ALBERT

Description: In this report, we will present the complete derivation of analytical expressions of the EM coupling torque in dependence on the parameters of the fields contributing to it. For this, we choose a special set of spherically harmonic (SH) base functions and present all major steps of the derivation. Our report will be (i) closer to a lecture note than to a scientific paper and should give all readers the possibility to follow the derivations with the related details in the appendix, and can be (ii) used as a formulary for scientists working on this special field of investigation.

Description: [...] We develop a regularizing solution procedure accounting for recent theoretical stability estimates. The capabilities of the procedure are shown for a single magnetic field component of the spherical harmonic field expansion beginning from the year 1900 by varying the mantle conductivity model and the degree of smoothness in the regularization. As an example, the radial component of a global (5,5) core-mantle boundary field is calculated for two epochs.

Description: For the motivation behind our investigations, we refer to the introduction of the first part of our report, Hagedoorn & Geiner-Mai (2008). In this part, we will (i) give an analytical description of the topographic surface of the core-mantle boundary (CMB) and derive an approximation for its normal unit vector containing information about the CMB topography, and (ii) derive an expression for the topographic torque as a function of the topographic height, h, and the velocity field, u. For this, we will check the assumptions made when applying the geostrophic approximation to the upper core-surface region. Finally, we will derive analytical expressions for the torque components depending upon the spherical harmonic (SH) coefficients of h and u in a cartesian geocentric coordinate system.

Description: The amplitude spectra of global geophysical phenomena were investigated to motivate research of physical connections between them. The suggested causality was derived from comparison of the spectra, and from cross correlation functions. The following global parameters were discussed: for the earth rotation by the variations of the length of day, for the gemagnetic variation by the global field intensity, changes of the dipole axis and the westward drift, and for climate change by the atmospheric excitation function derived from air pressure variations, and temperature variations. The model of atmospheric excitation, which can be proved most exactly for the annual variation of length of day, is responsible for the 11 and 22 years periods, too. It failed for longer periods e. g., partially for the 30 years periods and completely for the 60 to 80 years periods, which were also discovered in the mean temperature and geomagnetic field variations. Therefore, it was suggested that longer periods in climate change and in the variations of the earth's rotation are caused independently by the same process in the earth core, provided that a physical influence of the geomagnetic field on climate will be accepted in future. The investigation was completed by comparison with the spectra of some local temperature variations in Europe.

Description: This report continues a series of Scientific Technical Reports, in which the theoretical description of the electromagnetic (EM, see Hagedoorn & Greiner-Mai, 2008), topographic (TOP, see Greiner-Mai & Hagedoorn, 2008) and gravitational (GRAV, see Hagedoorn et al., 2012) core-mantle coupling torques are presented in detail. Based on these theoretical descriptions numerical codes were developed to compute individual coupling torques.

Description: 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.