  We present dynamos computed using a hybrid QG-3D numerical scheme in a thick
spherical shell geometry. Our model is based on a quasi-geostrophic convection
code extended with a 3D treatment of heat transport and magnetic induction. We
find a collection of self-sustained, multipolar, weak field dynamos with
magnetic energy one or two orders of magnitude lower than the kinetic energy.
The poloidal magnetic energy is weak and, by construction, there is a lack of
equatorially anti-symmetric components in the Buoyancy and Lorentz forces. This
leads to configurations where the velocity field is only weakly impacted by the
magnetic field, similar to dynamos found in 3D simulations where zonal flows
and the $\Omega$-effect dominate. The time-dependence of these dynamos is
characterised by quasi-periodic oscillations that we attribute to dynamo waves.
The QG-3D dynamos found so far are not Earth-like. The inability of our setup
to produce strong, dipole-dominated, magnetic fields likely points to a missing
ingredient in our QG flows, and a related lack of helicity and $\alpha$-effect.
The models presented here may be more relevant for studying stellar dynamos
where zonal flows are known to dominate. This study was carried out at modest
control parameters, however moving to lower Ekman numbers, when smaller values
of both the magnetic and hydrodynamic Prandtl numbers can be of interest, our
approach will be able to gain in efficiency by using relatively coarse grids
for the 3D magnetic and temperature fields and a finer grid for the QG velocity
field.
