Magnetic fields of low-mass stars and planets are thought to
originate from self-excited dynamo action in their
convective interiors. Observations reveal a variety of field
topologies ranging from large-scale, axial dipoles to more
structured magnetic fields. We investigate more than 200
three-dimensional, self-consistent dynamo
models in the Boussinesq and inelastic approximations obtained by
direct numerical simulations. The control parameters, the
aspect ratio, and the mechanical boundary conditions have been varied
to build up this sample of models. Both
strongly dipolar and multipolar models have been obtained. We show
that these dynamo regimes in general can be
distinguished by the ratio of a typical convective length scale to the
Rossby radius. Models with a predominantly
dipolar magnetic field were obtained, if the convective length scale
is at least an order of magnitude larger than
the Rossby radius. Moreover, we highlight the role of the strong shear
associated with the geostrophic zonal flow
for models with stress-free boundary conditions. In this case the
above transition disappears and is replaced by
a region of bistability for which dipolar and multipolar dynamos
coexist. We interpret our results in terms of
dynamo eigenmodes using the so-called test-field method. We can thus
show that models in the dipolar regime are
characterized by an isolated “single mode.” Competing overtones become
significant as the boundary to multipolar
dynamos is approached. We discuss how these findings relate to
previous models and to observations.