Anomalies of temperature and composition in the Earth's mantle induce anomalies of density and seismic velocities. The later are mapped by seismic tomography. However, seismic tomography models cannot be inverted simply for temperature and chemical anomalies. An alternative method is to solve the direct problem, i.e., to reconstruct seismic velocities and densities using laboratory measurements of elastic and thermal properties of the mantle minerals, and compare the results with seismological observations and models.
In a first stage, we aimed to constrain the average temperature and composition of the mantle. The later are of primary interest to infer mantle dynamics, but surprisingly little is known about it. We aim to constrain the lower mantle geotherm and average composition from 1D seismic models and experimental mineralogy data, explicitely accounting for possible sources of uncertainty. We employ a third order Birch-Murnaghan equation of state, and the modelling is in excellent agreement with recent ab initio calculations of density and bulk modulus for perovskite. Modelling the shear modulus is not as straightforward, but is needed because density and the bulk modulus alone do not sufficiently constrain temperature and composition. To correctly predict ab initio calculations for the shear modulus of Mg-perovskite, we needed to prescribe a cross-derivative, which we found by forward modelling. Special care is taken to assess uncertainties in our modelling, and we find uncertainties up to 260 K for temperature, 7.5% for perovskite and 2.6% for iron. Within these uncertainties, a strong (>2%) chemical density contrast in the mid-mantle is very unlikely, whereas a doming regime is possible. A purely adiabatic temperature profile can probably be excluded, but it is difficult to infer the number and location(s) of the non-adiabatic increase(s), D" being the most likely candidate. Major sources of uncertainty are the trade-off between thermal and compositional effects, the possible influence of aluminium perovskite, and poorly understood frequency effects.
We then propose a new method to constrain lateral variations of temperature and composition in the lower mantle from global tomographic models of shear- and compressional-wave speed. We assume that the mantle consists of a mixture of perovskite and magnesio-wstite. In a first stage, we directly invert V_{P}- and V_{S}-anomalies for variations of temperature and composition, using the appropriate partial derivatives (or sensitivities) of velocities to temperature and composition. However, uncertainties in the tomographic models and in the sensitivities are such that variations in composition are completely unconstrained. Inferring deterministic distributions of temperature and composition being currently not possible, we turn to a statistical approach, which allows to infer several robust features. Comparison between synthetic and predicted ratios of the relative shear-to-compressional velocity anomalies indicate that the origin of seismic anomalies cannot be purely thermal, but do not constrain the amplitude of the variations of temperature and composition. We show that histograms of the relative V_{P}- and V_{S}-anomalies at a given depth are able to estimate these variations. We computed histograms for a large variety of cases and found that at the bottom of the mantle, variations in the volumic fraction of perovskite from -14% to 10% are essential to explain seismic tomography. In the mid mantle, anomalies of perovskite are not required, but moderate variations (up to 6%) can explain the observed distributions equally well. These trade-offs between anomalies of temperature and composition cannot be resolved by relative velocity anomalies alone. An accurate determination of temperature and composition requires the knowledge of density variations as well. We show that anomalies of iron can then also be resolved.
Seismic velocities depend simultaneously on temperature, composition and pressure. Therefore the interpretation of tomographic models is not straightforward. Relative anomalies of S-wave velocity may be useful to constrain the temperature distribution, but they are not sufficient to determine the chemical variations. To infer the latter, one needs an additional set of data, such as density anomalies. Relative V_{s}-anomalies are converted in density anomalies using the scaling factor z=dlnr /dlnV_{s}.
Recently, we inverted gravity data and a seismic tomography model for a radial model of scaling factor (Deschamps et al., 2001). Gravity and global Vs-anomalies are provided by the recent models egm96 (Lemoine et al., 1996) and s16rlbm (Woodhouse and Trampert, 1995), respectively. By introducing a-priori errors in the seismological model, we estimate the variance in z. The gravity kernels account for possible radial viscosity variations (Forte and Peltier, 1991). Calculations are made separately for oceanic and continental regions, and data are filtered for the spherical harmonic orders 11 to 16. The sub-continental and sub-oceanic models of z are significantly different: z has positive values until z=250 km (z=150 km) below continents (oceans), with a typical value of 0.04. This value is small compared to experimental mineralogy estimates. At depths greater than 350 km, z is not well-constrained.
To map anomalies of temperature (dT) and iron (dFe) in the uppermost mantle, one can then invert the scaling factor and the relative velocity anomalies (dV_{s}). The choice to invert for anomalies of iron rather than for anomalies of olivine is driven by the observation that densities and seismic velocities are more sensitive to global iron content than to olivine fraction. Applying this method to the global S-wave model S16RLBM (Woodhouse and Trampert, 1995), we find that the mantle below old cratons is significantly colder than the average mantle and depleted in iron.
Our radial model of scaling factor has small values compared to that predicted by mineral physics. Accounting for anelasticity reduces most, but not all, of the discrepancy. In particular thermal anomalies alone fail to explain negative values of z: for a given variation of temperature, the thermal derivatives of density and velocity have the same sign. On the other hand, if the rock is enriched (depleted) in iron, the density increases (decreases), whereas the velocity decreases (increases). Simultaneous variations of temperature and iron content can therefore lead to negative values of z.
We have then performed inversions for dT and dFe following equations (1). First, one may note that positive dV_{s} are associated with negative temperature variations and iron depletion. For instance, the temperature and iron anomalies associated to dV_{s}=3% and z=0.03 are about dT=-150K and dFe=-1.25%, respectively. To compute values of dT and dFe in the uppermost mantle down to 300 km, we have used the global S-wave model S16RLBM (Woodhouse and Trampert, 1995). It turns out that the distributions of temperature and iron anomalies are well correlated to the observed surface tectonic. Below oceans and tectonic continents, the mantle is nearly homogeneous i.e., the mean values of dT and dFe are close to zero. On the other hand, down to z=250 km old cratons are significantly colder than average mantle and depleted in iron: the mean values of dT and dFe reach -300K and -2.7%, respectively. This result is important since depletion in iron induces positive buoyancy that may balance the negative buoyancy induced by low temperatures.
Results from the Galileo mission have increased the scientific interest in knowledge the internal structure of large icy satellites (e.g., Ganymede, Callisto, Titan, ...). An important question is the possible existence of a subsurface ocean, as it has been suggested in the case of Europa. Evolution of large icy satellites is controlled by heat transfer across the outer ice I layer. After the core overturn, a possible structure consists of a silicate core and a shell of molten ices. As the satellite cools down the primordial ocean crystallizes, and a solid cap grows at its top. If this outer layer is thick enough, convection is very likely to occur in it. To estimate the vigor and the efficiency of convection in the outer ice I layer, we have used the results of a recent two-dimensional numerical model of convection including variable viscosity (see thermal convection in a temperature-dependent viscosity fluid), and we have investigated the influence of parameters such as the rheological properties of ice I and the composition of the initial ocean.
Physical parameters (radius R, and mean density r) have only a limited influence: variations of 500 km on R and 0.5 g/cm^{3} on r do not induce significant differences in the values of the Rayleigh number (Ra) and the surface heat flux (F). On the other hand, the efficiency of convection depends strongly on rheological (reference viscosity, i.e., the viscosity close to the melting point m_{0}, and activation energy E) and compositional parameters. The values of m_{0} and E are not well constrained. With currently accepted values of E and m_{0} close to 60 kJ/mol and 5�0^{13} Pa s, respectively, convection is efficient enough to complete the crystallization of the initial ocean in about 4 Gyr. Higher values of E and/or m_{0} reduce significantly the vigor and the efficiency of convection. The freezing of an internal liquid shell is therefore not a rapid geological event (<100 Myr) but may have occurred very recently in the history of large icy satellites. The composition of the primordial ocean is also very important. For instance, the presence of ammonia reduces the convective strength and the heat flux. Therefore, the composition of the primordial ocean may significantly influence the tectonic activity and the present-day upper structure of the large icy satellites. If ammonia was present in the primordial ocean, a residual subsurface ocean may have subsisted, as it is expected for Callisto. Internal heating induced by tidal dissipation probably plays an important role as well and could also be a way to maintain a subsurface ocean.
Heat transfer in the Earth's mantle is controlled by convection. A classical image of convection is Rayleigh-Bénard convection, which consists in an upper thermal boundary layer (TBL), where cold instabilities form and sink, and a lower TBL, where hot instabilities grow and rise. In both boundary layers, the heat is transferred by conduction. Obviously, convection in the Earth's mantle is far from Rayleigh-Bénard convection. Complexities include the spherical geometry, the mode of heating and the rock rheology. Viscosity controls the amount of deformation supported by a rock submitted to a given stress. Within planetary mantles, it varies by several orders of magnitude due to temperature and pressure variations. In this work, we studied the influence of a strongly temperature-dependent viscosity on the strength and efficiency of convection.
Variable viscosity convection has been studied experimentally as well as numerically during the last two decades. The flow pattern is highly sensitive to viscosity variations. For instance, if the viscosity is strongly temperature-dependent and convection is not too vigorous, a quasi-stagnant lid develops at the top of the fluid, and convection is confined in a sublayer. This regime is known as the conductive-lid regime. In the stagnant lid, heat is transported by conduction. Therefore the heat transfer across the whole fluid is less efficient than in the case of an isoviscous fluid. The convective sublayer consists of cold sinking downwelling, hot rising plumes, thermal boundary layers, and a nearly adiabatic well-mixed interior. The hot plumes and the lower TBL are present only if the fluid is heated from below.
For a volumetrically heated fluid there is only one TBL (at the top), and laboratory and numerical experiments show that the temperature difference across the TBL is proportional to a viscous temperature scale. Moreover, the behavior of the convective sublayer is similar to that of an isoviscous fluid. In this study, we considered the case of a bottom heated fluid, in which a second TBL is present at the bottom. We have first carried out numerical experiments for a fluid with a strongly temperature-dependent viscosity. For each experiment the conservative equations of momentum, mass, and energy were solved for a 2-D cartesian fluid, assuming the Boussinesq approximation and an infinite Prandtl number. The fluid is heated from below, and internal generation of energy is neglected. The viscous law is exponential.
Thermal boundary layer analysis shows that the stability of thermal boundary layers is different from the case of a volumetrically heated fluid. In the conductive-lid regime, convection is controlled by instabilities in the lower TBL. In addition, the convective sublayer cannot be considered as an isoviscous fluid. The thickness of the stagnant lid is smaller than that predicted by viscous temperature scale (as in the case of a volumetrically heated fluid). A possible explanation is that hot plumes interact with the stagnant lid, so that this lid is thermally eroded.
Based on our results, we also proposed scaling relation to compute two important parameters: the temperature difference across the lower TBL, and the surface heat flux. The temperature difference across the lower TBL is a simple function of the viscous temperature scale, and the heat flux is determined with the help of a scaling relation between the thermal boundary layer Rayleigh number and the core Rayleigh number.These laws provide a convenient way to model the thermal evolution of planetary mantles.