Toroidal fluid motion at the top of the Earth's core

SUMMARY Geomagnetic secular variation is caused by flow of liquid iron in the core. Geomagnetic observations can be used to determine properties of the flow but such calculations in general have non‐unique solutions. We prove a uniqueness theorem: the flow is determined uniquely if it is toroidal (zero horizontal divergence), the mantle is an insulator, the core a perfect conductor (the frozen‐flux hypothesis), and there is no surface current in the boundary layer at the top of the core, and provided the magnetic field satisfies a simple point condition. The condition of no surface current allows use of the horizontal components of secular variation; previous studies have used only the radial component. Horizontal components allow simultaneous determination of the shear (radial derivatives of horizontal components of velocity). We have devised a new numerical method for determining core flows based on a discrete vector spherical transform (DVST) which exactly transforms between a grid of points on the surface of a sphere and a truncated spherical harmonic series. The transform method is faster than methods involving Gaunt and Elsasser integrals and eliminates most of the lengthy algebra. We determine a flow from spherical harmonic models of main field and secular variation for epoch 1970 which satisfies more than 95 per cent of the weighted radial component of secular variation and more than 94 per cent of the entire secular variation vector. This fit is considerably better than can be achieved by geostrophic flows and only slightly worse than by general steady motions; the improvement in adding poloidal motions is comparable with that in relaxing the frozen flux hypothesis. The main features of the flow are two gyres in the Atlantic hemisphere placed almost symmetrically about the equator with strong equatorial westerly flow and return easterly flows in high latitudes; it is similar to other published steady flows based on only the radial component of secular variation and severe damping. There is very little flow in the Pacific region. The shear indicates a radial length scale of about 600 km. It is very similar in form to the velocity and its sign is such that the velocity weakens with depth, as would be expected if the flow were driven from the core–mantle boundary rather than by convection from below. The overall pattern is consistent with flow in a density‐stratified layer driven by lateral temperature gradients in the lower mantle, although the result allows other scenarios.