  While the saturated magnetic energy is independent of viscosity in dynamo
experiments, it remains viscosity-dependent in state-of-the-art 3D direct
numerical simulations (DNS). Extrapolating such viscous scaling-laws to
realistic parameter values leads to an underestimation of the magnetic energy
by several orders of magnitude. The origin of this discrepancy is that fully 3D
DNS cannot reach low enough values of the magnetic Prandtl number $Pm$. To
bypass this limitation and investigate dynamo saturation at very low $Pm$, we
focus on the vicinity of the dynamo threshold in a rapidly rotating flow: the
velocity field then depends on two spatial coordinates only, while the magnetic
field consists of a single Fourier mode in the third direction. We perform
numerical simulations of the resulting set of reduced equations for $Pm$ down
to $2\cdot 10^{-5}$. This parameter regime is currently out of reach to fully
3D DNS. We show that the magnetic energy transitions from a high-$Pm$ viscous
scaling regime to a low-$Pm$ turbulent scaling regime, the latter being
independent of viscosity. The transition to the turbulent saturation regime
occurs at a low value of the magnetic Prandtl number, $Pm \simeq 10^{-3}$,
which explains why it has been overlooked by numerical studies so far.
