Radial shear in the flow at the Earth's core surface

The Earth's magnetic field at the core-mantle boundary is the gradient of a harmonic potential function if the mantle is electrically insulating, and the horizontal components of the field can be derived from its radial component in the mantle. Therefore, these components give no further observational information on the core dynamics. However, it can still be envisioned that the horizontal components of the induction equation at Earth's core surface yield further knowledge on the fluid motions at the top of the core independently of the observations. Here, we show that they provide a linear relationship between the surface velocity and the surface shear (strain shear) that depends on the mantle electrical conductivity. This offers a protocol to calculate the surface shear that we validate with synthetics obtained from dynamo simulations in the limit of a weak mantle conductance. First, using numerical simulations with stress-free boundary condition at the core surface, we retrieve the expected relationship between the horizontal flow u(Sigma) and the shear, u(Sigma)=r partial derivative(r)u(Sigma). Next, we investigate simulations with no-slip boundary condition and insulating mantle, and we obtain the same relationship, even though the shear is not imposed as a boundary condition. Finally, we calculate the flow shear at the top of the core from a magnetic field model based on satellite measurements. The application to geophysical data indicates larger values of the surface flow shear than in the synthetic case, suggesting a possible role of the mantle electrical conductivity. The surface flow shear, in the simulations, much differs from the radial shear in the flow, deeper in the core, which is influenced by the mostly quasi-geostrophic geometry. This implies that we cannot rely on the relationship between the flow and the radial shear for quasi-geostrophic motions to exploit the horizontal components of the induction equation and gain further information on the flow at the Earth's core surface.