  Dynamos in the Sun and other bodies tend to produce magnetic fields that
possess magnetic helicity of opposite sign at large and small scales,
respectively. The build-up of magnetic helicity at small scales provides an
important saturation mechanism. In order to understand the nature of the solar
dynamo we need to understand the details of the saturation mechanism in
spherical geometry. In particular, we want to understand the effects of
magnetic helicity fluxes from turbulence and meridional circulation. We
consider a model with just radial shear confined to a thin layer (tachocline)
at the bottom of the convection zone. The kinetic alpha owing to helical
turbulence is assumed to be localized in a region above the convection zone.
The dynamical quenching formalism is used to describe the build-up of mean
magnetic helicity in the model, which results in a magnetic alpha effect that
feeds back on the kinetic alpha effect. In some cases we compare with results
obtained using a simple algebraic alpha quenching formula. In agreement with
earlier findings, the magnetic alpha effect in the dynamical alpha quenching
formalism has the opposite sign compared with the kinetic alpha effect and
leads to a catastrophic decrease of the saturation field strength with
increasing magnetic Reynolds numbers. However, at high latitudes this quenching
effect can lead to secondary dynamo waves that propagate poleward due to the
opposite sign of alpha. Magnetic helicity fluxes both from turbulent mixing and
from meridional circulation alleviate catastrophic quenching.
