A NOTE ON DYNAMO ACTION AT ASYMPTOTICALLY SMALL EKMAN NUMBER

The physically realistic value of the Ekman number in the Earth's core is E approximate to 10(-15) which is much smaller than E approximate to 10(-4) achieved in the self-consistent numerical models of the geod ynamo. More recent models have used hyperdiffusivity to reduce E to ap proximate to 10(-6). We derive the dispersion relation for MHD waves i n an unbound domain and show that, as E --> 0, small scale rapidly osc illating waves develop which can cause numerical instability. We numer ically time step the magnetostrophic (E = 0) dynamo equations in the m ore realistic geometry of a sphere and confirm the existence of these waves. We show that hyperdiffusivity damps out the waves. Finally we d iscuss the results with reference to the use of this technique in nume rical investigations of the geodynamo problem.