Mean-field effects in the Galloway-Proctor flow

TL;DR

This paper analytically and numerically investigates the mean electromotive force in a helical, wobbling Galloway-Proctor flow, revealing complex dependencies of alpha and gamma effects on flow parameters and validating results with test-field simulations.

Contribution

It provides a detailed analytical theory and numerical validation of mean-field effects, including alpha and gamma, in a complex, helical flow with wobbling motion.

Findings

Alpha effect exists within the approximation range.

Gamma effect is generally nonzero, confirmed by numerical tests.

Alpha and gamma depend complexly on flow parameters.

Abstract

The coefficients defining the mean electromotive force in a Galloway-Proctor flow are determined. This flow shows a two-dimensional pattern and is helical. The pattern wobbles in its plane. Apart from one exception a circular motion of the flow pattern is assumed. This corresponds to one of the cases considered recently by Courvoisier, Hughes and Tobias (2006, Phys. Rev. Lett., 96, 034503). An analytic theory of the alpha effect and related effects in this flow is developed within the second-order correlation approximation and a corresponding fourth-order approximation. In the validity range of these approximations there is an alpha effect but no gamma effect, or pumping effect. Numerical results obtained with the test-field method, which are independent of these approximations, confirm the results for alpha and show that gamma is in general nonzero. Both alpha and gamma show a complex dependency on the magnetic Reynolds number and other parameters that define the flow, that is, amplitude and frequency of the wobbling motion. Some results for the magnetic diffusivity eta_t and a related quantity are given, too. Finally a result for alpha in the case of a randomly varying flow without the aforementioned circular motion is presented. This flow may be a more appropriate model for studying the alpha effect and related effects in flows that are statistical isotropic in a plane.