Orthogonality of harmonic potentials and fields in spheroidal and ellipsoidal coordinates: application to geomagnetism and geodesy

Abstract

SUMMARY We investigate the orthogonality of the potential distributions that are the basis solutions of Laplace’s equation appropriate to 3-D ellipsoidal (including spheroidal) coordinate systems, and also the orthogonality of the corresponding vector gradient fields, both over the surface of the ellipsoid, and for integration over the volume of the annular shell between two confocal ellipsoids. The only situation for which there is orthogonality is for the vector gradients when integratedovertheannularshell.Intheotherthreecases(potentialoversurfaceorannulus,and fieldoversurface)orthogonalitycanberestoredbyusinganappropriategeometricalweighting factor applied to the integrand; it is therefore still possible to perform the equivalent of a classical spherical harmonic analysis. In the special case of the sphere, there is real orthogonality in all four cases; in effect the weighting factors are all unity. In geodesy, spheroidal harmonic analysis is done using a method that relies on a particular result valid only for potential; it cannot be extended to the corresponding vector field, or to ellipsoidal geometry. The lack of orthogonality over the surface means that care must be taken when interpreting conventional geomagnetic ‘power spectra’, and geodetic ‘degree variance’, as these no longer correspond exactly to the mean-square values over the actual ellipsoidal surface. We illustrate some of the problems by comparing different versions of the power spectrum for a spheroidal analysis of the global lithospheric magnetic field. We use only simple vector algebra, and do not need to know the details of the actual basis solutions, only that they are the product of three functions, one for each coordinate and involving only that coordinate, and that they satisfy Laplace’s equation. Similarly, our results do not depend on the normalization used in the basis functions.