The region on the core--mantle boundary where a geostrophic velocity field can be determined from frozen-flux magnetic data

On the spherical surface S just below the core-mantle boundary layer, let u be the core-fluid velocity, Br the radial magnetic field, θ the colatitude, and ψ=Br sec θ. In the approximation where the flux is frozen and u is tangentially geostrophic, u can be determined from Br and ∂tBr at all points of S which are connected to the geographical equator by level lines of ψ. At the other points of S, u is determined by Br and ∂tBr only up to an unknown arbitrary tangentially geostrophic circulation around the level lines of ψ. The frozen flux approximation will always fail in a ‘leaky belt’ of approximate width | η∇2 (rBr) |/| ∇1∇1· (Bru) | radians centred on the ‘leaky curve’ where ∇1· (Bru) = 0. Here η is the magnetic diffusivity of the core and r-1∇1 is the surface gradient on S. The leaky belt includes those points at which null-flux curves appear or disappear. The geostrophic approximation will always fail in a narrow belt centred on the geographical equator. Neither failure interferes with the determination of u from Br and ∂tBr unless the belt widths are wider than the horizontal scale of u. Numerical calculation of u from Br and ∂tBr can be carried out using an explicit complete basis for the space of geostrophic motions on S. The basis fields are linear combinations of one, two or three surface vector spherical harmonics, and as predicted by Benton, they force the geographical equator to consist always of the same fluid particles. Each basis field, and every tangentially geostrophic flow, produces constant core fluid pressure on the geographical equator.