Convection in the Earth's core is affected by Lorentz and Coriolis forces. The relative importance of these is measured by the Elsasser number &Lgr;. Previous work suggested that the optimum condition for convection occurs when the Elsasser number &Lgr; is O(1); in particular, the critical modified Rayleigh number R* had a minimum value at some O(1) value of &Lgr;. This gave rise to the view that the size of the magnetic field generated by the dynamo would adjust to this &Lgr;-value because it optimised the convection. We have investigated convection in a rotating spherical shell with magnetic field distributions satisfying approximate boundary conditions in the form of the toroidal decay modes which we believe are more realistic for the Earth's core. Our results are rather different from the classical picture. We find no optimum Elsasser number that minimises R¿ but instead a monotonic decay of R* as &Lgr; increases.

At &Lgr;~10, R* goes negative, so that rolls are driven by magnetic instability rather than convection. Naturally, as this regime is entered the rolls must be destroying magnetic field rather than creating it by dynamo action. This suggests that in the Earth's fluid outer core, the field strength is such that 1<&Lgr;<10, so that the magnetic field is stable and the corresponding convection is dominated by large scales and is efficient. Another important feature of these solutions is that the azimuthal flow is primarily two-dimensional. This is surprising, as it has generally been believed that a magnetic field of this strength would break the Taylor-Proudman constraint. However, it appears that the magnetic field adjusts to preserve two-dimensionality even in the range 1<&Lgr;<20. ¿ American Geophysical Union 1994