We consider the consequence of strong stable stratification at the top of the Earth's fluid outer core, concentrating on the generation of the geomagnetic secular variation. We assume that the core near the core-mantle boundary is both strongly stably stratified and free of Lorentz forces: we find that this set of assumptions severely limits the class of possible motions, none of which is compatible with the geomagnetic secular variation. Relaxing either assumption is adequate: tangentially geostrophic flows are consistent with the secular variation if we relax the assumption that the core is strongly stably stratified (but retain the assumption that Lorentz forces are negligible); purely toroidal flows (consisting in this case of zonal flows together with horizontally polarized MAC-waves) may explain the secular variation if we include Lorentz forces. We are unable to discriminate between these two cases. ¿ American Geophysical Union 1990