Robustness of oscillatory α² dynamos in spherical wedges

¹ Department of Physics, Gustaf Hällströmin katu 2a, PO Box 64, University of Helsinki, 00014 Helsinki, Finland
e-mail: lizmcole@gmail.com
² Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, 10691 Stockholm, Sweden
³ Department of Astronomy, AlbaNova University Center, Stockholm University, 10691 Stockholm, Sweden
⁴ JILA and Department of Astrophysical and Planetary Sciences, Box 440, University of Colorado, Boulder, CO 80303, USA
⁵ Laboratory for Atmospheric and Space Physics, 3665 Discovery Drive, Boulder, CO 80303, USA
⁶ ReSoLVE Centre of Excellence, Department of Computer Science, Aalto University, PO Box 15400, 00076 Aalto, Finland
⁷ Max-Planck-Institut für Sonnensystemforschung, Justus-von-Liebig-Weg 3, 37077 Göttingen, Germany

Received: 20 January 2016
Accepted: 6 July 2016

Abstract

Context. Large-scale dynamo simulations are sometimes confined to spherical wedge geometries by imposing artificial boundary conditions at high latitudes. This may lead to spatio-temporal behaviours that are not representative of those in full spherical shells.

Aims. We study the connection between spherical wedge and full spherical shell geometries using simple mean-field dynamos.

Methods. We solve the equations for one-dimensional time-dependent α² and α²Ω mean-field dynamos with only latitudinal extent to examine the effects of varying the polar angle θ₀ between the latitudinal boundaries and the poles in spherical coordinates.

Results. In the case of constant α and η_t profiles, we find oscillatory solutions only with the commonly used perfect conductor boundary condition in a wedge geometry, while for full spheres all boundary conditions produce stationary solutions, indicating that perfect conductor conditions lead to unphysical solutions in such a wedge setup. To search for configurations in which this problem can be alleviated we choose a profile of the turbulent magnetic diffusivity that decreases toward the poles, corresponding to high conductivity there. Oscillatory solutions are now achieved with models extending to the poles, but the magnetic field is strongly concentrated near the poles and the oscillation period is very long. By changing both the turbulent magnetic diffusivity and α profiles so that both effects are more concentrated toward the equator, we see oscillatory dynamos with equatorward drift, shorter cycles, and magnetic fields distributed over a wider range of latitudes. Those profiles thus remove the sensitive and unphysical dependence on θ₀. When introducing radial shear, we again see oscillatory dynamos, and the direction of drift follows the Parker-Yoshimura rule.

Conclusions. A reduced α effect near the poles with a turbulent diffusivity concentrated toward the equator yields oscillatory dynamos with equatorward migration and reproduces best the solutions in spherical wedges. For weak shear, oscillatory solutions are obtained only for perfect conductor field conditions and negative shear. Oscillatory solutions become preferred at sufficiently strong shear. Recent three-dimensional dynamo simulations producing solar-like magnetic activity are expected to lie in this range.