Kinematic dynamos resulting from the interaction of high permeability material and flows of liquid sodium

We perform numerical simulations of the dynamo effect driven by various flow fields of a conducting liquid interacting with "magnetic material" characterized by a large relative permeability. The examinations are motivated by the key role of soft iron impellers for the Von-Kármán-Sodium (VKS) dynamo [1] and by the repeatedly expressed idea to make use of Oxide Dispersion Strengthened (ODS) ferritic/martensitic alloys in the core of a fast reactor which may exhibit a permeability much larger than one [2].

The results of our simulations that consider a localized distribution with finite permeability clearly differ from computations using simplyfying pseudo-vacuum boundary conditions (vanishing tangential field conditions) in order to mimic the impact of infinite permeability. Our kinematic simulations of an axisymmetric model of the VKS dynamo show a close connection between the exclusive occurrence of dynamo action in the presence of soft iron impellers and the observed axisymmetry of the magnetic field [3]. We qualitatively explain this effect by paramagnetic pumping at the fluid-disk interface and propose a simplified analytical model that quantitatively reproduces numerical results. In order to fully explain the observation of growing magnetic fields in the VKS dynamo we resort to mean-field dynamo theory [4] in terms of an α-effect caused by helical outflows between adjacent blades attached to the impeller disks.

In order to examine the properties of the α- and β-effect (which is closely related to the turbulent diffusivity) under influence of magnetic material [5] we use an idealized helical flow field (a modified Roberts flow). We compute the mean-field coefficients using the test-field method [6] and proof that the corresponding mean-field models are indeed capable to reproduce growth-rates and principle field structure of the fully resolved model by requiring much less computational efforts.

Further remarkable results are the observed reduction of the critical magnetic Reynolds number by roughly 30 percent independently of configuration or flow geometry when the permeability is sufficiently large. However, this universality is not reflected in the behavior of the mean-field coefficients. In particular, the β-effect strongly depends on the geometry and the permeability. A striking feature is the occurrence of negative β which has previously been observed in simulations [7] and, more recently, in experiments [8].

Our results for the mean-field coefficients allow the development of dynamo models for nearly arbitrary systems of various sizes consisting of a large number of helical small scale flow cells embedded into some large flow structure.

[1]Monchaux, R. et al., Phys. Rev. Lett. 98 (2007), 044502
[2]Dubuisson, P., de Carlan, Y., Garat, V. and Blat, M., J. Nucl. Mater. 428 (2012), 6–12
[3]Giesecke, A. et al., New J. Phys. 14 (2012), 053005
[4]Krause, F. and Rädler, K.-H. Mean-field Magnetohydrodynamics and dynamo theory, Pergamon Press 1980
[5]Giesecke, A. et al., New J. Phys. 16 (2014), 073034
[6]Schrinner, M. et al., Astron. Nachr. 326 (2005), 245-249
[7]Rädler, K.-H. and Brandenburg, A., Phys. Rev. E 67 (2003), 026401
[8]Frick, P., Noskov, V., Denisov, S. and Stepanov ,R., Phys. Rev. Lett. 105 (2010), 184502