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

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

Beeping a Maximal Independent Set

We consider the problem of computing a maximal independent set (MIS) in an extremely harsh broadcast model that relies only on carrier sensing. The model consists of an anonymous broadcast network in which nodes have no knowledge about the topology of the network or even an upper bound on its size. Furthermore, it is assumed that an adversary chooses at which time slot each node wakes up. At each time slot a node can either beep, that is, emit a signal, or be silent. At a particular time slot, beeping nodes receive no feedback, while silent nodes can only differentiate between none of its neighbors beeping, or at least one of its neighbors beeping. We start by proving a lower bound that shows that in this model, it is not possible to locally converge to an MIS in sub-polynomial time. We then study four different relaxations of the model which allow us to circumvent the lower bound and find an MIS in polylogarithmic time. First, we show that if a polynomial upper bound on the network size is known, it is possible to find an MIS in O(log^3 n) time. Second, if we assume sleeping nodes are awoken by neighboring beeps, then we can also find an MIS in O(log^3 n) time. Third, if in addition to this wakeup assumption we allow sender-side collision detection, that is, beeping nodes can distinguish whether at least one neighboring node is beeping concurrently or not, we can find an MIS in O(log^2 n) time. Finally, if instead we endow nodes with synchronous clocks, it is also possible to find an MIS in O(log^2 n) time.Comment: arXiv admin note: substantial text overlap with arXiv:1108.192

Visualization of Distributed Algorithms Based on Graph Relabelling Systems1 1This work has been supported by the European TMR research network GETGRATS, and by the “Conseil Régional d' Aquitane”.

AbstractIn this paper, we present a uniform approach to simulate and visualize distributed algorithms encoded by graph relabelling systems. In particular, we use the distributed applications of local relabelling rules to automatically display the execution of the whole distributed algorithm. We have developed a Java prototype tool for implementing and visualizing distributed algorithms. We illustrate the different aspects of our framework using various distributed algorithms including election and spanning trees