  Rapidly rotating spherical kinematic dynamos are computed using the
combination of a quasi geostrophic (QG) model for the velocity field and a
classical spectral 3D code for the magnetic field. On one hand, the QG flow is
computed in the equatorial plane of a sphere and corresponds to Rossby wave
instabilities of a geostrophic internal shear layer produced by differential
rotation. On the other hand, the induction equation is computed in the full
sphere after a continuation of the QG flow along the rotation axis.
Differential rotation and Rossby-wave propagation are the key ingredients of
the dynamo process which can be interpreted in terms of $\alpha\Omega$ dynamo.
Taking into account the quasi geostrophy of the velocity field to increase its
time and space resolution enables us to exhibit numerical dynamos with very low
Ekman (rapidly rotating) and Prandtl numbers (liquid metals) which are
asymptotically relevant to model planetary core dynamos.
