  We supplement the mean field dynamo growth equation with the total magnetic
helicity evolution equation. This provides an explicitly time dependent model
for alpha quenching in dynamo theory. For dynamos without shear, this approach
accounts for the observed large scale field growth and saturation in numerical
simulations. After a significant kinematic phase, the dynamo is resistively
quenched, i.e. the saturation time depends on the microscopic resistivity. This
is independent of whether or not the turbulent diffusivity is resistively
quenched. We find that the approach is also successful for dynamos that include
shear and exhibit migratory waves (cycles). In this case however, whether or
not the cycle period remains of the order of the dynamical time scale at large
magnetic Reynolds numbers does depend how on how the turbulent magnetic
diffusivity quenches. Since this is unconstrained by magnetic helicity
conservation, the diffusivity is presently an input parameter. Comparison to
current numerical experiments suggests a turbulent diffusivity that depends
only weakly on the magnetic Reynolds number, but higher resolution simulations
are needed.
