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Abstract:
It is of high interest to know the magnetic field, measured at the earth surface or by satellites, in the earth deep interior, especially at the core-mantle boundary (CMB). This knowledge is of relevance for the determination of fluid motions at the top of the outer core, the estimation of diffusion and the geomagnetic spectrum, as well as in calculations of the electromagnetic core-mantle coupling torques or in studying
the behaviour of geomagnetic jerk components near the CMB.
The presented procedure of nonharmonic downward continuation (NHDC) is a strong theoretical method, an illposed inverse initial boundary value problem, which determines the given outer geomagnetic field or the secular variation in the deep earth interior. It accounts for a prescribed mantle conductivity model depending on the radius. Boundary values are given only on one, the upper (outer) side of
the radial interval. We discuss the theoretical background of the method, referring to the intensively investigated inverse heat conduction problem in the field of parabolic differential equations, and adapt it to the geomagnetic downward continuation problem. Some historical remarks on the early trials in developing this method around the year 1980 are outlined.
After investigating the limited possibilities for analytical solutions, we present the numerical algorithm,
which uses the integral equation approach, combined with a special regularization variant. It can be
implemented on the basis of finite differences or the finite-element technique.
This algorithm enables simulations setting up simple function types (e.g. oscillations, time polynomials).
In addition, approximative approaches help to reveal the analytical dependence of the solution on
the conductivity function, e.g. its impact on the phase shifts or time shifts, which are different for radial
and tangential magnetic field components.
A couple of new applications are addressed, e.g. to check the divergence condition for the magnetic
field at the CMB and the way to make diffusion studies near the CMB. On the basis of NHDC, we derive
a new formula for the geomagnetic spectrum at the CMB, which shows in its approximated form the
influence of the mantle conductivity model. Finally, some remarks on future possibilities in the field of
geomagnetic downward continuation are added.
