Robustness of oscillatory $\alpha^2$ dynamos in spherical wedges

TL;DR

This study investigates how boundary conditions, diffusivity profiles, and shear influence oscillatory $oldsymbol{ extalpha}^2$ dynamo solutions in spherical wedge geometries, revealing conditions that produce oscillations and field migration patterns.

Contribution

It systematically analyzes the effects of boundary conditions, diffusivity, and shear on oscillatory dynamo solutions in spherical wedges, highlighting conditions for oscillations and field behavior.

Findings

Oscillatory solutions require wedge boundaries at least one degree from the poles.

Perfect conductor boundaries produce oscillations only in wedge geometries away from poles.

Radial shear and damping can induce oscillations with drift following Parker--Yoshimura rule.

Abstract

We study the connection between spherical wedge and full spherical shell geometries using simple mean-field $\alpha^2$ dynamos. We solve the equations for a one-dimensional time-dependent mean-field dynamo to examine the effects of varying the polar angle $\theta_0$ between the latitudinal boundaries and the poles in spherical coordinates. We investigate the effects of turbulent magnetic diffusivity and $\alpha$ effect profiles as well as different latitudinal boundary conditions to isolate parameter regimes where oscillatory solutions are found. Finally, we add shear along with a damping term mimicking radial gradients to study the resulting dynamo regimes. We find that the commonly used perfect conductor boundary condition leads to oscillatory $\alpha^2$ dynamo solutions only if the wedge boundary is at least one degree away from the poles. Other boundary conditions always produce stationary solutions. By varying the profile of the turbulent magnetic diffusivity alone, oscillatory solutions are 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 introducing radial shear and a damping term mimicking radial gradients, we again see oscillatory dynamos, and the direction of drift follows the Parker--Yoshimura rule. Oscillatory solutions in the weak shear regime are found only in the wedge case with $\theta_0 = 1^\circ$ and perfect conductor boundaries. A reduced $\alpha$ 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.