Poloidal and toroidal fields in geomagnetic field modeling

Abstract

The theory of surface operators is described and applied to four surface operators on spheres: the dimensionless surface gradient, ▽1=r▽‐r∂r; the dimensionless surface curl, ; the dimensionless surface Laplacian, ▽1²=▽1 · ▽1; and the Funk‐Hecke operators, integral operators with axisymmetric kernels. Three methods are given for solving ▽1²g=f as g=▽1−2f; one method works numerically when f has a rapidly convergent expansion in spherical harmonics, the second works when f is smooth in longitude but not latitude, and the third (a Funk‐Hecke operation) works when f is rough in all directions. With this apparatus, a complete proof is given of the Helmholtz representation of an arbitrary vector field v S (r), the spherical surface of radius r centered on the origin: there are unique scalar fields f, g, h on S(r) such that and 〈g〉r=〈h〉r=0. Here 〈g〉r is the average value of g on S (r). From the Helmholtz representation on spherical surfaces, the Mie or poloidal‐toroidal representation in spherical shells is deduced. Suppose S(a,c) is the spherical shell whose inner and outer boundaries are S(a) and S(c). Suppose B is solenoidal in S (a,c), i.e., ▽·B=0 and 〈Br〉a=0. Then there are unique scalar fields P and Q in S(a,c) such that B=▽ × Λ1P + Λ1Q and 〈P〉r=〈Q〉r=0 for a ⪕ r ⪕ c. The fields P = ▽ × Λ1P and Q=Λ1Q are the poloidal and toroidal parts of B. Applications of this formalism to geomagnetic field modeling are discussed. Gauss's resolution of the geomagnetic field B on S(b) into internal and external parts is generalized; if the radial current Jr does not vanish on S (b), then to Gauss's expression must be added a toroidal field on S(b) due entirely to Jr on S(b). A simple proof is given of Runcorn's theorem that to first order in susceptibility no external magnetic field results from magnetization in a horizontally homogeneous spherical shell polarized by sources inside the shell. A Funk‐Hecke‐based method of modeling ionospheric currents is described, which may be more accurate than truncated spherical harmonic expansions and easier to use than Biot‐Savart integrals. Finally, the formalism makes possible the modeling of satellite samples of the geomagnetic field in a spherical shell S (a,c) where electric currents cannot be neglected. Two approximation schemes are described. One is a truncated power series expansion in (c‐a)/H, where H is the radial length scale of the currents. The other assumes that most of B in S(a,c) is not due to the currents between S (a) and S(c), and that the currents in S(a,c) are field‐aligned. Then the collection of physically possible magnetic fields in S(a,c) is only 50% larger, in a well‐defined sense, than the collection of vacuum fields there. Methods of calculation are given explicitly.