Publications with abstracts

The same list without abstracts

[64]	Morison, A., Labrosse, S., Deguen, R., and Alboussière, T. Onset of thermal convection in a solid spherical shell with melting at either or both boundaries. Geophys. J. Int., 238(2):1121–1136, 2024. [ DOI | arXiv | http ] Thermal convection in planetary solid (rocky or icy) mantles sometimes occurs adjacent to liquid layers with a phase equilibrium at the boundary. The possibility of a solid–liquid phase change at the boundary has been shown to greatly help convection in the solid layer in spheres and plane layers and a similar study is performed here for a spherical shell with a radius-independent central gravity subject to a destabilizing temperature difference. The solid–liquid phase change is considered as a mechanical boundary condition and applies at either or both horizontal boundaries. The boundary condition is controlled by a phase change number, Φ, that compares the timescale for latent heat exchange in the liquid side to that necessary to build a topography at the boundary. We introduce a numerical tool, available at https://github.com/amorison/stablinrb, to carry out the linear stability analysis of the studied setup as well as other similar situations (Cartesian geometry, arbitrary temperature and viscosity depth-dependent profiles). Decreasing Φ makes the phase change more efficient, which reduces the importance of viscous resistance associated to the boundary and makes the critical Rayleigh number for the onset of convection smaller and the wavelength of the critical mode larger, for all values of the radii ratio, γ. In particular, for a phase change boundary condition at the top or at both boundaries, the mode with a spherical harmonics degree of 1 is always favoured for Φ ≲ 10−1. Such a mode is also favoured for a phase change at the bottom boundary for small (γ ≲ 0.45) or large (γ ≳ 0.75) radii ratio. Such dynamics could help explaining the hemispherical dichotomy observed in the structure of many planetary objects.
[63]	Lebec, L., Labrosse, S., Morison, A., and Tackley, P. J. Effects of salts on the exchanges through high-pressure ice layers of large ocean worlds. Icarus, 412:115966, 2024. [ DOI ] The habitability of large icy moons has long been disputed due to the presence of an high-pressure (HP) ice layer located between the core and the internal liquid ocean, preventing a direct contact and facilitated exchanges between both. In our previous study (Lebec et al. 2023), we have shown that under certain conditions and considering the phase change at the upper boundary of the HP ice shell, heat and mass transfer can occur and be efficient. Now, the aim of this work is to study the effects of impurities inside the HP ices on the flow dynamics and the mass transfer efficiency from the core to the ocean. These salts can enter the ice layer either by diffusion, which is a very inefficient process, or by partial melting at the bottom of the ice layer, if the temperature is high enough. In that case, the liquid water can interact with the core and get enriched in salts. This salty water, which can be lighter than high-pressure ices, could rise through the ice shell and refreeze. The resulting salts/ice mixture can be transported into the ocean by convection in the solid state but could also impede it due to its density higher than that of pure ice. The efficiency of convection depends on the buoyancy number , which is the ratio of density increase associated to salts compared to the decrease associated to temperature. To explore the impact of salts, we model a flux of salts from the core and the advection of these salts using Lagrangian tracers in order to study the effects of the Rayleigh number, the buoyancy number and , which is the partition coefficient of the salts between the ocean and the HP ices at the top interface, on the heat and salts transfer efficiency through the HP ice layer, as well as the implications for the evolution of salt concentration in the ocean of large icy moons and ocean worlds. We present a scaling of the bottom temperature and the top radial velocity as function of the Rayleigh number for various values of the buoyancy number and a scaling of the effective flux of salts across the top interface with the ocean as function of the buoyancy number for various values of the partition coefficient. We perform a numerical application for Ganymede and show that depending on the values of the system parameters, a significant percentage of the salts entering the HP ice layer from the core can efficiently flow into the ocean.
[62]	Labrosse, S., Morison, A., and Tackley, P. J. Solid-state mantle convection coupled with a crystallising basal magma ocean. C. R. Geosciences, 356:1–17, 2024. [ DOI ] Fractional crystallisation of a basal magma ocean (BMO) has been proposed to explain the formation of large scale compositional variations in the mantle and the persistence of partially molten patches in the lowermost mantle. We present a complete set of equations for the thermal and compositional evolution of the BMO and show that it can be implemented in a mantle convection code to solve the long term mantle evolution problem. The presence of the BMO modifies the dynamics of the mantle in several ways. The phase equilibrium at the bottom of the solid mantle implies a change of mechanical boundary condition, which helps solid state convection. The net freezing of the BMO implies a change of computational domain, which is treated by mapping the radial coordinate on a constant thickness domain. Fractional melting and freezing at the boundary makes the composition of the BMO and the solid mantle evolve, which is treated using Lagrangian tracers. A sample calculation shows that the persistence of the BMO and its long term evolution drastically changes the dynamics of the solid mantle by promoting downwelling currents and large scale flow. The gradual increase of the FeO content in the BMO and in the solid that crystallises from it leads to the stabilisation of large scale thermo-compositional piles at the bottom of the mantle, possibly explaining the observations from seismology.
[61]	Frasson, T., Labrosse, S., Nataf, H.-C., Coltice, N., and Flament, N. On the impact of true polar wander on heat flux patterns at the core–mantle boundary. Solid Earth, 15(5):617–637, 2024. [ DOI ]
[60]	Colin, L., Michaut, C., Labrosse, S., and Bourdon, B. Thermal evolution of the lunar magma ocean. Earth Planet. Sci. Lett., 648:119109, 2024. [ DOI ] The energy of the giant impact was large enough to generate an initially fully molten Moon. During the solidification of this lunar magma ocean (LMO), an anorthosite crust formed by flotation of light anorthite crystals. Lunar anorthosites show crystallization ages as young as 4.360 Gyr, suggesting a long-lived LMO or a rather young Moon. Existing models for LMO solidification are for a specific phase diagram based on one compositional model. However, the LMO solidification timescale depends on the lunar bulk composition and on the appearance of anorthite in the crystallization sequence. Here, we propose a physically robust 1D model for LMO evolution based on a simple anorthite/olivine-pyroxene eutectic phase diagram. Cumulates first settle at the ocean base for about a thousand years. This first stage results in an unstable thermal profile for the cumulates that can lead to their overturn. In the second stage, simultaneous crystallization of anorthite and cumulates leads to the formation of a buoyant lid that considerably slows down LMO cooling. We explore the impact of an initially hydrated composition, which reduces the stability of plagioclase, of the eutectic position and of the crust thermal conductivity. We show that cumulates overturn may reduce or extend the LMO solidification time depending on its duration. The total LMO solidification timescale ranges between 45 and 250 Myr. Given the most reliable age of 4.360 Gyr for FAN sample 60025, which derives from more than 99% of crystallization, we estimate an age of 4400 to 4560 Myr for the Moon. Keywords: Moon, Magma ocean, Crystallization time, Anorthosite crust
[59]	Alboussière, T., Ricard, Y., and Labrosse, S. Upper bound of heat flux in an anelastic model for Rayleigh–Bénard convection. J. Fluid Mech., 999:A94, 2024. [ DOI ]
[58]	Lebec, L., Labrosse, S., Morison, A., and Tackley, P. J. Scaling of convection in high-pressure ice layers of large icy moons and implications for habitability. Icarus, 396:115494, 2023. [ DOI ] 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. More recently, several studies challenged that hypothesis and showed that exchanges were possible under specific conditions, allowing transport of salts toward the ocean. In our study, we consider 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. 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. For the same conditions as standard convective systems, it also leads to faster mass transfer in the bulk. 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. This is controlled by a dimensionless parameter noted 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, in the case of a fixed heat flux from the core, a rigid or free-slip bottom boundary and various values of Φ. Keywords: Satellites, dynamics, Ices, Interiors, Ganymede
[57]	Ricard, Y., Alboussière, T., Labrosse, S., Curbelo, J., and Dubuffet, F. Fully compressible convection for planetary mantles. Geophys. J. Int., 230(2):932–956, 2022. [ DOI ] 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.
[56]	Hernandez, J.-A., Caracas, R., and Labrosse, S. Stability of high-temperature salty ice suggests electrolyte permeability in water-rich exoplanet icy mantles. Nature Com., 13(1):3303, 2022. [ DOI ] Electrolytes play an important role in the internal structure and dynamics of water-rich satellites and potentially water-rich exoplanets. However, in planets, the presence of a large high-pressure ice mantle is thought to hinder the exchange and transport of electrolytes between various liquid and solid deep layers. Here we show, using first-principles simulations, that up to 2.5 wt% NaCl can be dissolved in dense water ice at interior conditions of water-rich super-Earths and mini-Neptunes. The salt impurities enhance the diffusion of H atoms, extending the stability field of recently discovered superionic ice, and push towards higher pressures the transition to the stiffer ice X phase. Scaling laws for thermo-compositional convection show that salts entering the high pressure ice layer can be readily transported across. These findings suggest that the high-pressure ice mantle of water-rich exoplanets is permeable to the convective transport of electrolytes between the inner rocky core and the outer liquid layer.
[55]	Alboussière, T., Curbelo, J., Dubuffet, F., Labrosse, S., and Ricard, Y. A playground for compressible natural convection with a nearly uniform density. J. Fluid Mech., 940:A9, 2022. [ DOI ]
[54]	Ricard, Y., Labrosse, S., Terasaki, H., and Bercovici, D. Thermocapillary effects in two-phase medium and applications to metal-silicate separation. Phys. Earth Planet. Inter., 311:106640, 2021. [ DOI ] 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 multi-phase 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.
[53]	Morison, A., Labrosse, S., and Choblet, G. Sublimation-driven convection in Sputnik Planitia on Pluto. Nature, 600:419–423, 2021. [ DOI ]
[52]	Clarté, T. T., Schaeffer, N., Labrosse, S., and Vidal, J. The effects of a robin boundary condition on thermal convection in a rotating spherical shell. J. Fluid Mech., 918:A36, 2021. [ DOI ]
[51]	Bolrão, D. P., Ballmer, M. D., Morison, A., Rozel, A. B., Sanan, P., Labrosse, S., and Tackley, P. J. Timescales of chemical equilibrium between the convecting solid mantle and over-/underlying magma oceans. Solid Earth, 12:421–437, 2021. [ DOI ]
[50]	Hsieh, W.-P., Goncharov, A. F., Labrosse, S., Holtgrewe, N., Lobanov, S. S., Chuvashova, I., Deschamps, F., and Lin, J.-F. Low thermal conductivity of iron-silicon alloys at earth's core conditions with implications for the geodynamo. Nature Communications, 11:3332, 2020. [ DOI ] 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.
[49]	Morison, A., Labrosse, S., Deguen, R., and Alboussière, T. Timescale of overturn in a magma ocean cumulate. Earth Planet. Sci. Lett., 516:25 – 36, 2019. [ DOI ] 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. Keywords: magma ocean, overturn, mantle dynamics, linear stability
[48]	Menaut, R., Corre, Y., Huguet, L., Le Reun, T., Alboussière, T., Bergman, M. I., Deguen, R., Labrosse, S., and Moulin, M. Experimental study of convection in the compressible regime. Phys. Rev. Fluids, 4:033502, 2019. [ DOI ] 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 1012, 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 quasigeostrophic regime. Still, a classical power law (exponent 0.3±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.
[47]	Curbelo, J., Duarte, L., Alboussière, T., Dubuffet, F., Labrosse, S., and Ricard, Y. Numerical solutions of compressible convection with an infinite Prandtl number: comparison of the anelastic and anelastic liquid models with the exact equations Compressible convection with infinite Prandtl number. J. Fluid Mech, 873:646–687, 2019. [ DOI ] We developed a numerical method for the set of equations governing fully compressi-ble 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 q must be small compared to unity, where 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 αTD is small compared to unity, where α is the isobaric thermal expansion coefficient, T is the temperature (here αT = 1 for an ideal gas) and D is the dissipation number. †
[46]	Bouffard, M., Choblet, G., Labrosse, S., and Wicht, J. Chemical convection and stratification in the Earth's outer core. Frontiers Earth Sci., 7:99, 2019. [ DOI ] 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.
[45]	Agrusta, R., Morison, A., Labrosse, S., Deguen, R., Alboussière, T., Tackley, P. J., and Dubuffet, F. Mantle convection interacting with magma oceans. Geophys. J. Int., 220:1878–1892, 2019. [ DOI ] 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 two-dimensional 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.
[44]	Laneuville, M., Hernlund, J., Labrosse, S., and Guttenberg, N. Crystallization of a compositionally stratified basal magma ocean. Phys. Earth Planet. Inter., 276:86–92, 2018. [ DOI ] Earth's ∼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. Keywords: Earth, Magma ocean, Geodynamo
[43]	Labrosse, S., Morison, A., Deguen, R., and Alboussière, T. Rayleigh-Bénard convection in a creeping solid with a phase change at either or both horizontal boundaries. J. Fluid Mech., 846:5–36, 2018. [ DOI ] 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 τη, 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 τφ. The ratio Φ=τφ/τη controls whether the boundary condition is the classical non-penetrative one (Φ→∞) or allows for a finite flow through the boundary (small Φ). We study Rayleigh-Bénard 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 Φ, a non-deforming translation mode is possible with a critical Rayleigh number equal to 24Φ. At small values of Φ, this mode competes with a weakly deforming mode having a slightly lower critical Rayleigh number and a very long wavelength, λc∼8√(2)π/ 3√(Φ). 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 Rac=153 and the critical wavelength is λc=5. The Nusselt number increases about twice faster with Rayleigh number than in the classical case with non-penetrative conditions and the average temperature diverges from 1/2 when the Rayleigh number is increased, toward larger values when the bottom boundary is a phase change interface.
[42]	Deguen, R., Alboussière, T., and Labrosse, S. Double-diffusive translation of Earth's inner core. Geophys. J. Int., 214:88–107, 2018. [ DOI ] The hemispherical asymmetry of the inner core has been interpreted as resulting form a high-viscosity mode of inner core convection, consisting in a translation of the inner core. With melting on one hemisphere and crystallisation on the other one, inner core translation would impose a strongly asymmetric buoyancy flux at the bottom of the outer core, with likely important implications for the dynamics of the outer core and the geodynamo. The main requirement for convective instability in the inner core is an adverse radial density gradient. While older estimates of the inner core thermal conductivity favored a superadiabatic temperature gradient and the existence of thermal convection, the much higher values proposed in the last few years makes thermal convection very unlikely. Compositional convection might be a viable alternative to thermal convection: an unstable compositional gradient may arise in the inner core either because the light elements present in the core are predicted to 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 the magnitude of the destabilising effect of the compositional field is predicted to be similar to or smaller than the stabilising 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 decreases upward. We propose here a theoretical and numerical study of double-diffusive convection in the inner core that demonstrates that a translation mode can indeed exist if the compositional field is destabilising, even if the temperature profile is subadiabatic, and irrespectively of the relative magnitudes of the composition and potential temperature gradients. The predicted inner core translation rate is similar to the mean 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.
[41]	Hirose, K., Morard, G., Sinmyo, R., Umemoto, K., Hernlund, J. W., Helffrich, G., and Labrosse, S. Crystallization of silicon dioxide and compositional evolution of the Earth's core. Nature, 543(7643), 2017. [ DOI ] 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 ocean predicts that both silicon (Si) and oxygen (O) are among the impurities in the liquid outer core. 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 conductivity, from as early on as the Hadean eon. SiO2 saturation also sets limits on silicon and oxygen concentrations in the present-day outer core.
[40]	Bouffard, M., Labrosse, S., Choblet, G., Fournier, A., Aubert, J., and Tackley, P. J. A particle-in-cell method for studying double-diffusive convection in the liquid layers of planetary interiors. J. Comput. Phys., 346:552 – 571, 2017. [ DOI ] 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. Keywords: Particle-in-cell, Double-diffusive convection, Geodynamo
[39]	Labrosse, S. Thermal state and evolution of the Earth core and deep mantle, pages 43–56. AGU Geophysical Monograph Series, 2016.
[38]	Huguet, L., Alboussière, T., Bergman, M. I., Deguen, R., Labrosse, S., and Lesœur, G. Structure of a mushy layer under hypergravity with implications for earth's inner core. Geophys. J. Int., 204:1729–1755, 2016. [ DOI ] 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. Keywords: Core, outer core and inner core, Permeability and porosity, Composition of the core, Coda waves, Seismic attenuation, Wave scattering and diffraction
[37]	Lasbleis, M., Deguen, R., Cardin, P., and Labrosse, S. Earth's inner core dynamics induced by the Lorentz force. Geophys. J. Int., 202:548–563, 2015. [ DOI ] 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, whose 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 1012 Pa s. Keywords: Numerical solutions, Composition of the core, Seismic anisotropy
[36]	Labrosse, S., Hernlund, J. W., and Hirose, K. Fractional melting and freezing in the deep mantle and implications for the formation of a basal magma ocean, volume 212 of AGU Geophysical Monograph, pages 123–142. Wiley, 2015.
[35]	Labrosse, S. Thermal evolution of the core with a high thermal conductivity. Phys. Earth Planet. Inter., 247:36 – 55, 2015. Transport Properties of the Earth's Core. [ DOI ] 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 700km 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 700Myr 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 7000K must have prevailed 4.5Gyr 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 3.5Gyr. Implications regarding the evolution of a basal magma ocean are also considered. Keywords: Core thermodynamics, Core evolution, Core mantle boundary heat flow, Inner core age, Early geodynamo
[34]	Jaupart, C., Labrosse, S., Lucazeau, F., and Mareschal, J.-C. 7.06 - temperatures, heat, and energy in the mantle of the Earth. In Schubert, G. and Bercovici, D., editors, Treatise on Geophysics (Second Edition), pages 223 – 270. Elsevier, Oxford, second edition edition, 2015. [ DOI ] Keywords: Core heat flow; Geo-neutrinos; Heat flow; Heat generation; Mantle convection; Mantle energy budget; Plate tectonics; Secular cooling; Thermal evolution; Urey ratio
[33]	Ricard, Y., Labrosse, S., and Dubuffet, F. Lifting the cover of the cauldron: Convection in hot planets. Geochem. Geophys. Geosyst., 15:4617–4630, 2014. [ DOI ] 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. Keywords: Earth's interior: composition and state, convection, erosion, young planets, magma ocean
[32]	Labrosse, S. Thermal and compositional stratification of the inner core. C. R. Geosciences, 346:119–129, 2014. [ DOI ] 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. Keywords: Inner core, Thermal evolution, Compositional stratification, Convection
[31]	Hirose, K., Labrosse, S., and Hernlund, J. W. Composition and state of the core. Ann. Rev. Earth Planet. Sci., 41:657–691, 2013. [ DOI ] 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. Keywords: ultrahigh pressure, phase relation, composition, thermal evolution, stratification
[30]	Gomi, H., Ohta, K., Hirose, K., Labrosse, S., Caracas, R., Verstraete, M. J., and Hernlund, J. W. The high conductivity of iron and thermal evolution of the Earth's core. Phys. Earth Planet. Inter., 224:88 – 103, 2013. [ DOI ] We measured the electrical resistivity of iron and iron-silicon alloy to 100GPa. 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 90W/m/K. These values are significantly higher than conventional estimates, implying rapid secular core cooling, an inner core younger than 1Ga, 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 10TW is likely required to explain the observed dynamo characteristics. Keywords: Core, Electrical resistivity, High pressure, Thermal conductivity, Thermal evolution
[29]	Ulvrová, M., Labrosse, S., Coltice, N., Raback, P., and Tackley, P. J. Numerical modeling of convection interacting with a melting and solidification front: application to the thermal evolution of the basal magma ocean. Phys. Earth Planet. Inter., 206-207:51–66, 2012. [ DOI ] 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 fL as a function of temperature and a viscosity that varies strongly with fL. 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 (<0.1K), implying that the Earth's core must cool at the same pace as the BMO. Keywords: Melting, Solidification, Stefan problem, Phase change, Moving boundary, Convection, Core–Mantle dynamics
[28]	Šrámek, O., Milelli, L., Ricard, Y., and Labrosse, S. Thermal evolution and differentiation of planetesimals and planetary embryos. Icarus, 217:339–354, 2012. [ DOI ] 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 .R∝Rβ 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 ∼500km 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. Keywords: Planetesimals, Accretion, Thermal histories
[27]	Coltice, N., Rolf, T., Tackley, P. J., and Labrosse, S. Dynamic causes of the relation between area and age of the ocean floor. Science, 336:335–338, 2012. [ DOI ] The distribution of seafloor ages determines fundamental characteristics of Earth such as sea level, ocean chemistry, tectonic forces, and heat loss from the mantle. The present-day distribution suggests that subduction affects lithosphere of all ages, but this is at odds with the theory of thermal convection that predicts that subduction should happen once a critical age has been reached. We used spherical models of mantle convection to show that plate-like behavior and continents cause the seafloor area-age distribution to be representative of present-day Earth. The distribution varies in time with the creation and destruction of new plate boundaries. Our simulations suggest that the ocean floor production rate previously reached peaks that were twice the present-day value.
[26]	Ulvrová, M., Coltice, N., Ricard, Y., Labrosse, S., Dubuffet, F., and Šrámek, O. Compositional and thermal equilibration of particles, drops and diapirs in geophysical flows. Geochem. Geophys. Geosyst., 12(10):1–11, 2011. [ DOI ] 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 composition 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 Péclet 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.
[25]	Coltice, N., Moreira, M., Hernlund, J. W., and Labrosse, S. Crystallization of a basal magma ocean recorded by helium and neon. Earth Planet. Sci. Lett., 308:193–199, 2011. [ DOI ] Interpretation of the noble gas isotopic signature in hotspots is still controversial. It suggests that relatively primitive material remains untapped in the deepest mantle, even while mantle convection and sub-surface melting efficiently erase primordial heterogeneities. A recent model suggests that significant differentiation and fractionation affects the deepest mantle following the formation of a dense basal magma ocean (BMO) right after core segregation (Labrosse et al., 2007). Here we explore the consequences of the crystallization of a BMO for the noble gas evolution of the mantle. The crystals extracted from a BMO upon cooling generate dense chemical piles at the base of the mantle. We show that if the solid-melt partition coefficients of He and Ne are >0.01 at high pressure and temperature, He and Ne isotopic ratios in pile cumulates can be pristine like. Hence, the entrainment of modest amounts of BMO cumulate in mantle plumes (<10%) potentially explains the primitive-like He and Ne signatures in hotspots. Because pile material can be depleted in refractory elements while simultaneously enriched in noble gasses, our model forms a viable hypothesis to explain the complex relationship between He and refractory isotopic systems in Earth's interior. Keywords: mantle dynamics, noble gasses, early Earth, geochemistry, geophysics
[24]	Javoy, M., Kaminski, E., cois Guyot, F., Andrault, D., Sanloup, C., Moreira, M., Labrosse, S., Jambon, A., Agrinier, P., Davaille, A., and Jaupart, C. The chemical composition of the Earth: Enstatite chondrite models. Earth Planet. Sci. Lett., 293(3-4):259 – 268, 2010. [ DOI ] We propose a new model of Earth's bulk composition based on enstatite chondrites (E-chondrites), the only chondrite group isotopically identical to the Earth. This model allows a quantitative study of accretion and differentiation processes in the early Earth. Conditions for core formation are evaluated using data on silica–iron equilibrium at high pressure and temperature and the exchange budget equation SiO2+2Fe=Si+2FeO, which is the result of IW and Si-SiO2 oxygen buffers' interaction and controls the evolution of mantle fO2. Based on that equation, ranges for the compositions of the Bulk Silicate Earth, the lower mantle and the core are deduced from the compositions of E-chondrites and their constituents. For these ranges of compositions, we show that during core differentiation, the mantle fO2 evolves naturally from ≈IW-3.2 to IW-1.4±0.1. The model compositions are tightened using geophysical constraints on (1) the amount of light elements in the core, (2) the petrology of the upper and lower mantle and (3) the thermal and convective structure of the lower mantle. Our results indicate that the lower mantle is enriched in Si and Fe, which is consistent with recent geophysical studies, and depleted in highly refractory elements, notably in Uranium and Thorium. Keywords: chemical earth models, enstatite chondrites, Redox state, isotopic anomalies, core composition, heterogeneous mantle, early Earth, radioactive heating
[23]	Ichikawa, H., Labrosse, S., and Kurita, K. Direct numerical simulation of an iron rain in the magma ocean. J. Geophys. Res., 115(B1):B01404, 2010. [ DOI ] Core formation in terrestrial planets is a complex process, possibly involving several mechanisms. This paper presents a direct numerical simulation of one of these, the separation of an emulsion of metal in a magma ocean. The model, using a fully Lagrangian approach called the moving particle semi-implicit method, solves the equations of fluid dynamics, including a proper treatment of surface tension. It allows investigation of the balances controlling the distribution of drop size and velocity, in both two- and three-dimensional situations. A scaling analysis where buoyancy is balanced by both surface tension and inertia correctly predicts the average values in these quantities. The full calculation gives an average drop radius of 1.5 cm falling at a velocity of about 30 cm s-1. Analysis of the full distribution remains interesting and shows that a significant part of the smallest droplets is entrained upward by the return flow in molten silicate and might be entrained by succeeding thermal convection. In addition, we investigate the conversion of gravitational energy into viscous heating and the thermal equilibration between both phases. We find that viscous heating is essentially produced at the surface of iron drops and that thermal equilibration is dominated by advection. Scaling thermal diffusion to chemical diffusion leads to the estimation that the latter would happen in less than 100 m in the magma ocean. Keywords: magma ocean; core formation; Lagrangian method; 8124 Tectonophysics: Earth's interior: composition and state; 8125 Tectonophysics: Evolution of the Earth; 8130 Tectonophysics: Heat generation and transport; 5724 Planetary Sciences: Fluid Planets: Interiors; 5455 Planetary Sciences: Solid Surface Planets: Origin and evolution
[22]	Ichikawa, H. and Labrosse, S. Smooth particle approach for surface tension calculation in moving particle semi-implicit method. Fluid Dynamics Research, 42(3):035503, 2010. We present here an algorithm to solve three-dimensional multi-phase flow problems based on the moving particle semi-implicit (MPS) method. The method is fully Lagrangian and can treat flows with large deformations of the interface such as encountered in the break-up and coalescence of drops. The mean curvature and normal vector of the interface, needed for surface tension calculation, are estimated by a blending of smooth particle hydrodynamics (SPH) and MPS differential schemes. The method is applied to two problems: the free oscillation of a droplet and the transition of a falling drop into a vortex ring. The results are consistent with theory and experiment. This method can be successfully applied to the calculation of a process in planetary core formation, where centimetre scale liquid metal droplets form an emulsion in liquid silicate.
[21]	Breuer, D., Labrosse, S., and Spohn, T. Thermal evolution and magnetic field generation in terrestrial planets and satellites. Space Sci. Rev., 152(1):449–500, 2010. [ DOI ] Keywords: Magnetic field generation, Thermal evolution, Terrestrial planets, Satellites
[20]	Aubert, J., Labrosse, S., and Poitou, C. Modelling the palaeo-evolution of the geodynamo. Geophys. J. Int., 179:1414–1428, 2009. [ DOI ] Although it is known that the geodynamo has been operating for at least 3.2 Ga, it remains difficult to infer the intensity, dipolarity and stability (occurrence of reversals) of the Precambrian magnetic field of the Earth. In order to assist the interpretation of palaeomagnetic data, we produce models for the long-term evolution of the geodynamo by combining core thermodynamics with a systematic scaling analysis of numerical dynamo simulations. We update earlier dynamo scaling results by exploring a parameter space, which has been extended in order to account for core aspect ratios and buoyancy source distributions relevant to Earth in the Precambrian. Our analysis highlights the central role of the convective power, which is an output of core thermodynamics and the main input of our updated scalings. As the thermal evolution of the Earth's core is not well known, two end-member models of heat flow evolution at the core—mantle boundary (CMB) are used, respectively, terminating at present heat flows of 11 TW (high-power scenario) and 3 TW (low power scenario). The resulting models predict that until the appearance of the inner core, a thermal dynamo driven only by secular cooling, and without any need for radioactive heating, can produce a dipole moment of strength comparable to that of the present field, thus precluding an interpretation of the oldest palaeomagnetic records as evidence of the inner core presence. The observed lack of strong long-term trends in palaeointensity data throughout the Earth's history can be rationalized by the weakness of palaeointensity variations predicted by our models relatively to the data scatter. Specifically, the most significant internal magnetic field increase which we predict is associated to the sudden power increase resulting from inner core nucleation, but the dynamo becomes deeper-seated in the core, thus largely cancelling the increase at the core and Earth surface, and diminishing the prospect of observing this event in palaeointensity data. Our models additionally suggest that the geodynamo has lied close to the transition to polarity reversals throughout its history. In the Precambrian, we predict a dynamo with similar dipolarity and less frequent reversals than at present times, due to conditions of generally lower convective forcing. Quantifying the typical CMB heat flow variation needed for the geodynamo to cross the transition from a reversing to a non-reversing state, we find that it is unlikely that such a variation may have caused superchrons in the last 0.5 Ga without shutting down dynamo action altogether. Keywords: Dynamo: theories and simulations, Palaeointensity, Palaeomagnetic secular variation, Reversals: process, timescale, magnetostratigraphy
[19]	van Thienen, P., Benzerara, K., Breuer, D., Gillmann, C., Labrosse, S., Lognonné, P., and Spohn, T. Water, life, and planetary geodynamical evolution. Space Sci. Rev., 129(1):167–203, 2007. [ DOI ] In our search for life on other planets over the past decades, we have come to understand that the solid terrestrial planets provide much more than merely a substrate on which life may develop. Large-scale exchange of heat and volatile species between planetary interiors and hydrospheres/atmospheres, as well as the presence of a magnetic field, are important factors contributing to the habitability of a planet. This chapter reviews these processes, their mutual interactions, and the role life plays in regulating or modulating them.
[18]	Pouilloux, L., Kaminski, E., and Labrosse, S. Anisotropic rheology of a cubic medium and implications for geological materials. Geophys. J. Int., 170(2):876–885, 2007. [ DOI ] Dislocation creep, which is the dominant deformation mechanism in the upper mantle, results in a non-Newtonian anisotropic rheology. The implication of non-Newtonian rheology has been quite extensively studied in geodynamic models but the anisotropic aspect remains poorly investigated. In this paper, we propose to fill this gap by (1) introducing a simple mathematical description of anisotropic viscosity and (2) illustrating the link between plastic crystal deformation and bulk material rheology. The study relies on the highest symmetry of the anisotropic tensor, a cubic symmetry, for which anisotropy is characterized by one parameter only, δ. First-order implications of anisotropy are quantitatively explored as a function of δ. The effective rheology of the material is described as a function of the orientation of the crystals and of the imposed stress and the validity of the isotropic approximation is discussed. The model, applied to ringwoodite, a cubic crystal with spinel-type structure, predicts that the dynamics of the transition zone in the Earth's mantle is going to be strongly affected by mechanical anisotropy. Keywords: anisotropic rheology, dislocation creep, geophysical materials, mantle convection
[17]	Labrosse, S., Hernlund, J. W., and Coltice, N. A crystallizing dense magma ocean at the base of Earth's mantle. Nature, 450:866–869, 2007. [ DOI ] 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 mantle, together with estimates of melting temperatures for deep mantle phases and the amount of cooling of the underlying core required to maintain a geodynamo throughout much of the Earth's history, 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 rocks 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 ago
[16]	Labrosse, S. and Jaupart, C. Thermal evolution of the Earth: Secular changes and fluctuations of plate characteristics. Earth Planet. Sci. Lett., 260:465–481, 2007. [ DOI ] The average secular cooling rate of the Earth can be deduced from compositional variations of mantle melts through time and from rheological conditions at the onset of sub-solidus convection at the end of the initial magma ocean phase. The constraint that this places on the characteristics of mantle convection in the past are investigated using the global heat balance equation and a simple parameterization for the heat loss of the Earth. All heat loss parameterization schemes depend on a closure equation for the maximum age of oceanic plates. We use a scheme that accounts for the present-day distribution of heat flux at Earth's surface and that does not depend on any assumption about the dynamics of convection with rigid plates, which remain poorly understood. We show that heat supply to the base of continents and transient continental thermal regimes cannot be ignored. We find that the maximum sea floor age has not changed by large amounts over the last 3 Ga. Calculations lead to a maximum temperature at an age of about 3 Ga and cannot be extrapolated further back in time. By construction, these calculations are based on the present-day tectonic regime characterized by the subduction of large oceanic plates and hence indicate that this regime did not prevail until an age of about 3 Ga. According to this interpretation, the onset of rapid continental growth occurred when the current plate regime became stable. Keywords: Cooling of the Earth, Plate tectonics, Early Earth, Continental growth
[15]	Labrosse, S. Energy source for the geodynamo. In Gubbins, D. and Herrero-Bervera, E., editors, Encyclopedia Of Geomagnetism And Paleomagnetism, pages 300–302. Springer, 2007.
[14]	Labrosse, S. Heat flow across the core-mantle boundary. In Gubbins, D. and Herrero-Bervera, E., editors, Encyclopedia Of Geomagnetism And Paleomagnetism, pages 127–130. Springer, 2007.
[13]	Jaupart, C., Labrosse, S., and Mareschal, J.-C. 7.06 - temperatures, heat and energy in the mantle of the Earth. In Schubert, G. and Bercovici, D., editors, Treatise on Geophysics, pages 253 – 303. Elsevier, Amsterdam, 2007. [ DOI ] Keywords: Core heat flow; Geo-neutrinos; Heat flow; Heat generation; Mantle convection; Mantle energy budget; Plate tectonics; Secular cooling; Thermal evolution; Urey ratio
[12]	Hernlund, J. W. and Labrosse, S. Geophysically consistent values of the perovskite to post-perovskite transition Clapeyron slope. Geophys. Res. Lett., 34:L05309, 2007. [ DOI ] The double-crossing hypothesis posits that post-perovskite bearing rock in Earth's D″ layer exists as a layer above the core-mantle boundary bounded above and below by intersections between a curved thermal boundary layer geotherm and a relatively steep phase boundary. Increasing seismic evidence for the existence of pairs of discontinuities predicted to occur at the top and bottom of this layer motivates an examination of the consistency of this model with mineral physics constraints for the Clapeyron slope of this phase transition. Using independent constraints for a lower bound on temperature in Earth's deep mantle and the temperature of Earth's inner core boundary, we show that a post-perovskite double-crossing is inconsistent with plausible core temperatures for a Clapeyron slope less than about 7 MPa/K, with the higher range of experimental values yielding better agreement with recent estimates of the melting temperature of Earth's core. Keywords: Geotherm, Post-perovskite, Mantle, Core, Earth's interior: composition and state, Earth's interior: dynamics, Mantle processes, Dynamics: convection currents and mantle plumes
[11]	Grigné, C., Labrosse, S., and Tackley, P. J. Convection under a lid of finite conductivity in wide aspect ratio models: effect of continents on the wavelength of mantle flow. J. Geophys. Res., 112(B8):B08403, 2007. [ DOI ] Mantle convection in the Earth and Mars are dominated by large-scale flow, and this can be due at least in part to the effect of thermally insulating continent-like heterogeneities. Indeed, the presence of a finitely conducting lid on top of a convective isoviscous fluid at infinite Prandtl number induces the formation of a zone of hot upwelling centered beneath the lid and a large horizontal cellular circulation on each side of the lid. In a previous paper, this lateral cellular circulation was limited in size by the dimensions of the model. We now use very large aspect ratio models in order for this circulation to be unhampered. The large-scale circulation consists of the constructive superposition of hot and cold plumes attracted to and expelled from, respectively, the conductive lid. The width of the cellular circulation increases with the Rayleigh number of the fluid, and we propose a scaling law to explain this expansion. Scaling laws for the velocity of the plumes and heat transfer by this large-scale circulation are also developed. Although the model used is much simpler than actual planetary mantles, the driving mechanism for the large-scale circulation must operate in a similar fashion. Important implications in terms of heat transfer can therefore be predicted. Keywords: Mantle convection, Heat flux scaling, Continents, Heat flow, Dynamics: convection currents and mantle plumes, Heat generation and transport, Dynamics of lithosphere and mantle: general
[10]	Grigné, C., Labrosse, S., and Tackley, P. J. Convection under a lid of finite conductivity: Heat flux scaling and application to continents. J. Geophys. Res., 112(B8):B08402, 2007. [ DOI ] A scaling law for the heat flux out of a convective fluid covered totally or partially by a finitely conducting lid is proposed. This scaling is constructed in order to quantify the heat transfer out of the Earth's mantle, taking into account the effect of the dichotomy between oceans and continents, which imposes heterogeneous thermal boundary conditions at the surface of the mantle. The effect of these heterogeneous boundary conditions is studied here using simple two-dimensional models, with the mantle represented by an isoviscous fluid heated from below and continents represented by nondeformable lids of finite thermal conductivity set above the surface of the model. We use free-slip boundary conditions under the oceanic and continental zones in order to study in an isolated way the possible thermal effect of continents, independently of all mechanical effect. A systematic study of the heat transfer as a function of the Rayleigh number of the fluid, of the width of the lid, and of its thermal properties is carried out. We show that estimates of continental lithosphere thickness imply a strong insulating effect from continents on mantle heat loss, at least locally. The heat flux below continents was low in the past and of the order of the present one if the continental thickness has remained broadly constant over the Earth's history. Keywords: Mantle convection, Heat flux scaling, Continents, Heat flow, Heat generation and transport, Dynamics: convection currents and mantle plumes, Dynamics of lithosphere and mantle: general
[9]	Grigné, C., Labrosse, S., and Tackley, P. J. Convective heat transfer as a function of wavelength: Implications for the cooling of the Earth. J. Geophys. Res., 110(B3):B03409, 2005. [ DOI ] Attempting to reconstruct the thermal history of the Earth from a geophysical point of view has for a long time been in disagreement with geochemical data. The geophysical approach uses parameterized models of mantle cooling. The rate of cooling of the Earth at the beginning of its history obtained in these models is generally too rapid to allow a sufficient present-day secular cooling rate. Geochemical estimates of radioactive element concentrations in the mantle then appear too low to explain the observed present mantle heat loss. Cooling models use scaling laws for the mean heat flux out of the mantle as a function of its Rayleigh number of the form Q ∝Raβ. Recent studies have introduced very low values of the exponent β, which can help reduce the cooling rate of the mantle. The present study instead focuses on the coefficient C in the relation Q = C Raβ and, in particular, on its variation with the wavelength of convection. The heat transfer strongly depends on the wavelength of convection. The length scale of convection in Earth's mantle is that of plate tectonics, implying convective cells of wide aspect ratio. Taking into account the long wavelength of convection in Earth's mantle can significantly reduce the efficiency of heat transfer. The likely variations of this wavelength with the Wilson cycle thus imply important variations of the heat flow out of the Earth on a intermediate timescale of 100 Ma, which renders parameterized models of thermal evolution inaccurate for quantitative predictions. Keywords: Dynamics of lithosphere and mantle: general, Dynamics: convection currents and mantle plumes, Evolution of the Earth, Heat generation and transport, heat transfer, mantle convection, scaling law
[8]	Labrosse, S. and Macouin, M. The inner core and the geodynamo. C. R. Geosciences, 335(1):37–50, 2003. [ DOI ] English: 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. Français : Les contraintes thermodynamiques intégrales portant sur l'énergie et l'entropie dans le noyau de la Terre sont utilisées pour relier la dissipation dans le noyau au flux de chaleur à la frontière noyau-manteau (FNM). Des estimations raisonnables de la dissipation par effet Joule dans le noyau actuel nécessitent un flux de chaleur total entre 6 et 10 TW. Les données de paléointensités disponibles pour des âges inférieurs à 3,5 Ga peuvent être utilisées pour contraindre l'évolution de la dissipation par effet Joule au cours du temps, et en particulier pour la période avant l'existence de la graine, la dynamo fonctionnant alors uniquement par convection thermique. Les données étant très partielles, deux interprétations sont possibles : la dissipation est constante, indépendamment de la croissance de la graine, ou la dissipation par effet Joule était environ quatre fois plus faible avant l'apparition de la graine. Dans le premier cas, la convection thermique étant moins efficace que la convection compositionnelle, un flux de chaleur à la FNM entre 14 et 24 TW a été nécessaire il y a environ 1 Ga, alors que dans le second cas, un flux de chaleur proche du flux actuel était suffisant. Keywords: Core thermodynamics, Thermal evolution, Geomagnetism, Palaeomagnetism, Palaeointensity, Earth's magnetic field, Thermodynamique du noyau, Évolution thermique, Géomagnétisme, Paléomagnétisme, Paléointensité, Champ magnétique de la Terre
[7]	Labrosse, S. Thermal and magnetic evolution of the Earth's core. Phys. Earth Planet. Inter., 140:127–143, 2003. [ DOI ] 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. Keywords: Thermal evolution, Magnetic evolution, Earth's core, Inner core
[6]	Labrosse, S. Hotspots, mantle plumes and core heat loss. Earth Planet. Sci. Lett., 199:147–156, 2002. [ DOI ] 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. Keywords: Convection, Plume dynamics, Heat flux, Internal heating
[5]	Labrosse, S., Poirier, J.-P., and Le Mouël, J.-L. The age of the inner core. Earth Planet. Sci. Lett., 190:111–123, 2001. [ DOI ] 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. Keywords: Earth's core, Inner core, Thermal evolution
[4]	Grigné, C. and Labrosse, S. Effects of continents on Earth cooling: Thermal blanketing and depletion in radioactive elements. Geophys. Res. Lett., 28:2707–2710, 2001. [ DOI ] Estimate of mantle heat flow under continental shields are very low, indicating a strong insulating effect of continents on mantle heat loss. This effect is investigated with a simple approach: continents are introduced in an Earth cooling model as perfect thermal insulators. Continental growth rate has then a strong influence on mantle cooling. Various continental growth models are tested and are used to compute the mantle depletion in radioactive elements as a function of continental crust extraction. Results show that the thermal blanketing effect of continents strongly affects mantle cooling, and that mantle depletion must be taken into account in order not to overestimate mantle heat loss. In order to obtain correct oceanic heat flow for present time, continental growth must begin at least 3 Gy ago and steady-state for continental area must be reached for at least 1.5 Gy in our cooling model. Keywords: Planetology: Solid Surface Planets: Heat flow, Planetology: Fluid Planets: Interiors, Tectonophysics: Evolution of the Earth, Tectonophysics: Heat generation and transport
[3]	Bellanger, E., Le Mouël, J.-L., Mandea, M., and Labrosse, S. Chandler wobble and geomagnetic jerks. Phys. Earth Planet. Inter., 124:95–103, 2001. [ DOI ] 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. Keywords: Geomagnetic jerks, Chandler wobble, Earth's core, Core-mantle coupling
[2]	Sotin, C. and Labrosse, S. Three-dimensional thermal convection of an isoviscous, infinite–Prandtl–number fluid heated from within and from below: applications to heat transfer in planetary mantles. Phys. Earth Planet. Inter., 112:171–190, 1999. [ DOI ] 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[Hs3/4/Ra1/4], 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. Keywords: Internal heating, Thermal convection, Planetary mantles
[1]	Labrosse, S., Poirier, J.-P., and Le Mouël, J.-L. On cooling of the Earth's core. Phys. Earth Planet. Inter., 99:1–17, 1997. [ DOI ] 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.

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