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

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, Rmc, 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 F1 and F2). 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 Rm ∼ 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 dynamo model [7] by taking into account the helical nature of the flow that is ejected by the centrifugal force close to each impeller between successive blades. Relying on the mean flow alone to compute the kinematic dynamo, smoothes out these non axisymmetric velocity fluctuations and thus cannot generate an axisymmetric field according to Cowling theorem. A non axisymmetric field is obtained, dominated by an equatorial dipole [6, 8]. When the disks are counter-rotating at the same fre-