A note on dynamo action at asymptotically small Ekman number

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Abstract
The physically realistic value of the Ekman number in the Earth's core is E ≈ 10−15, which is much smaller than E ≈ 10-4 achieved in the self-consistent numerical models of the geodynamo. More recent models have used hyperdiffusivity to reduce E to ≈ 10-6. We derive the dispersion relation for MHD waves in an unbound domain and show that, as E = 0, small scale rapidly oscillating waves develop which can cause numerical instability. We numerically time step the magnetostrophic (E = 0) dynamo equations in the more realistic geometry of a sphere and confirm the existence of these waves. We show that hyperdiffusivity damps out the waves. Finally we discuss the results with reference to the use of this technique in numerical investigations of the geodynamo problem.