that the normal modes of the theory are a set of shearing waves, labelled by their sheared wavevectors. The integral kernels are expressed in terms of the velocity spectrum tensor, which is the fundamental quantity that needs to be specified to complete the integro-differential equation description of the time evolution of the mean magnetic field.