Kinetic and magnetic α-effects in non-Linear Dynamo theory

Abstract

The backreaction of the Lorentz force on the α-effect is studied in the limit of small magnetic and fluid Reynolds numbers, using the first-order smoothing approximation (FOSA) to solve both the induction and momentum equations. Both steady and time-dependent forcings are considered. In the low Reynolds number limit, the velocity and magnetic fields can be expressed explicitly in terms of the forcing function. The non-linear α-effect is then shown to be expressible in several equivalent forms in agreement with formalisms that are used in various closure schemes. On one hand, one can express completely in terms of the helical properties of the velocity field as in traditional FOSA, or, alternatively, as the sum of two terms, a so-called kinetic α-effect and an oppositely signed term proportional to the helical part of the small-scale magnetic field. These results hold for both steady and time-dependent forcing at arbitrary strength of the mean field. In addition, the t-approximation is considered in the limit of small fluid and magnetic Reynolds numbers. In this limit, the t closure term is absent and the viscous and resistive terms must be fully included. The underlying equations are then identical to those used under FOSA, but they reveal interesting differences between the steady and time-dependent forcing. For steady forcing, the correlation between the forcing function and the small-scale magnetic field turns out to contribute in a crucial manner to determine the net α-effect. However for delta-correlated time-dependent forcing, this force-field correlation vanishes, enabling one to write exactly as the sum of kinetic and magnetic α-effects, similar to what one obtains also in the large Reynolds number regime in the t-approximation closure hypothesis. In the limit of strong imposed fields, B₀, we find α∝B^-2₀ for delta-correlated forcing, in contrast to the well-known α∝B^-3₀ behaviour for the case of a steady forcing. The analysis presented here is also shown to be in agreement with numerical simulations of steady as well as random helical flows.