  Several one and two dimensional mean field models are analyzed where the
effects of current helicity fluxes and boundaries are included within the
framework of the dynamical quenching model. In contrast to the case with
periodic boundary conditions, the final saturation energy of the mean field
decreases inversely proportional to the magnetic Reynolds number. If a
nondimensional scaling factor in the current helicity flux exceeds a certain
critical value, the dynamo can operate even without kinetic helicity, i.e. it
is based only on shear and current helicity fluxes, as first suggested by
Vishniac & Cho (2001, ApJ 550, 752). Only above this threshold is the current
helicity flux also able to alleviate catastrophic quenching. The fact that
certain turbulence simulations have now shown apparently non-resistively
limited mean field saturation amplitudes may be suggestive of the current
helicity flux having exceeded this critical value. Even below this critical
value the field still reaches appreciable strength at the end of the kinematic
phase, which is in qualitative agreement with dynamos in periodic domains.
However, for large magnetic Reynolds numbers the field undergoes subsequent
variations on a resistive time scale when, for long periods, the field can be
extremely weak.
