Effect of magnetic boundary conditions on the dynamo threshold of von Kármán swirling flows

Abstract

Abstract. -We study the effect of different boundary conditions on the kinematic dynamo threshold of von Kármán type swirling flows in a cylindrical geometry. Using an analytical test flow, we model different boundary conditions: insulating walls all over the flow, effect of sodium at rest on the cylinder side boundary, effect of sodium behind the impellers, effect of impellers or side wall made of a high magnetic permeability material. We find that using high magnetic permeability boundary conditions decreases the dynamo threshold, the minimum being achieved when they are implemented all over the flow.Dynamo action, i.e., self-generation of magnetic field by the flow of an electrically conducting fluid, is at the origin of planetary, stellar and galactic fields [1]. Fluid dynamos have been observed only recently in laboratory experiments in Karlsruhe [2] and Riga [3] by geometrically constraining the flow lines in order to mimic laminar flows that were known analytically for their dynamo efficiency [4]. More recently, the VKS experiment displayed self-generation in a less constrained geometry, e.g. a von Kármán swirling flow generated between two counterrotating impellers in a cylinder [5]. However, until now, dynamo action in the VKS geometry has been found only when the impellers are made of soft iron. It is thus of primary importance to understand how the dynamo problem is modified by the presence of magnetic material at the flow boundaries. We address this problem here using a kinematic dynamo code in a cylindrical geometry. Two important approximations are made to simplify the study. First, an analytic test flow that mimics the geometry of the mean flow of the VKS experiment is considered. Second, the magnetic boundary conditions are taken in the limit of infinite magnetic permeability of the boundaries compared to the one of the fluid. This seems a reasonable approximation for soft iron compared to liquid sodium. Our main result is that the critical magnetic Reynolds number, Rm c , for dynamo generation is significantly decreased with boundaries of high magnetic permeability all over the flow.The VKS experimental set-up is sketched in Figure 1. A turbulent von Kármán flow of liquid sodium is generated by two counter-rotating impellers (rotation frequencies F 1 and F 2 ). The impellers are made of iron disks of radius 154 mm, fitted with 8 iron blades of height 41.2 mm, and are placed 371 mm apart in an inner cylinder of radius 206 mm and length 524 mm. It is surrounded by sodium at rest in another concentric cylindrical vessel, 578 mm in inner diameter. This has been shown to decrease the dynamo threshold in kinematic computations based on the mean flow velocity [6]. When the impellers are operated at equal and opposite rotation rates F , a statistically stationary magnetic field is generated above a magnetic Reynolds number R m ∼ 30 [5]. The large scale field involves an azimuthal component and a poloidal one which is dominated by an axial dipole. This geometry has been understood with a simple α − ω dy...

Liquid sodium models of the Earth’s core

“…Either or both mechanisms could be significant in the VKS dynamo. Gissinger et al (2008) found using a numerical model of VKS that both ferromagnetic disks and ferromagnetic blades decreased the threshold for dynamo action. Gissinger (2009) presented results of numerical simulations of VKS using a kinematic code, which by taking into account both the axisymmetric mean of the flow and its non-axisymmetric components could generate either an axial dipole or axial quadrupole.…”

Forward and inverse problems in fundamental and applied magnetohydrodynamics

“…This procedure has been proposed in two papers by Iskakov et al [24,25], and was recently modified and applied to various dynamo problems [26,27]. We will present this method in some more detail, since it seems to be most efficient for the simulation of dynamo problems in arbitrary domains.…”

“…The parameter σ div in (27) is chosen very small as it is a measure for the divergence the velocity solution is allowed to have. The parameter σ pen in (28) determines the trade-off between minimizing the mean squared residual deviation of the observed fields and minimizing the kinetic energy of the estimated velocity field.…”

Laboratory Dynamo Experiments

“…(b) Measurements of the induced axial magnetic field by BC-effect for a transverse applied field along the x-axis with both counter-rotating discs in von Kármán experiment, from Bourgoin et al (2004a) ferential rotation, and the effect has been accurately treated in several numerical simulations (Bourgoin et al 2004b;Stefani et al 2006;Xu et al 2008). Inhomogeneities in the magnetic boundary conditions, which appear to favour dynamo action ) have been considered in numerical studies (Avalos-Zuniga and Plunian 2005; Gissinger et al 2008) but need to be probed in more details experimentally.…”