Description: Based on the newly developed geomagnetic field model C3FM2 and a set of additional input models, like the electric conductivity of the Earth’s mantle and the topography of the core-mantle boundary (CMB), we compute the electromagnetic (EM) and topographic (TOP) core-mantle coupling torques. Those coupling torques cause variations in the Earth rotation and can be expressed by equivalent excitation functions. As in our previous work, for the determination of the EM coupling torque, the geomagnetic field is determined at the CMB by the nonharmonic downward continuation. Herein, we consider a set of different radial stratifications of the electrical conductivity of the Earth’s mantle. A requirement for the determination of the toroidal geomagnetic field at the CMB is the knowledge of the time variable surface fluid-flow velocity at the CMB in the outer core. Based on these velocities and the poloidal geomagnetic field at the CMB, we solve the initial boundary value problem for the time variable part of the toroidal geomagnetic field. The time-dependent fluid-flow velocities are also used to compute the TOP coupling torque consistently for different CMB topography models This investigation is restricted to this time interval 1962–2000, because the considered atmospheric (AAM) and oceanic angular momentum (OAM) functions are based on the ERA-40 reanalysis provided by the ECMWF, which are given for this time interval. We use a combination of the equivalent excitation functions, AAM and OAM time series, for the forward calculation of the variation of polar motion and length-of-day on decadal time scale. The contributions of the different excitation functions for different frequencies to the modelled variation of polar motion and length-of-day are analysed. They are focused on the decadal time scale, which is the characteristic one for coupling processes at the CMB.

Description: Considering the magnetic field energy and its temporal changes on the core-mantle boundary (CMB) has multivalent relevance: They can reflect and evaluate dynamical processes which are modelled on the basis of surface field data. On the other hand, these quantities form a significant link to features of the deeper field generation by a dynamo process. Our investigation is focused on the temporal changes of geomagnetic field energy on the CMB and their connection with sudden trend changes of the geomagnetic secular variation on the CMB (jerks). We show examples of the different components (radial and tangential) at different epoches. Further, we investigate the differently influenced degree parts. A measure for the total activity is introduced and compared for the whole CMB. As data we use the geomagnetic field model C3FM2 (Wardinski and Lesur, GFZ Potsdam, 2009) covering the years 1957-2006, which enables secular variation investigations with monthly values. Assuming a low, radially dependent mantle conductivity with a conductance of about 107S, we determine the geomagnetic field and the secular variation components on the CMB by the method of non-harmonic downward continuation.With these field quantities the geomagnetic energy and its time derivative on the CMB can be calculated. For the jerk detection, we use a simple straight-line approximation algorithm applied to each (#; ') position of a global 1 degree by 1 degree grid.

Description: Different processes influence the variation of the Earth’s rotation on decadal time scale. The consideration of surface processes, like the exchange of angular momentum of the fluid sub-systems of atmosphere and ocean, and coupling processes at the core-mantle boundary like electromagnetic and topographic coupling torques, lead still to significant differences between observed and modelled Earth rotation parameters (ERP). For the modelled ERP, we consider the electromagnetic (EM) and topographic (TOP) core-mantle coupling torques and compute equivalent excitation functions, which are combined with atmospheric (AAM) and oceanic (OAM) angular momentum functions. Our investigation is focused on an additional core-mantle coupling process, the gravitational coupling, which should partly explain the remaining differences between modelled and observed ERP. The influence of assumed geometrical settings, like flattening and topography of the core-mantle (CMB) and inner-core (ICB) boundary on the gravitational coupling torque are studied systematically. First, simplified geometrical settings are applied, where the CMB and ICB are represented by two-axial ellipsoids. In a second step, published CMB topographies based on seismic tomography are considered and the resulting torques are compared with the simplified case. Results from this study will be used for the further extension of our core-mantle coupling model by the gravitational core-mantle coupling.

Description: New geomagnetic field models (Wardinski and Lesur, GFZ Potsdam, 2009) covering the years 1957–2006 - here the model C3FM2 - enable new comparative secular variation investigations. Combined with the method of nonharmonic downward continuation and e.g. assuming a low, radially dependent mantle conductivity with a conductance of about 107 S, we can determine the three secular variation components also for the core-mantle boundary. For this time span, the jerks (which are sudden trend changes in the geomagnetic secular variation, e.g. around the years 1969, 1978, 1991) have been studied at the earth surface mainly by observatory data. We extend this to a common and uniform view of the jerks, both for the earth surface and the core-mantle boundary finding their different global topologies and their temporal characteristics. The jerk detection of a single secular variation component is done by a simple straight-line approximation algorithm applied to each (#; ') position of a global 1° by 1° grid. For the individual jerks, we show comparatively the time structure, i.e. the global/regional distribution of the jerk occurrence times in dependency of the time. It is clearly specific for each single component and each jerk and reveals also partly significant spectral differences between radial and tangential components.

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.

Description: For the computation of the electromagnetic (EM) core-mantle coupling torque, the geomagnetic field must be known at the core-mantle boundary (CMB). It can be divided into linearly independent poloidal and toroidal parts. As shown by previous investigations, the toroidal field produces more than 90% of the EM torque. It can be obtained by solving the associated (toroidal) induction equation for the electrically conducting part of the mantle, i.e. an initial boundary value problem (IBVP). The IBVP differs basically from that for the poloidal field by the boundary values at the interface between lower conducting and upper insulating parts of the mantle: the toroidal field vanishes, the poloidal field continues harmonically as potential field towards the Earth surface. The two major subjects are to find a suitable algorithm to solve the IBVP and to compute the toroidal geomagnetic field at the CMB. Compared to the poloidal field, the toroidal field at the CMB cannot be inferred from geomagnetic observations at the Earth’s surface. In this study, it is inferred from the velocity field of the fluid core flow and the poloidal field at the CMB using an approximation which is consistent with the frozen-field approximation of the geomagnetic secular variation. This investigation differs from earlier ones by: (i) inferring the poloidal field at the CMB from the observed geomagnetic field using a rigorous inversion of the associated (poloidal) induction equation on which the fluid-flow inversion is based to determine consistently the surface flow velocities at the CMB, (ii) applying orthonormal spherical harmonic functions for the representation of the fields and torques, (iii) solving the IBVP numerically by a modified Crank-Nicolson algorithm, which (iv) allows us to highlight the influence of this approach on the resulting EM coupling torques. In addition to an outline of the derivations of the theoretical formalism and numerical methods, the time-variable toroidal field at the CMB is presented for different conductivity models.

Description: We determine the occurrence times of geomagnetic impulses (jerks) around the year 1991 in the three geomagnetic secular variation components for the Earth’s surface by a simple optimization algorithm. The geomagnetic field models we use are the low-degree parts of the models CM4 and C3FM. We find that the temporal jerk pattern can be detected in fields (n ≤ 4), from which the spherical harmonic degrees n = 2 or n = 3 (tangential) and n = 4 (radial) are representative. To calculate the secular variation components at the core–mantle boundary (CMB) we apply the non-harmonic downward continuation (NHDC) method. For the mantle conductivity, three estimates, dependent on the radius, are assumed with conductances between 107 S and 2 × 108 S. The knowledge of the secular variation components at the CMB allows us to track the global distributions of jerk occurrence times in dependence on the mantle conductivity estimate. We find for each component a typically shaped, global topology for the location of the jerk occurrence times at the Earth’s surface and the CMB. For the tangential (ϑ, ϕ)-components, these global topologies show always the well-known temporal bimodality on each surface. Another characteristic feature is found for the jerk of the r-component. It displays a double jerk centred around 1991 consisting of a v-shaped and a reversely v-shaped part, which are significantly correlated. Between the CMB and the Earth’s surface, we find time delays in the range of 1–2 yr for the tangential and less than 1 yr for the radial jerk components. To understand these time delays, comparisons at fixed locations are carried out to check the influence of the respective conductivity function. For studying the time delay effects, we apply the inversion set-up of the NHDC to calculate simulated temporal oscillations and derive analytical expressions approximating the phase shifts. We find that jerk occurrence time delays and simulated phase shifts of temporal oscillations have a similar behaviour with respect to the influence of the conductivity and for the radial and the tangential components, respectively. In addition, a new concept for determining a jerk amplitude is presented briefly. This so-called dynamical jerk morphology, which forms a portion of the geomagnetic secular acceleration, is defined for each component by a time function on the considered surface. Its temporal motion patterns at the CMB are likely related with jerk originating processes in the fluid outer core.