Early in the history of terrestrial planets, the fractional crystallization of a magma ocean can lead to a mantle stratification characterized by a progressive enrichment in heavy elements from the core-mantle boundary to the surface. Such a configuration is gravitationally unstable; it causes mantle overturn and the formation of a stable chemical layering. Using simulations of thermochemical convection, we analyzed the consequences of overturn and subsequent layering on mantle dynamics assuming Mars' scaling parameters. We found that the time needed to achieve chemical homogenization via convective mixing scales exponentially with the buoyancy ratio , which measures the relative importance of chemical to thermal buoyancy. In addition, when using a strongly temperature-dependent viscosity, the formation of a stagnant lid prevents the uppermost crystallized layers from sinking into the mantle. In order to obtain their subduction, a yielding mechanism must be invoked. In the context of Mars' evolution, our results suggest that complete chemical mixing is unlikely to take place within time scales comparable with the planet's age. Magma ocean freezing could thus be responsible for the long-term preservation of compositional heterogeneities as required by meteoritic evidence. The inferred lack of a high density lid is difficult to reconcile with a stagnant-lid regime operating throughout Mars' history. An episode of surface mobilization induced by compositional overturn can resolve this difficulty provided that is large enough. Too large buoyancy ratios, however, tend to suppress convective heat transport, rendering it problematic to explain the late volcanic history of Mars.
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 The dynamics of and thermal transport in fluids undergoing thermal convection can be strongly affected by the presence of compositional gradients. When the chemical diffusivity κC differs significantly from the thermal diffusivity κ, the motion of the fluid is governed by the physics of double-diffusive convection (DDC). While in a purely thermal case, diffusion tends to stabilize the fluid in double-diffusive systems in which κ≫κC, the more rapid diffusion of heat can act as a source of buoyancy flux able to drive convection under conditions that would be subcritical for a system characterized by a single diffusion coefficient [Turner, 1979]. DDC has long been recognized to play a fundamental role in the dynamics of geophysical and astrophysical flows [Huppert and Turner, 1981a]. Originally developed in oceanographic context [Stommel et al., 1956; Stern, 1960; Turner and Stommel, 1964], the theory of DDC has been widely invoked to characterize various aspects of the dynamics of creeping flows in magma chambers, both from experimental [Huppert and Turner, 1981b; Turner and Campbell, 1986; Tait and Jaupart, 1990] and theoretical/numerical viewpoints [Spera et al., 1986; Hansen and Yuen, 1987, 1990, 1995; Hsui and Rihai, 2001]. In the context of solid planetary interiors, thermo-compositional convection is an important component for describing the dynamics and evolution of the Earth's mantle [e.g., Tan and Gurnis, 2007; Nakagawa et al., 2010; Deschamps et al., 2011; Tan et al., 2011] in particular when it comes to model the D′′region whose complexity is at least partly explained by the presence of chemically distinct materials above the core-mantle boundary [e.g., Hansen and Yuen, 1988; Kellogg et al., 1999; Hansen and Yuen, 2000; Montague and Kellogg, 2000; Garnero and McNamara, 2008; Nakagawa and Tackley, 2008].
 One possibility that may contribute to explain the existence of compositional heterogeneities is that of invoking one or multiple large-scale impacts that may have caused early melting of a significant portion of the Earth's mantle, leading to the formation of a magma ocean [e.g., Safronov, 1978; Sasaki and Nakazawa, 1986; Melosh, 1990]. Magma ocean freezing is a highly complex process involving the dynamic interaction of fluid and crystals [Suckale et al., 2012; Ulvrová et al., 2012]. From a qualitative point of view, the solidification of a liquid, strongly convecting, and hence isentropic, magma ocean proceeds from the bottom upward since the mantle liquidus and solidus are steeper than the adiabat in the convective fluid. It has been argued that freezing of the Earth's magma ocean consisted of two major phases (see Solomatov  for a thorough review of the whole subject). During the first phase, which occurs very rapidly (∼103 a), convection can prevent crystal settling and hence fractionation [Melosh, 1990] so that the rising solidification front leaves below a largely undifferentiated mantle. This phase continues until a critical crystal fraction of about 60% is reached, which marks a rheological transition from a low-viscosity suspension that behaves like a liquid, to a partially molten medium with a high effective viscosity more akin to a solid [Solomatov, 2000]. Afterward, fractional crystallization of the shallow magma ocean (i.e., the upper mantle) takes place over a much longer time scale (∼108 a) [Solomatov, 2000].
 Although it is possible that the Earth's mantle underwent a large- or even global-scale melting event early in its history, geochemical evidence in support of this scenario is not compelling since mantle convection and plate tectonics have contributed to mix and recycle any form of primordial crust that might have contained traces of a magma ocean [Elkins-Tanton, 2012]. Nevertheless, a basal magma ocean, which crystallized slowly enough to allow localized patches of dense melt to survive until present day, has been invoked to explain the existence of a deep primordial reservoir of incompatible species [Labrosse et al., 2007]. This scenario would help to solve the issue of the lack of geochemical compatibility between continental crust and a depleted mantle of chondritic composition [e.g., Hofmann, 1997]. Furthermore, for smaller bodies such as planetesimals [Taylor et al., 1993; Elkins-Tanton et al., 2011a], Mercury [Brown and Elkins-Tanton, 2009], the Moon [Wood et al., 1970; Borg et al., 1999; Elkins-Tanton et al., 2011b] and Mars [Righter et al., 1998; Elkins-Tanton et al., 2003], several arguments favor not only the scenario of a (possibly global) magma ocean but also of its fractional crystallization. In particular, as far as Mars is concerned, geochemical analyses based on Hf-W chronology, indicate that core-mantle differentiation took place within the first few million years of the Solar System [e.g., Kleine et al., 2002; Nimmo and Kleine, 2007], possibly as early as within 2–3 Ma [Dauphas and Pourmand, 2011]. An early core formation would concentrate the heat due to accretion, differentiation, and decay of short-lived isotopes over a short time period, thus facilitating complete mantle melting. In addition, analyses conducted on the Martian SNC meteorites (Shergottites, Nakhlites, Chassignites) require them to have originated from distinct source-reservoirs that differentiated during Mars' early history [e.g., Borg et al., 1997; Jagoutz, 1991; Borg et al., 2002]. The fractional crystallization of a magma ocean is compatible with this view and represents a viable way to achieve the needed primordial differentiation [Elkins-Tanton et al.2003].
 As described above, solidification starts at the bottom of the magma ocean and continues toward the surface. The first minerals to crystallize from the magma ocean are Mg-rich. This causes a progressive Fe-enrichment in the remaining liquid phase, whose content of incompatible heat-producing elements also tends to increase with the amount of solidification. Therefore, a gravitationally unstable cumulate stratigraphy eventually forms, which is prone to overturn because of Rayleigh-Taylor instability [Elkins-Tanton et al., 2003]. The overturn results then in a stable compositional stratification which involves the whole mantle if a global magma ocean is assumed. The problem of the stability of a dense basal layer in thermal convection has been investigated both numerically [e.g., Montague and Kellogg, 2000] and experimentally, particularly with regard to the onset of oscillatory convection in a two-layer fluid [Le Bars and Davaille, 2002]. However, the use in these studies of a step-like distribution of the compositional density and sometimes also of the viscosity [Le Bars and Davaille, 2002], render it somewhat difficult to directly compare these approaches with those like ours, based on a continuous density distribution as expected from a magma ocean scenario. In a few studies, numerical simulations have been used to analyze the overturn of Mars' magma ocean resulting from complex crystallization sequences derived from geochemical modeling [Elkins-Tanton et al., 2005a, 2005b; Debaille et al., 2007, 2009]. Nevertheless, the study of the subsequent thermochemical evolution has received relatively little attention so far. Zhong et al.  and Parmentier et al.  showed, for example, that the gravitational instability of an ilmenite-rich layer resulting from the fractionation of an anorthositic crust can provide a suitable explanation for the asymmetric magmatic history of the Moon, while de Vries et al.  described how such ilmenite-rich cumulates can lead to the formation of an outer lunar core surrounding a small inner metallic core or even to a fully ilmentic, non-metallic core. In a more general planetary context, Zaranek and Parmentier  analyzed numerically the convective cooling of a stably stratified fluid (i.e., starting from a post-overturn configuration) with temperature-dependent viscosity. The authors thoroughly characterized the way chemical mixing evolves as a consequence of cooling and derived scaling laws for the downward growth of a well-mixed upper mantle layer. However, convective cooling is only one of the processes controlling thermal evolution and chemical mixing of planetary mantles, which are also influenced by heat from the core as well as from internal heating due to radioactive decay. Furthermore, in the presence of a strongly temperature-dependent viscosity, complete overturn of the mantle may not take place since the uppermost and densest layers are located within the stagnant lid. If this is the case, it is important to investigate what are the conditions that allow such layers to sink into the mantle.
 The main goal of the present study is to analyze the dynamics of a Mars-like mantle including the initial overturn phase using numerical models of thermochemical convection. To this end, we investigated systematically different rheologies and heating modes. We focused in particular on quantifying the time evolution of mantle mixing in order to characterize the conditions that allow for the preservation or removal of compositional heterogeneities associated with the fractional solidification of a magma ocean. In section 2, we describe the mathematical and numerical aspects of the models. The results of the simulations are presented in section 3. This is subdivided in three subsections according to the rheology employed (isoviscous, temperature-dependent or viscoplastic), each of which is used to simulate systems heated from below and from within. The paper is concluded by the discussion and conclusions contained in sections 4 and 5, respectively.
2 Model and Methods
 Thermo-compositional convection is governed by the conservation equations of mass, linear momentum, thermal energy, and chemical composition. Assuming a Boussinesq fluid with negligible inertia, in nondimensional form, they read respectively:
All symbols and parameters reported in this section are listed and defined in Table 1. The system of equations (1)–(4) describes the motion of a fluid undergoing double-diffusive convection in the limit of negligible chemical diffusivity (i.e., infinite Lewis number) as appropriate for the mantle of rocky planets. The flow is driven by thermal and compositional buoyancy whose importance is determined respectively by the thermal Rayleigh number Ra and by the compositional Rayleigh number RaC, expressed in equation (2) through the buoyancy ratio (see equation (8) below).
Table 1. Variables and Parameters Used Accompanied by Their Scaling, Definition and/or Numerical Value (Either Dimensional for Material Parameters and Physical Quantities or Nondimensional for the Various Nondimensional Groups)
Scaling / Definition
unit vertical vector
second invariant of the strain-rate tensor
maximum nondimensional viscosity
0, 6×10−12 or 6×10−13 W/kg
box aspect ratio
2.5×1020 or 2.5×1021 Pas
temperature drop across the mantle
chemical density difference
thermal Rayleigh number
106 or 107
compositional Rayleigh number
internal heating Rayleigh number
0, 106 or 107
internal heating factor
0, 1 or 10
 We conducted three series of simulations (approximately 100 runs in total) in which we assumed the viscosity η to be constant (section 3.1), purely temperature dependent (section 3.2) or temperature and strain-rate dependent (section 3.3) in order to mimic the brittle behavior of the lithosphere. In the simulations with temperature-dependent viscosity, the temperature dependence is prescribed according to the Arrhenius law, which takes the following nondimensional form
where E, T0 and Tr are activation energy, surface- and reference temperature, respectively (see Table 1 for their dimensional values). A cutoff to the nondimensional viscosity resulting from equation (5) was applied when this exceeded ηmax=1010. When, in addition, strain-rate dependence is considered, this is obtained through a plastic viscosity
where σy is the yield stress and the second invariant of the strain-rate. In this case, the effective viscosity is simply calculated as
 We used our finite volume code YACC (Yet Another Convection Code, https://code.google.com/p/yacc-convection/, King et al. , Tosi et al. ) to solve equations (1)–(4) in a 2-D rectangular box with an aspect ratio of four. YACC is based on a primitive variables, staggered-grid discretization of equations (1) and (2) with direct solution of the mass conservation (1) for the dynamic pressure, which avoids the need to resort to iterative pressure-correction schemes [e.g., Gerya and Yuen, 2003]. The solution of the thermal energy equation (3) is obtained by combining a semi-implicit Crank-Nicholson scheme for the diffusion part with a semi-Lagrangian scheme for the advection part [Spiegelman and Katz, 2006]. Finally, the transport equation (4) is solved via particle-in-cell method [e.g., Gerya and Yuen, 2003], which minimizes the undesired numerical diffusion that typically arises when grid-based advection methods are employed.
 In all tests that we carried out, we used free-slip boundaries and reflective sidewalls, an isothermal top boundary and either an isothermal or insulating bottom boundary in order to simulate purely bottom-heated and internally heated systems, respectively.
 Most of the simulations were performed using a reference temperature of 1600 K at which we defined a Rayleigh number of 106, which is appropriate for the Mars' mantle when a dry rheology with reference viscosity of ∼2.5×1021 Pa s is assumed. We also conducted selected tests at Ra=107, which can be either interpreted as the Rayleigh number characteristic of a planet with a thicker mantle like Earth, or still of Mars' mantle but assuming a weaker (i.e., wetter) rheology. Note, however, that for an Earth-like planet the dimensional time scale is roughly three times longer than that of Mars'. For the simulations in which the mantle is heated only from within and cooled from above, we set the internal heating factor H=RaQ/Ra to 1 or 10, corresponding to heating rates of ∼6×10−13 and 6×10−12 W/kg, respectively. The tests with Ra=106 and 107 were run with a grid resolution of 450×150 and 600×200 nodes, respectively. In all cases we used 20 particles per cell.
 In order to simulate a scenario in which the fractional crystallization of a magma ocean has just taken place, the initial mantle temperature was set at the solidus of dry peridotite [Takahashi, 1990] over which an upper thermal boundary layer was superimposed (Figure 1a). The surface temperature was set to present-day values (250 K), consistently with the assumption that the primordial steam atmosphere resulting from degassing of the freezing magma ocean is lost within a time scale much smaller than that characteristic of solid-state deformation [Abe, 1993]. Although the crystallization sequence of a magma ocean can lead to a highly complex density stratification [Elkins-Tanton et al., 2003], for simplicity we initialized the composition field with a linear profile as in Zaranek and Parmentier , with the important difference that, instead of starting from a stable configuration, we assumed the material having the highest chemical density (i.e., Fe-rich) to be initially at the surface (C=1) and the lightest at the core-mantle boundary (C=0). A small-amplitude random perturbation was superimposed on the composition field to initiate the mantle overturn (Figure 1d). It should be noted that the fact that we considered a magma ocean scenario and took into account the overturn phase renders our simulations significantly different from those of Zaranek and Parmentier  who assumed an unperturbed stable stratification as initial condition. In particular, the overturn phase leads to the formation of an “inverted” temperature distribution characterized by cold and hot material above the core-mantle boundary and below the surface, respectively (see Figure 1), and to the introduction of lateral heterogeneities, both in temperature and composition, throughout the mantle. These induce local buoyancy instabilities that affect the time scales for the onset of convection and render the status of our post-overturn mantle qualitatively different from that employed by Zaranek and Parmentier.
 The relative importance of thermal and chemical buoyancy in driving or resisting convective motion is described by the buoyancy ratio (or buoyancy number), which is defined as the ratio between compositional and thermal Rayleigh numbers:
For any given Rayleigh number, rheology, and heating mode, we investigated a range of buoyancy numbers between 0.2 and 1.8. With the reference values of density, thermal expansivity, and temperature drop across the mantle reported in Table 1, this range corresponds to chemical density anomalies Δρ comprised between 27 and 244 kg/m3, yielding a maximum contrast with respect to the reference mantle density which ranges approximately from 0.8% to 7%.
 In order to assess the ability of the models to preserve or erase the primordial chemical heterogeneity associated with the assumed initial stratification, we analyzed the degree of chemical mixing by tracking the following quantity:
where 〈C〉 is the horizontally averaged profile of the composition field C, i.e.,
where L is the aspect ratio of the box. In the following, we will refer to as mixed fraction. The initial stratification of the composition field C corresponds to , i.e., to a fully heterogeneous mantle. If , instead, complete mixing has been achieved and the mantle is chemically homogeneous. We also determined a characteristic mixing time , which we defined as the time needed by a given system to reach a mixed fraction of 0.9. We ran the simulations until full mixing was achieved () or, when this did not occur, up to a maximum nondimensional time of 0.3, which corresponds to ∼30 Ga when scaling parameters appropriate for Mars' mantle are assumed.
 Admittedly, convective mixing and its time scales can be quantitatively characterized using significantly more sophisticated techniques that involve analyzing the temporal evolution of the separation between passive markers advected by the flow and between their trajectories and/or tracking of the strain-history of fluid elements [e.g., Ferrachat and Ricard, 1998; Farnetani and Samuel, 2003; Van Keken et al., 2003; Samuel et al., 2011]. Alternatively, simple parameterizations of chaotic mixing have been employed to determine the spatial distribution of heterogeneities in the framework of geochemical reservoir modeling [Kellogg et al., 2002; Jacobsen and Yu, 2012]. Our description based on equation (9), however, was sufficient for our purposes. Indeed, a comparison of the effects of the Rayleigh number on mixing times with the results of Samuel et al.  confirmed the validity of our approach (see Figure 3 and section 4.1).
3.1 Isoviscous Models
3.1.1 Heated From Below
 We start describing results from simulations of systems with constant viscosity heated from below. Figure 2a shows the time evolution of the mixed fraction for Ra=106 and nine different values of the buoyancy number . The unstable composition profile undergoes a rapid overturn that determines a steep increase of the mixed fraction (see Figure 1e), which is then followed by a decrease. This culminates with reaching a minimum for all buoyancy ratios, indicating that a temporary stable stratification has been achieved (see Figure 1f). As the top and bottom boundary layers become thermally unstable, increases again the more rapidly, the lower is the buoyancy ratio. Within the analyzed time interval, full mixing was observed for when Ra=106and for when Ra=107. As shown in Figure 3(black and red lines), the mixing time grows exponentially with the buoyancy number. Both series of experiments are well fitted by an exponential function:
where a=2.32×10−2, b=4.94×10−4, and c=4.38 for Ra=106 and a=7.08×10−3, b=1.59×10−4, and c=4.18 for Ra=107.
 As changes, a variety of convection regimes is observed. In Figure 4, we plot a series of snapshots of composition and temperature for , 1.4 and 1.8, which are representative of different planforms. For small buoyancy numbers up to 0.8, no stable layering of the system is observed. Upwellings and downwellings reach the deepest and shallowest parts of the domain, respectively (Figures 4a and 4d), causing relatively rapid and effective mixing of the whole mantle.
 Because of the choice of the initial temperature profile, the instability of the upper TBL is more significant than that of the bottom TBL. Thus, for larger values of the buoyancy ratio, layered convection is obtained with a well-mixed upper mantle and a lower mantle which either mixes more slowly, eventually leading to whole mantle mixing (e.g., , Figure 4b and 4e), or remains stably stratified throughout the simulation time (e.g., , Figure 4c and 4f). In the latter case, convection occurs only in the uppermost part of the mantle, while heat is transported by conduction in the lower part.
3.1.2 Heated From Within
 The dynamics of the overturn is completely dictated by the initial unstable chemical stratification and its strength. Therefore, this initial stage does not present any basic difference in systems heated from within with respect to systems heated from below. However, because of the lack of bottom boundary layer instabilities, in the latter case, mixing proceeds from above downward as cold downwellings progressively erode the stably stratified mantle [Zaranek and Parmentier, 2004]. It is important to note that the efficiency of such process depends on the amount of internal heating prescribed. In this view, considering systems simply cooled from above as in Zaranek and Parmentier  can lead to an underestimation of the actual ability of the mantle to mix chemical heterogeneities. A larger heating rate causes a higher mantle temperature and hence a larger temperature increase across the upper TBL. Its instability drives more vigorous convection, which leads to more rapid and efficient mixing. For the same range of buoyancy ratios used in the previous section, we compared two series of simulations with H=1 and H=10. Figures 5a and 5b show the evolution of the corresponding mixed fractions. For , the mixing time is nearly identical in both cases (). For , different regimes are obtained, with full mixing occurring for and eventually also for when H=10 but not when H=1. In Figure 6, we compared snapshots of the composition field for H=1 (Figures 6a, 6b, and 6c) and H=10 (Figures 6d, 6e, and 6f), taken at the same time, and different values of . It is evident that a larger heating rate promotes mixing. While with H=10 chemical piles tend to form atop the core-mantle boundary, and their longevity and thickness grow with the buoyancy ratio, with H=1, these are observed only for the smallest values of . In this case, in fact, the system evolves into a configuration characterized by a lower mantle which either poorly mixes (Figure 6b) or remains stably stratified (Figure 6c). These features are also easily recognizable in Figure 7 that shows laterally averaged profiles of composition at time t=0.3 for models with H=1 (Figures 6a and 6c) and H=10 (Figures 6b and 6d) (see also section 3.2.2).
3.2 Temperature-Dependent Viscosity Models
3.2.1 Heated From Below
 As it is well-known, the use of a strongly temperature-dependent viscosity leads to the formation of an immobile lid on top of the mantle which prevents the uppermost layers from taking part in the convective motion. As a consequence, only the portion of the mantle underlying such lid participates in the overturn, while the layers with the highest density remain near the surface. Red lines in Figure 1 portray profiles of temperature and composition during the overturn phase for . It should be noted that, in contrast to the isoviscous case (black lines), the overturn initially involves only the part of the mantle where the temperature is highest and thus the viscosity lowest.
 The value employed for the activation energy (E=300 kJ/mol) leads to extremely large viscosity contrasts, which guarantee that the system cannot escape the stagnant-lid regime and evolve into a mobile lid configuration. Therefore, we evaluated the mixed fraction by computing 〈C〉 below the stagnant-lid depth and not starting from the surface. Figure 2b shows the evolution of . Note that because of the low viscosity induced by the temperature, the overturn occurs very rapidly and cannot be seen on the linear time scale of Figure 2b. In addition, the fact that is evaluated below the stagnant lid explains why all models exhibit a mixed fraction of about 0.4 after the overturn. In contrast to the isoviscous case (Figure 2a), within the analyzed time interval, full mixing was achieved only for when using a reference Rayleigh number of 106. The blue dots in Figure 3 show the corresponding mixing time as a function of , which is also fitted by equation (11) (blue line) with a=2.88×10−2, b=1.09×10−3, and c=6.97. For , we note that is shorter than that for the isoviscous counterpart model, while for and 0.8, it becomes remarkably larger.
 In order to explain such behavior, we plotted in Figures 8 and 9 time series of the horizontally averaged profiles of viscosity, temperature, and composition for isoviscous and temperature-dependent viscosity models with and , respectively. With the smaller , the hampering effect of the compositional layering is weak and convection vigorous. The presence of the insulating stagnant lid causes the temperature of the mantle to increase (Figures 8b and 8d) and hence the viscosity to reduce by about 2 orders of magnitude (Figure 8a). The effective Rayleigh number of the system becomes larger and, as expected, full mixing occurs more rapidly when ηdepends on temperature than in the isoviscous case (Figures 8c and 8e). As increases, the tendency of the overturned stratification to suppress convection becomes more important. With , contrary to the isoviscous case (Figures 8b and 8c), with a temperature-dependent viscosity, heat transfer initially evolves into a nearly conductive mode. This causes a progressive cooling of the upper mantle (Figure 9d), a significant growth of the stagnant lid (Figure 9a), and therewith of the viscosity, which in turn reduces the effective Rayleigh number and hence the mixing efficiency (Figure 9e). However, because the system is heated from below, bottom boundary layer instabilities eventually grow, causing the temperature to increase again (Figure 9d) and the mantle to mix completely (Figure 9e).
3.2.2 Heated From Within
 The important effect due to the temperature-dependent viscosity of locking the uppermost chemical layers in the stagnant lid is clearly not influenced by the heating mode. As already seen in isoviscous simulations, systems heated solely from within and cooled from above exhibit a reduced mixing efficiency with respect to their bottom-heated counterparts. As shown in Figures 7c and 7d, at t=0.3, complete mixing does never occur, both for H=1 and H=10. Apart from the case with and H=10 (black line in Figure 7d), all other models present significant lower mantle stratifications, even though the upper mantle is well homogenized when using an internal heating rate of 10.
 It is interesting to note that, as evidenced by earlier numerical experiments and by the scaling laws derived to interpret them [see Zaranek and Parmentier, 2004, Figure 12], we also found, both in isoviscous and temperature-dependent calculations, that increasing the buoyancy number leads to thicker stable layers above the core-mantle boundary overlain by mixed upper layers in which the temperature converges to a constant value independently of (not shown here). However, the final temperature that we obtained differs from that predicted by Zaranek and Parmentier  because of the different initial conditions and of the presence of internal heating.
3.3 Viscoplastic Models
 With a strongly temperature-dependent viscosity, the upper most, Fe-rich layers that may have formed near the end of magma ocean freezing are destined to remain within the stagnant lid, independently of the buoyancy ratio. In order to investigate whether such layers can be eventually subducted into the mantle, we carried out a final set of simulations using temperature and strain-rate dependent viscosity that can mimic the brittle behavior of the lithosphere (see equation (7)). Figure 10 illustrates the dynamics of the overturn for a bottom-heated system with Ra=106, , and σy=1.25×105. Of the four upwellings that reach the bottom of the stagnant lid, the leftmost one generates sufficient stress to weaken the high-viscosity lid up to the surface (Figures 10a, 10e). The lithosphere starts to be subducted (Figures 10b, 10f). As larger portions of the mantle undergo plastic yielding, the surface is increasingly mobilized, which eventually causes all high-density layers to sink at the bottom of the mantle (Figures 10c, 10g). A stable chemical stratification is then obtained and a new stagnant lid is formed (Figures 10d, 10h).
 The possibility to observe the above behavior depends on the choice of the yield stress. The higher σy, the more difficult it is to break the lid. For each value of between 0.2 and 1.8, we determined a critical yield stress , defined as the highest σy for which mantle overturn results in breaking and subduction of the uppermost lid. The accuracy with which we estimated this value is δσy=0.5×104for Ra=106 and 0.5×105 for Ra=107, i.e., we verified that plastic yielding occurs for but not for . The results of this analysis are summarized in Figure 11 for bottom-heated simulations with Ra=106. They clearly show that increases linearly with . We conducted the same test also for an isochemical model () with the same initial temperature profile. In this case, we obtained (corresponding to 8.6 MPa) and observed a plate-like behavior only during a short initial phase after which the stagnant-lid regime was reestablished and persisted throughout a nondimensional simulation time of 0.3. A linear fit of the data, including the case for , yielded: .
 When and , the stagnant-lid regime is only broken because of light upwelling material that reaches the bottom of the lithosphere during mantle overturn. Therefore, in this context the heating mode does not play any role. We actually verified that the critical yield stress depends only on the buoyancy ratio and is identical both for systems heated from below and from within. Furthermore, since the thermal and chemical Rayleigh numbers as well as the yield stress scale with the reference viscosity, a lower (or higher) value of the latter implies a higher (or lower) nondimensional . The same tests carried out at a reference Rayleigh number of 107indeed showed that the linear relation derived above remains valid with larger by one order of magnitude, i.e., . It should be noted however that, for the same reason, the dimensional value of the critical yield stress is independent of the Rayleigh number.
4.1 Mixing and Compositional Layering
 The choice of the buoyancy number dictates whether convective mixing can lead or not to the complete homogenization of the mantle within a time scale comparable with the age of Mars (4.5 Ga or 0.049 in nondimensional units). While with isoviscous bottom-heated models this is achieved for (Figure 2a), with all other heating modes and rheologies analyzed, it was achieved only for . The use of larger values of can lead to a variety of configurations: a mixed upper mantle underlain by a less well-mixed lower mantle (e.g., Figure 4b), a mixed upper mantle underlain by a stably stratified chemical profile (e.g., Figure 6c), a homogeneous mantle with isolated chemical piles rising from the lower boundary layer (e.g., Figure 6e) or a stable chemical profile which completely prevents convection and mixing (e.g., with for bottom-heated models with temperature-dependent viscosity). All these planforms, but the last one, allow for the long-term preservation of fully primitive or partly mixed chemical heterogeneities that could be sampled and entrained in upwellings.
 As shown in Figure 3, the Rayleigh number significantly affects the mixing time. For instance, with , the scaling we derived for constant viscosity predicts that a reduction of Ra by one order of magnitude determines an increase of the mixing time by approximately a factor of 3.2. This result compares relatively well with the findings of Samuel et al.  who preformed a similar calculation (i.e., isoviscous fluid in 2-D Cartesian geometry with aspect ratio of four assuming ) and estimated mixing times through a more sophisticated Lagrangian technique. They found that, independently of the wavelength of heterogeneity considered, the mixing time obtained with Ra=106 is about four times larger than that obtained with Ra=107, which confirms the validity of our simple approach based on relative differences in the laterally averaged profile of the composition field.
 The heating mode also influences strongly the thermochemical evolution of the mantle. For a given buoyancy number, internally heated systems are characterized by mixing times which are remarkably longer than those obtained from bottom-heated simulations, even when using large heating rates (tests with H=10). When temperature-dependent viscosity is considered, complete mantle mixing does not occur in any of the tests with internal heating that we carried out. Convection in the terrestrial planets is driven by a combination of heat from below and from within, with the latter being the principal contributor. In addition, decay of radiogenic elements and increase of viscosity upon compression (both of which have been neglected in the present study) contribute to diminish convective vigor and hence mixing efficiency. Therefore, if a global magma ocean existed on Mars and crystallized fractionally, some form of layered convection is to be expected, which would also allow for the long-term preservation of large-scale primordial chemical heterogeneities. However, it must be noted that the currently accepted density profile proposed by Elkins-Tanton et al. [2005a] on the basis of geochemical modeling of the fractional crystallization of Mars' magma ocean implies an effective buoyancy ratio between approximately 3 and 6, depending on the thickness of the stagnant lid. As also pointed out by Elkins-Tanton et al. [2005a], with such values of , the overturn would lead to the formation of a highly stable chemical gradient that would be too large for convection to set in, let alone for any form of mixing to occur. A mantle in which heat has been transported only by conduction through the largest part of its history is difficult to reconcile with the large-scale volcanic features which characterize Mars' surface or with the evidence for recent volcanism, as young as a few tens of million years [Neukum et al., 2004], which can be instead explained in terms of isochemical thermal convection models [e.g., Kiefer, 2003]. Furthermore, although the Martian crustal dichotomy could be the result of a large-scale impact [Andrews-Hanna et al., 2008], the hypothesis according to which it originated from an endogenic process associated with a low-degree pattern of mantle convection has been considered in several studies [e.g., Roberts and Zhong, 2006; Keller and Tackley, 2009; Šrámek and Zhong, 2012]. This scenario however would only be feasible if such pattern was established during magma ocean overturn because the subsequent stratification would prevent any large-scale motion from taking place. One possibility that would help to lower the buoyancy ratio is to invoke equilibrium instead of fractional crystallization for at least part of the magma ocean. In fact, if crystal size and settling velocity in the partially molten mantle are sufficiently small, equilibrium solidification occurs before crystal-melt separation (i.e., fractionation) can take place [Solomatov, 2000], thus reducing the global chemical gradient resulting from magma ocean freezing.
 Finally, it is also worth spending a few words about the role of compositional layering in the generation of a magnetic field. It is well-known that magnetic field observations suggest that a hydrodynamic dynamo was active on Mars for ∼500 Ma [Acuña et al., 1999]. In order to account for this, it has been argued that Mars could have experienced an early phase of plate tectonics preceding the onset of the stagnant-lid regime [Nimmo and Stevenson, 2000] and/or that its evolution must have started with a superheated core [Breuer and Spohn, 2003]. In both cases, the heat flow escaping the core during Mars' early evolution would have been large enough to drive a thermal dynamo. Because of the assumed simplifications (e.g., absence of core-cooling and decaying heat sources and the use of end-member heating modes), our simulations do not permit us to make quantitative predictions about the thermal history of the planet. Yet we observed that the presence of an initially unstable stratification can play an important role in the generation of a magnetic field as already suggested by Elkins-Tanton et al. [2005a]. As a consequence of the overturn, in fact, upper mantle layers sink to the core-mantle boundary and cool the deep mantle. In the simulations featuring a bottom heating mode and even without considering plastic yielding, we found that during a time comprised between 200 Ma and 1 Ga (depending on the rheology and on the buoyancy number), the heat flow out of the core exceeds the critical range of ∼5−20 mW/m2characterizing the heat conducted along the core adiabat [Nimmo and Stevenson, 2000]. Afterward, such heat flow decreases the more rapidly the higher the is. Using buoyancy ratios that allow for the preservation of deep layering over several billion years, the core-mantle boundary heat flow remains constantly subcritical. The latter, however, when using smaller buoyancy ratios for which mantle mixing is possible within the planet's lifetime, can increase again, becoming sufficiently large to drive a thermal dynamo. Although internal heating and secular cooling would likely change this picture, it is interesting to observe how compositional layering could introduce some form of episodicity in the magnetic field history of a planet.
4.2 Different Mechanisms for Generating Compositional Layering
 The fractional crystallization of a magma ocean is not the only mechanism able to cause the formation of a stable chemical gradient early in Mars' history. Differentiation due to magmatism can lead to a similar situation [e.g., van den Berg et al., 1999; Schott et al., 2001; Ogawa and Yanagisawa, 2011]. Assuming in fact an initially hot and homogeneous interior, rapid heating of the mantle above the solidus can result in the generation of large amounts of magma and, as consequence, of a thick basaltic crust. The extraction of magma leaves a depleted upper mantle of harzburgitic composition that is positively buoyant with respect to the parent material. Furthermore, the crust can grow sufficiently thick for the transition to the denser ecologite to occur. This can be entrained and subducted into the mantle as a consequence of crustal recycling. Under these circumstances, a stable compositional layering can be established that is characterized by the accumulation above the core-mantle boundary of dense ecologite, overlain by lighter depleted mantle, and culminating with an even lighter basaltic crust, whose largest portion, however, is locked in the stagnant lid. Ogawa and Yanagisawa [2011, 2012] discussed in detail this scenario neglecting and accounting for the presence of water in the mantle, respectively. The authors modeled Mars' thermochemical evolution employing two-dimensional Cartesian simulations characterized by an accurate treatment of magma generation and migration based on two-phase flow theory. Even though the chemical stratification in this case is built progressively as more and more primordial material is processed via partial melting, the main compositional gradient forms within few hundred million years of evolution. Based on the maximum density difference between ecologite and basalt employed by Ogawa and Yanagisawa , a nominal buoyancy ratio of ∼1.2 is obtained. Among our simulations, those with temperature-dependent viscosity and H=10 (section 3.2.2) are probably the closest to those of Ogawa and Yanagisawa . From a qualitative point of view, the two studies agree well with each other despite fundamental differences related to the origin and formation of the compositional changes. In particular, as shown in Figure 7d, with , as well as with lower buoyancy ratios, we observe a homogeneous mantle overlying a highly stable layer that persists throughout the simulation time. Such configuration has also been identified by Ogawa and Yanagisawa , with the basal dense layer representing deeply subducted ecologite and layered mantle consisting of harzburgite and primitive material.
 Despite these similarities, it is important to point out that the generation of compositional layering through secondary volcanism as proposed by Ogawa and Yanagisawa [2011, 2012] represents a valuable alternative to the scenario of magma ocean fractionation but is hardly compatible with it. The stratification obtained when assuming the buoyancy ratios predicted by Elkins-Tanton et al. [2005a] is in fact too stable for any form of subsequent decompression melting associated with mantle plumes to take place.
4.3 Surface Mobilization
 Fractional crystallization preferentially occurs when the crystal fraction in the partially molten magma ocean exceeds a critical value that marks the transition from what Abe  calls a “soft” magma ocean to a “hard” magma ocean which deforms much more slowly. A primordial steam atmosphere can exert a blanketing effect able to keep the surface of the planet sufficiently hot for a soft and shallow magma ocean to be sustained. However, in order for this to occur, a very large energy flux is required (greater than 200 W/m2according to Abe and Matsui ) which can be maintained only during the main phase of accretion [Abe, 1993]. In a recent study, Lebrun et al.  described the evolution of a magma ocean coupled with a convective-radiative atmospheric model. They showed that the critical crystal fraction is reached in about 0.5 Ma, after which the surface temperature drops to ∼500 K and is expected to decrease steadily as a consequence of water vapor condensation [Lebrun et al., 2013]. Therefore, when the fractional crystallization of the magma ocean has completed, the surface temperature is unlikely to be large enough for solid-state convective mobilization to be possible. As a consequence, the temperature dependence of viscosity leads to the formation of a stagnant lid which is enriched in heavy and incompatible elements, and cannot participate in the overturn. The present-day existence of a thick lid possessing an anomalously high density is not supported by gravity data [e.g., Pauer and Breuer, 2008] and would be hardly compatible with the optimal density distribution predicted by interior structure models that well reproduce the moment of inertia factor [Sohl and Spohn, 1997]. As already hypothesized by Debaille et al. , an early episode of surface mobilization induced by plastic yielding can allow for the subduction of this lid. We investigated the conditions for this process to occur and determined the highest yield stress that allows for stagnant lid breaking induced by mantle overturn. We found that this critical yield stress grows linearly with the buoyancy number. On the one hand, this indicates that the large values expected for would facilitate this process, although they would also suppress convection after overturn. On the other hand, with lower values of that would be compatible with post-overturn convection, the critical yield stresses we found are generally small. Typical lithospheric strengths are several hundreds MPa [e.g., Kohlstedt et al., 1995], while the (dimensional) values we obtained are between 17 MPa for and 186 MPa for (see Figure 11). Furthermore, our estimates would have been even lower if we had used a Byerlee-type yielding with yield stress increasing with pressure. Nevertheless, it should be noted that the need to adopt a low yield stress in order to obtain a plate-like behavior is a common issue in numerical simulations of yielding-induced plate tectonics adopting a nondeformable free-slip surface [e.g., Foley and Becker, 2009; Tackley, 2000; Trompert and Hansen, 1998; van Heck and Tackley, 2008]. Recent simulations by Crameri  indicate that employing instead a free-surface allows for subduction initiation due to small-scale convection when assuming yield stresses about 25% higher than those needed to initiate subduction using a free-slip surface. These, however, still remain significantly lower than experimentally measured values of rock strengths and the use of a free-surface, albeit certainly more realistic, would not have an important impact on our results.
4.4 Technical Remarks
 From a numerical point of view, the domain aspect ratio can also affect the mixing efficiency and hence the mixing time. After carrying out convergence tests, we found that while the difference between aspect ratios three, four, or five was minor (10% at most), smaller aspect ratios of one or two delivered mixing times up to 30% longer, hence our choice of four.
 The choice of a 2-D Cartesian geometry clearly represents an important limitation of our models, in particular when it comes to drawing conclusions not only related to flow structures but also to quantitative estimates of mixing times. Consideration of the third spatial dimension is generally thought to enhance mixing through the combined action of poloidal and toroidal flows [Ferrachat and Ricard, 1998], the latter of which is intrinsically absent in two-dimensional configurations. However, while on Earth the surface poloidal and toroidal components are known to be approximately equally partitioned because of the presence of rigid and decoupled tectonic plates [e.g., Čadek and Ricard, 1992; Lithgow-Bertelloni et al., 1993], in a stagnant-lid planet like Mars, it is difficult to identify a plausible source of significant toroidal motion that can promote mixing through the interaction with poloidal flow. Furthermore, the picture is complicated by the fact that other 3-D simulations of constant-, high-viscosity fluids indicate instead that the presence of the third dimension tends to hinder mixing probably because three-dimensional flow structures resist stirring and stretching induced by boundary layer instabilities better than two-dimensional cells [Schmalzl et al., 1995, 1996]. In addition, our simulations assess mixing properties of double-diffusive systems, i.e., not of systems undergoing purely thermal convection like those treated in the above cited studies. It is thus difficult to establish how our findings would change in a 3-D setting. The possible effects of spherical geometry on mixing represent another source of complexity that has received little attention so far [van Keken and Zhong, 1999]. These are important aspects that will have to be considered in future studies.
 Despite a few inherent limitations imposed in particular by the choice of a 2-D Cartesian geometry, of end-member boundary conditions (i.e., of considering either purely bottom-heated or internally heated systems) and of a simplified initial profile of the chemical density, three robust conclusions can be drawn from our simulations:
 If Mars possessed a global magma ocean that crystallized fractionally according to the density profile proposed by Elkins-Tanton et al. [2005a], no thermal or chemical mantle convection is likely to have occurred throughout most of Mars' history because of the highly stable chemical gradient established after the overturn. Although this scenario can provide a suitable explanation for the long-term maintenance of chemical reservoirs as required by the analysis of the SNC meteorites [Elkins-Tanton et al., 2005a] and also for the formation of an ancient crust which is consistent with present-day observations [Elkins-Tanton et al., 2005b], it poses problems both with the volcanic morphology and with the recent volcanic history of the planet. Furthermore, it would prevent the generation of additional compositional differences resulting from secondary volcanism as proposed, e.g., by Ogawa and Yanagisawa[2011, 2012].
 In the presence of a less strong chemical gradient (i.e., of lower buoyancy ratios), the dynamics of the mantle is governed by double-diffusive convection. This can lead to a wide variety of convective planforms that generally involve some form of layering [e.g., Hansen and Yuen, 2000] and allow for the early establishment of chemical anomalies which, apart when using very small values of , do not subsequently mix within Mars' lifetime. In this view, an initial equilibrium crystallization of the magma ocean followed by a fractional phase could help to reduce the large gradients resulting from a pure fractionation.
 If the surface temperature present at the end of the crystallization phase is set to present-day values, the uppermost layers of the solidified magma ocean cannot be subducted as they remain locked in the stagnant lid, independently of the strength of the chemical gradient. The overturn in this case involves the mantle below the lid, the subduction of the latter being possible only if a mechanism of plastic yielding is assumed. However, the required yield stresses are generally lower than typical values of lithospheric strength when buoyancy ratios are assumed that are sufficiently low not to completely suppress mantle convection after the overturn.
 We are grateful to Gregor Golabek, an anonymous reviewer, and the associate editor for their comments, which helped to improve a previous version of the manuscript. We also thank Annika Stuke for her contribution in discussing and designing the numerical experiments. This work has been supported by the Deutsche Forschungs Gemeinschaft (grant TO 704/1-1), by the Helmholtz Gemeischaft through the alliance “Planetary evolution and life,” and by the High Performance Computing Center Stuttgart (HLRS) through the project Mantle Thermal and Compositional Simulations (MATHECO).