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SUMMARY The numerical simulations of convection inside the mantle of the Earth or of terrestrial planets have been based on approximate equations of fluid dynamics. A common approximation is the neglect of the inertia term which is certainly reasonable as the Reynolds number of silicate mantles, or their inverse Prandtl number, are infinitesimally small. However various other simplifications are made which we discuss in this paper. The crudest approximation that can be done is the Boussinesq approximation (BA) where the various parameters are constant and the variations of density are only included in the buoyancy term and assumed to be proportional to temperature with a constant thermal expansivity. The variations of density with pressure and the related physical consequences (mostly the presence of an adiabatic temperature gradient and of dissipation) are usually accounted for by using an anelastic approximation (AA) initially developed for astrophysical and atmospheric situations. The BA and AA cases provide simplified but self-consistent systems of differential equations. Intermediate approximations are also common in the geophysical literature although they are invariably associated with theoretical inconsistencies (non-conservation of total energy, non-conservation of statistically steady state heat flow with depth, momentum and entropy equations implying inconsistent dissipations). We show that, in the infinite Prandtl number case, solving the fully compressible (FC) equations of convection with a realistic equation of state (EoS) is however not much more difficult or numerically challenging than solving the approximate cases. We compare various statistical properties of the Boussinesq, AA and FC simulations in 2-D simulations. We point to an inconsistency of the AA approximation when the two heat capacities are assumed constant. We suggest that at high Rayleigh number, the profile of dissipation in a convective mantle can be directly related to the surface heat flux. Our results are mostly discussed in the framework of mantle convection but the EoS we used is flexible enough to be applied for convection in icy planets or in the inner core.

The core-mantle boundary (CMB) heat flux is an important variable of Earth's thermal evolution and dynamics. Seismic tomography enables access to seismic heterogeneities in the lower mantle, which can be related to present-day thermal heterogeneities. Alternatively, mantle convection models can be used to either infer the past CMB heat flux or to produce statistically realistic CMB heat flux distributions in self-consistent models. Mantle dynamics modifies the inertia tensor of the Earth, which implies a rotation of the Earth with respect to its rotation axis called True Polar Wander (TPW). This rotation has to be taken into account if mantle dynamics is to be linked to core dynamics. In this study, we explore the TPW and the CMB heat flux produced by a self-consistent mantle convection model. The geoid is also computed and investigated in order to determine the driving mechanism of TPW. This model includes continents, dense chemical piles at the bottom of the shell and plate-like behavior, providing the possibility to link TPW and the CMB heat flux with plate tectonics. A principal component analysis (PCA) of the CMB heat flux is computed to obtain the dominant heat flux patterns. The model shows a geoid dominated by upper mantle structures. Subduction zones and continents are correlated with positive geoid anomalies, about 20 times larger than the observed geoid anomalies. Chemical piles are mostly correlated with negative geoid anomalies because of the anti-correlation between the positions of subducting slabs and the piles. TPW thus tends to lock continents and subduction zones close to the equator, while chemical piles are shifted towards higher latitudes. The positive CMB heat flux anomalies are mostly located at low latitudes because of the equatorial slabs. The dominant heat flux patterns obtained by the PCA largely reflect the supercontinent cycle captured by the model, providing CMB heat flux patterns representative of the supercontinent formation and dispersal.

The existence of a high-pressure (HP) ice layer between the silicate core and the liquid ocean in large icy moons and ocean worlds is usually seen as a barrier to habitability, preventing a direct contact and therefore transfer of nutrients from the core to the liquid ocean (Figure 1). More recently, several studies [1-3] challenged that hypothesis and showed that exchanges were possible under specific conditions, allowing transport of salts toward the ocean. In the first part of our study, we considered an effect not taken into account in the previous works, which is the dynamical implications of the phase equilibrium at the ice-ocean interface allowing a non-zero vertical velocity at the surface of the HP ice layer. Convective stresses in the solid create a topography of the interface which can be erased by melting and freezing if flow on the liquid side is efficient. This effect, which can be modeled as a phase change boundary condition for the ice layer, has a significant impact on the flow dynamics and enables exchanges with the ocean by fusion and crystallization at the top interface of the HP ice layer, even without partial melting in the bulk of the ice layer [4]. For the same conditions as standard convective systems, it also leads to faster mass transfer in the bulk [5]. These exchanges are directly linked to the melting capacity of the ice at the interface between the HP ice layer and the core, depending on the efficiency of convection in the liquid ocean and quantified by a unique parameter, Φ. Considering this boundary condition at the interface between the HP ice layer and the liquid ocean, we propose a scaling of the bottom temperature and the top vertical velocity as function of the Rayleigh number (Figure 2), in the case of a fixed heat flux from the core, a rigid or free-slip bottom boundary and various values of the phase change parameter, Φ (Figure 1). In the interest of separating the problems, we first started with a model that does not include partial melting and compositional variations. But, in the case where the heat flux from the core would be sufficient to reach the melting temperature at the boundary between the core and the HP ice layer, and because the temperature at the interface is laterally variable, a thin film of melt, or localized melt pockets, could exist along the bottom of the ice shell. This melt, containing salts from the core, could be transported into the ocean through the ascending hot plumes into the HP ice layer by successive melting and refreezing episodes. Then, we performed preliminary calculations based on the results of this first study, applied to Ganymede, and the conclusion is that in several conditions melt occurs at the bottom boundary and even in the bulk of the HP ice layer. Then, the partial melting and two-phase convection into the HP ice layer will be considered in further studies. The second part of this study is to include impurities into the HP ice layer in order to study the ability of salt transfer by convection in the solid. Salts can enter the ice layer by diffusion, a very inefficient process. Otherwise, if melting occurs at the bottom of the ice layer, as mentioned above, liquid water can get enriched in salt and, since water is lighter than high pressure ice, it can rise by porous flow or fracking and refreeze. This results in a high concentration of salts in a layer of ice some height above the bottom boundary. This ice is denser than pure ice at the same temperature and can impede convection depending on the ratio of density increase associated to salts compared to the decrease associated to temperature, a ratio called buoyancy number Bi. We study this problem using Lagrangian tracers in the convection code and adding a flux of salts in a layer at the bottom of the ice layer. We study the effects of the buoyancy number on the heat and salt transfer efficiency in the layer, and the implications for the evolution of salt concentration in the ocean. Figure 1 - Model illustration for the interior of an ocean world with an HP ice layer. Figure 2 - Left panel shows the dimensionless radial velocity at the top boundary of the HP ice layer. Right panel shows the dimensionless Nusselt number. Both in function of the Rayleigh Number and for various values of Φ. References: [1] G. Choblet, G. Tobie, C. Sotin, K. Kalousová, O. Grasset (2017). Heat transport in the high-pressure ice mantle of large icy moons. Icarus, 285, 252-262 [2] K. Kalousová, C. Sotin, G. Choblet, G. Tobie, O. Grasset (2018). Two-phase convection in Ganymede’s high-pressure ice layer — Implications for its geological evolution. Icarus, 299, 133-147 [3] K.Kalousová, C. Sotin (2018). Melting in High-Pressure Ice Layers of Large Ocean Worlds—Implications for Volatiles Transport. Geophys. Res. Lett., 45, 8096-8103. [4] Labrosse, S., Morison, A., Deguen, R., & Alboussière, T. (2018). Rayleigh–Bénard convection in a creeping solid with melting and freezing at either or both its horizontal boundaries. J. Fluid Mech., 846, 5–36. https://doi.org/10.1017/jfm.2018.258 [5] Agrusta, R., Morison, A., Labrosse, S., Deguen, R., Alboussière, T., Tackley, P. J., & Dubuffet, F. (2020). Mantle convection interacting with magma oceans. Geophys. J. Int, 220, 1878–1892. https://doi.org/10.1093/gji/ggz549

In the quest to understand the basic universal features of compressible convection, one would like to disentangle genuine consequences of compression from spatial variations of transport properties. In the present work, we consider a very peculiar equation of state, whereby entropy is solely dependent on density, so that a nearly isentropic fluid domain is nearly isochoric. Within this class of equations of state, there is a thermal adiabatic gradient and a key property of compressible convection is still present, namely its capacity to viscously dissipate a large fraction of the thermal energy involved, of the order of the well-named dissipation number. In a series of anelastic approximations, under the assumption of an infinite Prandtl number, the number of governing parameters can be brought down to two, the Rayleigh number and the dissipation number. This framework is proposed as a playground for compressible convection, an opportunity to extend the vast corpus of theoretical analyses on the Oberbeck-Boussinesq equations regarding stability, bifurcations or the determination of upper bounds for the turbulent heat transfer. Here, in a two-dimensional geometry, we concentrate on the structure of upward and downward plumes depending on the dissipation number, on the heat flux dependence on the dissipation number and on the ratio of dissipation to convective heat flux. For dissipation numbers of order unity, in the limit of large Rayleigh numbers, dissipation becomes related to the entropy heat flux at each depth, so that the vertical dissipation profile can be predicted, and consequently so does the total ratio of dissipation to convective heat flux., 42 pages, 17 figures

Earth’s core is composed of iron (Fe) alloyed with light elements, e.g., silicon (Si). Its thermal conductivity critically affects Earth’s thermal structure, evolution, and dynamics, as it controls the magnitude of thermal and compositional sources required to sustain a geodynamo over Earth’s history. Here we directly measured thermal conductivities of solid Fe and Fe–Si alloys up to 144 GPa and 3300 K. 15 at% Si alloyed in Fe substantially reduces its conductivity by about 2 folds at 132 GPa and 3000 K. An outer core with 15 at% Si would have a conductivity of about 20 W m−1 K−1, lower than pure Fe at similar pressure–temperature conditions. This suggests a lower minimum heat flow, around 3 TW, across the core–mantle boundary than previously expected, and thus less thermal energy needed to operate the geodynamo. Our results provide key constraints on inner core age that could be older than two billion-years., Thermal conductivity of Earth’s core affects Earth’s thermal structure, evolution and dynamics. Based on thermal conductivity measurements of iron–silicon alloys at high pressure and temperature conditions, the authors here propose Earth’s inner core could be older than previously expected.

We developed a numerical method for the set of equations governing fully compressible convection in the limit of infinite Prandtl numbers. Reduced models have also been analysed, such as the anelastic approximation and the anelastic liquid approximation. The tests of our numerical schemes against self-consistent criteria have shown that our numerical simulations are consistent from the point of view of energy dissipation, heat transfer and entropy budget. The equation of state of an ideal gas has been considered in this work. Specific effects arising because of the compressibility of the fluid are studied, like the scaling of viscous dissipation and the scaling of the heat flux contribution due to the mechanical power exerted by viscous forces. We analysed the solutions obtained with each model (fully compressible model, anelastic and anelastic liquid approximations) in a wide range of dimensionless parameters and determined the errors induced by each approximation with respect to the fully compressible solutions. Based on a rationale on the development of the thermal boundary layers, we can explain reasonably well the differences between the fully compressible and anelastic models, in terms of both the heat transfer and viscous dissipation dependence on compressibility. This could be mostly an effect of density variations on thermal diffusivity. Based on the different forms of entropy balance between exact and anelastic models, we find that a necessary condition for convergence of the anelastic results to the exact solutions is that the product $\unicode[STIX]{x1D716}q$ must be small compared to unity, where $\unicode[STIX]{x1D716}$ is the ratio of the superadiabatic temperature difference to the adiabatic difference, and $q$ is the ratio of the superadiabatic heat flux to the heat flux conducted along the adiabat. The same condition seems also to be associated with a convergence of the computed heat fluxes. Concerning the anelastic liquid approximation, we confirm previous estimates by Anufriev et al. (Phys. Earth Planet. Inter., vol. 152, 2005, pp. 163–190) and find that its results become generally close to those of the fully compressible model when $\unicode[STIX]{x1D6FC}T{\mathcal{D}}$ is small compared to unity, where $\unicode[STIX]{x1D6FC}$ is the isobaric thermal expansion coefficient, $T$ is the temperature (here $\unicode[STIX]{x1D6FC}T=1$ for an ideal gas) and ${\mathcal{D}}$ is the dissipation number.

International audience; Sputnik Planitia is a nitrogen-ice-filled basin on Pluto1. Its polygonal surface patterns2 have been previously explained as a result of solid-state convection with either an imposed heat flow3 or a temperature difference within the 10-km-thick ice layer4. Neither explanation is satisfactory, because they do not exhibit surface topography with the observed pattern: flat polygons delimited by narrow troughs5. Internal heating produces the observed patterns6, but the heating source in such a setup remains enigmatic. Here we report the results of modelling the effects of sublimation at the surface. We find that sublimation-driven convection readily produces the observed polygonal structures if we assume a smaller heat flux (~0.3 mW m−2) at the base of the ice layer than the commonly accepted value of 2-3 mW m−2 (ref. 7). Sustaining this regime with the latter value is also possible, but would require a stronger viscosity contrast (~3,000) than the nominal value (~100) considered in this study.

The early evolution of the Earth was likely affected by a large scale magma ocean, in particular in the aftermath of the giant impact that formed the Moon. The exact structure and dynamics of the Earth following that event is unknown but several possible scenarios feature the existence of a basal magma ocean (BMO), whose last remaining drops may explain the current seismically detected ultra low velocity zones. The presence of a BMO covering the core carries many implications for the dynamics and evolution of the overlying solid mantle. The phase equilibrium between the magma and the solid mantle allows matter to flow through the boundary by melting and freezing. In practice, convective stresses in the solid create a topography of the interface which displaces the equilibrium. Heat and solute transfer in the liquid acts to erase this topography and, if this process is faster than that the producing topography, the boundary appears effectively permeable to flow. This leads to convective motions much faster than in usual mantle convection. We developed a mantle convection model coupled to a model for the thermal and compositional evolution of the BMO and the core that takes into account the phase equilibrium at the bottom of the solid mantle. It also includes the fractional crystallisation at the interface and net freezing of the magma ocean. Early in the history, convection in the mantle is very fast and dominated by down-welling currents. As fractional crystallisation proceeds, the magma ocean gets enriched in FeO which makes the cumulate to also get richer. Eventually, it becomes too dense to get entrained by mantle convection and starts to pile up at the bottom of the mantle, which inhibits direct mass flow through the phase change boundary. This allows a thermal boundary layer and hot plumes to develop.This model therefore allows to explain the present existence of both residual partial melt and large scale compositional variations in the lower mantle, as evidenced by seismic velocity anomalies. It also predicts a regime change between early mantle convection dominated by down-welling flow to the onset of hot plumes in the more recent past.

The light plagioclase-enriched crust as well as the KREEP layer at the surface of the Moon are believed to be remnants of the bottom-up crystallization of a global Lunar Magma Ocean. In such a setup, the primitive Lunar solid mantle is coated by a liquid magma ocean of similar composition. We propose here to study the dynamic and evolution of the primitive Lunar solid mantle, accounting for the presence of the Lunar Magma Ocean.We solve numerically the equations of solid-state convection in the solid part of the mantle. This model is coupled to 1D models of crystallization of the magma oceans to self-consistently compute the thickening of the solid part as heat is evacuated from the mantle. We take into account fractional crystallization at the freezing front.Moreover, the boundaries between the solid and the magma oceans are phase-change interfaces. Convecting matter in the solid arriving near the boundary or getting away from it forms a topography which can be erased by melting or freezing. Hence, provided the melting and freezing occurs rapidly compared to the time needed to build the topographies by viscous forces, dynamical exchange of matter can occur between the solid mantle and the magma oceans. We take this effect into account in our model with a boundary condition applied to the solid.We find that the boundary condition allowing matter to cross the interfaces between the solid and the magma oceans greatly affects the convection patterns in the solid as well as its heat flux. Larger-scale convection patterns are selected compared to the classical case with non-penetrative boundary conditions; and the heat transfert in the solid is more efficient with these boundary conditions. This affects the long term thermal evolution of the mantle as well as the shape of chemical heterogeneities that can be built by fractional crystallization of magma oceans.

The existence of a high pressure ice layer between the silicate core and the liquid ocean in large icy moons and ocean worlds is usually seen as a barrier to habitability, preventing the compounds needed for life to flow into the ocean. More recently, three studies from Choblet et al [1] and Kalousová et al [2, 3] challenged that hypothesis and showed that, in certain conditions, exchanges were possible between the core and the ocean, allowing transport of salts toward the ocean. Here, we consider an effect not taken into account in these previous studies: the possibility of mass exchange between the ice and ocean layers by phase change. Convective stresses in the solid create a topography of the interface which can be erased by melting and freezing if flow on the liquid side is efficient. This effect is included in a convection model as a phase change boundary condition, allowing a non-zero vertical velocity at the surface of the HP ice layer, which has a significant impact on the flow dynamics and enables exchanges with the ocean by fusion and crystallization of the ice at the top interface, even without partial melting in the bulk of the ice layer. These exchanges are directly linked to the melting capacity of the ice at the interface between the HP ice layer and the core, depending on the Rayleigh number and the efficiency of convection. Then, considering this new condition at the interface between the HP ice layer and the liquid ocean, we propose a scaling of the bottom temperature and the vertical velocity. Applied to a specific celestial body, as Ganymede or Titan, it would be the first step to conclude about its habitability. References:[1] G. Choblet, G. Tobie, C. Sotin, K. Kalousová, O. Grasset (2017). Heat transport in the high-pressure ice mantle of large icy moons. Icarus, 285, 252-262[2] K. Kalousová, C. Sotin, G. Choblet, G. Tobie, O. Grasset (2018). Two-phase convection in Ganymede’s high-pressure ice layer — Implications for its geological evolution. Icarus, 299, 133-147[3] K.Kalousová, C. Sotin (2018). Melting in High-Pressure Ice Layers of Large Ocean Worlds—Implications for Volatiles Transport. Geophys. Res. Lett., 45, 8096-8103.

International audience; The separation of a liquid phase from a solid but deformable matrix made of mineral grains is controlled at small scale by surface tension. The role of interfacial surface tension is twofold as it explains how a small volume of liquid phase can infiltrate the grain boundaries, be distributed and absorbed in the matrix, but after complete wetting of the grains, surface tension favors the self-separation of the liquid and solid phases. Another consequence of surface tension is the existence of Marangoni forces, which are related to the gradients of surface tension that are are usually due to temperature variations. In this paper, using a continuous multiphase formalism we clarify the role of these different effects and quantify their importances at the scale of laboratory experiments and in planets. We show that Marangoni forces can control the liquid metal-solid silicate phase separation in laboratory experiments. The Marangoni force might help to maintain the presence of metal at the surface of asteroids and planetesimals that have undergone significant melting.

International audience; Convection in a spherical shell is widely used to model fluid layers of planets and stars. The choice of thermal boundary conditions in such models is not always straightforward. To understand the implications of this choice, we report on the effects of the thermal boundary condition on thermal convection, in terms of instability onset, fully developed transport properties and flow structure. We use the Boussinesq approximation, and impose a Robin boundary condition at the top. This enforces the temperature anomaly and its radial derivative to be linearly coupled with a proportionality factor β. Using the height H of the fluid layer, we introduce the non-dimensional Biot number Bi* = βH. Varying Bi* allows us to transition from fixed temperature for Bi* = +∞, to fixed thermal flux for Bi* = 0. The bottom boundary of the shell is kept isothermal. We find that the onset of convection is only affected by Bi* in the non-rotating case. Far from onset, considering an effective Rayleigh number and a generalized Nusselt number, we show that the Nusselt and Péclet numbers follow standard universal scaling laws, independent of Bi* in all cases considered. However, the large-scale flow structure keeps the signature of the boundary condition with more vigorous large scales for smaller Bi* , even though the global heat transfer and kinetic energy are the same. Finally, for all practical purposes, the Robin condition can be safely replaced by a fixed flux when Bi* < 0.03 and by a fixed temperature for Bi* > 30.

After accretion and formation, terrestrial planets go through at least one magma ocean episode. As the magma ocean crystallises, it creates the first layer of solid rocky mantle. Two different scenarios of magma ocean crystallisation involve that the solid mantle either (1) first appears at the core-mantle boundary and grows upwards, or (2) appears at mid-mantle depth and grows in both directions. Regardless of the magma ocean freezing scenario, the composition of the solid mantle and liquid reservoirs continuously change due to fractional crystallisation. This chemical fractionation has important implications for the long-term thermo-chemical evolution of the mantle, as well as its present-day dynamics and composition. In this work we use numerical models to study convection in a solid mantle bounded at either or both boundaries by magma ocean(s), and in particular, the related consequences for large-scale chemical fractionation. We use a parameterisation of fractional crystallisation of the magma ocean(s) and (re-)melting of solid material at the interface between these reservoirs. When these crystallisation/re-melting processes are taken into account, convection in the solid mantle occurs readily and is dominated by large wavelengths. Related material transfer across the mantle magma-ocean boundaries promotes chemical equilibrium, and prevents extreme enrichment of the last-stage magma ocean (as would otherwise occur due to pure fractional crystallisation). The timescale of equilibration depends on the convective vigour of mantle convection and on the efficiency of material transfer between the solid mantle and magma ocean(s). For Earth, this timescale is comparable to that of magma ocean crystallisation suggested in previous studies (Lebrun et al., 2013), which may explain why the Earth's mantle is rather homogeneous in composition, as supported by geophysical constraints.

Several studies suggest that Mars went through an episode of Magma Ocean (MO) early in its history. When the MO crystallises, solid mantle appears. The crystallisation of this MO starts at the Core-Mantle Boundary (CMB) and continues upwards to the surface of the planet. Assuming that this process occurs by fractional crystallisation, the solid cumulates that form are progressively enriched in incompatible elements, including iron, and an unstable density stratification is developed. This stratification is thought to have resulted in a planetary-scale mantle overturn after MO crystallisation, potentially explaining the early magnetic field, crustal dichotomy and chemical heterogeneities present on martian mantle.However, previous studies on the thermo-chemical evolution of Mars consider only fractional crystallisation of the MO, and lack the possibility of re-melting/re-freezing of material at the mantle-MO interface, before the MO is fully crystallised.In this study we investigate the effect of re-melting/re-freezing of material at the mantle-MO interface during MO crystallisation, on the dynamics and composition of the solid mantle. We use a numerical method with the convection code StagYY. The solid mantle is represented by a 2D spherical annulus geometry, and the MO by a 0D object at top of the mantle. The boundary condition applied to the solid domain allows the parameterisation of fractional crystallisation/re-melting of material at the mantle-MO interface. We model the growth of the solid mantle from the CMB up to the surface of the planet, and we account for core cooling and the presence of an atmosphere.We show that by taking re-melting/re-freezing of material into account, the onset of convection can start earlier in Mars history. These results bring implications for the density stratification and overturn, and to the existence of isotopically distinct reservoirs on the mantle. Moreover, our results show that the mode of convection is preferentially degree-1, which can potentially explain the crustal dichotomy.

International audience; Many planetary bodies contain internal liquid layers in their metallic cores or as buried water oceans. Convection in these layers is usually driven by buoyancy sources of thermal or compositional origin, with very different molecular diffusivities. Such conditions can potentially trigger double-diffusive instabilities and fundamentally affect the convective features. In numerical models, the weak diffusivity of the compositional field requires the use of a semi-Lagrangian description to produce minimal numerical diffusion. We implemented a "particle-in-cell" (PIC) method into a pre-existing geodynamo code in 3D spherical geometry to describe the compositional field properly. We developed several numerical strategies to solve various problems inherent to the implementation of a PIC method for convection in spherical geometry and coded a hybrid scheme suitable for massively parallel platforms. We tested our new code on two benchmark cases which validate its applicability to the study of double-diffusive convection in the internal liquid layers of planets. As a first application, we study a case of non-magnetic double-diffusive convection at infinite Lewis number. Major differences emerge both in the compositional field and the convective pattern when the compositional diffusivity is neglected. (C) 2017 Elsevier Inc. All rights reserved.

SUMMARY The presence of a magma ocean may have characterized the beginning of terrestrial planets and, depending on how the solidification has proceeded, the solid mantle may have been in contact with a magma ocean at its upper boundary, its lower boundary, or both, for some period of time. At the interface where the solid is in contact with the liquid the matter can flow through by changing phase, and this affects convection in the solid during magma ocean crystallization. Linear and weakly non-linear analyses have shown that Rayleigh–Bénard flow subject to two liquid–solid phase change boundary conditions is characterized by a non-deforming translation or weakly deforming long wavelength mode at relatively low Rayleigh number. Both modes are expected to transfer heat very efficiently, at least in the range of applicability of weakly non-linear results for the deforming mode. When only one boundary is a phase change, the critical Rayleigh number is also reduced, by a factor of about 4, and the heat transfer is also greatly increased. In this study we use direct numerical simulations in 2-D Cartesian geometry to explore how the solid convection may be affected by these boundary conditions for values of the Rayleigh number extending beyond the range of validity of the weakly non-linear results, up to 103 times the critical value. Our results suggest that solid-state convection during magma ocean crystallization may have been characterized by a very efficient mass and heat transfer, with a heat flow and velocity at the least twice the value previously thought when only one magma ocean is present, above or below. In the situation with a magma ocean above and below, we show that the convective heat flow through the solid layer could reach values of the same order as that of the black-body radiation at the surface of the magma ocean.

International audience; The formation and differentiation of planetary bodies are thought to involve magma oceans stages. We study the case of a planetary mantle crystallizing upwards from a global magma ocean. In this scenario, it is often considered that the magma ocean crystallizes more rapidly than the time required for convection to develop in the solid cumulate. This assumption is appealing since the temperature and composition profiles resulting from the crystallization of the magma ocean can be used as an initial condition for convection in the solid part. We test here this assumption with a linear stability analysis of the density profile in the solid cumulate as crystallization proceeds. The interface between the magma ocean and the solid is a phase change interface. Convecting matter arriving near the interface can therefore cross this boundary via melting or freezing. We use a semi-permeable condition at the boundary between the magma ocean and the solid to account for that phenomenon. The timescale with which convection develops in the solid is found to be several orders of magnitude smaller than the time needed to crystallize the magma ocean as soon as a few hundreds kilometers of cumulate are formed on a Mars-to Earth-size planet. The phase change boundary condition is found to decrease this timescale by several orders of magnitude. For a Moon-size object, the possibility of melting and freezing at the top of the cumulate allows the overturn to happen before complete crystallization. The convective patterns are also affected by melting and freezing at the boundary: the linearly most-unstable mode is a degree-1 translation mode instead of the approximately aspect-ratio-one convection rolls found with classical non-penetrative boundary conditions. The first overturn of the crystallizing cumulate on Mars and the Moon could therefore be at the origin of their observed degree-1 features.

International audience; Convection in the Earth's outer core is driven by buoyancy sources of both thermal and compositional origin. The thermal and compositional molecular diffusivities differ by several orders of magnitude, which can affect the dynamics in various ways. So far, the large majority of numerical simulations have been performed within the codensity framework that consists in combining temperature and composition, assuming artificially enhanced diffusivities for both variables. In this study, we use a particle-in-cell method implemented in a 3D dynamo code to conduct a first qualitative exploration of pure compositional convection in a rotating spherical shell. We focus on the end-member case of infinite Schmidt number by totally neglecting the compositional diffusivity. We show that compositional convection has a very rich physics that deserves several more focused and quantitative studies. We also report, for the first time in numerical simulations, the self-consistent formation of a chemically stratified layer at the top of the shell caused by the accumulation of chemical plumes and blobs emitted at the bottom boundary. When applied to likely numbers for the Earth's core, some (possibly simplistic) physical considerations suggest that a stratified layer formed in such a scenario would be probably weakly stratified and may be compatible with magnetic observations.

An experiment of thermal convection with significant compressible effects is presented. The high-gravity environment of a centrifuge and the choice of xenon gas enable us to observe an average adiabatic temperature gradient up to 3.5 K cm$^{-1}$ over a 4 cm high cavity. At the highest rotation rate investigated, 9990 rpm, the superadiabatic temperature difference applied to the gas layer is less than the adiabatic temperature difference. The convective regime is characterized by a large Rayleigh number, about 10$^{12}$, and dominant Coriolis forces (Ekman number of order 10$^{-6}$). The analysis of temperature and pressure fluctuations in our experiments shows that the dynamics of the flow is in a quasi-geostrophic regime. Still, a classical power law (exponent 0.3 $\pm$ 0.04) is observed between the Nusselt number (dimensionless heat flux) and the superadiabatic Rayleigh number (dimensionless superadiabatic temperature difference). However, a potential hysteresis is seen between this classical high flux regime and a lower heat flux regime. It is unclear whether this is due to compressible or Coriolis effects. In the transient regime of convection from an isothermal state, we observe a local decrease of temperature which can only be explained by adiabatic decompression.

Earth's similar to 3.45 billion year old magnetic field is regenerated by dynamo action in its convecting liquid metal outer core. However, convection induces an isentropic thermal gradient which, coupled with a high core thermal conductivity, results in rapid conducted heat loss. In the absence of implausibly high radioactivity or alternate sources of motion to drive the geodynamo, the Earth's early core had to be significantly hotter than the melting point of the lower mantle. While the existence of a dense convecting basal magma ocean (BMO) has been proposed to account for high early core temperatures, the requisite physical and chemical properties for a BMO remain controversial. Here we relax the assumption of a well mixed convecting BMO and instead consider a BMO that is initially gravitationally stratified owing to processes such as mixing between metals and silicates at high temperatures in the core-mantle boundary region during Earth's accretion. Using coupled models of crystallization and heat transfer through a stratified BMO, we show that very high temperatures could have been trapped inside the early core, sequestering enough heat energy to run an ancient geodynamo on cooling power alone. (C) 2017 Elsevier B.V. All rights reserved.

International audience; The hemispherical asymmetry of the inner core has been interpreted as resulting from a high-viscosity mode of inner core convection, consisting in a translation of the inner core. A thermally driven translation, as originally proposed, is unlikely if the currently favoured high values of the thermal conductivity of iron at core conditions are correct. We consider here the possibility that inner core translation results from an unstable compositional gradient, which would develop either because the light elements present in the core become increasingly incompatible as the inner core grows, or because of a possibly positive feedback of the development of the F-layer on inner core convection. Though themagnitude of the destabilizing effect of the compositional field is predicted to be similar to or smaller than the stabilizing effect of the thermal field, the huge difference between thermal and chemical diffusivities implies that double-diffusive instabilities can still arise even if the net buoyancy increases upward. Using linear stability analysis and numerical simulations, we demonstrate that a translation mode can indeed exist if the compositional field is destabilizing, even if the temperature profile is subadiabatic, and irrespectively of the relative magnitudes of the composition and potential temperature gradients. The existence of this double diffusive mode of translation requires that the following conditions are met: (i) the compositional profile within the inner core is destabilizing, and remains so for a duration longer than the destabilization timescale (on the order of 200 Myr, but strongly dependent on the magnitude of the initial perturbation); and (ii) the inner core viscosity is sufficiently large, the required value being a strongly increasing function of the inner core size (e. g. 10(17) Pa s when the inner core was 200 km in radius, and similar or equal to 3 x 10(21) Pa s at the current inner core size). If these conditions are met, the predicted inner core translation rate is found to be similar to the inner core growth rate, which is more consistent with inferences from the geomagnetic field morphology and secular variation than the higher translation rate predicted for a thermally driven translation.

International audience; The rate at which heat is extracted across the core mantle boundary (CMB) is constrained by the requirement of dynamo action in the core. This constraint can be computed explicitly using the entropy balance of the core and depends on the thermal conductivity, whose value has been revised upwardly. A high order model (fourth degree polynomial of the radial position) for the core structure is derived and the implications for the core cooling rate and thermal evolution obtained, using the recent values of the thermal conductivity. For a thermal conductivity increasing with depth as proposed by some of these recent studies, a CMB heat flow equal to the isentropic value (13.25TW at present) leads to a 700 km thick layer at the top of the core where a downward convective heat flow is necessary to maintain an isentropic and well mixed average state. Considering a CMB heat flow larger than the well mixed isentropic value leads to an inner core less than 700 Myr old and the thermal evolution of the core is largely constrained by the conditions for dynamo action without an inner core. Analytical calculations for that period show that a CMB temperature larger than 7000 K must have prevailed 4.5 Gyr ago if the geodynamo has been driven by thermal convection for that whole time. This raises questions regarding the onset of the geodynamo and its continuous operation for the last 3.5 Gyr. Implications regarding the evolution of a basal magma ocean are also considered.

Solid state convection can take place in the rocky or icy mantles of planetary objects and these mantles can be surrounded above or below or both by molten layers of similar composition. A flow toward the interface can proceed through it by changing phase. This behaviour is modeled by a boundary condition taking into account the competition between viscous stress in the solid, that builds topography of the interface with a timescale $\tau_\eta$, and convective transfer of the latent heat in the liquid from places of the boundary where freezing occurs to places of melting, which acts to erase topography, with a timescale $\tau_\phi$. The ratio $\Phi=\tau_\phi/\tau_\eta$ controls whether the boundary condition is the classical non-penetrative one ($\Phi\rightarrow \infty$) or allows for a finite flow through the boundary (small $\Phi$). We study Rayleigh-B\'enard convection in a plane layer subject to this boundary condition at either or both its boundaries using linear and weakly non-linear analyses. When both boundaries are phase change interfaces with equal values of $\Phi$, a non-deforming translation mode is possible with a critical Rayleigh number equal to $24\Phi$. At small values of $\Phi$, this mode competes with a weakly deforming mode having a slightly lower critical Rayleigh number and a very long wavelength, $\lambda_c\sim 8\sqrt{2}\pi/ 3\sqrt{\Phi}$. Both modes lead to very efficient heat transfer, as expressed by the relationship between the Nusselt and Rayleigh numbers. When only one boundary is subject to a phase change condition, the critical Rayleigh number is $\Ray_c=153$ and the critical wavelength is $\lambda_c=5$. The Nusselt number increases about twice faster with Rayleigh number than in the classical case with non-penetrative conditions when the bottom boundary is a phase change interface., Comment: Submitted to the Journal of Fluid Mechanics. 28 pages, 16 figures

Melting experiments with liquid Fe–Si–O alloy at the pressure of the Earth’s core reveal that the crystallization of silicon dioxide leads to core convection and a dynamo. The Earth's core contains large amounts of iron (Fe), but its density, about ten per cent less than that of pure iron, indicates the presence of lighter elements in the outer core, potentially including silicon (Si) and oxygen (O). To simulate the early Earth, Kei Hirose and co-authors present melting experiments on liquid Fe–Si–O alloy at the pressures of the Earth's core in a laser-heated diamond-anvil cell. They find that an initial Fe–Si–O core would be able to crystallize silicon dioxide (SiO2) as it cools. The authors conclude that if crystallization proceeds from the top of the core, the sinking of SiO2-depleted Fe–Si–O liquid would have been more than enough to power core convection and a dynamo in the early Earth. The Earth’s core is about ten per cent less dense than pure iron (Fe), suggesting that it contains light elements as well as iron. Modelling of core formation at high pressure (around 40–60 gigapascals) and high temperature (about 3,500 kelvin) in a deep magma ocean1,2,3,4,5 predicts that both silicon (Si) and oxygen (O) are among the impurities in the liquid outer core6,7,8,9. However, only the binary systems Fe–Si and Fe–O have been studied in detail at high pressures, and little is known about the compositional evolution of the Fe–Si–O ternary alloy under core conditions. Here we performed melting experiments on liquid Fe–Si–O alloy at core pressures in a laser-heated diamond-anvil cell. Our results demonstrate that the liquidus field of silicon dioxide (SiO2) is unexpectedly wide at the iron-rich portion of the Fe–Si–O ternary, such that an initial Fe–Si–O core crystallizes SiO2 as it cools. If crystallization proceeds on top of the core, the buoyancy released should have been more than sufficient to power core convection and a dynamo, in spite of high thermal conductivity10,11, from as early on as the Hadean eon12. SiO2 saturation also sets limits on silicon and oxygen concentrations in the present-day outer core.

The improvements of the knowledge of the seismic structure of the inner core and the complexities thereby revealed ask for a dynamical origin. Sub-solidus convection was one of the early suggestions to explain the seismic anisotropy, but it requires an unstable density gradient either from thermal or compositional origin, or from both. Temperature and composition profiles in the inner core are computed using a unidimensional model of core evolution including diffusion in the inner core and fractional crystallisation at the inner core boundary (ICB). The thermal conductivity of the core has been recently revised upwardly and, moreover, found to increase with depth. Values of the heat flow across the core mantle boundary (CMB) sufficient to maintain convection in the whole outer core are not sufficient to make the temperature in the inner core super-isentropic and therefore prone to thermal instability. An unreasonably high CMB heat flow is necessary to this end. The compositional stratification results from a competition of the increase of the concentration of light elements in the outer core with inner core growth, which makes the inner core concentration also increase, and of the decrease of the liquidus, which makes the partition coefficient decrease as well as the concentration of light elements in the solid. While the latter (destabilizing) effect dominates at small inner core sizes, the former takes over for a large inner core. The turnover point is encountered for an inner core about half its current size in the case of S, but much larger for the case of O. The combined thermal and compositional buoyancy is stabilizing and solid-state convection in the inner core appears unlikely, unless an early double-diffusive instability can set in., 20 pages, 5 figures

We measured the electrical resistivity of iron and iron-silicon alloy to 100 GPa. The resistivity of iron was also calculated to core pressures. Combined with the first geophysical model accounting for saturation resistivity of core metal, the present results show that the thermal conductivity of the outermost core is greater than 90 W/m/K. These values are significantly higher than conventional estimates, implying rapid secular core cooling, an inner core younger than 1 Ga, and ubiquitous melting of the lowermost mantle during the early Earth. An enhanced conductivity with depth suppresses convection in the deep core, such that its center may have been stably stratified prior to the onset of inner core crystallization. A present heat flow in excess of 10 TW is likely required to explain the observed dynamo characteristics.

The composition and state of Earth's core, located deeper than 2,900 km from the surface, remain largely uncertain. Recent static experiments on iron and alloys performed up to inner core pressure and temperature conditions have revealed phase relations and properties of core materials. These mineral physics constraints, combined with theoretical calculations, continue to improve our understanding of the core, in particular the crystal structure of the inner core and the chemical composition, thermal structure and evolution, and possible stratification of the outer core.

International audience; Crystallization experiments in the dendritic regime have been carried out in hypergravity conditions (from 1 to 1300 g) from an ammonium chloride solution (NH4Cl and H2O). A commercial centrifuge was equipped with a slip ring so that electric power (needed for a Peltier device and a heating element), temperature and ultrasonic signals could be transmitted between the experimental setup and the laboratory. Ultrasound measurements (2-6 MHz) were used to detect the position of the front of the mushy zone and to determine attenuation in the mush. Temperature measurements were used to control a Peltier element extracting heat from the bottom of the setup and to monitor the evolution of crystallization in the mush and in the liquid. A significant increase of solid fraction and attenuation in the mush is observed as gravity is increased. Kinetic undercooling is significant in our experiments and has been included in a macroscopic mush model. The other ingredients of the model are conservation of energy and chemical species, along with heat/species transfer between the mush and the liquid phase: boundary-layer exchanges at the top of the mush and bulk convection within the mush (formation of chimneys). The outputs of the model compare well with our experiments. We have then run the model in a range of parameters suitable for the Earth's inner core. This has shown the role of bulk mush convection for the inner core and the reason why a solid fraction very close to unity should be expected. We have also run melting experiments: after crystallization of a mush, the liquid has been heated from above until the mush started to melt, while the bottom cold temperature was maintained. These melting experiments were motivated by the possible local melting at the inner core boundary that has been invoked to explain the formation of the anomalously slow F-layer at the bottom of the outer core or inner core hemispherical asymmetry. Oddly, the consequences of melting are an increase in solid fraction and a decrease in attenuation. It is hence possible that surface seismic velocity and attenuation of the inner core are strongly affected by melting.

Old Plates and the Sea Estimates for the area and age of the ocean floor are at odds with assumptions for mantle convection, which imply that an older sea floor—rather than a new one—would be preferentially subducted over time. Previous efforts to explain these relationships have been based on geologic evidence and simple models. Coltice et al. (p. 335 ) created numerical three-dimensional convection models representing more realistic physical boundaries, including a spherical Earth, the existence of continents and supercontinents over time, and realistic rheologies. A combination of continents and plate-like behavior of the ocean floor sufficed to produce the observed relationship between plate area and plate age, which explains why some old oceanic crust still remains.

International audience; Of the terrestrial planets, Earth and Mercury have self-sustained fields while Mars and Venus do not. Magnetic field data recorded at Ganymede have been interpreted as evidence of a self-generated magnetic field. The other icy Galilean satellites have magnetic fields induced in their subsurface oceans while Io and the Saturnian satellite Titan apparently are lacking magnetic fields of internal origin altogether. Parts of the lunar crust are remanently magnetized as are parts of the crust of Mars. While it is widely accepted that the magnetization of the Martian crust has been caused by an early magnetic field, for the Moon alternative explanations link the magnetization to plasma generated by large impacts. The necessary conditions for a dynamo in the terrestrial planets and satellites are the existence of an iron-rich core that is undergoing intense fluid motion. It is widely accepted that the fluid motion is caused by convection driven either by thermal buoyancy or by chemical buoyancy or by both. The chemical buoyancy is released upon the growth of an inner core. The latter requires a light alloying element in the core that is enriched in the outer core as the solid inner core grows. In most models, the light alloying element is assumed to be sulfur, but other elements such as, e.g., oxygen, silicon, and hydrogen are possible. The existence of cores in the terrestrial planets is either proven beyond reasonable doubt (Earth, Mars, and Mercury) or the case for a core is compelling as for Venus and the Moon. The Galilean satellites Io and Ganymede are likely to have cores judging from Galileo radio tracking data of the gravity fields of these satellites. The case is less clear cut for Europa. Callisto is widely taken as undifferentiated or only partially differentiated, thereby lacking an iron-rich core. Whether or not Titan has a core is not known at the present time. The terrestrial planets that do have magnetic fields either have a well-established inner core with known radius and density such as Earth or are widely agreed to have an inner core such as Mercury. The absence of an inner core in Venus, Mars, and the Moon (terrestrial bodies that lack fields) is not as well established although considered likely. The composition of the Martian core may be close to the Fe-FeS eutectic which would prevent an inner core to grow as long as the core has not cooled to temperatures around 1500 Kelvin. Venus may be on the verge of growing an inner core in which case a chemical dynamo may begin to operate in the geologically near future. The remanent magnetization of the Martian and the lunar crust is evidence for a dynamo in Mars' and possibly the Moon's early evolution and suggests that powerful thermally driven dynamos are possible. Both the thermally and the chemically driven dynamo require that the core is cooled at a sufficient rate by the mantle. For the thermally driven dynamo, the heat flow from the core into the mantle must by larger than the heat conducted along the core adiabat to allow a convecting core. This threshold is a few mW m(-2) for small planets such as Mercury, Ganymede, and the Moon but can be as large as a few tens mW m(-2) for Earth and Venus. The buoyancy for both dynamos must be sufficiently strong to overcome Ohmic dissipation. On Earth, plate tectonics and mantle convection cool the core efficiently. Stagnant lid convection on Mars and Venus are less efficient to cool the core but it is possible and has been suggested that Mars had plate tectonics in its early evolution and that Venus has experiened episodic resurfacing and mantle turnover. Both may have had profound implications for the evolution of the cores of these planets. It is even possible that inner cores started to grow in Mars and Venus but that the growth was frustrated as the mantles heated following the cessation of plate tectonics and resurfacing. The generation of Ganymede's magnetic field is widely debated. Models range from magneto-hydrodynamic convection in which case the field will not be self-sustained to chemical and thermally-driven dynamos. The wide range of possible compositions for Ganymede's core allows models with a completely liquid near eutectic Fe-FeS composition as well as models with Fe inner cores or cores in with iron snowfall.

If a stable layer of dense melt formed at the base of the mantle early in Earth's history, it would have undergone slow fractional crystallization and could provide an unsampled geochemical reservoir hosting a variety of incompatible geochemical species (most notably the missing budget of heat producing elements). The distribution of geochemical species in the Earth’s interior is largely controlled by fractional melting and crystallization processes that are intimately linked to the thermal state and evolution of the mantle. The existence of patches of dense partial melt at the base of the Earth’s mantle1, together with estimates of melting temperatures for deep mantle phases2 and the amount of cooling of the underlying core required to maintain a geodynamo throughout much of the Earth’s history3, suggest that more extensive deep melting occurred in the past. Here we show that a stable layer of dense melt formed at the base of the mantle early in the Earth’s history would have undergone slow fractional crystallization, and would be an ideal candidate for an unsampled geochemical reservoir hosting a variety of incompatible species (most notably the missing budget of heat-producing elements) for an initial basal magma ocean thickness of about 1,000 km. Differences in 142Nd/144Nd ratios between chondrites and terrestrial rocks4 can be explained by fractional crystallization with a decay timescale of the order of 1 Gyr. These combined constraints yield thermal evolution models in which radiogenic heat production and latent heat exchange prevent early cooling of the core and possibly delay the onset of the geodynamo to 3.4–4 Gyr ago5.

Processes that operated in the early Earth have largely been erased or overprinted by subsequent evolution. However, some traces may persist in the deep Earth as imaged by seismology. Large‐scale features with reduced seismic velocities are simply explained as variations of composition, and small‐scale ultra‐low velocity zones are explained by the presence of Fe‐rich material that may be partially molten. Both can originate from fractional crystallization of an originally thick basal magma ocean (BMO). Many questions are raised by this scenario regarding properties of melts with various compositions, in particular the partition coefficients between melt and crystals of various elements and their relative densities. After reviewing recent progress on both the structure of the lower mantle and the mineral physics associated with partial melting/freezing of silicates at high pressure, we discuss several ways in which a BMO can be produced. We argue that, in most cases, independently of whether a melt of composition similar to that of the bulk mantle is more or less dense than crystals in equilibrium, the compositional evolution of both the magma and the solid should lead to the formation of a dense, Fe‐rich BMO whose subsequent slow evolution would explain some features of the present lower mantle. 1 Laboratoire de geologie de Lyon, ENS de Lyon, Universite Lyon-1, CNRS, Lyon, France 2 Earth‐Life Science Institute, Tokyo Institute of Technology, Meguro, Tokyo, Japan c07.indd 123 8/24/2015 4:02:55 PM

Seismic studies indicate that the Earth's inner core has a complex structure and exhibits a strong elastic anisotropy with a cylindrical symmetry. Among the various models which have been proposed to explain this anisotropy, one class of models considers the effect of the Lorentz force associated with the magnetic field diffused within the inner core. In this paper we extend previous studies and use analytical calculations and numerical simulations to predict the geometry and strength of the flow induced by the poloidal component of the Lorentz force in a neutrally or stably stratified growing inner core, exploring also the effect of different types of boundary conditions at the inner core boundary (ICB). Unlike previous studies, we show that the boundary condition that is most likely to produce a significant deformation and seismic anisotropy is impermeable, with negligible radial flow through the boundary. Exact analytical solutions are found in the case of a negligible effect of buoyancy forces in the inner core (neutral stratification), while numerical simulations are used to investigate the case of stable stratification. In this situation, the flow induced by the Lorentz force is found to be localized in a shear layer below the ICB, which thickness depends on the strength of the stratification, but not on the magnetic field strength. We obtain scaling laws for the thickness of this layer, as well as for the flow velocity and strain rate in this shear layer as a function of the control parameters, which include the magnitude of the magnetic field, the strength of the density stratification, the viscosity of the inner core, and the growth rate of the inner core. We find that the resulting strain rate is probably too small to produce significant texturing unless the inner core viscosity is smaller than about $10^{12}$ Pa.s., submitted to Geophysical Journal International

The magnetic field of the Earth is generated by convection in the liquid-core and the energy necessary for this process comes from the cooling of the core which provide several buoyancy sources. The thermodynamics of this system is used to relate the Ohmic dissipation in the core to all energy sources and to model the thermal evolution of the core. If the same dissipation is maintained just before the onset of inner-core crystallization, and the associated compositional convection, as at present, a much larger heat flow at the core mantle boundary (CMB) is necessary which, if extrapolated backward, may require a very high initial temperature. Two solutions to that problem are studied: either the Ohmic dissipation was smaller then, which could be maintained with the same heat flow as at present or an important radioactivity is present in the core. The presence of radioactivity in the core makes the inner core only a few hundred million years (Ma) older than non-radioactive cases with the same dissipation, because the low efficiency of radioactive heating requires a much larger heat flow at the core mantle boundary. Although the age of the inner core is controlled by the heat flow at the CMB, the Ohmic dissipation to be maintained is the constraint that makes it low.

Using energy and entropy constraints applicable to the Earth's core, the heat flow at the core–mantle boundary (CMB) needed to sustain a given total dissipation in the core can be computed. Reasonable estimates for the present Joule dissipation in the core gives a present heat flow of 6 to 10 TW at the CMB. Palaeointensity data acquired from rocks younger than 3.5 Ga provide support that the Joule dissipation in the core before inner core crystallization was between today's value and four times lower than today. Prior to inner core crystallization (around 1 Ga), the magnetic field was maintained by thermal convection driven by core cooling, and our calculations of the two extreme cases predict that the heat flow at the CMB at that time was either 14 to 24 TW in the case of constant dissipation, or essentially the same as today in the lower field intensity case.

International audience; The heat flux at the core-mantle boundary (CMB) is a key parameter for core dynamics since it controls its cooling. However, it is poorly known and estimates range from 2 TW to 10 TW. The lowest bound comes from estimates of buoyancy fluxes of hotspots under two assumptions: that they are surface expression of mantle plumes originating from the base of the mantle, and that they are responsible for the totality of the heat flux at the CMB. Using a new procedure to detect plumes in a numerical model of Rayleigh-Bénard convection (convection between isothermal horizontal planes) with internal heating, it is shown that many hot plumes that start from the bottom boundary do not reach the top surface and that the bottom heat flux is primarily controlled by the arrival of cold plumes. Hot plumes easily form at the bottom boundary but they are mostly due to the spreading of cold plume heads that allow the concentration of hot matter. These plumes are generally not buoyant enough to cross the whole system and the hot plumes that reach the top surface result from an interaction between several hot plumes. According to this simple dynamical behavior, the heat flux at the bottom boundary is shown to be strongly correlated with the advection due to cold plumes and not with advection by hot plumes that arrive at the surface. It is then inferred that the heat flux out of hotspots can only give a lower bound to the heat flow at the CMB and that knowing the advection by subducted plates would give a better estimate.

cited By 3; International audience; Convection models of planetary mantles do not usually include a specific treatment of near-surface dynamics. In all situations where surface dynamics is faster than internal dynamics, the lateral transport of material at the surface forbids the construction of a topography that could balance the internal convective stresses. This is the case if intense erosion erases the topography highs and fills in the depressions or if magma is transported through the lithosphere and spreads at the surface at large distances. In these cases, the usual boundary condition of numerical simulations, that the vertical velocity cancels at the surface should be replaced by a condition where the vertical flux on top of the convective mantle equilibrates that allowed by the surface dynamics. We show that this new boundary condition leads to the direct transport of heat to the surface and changes the internal convection that evolves toward a heat-pipe pattern. We discuss the transition between this extreme situation where heat is transported to the surface to the usual situation where heat diffuses through the lithosphere. This mechanism is much more efficient to cool a planet and might be the major cooling mechanism of young planets. Even the modest effect of material transport by erosion on Earth is not without effect on mantle convection and should affect the heat flow budget of our planet. Key Points: Convection and erosion can be strongly coupled on young planets A heat pipe mechanism can cool a young planet very rapidly Free slip conditions may not be appropriate in mantle convection models. © 2014. American Geophysical Union. All Rights Reserved.

The energy conservation law, when applied to the Earth’s core and integrated between the onset of the crystallization of the inner core and the present time, gives an equation for the age of the inner core. In this equation, all the terms can be expressed theoretically and, given values and uncertainties of all relevant physical parameters, the age of the inner core can be obtained as a function of the heat flux at the core–mantle boundary and the concentrations in radioactive elements. It is found that in absence of radioactive elements in the core, the age of the inner core cannot exceed 2.5 Ga and is most likely around 1 Ga. In addition, to have an inner core as old as the Earth, concentrations in radioactive elements needed in the core are too high to be acceptable on geochemical grounds.

Some features of the polar motion may be due to core–mantle coupling, but no convincing quantitative mechanism has yet been proposed. Considering phase jumps in the Chandler wobble and noticing their correlation with geomagnetic jerks [J. Geophys. Res. 103 (B11) (1998) 27069–27089], we suggest that the instability of a layer at the top of the core and its downward propagation induce a step in the core–mantle torque strong enough to explain phase jumps in the Chandler wobble. The surface magnetic signature of this instability is comparable with the typical evolution of the geomagnetic field during a jerk.

Numerical experiments have been carried out to explore the efficiency of heat transfer through a three-dimensional layer heated from both within and below as it is the case for the mantle of earth-like planets. A systematic study for Rayleigh numbers (Ra) between 105 and 107 and non-dimensional internal heating rate (Hs) between 0 and 40 allows us to investigate the pattern of convection and the thermal characteristics of the layer in a range of parameters relevant to mantle convection in earth-like planets. Inversion of the results for the mean temperature and non-dimensional heat flux at the top and the bottom boundaries yields simple parameterization of the heat transfer. It is shown that the mean temperature of the convective fluid (θ) is the sum of the temperature that would exist with no internal heating and a contribution of the non-dimensional internal heating rate (Hs). As predicted by thermal boundary layer analysis, the non-dimensional heat flux at the upper boundary layer can be described by Q=[(Ra)/(Raδ)]1/3θ4/3 with θ=0.5+1.236[(Hs)3/4/(Ra)1/4], and Raδ being the thermal boundary layer Rayleigh number equal to 24.4. In agreement with laboratory experiments, this value slightly increases with the value of the Rayleigh number. This value is identical to that obtained for fluids heated from within only. In most cases, the hot plumes that form at the lower thermal boundary layer do not reach the upper boundary layer. No simple law has been found to describe the heat transfer through the lower thermal boundary layer, but the bottom heat flux can be determined using the global energy balance. The thermal boundary layer analysis performed in this study allows us to extrapolate our results to 3D spherical geometry and our predictions are in good agreement with numerical experiments described in the literature. A simple case of spherical 3D convection has been performed and provides the same thermal history of planetary mantles than that obtained from 3D numerical runs. Compared to previous parameterized analysis, this study shows that the behaviour of the thermal boundary layers is much different than that predicted by experiments for a fluid heated only from below: at similar Rayleigh numbers, the mean temperature is larger and the surface heat flux is much larger. It seems therefore necessary to reconsider previous models of the thermal evolution of planetary mantles.

We have constructed a self-consistent model for cooling of the Earth's core in which the thermal history of the core is computed as a function of the time evolution of the heat flux delivered to the mantle across the core-mantle boundary. The temperature profile in the convecting core is first assumed to be adiabatic, and its evolution in time is calculated with the only constraint that energy be globally conserved. When the temperature at the centre drops below the freezing point of the core alloy, the inner core starts growing and cools by conduction; it is found that it cannot have reached its present size in more than 1.7 billion years. If the heat flux delivered to the mantle becomes less than that conducted down the adiabat, the temperature profile becomes subadiabatic in a shell at the top of the core, through which heat is evacuated by conduction. Although it is stable against thermal convection, this shell is not necessarily stagnant and may be the seat of motions owing to compositional convection.

Melting and solidification are fundamental to geodynamical processes like inner core growth, magma chamber dynamics, and ice and lava lake evolution. Very often, the thermal history of these systems is controlled by convective motions in the melt. Computing the evolution of convection with a solid–liquid phase change requires specific numerical methods to track the phase boundary and resolve the heat transfer within and between the two separate phases. Here we present two classes of method to model the phase transition coupled with convection. The first, referred to as the moving boundary method, uses the finite element method and treats the liquid and the solid as two distinct grid domains. In the second approach, based on the enthalpy method, the governing equations are solved on a regular rectangular grid with the finite volume method. In this case, the solid and the liquid are regarded as one domain in which the phase change is incorporated implicitly by imposing the liquid fraction f L as a function of temperature and a viscosity that varies strongly with f L . We subject the two modelling frameworks to thorough evaluation by performing benchmarks, in order to ascertain their range of applicability. With these tools we perform a systematic study to infer heat transfer characteristics of a solidifying convecting layer. Parametrized relations are then used to estimate the super-isentropic temperature difference maintained across a basal magma ocean (BMO) ( Labrosse et al., 2007 ), which happens to be minute ( 0.1 K ), implying that the Earth’s core must cool at the same pace as the BMO.

International audience; In early Solar System during the runaway growth stage of planetary formation, the distribution of planetary bodies progressively evolved from a large number of planetesimals to a smaller number of objects with a few dominant embryos. Here, we study the possible thermal and compositional evolution of these planetesimals and planetary embryos in a series of models with increasing complexities. We show that the heating stages of planetesimals by the radioactive decay of now extinct isotopes (in particular 26Al) and by impact heating can occur in two stages or simultaneously. Depending on the accretion rate, melting occurs from the center outward, in a shallow outer shell progressing inward, or in the two locations. We discuss the regime domains of these situations and show that the exponent β that controls the planetary growth rate View the MathML source of planetesimals plays a crucial role. For a given terminal radius and accretion duration, the increase of β maintains the planetesimals very small until the end of accretion, and therefore allows radioactive heating to be radiated away before a large mass can be accreted. To melt the center of ∼500 km planetesimal during its runaway growth stage, with the value β = 2 predicted by astrophysicists, it needs to be formed within a couple of million years after condensation of the first solids. We then develop a multiphase model where the phase changes and phase separations by compaction are taken into account in 1-D spherical geometry. Our model handles simultaneously metal and silicates in both solid and liquid states. The segregation of the protocore decreases the efficiency of radiogenic heating by confining the 26Al in the outer silicate shell. Various types of planetesimals partly differentiated and sometimes differentiated in multiple metal-silicate layers can be obtained.

Core formation, crystal/melt separation, mingling of immiscible magmas, and diapirism are fundamental geological processes that involve differential motions driven by gravity. Diffusion modifies the compo- sition or/and temperature of the considered phases while they travel. Solid particles, liquid drops and viscous diapirs equilibrate while sinking/rising through their surroundings with a time scale that depends on the physics of the flow and the material properties. In particular, the internal circulation within a liquid drop or a diapir favors the diffusive exchange at the interface. To evaluate time scales of chemical/thermal equilibration between a material falling/rising through a deformable medium, we propose analytical laws that can be used at multiple scales. They depend mostly on the non-dimensional Peclet and Reynolds numbers, and are consistent with numerical simulations. We show that equilibration between a particle, drop or diapir and its host needs to be considered in light of the flow structure complexity. It is of fundamental importance to identify the dynamic regime of the flow and take into account the role of the inner circulation within drops and diapirs, as well as inertia that reduces the thickness of boundary layers and enhances exchange through the interface. The scaling laws are applied to predict nickel equilibration between metals and silicates that occurs within 130 m of fall in about 4 minutes during the metal rain stage of the Earth's core formation. For a mafic blob (10 cm diameter) sinking into a felsic melt, trace element equilibration would occur over 4500 m and in about 3 years.