Numerical Simulations of the von Karman Sodium dynamo

In the Cadarache von-Karman-Sodium (VKS) experiment a flow of liquid sodium is driven by two counterrotating impellers located at the top and the bottom of a cylindrical vessel. Dynamo action is obtained at a critical magnetic Reynolds number Rm_c=32. Striking property of the self-generated field is the high degree of axisymmetry. Furthermore, dynamo action is obtained only with impellers made of soft iron with a relative permeability of the order of mu_r ~ 100...1000. So far, no satisfying explanation is available that explains the failure of field generation when using steel impellers. Therefore, the role of the ferromagnetic material to obtain a dynamo, appears to be a critical issue and deserves further experimental and numerical investigations.

Numerical simulations of the kinematic induction equation have been carried out in a cylindrical domain that resembles the VKS setup. In case of a prescribed axisymmetric velocity distribution the resulting magnetic field is always determined by an azimuthal m=1 -- mode. Axisymmetric fields can be obtained applying a (localized) alpha-effect that might arise from the induction action of radially oriented helical outflow trapped between the impeller blades. However, it turns out, that the amplitude of alpha, which is necessary to generate an axisymmetric field, is far above realistic values. Therefore, a simple alpha-omega-model can be ruled out as the single explanation for the dynamo mechanism in the VKS experiment. Additional support of dynamo action stems from the presence of a high permeability domain within the cylindrical domain. In numerical simulations with a non-uniform permeability distribution that resembles the shape of the impeller disk (including the flow driving blades) the axisymmetric field mode is significantly enhanced, whereas the first non-axisymmetric mode remains nearly unaffected. To circumvent the restrictions of Cowling's theorem, still an alpha-effect is required for a growing axisymmetric field. However, the necessary magnitude of alpha is significantly reduced.