We study axisymmetric mean-field dynamo models containing differential rotation, the $alpha$ effect and the additional turbulent induction effects. The additional effects result from the combined action of rotation and an inhomogeneity of the large-scale magnetic field. The best known of them is the $vec{Omega}timesvec{J}$ effect. We also include anisotropic diffusion and a new dynamo term which is of third order in the rotation vector $vec{Omega}$ The model calculations are carried out using the rotation profile of the Sun as obtained from helioseismic measurements and radial profiles of other quantities according to a standard model of the solar interior. In addition, we consider a dynamo model for a full sphere which is solely based on the joint induction effects of rotation and an inhomogeneity of the large-scale magnetic field, without differential rotation and the $alpha$ effect (a $delta^{2}$ dynamo model). This kind of dynamo model may be relevant for fully convective stars.
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