Do spherical (alpha)^2-dynamos oscillate?

The question of whether kinematic (alpha)^2-shell-dynamos are able to produce a cyclic activity is addressed. The (alpha)-effect is allowed to be latitudinally inhomogeneous and anisotropic but it is assumed as radially uniform in the turbulent shell. For a symmetric (alpha)-tensor we only find oscillatory solutions if i) the (alpha)-tensor component (alpha)%% vanishes or is of the opposite sign as (alpha)(phi)(phi), ii) the (alpha)-effect is strongly concentrated in the equatorial region and iii) the (alpha)-effect is concentrated in a rather thin outer shell. In the other cases almost always the nonaxisymmetric field mode S1 (which slowly drifts along the azimuthal direction) possesses the lowest critical dynamo number. The uniform but anisotropic (alpha)-effect (e.g. (alpha)%%=0) leads to nonaxisymmetric solutions as is experimentally confirmed by the Karlsruhe dynamo installation. One of the antisymmetric parts of the (alpha)-tensor, however, plays the role of differential rotation in the induction equation. Using the results of a numerical simulation for the (alpha)-tensor of the solar convection zone for the radial profile of this effect, one finds the possibility of oscillating (alpha)^2dynamos even without the existence of a real nonuniform rotation law. Compared with the results of numerical simulations of the full (alpha)-tensor, the amplitude of the antisymmetric part of the tensor must be artifically increased by only a factor of three in order to find oscillating solutions.