  The onset of dynamo action is investigated within the context of a newly
developed low Rossby, low magnetic Prandtl number, convection-driven dynamo
model. This multiscale model represents an asymptotically exact form of an
$\alpha^2$ mean field dynamo model in which the small-scale convection is
represented explicitly by finite amplitude, single mode solutions. Both steady
and oscillatory convection are considered for a variety of horizontal
planforms. The kinetic helicity is observed to be a monotonically increasing
function of the Rayleigh number. As a result, very small magnetic Prandtl
number dynamos can be found for sufficiently large Rayleigh numbers. All
dynamos are found to be oscillatory with an oscillation frequency that
increases as the strength of the convection is increased and the magnetic
Prandtl number is reduced. Kinematic dynamo action is strongly controlled by
the profile of the helicity; single mode solutions which exhibit boundary layer
behavior in the helicity show a decrease in the efficiency of dynamo action due
to the enhancement of magnetic diffusion in the boundary layer regions. For a
given value of the Rayleigh number, lower magnetic Prandtl number dynamos are
excited for the case of oscillatory convection in comparison to steady
convection. With regards to planetary dynamos, these results suggest that the
low magnetic Prandtl number dynamos typical of liquid metals are more easily
driven by thermal convection than by compositional convection.
