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

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).