Mapping the fluid flow and shear near the core surface using the radial and horizontal components of the magnetic field

SUMMARY We examine the problem of determining the fluid flow and the shear (the radial derivative of the flow) at the core surface given a model of the temporal variation of the magnetic field. Whereas most previous work has focused on determining only the flow, which requires only the use of the radial component of the magnetic field, here, in addition, we determine the shear for which we must use the horizontal component of the magnetic field. Estimates of the jump in the value of the horizontal magnetic field B h across the boundary layer between the top of the free stream and the base of the mantle are small, and suggest that to a high level of accuracy the mantle values of B h can be used at the top of the core. Except in the special case of an insulating mantle, only the horizontal poloidal field is known at the core‐mantle boundary and supplies one extra equation for the determination of velocity and shear. We show how the matrix elements relating the coefficients of the spectral expansion of the flow and shear are related to the geomagnetic secular variation coefficients in closed form. We examine the uniqueness of the resulting inverse problem, and show that one part of the non‐uniqueness from which the shear suffers is particularly easy to describe: it takes the same form as that which affects the flow, namely a toroidal ambiguity in the field u′ B r . However, certain uniqueness theorems can be derived: we extend the steady motions theorem of Voorhies & Backus (1985) and the geostrophic motions theorem of Hills (1979) and Backus & LeMouël (1986) to the determination of the flow and shear, and derive closely analogous results. Uniqueness in the steady case depends on the value of the same discriminant as the velocity, and in the geostrophic case the shear can be determined uniquely in the same areas as can the velocity (i.e. outside certain ambiguous patches). For the geostrophic regime, the lateral density (or temperature) variations at the top of the core can be found in a self‐consistent manner. We apply our method to the temporal evolution of the field over the period 1960–1980, and produce solutions for each of the assumptions of unconstrained steady motions, geostrophic motions, and purely toroidal motions. We find that the form of the flow changes very little from solutions based only on the radial induction equation, and that the shear is weak and aligned with the flow, with a sense such that the strength of the flow decreases with depth with a length‐scale for linear decay of half the core radius. This suggests that the flow near the core surface is indicative of whole core flow, rather than a flow confined to a layer near the core surface.