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Mantle convection interacting with magma oceans
Abstract
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.
... Compositional changes associated with the phase change are also required. Both aspects have already been reported and the implications of such a boundary have been explored , Agrusta et al., 2019, Bolrão et al., 2021, Lebec et al., 2023. The possibility of melting and freezing at one of the horizontal boundaries helps convection in the solid. ...
... An important feature of the model, already included in StagYY for a few previous studies [Agrusta et al., 2019, Bolrão et al., 2021, Lebec et al., 2023, is the solid-liquid phase change boundary condition at the bottom of the solid shell [e.g. Labrosse et al., 2018], ...
... With this equation, the boundary condition can be the classical no-penetrative one for Φ → ∞, in which case the phase change is effectively prohibited (u r = 0), or of the flow-through type for small values of Φ. This boundary condition strongly affects convection: for small values of Φ, convection is easier to start [i.e. the critical Rayleigh number for the onset of convection is reduced, see Deguen, 2013, heat and mass transfer are increased and the wavelength of convection is increased compared to the situation usually considered in mantle convection [Agrusta et al., 2019]. For a purely thermal problem , the phase change number can be expressed as ...
... Thus, it is necessary to simply review the dynamical structures and scaling law estimation of a turbulent magma ocean. Although the turbulent convection of MO should be addressed in more detail in future researches, recent studies of MO considered the turbulent magma as a translation layer for the underlying solid mantle (Labrosse et al. 2007;Agrusta et al. 2020) or the overlying atmospheric layer ). We will go through these coupled layerings in Sects. ...
... The pure turbulent stage starts after the accretion and ends at crystalization, which can last around 10 4 year (Fig. 1d) and is very transient in a geological time scale. Although the turbulent convection of MO should be addressed in more detail in future researches, recent studies of MO considered the turbulent magma as a translation layer for the underlying solid mantle (Labrosse et al. 2007;Agrusta et al. 2020) or the overlying atmospheric layer ). We will go through these coupled layerings in Sects. ...
... Different from present mantle convection (crust as its upper boundary), it is important to consider the interaction of mantle with the residual MO. Two types of model have been developed to account for this interaction: one is solidstate convection with a MO boundary condition at the meltcumulate interface (Deguen 2013;Agrusta et al. 2020); the other is two-layer miscible convection (Le Bars and Davaille 2002;Höink and Lenardic 2010;Wilczynski and Hughes 2019). ...
Geological and astronomical observations on the ''lava world'' of the rocky planet, with additional theoretical interpretation of Moon's crustal formation, bring up to the occurrence of the magma ocean and lava ponds, which inherits accretion energy of rocky planetesimal and evolves with subsequent energy releases. Hemispherical or global oceans of silicate melt could be a widespread lava phase after rocky planet accretion as well as large impact and could persist on planets on orbits around other stars for various time scales. The processes of magma ocean for mation and solidification change the phases, cause element segregations, and strongly affect the earliest compositional differentiation and volatile content of the terrestrial planets. They form the starting point for cooling to mildly habitable conditions and for the onset of thermally driven solid state mantle convection. The formation and crystallization of magma oceans also influence the assembly of a core, the origin of a crust, initiation of tectonics, and formation of an atmosphere. It is inevitable to investigate the magma ocean dynamics of such an early period of Earth evolution. This review focuses on the internal dynamics of magma oceans after planetesimal accretion and planetary formation including turbulence, particle motion, and solid-state convection, which determine the associated processes of cooling, crystallization, and convection of magma ocean. Geochemical differentiation is discussed correspondingly. The thermodynamics of equilibration between a magma ocean and an overlying, outgassed atmosphere is also discussed, highlighting the need for more data on volatile solubility in silicate melts. The effect of coupling between magma ocean and solid-state mantle convection is also discussed.
... While any such convection would imply remelting of solid cumulates, the related consequences for mantle evolution are poorly understood. Only a few numerical modelling studies have explicitly incorporated coupled remelting and crystallisation at the magma ocean-mantle boundary or boundaries Morison et al., 2019;Agrusta et al., 2019), and none of these studies have explored the consequences for chemical evolution. ...
... In principle, when both magma oceans are present, the critical Rayleigh number can even be arbitrarily low as ± decreases towards zero . Moreover, Agrusta et al. (2019) showed that the heat flow and root mean square (RMS) velocity in the solid mantle vary linearly with Ra when both magma oceans are present, whereas heat flow and RMS velocity in the solid mantle vary as Ra 1/3 and Ra 2/3 respectively when only one magma ocean present. This further increases the difference between the two scenarios at a given value of the Rayleigh number. ...
... On the other hand, it is conceivable that thermally coupled TMO and BMO crystallise more slowly than expected for a thermally isolated TMO (Agrusta et al., 2019). The presence of a BMO makes heat transfer across the mantle and out of the thermally coupled BMO and core more efficient than for cases without a BMO and with a boundary layer at the CMB instead. ...
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 one 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 and remelting 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 , which may explain why the Earth's mantle is rather homogeneous in composition, as supported by geophysical constraints.
... Only a few numerical modelling studies have explicitly incorporated coupled re-melting and crystallisation at the magma ocean mantle boundary or boundaries Morison et al., 2019;Agrusta et al., 2019), and none of these studies have explored the consequences for chemical evolution. ...
... In principle, when both magma oceans are present, the critical Rayleigh number can even be arbitrarily low as Φ ± decreases towards 0 . Moreover, Agrusta et al. (2019) show that the heat flow and RMS velocity in the solid mantle vary linearly with Ra when both magma oceans are 270 present, whereas heat flow and RMS velocity in the solid mantle vary as Ra 1/3 and Ra 2/3 , respectively, in the case of only one magma ocean present. This further increases the difference between the two scenarios at a given value of the Rayleigh number. ...
... On the other hand, it is conceivable that a thermally-coupled TMO and BMO crystallise more slowly than expected for a thermally-isolated TMO (Agrusta et al., 2019). The presence of a BMO makes heat transfer across the mantle and out of 325 the thermally coupled BMO and core more efficient than for cases without a BMO and with a boundary layer at the CMB instead. ...
Abstract. 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.
... One caveat with this model is that it does not account for melting of the deep mantle, which is expected to occur in the cases with hot initial cores. The effect of a BMO on core cooling is unsettled, with some predicting an insulating effect Labrosse et al., 2007;Laneuville et al., 2018;Monteux et al., 2016) and others predicting faster cooling (Agrusta et al., 2020;Labrosse et al., 2018). Adding the physics and composition of the BMO requires the addition of a number of poorly determined parameters. ...
... One implication of such a hot initial CMB temperature is that the base of the mantle would be above its solidus, creating a BMO (Labrosse et al., 2007). The presence of a BMO could effect the rate of mantle and core cooling, possibly causing it to cool slower (Laneuville et al., 2018) or faster (Agrusta et al., 2020) depending on its composition and solidification style. Davies et al. (2020) finds that the presence of a long-lived BMO can modulate the CMB heat flow so that it is nearly constant, or even increasing over time, as the mantle cools. ...
The “new core paradox” suggests that the persistence of the geomagnetic field over nearly all of Earth history is in conflict with the core being highly thermally conductive, which makes convection and dynamo action in the core much harder prior to the nucleation of the inner core. Here we revisit this issue by exploring the influence of six important parameters on core evolution: upper/lower mantle viscosity ratio, core thermal conductivity, core radiogenic heat rate, mantle radiogenic heating rate, central core melting temperature, and initial core‐mantle boundary (CMB) temperature. Each parameter is systematically explored by the model, which couples mantle energy and core energy‐entropy evolution. A model is “successful” if the correct present‐day inner core size is achieved and the dynamo remains alive, as implied by the paleomagnetic record. In agreement with previous studies, we do not find successful thermal evolutions using nominal parameters, which includes a core thermal conductivity of 70 Wm⁻¹K⁻¹, zero core radioactivity, and an initial CMB temperature of 5,000 K. The dynamo can be kept alive by assuming an unrealistically low thermal conductivity of 20 Wm⁻¹K⁻¹ or an unrealistically high core radioactive heat flow of 3 TW at present‐day, which are considered “unsuccessful” models. We identify a third scenario to keep the dynamo alive by assuming a hot initial CMB temperature of ∼6,000 K and a central core liquidus of ∼5,550 K. These temperatures are on the extreme end of typical estimates, but should not be ruled out and deserve further scrutiny.
... L'océan est alors divisé en deux couches distinctes séparées par un solide. La dynamique de ce nouveau système peut s'avérer complexe, le solide développant éventuellement des modes de convection de translation (Morison et al., 2019) dès lors qu'il est autorisé à changer de phase à chaque interface (Agrusta et al., 2020;Labrosse et al., 2018;Morison, 2020). ...
... In the long term, nevertheless, it could be beneficial to take into account the behavior of these interactions with the solidified mantle. Previous studies (see for example Agrusta et al. (2020); Morison (2020)) have shown that convection in a mantle bounded above and below by magma oceans could produce translation modes, with melting-freezing at their interfaces. This mechanism would authorize indirect exchanges between a basal and a surface magma oceans. ...
This thesis manuscript main focus is the thermo-compositional convection in a terrestrial magma ocean. This latter is a spherical shell of liquid silicates, similar to a rocky planets' melt mantle. Earth may have undergone such a phasis during the end of its accretion period because of, for example, moon forming impact.Up to now, most of physical models for terrestrial magma ocean remained quite simple, postulating a well-mixed ocean by vigorous thermal convection. This hypothesis could be mitigated by a basal incongruent crystallization. This work's objective is to provide a more detail model via hydrodynamical approach helped by numerical tool.First, we study thermal convection in a rotating spherical shell submitted to a Robin boundary condition. This one is intermediate between Neumann and Dirichlet ones. It enables to model radiative equilibrium between atmosphere and surface of a terrestrial magma ocean. We show, in that pure thermal setup, that the Robin boundary condition plays little role in a rotation dominant context (except regarding convective structures). More generally, we give the range of parameters in which Robin boundary condition can be safely replaced by more usual ones.Thereafter, we focus on the thermo-compositional convection inside a non-rotating terrestrial magma ocean crystallizing from the bottom upward. We list the different transport regimes developped by the system as a function of the input parameters, particularly we discuss the sustainability of a basal stably straified chemical layer. We show that a terrestrial magma ocean under thick H2O atmosphere may not be able to erode such a pre-existant basal layer.
... A large value of B leads to a strong effect of composition and possible stratification, while efficient entrainment is possible for B ( 1. The exact value for the transition in regimes has not been studied for the problem discussed here but is generally around 1. For convection in the solid ice layer surrounded by an ocean, ΔT is the temperature drop across the bottom boundary layer 46 and can be estimated from the scaling of heat transfer as function of the Rayleigh number (see Methods). The value of Δρ χ =ρ must be lower than the solubility limit, δρ m ¼ 0:7%, but also depends on the mode of transfer from the rocky core. ...
... The value of D was estimated from DFT-MD as detailed in the previous section. To estimate the temperature difference ΔT across the boundary layer, we use the scaling of the dimensionless heat flux as function of the Rayleigh number 46 as Nu ¼ 0:37Ra 1=3 which provides a scaling for the thickness of the boundary layer as d BL ¼ d=ð0:37Ra 1=3 Þ. The temperature difference is then ΔT ¼ q c d BL =k. ...
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.
... Material can cross this boundary provided the necessary latent heat of crystallization is released or absorbed (Labrosse et al., 2018). Agrusta et al. (2020) showed that the presence of a magma ocean can significantly enhanced convective heat transport in the underlying solid mantle, especially in case where a solid layer is sandwiched between upper and lower (basal) magma oceans. This essentially means the entire mantle contributes to the thermal budget of a magma ocean, which could greatly increase its longevity against complete crystallization (Agrusta et al., 2020). ...
... Agrusta et al. (2020) showed that the presence of a magma ocean can significantly enhanced convective heat transport in the underlying solid mantle, especially in case where a solid layer is sandwiched between upper and lower (basal) magma oceans. This essentially means the entire mantle contributes to the thermal budget of a magma ocean, which could greatly increase its longevity against complete crystallization (Agrusta et al., 2020). Such a scenario for delayed crystallization has been used to estimate the lifetime of the lunar magma ocean and to infer the Moon's formation time (Maurice et al., 2020). ...
The magma ocean concept was first conceived to explain the geology of the Moon, but hemispherical or global oceans of silicate melt could be a widespread lava world” phase of rocky planet accretion, and could persist on planets on short-period orbits around other stars. The formation and crystallization of magma oceans could be a defining stage in the assembly of a core, origin of a crust, initiation of tectonics, and formation of an atmosphere. The last decade has seen significant advances in our understanding of this phenomenon through analysis of terrestrial and extraterrestrial samples, planetary missions, and astronomical observations of exoplanets. This review describes the energetic basis of magma oceans and lava worlds and the lava lake analogs available for study on Earth and Io. It provides an overview of evidence for magma oceans throughout the Solar System and considers the factors that control the rocks these magma oceans leave behind. It describes research on theoretical and observed exoplanets that could host extant magma oceans and summarizes efforts to detect and characterize them. It reviews modeling of the evolution of magma oceans as a result of crystallization and evaporation, the interaction with the underlying solid mantle, and the effects of planetary rotation. The review also considers theoretical investigations on the formation of an atmosphere in concert with the magma ocean and in response to irradiation from the host star, and possible end-states. Finally, it describes needs and gaps in our knowledge and points to future opportunities with new planetary missions and space telescopes to identify and better characterize lava worlds around nearby stars.
... Material can cross this boundary provided the necessary latent heat of crystallization is released or absorbed (Labrosse et al., 2018). Agrusta et al. (2020) showed that the presence of a magma ocean can significantly enhanced convective heat transport in the underlying solid mantle, especially in case where a solid layer is sandwiched between upper and lower (basal) magma oceans. This essentially means the entire mantle contributes to the thermal budget of a magma ocean, which could greatly increase its longevity against complete crystallization (Agrusta et al., 2020). ...
... Agrusta et al. (2020) showed that the presence of a magma ocean can significantly enhanced convective heat transport in the underlying solid mantle, especially in case where a solid layer is sandwiched between upper and lower (basal) magma oceans. This essentially means the entire mantle contributes to the thermal budget of a magma ocean, which could greatly increase its longevity against complete crystallization (Agrusta et al., 2020). Such a scenario for delayed crystallization has been used to estimate the lifetime of the lunar magma ocean and to infer the Moon's formation time (Maurice et al., 2020). ...
The magma ocean concept was first conceived to explain the geology of the Moon, but hemispherical or global oceans of silicate melt could be a widespread "lava world" phase of rocky planet accretion, and could persist on planets on short-period orbits around other stars. The formation and crystallization of magma oceans could be a defining stage in the assembly of a core, origin of a crust, initiation of tectonics, and formation of an atmosphere. The last decade has seen significant advances in our understanding of this phenomenon through analysis of terrestrial and extraterrestrial samples, planetary missions, and astronomical observations of exoplanets. This review describes the energetic basis of magma oceans and lava worlds and the lava lake analogs available for study on Earth and Io. It provides an overview of evidence for magma oceans throughout the Solar System and considers the factors that control the rocks these magma oceans leave behind. It describes research on theoretical and observed exoplanets that could host extant magma oceans and summarizes efforts to detect and characterize them. It reviews modeling of the evolution of magma oceans as a result of crystallization and evaporation, the interaction with the underlying solid mantle, and the effects of planetary rotation. The review also considers theoretical investigations on the formation of an atmosphere in concert with the magma ocean and in response to irradiation from the host star, and possible end-states. Finally, it describes needs and gaps in our knowledge and points to future opportunities with new planetary missions and space telescopes to identify and better characterize lava worlds around nearby stars.
... However, recent studies have shown that if the magma ocean lifetime exceeds 10 5 -10 6 years, both Rayleigh-Bénard and Rayleigh-Taylor instabilities can trigger solid-state convection in the growing cumulates of Mars (Maurice et al., 2017), of the Earth (Ballmer et al., 2017), as well as of the Moon (Boukaré et al., 2018;Morison et al., 2019). The existence of melting/freezing boundaries between the magma ocean and its cumulates can further help the onset of convection (Agrusta et al., 2019;Morison et al., 2019). The long lifetime of the LMO (∼200 Myr) recently inferred by Maurice et al. (2020) would strongly favor this scenario. ...
The fractional crystallization of the lunar magma ocean (LMO) results in a gravitationally unstable layering, with dense Fe‐ and Ti‐oxides overlying lighter mafic cumulates. Due to their high density, these are prone to overturn via Rayleigh‐Taylor instability. However, this instability competes with the downward growth of the cold and stiff stagnant lid, which tends to trap the cumulates preventing their overturn. The fate of high‐Ti cumulates (HTC) plays a major role in several aspects of the Moon's history, including mare volcanism, early magnetism, and the presence of a partially molten layer at the core‐mantle boundary (CMB). To assess the extent of the overturn of HTC, we use 2D simulations of thermo‐chemical mantle convection in the presence of a solidifying LMO. The long lifetime of the magma ocean caused by the insulating effect of the plagioclase crust delays the growth of the stagnant lid and promotes the onset of solid‐state convection during magma ocean solidification. Both phenomena favor a high degree of mobilization of the HTC. Independent of the rheology of the mafic and HTC, the overturn is always characterized by small‐scale instabilities and is completed within ∼300–600 Myr. High‐Ti material accumulates at the CMB where it undergoes partial melting until present‐day, in agreement with the existence of a deep, partially molten layer inferred from geodetic data. Part of the overturned cumulates is entrained by mantle flow and can participate in secondary melting until ∼1 Ga, in agreement with the age of high‐Ti mare basalts.
... Moore and Weiss [7], McKenzie et al [8], Jarvis and McKenzie [9], Jarvis and Peltier [10] and recently Agrustra et. al. [11] performed various numerical simulations that confirmed the predictions of these analyses. At all Rayleigh numbers, boundary layer analysis reveals that vertical advection is important in the energy balance within horizontal boundary layers, whereas the energy balance within vertical plumes is dominated by horizontal and vertical advection, with horizontal diffusion playing a minor role [10]. ...
... Alternatively, gravitational instability can be triggered by the Fe enrichment of the upper layers during crystallization 43,44 . Recent studies have also shown that convection within the solidified mantle could become more efficient as the magma ocean crystallizes [45][46][47] . However, the TMMs we focus on here are incorporated into the solid lower layer, which has a lower potential temperature than the overlying molten layer. ...
Seismic images of Earth’s interior have revealed two continent-sized anomalies with low seismic velocities, known as the large low-velocity provinces (LLVPs), in the lowermost mantle¹. The LLVPs are often interpreted as intrinsically dense heterogeneities that are compositionally distinct from the surrounding mantle². Here we show that LLVPs may represent buried relics of Theia mantle material (TMM) that was preserved in proto-Earth’s mantle after the Moon-forming giant impact³. Our canonical giant-impact simulations show that a fraction of Theia’s mantle could have been delivered to proto-Earth’s solid lower mantle. We find that TMM is intrinsically 2.0–3.5% denser than proto-Earth’s mantle based on models of Theia’s mantle and the observed higher FeO content of the Moon. Our mantle convection models show that dense TMM blobs with a size of tens of kilometres after the impact can later sink and accumulate into LLVP-like thermochemical piles atop Earth’s core and survive to the present day. The LLVPs may, thus, be a natural consequence of the Moon-forming giant impact. Because giant impacts are common at the end stages of planet accretion, similar mantle heterogeneities caused by impacts may also exist in the interiors of other planetary bodies.
... An accurate parameterization of the secular cooling term involves resolving various complex heat transfer mechanisms within the young Earth, which are, for some, not well understood (e.g., the heat flux at the melting interface between the MO and solid mantle; Labrosse et al. 2018;Agrusta et al. 2020). For the sake of simplicity, here we consider that the thermal energy of the planet decreases linearly with T pot between the initial and final thermal states (see Appendix B.1). ...
Magma oceans (MOs) are episodes of large-scale melting of the mantle of terrestrial planets. The energy delivered by the Moon-forming impact induced a deep MO on the young Earth, corresponding to the last episode of core-mantle equilibration. The crystallization of this MO led to the outgassing of volatiles initially present in the Earth’s mantle, resulting in the formation of a secondary atmosphere. During outgassing, the MO acts as a chemical buffer for the atmosphere via the oxygen fugacity, set by the equilibrium between ferrous- and ferric-iron oxides in the silicate melts. By tracking the evolution of the oxygen fugacity during MO solidification, we model the evolving composition of a C-O-H atmosphere. We use the atmospheric composition to calculate its thermal structure and radiative flux. This allows us to calculate the lifetime of the terrestrial MO. We find that, upon crystallizing, the MO evolves from a mildly reducing to a highly oxidized redox state, thereby transiting from a CO- and H 2 -dominated atmosphere to a CO 2 - and H 2 O-dominated one. We find the overall duration of the MO crystallization to depend mostly on the bulk H content of the mantle, and to remain below 1.5 millions yr for up to nine Earth’s water oceans’ worth of H. Our model also suggests that reduced atmospheres emit lower infrared radiation than oxidized ones, despite of the lower greenhouse effect of reduced species, resulting in a longer MO lifetime in the former case. Although developed for a deep MO on Earth, the framework applies to all terrestrial planet and exoplanet MOs, depending on their volatile budgets.
... An accurate parametrization of the secular cooling term involves resolving various complex heat transfer mechanisms within the young Earth which are, for some, not well understood (e.g. the heat flux at the melting interface between magma ocean and solid mantle (Labrosse et al. 2018;Agrusta et al. 2020)). For the sake of simplicity, here we consider that the thermal energy of the planet decreases linearly with T pot between the initial and final thermal states (see Section B.1). ...
Magma oceans are episodes of large-scale melting of the mantle of terrestrial planets. The energy delivered by the Moon-forming impact induced a deep magma ocean on the young Earth, corresponding to the last episode of core-mantle equilibration. The crystallization of this magma ocean led to the outgassing of volatiles initially present in the Earth's mantle, resulting in the formation of a secondary atmosphere. During outgassing, the magma ocean acts as a chemical buffer for the atmosphere via the oxygen fugacity, set by the equilibrium between ferrous- and ferric-iron oxides in the silicate melts. By tracking the evolution of the oxygen fugacity during magma ocean solidification, we model the evolving composition of a C-O-H atmosphere. We use the atmosphere composition to calculate its thermal structure and radiative flux. This allows us to calculate the lifetime of the terrestrial magma ocean. We find that, upon crystallizing, the magma ocean evolves from a mildly reducing to a highly oxidized redox state, thereby transiting from a CO- and H2-dominated atmosphere to a CO2- and H2O-dominated one. We find the overall duration of the magma ocean crystallization to depend mostly on the bulk H content of the mantle, and to remain below 1.5 millions years for up to 9 Earth's water oceans' worth of H. Our model also suggests that reduced atmospheres emit lower infrared radiation than oxidized ones, despite of the lower greenhouse effect of reduced species, resulting in a longer magma ocean lifetime in the former case. Although developed for a deep magma ocean on Earth, the framework applies to all terrestrial planet and exoplanet magma oceans, depending on their volatile budgets.
... Depiction of our Hadean planet. The crust was completely submerged with a ~5km deep ocean as the magma ocean was still too mushy to support significant bulges even at the apices of mantle plumes[98][99][100][101][102]106,[233][234][235] EoL: emergence of life. ...
The assumption that there was a “water problem” at the emergence of life—that the Hadean Ocean was simply too wet and salty for life to have emerged in it—is here subjected to geological and experimental “reality checks”. The “warm little pond” that would take the place of the submarine alkaline vent theory (AVT), as recently extolled in the journal Nature, flies in the face of decades of geological, microbiological and evolutionary research and reasoning. To the present author, the evidence refuting the warm little pond scheme is overwhelming given the facts that (i) the early Earth was a water world, (ii) its all-enveloping ocean was never less than 4 km deep, (iii) there were no figurative “Icelands” or “Hawaiis”, nor even an “Ontong Java” then because (iv) the solidifying magma ocean beneath was still too mushy to support such salient loadings on the oceanic crust. In place of the supposed warm little pond, we offer a well-protected mineral mound precipitated at a submarine alkaline vent as life’s womb: in place of lipid membranes, we suggest peptides; we replace poisonous cyanide with ammonium and hydrazine; instead of deleterious radiation we have the appropriate life-giving redox and pH disequilibria; and in place of messy chemistry we offer the potential for life’s emergence from the simplest of geochemically available molecules and ions focused at a submarine alkaline vent in the Hadean—specifically within the nano-confined flexible and redox active interlayer walls of the mixed-valent double layer oxyhydroxide mineral, fougerite/green rust comprising much of that mound.
... But even in this case, if the heat flux variations from the ocean are large, the mass flux through the interface is significant and may influence the flow in the ice shell. The effect of phase transitions on the thermal state of the shell has not yet been studied in detail but its potential importance has been highlighted in recent studies dealing with early magma oceans (Morison et al., 2019;Agrusta et al., 2020). ...
We use the recent models of Titan’s shape and gravity to estimate the large-scale density structure of its outer ice shell. We assume that the bottom boundary of the ice shell is an equipotential surface, in agreement with the decrease in ice viscosity expected near the ice/water interface, and the topography is supported by density variations within the ice shell. The density model shows strong degree 2 and 4 zonal components, indicating an important role of atmospheric and/or oceanic processes. We discuss three mechanisms that may explain the derived density anomalies: ethane precipitation, temperature anomalies and variations in porosity. While ethane precipitation provides a plausible explanation for positive density anomalies associated with polar depressions, large-scale variations in temperature are likely to play a role at mid- and low latitudes. These variations can be associated with a laterally varying thickness of the methane clathrate crust, which controls the heat transport in the ice shell, and/or with variations in the heat flux from the ocean. The density of the ice shell can also be affected by variations in porosity of the near-surface material, but these variations are unlikely to be the main cause of the large-scale density anomalies identified from the shape and gravity data.
... The initial CMB temperature stays below 8500 K for all our best-fit simulations (red lines on the right panel of Fig. 12) and for f ν ≥ 5 it stays even below 6500 K. Such temperatures, however, still indicate widespread melting in the lower mantle and would result in a basal magma ocean, a plausible scenario , with potentially important implications for heat transfer in the lower mantle that have only started to be investigated (Agrusta et al., 2019). Under such conditions, Eq. (20) no longer holds. ...
The role of heat flow coming from the core is often overlooked or underestimated in simple models of Earth's thermal evolution. Throughout most of Earth's history, the mantle must have been extracting from the core at least the amount of heat that is required to operate the geodynamo. In view of recent laboratory measurements and theoretical calculations indicating a higher thermal conductivity of iron than previously thought, the above constraint has important implications for the thermal history of the Earth's mantle. In this work we construct a paramaterized mantle convection model that treats both the top and the core-mantle boundary heat fluxes according to the boundary layer theory, or alternatively employs the model of Labrosse (2015) to compute the thermal evolution of the Earth's core. We show that the core is likely to provide all the missing heat that is necessary in order to avoid the so-called “thermal catastrophe” of the mantle. Moreover, by analyzing the mutual feedback between the core and the mantle, we provide the necessary ingredients for obtaining thermal histories that are consistent with the petrological record and have reasonable initial conditions. These include a sufficiently high viscosity contrast between the lower and upper mantle, whose exact value is sensitive to the activation energy that governs the temperature dependence of the viscosity.
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.
The current state and surface conditions of the Earth and its twin planet Venus are drastically different. Whether these differences are directly inherited from the earliest stages of planetary evolution, when the interior was molten, or arose later during the long-term evolution is still unclear. Yet, it is clear that water, its abundance, state, and distribution between the different planetary reservoirs, which are intimately related to the solidification and outgassing of the early magma ocean, are key components regarding past and present-day habitability, planetary evolution, and the different pathways leading to various surface conditions.
In this chapter we start by reviewing the outcomes of the accretion sequence, with particular emphasis on the sources and timing of water delivery in light of available constraints, and the initial thermal state of Venus at the end of the main accretion. Then, we detail the processes at play during the early thermo-chemical evolution of molten terrestrial planets, and how they can affect the abundance and distribution of water within the different planetary reservoirs. Namely, we focus on the magma ocean cooling, solidification, and concurrent formation of the outgassed atmosphere. Accounting for the possible range of parameters for early Venus and based on the mechanisms and feedbacks described, we provide an overview of the likely evolutionary pathways leading to diverse surface conditions, from a temperate to a hellish early Venus. The implications of the resulting surface conditions and habitability are discussed in the context of the subsequent long-term interior and atmospheric evolution. Future research directions and observations are proposed to constrain the different scenarios in order to reconcile Venus’ early evolution with its current state, while deciphering which path it followed.
Recent studies of ocean dynamics suggest that the long-wavelength topography of some icy moons may reflect the phase transitions (melting/freezing) at the interface between the ice shell and the water ocean. Despite the obvious importance of phase changes in the evolution of icy moons, very little is known about how these processes influence their shape, gravity and near-surface stress. Here we address this issue by performing a series of numerical experiments in which we explore the thermo-mechanical response of an ice layer to heat flux variations imposed at the bottom (phase) boundary. We assume that the heat flux from the ocean consists of two components: the heat flux originating in the deep ocean and associated with the global ocean circulation, and the heat flux due to the flow of water generated by variations in the melting temperature along the deformed ice-water interface. The effect of salinity on the heat flux from the ocean is neglected. We demonstrate that the mass exchange between the ocean and the ice layer is a natural consequence of the ice-water phase transition and it occurs in both convection and conduction modes, regardless of whether the system is in equilibrium or not. The magnitude of the heat-flux induced topography strongly depends on the viscosity of ice and the flow of water controlled by the melting temperature. When heat transfer in the ice shell occurs by convection, the surface topography is dominated by small-scale convective features varying in time and its large-scale component does not exceed 10 m. When the viscosity of the ice shell is high (≳1016 Pa s) and the heat is transferred by conduction, the topography is negatively correlated with the heat flux from the ocean and its amplitude increases with increasing viscosity. Topographic amplitudes comparable to those observed on Titan, Enceladus and Dione are obtained only if the water flow associated with lateral variations in the melting temperature is neglected. This suggests that this flow may be too weak to reduce the variations in ice shell thickness and the motion of water along the phase boundary is more likely to be controlled by other factors, such as variations in salinity and the presence of non-ice material. In addition, we show that the standard formulation of Airy isostasy can lead to an error of 5%–15% in determining the variations in ice thickness and we propose a new formulation that takes into account the effect of thermal expansion.
It has been suggested that the long-wavelength topography of Titan is related to lateral variations in the heat flux from the ocean. Recent studies of the heat transfer in Titan’s ocean agree that the time-averaged heat flux can vary in latitude by tens of percent, but they predict different distributions of heat flux anomalies at the upper boundary of the ocean. In order to clarify this issue, we perform 115 numerical simulations of thermal convection in a rotating spherical shell, varying the mechanical boundary conditions and dimensionless input parameters (Rayleigh, Ekman and Prandtl numbers) by at least one order of magnitude. The results of the simulations are examined in terms of the modified transitional number, RG∗=RaEk12/7Pr−1. Depending on the relative importance of rotation, the heat flux maximum is located either at the equator (equatorial cooling) or at the poles (polar cooling). We demonstrate that equatorial cooling occurs when RG∗10 while polar cooling occurs when RG∗∈〈1,10〉. Based on this result, we predict that Titan’s ocean is in the polar cooling mode and the heat flux distribution is controlled by zonal degree 2 and 4 harmonics. The predicted heat flux shows a high degree of similarity with the axisymmetric part of Titan’s long-wavelength topography, indicating a strong relationship between ocean dynamics and the processes in the ice shell.
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.
The Earth is in a plate tectonics regime with high surface heat flow concentrated at constructive plate boundaries. Other terrestrial bodies that lack plate tectonics are thought to lose their internal heat by conduction through their lids and volcanism: hotter planets (Io and Venus) show widespread volcanism whereas colder ones (modern Mars and Mercury) are less volcanically active. However, studies of terrestrial magmatic processes show that less than 20% of melt volcanically erupts, with most melt intruding into the crust. Signatures of large magmatic intrusions are also found on other planets. Yet, the influence of intrusive magmatism on planetary cooling remains unclear. Here we use numerical magmatic-thermo-mechanical models to simulate global mantle convection in a planetary interior. In our simulations, warm intrusive magmatism acts to thin the lithosphere, leading to sustained recycling of overlying crustal material and cooling of the mantle. In contrast, volcanic eruptions lead to a thick lithosphere that insulates the upper mantle and prevents efficient cooling. We find that heat loss due to intrusive magmatism can be particularly efficient compared to volcanic eruptions if the partitioning of heat-producing radioactive elements into the melt phase is weak. We conclude that the mode of magmatism experienced by rocky bodies determines the thermal and compositional evolution of their interior. Rocky planets dominated by intrusive magmatism can cool more efficiently than those dominated by extrusive volcanism, according to numerical simulations of mantle convection.
Just after accretion, the Earth's mantle was significantly molten by the heat dissipation due to large impacts and to the segregation of the core. The mineralogical observations and thermodynamics models of solid-liquid equilibrium of silicates show that several types of crystallization may have happened at different depths in the mantle. Solids were probably formed first at the bottom of the lower mantle or at mid mantle leaving two possible magma oceans, a shallow one and an abyssal one. Near the bottom of the mantle, the liquid phase might become denser than solids due to iron enrichment. In the shallow magma ocean, the crystallizing solid phase was denser and sank through the magma to settle and compact at depth. To understand these complex dynamics, we develop a two phase numerical code that can handle simultaneously convection in each phase and in the slurry, and the compaction or decompaction of the two phases. Although our code can only run in a parameter range (Rayleigh number, viscosity contrast between phases, Prandlt number) far from what would be realistic, we think it already provides a rich dynamics that illustrates what could have happened. We show situations in which the crystallization front is gravitationally stable and situations were the newly formed solids are gravitationally unstable and can snow across the magma. Our study suggests that the location of a density contrast between solid and magma must be considered of equal importance with that of the intersection between liquidus and isentrope for what concerns mantle solidification.
How the volatile content influences the primordial surface conditions of terrestrial planets, and thus, their future geodynamic evolution is an important question to answer. We simulate the secular convective cooling of a 1D magma ocean (“ MO”) in interaction with its outgassed atmosphere. The heat transfer in the atmosphere is computed either using the grey approximation or using a k-correlated method. We vary the initial CO2 and H2O contents (respectively from 0.1×10−2 to 14×10−2 wt% and from 0.03 to 1.4 times the Earth Ocean current mass (MEO)) and the solar distance - from 0.63 to 1.30 AU. A first rapid cooling stage, where efficient MO cooling and degassing take place, producing the atmosphere, is followed by a second quasi-steady-state where the heat flux balance is dominated by the solar flux. The end of the rapid cooling stage (“ ERCS”) is reached when the mantle heat flux becomes negligible compared to the absorbed solar flux. The resulting surface conditions at ERCS, including water ocean's formation, strongly depend both on the initial volatile content and solar distance D. For D > DC, the “ critical distance”, the volatile content controls water condensation and a new scaling law is derived for the water condensation limit. Although today's Venus is located beyond DC due to its high albedo, its high CO2/H2O ratio prevents any water ocean formation. Depending on the formation time of its cloud cover and resulting albedo, only 0.3 Earth ocean mass might be sufficient to form a water ocean on early Venus.
Polycrystalline samples of Mg2SiO4 forsterite and wadsleyite were synthesized and then dynamically loaded to pressures of 39–200 GPa. Differences in initial density and internal energy between these two phases lead to distinct Hugoniots, each characterized by multiple phase regimes. Transformation to the high-pressure phase assemblage MgO + MgSiO3 perovksite is complete by 100 GPa for forsterite starting material but incomplete for wadsleyite. The datum for wadsleyite shocked to 136 GPa, however, is consistent with the assemblage MgO + MgSiO3 post-perovksite.
Marked increases in density along the Hugoniots of both phases between 130 and 150 GPa are inconsistent with any known solid-solid phase transformation in the Mg2SiO4 system but can be explained by melting. Density increases upon melting are consistent with a similar density increase observed in the MgSiO3 system. This implies that melts with compositions over the entire Mg/Si range likely for the mantle would be negatively or neutrally buoyant at conditions close to the core-mantle boundary, supporting the partial melt hypothesis to explain the occurrence of ultra-low velocity zones at the base of the mantle. From the energetic difference between
the high-pressure segments of the two Hugoniots, we estimate a Gru ̈neisen parameter (g) of 2.6 ± 0.35 for Mg2SiO4-liquid between 150 and 200 GPa. Comparison to low-pressure data and fitting of the absolute pressures along the melt Hugoniots both require that g for the melt increases with increasing density. Similar behavior was recently predicted in MgSiO3 liquid via molecular dynamics simulations. This result changes estimates of the temperature profile, and hence the dynamics, of a deep terrestrial magma ocean.
At the end of Earth's accretion and after the core-mantle segregation, the existence of a basal magma ocean at the top of the CMB depends on the physical properties of mantle materials at relevant pressure and temperature. Present-day deep mantle structures such as ultralow-velocity zones (ULVZs) and low-shear velocity provinces (LLSVPs) might be directly linked to the still ongoing crystallization of a primordial magma ocean. We provide the first steps towards a self-consistent thermodynamic model of magma ocean crystallization at high-pressure. We build a solid-liquid thermodynamic database for silicates in the MgO-FeO-SiO2 system from 20 GPa to 140 GPa. We use already published chemical potentials for solids, liquid MgO and SiO2. We derive standard state chemical potential for liquid FeO and mixing relations from various indirect observations. Using this database, we compute the ternary phase diagram in the MgO-FeO-SiO2 system as a function of temperature and pressure. We confirm that the melt is lighter than the solid of same composition for all mantle conditions but at thermodynamic equilibrium, the iron-rich liquid is denser than the solid in the deep mantle. We compute a whole fractional crystallization sequence of the mantle and show that an iron rich and fusible layer should be left above the CMB at the end of the crystallization.
In a number of geophysical or planetological settings (Earth's inner
core, a silicate mantle crystallizing from a magma ocean, or an ice
shell surrounding a deep water ocean) a convecting crystalline layer is
in contact with a layer of its melt. Allowing for melting/freezing at
one or both of the boundaries of the solid layer is likely to affect the
pattern of convection in the layer. We study here the onset of thermal
convection in a viscous spherical shell with dynamically induced
melting/freezing at either or both of its boundaries. It is shown that
the behavior of each interface depends on the value of a dimensional
number P, which is the ratio of a melting/freezing timescale over a
viscous relaxation timescale. A small value of P corresponds to
permeable boundary conditions, while a large value of P corresponds to
impermeable boundary conditions. The linear stability analysis predicts
a significant effect of semi-permeable boundaries when the number P
characterizing either of the boundary is small enough: allowing for
melting/freezing at either of the boundary results in the emergence of
larger scale convective modes. The effect is particularly drastic when
the outer boundary is permeable, since the degree 1 mode remains the
most unstable even in the case of thin spherical shells. In the case of
a spherical shell with permeable inner and outer boundaries, the most
unstable mode consists in a global translation of the solid shell, with
no deformation. In the limit of a full sphere with permeable outer
boundary, this corresponds to the "convective translation" mode recently
proposed for Earth's inner core. As an example of possible application,
we discuss the case of thermal convection in Enceladus' ice shell
assuming the presence of a global subsurface ocean, and found that
melting/freezing could have an important effect on the pattern of
convection in the ice shell.
The thermal evolution of magma oceans produced by collision with giant
impactors late in accretion is expected to depend on the composition and
structure of the atmosphere through the greenhouse effect of
CO2 and H2O released from the magma during its
crystallization. In order to constrain the various cooling timescales of
the system, we developed a 1-D parameterized convection model of a magma
ocean coupled with a 1-D radiative-convective model of the atmosphere.
We conducted a parametric study and described the influences of the
initial volatile inventories, the initial depth of the magma ocean, and
the Sun-planet distance. Our results suggest that a steam atmosphere
delays the end of the magma ocean phase by typically 1 Myr. Water vapor
condenses to an ocean after 0.1, 1.5, and 10 Myr for, respectively,
Mars, Earth, and Venus. This time would be virtually infinite for an
Earth-sized planet located at less than 0.66 AU from the Sun. Using a
more accurate calculation of opacities, we show that Venus is much
closer to this threshold distance than in previous models. So there are
conditions such as no water ocean is formed on Venus. Moreover, for Mars
and Earth, water ocean formation timescales are shorter than typical
time gaps between major impacts. This implies that successive water
oceans may have developed during accretion, making easier the loss of
their atmospheres by impact erosion. On the other hand, Venus could have
remained in the magma ocean stage for most of its accretion.
1] We have conducted new equation of state measurements on liquid Fe 2 SiO 4 in a collaborative, multi-technique study. The liquid density (r), the bulk modulus (K), and its pressure derivative (K′) were measured from 1 atm to 161 GPa using 1-atm double-bob Archimedean, multi-anvil sink/float, and shock wave techniques. Shock compression results on initially molten Fe 2 SiO 4 (1573 K) fitted with previous work and the ultrasonically measured bulk sound speed (C o) in shock velocity (U S)-particle velocity (u p) space yields the Hugoniot: U S = 1.58(0.03) u p + 2.438(0.005) km/s. Sink/float results are in agreement with shock wave and ultrasonic data, consistent with an isothermal K T = 19.4 GPa and K′ = 5.33 at 1500 C. Shock melting of initially solid Fe 2 SiO 4 (300 K) confirms that the Grüneisen parameter (g) of this liquid increases upon compression where g = g o (r o /r) q yields a q value of –1.45. Constraints on the liquid fayalite EOS permit the calculation of isentropes for silicate liquids of general composition in the multicomponent system CaO-MgO-Al 2 O 3 -SiO 2 -FeO at elevated temperatures and pressures. In our model a whole mantle magma ocean would first crystallize in the mid-lower mantle or at the base of the mantle were it composed of either peridotite or simplified "chondrite" liquid, respectively. In regards to the partial melt hypothesis to explain the occurrence and characteristics of ultra-low velocity zones, neither of these candidate liquids would be dense enough to remain at the core mantle boundary on geologic timescales, but our model defines a compositional range of liquids that would be gravitationally stable. (2012), Multi-technique equation of state for Fe 2 SiO 4 melt and the density of Fe-bearing silicate melts from 0 to 161 GPa,
Received 24 February 2006; revised 22 November 2006; accepted 1 March 2007; published 30 June 2007. (1) Polycrystalline samples of Mg2SiO4 forsterite and wadsleyite were synthesized and then dynamically loaded to pressures of 39-200 GPa. Differences in initial density and internal energy between these two phases lead to distinct Hugoniots, each characterized by multiple phase regimes. Transformation to the high-pressure phase assemblage MgO + MgSiO3 perovksite is complete by 100 GPa for forsterite starting material but incomplete for wadsleyite. The datum for wadsleyite shocked to 136 GPa, however, is consistent with the assemblage MgO + MgSiO3 post-perovksite. Marked increases in density along the Hugoniots of both phases between � 130 and 150 GPa are inconsistent with any known solid-solid phase transformation in the Mg2SiO4 system but can be explained by melting. Density increases upon melting are consistent with a similar density increase observed in the MgSiO3 system. This implies that melts with compositions over the entire Mg/Si range likely for the mantle would be negatively or neutrally buoyant at conditions close to the core-mantle boundary, supporting the partial melt hypothesis to explain the occurrence of ultra-low velocity zones at the base of the mantle. From the energetic difference between the high-pressure segments of the two Hugoniots, we estimate a Gruneisen parameter (g )o f 2.6 ±0 .35 for Mg 2SiO4-liquid between 150 and 200 GPa. Comparison to low-pressure data and fitting of the absolute pressures along the melt Hugoniots both require that g for the melt increases with increasing density. Similar behavior was recently predicted in MgSiO3 liquid via molecular dynamics simulations. This result changes estimates of the temperature profile, and hence the dynamics, of a deep terrestrial magma ocean.
Phase relations of the olivine-wadsleyite transition in the system (Mg,Fe)2SiO4 have been determined at 1600 and 1900 K using the quench method in a Kawai-type high-pressure apparatus. Pressure was determined at a precision better than 0.2 GPa using in situ X-ray diffraction with MgO as a pressure standard. The transition pressures of the end-member Mg2SiO4 are estimated to be 14.2 and 15.4 GPa at 1600 and 1900 K, respectively. Partition coefficients for Fe and Mg between olivine and wadsleyite are 0.51 at 1600 K and 0.61 at 1900 K. By comparing the depth of the discontinuity with the transition pressure, the temperature at 410 km depth is estimated to be 1760 ± 45 K for a pyrolitic upper mantle. The mantle potential temperature is estimated to be in the range 1550–1650 K. The temperature at the bottom of the upper mantle is estimated to be 1880 ± 50 K. The thickness of the olivine-wadsleyite transition in a pyrolitic mantle is determined to be between 7 and 13 km for a pyrolitic mantle, depending on the efficiency of vertical heat transfer. Regions of rapid vertical flow (e.g., convection limbs), in which thermal diffusion is negligible, should have a larger transition interval than stagnant regions, where thermal diffusion is effective. This is in apparent contradiction to short-period seismic wave observations that indicate a maximum thickness of
Previous experimental studies of convection in fluids with temperature-dependent viscosity reached viscosity contrasts of the order of 105. Although this value seems large, it still might not be large enough for understanding convection in the interiors of Earth and other planets whose viscosity is a much stronger function of temperature. The reason is that, according to theory, above 104–105 viscosity contrasts, convection must undergo a major transition—to stagnant lid convection. This is an asymptotic regime in which a stagnant lid is formed on the top of the layer and convection is driven by the intrinsic, rheological, temperature scale, rather than by the entire temperature drop in the layer. A finite element multigrid scheme appropriate for large viscosity variations is employed and convection with up to 1014 viscosity contrasts has been systematically investigated in a 2D square cell with free-slip boundaries. We reached the asymptotic regime in the limit of large viscosity contrasts and obtained scaling relations which are found to be in good agreement with theoretical predictions.
We present new equation-of-state (EoS) data acquired by shock loading to pressures up to 245 GPa on both low-density samples (MgSiO3 glass) and high-density, polycrystalline aggregates (MgSiO3 perovskite + majorite). The latter samples were synthesized using a large-volume press. Modeling indicates that these materials transform to perovskite, postperovskite, and/or melt with increasing pressure on their Hugoniots. We fit our results together with existing P-V-T data from dynamic and static compression experiments to constrain the thermal EoS for the three phases, all of which are of fundamental importance to the dynamics of the lower mantle. The EoS for perovskite and postperovskite are well described with third-order Birch-Murnaghan isentropes, offset with a Mie-Grüneisen-Debye formulation for thermal pressure. The addition of shock data helps to distinguish among discrepant static studies of perovskite, and for postperovskite, constrain a value of K' significantly larger than 4. For the melt, we define for the first time a single EoS that fits experimental data from ambient pressure to 230 GPa; the best fit requires a fourth-order isentrope. We also provide a new EoS for Mg2SiO4 liquid, calculated in a similar manner. The Grüneisen parameters of the solid phases decrease with pressure, whereas those of the melts increase, consistent with previous shock wave experiments as well as molecular dynamics simulations. We discuss implications of our modeling for thermal expansion in the lower mantle, stabilization of ultra-low-velocity zones associated with melting at the core-mantle boundary, and crystallization of a terrestrial magma ocean.
Inner core translation, with solidification on one hemisphere and melting on
the other, provides a promising basis for understanding the hemispherical
dichotomy of the inner core, as well as the anomalous stable layer observed at
the base of the outer core - the F-layer - which might be sustained by
continuous melting of inner core material. In this paper, we study in details
the dynamics of inner core thermal convection when dynamically induced melting
and freezing of the inner core boundary (ICB) are taken into account. If the
inner core is unstably stratified, linear stability analysis and numerical
simulations consistently show that the translation mode dominates only if the
viscosity is large enough, with a critical viscosity value, of order Pas, depending on the ability of outer core convection to supply or
remove the latent heat of melting or solidification. If is smaller, the
dynamical effect of melting and freezing is small. Convection takes a more
classical form, with a one-cell axisymmetric mode at the onset and chaotic
plume convection at large Rayleigh number. [...] Thermal convection requires
that a superadiabatic temperature profile is maintained in the inner core,
which depends on a competition between extraction of the inner core internal
heat by conduction and cooling at the ICB. Inner core thermal convection
appears very likely with the low thermal conductivity value proposed by Stacey
& Davis (2007), but nearly impossible with the much higher thermal conductivity
recently put forward. We argue however that the formation of an iron-rich layer
above the ICB may have a positive feedback on inner core convection: it implies
that the inner core crystallized from an increasingly iron-rich liquid,
resulting in an unstable compositional stratification which could drive inner
core convection, perhaps even if the inner core is subadiabatic.
Theory and observations point to the occurrence of magma ponds or oceans in the early evolution of terrestrial planets and in many early-accreting planetesimals. The apparent ubiquity of melting during giant accretionary impacts suggests that silicate and metallic material may be processed through multiple magma oceans before reaching solidity in a planet. The processes of magma ocean formation and solidification, therefore, strongly influence the earliest compositional differentiation and volatile content of the terrestrial planets, and they form the starting point for cooling to clement, habitable conditions and for the onset of thermally driven mantle convection and plate tectonics. This review focuses on evidence for magma oceans on planetesimals and planets and on research concerning the processes of compositional differentiation in the silicate magma ocean, distribution and degassing of volatiles, and cooling.
Mantle Convection in the Earth and Planets is a comprehensive synthesis of all aspects of mantle convection within the Earth, the terrestrial planets, the Moon, and the Galilean satellites of Jupiter. The book includes up-to-date discussions of the latest research developments that have revolutionized our understanding of the Earth and the planets. It is suitable as a text for graduate courses in geophysics and planetary physics, and as a supplementary reference for use at the undergraduate level. It is also an invaluable review for researchers in the broad fields of the Earth and planetary sciences including seismologists, tectonophysicists, geodesists, mineral physicists, volcanologists, geochemists, geologists, mineralogists, petrologists, paleomagnetists, planetary geologists, and meteoriticists. The book features a comprehensive index, an extensive reference list, numerous illustrations (many in color) and major questions that focus the discussion and suggest avenues of future research.
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. 10¹⁷ Pa s when the inner core was 200 km in radius, and ≃3 × 10²¹ 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. © The Author(s) 2018. Published by Oxford University Press on behalf of The Royal Astronomical Society.
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 towards the interface can proceed through it by changing phase. This behaviour is modelled by a boundary condition taking into account the competition between viscous stress in the solid, which builds topography of the interface with a time scale \unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}} , 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 time scale \unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D719}} . The ratio \unicode[STIX]{x1D6F7}=\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D719}}/\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}} controls whether the boundary condition is the classical non-penetrative one ( \unicode[STIX]{x1D6F7}\rightarrow \infty ) or allows for a finite flow through the boundary (small \unicode[STIX]{x1D6F7} ). We study Rayleigh–Bénard convection in a plane layer subject to this boundary condition at either or both its boundaries using linear and weakly nonlinear analyses. When both boundaries are phase-change interfaces with equal values of \unicode[STIX]{x1D6F7} , a non-deforming translation mode is possible with a critical Rayleigh number equal to 24\unicode[STIX]{x1D6F7} . At small values of \unicode[STIX]{x1D6F7} , this mode competes with a weakly deforming mode having a slightly lower critical Rayleigh number and a very long wavelength, \unicode[STIX]{x1D706}_{c}\sim 8\sqrt{2}\unicode[STIX]{x03C0}/3\sqrt{\unicode[STIX]{x1D6F7}} . 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 and the critical wavelength is \unicode[STIX]{x1D706}_{c}=5 . The Nusselt number increases approximately two times faster with the 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, towards larger values when the bottom boundary is a phase-change interface.
Solidification of magma oceans (MOs) formed early in the evolution of planetary bodies sets the initial condition for their evolution on much longer time scales. Ideal fractional crystallization would generate an unstable chemical stratification that subsequently overturns to form a stably stratified mantle. The simplest model of overturn assumes that cumulates remain immobile until the end of MO solidification. However, overturning of cumulates and thermal convection during solidification may act to reduce this stratification and introduce chemical heterogeneity on scales smaller than the MO thickness. We explore overturning of cumulates before the end of MO crystallization and the possible consequences for mantle structure and composition. In this model, increasingly dense iron-rich layers, crystallized from the overlying residual liquid MO, are deposited on a thickening cumulate layer. Overturn during solidification occurs if the dimensionless parameter, Rc, measuring the ratio of the MO time of crystallization τMO to the timescale associated with compositional overturn τov=μ/ΔρgH exceeds a threshold value. If overturn did not occur until after solidification, this implies that the viscosity of the solidified mantle must have been sufficiently high (possibly requiring efficient melt extraction from the cumulate) for a given rate of solidification. For the lunar MO, possible implications for the generation of the Mg-suites and mare basalt are suggested.
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Crystallization and chemical differentiation of the early Earth were governed by a combination of various processes. Within the uncertainties of physical parameters, the two end-member models, equilibrium crystallization and fractional crystallization, are both possible. In terms of crystal size, the boundary between these models is ~1mm. Analysis of nucleation and crystal growth in the convecting magma ocean suggests that the crystals in the magma ocean are approximately this size. The equilibrium model is preferred because it seems to satisfy better the geochemical constraints. According to this model, at the early stages, crystallization proceeds from bottom up without any substantial crystal-melt segregation. In a vigorously convecting magma ocean, the thermal profile is approximately adiabatic and the crystal fraction decreases gradually with depth. The basic physical reason for equilibrium crystallization is a fast convective cooling during which the solid and the liquid phases had a very limited time window for crystal-melt segregation, about 1000 years. Chemical differentiation due to crystal-melt segregation is more significant at low pressures corresponding - with large uncertainties - to the upper mantle. It is caused by several factors. When the temperature drops below liquidus everywhere, the nucleation-growth-dissolution cycle of crystals changes to continuous crystal growth. This increases the crystal size to ~1cm indicating the beginning of crystal-melt segregation. When the crystal fraction increases to about 60% near the surface, the convective heat transport is controlled by solid-state creep, which is many orders of magnitude slower. Crystal-melt segregation occurs via melt percolation, which is consistent with the geochemical constraints. Crystallization of the remaining partially molten layers takes about 10⁷-10⁹ years. This stage merges with the subsequent planetary evolution controlled by mantle convection and radiogenic heating. The long lifetime of the shallow magma ocean suggests that this might be the place where liquid iron equilibrates with the mantle before it sinks down to the core.
Terrestrial planets are thought to experience episode(s) of large-scale melting early in their history. Fractionation during magma-ocean freezing leads to unstable stratification within the related cumulate layers due to progressive iron enrichment upwards, but the effects of incremental cumulate overturns during MO crystallization remain to be explored. Here, we use geodynamic models with a moving-boundary approach to study convection and mixing within the growing cumulate layer, and thereafter within the fully-crystallized mantle. For fractional crystallization, cumulates are efficiently stirred due to subsequent incremental overturns, except for strongly iron-enriched late-stage cumulates, which persist as a stably stratified layer at the base of the mantle for billions of years. Less extreme crystallization scenarios can lead to somewhat more subtle stratification. In any case, the long-term preservation of at least a thin layer of extremely enriched cumulates with Fe#>0.4, as predicted by all our models, is inconsistent with seismic constraints. Based on scaling relationships, however, we infer that final-stage Fe-rich magma-ocean cumulates originally formed near the surface should have overturned as small diapirs, and hence undergone melting and reaction with the host rock during sinking. The resulting moderately iron-enriched metasomatized/hybrid rock assemblages should have accumulated at the base of the mantle, potentially fed an intermittent basal magma ocean, and be preserved through the present-day. Such moderately iron-enriched rock assemblages can reconcile the physical properties of the large low shear-wave velocity provinces in the present-day lower mantle. Thus, we reveal Hadean melting and rock-reaction processes by integrating magma-ocean crystallization models with the seismic-tomography snapshot.
The energy sources involved in the early stages of the formation of terrestrial bodies can induce partial or even complete melting of the mantle, leading to the emergence of magma oceans. The fractional crystallization of a magma ocean can cause the formation of a compositional layering that can play a fundamental role for the subsequent long-term dynamics of the interior and for the evolution of geochemical reservoirs. In order to assess to what extent primordial compositional heterogeneities generated by magma ocean solidification can be preserved, we investigate the solidification of a whole-mantle Martian magma ocean, and in particular the conditions that allow solid-state convection to start mixing the mantle before solidification is completed. To this end, we performed 2-D numerical simulations in a cylindrical geometry. We treat the liquid magma ocean in a parametrized way while we self-consistently solve the conservation equations of thermochemical convection in the growing solid cumulates accounting for pressure-, temperature- and, where it applies, melt-dependent viscosity. By testing the effects of different cooling rates and convective vigor, we show that for a lifetime of the liquid magma ocean of 1 Myr or longer, the onset of solid-state convection prior to complete mantle crystallization is likely and that a significant part of the compositional heterogeneities generated by fractionation can be erased by efficient mantle mixing. We discuss the consequences of our findings in relation to the formation and evolution of compositional reservoirs on Mars and on the other terrestrial bodies of the Solar System.
Energy sources involved in the early stages of planetary formation can cause partial or even complete melting of the mantle of terrestrial bodies leading to the formation of magma oceans. Upon planetary cooling, solidification is expected to take place from the bottom upwards because of the steeper slope of the liquid adiabat with respect to the liquidus (Elkins-Tanton, 2012; Solomatov, 2015). Fractional solidification, in particular, can lead to the formation of a compositional layering that can play a fundamental role for the subsequent long-term dynamics and evolution of the interior (Tosi et al., 2013; Plesa et al., 2014). In order to assess to what extent primordial compositional heterogeneities generated upon magma ocean solidification can be preserved, we investigate the cooling and solidification of a whole-mantle magma ocean along with the conditions that allow solid-state convection to start mixing the mantle before solidification has completed. To this end, we run 2-D numerical simulations in cylindrical geometry using the finite-volume code GAIA (Hüttig et al., 2013). We treat the liquid magma ocean in a parametrized fashion while we self-consistently solve the conservation equations of thermochemical convection in the growing solid mantle accounting for pressure-, temperature- and melt-dependent rheology. We consider two end-member cases: fractional crystallization, where melt is instantaneously extracted into the overlying liquid leaving beneath a differentiated mantle, and batch crystallization where melt remains in contact with the silicate matrix throughout solidification causing no differentiation. By testing the effects of different cooling rates and Rayleigh numbers, we show that for a lifetime of the liquid magma ocean between 1 and 10 Myr (Lebrun et al., 2013), the onset of solid state convection prior to complete mantle crystallization is possible and that part or even all of the compositional heterogeneities generated upon fractionation can be erased by efficient mantle stirring (Figure 1). We discuss the consequences of our findings in relation to the early and long-term evolution of compositional heterogeneities generated via fractional crystallization of magma oceans in terrestrial bodies with emphasis on Mars' thermochemical history.
The intense plume activity at the South Pole of Enceladus together with the recent detection of libration hints at an internal water ocean underneath the outer ice shell. However, the interpretation of gravity, shape, and libration data leads to contradicting results regarding the depth of ocean/ice interface and the total volume of the ocean. Here we develop an interior structure model consisting of a rocky core, an internal ocean, and an ice shell, which satisfies simultaneously the gravity, shape, and libration data. We show that the data can be reconciled by considering isostatic compensation including the effect of a few hundred meter thick elastic lithosphere. Our model predicts that the core radius is 180–185 km, the ocean density is at least 1030 kg/m3, and the ice shell is 18–22 km thick on average. The ice thicknesses are reduced at poles decreasing to less than 5 km in the south polar region.
Numerical models show that small-scale convection (SSC) occurring atop a mantle plume is a plausible mechanism to rejuvenate the lithosphere. The triggering of SSC depends on the density contrast and on the rheology of the unstable layer underlying the stagnant upper part of the thermal boundary layer (TBL). Both properties may be changed by partial melting. We analyze, using 2D numerical simulations, how partial melting influences the dynamics of time-dependent SSC instabilities and the resulting thermo-mechanical rejuvenation of an oceanic plate moving atop of a plume. Our simulations show a complex behavior, with acceleration, no change, or delay of the SSC onset, due to competing effects of the latent heat of partial melting, which cools the plume material, and of the buoyancy increase associated with both melt retention and depletion of residue following melt extraction. The melt-induced viscosity reduction is too localized to affect significantly SSC dynamics. Faster SSC triggering is promoted for low melting degrees (low plume temperature anomalies, thick lithosphere, or fast moving plates), which limit both the temperature reduction due to latent heating and the accumulation of depleted buoyant residue to the upper part of the unstable layer. In contrast, high partial melting degrees lead to a strong temperate decrease due to latent heat of melting and development of a thick depleted layer within the sublithospheric convecting layer, which delays the development of gravitational instabilities. Despite differences in SSC dynamics, the thinning of the lithosphere is not significantly enhanced relatively to simulations that neglect partial melting. This article is protected by copyright. All rights reserved.
We review both the constraints and the models pertaining to the global energy budget of the mantle.
We consider a fluid crossing a zone of rapid density change, so thin that it can be considered as a density jump interface. In this case, the normal velocity undergoes a jump. For a Newtonian viscous fluid with low Reynolds number (creeping flow) that keeps its rheological properties within the interface, we show that this implies that the traction cannot be continuous across the density jump because the tangential stress is singular. The appropriate jump conditions are established by using the calculus of distributions, taking into account the curvature of the interface as well as the density and viscosity changes. Independently of any intrinsic surface tension, a dynamic surface tension appears and turns out to be proportional to the mass transfer across the interface and to a coefficient related to the variations of density and viscosity within the interface. Explicit solutions are exhibited to illustrate the importance of these new jump conditions. The example of the Earth's inner core crystallisation is questioned.
Several quantities measured by the Cassini-Huygens mission provide insight into the interior of Titan: the second-degree gravity field coefficients, the shape, the tidal Love number, the electric field, and the orientation of its rotation axis. The measured obliquity and tides, as well as the electric field, are evidence for the presence of an internal global ocean beneath the icy shell of Titan. Here we use these different observations together to constrain the density profile assuming a four-layer interior model (ice I shell, liquid water ocean, high pressure ice mantle, and rock core). Even though the observed second degree gravity field is consistent with the hydrostatic relation J2=10C22/3J2=10C22/3, which is a necessary but not sufficient condition for a synchronous satellite to be in hydrostatic equilibrium, the observed shape of the surface as well as the non-zero degree-three gravity signal indicate some departure from hydrostaticity. Therefore, we do not restrain our range of assumed density profiles to those corresponding to the hydrostatic value of the moment of inertia (0.34). From a range of density profiles consistent with the radius and mass of the satellite, we compute the obliquity of the Cassini state and the tidal Love number k2k2. The obliquity is computed from a Cassini state model for a satellite with an internal liquid layer, each layer having an ellipsoidal shape consistent with the measured surface shape and gravity field. The observed (nearly hydrostatic) gravity field is obtained by an additional deflection of the ocean–ice I shell interface, assuming that the layers have uniform densities. We show that the measured obliquity can be reproduced only for internal models with a dense ocean (between 1275 and 1350 kg m−3) above a differentiated interior with a full separation of rock and ice. We obtain normalized moments of inertia between 0.31 and 0.33, significantly lower than the expected hydrostatic value (0.34). Evolutionary mechanisms leading to a significant departure from hydrostatic equilibrium while J2=10C22/3J2=10C22/3 remain an open issue. The tidal Love number is found to be mostly sensitive to the ocean density and to a lesser extent to the ice shell thickness. By combining obliquity and tidal Love number constraints, we show that the thickness of the outer ice shell is at least 40 km and the ocean thickness is less than 100 km, with an averaged density of 1300–1350 kg m−3. The elevated density (>3400 kg m−3) found for the rocky core further suggests that it might possess a significant fraction of iron.
Mantle Convection in the Earth and Planets is a comprehensive synthesis
of all aspects of mantle convection within the Earth, the terrestrial
planets, the Moon, and the Galilean satellites of Jupiter. The authors
include up-to-date discussions of the latest research developments that
have revolutionized our understanding of the Earth and the planets. The
book features a comprehensive index, an extensive reference list,
numerous illustrations (many in color) and major questions that focus
the discussion and suggest avenues of future research. It is suitable
as a text for graduate courses in geophysics and planetary physics, and
as a supplementary reference for use at the undergraduate level. It is
also an invaluable review for researchers in the broad fields of the
Earth and planetary sciences.
[1] We performed shock compression experiments on preheated forsterite liquid (Mg2SiO4) at an initial temperature of 2273 K and have revised the equation of state (EOS) that was previously determined by shock melting of initially solid Mg2SiO4 (300 K). The linear Hugoniot, US = 2.674 ± 0.188 + 1.64 ± 0.06 up km/s, constrains the bulk sound speed within a temperature and composition space as yet unexplored by 1 bar ultrasonic experiments. We have also revised the EOS for enstatite liquid (MgSiO3) to exclude experiments that may have been only partially melted upon shock compression and also the EOS for anorthite (CaAl2SiO6) liquid, which now excludes potentially unrelaxed experiments at low pressure. The revised fits and the previously determined EOS of fayalite and diopside (CaMg2SiO6) were used to produce isentropes in the multicomponent CaO-MgO-Al2O3-SiO2-FeO system at elevated temperatures and pressures. Our results are similar to those previously presented for peridotite and simplified “chondrite” liquids such that regardless of where crystallization first occurs, the liquidus solid sinks upon formation. This process is not conducive to the formation of a basal magma ocean. We also examined the chemical and physical plausibility of the partial melt hypothesis to explain the occurrence and characteristics of ultra-low velocity zones (ULVZ). We determined that the ambient mantle cannot produce an equilibrium partial melt and residue that is sufficiently dense to be an ultra-low velocity zone mush. The partial melt would need to be segregated from its equilibrium residue and combined with a denser solid component to achieve a sufficiently large aggregate density.
The structure and time dependence of 3-D thermal convection in a volumetrically heated, infinite Prandtl number fluid is examined for high values of the Rayleigh number. The methods employed allow the numerical experiments to proceed for long-enough times to derive good estimates of mean and fluctuating parts of the structure. An iterative multirigid method to solve for the buoyant, incompressible viscous flow at each time step of the energy equation is a novel aspect of the methodology. A simple explicit time step of the energy equation is utilized that vectorizes well on serial computers and which is ideally suited to massively parallel computers. Numerical experiments were carried out for Rayleigh numbers from 3 × 106 to 3 × 107 in a cartesian region with a prescribed temperature at the top boundary and an adiabatic bottom boundary. Over this complete range of Rayleigh number, the flow structure consists dominantly of cold, nearly axisymmetric plumes that migrate horizontally sweeping off the cold thermal-boundary layer that forms at the top of the convecting fluid. Plumes disappear by coalescing with other plumes; new plumes are created by thermal-boundary-layer instability. Sheet plumes form only occasionally and do not penetrate to significant depths in the fluid. Plumes have sizes comparable to the thickness of the thermal-boundary layer and an average spacing comparable to the fluid depth. No persistent large-scale motion in the fluid can be identified. Its absence may reflect the large subadiabatic stratification that develops beneath the thermal-boundary layer as cold plumes penetrate to the bottom boundary without thermally equilibrating with surrounding fluid. We consider the possible implications for convection in planetary mantles and for the existence of plate tectonics.
The large temperature excess in young planets is probably evacuated as much by intense volcanic activity as by thermal diffusion across a lithosphere. However, most fluid dynamic codes in Earth Science are implementing a zero vertical velocity at the surface of their models. This condition forbids any direct transport of material and therefore any direct extraction of heat from depth. We propose a new set of boundary conditions on the top surface for the momentum and energy equations. The vertical velocity is not imposed to zero but we compute the topography generated from this velocity. A diffusion term mimics the various processes that can redistribute the topography (mechanical and chemical erosion, gravity sliding, magma spreading on the surface...). The resulting topography affects the internal flow by imposing an equivalent vertical stress to the mantle. We show that with minimal approximations the new condition can be very easily implemented. In the energy equation we only impose a surface temperature when the surface velocity is downward and a zero temperature gradient elsewhere. Our 2D numerical simulations of bottom heated convection show that when increasing the Rayleigh number, the model evolves continuously from the typical pattern of convection and heat diffusion through a thermal lithosphere to a planform where all the heat is brought to the surface by large plumes. The extracted heat flux increases with the Rayleigh number from Ra^.3 at low Rayleigh number to Ra^.5 at high Rayleigh number where simultaneously, the planet temperature decreases from .5 to very low values. We discuss the implication of this model for the early evolution of the Earth and other solid planets, and the present state of Io.
Previous experimental studies of convection in fluids with temperature-dependent viscosity reached viscosity contrasts of the order of 105. Although this value seems large, it still might not be large enough for understanding convection in the interiors of Earth and other planets whose viscosity is a much stronger function of temperature. The reason is that, according to theory, above 104-105 viscosity contrasts, convection must undergo a major transition-to stagnant lid convection. This is an asymptotic regime in which a stagnant lid is formed on the top of the layer and convection is driven by the intrinsic, rheological, temperature scale, rather than by the entire temperature drop in the layer. A finite element multigrid scheme appropriate for large viscosity variations is employed and convection with up to 1014 viscosity contrasts has been systematically investigated in a 2D square cell with free-slip boundaries. We reached the asymptotic regime in the limit of large viscosity contrasts and obtained scaling relations which are found to be in good agreement with theoretical predictions.
Understanding the origins of the diversity in terrestrial planets is a fundamental goal in Earth and planetary sciences. In the Solar System, Venus has a similar size and bulk composition to those of Earth, but it lacks water. Because a richer variety of exoplanets is expected to be discovered, prediction of their atmospheres and surface environments requires a general framework for planetary evolution. Here we show that terrestrial planets can be divided into two distinct types on the basis of their evolutionary history during solidification from the initially hot molten state expected from the standard formation model. Even if, apart from their orbits, they were identical just after formation, the solidified planets can have different characteristics. A type I planet, which is formed beyond a certain critical distance from the host star, solidifies within several million years. If the planet acquires water during formation, most of this water is retained and forms the earliest oceans. In contrast, on a type II planet, which is formed inside the critical distance, a magma ocean can be sustained for longer, even with a larger initial amount of water. Its duration could be as long as 100 million years if the planet is formed together with a mass of water comparable to the total inventory of the modern Earth. Hydrodynamic escape desiccates type II planets during the slow solidification process. Although Earth is categorized as type I, it is not clear which type Venus is because its orbital distance is close to the critical distance. However, because the dryness of the surface and mantle predicted for type II planets is consistent with the characteristics of Venus, it may be representative of type II planets. Also, future observations may have a chance to detect not only terrestrial exoplanets covered with water ocean but also those covered with magma ocean around a young star.
The thickness of Europa's ice shell is constrained with numerical experiments of thermal convection, including heterogeneous tidal heating. Thermal convection occurs in the stagnant lid regime with most of the tidal heating located in the bottom half of the layer. The addition of tidal heating mainly results in the increase of the temperature of the well-mixed interior and in the decrease of the heat flux at the base of the ice layer. In many simulations, the ice in hot plumes is heated up to its melting point. This induces episodic upwellings (0.5 Ma) of partially molten ice up to the base of the conductive lid, with possible implications for the formation of lenticulae and chaos regions. The thickness of the convective ice shell in equilibrium with the heat flow from the silicate core is estimated to be about 20–25 km. Tidal dissipation and surface temperature variations create lateral variations of the ice shell thickness of about 5 km, with maxima near the equator at the Jovian and anti-Jovian points and minima at midlatitudes. Surface heat flux is about 35–40 mW.m−2; it is almost constant all over Europa's surface, even though the tidal dissipation rate is four times larger at the poles than at the equator.
A fundamental property of a convecting fluid is its planformthe distribution in the horizontal plane of hot rising regions and cold sinking regions. For the Earth's mantle the planform might he visualized as a map of subduction zones, hotspots and possibly ocean ridges. Here I report numerical experiments of convection at high Rayleigh number which show a strong dependence of planform on heating mode. When heat generation is distributed uniformly through the box the preferred planform consists of an ensemble of time-dependent cold axial sinkers distributed in a hot diffuse upward flow. When half of the heat is generated within the box and the other half is input through the base, the preferred planform consists of an array of hot axial plumes and elongated cold sheets. In the former case the mean horizontal wavelength is about equal to the layer depth; for the latter it is about twice the layer depth.
We investigated the melting properties of a synthetic chondritic primitive mantle up to core-mantle boundary (CMB) pressures, using laser-heated diamond anvil cell. Melting criteria are essentially based on the use of X-rays provided by synchrotron radiation. We report a solidus melting curve lower than previously determined using optical methods. The liquidus curve is found between 300 and 600. K higher than the solidus over the entire lower mantle. At CMB pressures (135. GPa), the chondritic mantle solidus and liquidus reach 4150 (±. 150) K and 4725 (±. 150) K, respectively. We discuss that the lower mantle is unlikely to melt in the D"-layer, except if the highest estimate of the temperature profile at the base of the mantle, which is associated with a very hot core, is confirmed. Therefore, recent suggestions of partial melting in the lowermost mantle based on seismic observations of ultra-low velocity zones indicate either (1) a outer core exceeding 4150. K at the CMB or (2) the presence of chemical heterogeneities with high concentration of fusible elements. Our observations of a high liquidus temperature as well as a large gap between solidus and liquidus temperatures have important implications for the properties of the magma ocean during accretion. Not only complete melting of the lower mantle would require excessively high temperatures, but also, below liquidus temperatures partial melting should take place over a much larger depth interval than previously thought. In addition, magma adiabats suggest very high surface temperatures in case of a magma ocean that would extend to more than 40. GPa, as suggested by siderophile metal-silicate partitioning data. Such high surface temperature regime, where thermal blanketing is inefficient, points out to a transient character of the magma ocean, with a very fast cooling rate.
Physics of the Earth and Planetary Interiors j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / p e p i a b s t r a c t Most methods for modeling mantle convection in a two-dimensional (2D) circular annular domain suffer from innate shortcomings in their ability to capture several characteristics of the spherical shell geometry of planetary mantles. While methods such as rescaling the inner and outer radius to reduce anomalous effects in a 2D polar cylindrical coordinate system have been introduced and widely implemented, such fixes may have other drawbacks that adversely affect the outcome of some kinds of mantle convection studies. Here we propose a new approach that we term the "spherical annulus," which is a 2D slice that bisects the spherical shell and is quantitatively formulated at the equator of a spherical polar coordinate system after neglecting terms in the governing equations related to variations in latitude. Spherical scaling is retained in this approximation since the Jacobian function remains proportional to the square of the radius. We present example calculations to show that the behavior of convection in the spherical annulus compares favorably against calculations performed in other 2D annular domains when measured relative to those in a fully three-dimensional (3D) spherical shell.
Abstract— Crystallization of a magma ocean on a large terrestrial planet that is significantly melted by the energy of accretion may lead to an unstable cumulate density stratification, which may overturn to a stable configuration. Overturn of the initially unstable stratification may produce an early basaltic crust and differentiated mantle reservoirs. Such a stable compositional stratification can have important implications for the planet's subsequent evolution by delaying or suppressing thermal convection and by influencing the distribution of radiogenic heat sources. We use simple models for fractional crystallization of a martian magma ocean, and calculate the densities of the resulting cumulates. While the simple models presented do not include all relevant physical processes, they are able to describe to first order a number of aspects of martian evolution. The models describe the creation of magma source regions that differentiated early in the history of Mars, and present the possibility of an early, brief magnetic field initiated by cold overturned cumulates falling to the coremantle boundary. In a model that includes the density inversion at about 7.5 GPa, where olivine and pyroxene float in the remaining magma ocean liquids while garnet sinks, cumulate overturn sequesters alumina in the deep martian interior. The ages and compositions of source regions are consistent with SNC meteorite data.
Crystallization of the lunar magma ocean creates a chemically stratified Moon consisting of an anorthositic crust and magma ocean cumulates overlying the primitive lunar interior. Within the magma ocean cumulates the last liquids to crystallize form dense, ilmenite-rich cumulates that contain high concentrations of incompatible radioactive elements. The underlying olivine-orthopyroxene cumulates are also stratified with later crystallized, denser, more Fe-rich compositions at the top. This paper explores the chemical and thermal consequences of an internal evolution model accounting for the possible role of these sources of chemical buoyancy. Rayleigh-Taylor instability causes the dense ilmenite-rich cumulate layer and underlying Fe-rich cumulates to sink toward the center of the Moon, forming a dense lunar core. After this overturn, radioactive heating within the ilmenite-rich cumulate core heats the overlying mantle, causing it to melt. In this model, the source region for high-TiO2 mare basalts is a convectively mixed layer above the core-mantle boundary which would contain small and variable amounts of admixed ilmenite and KREEP. This deep high-pressure melting, as required for the generation of mare basalts, occurs after a reasonable time interval to explain the onset of mare basalt volcanism if the content of radioactive elements in the core and the chemical density gradients above the core are sufficiently high but within a range of values that might have been present in the Moon. Regardless of details implied by particular model parameters, gravitational overturn driven by the high density of magma ocean Fe-rich cumulates should concentrate high-TiO2 mare basalt sources, and probably a significant fraction of radioactive heating, toward the center of the Moon. This will have important implications for both the thermal evolution of the Moon and for mare basalt genesis.
The melting curve of perovskite MgSiO3 and the liquidus and solidus curves of the lower mantle were estimated from thermodynamic data and the results of experiments on phase changes and melting in silicates.The initial slope of the melting curve of perovskite MgSiO3 was obtained as at 23 GPa. The melting curve of perovskite was expressed by the Kraut-Kennedy equation as , where Tm≳2900 K and P≳23 GPa; and by the Simon equation, .The liquidus curve of the lower mantle was estimated as (perovskite) and this gives the liquidus temperature Tliq=7000 ±500 K at the mantle-core boundary. The solidus curve of the lower mantle was also estimated by extrapolating the solidus curve of dry peridotite using the slope of the solidus curve of magnesiowüstite at high pressures. The solidus temperature is ∼ 5000 K at the base of the lower mantle. If the temperature distribution of the mantle was 1.5 times higher than that given by the present geotherm in the early stage of the Earth's history, partial melting would have proceeded into the deep interior of the lower mantle.Estimation of the density of melts in the MgOFeOSiO2 system for lower mantle conditions indicates that the initial melt formed by partial fusion of the lower mantle would be denser than the residual solid because of high concentration of iron into the melt. Thus, the melt generated in the lower mantle would tend to move downward toward the mantle-core boundary. This downward transportation of the melt in the lower mantle might have affected the chemistry of the lower mantle, such as in the D″ layer, and the distribution of the radioactive elements between mantle and core.
Molten komatiite and peridotite have been compressed in an octahedral multi-anvil device up to 10 GPa. Densities of the melts were measured at pressure intervals in the range 7 to 10 GPa by observing sinking and floating San Carlos olivines and synthetic forsterite marker spheres. The multi-anvil results for komatiite, when combined with piston-cylinder measurements done at 4 to 6 GPa and a calculated reference density at 105 Pa, yield a Birch-Murnaghan isothermal bulk modulus of(K1900C) = 26 GPa and pressure derivativeK′ = 4.25. The pressure of neutral buoyancy for olivine in komatiite is confirmed to be near 8 GPa as predicted in earlier work. Olivine flotation in the experimental komatiite commences at a pressure close to where the liquidus phase changes from olivine to denser garnet, leading to the possibility of density driven crystal sorting during fractionation. Molten peridotite (KLB-1) shows an isothermal compression (2000°C) of 0.065 g cm−3 GPa−1 in the interval 105 Pa to 8.2 GPa. The olivine/liquid peridotite density crossover is predicted to lie between 9 and 11 GPa, indicating that olivine flotation can operate at depths of 300–500 km in a molten peridotitic mantle.
The Earth is likely to have experienced a magma ocean stage during accretion. Thermal and chemical evolution of magma ocean is investigated based on a one-dimensional two-phase-flow heat and mass transfer model. Differentiation at lower mantle pressure depends on the type of magma ocean and surrounding atmosphere. If the magma ocean is formed by the blanketing effect of a solar-type proto-atmosphere, extensive differentiation proceeds at lower mantle pressure. If the magma ocean is formed by the blanketing effect of an impact-induced steam atmosphere, no differentiation at lower mantle pressure is likely. If a very deep magma ocean is formed by a giant impact, whether differentiation proceeds at lower mantle pressure or not depends on grain size, viscosity of melt and/or properties of a transient atmosphere. On the contrary, chemical differentiation likely proceeds at upper mantle pressure irrespective of magma ocean type. A shallow magma ocean can remain for 100 ∼ 200 My without any heating processes.
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 10^5 and 10^7 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.
A surprisingly simple and precise major element mass balance is consistent with derivation of average upper mantle peridotite from a partially molten chondritic Earth by subtraction of perovskite and addition of olivine. Majorite involvement is precluded unless some as yet unidentified components play a role. Perovskite subtraction during a primordial melting event is expected to occur by crystal fractionation at depth, while olivine addition is accomplished by a combination of buoyancy mechanisms: crystal flotation from a deep layer of melt buried by its own compressibility to the base of the solidifying upper mantle and subsequent solid state convection of this buoyant magnesian olivine upward. These processes are consistent with known density relations of crystals and liquid at very high pressure. Mass balance predicts that the residual magma body at depth after supplying olivine by flotation upward can be komatiitic. Distribution of originally C1 chrondritic bulk Earth material a few 100 m.y. after primordial differentiation is solid peridotite upper mantle, perovskite lower mantle, and a komatiitic liquid sandwich horizon.