Determining the magnetic field in the core-mantle boundary zone by non-harmonic downward continuation

The length of day and the geomagnetic field are clearly correlated over decadal periods (10-100 yr). Provided the electrical conductivity of the lower mantle is sufficiently high, a considerable part of this correlation can be explained by electromagnetic core-mantle coupling. Investigating the associated core-mantle coupling torque and fluid velocity fields near the core surface, as well as the interpretation of the observed time lag between length of day and geomagnetic field variations requires the calculation of the temporally variable magnetic field near both sides of and on the core-mantle boundary by solving the magnetic induction equation. Such a solution presents a downward field continuation that has a non-harmonic character if the electrical conductivity is accounted for. In this paper the Earth's mantle is assumed to be a two-layer spherical shell, whose inner layer is electrically conductive. We only consider the poloidal part of the magnetic field with boundary values that are conventionally given by a spherical harmonic expansion of the observed geomagnetic potential field at the Earth's surface. Thus, we are concerned with a one-side-data supported problem, analogous to the inverse heat conduction problem (or sideways heat equation problem), well-known as an ill-posed inverse boundary value problem for a parabolic partial differential equation (diffusion equation). We develop a several-step solution procedure for this inverse problem in its integral form and use a special regularization method for the final solution. The capabilities of this downward continuation method (which includes varying the mantle conductivity model, quality of data approximation in the regularization and two different depths) are presented and discussed in comparison with the perturbation approach and the usual harmonic downward continuation. The data series used are the single magnetic field components (Gauss coefficients) of the spherical harmonic field expansion (, J. geophys. Res., 97, 19 537-19 563) beginning from the year 1840 to 1990. In addition, to investigate the spectral effects (changing amplitudes, phase shifting), an artificial data series is used. The main result is the downward continuation of the radial component of a global (8,8) field onto the core-mantle boundary, produced using the different methods and for the two epochs, 1910 and 1960. Comparing the results with the perturbation solution reveals temporally and locally variable differences up to the order of 5000 nT, while the difference to the harmonic downward continuation amounts to 15000 nT.