Dynamical models for the Earth's geoid

The Earth's largest geoid anomalies occur at the lowest spherical harmonic degrees, or longest wavelengths, and are primarily the result of mantle convection. Thermal density contrasts due to convection are partially compensated by boundary deformations due to viscous flow whose effects must be included in order to obtain a dynamically consistent model for the geoid. These deformations occur rapidly with respect to the timescale for convection, and we have analytically calculated geoid response kernels for steady-state, viscous, incompressible, self-gravitating, layered Earth models which include the deformation of boundaries due to internal loads. Both the sign and magnitude of geoid anomalies depend strongly upon the viscosity structure of the mantle as well as the possible presence of chemical layering.

Correlations of various global geophysical data sets with the observed geoid can be used to construct theoretical geoid models which constrain the dynamics of mantle convection. Surface features such as topography and plate velocities are not obviously related to the low-degree geoid, with the exception of subduction zones which are characterized by geoid highs (degrees 4-9). Recent models for seismic heterogeneity in the mantle provide additional constraints, and much of the low-degree (2-3) geoid can be attributed to seismically inferred density anomalies in the lower mantle. The Earth's largest geoid highs are underlain by low density material in the lower mantle, thus requiring compensating deformations of the Earth's surface. A dynamical model for whole mantle convection with a low viscosity upper mantle can explain these observations and successfully predicts more than 80% of the observed geoid variance.

Temperature variations associated with density anomalies in the man tie cause lateral viscosity variations whose effects are not included in the analytical models. However, perturbation theory and numerical tests show that broad-scale lateral viscosity variations are much less important than radial variations; in this respect, geoid models, which depend upon steady-state surface deformations, may provide more reliable constraints on mantle structure than inferences from transient phenomena such as postglacial rebound. Stronger, smaller-scale viscosity variations associated with mantle plumes and subducting slabs may be more important. On the basis of numerical modelling of low viscosity plumes, we conclude that the global association of geoid highs (after slab effects are removed) with hotspots and, perhaps, mantle plumes, is the result of hot, upwelling material in the lower mantle; this conclusion does not depend strongly upon plume rheology. The global distribution of hotspots and the dominant, low-degree geoid highs may correspond to a dominant mode of convection stabilized by the ancient Pangean continental assemblage.

Tectonic and geodynamic pattern of marginal seas on the West Pacific inferred from multi-satellite altewetry geoid

With the rapid development of satellite observations, we can use the altimetry geoid to study submarine tectonics and geodynamics. On the basis of the 4' x 4' geoid undulation calculated from altimeter data of Geosat, ERS-1/2 and Topex/Poseidon on the West Pacific, located between 0degreesN similar to 45degreesN, 100degreesC similar to 150degreesE, Bouguer, Glenni and isostatic geoid undulation are obtained from correction of gravitational potential of the global topography and isostacy. Moho discontinuity depth is inversed by the Glenni geoid undulation, and the stress field from small-scale mantle convection is reasonably calculated from the isostatic geoid undulation. The results show that within the Philippine Sea and the South China Sea, short-wavelength lineations of the geoid undulation are parallel or cross to magnetic lineations and rifting ridges. The Moho depth of marginal sea basins becomes shallow southward, and its values are similar to that of the Philippine Sea. These facts show that strength of tectonic activities are almost the same on the both sides of the Ryukyu-Taiwan-Philippine are. Various kinds of tectonic features with different driving mechanisms of small-middle and large-scale of mantle convection, however, display a special pattern of tectonics and geodynamics of the continental marginal seas distinguished from oceans and continents.

Calculating the ocean's mean dynamic topography from a mean sea surface and a geoid

In principle the global mean geostrophic surface circulation of the ocean can be diagnosed by subtracting a geoid from a mean sea surface (MSS). However, because the resulting mean dynamic topography (MDT) is approximately two orders of magnitude smaller than either of the constituent surfaces, and because the geoid is most naturally expressed as a spectral model while the MSS is a gridded product, in practice complications arise. Two algorithms for combining MSS and satellite-derived geoid data to determine the ocean’s mean dynamic topography (MDT) are considered in this paper: a pointwise approach, whereby the gridded geoid height field is subtracted from the gridded MSS; and a spectral approach, whereby the spherical harmonic coefficients of the geoid are subtracted from an equivalent set of coefficients representing the MSS, from which the gridded MDT is then obtained. The essential difference is that with the latter approach the MSS is truncated, a form of filtering, just as with the geoid. This ensures that errors of omission resulting from the truncation of the geoid, which are small in comparison to the geoid but large in comparison to the MDT, are matched, and therefore negated, by similar errors of omission in the MSS. The MDTs produced by both methods require additional filtering. However, the spectral MDT requires less filtering to remove noise, and therefore it retains more oceanographic information than its pointwise equivalent. The spectral method also results in a more realistic MDT at coastlines. 1. Introduction An important challenge in oceanography is the accurate determination of the ocean’s time-mean dynamic topography (MDT). If this can be achieved with sufficient accuracy for combination with the timedependent component of the dynamic topography, obtainable from altimetric data, then the resulting sum (i.e., the absolute dynamic topography) will give an accurate picture of surface geostrophic currents and ocean transports.

An ocean modelling and assimilation guide to using GOCE geoid products

We review the procedures and challenges that must be considered when using geoid data derived from the Gravity and steady-state Ocean Circulation Explorer (GOCE) mission in order to constrain the circulation and water mass representation in an ocean 5 general circulation model. It covers the combination of the geoid information with timemean sea level information derived from satellite altimeter data, to construct a mean dynamic topography (MDT), and considers how this complements the time-varying sea level anomaly, also available from the satellite altimeter. We particularly consider the compatibility of these different fields in their spatial scale content, their temporal rep10 resentation, and in their error covariances. These considerations are very important when the resulting data are to be used to estimate ocean circulation and its corresponding errors. We describe the further steps needed for assimilating the resulting dynamic topography information into an ocean circulation model using three different operational fore15 casting and data assimilation systems. We look at methods used for assimilating altimeter anomaly data in the absence of a suitable geoid, and then discuss different approaches which have been tried for assimilating the additional geoid information. We review the problems that have been encountered and the lessons learned in order the help future users. Finally we present some results from the use of GRACE geoid in20 formation in the operational oceanography community and discuss the future potential gains that may be obtained from a new GOCE geoid.

Magnetotelluric deep soundings, gravity and geoid in the south Sao Francisco craton: Geophysical indicators of cratonic lithosphere rejuvenation and crustal underplating

In the south Sao Francisco craton a circular and 8-m amplitude geoid anomaly coincides with the outcropping terrain of an Archean-Paleoproterozoic basement. Broadband magnetotelluric (MT) data inversions of two radial profiles within the positive geoid and Bouguer gravity anomaly yield geo-electrical crustal sections, whereby the lower crust is locally more conductive (10 to 100 Omega m) in spatial coincidence with a denser lower crust modeled by the gravity data. This anomalous lower crust may have resulted from magmatic underplating, associated with Mesoarchean and Proterozoic episodes of tholeiitic dike intrusion. Long-period MT soundings reveal a low electrical resistivity mantle (20 to 200 Omega m) from depths beyond 120 km. Forward geoid modeling, using the scope of the low electrical resistivity region within the mantle as a constraint, entails a density increase (40 to 50 kg/m(3)) possibly due to Fe enrichment of mantle minerals. However, this factor alone does not explain the observed resistivity. A supplemented presence of small amounts of percolated carbonatite melting (similar to 0.005 vol.%), dissolved water and enhanced oxygen fugacity within the peridotitic mantle are viable agents that could explain the less resistive upper mantle. We propose that metasomatic processes confined in the sub-continental lithospheric mantle foster the conditions for a low degree melting with variable CO(2), H(2)O and Fe content. Even though the precise age of this metasomatism is unknown it might be older than the Early Cretaceous based on the evidence that a high-degree of melting in a lithospheric mantle impregnated with carbonatites originated the tholeiitic dike intrusions dispersed from the southeastern border of the Sao Francisco craton, during the onset of the lithosphere extension and break-up of the western Gondwana. The proxies are the NE Parana and Espinhaco (130 Ma, Ar/Ar ages) tholeiitic dikes, which contain (similar to 3%) carbonatites in their composition. The occurrence of a positive geoid anomaly (+ 10 m) and pre-tholeiites (age > 138 Ma), carbonatites and kimberlites along the west African continental margin (Angola and Namibia) reinforces the presumed age of the Sao Francisco-Congo craton rejuvenation to be prior to its fragmentation in the Lower Cretaceous. (C) 2010 Elsevier B.V. All rights reserved.

Use of recent geoid models to estimate mean dynamic topography and geostrophic currents in South Atlantic and Brazil Malvinas confluence

The use of geoid models to estimate the Mean Dynamic Topography was stimulated with the launching of the GRACE satellite system, since its models present unprecedented precision and space-time resolution. In the present study, besides the DNSC08 mean sea level model, the following geoid models were used with the objective of computing the MDTs: EGM96, EIGEN-5C and EGM2008. In the method adopted, geostrophic currents for the South Atlantic were computed based on the MDTs. In this study it was found that the degree and order of the geoid models affect the determination of TDM and currents directly. The presence of noise in the MDT requires the use of efficient filtering techniques, such as the filter based on Singular Spectrum Analysis, which presents significant advantages in relation to conventional filters. Geostrophic currents resulting from geoid models were compared with the HYCOM hydrodynamic numerical model. In conclusion, results show that MDTs and respective geostrophic currents calculated with EIGEN-5C and EGM2008 models are similar to the results of the numerical model, especially regarding the main large scale features such as boundary currents and the retroflection at the Brazil-Malvinas Confluence.

Use of recent geoid models to estimate mean dynamic topography and geostrophic currents in South Atlantic and Brazil Malvinas confluence

The use of geoid models to estimate the Mean Dynamic Topography was stimulated with the launching of the GRACE satellite system, since its models present unprecedented precision and space-time resolution. In the present study, besides the DNSC08 mean sea level model, the following geoid models were used with the objective of computing the MDTs: EGM96, EIGEN-5C and EGM2008. In the method adopted, geostrophic currents for the South Atlantic were computed based on the MDTs. In this study it was found that the degree and order of the geoid models affect the determination of TDM and currents directly. The presence of noise in the MDT requires the use of efficient filtering techniques, such as the filter based on Singular Spectrum Analysis, which presents significant advantages in relation to conventional filters. Geostrophic currents resulting from geoid models were compared with the HYCOM hydrodynamic numerical model. In conclusion, results show that MDTs and respective geostrophic currents calculated with EIGEN-5C and EGM2008 models are similar to the results of the numerical model, especially regarding the main large scale features such as boundary currents and the retroflection at the Brazil-Malvinas Confluence.

Regional geoid model for the area of the subglacial Lake Vostok

We present a regional geoid model for the area of Lake Vostok, Antarctica, from a combination of local airborne gravity, ice-surface and ice-thickness data and a lake bathymetry model. The topography data are used for residual terrain modelling (RTM) in a remove-compute-restore approach together with the GOCE satellite model GOCO03S. The disturbing potential at the Earth's surface, i.e. the quasigeoid, is predicted by least-squares collocation (LSC) and subsequently converted to geoid heights. Compared to GOCO03S our regional solution provides an additional short-wavelength signal of up to 1.48 m, or 0.56 m standard deviation, respectively. More details can be found in Schwabe et. al (2014).

Improved geoid solution for the Weddell Sea region

We present a geoid solution for the Weddell Sea and adjacent continental Antarctic regions. There, a refined geoid is of interest, especially for oceanographic and glaciological applications. For example, to investigate the Weddell Gyre as a part of the Antarctic Circumpolar Current and, thus, of the global ocean circulation, the mean dynamic topography (MDT) is needed. These days, the marine gravity field can be inferred with high and homogeneous resolution from altimetric height profiles of the mean sea surface. However, in areas permanently covered by sea ice as well as in coastal regions, satellite altimetry features deficiencies. Focussing on the Weddell Sea, these aspects are investigated in detail. In these areas, ground-based data that have not been used for geoid computation so far provide additional information in comparison with the existing high-resolution global gravity field models such as EGM2008. The geoid computation is based on the remove-compute-restore approach making use of least-squares collocation. The residual geoid with respect to a release 4 GOCE model adds up to two meters and more in the near-coastal and continental areas of the Weddell Sea region, also in comparison with EGM2008. Consequently, the thus refined geoid serves to compute new estimates of the regional MDT and geostrophic currents.

Dynamics of the North American continent

The forces that cause deformation of western North America have been debated for decades. Recent studies, primarily based on analysis of crustal stresses in the western United States, have suggested that the deformation of the region is mainly controlled by gravitational potential energy (GPE) variations and boundary loads, with basal tractions due to mantle flow playing a relatively minor role. We address these issues by modelling the deviatoric stress field over western North America from a 3-D finite element mantle circulation model with lateral viscosity variations. Our approach takes into account the contribution from both topography and shallow lithosphere structure (GPE) as well as that from deeper mantle flow in one single model, as opposed to separate lithosphere and circulation models, as has been done so far. In addition to predicting the deviatoric stresses we also jointly fit the constraints of geoid, dynamic topography and plate motion both globally and over North America, in order to ensure that the forces that arise in our models are dynamically consistent. We examine the sensitivity of the dynamic models to different lateral viscosity variations. We find that circulation models that include upper mantle slabs yield a better fit to observed plate velocities. Our results indicate that a model of GPE variations coupled with mantle convection gives the best fit to the observational constraints. We argue that although GPE variations control a large part of the deformation of the western United States, deeper mantle tractions also play a significant role. The average deviatoric stress magnitudes in the western United States range 30-40 MPa. The cratonic region exhibits higher coupling to mantle flow than the rest of the continent. We find that a relatively strong San Andreas fault gives a better fit to the observational constraints, especially that of plate velocity in western North America.

Joint modeling of lithosphere and mantle dynamics: Evaluation of constraints from global tomography models

With the advances in technology, seismological theory, and data acquisition, a number of high-resolution seismic tomography models have been published. However, discrepancies between tomography models often arise from different theoretical treatments of seismic wave propagation, different inversion strategies, and different data sets. Using a fixed velocity-to-density scaling and a fixed radial viscosity profile, we compute global mantle flow models associated with the different tomography models and test the impact of these for explaining surface geophysical observations (geoid, dynamic topography, stress, and strain rates). We use the joint modeling of lithosphere and mantle dynamics approach of Ghosh and Holt (2012) to compute the full lithosphere stresses, except that we use HC for the mantle circulation model, which accounts for the primary flow-coupling features associated with density-driven mantle flow. Our results show that the seismic tomography models of S40RTS and SAW642AN provide a better match with surface observables on a global scale than other models tested. Both of these tomography models have important similarities, including upwellings located in Pacific, Eastern Africa, Iceland, and mid-ocean ridges in the Atlantic and Indian Ocean and downwelling flows mainly located beneath the Andes, the Middle East, and central and Southeast Asia.

Three-dimensional models of mantle convection : influence of phase transitions and temperature-dependent viscosity

Two of the most important questions in mantle dynamics are investigated in three separate studies: the influence of phase transitions (studies 1 and 2), and the influence of temperature-dependent viscosity (study 3).

(1) Numerical modeling of mantle convection in a three-dimensional spherical shell incorporating the two major mantle phase transitions reveals an inherently three-dimensional flow pattern characterized by accumulation of cold downwellings above the 670 km discontinuity, and cylindrical 'avalanches' of upper mantle material into the lower mantle. The exothermic phase transition at 400 km depth reduces the degree of layering. A region of strongly-depressed temperature occurs at the base of the mantle. The temperature field is strongly modulated by this partial layering, both locally and in globally-averaged diagnostics. Flow penetration is strongly wavelength-dependent, with easy penetration at long wavelengths but strong inhibition at short wavelengths. The amplitude of the geoid is not significantly affected.

(2) Using a simple criterion for the deflection of an upwelling or downwelling by an endothermic phase transition, the scaling of the critical phase buoyancy parameter with the important lengthscales is obtained. The derived trends match those observed in numerical simulations, i.e., deflection is enhanced by (a) shorter wavelengths, (b) narrower up/downwellings (c) internal heating and (d) narrower phase loops.

(3) A systematic investigation into the effects of temperature-dependent viscosity on mantle convection has been performed in three-dimensional Cartesian geometry, with a factor of 1000-2500 viscosity variation, and Rayleigh numbers of 10^5-10^7. Enormous differences in model behavior are found, depending on the details of rheology, heating mode, compressibility and boundary conditions. Stress-free boundaries, compressibility, and temperature-dependent viscosity all favor long-wavelength flows, even in internally heated cases. However, small cells are obtained with some parameter combinations. Downwelling plumes and upwelling sheets are possible when viscosity is dependent solely on temperature. Viscous dissipation becomes important with temperature-dependent viscosity.

The sensitivity of mantle flow and structure to these various complexities illustrates the importance of performing mantle convection calculations with rheological and thermodynamic properties matching as closely as possible those of the Earth.

Aplicação das fórmulas de Vincenty nos cálculos das correções dos efeitos do relevo na gravidade e na altura geoidal.

O presente trabalho apresenta a aplicação das fórmulas de Vincenty nos cálculos das correções do terreno e do efeito indireto, que desempenham papel relevante na construção de cartas geoidais. Implementa-se um programa de processamento que realiza a integração numérica sobre o modelo digital do terreno, discretizado em células triangulares de Delaunay. O sistema foi desenvolvido com a linguagem de programação FORTRAN, para a execução de intensos algoritmos numéricos usando compiladores livres e robustos. Para o cálculo do efeito indireto, considera-se a redução gravimétrica efetuada com base no segundo método de condensação de Helmert, face ao pequeno valor de efeito indireto no cálculo do geóide, em função da mudança que este produz no potencial da gravidade devido ao deslocamento da massa topográfica. Utiliza-se, o sistema geodésico SIRGAS 2000 como sistema de referência para o cômputo das correções. Simplificando o exame dos resultados alcançados, distingue-se o processamento e desenvolvimento do trabalho em etapas como a escolha de ferramentas geodésicas para máxima precisão dos resultados, elaboração de subrotinas e comparação de resultados com cálculos anteriores. Os resultados encontrados foram de geração sadia e satisfatória e podem ser perfeitamente empregados no cálculo do geóide em qualquer área do globo.