For a spherically symmetric viscoelastic earth model, the movement of the rotation vector due to surface and internal mass redistribution during the Pleistocene glaciation cycle has conventionally ...been computed in the Laplace-transform domain. The method involves multiplication of the Laplace transforms of the second-degree surface-load and tidal-load Love numbers with the time evolution of the surface load followed by inverse Laplace transformation into the time domain. The recently developed spectral finite-element method solves the field equations governing glacial-isostatic adjustment (GIA) directly in the time domain and, thus, eliminates the need of applying the Laplace-domain method. The new method offers the possibility to model the GIA-induced rotational response of the Earth by time integration of the linearized Liouville equation. The theory presented here derives the temporal perturbation of the inertia tensor, required to be specified in the Liouville equation, from time variations of the second-degree gravitational-potential coefficients by the MacCullagh's formulae. This extends the conventional approach based on the second-degree load Love numbers to general 3-D viscoelastic earth models. The verification of the theory of the GIA-induced rotational response of the Earth is performed by using two alternative approaches of computing the perturbation of the inertia tensor: a direct numerical integration and the Laplace-domain method. The time-domain solution of both the GIA and the induced rotational response of the Earth is readily combined with a time-domain solution of the sea level equation with a time-varying shoreline geometry. In a follow-up paper, we derive the theory for the case when GIA-induced perturbations in the centrifugal force affect not only the distribution of sea water, but also deformations and gravitational-potential perturbations of the Earth.
The Fennoscandian relaxation-time spectrum is an observational data set representing post-glacial land uplift and employed in interpretations of glacial-isostatic adjustment. We invert this data set ...by adapting the spectral finite-element method, originally developed for a 3D self-gravitating spherical earth model Martinec, Z., 2000. Spectral–finite element approach to three-dimensional viscoelastic relaxation in a spherical earth. Geophys. J. Int. 142, 117–142, to an axisymmetric viscosity distribution. The free parameters used in the inversion are either the central-lithosphere thickness (below the former Fennoscandian ice sheet) and the upper-mantle viscosity or the peripheral-lithosphere thickness and the peripheral-asthenosphere viscosity (surrounding the former Fennoscandian ice sheet). We demonstrate that a model featuring a central lithosphere with a thickness of 200 km and a peripheral lithosphere with a thickness of 80 km underlain by an asthenosphere satisfies the relaxation-time spectrum. We also show that the spectrum can be explained on the basis of a spherically symmetric model featuring a 100-km thick lithosphere, but no asthenosphere. Whereas this result is essentially consistent with previous interpretations of glacial-isostatic adjustment, the result obtained for the axisymmetric viscosity model also agrees with a large number of interpretations based on seismic, geomagnetic, thermal or rheological evidence for Fennoscandia.
In the first attempt to solve the Stokes boundary-value problem in ellipsoidal coordinates numerically (Ardestani and Martinec, 2000), we focused on the near-zone contribution since the effect of the ...ellipsoidal Stokes function in the far-zone contribution is not considered. We present a method for solving the ellipsoidal Stokes integral in far-zone contribution. The numerical results of computing the magnitude of this term for an area in north of Canada are presented.
We present a new algorithm for a fast iterative improvement of the shallow-ice approximation (SIA) for the modeling of glacier flow. Based on the traditional SIA scaling assumptions, the solution of ...the Stokes problem is found by an operator-splitting iterative technique. The SIA solution obtained in the first step is successively improved to obtain a higher-order approximation. Each iterative step has computational demands comparable to solving the SIA, which makes the algorithm substantially faster than other higher-order or full-Stokes solvers. The performance of the algorithm is tested on a model example taken from the ISMIP-HOM intercomparison project.
Martinec and Grafarend (1997) have shown how the construction of Green's function in the Stokes boundary-value problem with gravity data distributed on an ellipsoid of revolution is approached in the ...O(e02)-approximation. They have also expressed the ellipsoidal Stokes function describing the effect of ellipticity of the boundary as a finite sum of elementary functions. We present an effective method of avoiding the singularity of spherical and the ellipsoidal Stokes functions, and also an analytical expression for the ellipsoidal Stokes integral around the computational point suitable for numerical solution. We give the numerical results of solving the ellipsoidal Stokes boundary-value problem and their difference with respect to the spherical Stoke boundary-value problem.
The first results on the modelling of magnetic signals induced by transient equatorial ring currents in the magnetosphere with a timescale of the order of days and recorded at satellite altitudes are ...presented. The input of modelling consists of the X-component of the time-series recorded by the CHAMP vector magnetometer along individual night-time, mid-latitude satellite tracks. We have not considered the magnetic signals measured along day-time tracks and above polar regions because they are disturbed by signals with sources different from magnetospheric ring currents. The modelling procedure is divided into two parts. First, the X-components of satellite magnetic signals are processed by a two-step least-squares analysis. As a result, the X-data are represented in terms of series of Legendre polynomial derivatives. Four examples of the two-step analysis of the signals recorded by the CHAMP vector magnetometer are presented. Second, forward modelling of the electromagnetic induction response of a 2-D heterogeneous conducting sphere to a transient external current excitation is carried out in the time domain using a recently developed spectral finite-element technique (Martinec 2003), which has been modified to include satellite magnetic data. The modified approach has been verified against a semi-analytical solution for a 2-D geometry: a spherical inclusion eccentrically nested in a homogeneous sphere. The output of the forward modelling of electromagnetic induction, that is, the predicted Z-component at satellite altitude, can then be compared with the satellite observations to obtain data to be minimized when adjusting electrical conductivity models in future inversion modelling. To outline such a procedure, we use a 5-layer spherically symmetric model of electrical conductivity and compare the predicted and observed Z-data for the magnetic storm recorded by the CHAMP satellite between 2001 September 25 and 2001 October 7. The agreement between observations and predictions is quantitatively satisfactory. The main advantage of the present method is its ability to use the satellite magnetic data directly, without the need to continue them from the satellite altitude to the ground or to decompose them into the inducing and induced parts by spherical harmonic analysis.
The interpretation of long wavelength geoid and plate motions on the basis of dynamic Earth models has usually been done assuming linear viscous rheologies in the mantle. In this paper, we develop ...spherical 3-D models of mantle circulation using power-law creep rheologies with an exponent n= 3. We found that the stress-dependent rheologies only modify the amplitude of the topography supported by an internal load, by a small percentage with respect to the linear predictions. the geoid anomalies induced by internal loads can be affected by around 20 per cent. These changes are also occurring at degrees and orders different from those of the mass anomaly itself. As the geoid spectrum is strongly decreasing with degree, the dynamic topography induced at high degrees can be contaminated in a nonnegligible way by the low degree loads. the main contamination occurs at a harmonic triple of that of the most important load. the flow structure is much more dependent on the creep law than are the dynamic topography and the geoid. In contrast to linear rheology, a power-law creep is able to sustain a toroidal velocity field. However, this toroidal component only carries a small percentage of the kinetic energy and thus the non-linear creep with n= 3 cannot by itself explain the observed quasi-equipartition of plate tectonic energy between toroidal and poloidal components.
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