A new approach to computing accurate gravity time variations for a realistic earth model with lateral heterogeneities

International audienceWe have developed a new elasto-gravitational earth model able to take into account lateral variations, deviatoric pre-stresses and topographies. As a first application, we assume an el-lipsoidal earth with hydrostatic pre-stresses, and validate and discuss our numerical model by comparison with previous studies on the M 2 body tide. We then study the response of the ellipsoidal earth to zonal atmospheric loads, and find that global lateral variations within the Earth, such as ellipticity, have a weak impact (about 1 per cent) on the elasto-gravitational deformations induced by atmospheric loading. At low frequencies, the Earth is deformed mainly by luni-solar tides and by surface loads, including ocean, atmosphere, ice volumes and post-glacial rebound. In this work, we focus our attention on the Earth's body tides and atmospheric loadings. The most accepted Earth body-tide models presently deal with an ellipsoidal, rotating earth, containing a liquid core and an anelastic mantle with hydrostatic pre-stresses (Wahr 1981; Wahr & Bergen 1986). The Earth, however, is not an exact ellipsoid, but presents lateral variations and deviatoric pre-stresses: there are long-wavelength density anomalies within the mantle, as shown by geoid anomalies and tomography studies (e.g. Romanowicz & Gung 2002). Wang (1994) and Dehant et al. (1999) studied the influence of lateral heterogeneities on Earth tides and showed that this effect is small but not necessarily negligible. They did not, however, take into account possible deviatoric pre-stresses: these effects on the Earth's body tides are totally unknown. In addition to tidal forces, mass changes in the atmosphere also cause deformation and mass redistribution inside the planet, involving both local and global surface motions and variations in the gravity field, which may be observed in geodetic experiments. For several decades, satellite geodesy has provided information on the temporal variation of the Earth's geopotential, and especially on the low-degree zonal harmonics (J 2 , J 3. . .) (Gegout & Cazenave 1993), which are essentially controlled by surface loads. These hydrological , atmospheric or oceanic effects on the Earth's gravity field are usually modelled assuming a spherical earth with hydro-static pre-stress (e.g. Farrell 1972; Wahr et al. 1998). With the advent of the new generation of gravity measurements, one of the challenges of the coming decade will be to provide more realistic earth models that show the variation of gravity with time. In particular, global studies based on gravity data from satellites such as GRACE, GOCE, and future GRACE/GOCE follow-on ones require accurate body-tide deformation models. More realistic gravity variation models are also needed for local and ground measurements, particularly for the very accurate superconducting gravimeters and the associated gravimetric observatory network such as the Global Geodynamic Project (Crossley et al. 1999). The formalism developed to compute this elasto-gravitational model is usually based on spherical harmonic analysis. The addition of lateral variations leads to couplings between spherical harmonics , i.e. to a more complex formalism that requires a large numerical effort (e.g. Wang 1994; Plag et al. 1996). We develop here a new approach for a non-radially symmetrical earth model using a finite-element method known as the spectral element method. The efficiency of this method is less dependent on the shape of the lateral heterogeneities than the spherical harmonic method. Our method is therefore well adapted to studying the impact of global and local lateral variations on the Earth deformation. We solve the elasto-gravitational equations taking into consideration the lateral variations within the Earth by using a first-order perturbation theory (Smith 1974; Dahlen & Tromp 1998). This new model allows us to take into account lateral variations of density and rheological parameters, deviatoric pre-stresses and interface topography. In order to validate our calculations, we tackle a well-known problem: the impact of the hydrostatic ellipticity on the Earth body tides. An analytical solution for this problem can be derived for a simple model in which the earth is assumed to be homogeneous and incompressible. The gravitational potential and the vertical displacement on the surface of the deformed ellipsoid were first derived by Love (1911) and then corrected by Wang (1994). We have recently extended these analytical results to the tangential surface displacement (Greff-Lefftz et al. 2005). We first validate our model with our analytical solutions, and then compare our results wit

Mantle lateral variations and elastogravitational deformations – I. Numerical modelling

International audienceThe Earth response (deformation and gravity) to tides or to surface loads is traditionally computed assuming radial symmetry in stratified earth models, at the hydrostatic equilibrium. The present study aims at providing a new earth elastogravitational deformation model which accounts for the whole complexity of a more realistic earth. The model is based on a dynamically consistent equilibrium state which includes lateral variations in density and elastic parameters, and interface topographies. The deviation from the hydrostatic equilibrium has been taken into account as a first-order perturbation. We use a finite element method (spectral element method) and solve numerically the gravitoelasticity equations. As a validation application, we investigate the deformation of the Earth to surface loads. We first evaluate the classical loading Love numbers with a relative precision of about 0.3 per cent for PREM earth model. Then we assume an ellipsoidal homogeneous incom-pressible earth with hydrostatic pre-stresses. We investigate the impact of ellipticity on loading Love numbers analytically and numerically. We validate and discuss our numerical model. At periods greater than 1 hr, the solid earth is mainly deformed by luni-solar tides and by surface loads induced by different external fluid layers (ocean, atmosphere, continental hydrology, ice volumes). This work is devoted to the analytical and numerical development to compute the response of the Earth to such forcing. The body tides have been investigated since the 19th century. In 1862, Lord Kelvin (Sir William Thomson) made the first calculation of the elastic deformation of a homogeneous incompressible earth under the action of the tidal gravitational potential (Thomson 1862). Some years later, Love (1911) studied a compressible homogeneous earth model and showed that the tidal effects could be represented by a set of dimensionless numbers, the so-called Love numbers. Takeuchi (1950) obtained a first estimation of the Love numbers by a numerical integration of the equations using a reference earth model deduced from seismology. These results have been later extended (Smith 1974; Wahr 1981) to an ellipsoidal, rotating Earth with hydrostatic pre-stresses and a liquid core, and finally the effects of mantle anelasticity have been included (Wahr & Bergen 1986; Dehant 1987). In addition to tidal forces, mass changes in the atmosphere cause deformation and mass redistribution inside the planet. The Earth's response to such forcing involves both local and global surface motions and variations in the gravity field, which may be observed in geodetic experiments. These hydrological, atmospheric or oceanic effects on the Earth's gravity field are usually modelled for a spherical Earth with hydrostatic pre-stress (e.g. Farrell 1972; Wahr et al. 1998), generally identified to the preliminary reference earth model (PREM) developed by Dziewonski & Anderson (1981). However, the internal structure of the Earth is more complex than in a spherical non-rotating elastic isotropic (SNREI) earth model like PREM. Seismology and fluid dynamic studies show that the mantle presents heterogeneous structure induced by a thermochemical convection (Davaille 1999; Gu et al. 2001; Forte & Mitrovica 2001) and a bias from hydrostatic state. Large lateral heterogeneities have taken place on a million year timescale (Courtillot et al. 2003), like the two supposed superplumes under the Pacific and South Africa superswells, or like descending slabs. These aspects of the mantle structure are classically not taken into account in the deformation models. The elastogravitational deformations are presently observed with very high accuracy. The accuracy of superconducting gravimeter and of positioning techniques (GPS, VLBI) has seen a large improvement in the last decade. Moreover, the global gravity field will be of interest in the next 10 yr with the launch of the missions GRACE (in 2002) and GOCE (in 2007), which are dedicated to gravimetry and gradiometry 106

Dissipation at the core-mantle boundary on a small-scale topography

International audienceThe parameters of the nutations are now known with a good accuracy, and the theory accounts for most of their values. Dissipative friction at the core-mantle boundary (CMB) and at the inner core boundary is an important ingredient of the theory. Up to now, viscous coupling at a smooth interface and electromagnetic coupling have been considered. In some cases they appear hardly strong enough to account for the observations. We advocate here that the CMB has a small-scale roughness and estimate the dissipation resulting from the interaction of the fluid core motion with this topography. We conclude that it might be significant

Tracking the Late Jurassic apparent (or true) polar shift in U-Pb-dated kimberlites from cratonic North America (Superior Province of Canada)

Different versions of a composite apparent polar wander (APW) path of variably selected global poles assembled and averaged in North American coordinates using plate reconstructions show either a smooth progression or a large (∼30°) gap in mean paleopoles in the Late Jurassic, between about 160 and 145 Ma. In an effort to further examine this issue, we sampled accessible outcrops/subcrops of kimberlites associated with high-precision U-Pb perovskite ages in the Timiskaming area of Ontario, Canada. The 154.9 ± 1.1 Ma Peddie kimberlite yields a stable normal polarity magnetization that is coaxial within less than 5° of the reverse polarity magnetization of the 157.5 ± 1.2 Ma Triple B kimberlite. The combined ∼156 Ma Triple B and Peddie pole (75.5°N, 189.5°E, A95 = 2.8°) lies about midway between igneous poles from North America nearest in age (169 Ma Moat volcanics and the 146 Ma Ithaca kimberlites), showing that the polar motion was at a relatively steady yet rapid (∼1.5°/Myr) pace. A similar large rapid polar swing has been recognized in the Middle to Late Jurassic APW path for Adria-Africa and Iran-Eurasia, suggesting a major mass redistribution. One possibility is that slab breakoff and subduction reversal along the western margin of the Americas triggered an episode of true polar wander

Mantle lateral variations and elastogravitational deformations - II. Possible effects of a superplume on body tides

The body tides response (deformation and gravity) of the Earth is generally computed assuming radial symmetry in stratified earth models, at the hydrostatic equilibrium. We present in this paper numerical experiments with the aim to evaluate the impact of very large mantle heterogeneities of density on body tides. In a companion paper, we have developed a new earth elasto-gravitational deformation model able to take into account the heterogeneous structure of the mantle. We use this model to calculate the theoretical perturbation induced by three types of spherical heterogeneities in the mantle on M2 body tides response. The heterogeneities are: (1) our limit case, a heterogeneity of 1000 km of radius in the lower mantle; (2) a heterogeneity of 500 km of radius at the bottom of the lower mantle and (3) a heterogeneity of 285 km of radius in the upper mantle. The density variation has been set to -50 kg m-3. For each experiment, we first calculate the equilibrium state of the Earth when it contains a heterogeneity, including non-hydrostatic pre-stresses, dynamical topography and lateral variation of density. Then we compute the M2 tidal perturbation. We find that the surface tidal displacement perturbation is smaller than 1 mm, and that the gravity perturbation has a maximum amplitude of 525 nanoGal (nGal). Regarding to the present precision in position measurement, the displacement is too small to be detected. The gravity perturbation should be measurable with superconducting gravimeters, which have a nGal instrumental precision. In experiment 2, the maximum gravity perturbation is about 120 nGal, and in experiment 3, the maximum perturbation is about 33 nGal. Finally, we investigate the maximum theoretical impact of the Pacific and the African superplumes on the M2 body tide. The superplumes have been modelled as two spherical heterogeneities with a radius of 1000 km in the lower mantle. We find that these superplumes induce a maximum perturbation in gravity of about 370 nGal with a large part corresponding to a mean variation of gravity. We conclude that we can expect to have a gravity perturbation of body tide with an order of magnitude of about hundred of nGal induced by the biggest mantle heterogeneities of density. This perturbation in gravity should be measurable with superconducting gravimeters if all other contributions in the signal could be extracted with a sufficient precision

Mantle dynamics, geoid, inertia and TPW since 120 Myr

International audienceWe investigate the effect of internal masses redistributions on the position of the Earth rotational pole for the last 120 Myr. We use a geodynamic model based on plate reconstructions that estimates the location and rate of subducted slabs under the assumption that they sink vertically into the mantle (Ricard et al., 1993b). Our model also takes into account the effect of large-scale upwellings (domes) derived from an analysis of seismic tomography. Their location is assumed to remain stable with time. We then compute the geoid associated with the time-dependent mantle density heterogeneities. In order to reconcile the computed and observed geoids, we investigate the influence of the depth down to which the subducted Pacific plates beneath the Americas present a significant density contrast with respect to the surrounding mantle on the present-day geoid, and propose a plate model in which we obtain a variance reduction greater than 0.9 for the degree 2. We show that the vertical oscillation of domes within mantle only modulates the amplitude of the associated geoid. The temporal variation of the mantle density heterogeneities is consequently essentially due to changes in the subduction history. The temporal evolution of the Principal Inertia Axis (PIA) of the Earth derived from our model (the rotational axis is aligned to the maximum PIA) is then investigated, and finally compared to estimations of TPW. Both the maximum and intermediate PIAs have moved in a plane perpendicular to Africa, along a circle corresponding to the low of geoid induced by subduction around the Pacific. The minimum PIA seems to be relatively stable since 120 Myr, and close to the maximum degree 2 geoid high under Africa. This can account for observed TPW or African APW in the last 200 Myr

Atmospheric and oceanic excitation of the rotationof a three-layer Earth

In this paper, we evaluate the nutational Earth response to the excitation exerted by a surface fluid (atmosphere, ocean, and hydrology) for a simple Earth model, constituted of three homogeneous layers: a solid deformable inner-core, a liquid outer core, and an elastic mantle. Our formula, valid in the quasi-diurnal frequency band, includes two resonances, at the Free Core Nutation (FCN) and the Free Inner-Core Nutation (FICN). Additionally, we have evaluated the amplitudes of those wobbles in response to a random noise excitation. We show that, compared with the FCN signal, the resonance at the FICN frequency induced by a surface fluid layer only induces a very small signal in the Earth rotation, and that, with an excitation comparable to the one available at the FCN, the FICN would generate a signal at the Earth surface at the sub-micrometer level

The effects of the atmospheric pressure on nutations

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