Earth and Planetary Science Letters

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1 Earth and Planetary Science Letters 491 (2018) Contents lists available at ScienceDirect Earth and Planetary Science Letters Timing of mantle overturn during magma ocean solidification C.-E. Boukaré, E.M. Parmentier, S.W. Parman Department of Earth, Environmental and Planetary Sciences, Brown University, 324 Brook Street, Providence, RI, 02912, USA a r t i c l e i n f o a b s t r a c t Article history: Received 15 September 2017 Received in revised form 16 March 2018 Accepted 17 March 2018 Available online xxxx Editor: W.B. McKinnon Keywords: magma ocean cumulate overturn lunar interior early planetary mantle dynamics 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, R c, measuring the ratio of the MO time of crystallization τ MO to the timescale associated with compositional overturn τ ov = μ/ ρ gh 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 Elsevier B.V. All rights reserved. 1. Introduction Models of planetary accretion have shown that giant impacts during planet formation may likely melt terrestrial planets substantially (Safronov, 1964, 1978; Hostetler and Drake, 1980; Chambers and Wetherill, 1998; Canup, 2008; Nakajima and Stevenson, 2015). Evolution of such magma ocean(s) (MO) sets the initial conditions for long-term evolution of planetary interiors. Better understanding MO evolution is also motivated by geochemical evidence showing very early fractionation of the mantle of the Moon and Mars (Smith et al., 1970; Warren and Wasson, 1979; Blichert-Toft et al., 1999; Borg et al., 2003; Borg and Draper, 2003). Interestingly, it has been reported that the mantle sources of martian meteorites show strong similarities with lunar mantle sources. Sm/Nd, Rb/Sr and Lu/Hf systematics indicates that the composition of martian meteorites sources could be explained by a mixing of lunar-like compositions, similar to the depleted lunar mafic materials and the enriched KREEPy components (Borg et al., 2003). One of the key question regarding mantle structure is the degree of mantle mixing and how dense components and heterogeneities containing heat producing elements are spatially distributed. Is the mantle * Corresponding author. address: (C.-E. Boukaré). layered? If so, is it stably stratified? Mantle dynamics governs core cooling which is key for magnetic field generation by core dynamo. Surface and subsurface of planets are shaped by mantle dynamics through magmatism and tectonics. Constraining the structure of solid mantles that emerge from MO solidification is thus important but remains poorly understood. The initial solid mantle structure depends on the details of how the MO (MO) solidified. Two key aspects are: (1) bottomup/top down crystallization which controls where the solid mantle accumulate and (2) batch/fractional crystallization which controls extent to which solidification generates compositional heterogeneities. For planetary bodies smaller than the Earth or for those with shallow MOs, silicate MOs would crystallize from the bottom to the top as the liquidus first intersects the adiabat at the bottom of the mantle (Elkins-Tanton, 2012). For large silicate planets (Earth-size and larger), crystallization of a sufficiently deep MO might occur in a top-down fashion due either to the competition between liquidus and adiabat, or the density contrast between melts and solids (Labrosse et al., 2007; Stixrude et al., 2009; Boukaré et al., 2015; Boukaré and Ricard, 2017). In this study, we focus on the structure of solid mantle crystallizing from the bottom to the top by batch or fractional crystallization. As the crystallization front moves towards the surface, the composition of mineral phases added on top of the solidified mantle evolves with pressure and composition of the residual liquid MO X/ 2018 Elsevier B.V. All rights reserved.

2 C.-E. Boukaré et al. / Earth and Planetary Science Letters 491 (2018) (Solomon and Longhi, 1977; Warren, 1985). For the Moon, bottomup fractional crystallization has been proposed to generate an unstable chemical stratification in the solid mantle susceptible to overturn (Hess and Parmentier, 1995 and references therein). The lunar overturn model is a good candidate for explaining the deep origin of early magmatism (Hess and Parmentier, 1995; Shearer et al., 2006), the hemispheric asymmetry of mare basalts (Parmentier et al., 2002), and might be consistent with an early generation of a lunar core dynamo (Stegman et al., 2003). The cumulate overturn hypothesis has also been applied to other planetary bodies such as the Earth (Elkins-Tanton, 2008) and Mars (Elkins-Tanton et al., 2003, 2005). Current models of lunar mantle overturn considered generally post MO overturn where the unstably stratified cumulates remain immobile until the end of MO solidification. However, cumulates are unstably stratified as they form. Whether it waits until after complete solidification to overturn, or overturns (or convects) during solidification is governed by a number of factors that we aim to determine in this study. Models of cumulate overturn have pointed out that the growth time of Rayleigh Taylor instabilities associated with unstable chemical stratification may be smaller than the time of MO crystallization (Hess and Parmentier, 1995). If the cumulate overturns during its solidification, the early lunar mantle dynamics would differ from the post-solidification overturn models. In particular, the chemical layering created by fractional crystallization would be partially homogenized by mantle stirring. The driving forces for a post-mo solidification degree one overturn would be reduced. This study aims to quantify the timing of mantle overturn relative to the time of MO crystallization and the amount of mantle mixing that occurs. We explore the behavior of a simple model in which a viscous cumulate layer thickens progressively by solidification of the overlying well-mixed liquid MO. In the case of fractional crystallization, fractionation of Fe/Mg between solid and liquid results in iron enrichment of residual liquid. Progressively iron enriched MO increases the density of solids forming from the liquid. We solve the Stokes flow in the solidifying MO cumulate in 2D Cartesian geometry. Recently, two studies have investigated the onset of thermochemical convection in the cumulate for the case of Mars (Maurice et al., 2017) and the Earth (Ballmer et al., 2017b). These studies have shown that thermo-chemical convection in the cumulate layer can start before the end of MO solidification affecting the initial degree of mantle mixing. By varying the cumulate viscosity, the rate of MO solidification, the modes of crystallization (i.e., equilibrium vs. fractional) and the compositional density contrast between Mg-rich and Fe-rich mineral phases, Maurice et al. (2017) and Ballmer et al. (2017b) have shown that solid-state mantle convection during MO solidification can lead to various pictures of primitive mantle mixing. Maurice et al. (2017) have quantified the effects of convective vigor on mantle mixing during MO solidification. Such early mantle dynamics can affect the long-term preservation of deep geochemical reservoir (Ballmer et al., 2017b). In contrast to these previous works that have explored a relevant range of parameters, the present study aims to determine dimensionless criterion that characterize the timing of thermo-chemical convection relative to MO solidification. The competitive effects of cumulate viscosity and MO cooling rate, that has been pointed in these previous study (Maurice et al., 2017; Ballmer et al., 2017b), are here quantified. This dimensionless analysis allows to better illustrate the role of planetary size, MO cooling history and cumulate viscosity regarding the issue of early mantle mixing. The simplicity of this approach allows the identification of two crucial aspects of the dynamics affecting the initial degree of mantle mixing. The degree of mantle mixing is controlled by the vigor of syn-solidification convection but also by the rate at which materials of distinct composition are added to the solid mantle. These two dynamic processes can be captured by the thermal Rayleigh number, Ra which is a measure of the convective vigor and a dimensionless parameter R c, which is a measure of the ratio of MO crystallization time, τ MO, and the timescale associated with chemical overturn. The dimensionless parameter R c must exceed a critical value for having a syn-solidification overturn. For a given convective vigor, syn-solidification convection increases the degree of mantle mixing. Indeed, mixing progressively small amount of dense components is much easier than mixing instantaneously large amounts of dense materials. By quantitatively linking mantle viscosity, MO crystallization time and degree of mantle mixing, the present work offers a basic model to relate some aspects of early planet evolution such as timing and spatial distribution of magmatism, compositional variability of lavas and the amount of heat extracted from the mantle. 2. Model 2.1. General approach To explore the role of mantle flow during MO solidification, a simple model is used in which a partially molten viscous cumulate layer thickens progressively by solidification of an overlying well-mixed, liquid MO. The top of the cumulate layer, defined by a freezing temperature that increases with depth, thickens with time as the remaining liquid layer cools. The freezing temperature being the liquidus temperature of the residual liquid layer. Here, the case of fractional solidification is adopted where Fe/Mg fractionation between solid and liquid results in iron enrichment of the residual liquid, thus progressively increasing the density of solids forming from this liquid. The case of batch crystallization (with no chemical fractionation) associated with the issue of retained melt in the cumulate is discussed later. Viscous flow in the cumulate layer occurs in response to the unstable compositional and/or thermal stratification. In turn, the viscous flow causes compositional mixing and convective heat transport. Rather than modeling a particular MO scenario, the goal of the current study is to investigate the effects of mantle thermal stratification resulting from the evolving freezing temperature, chemical stratification (generated by the mode of solidification) and MO solidification rate on the solid-state cumulate convection. In this model, the MO cumulate grows by adding layers of small but finite thickness at a prescribed, constant rate. τ MO is the time for solidification when the thickness of the cumulate layer, h, reaches the initial MO thickness, H. In an actual MO, this rate would vary and reflect the rate of heat loss from the planetary surface. The top of the cumulate layer corresponds to the rheologically critical melt fraction (RCMF) at which the cumulate viscosity becomes close to that of the solid (Goetze, 1977; Arzi, 1978; Costa, 2005). The upward velocity of this crystallization front is controlled by both the deposition rate and compaction rate of cumulates. If the crystallization front velocity is less than a critical value, compaction occurs in two zones, one near the crystallization front and the other at the bottom of the cumulates. These zones separated by a layer of uniform melt fraction (Shirley, 1986). The critical crystallization front velocity, v i is controlled by the buoyant percolation velocity, v i K ρ s l g/μ l where K is the permeability, ρ s l is the density contrast between melt and solid, g is the gravity acceleration and μ l is the melt viscosity. The thickness of the compaction zone just beneath crystallization front depends on the compaction length, and the residual melt fraction beneath it is controlled by buoyant melt percolation. The segregation of crystals and melt either by settling of dense crystals above the RCMF boundary or compaction in partially molten mantle beneath it is a process not treated explicitly in this model. Here, the upward

3 218 C.-E. Boukaré et al. / Earth and Planetary Science Letters 491 (2018) velocity of this moving top boundary is imposed and set to a constant that reflects an idealized time of MO solidification, τ MO. The cumulate viscosity is a constant interpreted as the effective cumulate viscosity in the presence of retained melt. The application of this simplified approach to more realistic case where both cumulate viscosity and τ MO change with time is discussed in section 4. It is important to note that this study considers the solidifying mantle as isolated from the overlying liquid MO. The crystallization front is a complex interface which may allow interactions between solid mantle dynamics and liquid MO dynamics. For instance, syn-solidification overturn should be associated with solid mantle remelting near the crystallization front (see Fig. 2). Solid mantle remelting may change the composition of the liquid MO which would affect the composition of the crystallizing phases. Moreover, in the case of impermeable boundary conditions, overturn dynamics is limited by viscous deformation in the upper and bottom part of the convective system. Permeable top boundary might facilitate overturn as material can melt, i.e., cross the top boundary, instead of being deformed (Labrosse et al., 2017). The composition of the layers added on top of the solidifying mantle is controlled as follows. We assume a homogeneous liquid MO overlying a crystallizing cumulate layer. At the crystallization front, C s id (h) = DCl (1) where D represents the iron Nerst partition coefficient between melt and solid, h is the thickness of the solidifying mantle, C l is the iron composition of the MO and C s is the iron composition id of the crystallizing layer. Using mass conservation of iron during crystallization of an infinitely thin layer, we can show that, ( ) 1 D C s id (h ) = DC l h 3 (2) where C l 0 is the iron-content of the MO before solidification. Equation (2) represents the idealized iron-content profile of the cumulate in the case where cumulate remains immobile during MO solidification. Note that equation (2) is based on iron mass conversation in a spherical geometry. Even though this model treats the dynamics in a 2D Cartesian geometry, we account for the effect of spherical geometry on fractional crystallization. Perfect fractional solidification assumes that, once deposited, solid no longer maintains equilibrium with the residual MO liquid. This will be valid if the compaction zone is thin and the residual melt fraction in the cumulate is small as discussed above. Retained melt within the portion of the still partially molten cumulate layer, where temperature is above the solidus, can provide transport pathways that if sufficiently rapid could allow equilibration with overlying MO liquid. The effect of retained melt alone can be described by a reduced effective value of the partition coefficient. In contrast to the fractional crystallization case, perfect batch crystallization will rule out the occurrence of a compositional overturn. In this case, early convection in the cumulate will be driven by thermal density variation only. Potential application of this analysis to the case of batch crystallization is discussed later. The model intentionally uses a uniform viscosity to clearly illustrate the influence of viscosity on the overturn timing relative to MO solidification. However, viscosity of partially molten mantle is expected to depend strongly on the melt fraction present, particularly for melt fractions near the RCMF. The appropriate viscosity value in models which meet the above criteria is that at the residual melt fraction, which is expected to approach the one at the RCMF for very rapid solidification. There is abundant room for further investigation of the effect of viscosity variation due to stress, pressure, temperature or composition. The study of Maurice et al. (2017) for example treats more realistic mantle rheologies. One example where the effect of composition may be important is the possible early isolation of highly viscous Si-rich components during the solidification of a putative Earth s MO which might control the structure of the Earth s mantle convection (Ballmer et al., 2017a). We argue here that solid mantle viscosity during mantle solidification would be controlled to first order by the fraction of interstitial melt. The temperature of the layers added on top of the solidifying mantle is set to a freezing temperature. The freezing temperature is the liquidus temperature of the residual liquid. In the case of batch crystallization, the relevant freezing temperature should be associated with the temperature of the RCMF. In the case of fractional crystallization, the freezing temperature should also account for the evolution of the MO liquid composition Governing parameters In the solidifying layer, Fe-free silicate and pure Fe-silicate are present in volume proportion 1 C s and C s. They have properties denoted by the subscript f and m. We consider thermo-chemical convection, so we reduce the density ρ f and ρ m to, ρ f = ρ 0 (1 αt ) ρ, ρ m = ρ 0 (1 αt ) 1 2 ρ, (3) where ρ 0 is the reference density, α is the coefficient of thermal expansion (we assume that the two chemical end-members have the same α). ρ is the density contrast between Fe-free silicate and pure Fe silicate. Behavior in the thickening cumulate layer depends on a competition between τ MO and the combined effects of overturn of an unstable compositional stratification and thermal convection. The timing of overturn of an unstable compositional or thermal stratification is governed by dimensionless parameters, R c, that measure the ratio of τ MO to the timescale associated with compositional or thermal overturn. The vigor of compositional and thermal convection are described by Rayleigh numbers that measure advection timescale due to compositional or thermal buoyancy relative to the respective diffusion timescale. We non-dimensionalize lengths by the thickness of the mantle H, time by τ MO the time of MO crystallization, velocity by H/τ MO and pressures by μ/τ MO. The temperature difference of reference, T ref accounts for the slope of the freezing temperature and may require to be corrected by the adiabat, T ref = ( z T F z T ad) H, where T F is the freezing temperature and T ad is the adiabatic temperature. We assume that T F and T ad vary linearly with depth (we use z T ad = 0for the sake of simplicity). Dimensionless quantities are denoted with the superscript. Using this scaling, mass, momentum and energy conservation equations write, v = 0 (4) + 2 v = C s R c h + T R T h (5) t T + v T = R T d h 2 2 T, (6) t C s + v C s = Rc d h 2 2 C s, (7) where is the deviation from lithostatic pressure, v is the velocity, h is the variable thickness of the cumulate, T is the temperature and,

4 C.-E. Boukaré et al. / Earth and Planetary Science Letters 491 (2018) R c = τ MO μ ρ gh, R T = τ MO μ ρ 0α T ref gh, R c d = τ MO H 2 κ c, R d = τ MO κ H 2 T κ c and κ T are the chemical and thermal diffusivity, respectively. g is the gravity acceleration. R c and R T have the physical significance discussed above. The R d s are the chemical (or thermal) diffusion timescale normalized by τ MO. These four dimensionless numbers describe a general case where both temperature and composition diffuse. Depending on the situation of interest, various assumptions can be made. For instance, it is here assumed that chemical diffusion in solid silicate rocks is negligible compared to thermal diffusion. Here, R c is thus d set to 0. Then, the density of the heaviest chemical components, their partitioning coefficient and the mode of crystallization (i.e., batch vs. fractional) set the extent of compositional density stratification. If the solidifying system undergoes batch solidification or there is a small enough density difference between chemical endmembers, ρ can be set to 0, so R c = 0. In this case, convection in the cumulate is driven only by thermal density variation imposed by the freezing temperature. Alternatively, for a large enough fractionation of dense element (e.g., iron), both thermal and compositional effects must be accounted, i.e., both R c and R T are required. This work aims to quantify the timing of cumulate overturn in the context of silicate MO where both thermal and compositional stratification are potentially unstable. Here, R c, R T and R d are used to characterize the convective regime in the cumulate. Adopting the case fractional crystallization, the iron partitioning generates a compositional density stratification more unstable than the thermal stratification (R c /R T = ρ/ρ 0 α T ref 30). Therefore, R c is used here to characterize the timing of cumulate convection relative to MO solidification. The goal is to determine the critical R c value at which the overturn occur before the end of MO solidification. The present work may also be applied to batch crystallization. For this case, this analysis provides a necessary condition for having syn-solidification thermal convection: R T must be supercritical. The second necessary condition is that the thermal diffusion timescale must be lower than the thermal overturn timescale, i.e., thermal Rayleigh number must be super-critical. Generally, non-dimensionalization of thermal convection problems in geophysics uses thermal diffusion timescale as reference for time (Schubert et al., 2001). This non-dimensionalization results in the appearance of the thermal Rayleigh number, Ra = ρ 0 α T ref H 3 Dμ. In this model, the thermal Rayleigh number can be computed as it follows, Ra = R T R d. (9) While the thermal Rayleigh number Ra is a good proxy for estimating thermal convective vigor, R c and R T describe the rate at which the cumulate layer thickens relative to a chemical or thermal overturn timescale. The numerical methods used to solve equations (5), (6), (7) are presented in the supplementary materials Quantifying mantle chemical stratification during solidification In order to follow the evolution of the iron stratification in the solidifying layer, we calculate the spatial distribution of iron using the following potential, (8) Fig. 1. Thermo-chemical structures after magma ocean (MO) solidification. The differences between the two cases are due to differences between mantle overturn timing relative to MO solidification. a) Classic overturn (i.e., late overturn occurring after MO solidification) with the solidification timescale shorter than overturn timescale. The dimensionless number are R c = and R d = (see equation (8)) and correspond to a MO s time of crystallization, τ MO, of 0.5 Myrs and a MO cumulate viscosity, μ, of Pa s in the case of the Moon. b) Early convection model with solidification timescale larger than overturn timescale. The dimensionless number are R c = and R d = 10 3 and correspond to τ MO = 50 Myrs and μ = Pa s. For a given cumulate layer viscosity, early convection during MO crystallization increases the degree of mixing. The snapshots are taken 100 Myrs after the onset of solidification (at τ = 200 τ MO for the late overturn case and at τ = 2 τ MO for the early convection model). Movies are available in supplementary materials. (h ) = h 0 C s (x)xdx (10) where C s is the horizontal averaged iron-content at a given depth in the numerical simulations. As the iron content C s substantially affects density, the quantity can be seen as the gravitational potential energy built-up during mantle solidification. By combining equations (2) and (10), we can predict the ideal maximum value of the potential. This maximum is reached if the mantle remains immobile during mantle solidification (i.e., C s (x) = C s (x)). Similarly, the minimum is reached if the ideal mantle stratification is id exactly inverted (i.e., C s (x) = C s (1 x)). id 3. Results Fig. 1 shows post-overturn mantle stratification of two endmember cases. The movies corresponding to the snapshots of Fig. 1 can be downloaded with the supplementary materials. The canonical late overturn case (Fig. 1.a) corresponds to an MO solidification timescale shorter than overturn timescale (R c = , R d = , R T = R c /30 and Ra 10 7 ) at τ =

5 220 C.-E. Boukaré et al. / Earth and Planetary Science Letters 491 (2018) Fig. 2. Mantle thermo-chemical stratification produced during MO solidification for the late overturn (left, a and b) and the early convection cases (right, c and d). The top row (a and c) corresponds to the end of MO solidification. The bottom row (b and d) is representative of the post-overturn stratification. Orange dots depict the grid cells values. Thin black curves show the standard deviation from the horizontal averaged values (blue and red squares). (a) The unstable density stratification does not overturn before the end of MO solidification. The temperature stratification follows closely the freezing temperature. (b) Post-overturn chemical stratification follows exactly the ideal solution. Near the bottom of the solidified layer, temperature is only diffusing as thermal convection is inhibited by the stable chemical stratification. (c) Thermal convection starts during mantle solidification and mix the layer. Just before the end of MO solidification, lateral heterogeneities are present in the solidified MO. (d) Post-overturn chemical stratification differs slightly from the ideal solution. However, much larger lateral compositional variations are present than in late overturn case (see histograms in b and d). (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.) 200 τ MO = 100 Myrs. The early convection case (Fig. 1.b) corresponds to MO solidification timescale larger than overturn timescale (R c = , R d = 10 3, R T = R c /30 and Ra 10 7 ) at τ = 2 τ MO = 100 Myrs. Applied to a deep lunar MO with H = 1100 km, g = 1.6 ms 2 and ρ = 1000 g cm 3, the late overturn case corresponds to τ MO = 0.5 Myr and the early convection case corresponds to τ MO = 50 Myr. The two end-members simulations are computed with the same viscosity (μ = ) and therefore have the same convective vigor. The time at which each case is shown was chosen so that compositional overturn is reasonably complete and the continuing evolution is dominated by thermal convection modulated by stable compositional stratification. Temporal evolution of these two cases is given by animations in the supplementary materials. In Fig. 2, we compare post-overturn stratification to the one at the end of MO solidification. Fig. 2.b and Fig. 2.d correspond to the post-overturn stratifications presented in Fig. 1. Fig. 2.a and Fig. 2.c correspond to the situations at the end of MO solidification (τ = 1). For the late overturn case, the unstable density stratification does not overturn before the end of MO solidification (Fig. 2.a). The pre-overturn compositional profile corresponds to the ideal stratification predicted by stagnant fractional crystallization. There are no lateral heterogeneities at the end of MO solidification. The temperature stratification follows closely the freezing temperature. After overturn, the degree of mixing in the solidified layer remains low (Fig. 1.a). Flat layers are piled up in monotonically decreasing density upwards. The compositional stratification closely follows the ideal solution (Fig. 2.b). The compositional overturn slightly mixes solidified layer but the iron number standard deviation remains small (thin black lines, Fig. 2.b). Temperature has been almost passively advected during compositional overturn. Originally deep hot material has been brought to the top and cold shallow material has sunk towards the core mantle boundary. Thermal convection driven by heating from below and cooling from the top tends to erase the post-overturn thermal stratification. In the deep layer, stable compositional stratification is so strong that thermal plumes cannot form and heat is mostly transported by diffusion. The thermal structure of the deep mantle primarily reflects the freezing temperature recorded during MO solidification. In the upper mantle where the strength of the stable compositional stratification is small, heat is extracted by thermal convection. For the early convection case, thermo-chemical convection starts during solidification and mixes the solidified layer. More precisely, thermal convection starts before compositional overturn

6 C.-E. Boukaré et al. / Earth and Planetary Science Letters 491 (2018) Fig. 3. Relationship between the key dimensionless parameter R c and the timing of the mantle overturn. R c relates the timescale of MO crystallization to the timescale of chemical overturn, R c = τmo μ ρ gh. (a) Build-up of the gravitational potential energy with time (equation (10)). Cases shown by solid lines do not include thermal convection (R c = 10 3, blue, R c = 10 4, blue and R c = 10 5, red). The dotted lines include thermal convection (R T = R c /30) and correspond to the models presented in Figs. 1 and 3: late overturn (blue, R c = and R d = ) and early convection (red, R c = and R d = 10 3 ). We bracket the critical R c between 10 4 and For super-critical R c values, thermal convection homogenizes chemical stratification produced during solidification (red dotted curve). (b) For a given planetary body, Late overturn vs. Early convection regime depends mainly on the ratio of the time of MO crystallization and mantle viscosity. At the intersection of the two domains, R c = The location of the boundary between the two regimes (early convection and late overturn) depends on the relevant gravity acceleration (planet size), g, and characteristic length scale H (MO depth) (see equation (8)). Circles and squares correspond to the situations discussed in Figs. 1 and 3. because the gradient of the thermal density stratification is greater than that of the chemical density stratification in the early stages of the solidification (see Fig. 2). Just before the end of MO solidification, the cumulate layer is already strongly heterogeneous (Fig. 2.c) and the signature of the thermal stratification imposed by freezing temperature is erased. Later after MO solidification, the compositional stratification differs slightly from the ideal solution (blue dots, Fig. 2.d). However, we observe substantial lateral variations of the iron number (see the histograms in Fig. 2.d). Large buoyancy number B = R c /R T forces active heterogeneities to reorganize in monotonically descending density order that decreases lateral variability with time (see thin black lines in Fig. 2.c and 2.d). Late overturn vs. early convection behavior is governed by the relative rates of solidification and overturn. In this simplified MO model that assumes fractional crystallization, the compositional density stratification near the end of the solidification is more unstable than the thermal density stratification. Therefore, R c can be interpreted as a conservative measure of the overturn timing. In Fig. 3.a, we illustrate the relationship between R c and timing of mantle overturn. To follow the evolution of the compositional stratification during solidification, we plot the potential (equation (10)) as a function of the dimensionless time τ. The end of MO solidification is at τ = 1. Solid lines are for cases that do not include thermal convection (R c = 10 3, blue, R c = 10 4, blue, R c = 10 5, red and R T = 0for these three cases). The models presented in Figs. 1 and 2 are depicted in dotted lines and include thermal convection (R T = R c /30). Thin horizontal black lines correspond to the ideal maximum and minimum values of the potential (see section 2.3). The two following aspects should be noted. (1) The potential always reaches the ideal maximum for R c values lower than reaching the ideal maximum shows that the canonical compositional layering has time to build-up before mantle convection starts. In other words, the overturn occurs after the end of MO solidification. (2) The potential always reaches the ideal minimum for purely compositional models. reaching the ideal minimum shows that canonical compositional layering is perfectly reversed. Indeed, with no thermal convection, the only driving force is the chemical density contrast produced by fractional crystallization. During solidification, evolution of compositional stratification in the layer can be described in four phases: (1) build-up, (2) incubation time, (3) overturn and (4) post-overturn dynamics (see Fig. 3.a). During the build-up phase, progressively iron enriched components are deposited on top of the solidifying layer which increases the potential. During the incubation stage, the gravitational potential has reached its maximum and the density stratification of the system is unstable. The unstable mantle is waiting to overturn. During overturn, the mantle tends to minimize its gravitational potential energy. As the dimensionless R c value is increased, the incubation time decreases (blue curves, Fig. 3.a). For R c values higher than 10 4, incubation time is small enough to allow mantle overturn during MO solidification. For such cases, the potential cannot reach the ideal maximum value (red curves, Fig. 3.a). According to Fig. 3.a, we can bracket the critical dimensionless value Rc C at which overturn occurs during solidification: 104 < Rc C < 105. Rc C is arbitrarily chosen to be If thermal diffusion is negligible (i.e., R d 0), the general behavior of thermochemical overturn is equivalent to the purely compositional case. Nevertheless, in the case of thermo-chemical overturn, the unstable thermal stratification (associated with the slope of the freezing temperature) can be built before the generation of a more unstable compositional stratification. A more unstable compositional density

7 222 C.-E. Boukaré et al. / Earth and Planetary Science Letters 491 (2018) stratification would only appear near the end of fractional crystallization for the canonical silicate MOs of the Moon, Earth and Mars (Elkins-Tanton, 2012). However, this compositional density stratification depends on the partitioning coefficients in the general case. Very early overturn can thus be generated by unstable thermal stratification as the effective R T can be larger than the effective R c in the early stage of MO solidification. In the case of batch crystallization, R c = 0. The timing and vigor of thermal convection are governed by R T and R d (Ra = R T /R d ). The analysis developed here for R c can be transferable to R T where the slope of the freezing temperature only drives thermal convection. However, the thermal diffusion timescale must be low enough such that thermal variations do not diffuse too fast during overturn (i.e., the effective Rayleigh number must be supercritical). There are two necessary conditions for having purely thermal syn-solidification convection, R T > Rc C and R T /R d > Ra c where Ra c is the critical Rayleigh number. A more detailed estimation of the effect of thermal diffusion on thermal overturn timing can be found in the supplementary materials. This dimensionless analysis offers a simple framework for predicting the onset of solid-state mantle convection relative to MO solidification (see Fig. 4). Fig. 3.b illustrates how the dimensionless number R c can be used to constrain the syn-solidification convective regime of Earth-sized (purple), Moon-sized (green) and Vesta-sized (yellow) bodies. In the μ τ MO space, the early convection and late overturn regime are separated by a line where R c = For sub-critical R c value (bottom right in Fig. 3.b), the dynamics follows the classic overturn model where the cumulate layer remains immobile during MO solidification. For supercritical R c (upper left in Fig. 3.b), mantle convection starts during solidification. As R c gh and g ρ m R (where R is the planet s radius and ρ m its mean density), the early convection regime may be more easily reached in large planetary bodies. For instance, Fig. 3.b shows that Vesta-sized body requires a cumulate viscosity about two orders of magnitude lower than Earth-sized body to reach the early convection regime. It is important to bear in mind that the boundary conditions (here, impermeable free-slip on the four sides) affect the critical R c. Cumulate materials might flow through the solidification front by remelting. In this case, such permeable boundary conditions are expected to decrease the critical R c (Labrosse et al., 2017). 4. Discussion In this study, overturn timing relative to MO solidification is quantified using the dimensionless parameter R c (see Figs. 3.b and 4). The overturn timing can be seen as the rate at which heterogeneities are incorporated in the mantle. The amount of mantle mixing during MO solidification depends on both the rate of heterogeneity incorporation and syn-solidification convective vigor. While syn-solidification convective vigor depends mainly on mantle viscosity (i.e., Ra), the rate of heterogeneity incorporation depends on the ratio of MO solidification timescale and mantle viscosity (i.e., R c ). However, the exact estimation of MO solidification timescale and effective mantle viscosity is complex as these parameters are likely to vary during MO solidification. In the following, the effects of a range of cumulate viscosity and MO cooling rate are discussed. Potential applications of the present study to the lunar magma ocean (LMO) are then suggested. MO solidification is not likely to occur at a constant uniform rate as assumed in this study. MO cooling can be limited by pure black-body radiation, black-body radiation in the presence of atmosphere, thermal conduction through a solidifying lid and/or tidal heating (Solomon and Longhi, 1977; Abe and Matsui, 1986; Abe, 1993; Elkins-Tanton, 2008; Meyer et al., 2010). The relative importance of these cooling mechanisms should affect the MO thermal history by many orders of magnitude. In the absence of atmosphere degassing or solid conductive lid, MO would crystallize in several kyrs. In the presence of a solid conductive lid, tidal heating may delay solidification of the underlying MO residual by several hundred of Myr (Meyer et al., 2010). Complexity arises since MOs are expected to evolve from one thermal regime to another. For MO formed by giant impact, post-impact thermal and dynamic evolution of the synestia would affect the volatile budget of the planet and the size of an initial thermally insulating atmosphere (e.g. Canup et al., 2015; Lock and Stewart, 2017). Depending on the amount of volatiles retained in the magma, volatile degassing can also occur upon cooling of the MO (e.g. Elkins-Tanton, 2008). Later in the solidification, formation of buoyant solids phases such as plagioclase in the LMO, further slows down MO solidification. Large planetary bodies might also have a thicker insulating atmosphere than small bodies as they should retain a larger mass of volatiles. The latter would reinforce the fact that the early convection regime may be more easily reached in large planetary body as R c gh. The regime diagram illustrated in Fig. 3.b proposes a first-order constraint to predict the onset of cumulate mantle convection depending on the relevant MO cooling timescale. However, more work is needed to thoroughly assess the effects of a temporally changing MO cooling rate on MO evolution. In particular, the timescale of MO solidification can influence the amount of retained melt in the cumulate that effects its viscosity as discussed below. For a LMO on the Moon with a melt viscosity μ l = 10 Pa s, ρ = 300 kg m 3, g = 1.6 ms 2, a grain size d = m, and RCMF = 60%, the critical velocity above which solid liquid segregation does not have time to occur is calculated to be about 50 km/myr (Shirley, 1986, their equation (1)). For a deep LMO with H = 1100 km, this corresponds to τ MO 20 Myr. Thus the early stages of LMO solidification when crystallization is expected to be fast (τ MO < 20 Myr), cumulates should retain a melt fraction comparable to RCMF. This would result in cumulate layer with a very low effective viscosity and a weak degree of fractionation. Equilibrium or batch crystallization may be a good approximation so that fractionation will be restricted to the most highly incompatible elements. Later in the evolution a stable conductive lid is expected to substantially reduce the cooling rate, and the final stages of the LMO solidification may take several hundred Myr. With solidification front velocity below the critical velocity, the melt fraction retained in the cumulates is expected to be substantially reduced by melt migration and compaction. Cumulates containing a high melt fraction will be restricted to a layer near the crystallization front. Small retained melt fractions will result in more nearly fractional crystallization. Even at small retained melt fraction, the viscosity of cumulates could be substantially reduced. For instance, a retained melt fraction by 5% is expected decrease the effective viscosity of the cumulate by about one order of magnitude (Hirth and Kohlstedt, 2003). To some extent, the effects of timescale of MO solidification and cumulate viscosity are linked. Fast crystallizing MO would have high retained melt fraction that decreases the overturn timescale. Whereas the opposite is true for slow crystallizing MO. Further studies that explore detailed evolution scenario are required to better quantify these competitive effects. The Moon is a planetary body for which we have the strongest evidence for the existence of a MO and the generation of highly fractionated mantle compositions. The anorthositic crust, a primary motivation of the LMO hypothesis can provide important constraints on MO evolution. Wide range of lunar anorthositic crustal ages can be interpreted as a signature of long MO solidification time (about 200 Myrs) (Elkins-Tanton, 2012). For such long MO solidification, anorthosite crystallization becomes contemporaneous with the Mg-suites generation (Nyquist et al., 1995; Edmunson et al., 2009). The sources of Mg-suite materials are distinguished

8 C.-E. Boukaré et al. / Earth and Planetary Science Letters 491 (2018) Fig. 4. Cartoon illustrating two styles of solidifying planets evolution. The kind of evolution depends on the competition between the rate of cooling and the viscosity of the cumulate layer. In our study, the dimensionless parameter R c allows to distinguish the two regimes. Differences in rate of cooling is illustrated here with different atmosphere thickness (in blue). Nevertheless, planet s solidification history are much more complex and may involve processes such as primitive crust formation, tidal heating or stellar radiation in variable contribution. Planetary mantle solidifying slowly and having a low enough viscosity would experience early mantle dynamics and possibly an early mixing (orange components). As the opposite, fast cooling and high cumulate viscosity would favor a late onset of solid-state mantle dynamics which might allow to preserve a compositional layering through the planet history. from the ferroan anorthosites by their high Mg# and high concentration of incompatible elements (Shearer et al., 2006 and reference therein). If Mg-suite magmas are formed by decompression melting associated with cumulate overturn (Hess, 1994), the lunar overturn timing may be critical. In the absence of an atmosphere, the LMO cooling history is here divided into two main stages. Before plagioclase precipitation (<80% of MO solidification), solidification is expected to be fast and occurs in 1 Myr. The last 20% of solidification is thought to take 200 Myrs. During this second stage, the iron and KREEPy enriched residual liquid is overlain by the accumulating anorthosite crust and solidifies slowly on top of earlier cumulate. As the thermal profile in the cumulate would be close to the freezing temperature, any upwelling would result in decompression melting. If the first formed Mg-rich cumulates must be preserved, convection in the cumulate should not occur during the first stage of solidification. One might assume that chiefly thermal density contrast drives convection during the first solidification stages. As R c /R T 30, Fig. 2 shows that the effective cumulate viscosity must be larger than / Pa s to prevent thermal convection on a solidification timescale of 1 Myr (as the thermal Rayleigh number is expected to be super-critical). On the other hand, if material of the Mg-suite is contemporaneous with anorthosites solidification, cumulate overturn should occur in less than 200 Myrs. According to Fig. 2.b, the effective cumulate viscosity must be lower than Both constraints seem to lie in a realistic viscosity range but the question of whether these viscosity values are consistent with the timescale of cumulate compaction and flow laws of partially molten rock is left for further study. The simplified analysis should not be regarded as a definitive answer but aims to better illustrate how the present study can be used. In addition to the anorthositic crust, highly fractionated mantle source compositions are required for mare basalt magmatism that occurred on time scales substantially longer than the youngest anorthosites. Both mare basalt volcanism and an earlier formed U Th anomaly centered on the lunar near side are fundamental features of the Moon. The generation of mare basalts relies on remelting of Ti-bearing cumulates (IBC) at substantial depths (>400 km) in the lunar mantle. Two models have been suggested as possible explanations for this hemispheric asymmetry. The implications of the present study are important to consider for both models. The degree-one downwelling model is based on post MO overturn where an appropriate viscosity stratification of cumulates (the overlying iron rich layer is 3 orders of magnitude less viscous than the cumulates below) leads to a degree-one compositional overturn of dense MO residual mantle. Radioactive heating and thermal expansion in dense IBC that may have sunk only part way through the mantle generate the mare basalts. The downwelling model has the potential to explain both the U Th anomaly and the presence of mare basalts on the near side. A sufficiently thick dense IBC layer ( 50 km) is required for generating a degree-one convection pattern. However, slow MO crystallization and a cumulate viscosity lower than 10 20, as suggested above, could tend to dilute or erode the IBC layer by solid state convection. Alternatively, the degree-one upwelling model relies on the subsequent radioactive heating of an IBC layer that has sunk through the entire the mantle, forming a layer at the top of the core (Zhong et al., 2000; Parmentier et al., 2002; Stegman et al., 2003). The subsequent radioactive heating in this layer would have made the dense layer buoyant relative to the overlying mantle. It is interesting to note that thermal convection during solidification should not prevent the formation of a dense layer at the base of the lunar mantle (see Fig. 1.b). Although the compositional density contrast is high enough such that IBC components cannot be fully homogenized by thermal convection, the IBC concentration at depth may be significantly affected. Early Mg-rich cumulates could provide mantle source for the Mg-suite (Hess and Parmentier, 1995) that appear to have formed contemporaneously with the anorthositic crust. An initial compositional driven overturn that moves the dense IBC layer to greater depth could cause decompression melting of this hot Mg-rich cumulate. By constraining the timing of cumulate overturn, dimensionless analysis like that described here would contribute to better assess possible geodynamic links between Mg-suite formation and subsequent mare basalt magmatism.

9 224 C.-E. Boukaré et al. / Earth and Planetary Science Letters 491 (2018) Conclusion The timing of MO cumulate overturn relative to MO solidification depends mainly on the competition between MO solidification time and the viscosity of the cumulate layer. Early overturn tends to mix the young mantle and homogenized the idealized cumulate stratigraphy considered in previous models (Hess and Parmentier, 1995). It is physically possible that the Moon s cumulate was convecting and mixed before complete solidification. If the lunar mantle dynamics started after complete MO solidification, it implies that, for a given rate of MO solidification, the viscosity of the cumulate was high enough. Such high viscosity could mean that the retained melt fraction was low (Hirth and Kohlstedt, 2003). Early rapid solidification, controlled primarily by heat flux radiated from the surface may later be reduced substantially by heat conduction through a primitive crust. The extent of overturn may thus vary substantially during the evolution allowing a new perspective of the role of overturn in early planetary evolution. Two dynamic aspects that affect the degree of mantle mixing at the end of MO solidification have been identified. The degree of mantle mixing is controlled by the vigor of syn-solidification convection but also by the rate at which heterogeneities are incorporated into the solid mantle. These two processes can be captured by the thermal Rayleigh number, Ra which is a measure of the convective vigor, and the dimensionless parameter R c, which is a measure of the ratio of MO crystallization time and the timescale associated with chemical overturn. The dimensionless parameter R c must exceed a critical value for having a syn-solidification overturn. For a given convective vigor, syn-solidification convection increases the degree of mantle mixing and reduce the scale of heterogeneities. All together, this model aims at linking through a simple geodynamic approach: (1) the timescale of MO crystallization, (2) the rheology of planetary materials and (3) both the composition and the spatial distribution of mantle heterogeneities. Therefore, it represents a step towards a better understanding of the early evolution of other terrestrial mantles as MO episode should not be restricted only to the case of the Moon. Acknowledgements We wish to thank William B. McKinnon and two anonymous reviewers for their very constructive comments that helped to improve this paper. This research was funded by the NASA grant and SSERVI-SEEED grant NNA14AB01A. Appendix A. Supplementary material Supplementary material related to this article can be found online at / /j.epsl References Abe, Y., The evolving Earth physical state of the very early Earth. Lithos 30 (3), /science /article /pii / D. 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