  We performed kinematic studies of the evolution of small-scale magnetic
fields in the surface laters of M-dwarfs. We solved the induction equation for
a prescribed velocity field, magnetic Reynolds number, and boundary conditions
in a Cartesian box, representing a volume comprising the optically thin stellar
atmosphere and the uppermost part of the optically thick convective envelope.
The velocity field is spatially and temporally variable, and stems from
detailed radiation-hydrodynamics simulations of convectione flows in a
proto-typical late-type M-dwarf. We find dynamo action for large magnetic
Reynolds numbers. Growth time scales of the magnetic field is comparable to the
convective turn-over time scale (approximately 150 seconds). The convective
velocity field concentrates the magnetic field in sheets and tubular structures
in the inter-granular down-flows. Scaling from solar conditions suggests that
field strengths as high as 20 kG might be reached locally. Perhps surprisingly,
the magnetic Reynolds number is of the order unity in the surface layers of
cooler M-dwarfs, rendering the dynamo inoperative. In all studied cases we find
a rather low spatial filling factor of the magnetic field.
