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Mantle convection is the process by which heat from the Earth’s core is transferred upwards to the surface and it is accepted to explain the dynamics of the Earth’s interior. On geological time-scales, mantle material flows like a viscous fluid as a consequence of the buoyancy forces arising from thermal expansion. Indeed, mantel convection provides a framework which links together the major disciplines, such as seismology, mineral physics, geochemistry tectonic and geology. The numerical model has been applied to understand the dynamic, structure and evaluation of the Earth, and other terrestrial planets and the investigations continue to explore, different aspects of the mantle convection.
In fact, to model this phenomenon, two complementary approaches are possible. On the one hand, one can solve self-consistently the equations of thermal convection, including parameters and employing physical relationships derived from mineral physics. Our understanding of mantle convection depends ultimately upon the success of such fully self-consistent dynamic models in explaining observable features of the flow. Although, these models presently unable to predict the actual convection pattern of the Earth, they are extremely useful to investigate general characteristics of given physical systems. On the other hand, to permit comparison with specific observables associated with the flow, one can consider a more restricted problem. Instead of focusing on the time evolution of mantle flow, if we know a priori the temperature - and hence presumably the density - anomalies that drive the convection, we can try to build a snapshot of the present-day flow pattern, consistent with those anomalies, that can successfully predict the observables. As matter of fact, the aim of this study is to investigate both approaches in comparison with the main geophysical constraints on mantle structure. These constraints include the geoid anomalies, the dynamic surface and core-mantle boundary topography and tectonic plate motions.
The most appropriate mathematical basis functions for describing a bounded and continuous function on a spherical surface are spherical harmonics. We may therefore expand the geodynamic observables in terms of spherical harmonics. We have investigated two methods of the global spherical harmonic analysis by specific attention to the dynamic geoid computation of the geodynamic models. The first method is the quadrature method in which the loss of the orthogonality of the Legendre functions in transition from continues to discrete case is the major drawback to the method. Particularly, we showed that in the absence of the tesseral harmonics, quadrature formulation leads to obtain inaccurate results. The second method is the least-squares which can be considered as the best linear unbiased estimator that provides the exact results. We showed that even with a low resolution grid data it is possible to reconstruct the data and achieve an accurate result by using this method, which is extremely remarkable in three-dimensional global convection studies. However, special care has to be taken since there is some source of errors that might influence the efficiency of this method.
In general, to better understanding of the properties of the mantle, it is useful to assess observable characteristics of plumes in the mantle, including geoid, topography and heat flow anomalies. However, only few studies exist on geoid and topography for axi-symmetric convection and their models were restricted to isoviscous (or stratified) mantle and low Rayleigh numbers. We studied fully coupled depth and temperature dependent Arrhenius type of viscosity in axi-symmetric spherical shell geometry in order to investigate the shape of geoid anomalies and dynamic topography above a plume. Indeed, the topography and geoid anomalies produced from plumes are sensitive to rheology of the mantle and rheology of the plume; both have effects on shape and amplitude of the geoid anomalies. As results we are able to define different classes of plumes by their geoid signals.
Mainly depth-dependent viscosity models show a geoid with negative sign above the plume which can turn to the positive sign by decrease the viscosity contrast. This can be considered as a transition between the strongly depth dependent and the constant viscosity case. Our results basically support the idea by Morgan [1965] and McKenzie [1977]. They have shown the magnitude and even the sign of the total gravity anomaly depend on the spatial variation in effective viscosity. In addition, Hager [1984] has concluded that the total gravity field is depend on the radial distribution of effective viscosity, and a small change in viscosity contrast leads to varying sign of the response function.
In the case of temperature-dependent viscosity, the formation of an immobile lithosphere is a natural result, and the flow as well as the total geoid becomes strongly time dependent. When we increase the activation energy, all geoids associated with the first arriving plumes look like bell shaped whereas for typical plumes, after reaching a statistical steady state, bell-shaped geoids with decreasing amplitude as well as linear flank shaped geoids are observed. It is surprising that in spite of large differences in lateral and depth varying viscosities, the shapes of the geoid anomalies remained rather similar. We also identified different behaviors in the combined model with temperature-and pressure-dependent viscosity. In fact, in spite of the strongly different rheology, the geoid anomalies in all cases were surprisingly similar. Furthermore, we proposed a scaling law for the geoid which makes our results directly applicable to other planets. Moreover, we can apply the results of our calculation to find relations between different rheology and sub-lid temperature, since we know that the mantle temperature can change significantly with variation in pressure-temperature dependent viscosity. It is also possible to define a range of stagnant lid thickness related to the amplitude of the geoid which can be reasonable for study of the lid thickness in Venus or Mars.
Nevertheless, in these series of models, we simplified a number of complexities within the Earth. One of the most important of such simplification is the Boussinesq approximation. This approximation is valid if the temperature scale height (i.e. the depth over which temperature increases by a factor of “ ” due to adiabatic compression) is much greater than the convection depth. However, a temperature scale height in the Earth’s mantle is at best only slightly greater than the mantle depth. Hence, the Boussinesq approximation could mask some very important stratification and compressibility effects that influence both the spatial and temporal structure of the convection. Therefore, in more advance models we considered compressibility in our mantle convection models, assuming that density vary both radially and laterally, being determined as a function of pressure and temperature through an appropriate equation of the state. Moreover, thermodynamic properties assumed to be a function of depth.
We examined the details of the structure of the spherical axi-symmetric Anelastic Liquid Approximation model (ALA) with special attention to the Arrhenius rheology, and compare it to the cases of compressible convection without depth dependent thermodynamical properties, and to cases of the extended Boussinesq approximation. At the same time, the effects of the interaction between temperature and pressure-dependent viscosity and thermodynamic parameters in the compressible mantle convection on the geoid and topography have been studied. We showed that assuming compressible convection with depth-dependent thermodynamic properties strongly influence the geoid undulations. Using compressible convection with constant thermodynamic properties is physically inconsistent and may lead to spurious results for the geoid and convection pattern. Indeed, by a systematic study of different approaches of compressibility in the spherical shell convection for different Arrhenius viscosity laws we proved that only in the unrealistic case of zero activation energy the different compressibility modes result in comparable convection and geoid patterns. In all other rheological cases, large differences have been obtained, that stressing the important role of consistent compressible thermodynamic properties for mantle convection.
In addition, we examine the impact of compressibility as well as different rheologies on the power law relation that connects the Nusselt number to the Rayleigh number. We have discovered that the power law index of the relationship is controlled by the rheology, independent of which approximation is used. Instead, the bound of this relation is controlled by a combination of different approximation and rheology.
Next, instead of focusing on the time evolution of mantle flow, we have carried out three-dimensional spherical shell models of mantle circulation to investigate the effects of joint radial and lateral viscosity variations on the Earth’s non-hydrostatic geoid, surface and core-mantle boundary topographies. These models include realistic lateral viscosity variations (LVV) in the lithosphere, upper mantle and lower mantle in combination with different stratified viscosity structures. We have demonstrated that the contradictory results concerning the effects of LVV can be clarified by the most straight-forward problem in geoid modeling; namely, rather poorly known stratified viscosity structure. We explored three classes of dynamic geoid models due to lateral viscosity variations. In the first class, the LVV strongly improved the fit to the observed geoid. Indeed, when the viscosity contrast between lower and upper mantles is not large enough to produce a good fit to geoid the LVVs are able to perform this action by adjusting amplitudes, so that it becomes comparable with observation. In the second class, inducing the LVV moderately improved the fit. Actually, when the geoid induced by a stratified viscosity structure already has a good correlation with observation, then the LVV causes its amplitude to further improve. In the last class, if the viscosity contrast between upper and lower mantle would be high enough, inducing LVV deteriorate the fit to the observed geoid.. Indeed, depending on the stratified viscosity, inducing the LVV may take place in one of these categories.
We also quantified the effects of LVV in the mantle and lithosphere individually. We found that the presence of LVV in the mantle (upper and lower) improves the fit to the observed geoid regardless of stratified viscosity. While LVV in the lithosphere is a crucial parameter, and dependent of the stratified viscosity, may increase or decrease the geoid fit. In fact, when the lower mantle considers being viscous enough, it would support the negative buoyancy of subducting slabs. Thus, it transmits some of the stress back to the top boundary and causes a weak coupling between slab and surface. Therefore, by including the low viscous plate boundaries in this model, the slabs and overriding plates decouples and the fit to the observed geoid degrades. In contrast, when the lower mantle viscosity is not sufficiently stiff, the presence of the low viscous plate boundaries assists to weaken the strong mechanical coupling between slab and surface. Hence, a better fit achieved.