Mean-field effects in the Galloway–Proctor flow

Karl-Heinz R adler 1 Axel Brandenburg 0 0 NORDITA , Roslagstullsbacken 23, SE-106 91 Stockholm , Sweden 1 Astrophysical Institute Potsdam , An der Sternwarte 16, D-14482 Potsdam , Germany A B S T R A C T In the framework of mean-field electrodynamics the coefficients defining the mean electromotive force in Galloway-Proctor flows are determined. These flows show a two-dimensional pattern and are helical. The pattern wobbles in its plane. Apart from one exception a circularly polarized Galloway-Proctor flow, i.e. a circular motion of the flow pattern is assumed. This corresponds to one of the cases considered recently by Courvoisier, Hughes & Tobias. An analytic theory of the 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 effect but no effect, or pumping effect. Numerical results obtained with the test-field method, which are independent of these approximations, confirm the results for and show that is in general non-zero. Both and show a complex dependency on the magnetic Reynolds number and other parameters that define the flow, that is, amplitude and frequency of the circular motion. Some results for the magnetic diffusivity t and a related quantity are given, too. Finally, a result for in the case of a randomly varying linearly polarized Galloway-Proctor flow, without the aforementioned circular motion, is presented. The flows investigated show quite interesting effects. There is, however, no straightforward way to relate these flows to turbulence and to use them for studying properties of the effect and associated effects under realistic conditions. 1 I N T R O D U C T I O N In astrophysical context, turbulent flows are of interest as occurring for example in stellar convection zones, accretion discs and galaxies. They deviate in general markedly from isotropic turbulence, and the time-dependence of the flow is important. Although not much direct observational evidence for the nature of the turbulence in these objects is available, the reasons for anisotropies are compelling. They include strong density stratification and rapid rotation. Both tend to make the turbulence two-dimensional in the sense that the variation of the velocity components is negligible along the directions of the density gradient and/or the axis of rotation. Such flows are referred to as geostrophic (Liao, Zhang & Feng 2005). As for the Sun, observations of the supergranulation and local helioseismology provide direct evidence for time dependence ( Svanda, Kosovichev & Zhao 2007). Simple models of flows showing such anisotropies and time dependencies are the circularly and linearly polarized flows of Galloway & Proctor (1992). These flows are two-dimensional in the sense that they depend only on two Cartesian coordinates, e.g. x and y. This can simplify the analysis significantly, even in dynamo problems that are inherently three-dimensional. The GallowayProctor (GP) flows are related to a flow considered by Roberts (1972). By contrast to the GP flows the Roberts flow1 is steady. It is an early example of a spatially periodic flow that acts as a dynamo and produces a magnetic field with a significant non-zero xy average. In mean-field electrodynamics this is understood as a consequence of the effect. The term in the mean-field induction equation is crucial to model the generation of large-scale magnetic fields from small-scale helical fluid motions in stars and galaxies; see, for example, Moffatt (1978), Parker (1979) and Krause & Radler (1980) for standard references. Like the Roberts flow the GP flow has frequently been used to study dynamo action and mean-field transport coefficients. A comprehensive review of these and other aspects of the GP flow is given in the thesis of Wilkinson (2004). Particularly important has been the study of the dependence of the quenching of the effect on the field strength and the value of the magnetic Reynolds number (Cattaneo & Hughes 1996). The GP flow has also been used to study the decay of a magnetic field in two dimensions as a function of field strength and magnetic Reynolds number (Silvers 2005). 1 As usual, the term Roberts flow refers to the flow given by equation (5.1) of Roberts (1972). The main motivation for the present paper were the results reported by Courvoisier, Hughes & Tobias (2006, hereafter CHT06) and Courvoisier (2008) on and effects in GP flows. They found that there is a complicated dependence of on the magnetic Reynolds number Rm and on parameters defining the flow, with sign changes and no indication of convergence with increasing Rm, and no clear relation between and the helicity of the flow, contrary to what is often assumed for the parametrization of meanfield dynamo models. In addition to the effect a effect has been observed, too. At first glance these results are surprising and very different from those for comparable cases. For the Roberts flow a smooth dependence of on Rm was found without any sign changes and with tending to zero as Rm (Soward 1987, 1989; Radler et al. 2002a,b). There are further recent studies in which astrophysical turbulence is modelled by random forcing. In the case of helical isotropic turbulence such investigations show that approaches a finite value as soon as Rm exceeds a value of the order of unity. This has been observed at least for Reynolds numbers up to 200 (Sur, Brandenburg & Subramanian 2008). Both examples provide no reasons to doubt the proportionality of with the helicity of the flow, and any effect can be excluded. The purpose of the paper is to study the mean-field effects of specific GP flows in more detail. We focus attention here on the simplest case considered in CHT06 with a flow being purely periodic in time and add a few results for a simple flow with random time dependence. The main tool of our investigations is the test-field method which allows to calculate numerically all components of the and t tensors defining the mean electromotive force E for a given flow (Schrinner et al. 2005, 2007). If, as we assume here, too, the mean magnetic field B depends only on one of the Cartesian coordinates, say z, only two 2 2 tensors for and t are of interest. The testfield method has recently been used to calculate diagonal and offdiagonal components of t (Brandenburg et al. 2008a), the magnetic Reynolds number dependence of and t (Sur et al. 2008), as well as their scale dependence (Brandenburg, Radler & Schrinner 2008b). We begin by exploring general properties of the mean electromotive force in the GP flow and present analytical results for coefficients like and , which are crucial for the electromotive force, gained in the second-order correlation approximation (SOCA) and in a corresponding fourth-order approximation. After explaining the test-field method we give a series of numerica (...truncated)