A numerical model of the VKS experiment

We present numerical simulations of the magnetic field generated by the flow of liquid sodium driven by two counter-rotating impellers (VKS experiment). Using a kinematic code in cylindrical geometry, it is shown that different magnetic modes can be generated depending on the flow configuration. While the time averaged axisymmetric mean flow generates an equatorial dipole, our simulations show that an axial field of either dipolar or quadrupolar symmetry can be generated by taking into account non-axisymmetric components of the flow. Moreover, we show that by breaking a symmetry of the flow, the magnetic field becomes oscillatory. This leads to reversals of the axial dipole polarity, involving a competition with the quadrupolar component. Introduction. – The dynamo effect is a process by which a magnetic field is generated by the flow of an electrically conducting fluid. It is believed to be responsible for magnetic fields of planets, stars and galaxies [1]. Fluid dynamos have been observed in laboratory experiments in Karlsruhe [2] and Riga [3]. More recently, the VKS experiment displayed self-generation in a less constrained geometry, i.e., a von Kármán swirling flow generated between two counter-rotating disks in a cylinder [4]. In contrast with Karlsruhe and Riga experiments, the observed magnetic field strongly differs from the one computed taking into account the mean flow alone. Previous simulations, using the mean flow (time averaged) of the VKS experiment or an analytical velocity field with the same geometry, predicted an equatorial dipole [5–8] in contradiction with the axial dipole observed in the experiment. Understanding the geometry of the magnetic field observed in the experiment is still an open problem. In addition, time-dependent regimes, including field reversals are observed when the impellers rotate at different frequencies [9]. No numerical study of this effect have been performed so far. We address these problems using a kinematic dynamo code in a cylindrical geometry. By considering time dependent and non-axisymmetric fluctuations of the velocity field, we show that the system is able to generate a nearly axisymmetric dipolar field. Another result of this study is that when the analytic flow mimics two disks counter-rotating with different frequencies, the system bifurcates to a regime of oscillations between dipole and quadrupole, illustrating a recent model proposed in [10] in order to explain reversals of the magnetic field in the VKS experiment or in the Earth. We will see that an α−ω mechanism can explain the generation of the axial field and we understand the transition to oscillations as a saddle node bifurcation associated with the breaking of a symmetry in the flow. Numerical model. – In the VKS experiment, a turbulent von Kármán flow of liquid sodium is generated by two counter-rotating impellers (with 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. The magnetic Reynolds numbers are defined as Rmi = 2πμ0σR Fi where μ0 is the magnetic permeability of vacuum. When the impellers are operated at equal and opposite rotation rates F , a statistically stationary magnetic field with either polarity is generated above Rm ∼ 30 [4]. The mean field involves an azimuthal component and a poloidal one which is dominated by an axial dipole. When the disks are counter-rotating at the same frequency, the structure of the mean flow (averaged in time) has the following characteristics: the fluid is ejected radially from the disks by centrifugal force and loops back