Influence of rotation on the metal rain in a Hadean magma ocean

Authors


Corresponding author: A. Moeller, Institute for Geophysics, University of Muenster, Correnstr. 24, 48149 Muenster, Germany (alexam@uni-muenster.de)

Abstract

[1] Today, it is widely accepted that during its early evolution, the Earth experienced a magma ocean that covered most of its surface. The separation of metal from silicate was much facilitated in the environment of such a magma ocean. The differentiation mechanism is known as the “metal‒rain scenario”. Our study will focus on the settling dynamics of these metal droplets. Because of the low viscosity of molten silicate and a higher rotation period of the Earth at that time the rotation has a potentially strong influence on the dynamics of the magma ocean. We use numerical 3D fluid simulations to analyze the combined effects of strong rotation and convection on the settling of the iron droplets. We show that the influence of rotation on the settling depends on the latitude. At the poles, the influence of rotation is only marginal. At the equator, three different scenarios can be distinguished. First, at low rotation rates, the particles form a dense layer at the bottom. Second, for strong rotation, the particles stay mostly suspended and layers form in the temperature field. Third, at higher rotation rates, the particles form a ribbon‒like structure in the middle of the box. The influence of rotation on the iron droplets may lead to a scenario where part of the iron is kept in the mantle instead of transported to the core. This would have a strong influence on the later states of the differentiation process and the amount of siderophile elements in the mantle.

1 Introduction

[2] Numerical simulations of Canup [2004] show that the formation of the Moon can best be explained by an impact of a Mars size object in the later state of the Earth's accretion. The impactor melted and vaporized a great part of the Earth's mantle. Large amounts of the Earth's mantle were also ejected into the near orbits of the Earth. Under the appropriate impact conditions with respect to angular momentum, mass and velocity of the impacting planetesimal, this debris can form a stable moon around the planet [Canup, 2004]. On the Earth itself, the vigor of the impact leaves most of the mantle silicate in a molten state. This leads to a magma ocean extending from a depth of approximately 1000 km to the surface, covering the whole Earth [Tonks and Melosh, 1990; Solomatov and Stevenson, 1993; Abe, 1997].

[3] Evidence for a largely molten Earth's mantle at some time in the late Hadean, independent of the large impact hypothesis, can be found by analyzing different isotopic systems like 142Nd/ 144Nd and 182Hf/ 182W. Indications for a nearly molten state of the Earth are given especially through the measurement of the 142Nd/ 144Nd ratio. Boyet et al. [2003], Boyet and Carlson [2005] and Caro et al. [2003] show that the Earth must have experienced a major differentiation event within the first 150 Myrs of planetary accretion. This is a strong indication for a magma ocean of at least some extent at that time.

[4] The role that a magma ocean plays for the separation of metal and silicate can best be derived from the results of a 182Hf‒ 182W ratio analysis. Hafnium (182Hf) is lithophile, which means it stays in the mantle during core forming processes, and decays with a half‒life of about 9 Myrs to Tungsten (182W). Tungsten is siderophile and is therefore removed from the mantle together with the iron during core formation. The ratio between Hafnium and Tungsten isotopes depends on whether core formation takes place before the Hafnium could decay. Measurements and analyses of this ratio show that core formation on Earth must have been a very rapid process [Lee and Halliday, 1995; Halliday et al., 1996; Kleine et al., 2002; Jacobsen, 2005]. After 30–50 Myrs, the core forming process was completed [Halliday, 2004; Kleine et al., 2005]. The separation of solid metal from solid silicate is too slow to match the timescales given by the 182Hf‒ 182W ratio analyses. This gives strong evidence for the existence of a magma ocean on Earth during the late stage of its accretion [Stevenson, 1990]. In a molten state, the separation of both phases can happen fast and efficiently, due to the great density difference between metal and silicate.

[5] The metal in the magma ocean could come from three potential sources. First, a Mars sized impactor would bring a massive amount of iron into the Earth. This iron, however, may not be important for the “metal‒rain” scenario as studies from Cameron [2000] show. The core of such an impactor can fall directly through the Earth's mantle after the impact, so that this iron is directly transferred to the Earth's core. The other two sources are more important for the “metal‒rain” scenario in a magma ocean that is studied here. The metal in the mantle can exist in the form of sub‒millimeter particles originating from undifferentiated chondritic material. Also, the metal can originate from the iron cores of small, but already differentiated impactors. These protocores would have an extent of meter to hundreds of kilometers in size [Rubie et al., 2007]. This iron material can either be trapped in the mantle before the mantle was molten or be brought into the magma ocean during its existence.

[6] In the molten silicate of a magma ocean, the small iron particles from the undifferentiated material will tend to merge and form larger droplets. On the contrary, large metal bodies like the protocores of impactors tend to become unstable during their fall through the fluid silicate and will break up into small droplets [Rubie et al., 2007; Samuel, 2012]. The size of a metal droplet being stable can be estimated by the dimensionless Weber number [Young, 1965]:

display math(1)

[7] Here, ρm stands for the density of the metal, ρs for the density of the silicate, dm is the diameter of a metal droplet, vsthe velocity of the molten silicate surrounding the droplet, and σ is the surface energy of the metal‒silicate interface. For a stable droplet size, the Weber number is approximately 10. From this, Rubie et al. [2003] calculate a stable iron droplet size of about dm=1 cm for a convecting magma ocean. Ichikawa et al. [2010] verify this result with numerical simulations. Additionally, Rubie et al. [2003] estimated that large masses of iron in a convecting magma ocean would break up into droplet sizes of about dm=1 cm after a falling distance of only a few diameters of their original size. To validate this assumption, Samuel [2012] used numerical calculations to study the break‒up of iron diapirs in a magmatic environment. His results show that for a wide range of Weber numbers (10−1−104) and Reynolds numbers (10−2−103) iron bodies smaller than the thickness of the magma layer break up into smaller units within a distance comparable to their radius. The new estimates in Samuel [2012], see stable iron droplet sizes lower than dm<0.2 m as plausible for the magma ocean. Dahl and Stevenson [2010] also reached the conclusion that iron impactors smaller than 10 km in diameter break up into small droplets, but contrary to Samuel [2012] that larger impactors do not emulsify in their model. Unfortunately, only a limited range of Weber and Reynolds numbers can be reached in numerical models. Therefore, the question of the fragmentation of large iron masses and the resulting droplet size in a magma ocean can still not be answered completely.

[8] The main conclusion from the studies mentioned above is that in a magma ocean the iron will quickly separate from the silicate and exist in form of small metal droplets falling through the magma to its bottom where the mantle is still solid. This scenario is called the “metal‒rain” scenario. Figure 1visualizes the model of this first stages of core formation and incorporates the works of Stevenson [1990], Rubie et al. [2003], and Wood et al. [2006]. First, the iron is segregated from the silicate forming small droplets. These droplets sink through the convecting magma ocean to the bottom, where they form a metal pond. This configuration is unstable due to the high density contrast between the iron and the silicate. Small perturbations will lead to large‒scale instabilities (Rayleigh‒Taylor‒instabilities) and form large diapirs falling through the solid part of the mantle into the protocore [Samuel and Tackley, 2008; Golabek et al., 2008].

Figure 1.

Schematic view of the Earth 4.5 billion years ago after the giant impact. It shows the metal rain scenario and the following core formation processes. This figure is drawn in reference to the works of Stevenson [1990], Rubie et al. [2003], and Wood et al. [2006].

[9] In this paper, we focus on the “metal‒rain” scenario, because it is yet not clear how exactly the iron droplets behave in the magma ocean. The low viscosity of the silicate magma leads to strong convection with Rayleigh numbers between Ra=1026 to Ra=1029 [Höink et al., 2006; Solomatov, 2007]. Despite this high Rayleigh numbers and therefore strong convective fluid flows, numerical studies like Höink et al. [2006] and Melosh and Rubie [2007] suggest that the iron droplets are not strongly affected by convection and mostly fall with Stokes' velocity through the fluid to the bottom.

[10] In all these studies, the effect of rotation on the differentiation process is mostly neglected. Given the low viscosity of the molten magma and the high rotation rate of the Earth after the giant impact (as calculated from studies of the angular momentum distribution of the Earth‒Moon system [Canup, 2004]), rotation can have played a major role for the dynamics of the magma ocean and thus for the settling of the iron droplets. To estimate the strength of the rotational forces in a fluid system, the Taylor number (for a mathematical definition see section 'The Fluid Model') can be employed. A higher Taylor number means a strong influence of the Coriolis force, while low Taylor numbers mean a low influence of the Coriolis force. Assuming parameters for this period of the Earth's formation (an Earth day had a length of day of about 5 h and the magma ocean a viscosity near the viscosity of water [Canup, 2004; Solomatov, 2000]) leads to a value for the Taylor number between 1025 and 1027[Solomatov, 2000; Höink et al., 2006]. The effect of such a strong rotation can potentially influence the dynamics of the magma ocean, which makes it worthy studying.

[11] The convective Rossby number, which is the ratio between the convectional and rotational forces in the system (see section 'The Fluid Model'), is appropriate to measure the effect of rotation in a rotating and convecting system. It is this force ratio that defines the dynamics of the fluid fundamentally. A strong effect of rotation is expected for Rossby numbers below Ro=1. The Rossby number of the magma ocean following the estimations of the Rayleigh and Taylor numbers above is in a range of Ro = 0.03 up to Ro = 100. This means that the dynamics of the magma ocean ranges from a strongly rotation dominated system to a system that is clearly dominated by convection. To fully understand the complex dynamics of the magma ocean, it is important to study the possible influence of rotation and its consequences for the “metal‒rain” scenario.

[12] While the individual parameters like the Rayleigh number and the Taylor number cannot be reached in numerical simulations, the ratio of these two forces represented by the Rossby number falls in the range feasible with today's computational systems. Our study focuses on the possible influence of different rotation rates and therefore will use Rossby numbers between Ro=0.1 and Ro=10.

[13] If rotation affects the magma ocean, then its influence will most likely depend on the latitude. At the poles, the gravitational force and the rotation axis are parallel to each other. Such a scenario can easily be studied in a laboratory, but is much more complicated to study when gravity and rotation vector are perpendicular to each other, which is the case for the equatorial scenario. Griffiths [1987] points out that it is unknown how rotation in different latitudes affects magma chambers even on today's Earth. Our numerical model allows for the investigation of the effect that different orientations of the rotation axes have on the particles and the fluid flow. Thus, we will use the advantage of numerical simulations to investigate the Hadean magma ocean not only at the poles but also at the equator, and we will show that different flow patterns for different latitudes lead to different particle dynamics.

2 The Model

[14] To investigate the effects of the combined convection and rotation on metal droplets, we use a numerical fluid model that is capable of dealing with high Rayleigh and Taylor numbers in a 3D box [Schmalzl and Hansen, 2000] and a discrete element method to characterize the metal droplets [Verhoeven and Schmalzl, 2009].

2.1 The Fluid Model

[15] The governing equations of a Boussinesq fluid in three dimensions with both finite Prandtl number and rotation are non‒dimensionalized by introducing the characteristic scales. For the length scale, we use the depth of the fluid layer d. As characteristic timescale for this problem, we use d2/κ with the thermal diffusivity math formula, where k is the heat conductivity coefficient, cp the specific heat at constant pressure and ρ0is the density. Applying these scales, we obtain the set of governing equations in non‒dimensional form (a detailed derivation can be found for example in [Chandrasekhar, 1961]).

[16] The continuity equation

display math(2)

where math formula is the velocity vector of the fluid. The heat transport equation

display math(3)

with T being the temperature. And the momentum equation

display math(4)

where p is the pressure. The chemical component in the buoyancy term, C, describes the influence of the particles on the fluid motion. It is calculated by means of the particle concentration. The influence of the chemical component on the fluid motion is described with the buoyancy number B:

display math(5)

[17] Here, α is the thermal expansion coefficient, ρpthe particle density, and ρfl the density of the fluid. This number is a measure for the density difference between the particles and the fluid. For iron and molten silicate, the buoyancy number can be calculated as B=37.5 [Höink et al., 2006]. We will use this number in all simulations discussed below.

[18] Together with the buoyancy number, three more dimensionless control parameters appear in equation (4). First, the Rayleigh number

display math(6)

describes the ratio between forces supporting convection to forces preventing it. The higher the Rayleigh number, the stronger the convection. In the Hadean magma ocean, the Rayleigh number is estimated to be between Ra=1027and Ra=1029[Höink et al., 2006; Solomatov, 2007]. Rayleigh numbers in this range are not achievable for computational or experimental studies today. In this study, we use a Rayleigh number of Ra=108, which is in a comparable range to the study of Höink et al. [2006]. The Prandtl number

display math(7)

represents the ratio between the thermal and the momentum diffusion. For the magma ocean, the Prandtl number is estimated to be between Pr=1 and Pr=100 [Höink et al., 2006; Solomatov, 2007]. The strongest dependence of convection on the Prandtl number has been reported for Prandtl values around unity. For larger Prandtl numbers, the difference becomes less significant [Schmalzl et al., 2001]. In this study, the Prandtl number is fixed to a value of Pr=1. The Taylor number

display math(8)

describes the strength of the Coriolis force compared to the viscous forces. Here, ν is the dynamic viscosity of the fluid and Ωis the rotation rate. For the magma ocean, the Taylor number is in a range between Ta=1025 and Ta=1027[Solomatov, 2000; Höink et al., 2006]. In our simulations we use Taylor numbers ranging from Ta=106 to Ta=1010 and for comparison a non‒rotating case with Ta=0.

[19] Like in many other cases in Geodynamics, the realistic parameter range for the magma ocean is not within reach of today's computational capacities. As in these other cases, our study is guided by the idea that not the exact values of the parameters are important for the dynamics but the balance between the different forces that act on the fluid. A measure for this balance is the convective Rossby number Ro[Gilman, 1977]:

display math(9)

[20] For Rossby numbers Ro>1, the system is dominated by the buoyant force. When the Rossby number is lower than unity (Ro<1), the system is dominated by the Coriolis force and is expected to be strongly affected by rotation. Using the estimated parameters for the magma ocean from above, one can calculate a Rossby number range from Ro=0.03 to Ro=100. This leads to the conclusion that the dynamics of the magma ocean can range from a strong rotational dominated to a purely convective regime and, in fact, the system may traverse between one regime to the other during its evolution.

[21] The fluid equations are solved numerically using an algorithm described in Schmalzl and Hansen [2000]. It uses the primitive variable formulation on a uniform collocated 3D grid. For the spacial discretization, we use a finite volume discretization method [Patankar, 1980] with a second order central scheme for the diffusive fluxes and a QUICK [Leonardo, 1979] scheme for all other terms.

[22] For the time evolution we use a linear multi‒step method, wherein diffusive terms are treated implicitly and convective explicitly. For this so called IMEX method [Frank et al., 1997], stability is obtained by a step‒size limitation to the Levy‒Courant‒criterion.

[23] The fluid model is capable of handling a rotation axis with an arbitrary angle with respect to the gravitational acceleration. Figure 2 is a schematic picture of the configurations. In the numerical model, other than in Figure 2 and in reality, not the gravitation axis, but the rotation axis is tilted when the equator is simulated. This is physically equivalent but numerically much easier to implement. At the pole, the rotation axis is parallel to the z‒axis. At the equator, it is parallel to x‒axis. The gravitation vector remains parallel to the z‒axis for all cases. This setting allows to systematically explore the effect of the latitude on the settling of the iron droplets. In this study, we will focus on the most extreme cases, the polar and equatorial regions of the magma ocean.

Figure 2.

The numerical model in the magma ocean context. The black squares represent the model area. Vectors show how gravitational acceleration and rotational axis change from the polar to the equatorial regions.

2.2 Tracer Model

[24] The iron in the magma ocean exists as small, separated droplets of approximately 1 cm to 20 cm in size, as indicated by the Weber number [Young, 1965; Rubie et al., 2003; Ichikawa et al., 2010] and the studies of Samuel [2012]. According to these studies, we can assume that the droplets do not merge but stay separated as individual particles (see section 'Introduction'). Therefore, as a first approach, we simulate the droplets as individual tracer particles in the fluid.

[25] We use a discrete element model to describe the behavior of the iron particles in the fluid. Due to the high density of the iron, as compared to the silicates, the particles will influence the dynamics of the flow itself. We use the particle concentration to define the chemical component C. According to equation (4), this chemical component influences the fluid by the buoyancy term. The particle algorithm follows in most parts that of Verhoeven and Schmalzl [2009].

[26] Our numerical particles are not point‒shaped but rather have a certain extension and a spherical shape. They experience inertia due to their mass, are not deformable and too big to be influenced by Brownian motion. In our simulations, the particles are chosen to be much smaller than the size of a fluid cell, thus avoiding to resolve the fluid flow around the particles explicitly for numerical reasons. The flow around the particle is assumed to be laminar. In this case, the force acting on a moving spherical particle in a viscous fluid is the Stokes friction force. Clearly, this means a simplification with respect to iron droplets in a magma ocean. However, studies of Ziethe [2009] have shown that particles in a magma ocean subject to a fully resolved turbulent flow and those subject to simple Stokes drag show very similar sinking behavior. Thus, assuming Stokes drag seems a reasonable approximation and has been employed in several studies [Höink et al., 2006; Rubie et al., 2003]. From this follows the force of the fluid acting on the particles:

display math(10)

where rp is the particle radius, vp is the velocity of the particle and vi is the velocity of the fluid at the location of the particle xi. The stronger the velocity difference between the particle and the fluid, the stronger the force acting on the particle. This leads to particles following the fluid flow. Further external forces do act on a particle. In the early Earth's ”metal‒rain” scenario, the gravity and the Coriolis force are of importance.

display math(11)
display math(12)

[27] The position and velocity of each particle as resulting from the action of these forces is calculated during each time step of the simulation.

[28] With the gravitational force and the frictional force, the Stokes' sinking velocity can easily be calculated [Kundu and Cohen, 2008]. In a non‒dimensional form, it can be expressed by the Rayleigh and the buoyancy number leading to

display math(13)

[29] This is the velocity of a spherical particle sinking in a viscous motionless fluid neglecting the influence of turbulent fluid motion around the particle. It also defines a typical timescale, the so‒called Stokes' time for the particles.

display math(14)

[30] The Stokes' time is the time in which a particle falls from the top of the numerical box through the motionless fluid to the bottom. We will use this Stokes' time to compare the timescales of the different simulations.

[31] To simulate spatially extended spherical particles, one has to apply a mechanism to prevent the particles from occupying the same space. In our study, this is done through a collision algorithm described in detail in [Verhoeven and Schmalzl, 2009].

2.3 Boundary Conditions

[32] The equations are solved in a 3D Cartesian box where the bottom of the box represents the bottom of the magma ocean and the top the surface of the magma ocean (Figure 2). We use isothermal boundary conditions at the top and the bottom (Figure 3). Fixing the temperature at the lower boundary resembles a scenario where the magma ocean is heated from below. The heat stems from differentiation processes that took place prior to the formation of the magma ocean and from the core of the giant impactor [Cameron, 2000]. Assuming a zero non‒dimensional temperature at the top resembles a non‒insulating atmosphere. Free‒slip boundary conditions are used at the top and the bottom for the velocity field. At the top, the free‒slip boundary condition is used to simulate the free surface of the magma ocean. At the bottom, a mushy layer with uncertain properties may have existed. It is not clear if, in a simple model, this can be better resembled by a stress‒free or by a rigid boundary condition. Studies on the settling history in magma chambers have employed rigid boundaries [Martin and Nokes, 1988; Lavorel and Le Bars, 2009] and it has been demonstrated that the settling of the particles took place on an exponential timescale. In a rotating system, the boundary layers have a very different behavior. A no‒slip condition in a rotating system results in the formation of an Ekman‒layer. Differently from a non‒rotating system, the Ekman‒layer is not passive. It is rather an active boundary layer in which the so called Ekman‒pumping takes place. For a real magma ocean the Ekman number is so small (Ek=10−12 to Ek=10−14) that it could not be reached in any of today's models. That means that assuming a no‒slip condition at the bottom would necessarily result in a strong overestimation of the role of the Ekman‒pumping. Thus, it seems more realistic to assume a stress‒free boundary and thus neglect the Ekman‒layer, rather than heavily overestimating its influence. A similar argument has been pushed forward by Busse and Simitev [2011]. For the sidewalls, we use periodic boundary conditions to reduce the influence of walls on the fluid and the particles. The top and bottom are impermeable for the particles. To hinder the particles from falling through the bottom, we use the same collision algorithm as described in [Verhoeven and Schmalzl, 2009]. In our numerical model, this can technically lead to numerical artifacts in the chemical field at the boundary points. Therefore, the first and last points in z‒direction of the chemical field are removed from all graphs shown here.

Figure 3.

Exemplary starting and boundary conditions for a polar run. The simulation is started from a statistically steady state. The particles are randomly distributed throughout the whole box (yellow droplets).

[33] In order to avoid the transient and unrealistic fluid behavior in the first time steps, we add the particles to a fluid simulation that has reached a statistically steady state. The particles start randomly, but homogeneously spread over the Cartesian box (Figure 3). Although this might not have been the case in the magma ocean, this makes our study more comparable to previous studies like Höink et al. [2006]. In this study, we use N=200000 particles with a non‒dimensional radius of rp=0.002.

3 Results

[34] In this section, we will show that the dynamics of particles in a rotating fluid is strongly dependent on the angle between the rotation and the gravitation vector. First, we will discuss the polar case where the rotation axis is parallel to the gravitational acceleration. Second, we will discuss the situation at the equator where the rotation axis is perpendicular to the gravitation. For a better comparison of the results, we use a fixed Rayleigh and Prandtl number with Ra=108 and Pr=1 and different Rossby numbers between Ro=0.1 and Ro=1 (see section 'The Fluid Model').

3.1 Settling at the Poles

[35] At the poles, the rotation axis is parallel to the gravitational force. The dynamics of a fluid without particles under these circumstances has been broadly investigated and analyzed (e.g., Chandrasekhar [1961]). As described by the Taylor‒Proudman‒Theorem, a flow subject to strong rotation develops a mostly two dimensional pattern in a plane perpendicular to the rotation axis, while motion in direction of the rotation axis is suppressed. In a thermally driven convecting fluid, rotation is known to lower the flow amplitude and to effectively increase the critical Rayleigh number [Chandrasekhar, 1961].

[36] The motion of particles in the convecting and rotating fluid results from a superposition of the gravity force acting in z‒direction and the forces exerted by the motion of the surrounding fluid. A particle subject only to the gravitational force only would sink with Stokes' velocity straight to the bottom. However, convectively driven upward and downward currents do influence the sinking velocity of the droplet. If convection is sufficiently strong, the particles can even stay suspended or can sink significantly slower as compared to the non‒convecting case. Since rotation hinders convection, it can be expected that particles will be less influenced by convection and thus will sink faster.

[37] In order to get the first impression of the particle dynamics, we display the fraction of particles being in the upper 2/3 of the domain as a function of time (Figure 4). All calculations were performed at the same value of the Rayleigh number (Ra=108) but at different Rossby numbers, i.e., at different rotation rates. The colors indicate the strength of rotation, ranging from no rotation (red, Ro=) to the highest rotation rate (blue, Ro=0.1). As a timescale, we use the non‒dimensional Stokes' time that is defined in equation (14). This timescale allows us to compare the settling behavior of particles for different fluid parameters. A particle sinking through a quiescent fluid would reach the bottom of the box after a Stokes' time of ts=1. For pure Stokes' sinking, the upper 2/3 of the box would be cleared of particles after ts=0.6667 Stokes' time. Due to convection, the particles can fall slower or faster than the Stokes' velocity. This results in different mean sinking velocities that are represented through different slopes of the curves in Figure 4. Thus, we are able to monitor many aspects of the settling history of the particles even when the figure focuses only on 2/3 of the box.

Figure 4.

Time history of the particle concentration in the upper 2/3 of the box. All cases have a fixed Rayleigh number with Ra=108 and Rossby numbers ranging from Ro= to Ro=0.1. At the beginning, the particles are randomly but homogeneously spread over the whole box, therefore, all curves start at a particle concentration of 0.6667. The slope of the curves gives the mean velocity of all falling particles.

[38] Figure 4 clearly shows that, in all cases, the particle fraction is decreasing with time. However, the mean sinking velocity depends on the rate of rotation. For fast rotation (blue line) and thus small Rossby number, the particle fraction decreases the fastest, while at no rotation (red line, infinite Rossby number) the slowest decrease is observed. Without rotation, it takes about six Stokes' times to clear the upper 2/3 of the box, while this is reached after only about four Stokes' times at the highest rotation rate. Obviously, convection is able to prevent the particles from falling with Stokes' velocity in all cases under investigation, even in the case with strongest rotation. For strong rotation (violet and blue line), finally, the particles settle at the ground and the upper 2/3 are clear. Differently, at no or low rotation rates (red and green line), a statistically stationary flow develops in which a portion of the particles stay suspended.

[39] Thus, Figure 4 displays the expected behavior, the stronger the rotation, the faster do the particles sink. If, on the other hand, convection is strong enough, a certain amount of particles can be kept in suspension.

[40] After a transient period, during which the particles settle, all cases display a statistically stationary Nusselt number and root‒mean‒square velocity for more than 40 Stokes' times. Therefore, we find it appropriate to treat the system as stationary and to employ time‒averaged quantities for further analysis.

[41] The temperature and particle concentration profiles in Figures 5a and 5b are useful to determine the behavior of the particles and their influence on the fluid flow. Figure 5a shows the averaged particle concentration profiles in the statistically steady state. The profiles correspond to the four cases in Figure 4. The same colors are used for the same cases. The particle concentration profiles in all four cases show mostly the same characteristics. Independent of the rotational strength, the particle concentration in the bulk of the box is nearly zero. At the bottom, the particles accumulate and form a layer extending from z=0 to z=0.1 with a maximum concentration of Cmax=0.36 (for a better visibility, we show the profiles only for a concentration up to C=0.2). The enlarged part of the graph clearly shows that the differences in the concentration profiles are marginal, only the case with Ro=0.1 (light blue line) shows a slight difference in the increase of the particle concentration. In every case, the mean particle concentration seems hardly influenced by the rotation rate. All particles settle in a thin layer at the bottom after some time, as being reflected by the particle concentration depth profiles (Figure 5a).

Figure 5.

(a) Particle concentration profiles and (b) temperature profiles averaged over time for the polar case with Ra=108 and different Rossby numbers ranging from Ro=1 to Ro=0.1.

[42] The corresponding temperature depth profiles are displayed in Figure 5b. Again, the colors denote the strength of rotation. The profiles change from a typical convective profile with averaged temperature of T=0.5 at no rotation (red line) to an almost conductive profile at high rotation (light blue line). Since the particle distribution is virtually independent of rotation (Figure 5a), the change in the temperature profile reflects the influence of rotation on the temperature field, rather than the influence of the particles. However, the presence of the particle layer can be inferred from the asymmetry of the temperature depth profiles. In all four cases, a bottom boundary layer significantly thicker than the top boundary layer can be observed (Figure 5b). The particles which have settled at the bottom, stabilize the lower layer. Due to the heavy particles, plumes emerging from the bottom boundary are slowed down, which hinders an effective upward heat transport. This leads to a thick, hot lower boundary layer.

[43] In order to highlight the effect of the dense particles, we have compared the case as discussed above, with a scenario having identical conditions, except that there are no particles. In Figures 6a and 6b, we show snapshots of the particle concentration and the temperature field for a case with particles and in Figure 6c, a snapshot of a temperature field for a case without particles. Figure 6a shows that the particles form a layer at the bottom, and a comparison of Figures 6b and 6c allows to isolate the effect of the particles on the temperature field. While in the particle‒free case (Figure 6c), the upper and lower boundaries are symmetric, the particles decrease convective heat transport in the lower layer, leading to a significant thickening of the lower thermal boundary layer (Figure 6b).

Figure 6.

Snapshots of the particle concentration and the temperature field for a polar case with Ra=108 and Ro=0.3. The particle concentration field is shown as contour plot, where the yellow plane represents the contours of the chemical field at a concentration of C=0.05. In the volume plot of the temperature field, red represents hot temperatures and blue represents low temperatures. The parts representing intermediate temperatures between T=0.4 and T=0.6 are removed for better visibility. At the right, a temperature field of a run with the same parameters (Ra=108 and Ro=0.3), but without particles, is shown for a comparison.

[44] The dynamics within the particle layer is complex and results from (a) heating of the particle layer from the bottom and (b) the impact of the existing large‒scale circulation on the particle layer. Snapshots of temperature, particle distribution and velocities in the xy‒plane at a depth of d=0.02 within the lower boundary layer display details of the dynamics (Figures 7a–7c). In general, positive velocities (Figure 7c) correlate with hot patches (red spots in Figure 7a) and high particle concentrations (red patches in Figure 7b). Within the thermal boundary layer, a circulation exists. Upwelling plumes collect particles, rise, and circulate back to the bottom. These regions with negative velocities (blue in Figure 7c) show relatively low temperatures (yellow in Figure 7a). In the middle of the box, significantly above the boundary layer, the velocity field is no longer influenced by the particles. This can readily be seen from Figure 7d, displaying the vertical velocity in the xy‒plane at mid‒depth. Here, the velocity field only shows variations on much larger scales, as compared to the boundary layer (Figure 7c). The influence of the particles on the flow is thus limited to the boundary layer and is not visible at mid‒depth of the box (Figure 7d). On the other hand, the large scale motion does shape the particle layer. In the middle of the left edge in Figure 7a, a green spot denotes a region of cold temperature, originating from a strong downwelling, which reaches the bottom. The region is virtually particle‒free (deep blue in Figure 7b) and there is no small‒scale velocity fluctuation within that region. A similar phenomenon, however less pronounced, can be observed on the right side of Figures 7a and 7b.

Figure 7.

These snapshots show cross‒sections of the temperature field, particle concentration and z‒component of the velocity field in the xy‒plane within the bottom boundary layer. They are typical for all polar cases. For comparison, a snapshot of the z‒component of the velocity in the middle of the box is shown at the right side. For the temperature and the particle concentration, the values range from zero (dark blue) to one (dark red). For the velocities, blue represents negative velocity and red represents positive velocity in z direction.

[45] In summary, if rotation axis and gravity are aligned, the particles establish a stable, dense bottom boundary layer after a few Stokes' times, independent of the strength of rotation. In this constellation, our results with strong rotation resemble qualitatively those of Höink et al. [2006], which were obtained without rotation.

3.2 Settling at the Equator

[46] At the equator, the rotation axis is perpendicular to the gravitational force and therefore the Coriolis force can act parallel to the gravitational acceleration. Since in this case the rotation axis is parallel to the x‒direction, this leads to a decrease of gradients in the x‒velocity with increasing rotation rate (see Figure 2 and section 'The Fluid Model'). Under the additional influence of thermal forcing, other phenomena, like zonal and thermal winds, can occur [Jones, 2007]. Zonal winds arise through fluid motions perpendicular to the rotation axis and lead to flows in y‒direction. Thermal winds are a result of temperature gradients in the direction of the rotation axis [Jones, 2007]. Although the detailed mechanism behind the generation is still subject of current research, it seems clear that for our setup the zonal and thermal winds create shear flows in the fluid, which could significantly influence the particle dynamics.

[47] In our case, strong shear flows develop in the yz‒plane perpendicular to the rotation axes. This phenomenon is shown in Figure 8a for a case without particles for Ra=108 and a Rossby number of Ro=0.1. The colors indicate flow in different directions at the top and the bottom. In between these two flow directions, the velocity reaches nearly zero. The temperature field in Figure 8b for the same case is typical of a temperature field at high rotation rates without particles (red represents high temperatures, blue represents low temperatures). The dynamics of the fluid is dominated by the strong shear flow. This flow prevents plumes from rising or sinking and therefore suppresses an effective transportation of heat through the system. It is difficult to predict how particles will behave under such circumstances, especially if they actively do influence the flow pattern, due to their excess density. This issue will be discussed further in this section.

Figure 8.

Examples of the fluid flow in a strongly rotating fluid, without particles, at the equator. The y velocity shows the strong shear flow that develops under these circumstances (blue represents negative velocities and red positive velocities). The snapshot of the temperature shows a typical temperature field of a strong rotating fluid (red represents hot temperatures and blue represents cold temperatures).

[48] The time history of the fraction of particles in the upper 2/3 of the box, as portrayed in Figure 9, reveals a behavior clearly different from that observed in the polar case. All time history plots are given in terms of Stokes' time. At low rotation rates, with Rossby numbers Ro>1, the particles show a similar settling characteristic as in the polar case (dark blue line). The fraction of particles in the upper 2/3 of the box decreases and after about five Stokes' times there are nearly no particles left in the upper part of the box. For a further increase of the rotation rate, the effect of the tilted rotation axis becomes significant. At a Rossby number of Ro=0.2 (black line in Figure 9) the settling behavior is clearly distinct from the case with low rotation. A significant fraction of the particles stays suspended in the upper 2/3 of the box, rather than settle at the lower boundary. Even in the statistically stationary state, about 40% of the particles are suspended in the upper 2/3. A further increase of the rotation rate to a value of Ro=0.1 (light blue line in Figure 9) leads to a situation in which virtually all particles stay suspended in the upper 2/3 of the model box. According to our simulations, these “suspension states” are stable for more than 100 Stokes' times.

Figure 9.

Time history of the particle concentration in the upper 2/3 of the box for the equatorial cases with Ra=108 and Rossby numbers ranging from no rotation to Rossby numbers between Ro=1 to Ro=0.1. Due to the homogeneous distribution of particles at t=0, all graphs show a starting particle concentration of 0.6667. At high rotation rates, the light blue line shows that nearly all particles are suspended in the upper part of the box contrary to at low rotation rates which correspond to the red and dark blue line.

[49] Different from the polar case, the settling history at the equator is crucially influenced by rotation. In what follows, we separately describe the dynamics of the cases with both low and high rotation.

3.3 The Low Rotational Case

[50] At low rotation rates with Rossby numbers greater than one (Ro>1), we find a regime similar to the polar scenario at likewise low rotation rates. Figures 10a and 10b show the time history of the particle concentration and the temperature. A depth profile of each quantity was taken at every time instant. The profiles were arranged as a sequence to show the temporal evolution of the particle concentration and the temperature. Dark blue areas in Figure 10a denote virtually particle‒free regions, while red areas denote a particle concentration higher than C=0.05. This value is chosen in order to achieve a good visibility of the regions with low particle concentrations, like those developing during the particle settling period. Figure 10a shows that the particles are initially evenly distributed throughout the fluid (light blue at all depth levels). After a few Stokes' times (about ts=5) the settling particles have formed a dense bottom layer (red layer in Figure 10a). After this initial cumulation period, all particles have settled at the bottom and the system reaches a virtually stationary state. In Figure 10b we show the evolution of the thermal profile. Corresponding to the appearance of the particle layer in Figure 10a, we observe the formation of a thick thermal boundary layer in Figure 10b. The heavy particles stabilize a rather thick boundary at the bottom of the system, while an upper thermal boundary layer is much less developed. In Figures 10c and 10(d), the depth profiles of the particle concentration and the temperature, averaged over the statistically stationary period, are displayed. The presence of the particle layer and the asymmetry of the temperature field is clearly exhibited. As in the polar case, the increase of the temperature in the bulk of the system indicates that convection is rather weak, caused by the influence of rotation. In summary, the settling behavior at low rotation rates is similar at the poles and at the equator.

Figure 10.

(a–b) Time history of temperature and particle concentration profiles of a case with Ra=108 and Ro=1 representing the low rotational case. For every time step, a particle concentration and temperature profile is shown color‒coded, to illustrate the development of these profiles with time. For example, the time history of the particle concentration profile shows how, at the beginning, the particle density is low but evenly distributed over the hole depth. Due to the sinking of the particles, the concentration of particles at the bottom begins to rise. After about five Stokes' times a particle layer at the bottom has formed and the concentration in the rest of the box is zero. (c–d) The time‒averaged depth profiles in the statistically steady state for the particle concentration and temperature profile.

3.4 The Layer Case

[51] With increasing influence of rotation, the particles can stay more and more in suspension. Thus, a particle layer forms, which ranges from the bottom to almost mid‒depth. As a consequence of the presence of the dense particles, a hot layer of similar extension develops. This phenomenon is demonstrated for a case with Ro=0.3 and Ra=108. As in the previous case, Figure 11 describes the temporal evolution of the depth profiles of the particle concentration (Figure 11a) the temperature profile (Figure 11b), the time‒averaged depth profiles of particle concentration (Figure 11c), and the temperature (Figure 11d). In Figure 11a, the concentration is scaled between C=0 (dark blue, no particles) and C=0.02 (red). Initially, all particles are evenly distributed. Already after about two Stokes' times, we observe a particle layer extending from the bottom to a depth of about d=0.4. Interestingly, the depth of the particle layer does not vary with time, while there are substantial density fluctuations within the layer, which must be an expression of the internal dynamics of that layer. From the thermal profile (Figure 11b), it is noticeable that after the formation of the particle layer, a thermal layer of virtually the same thickness evolves. Starting from a thin thermal boundary layer, the layer heats up until it reaches a value of about T=0.98 in the statistically stationary state.

Figure 11.

(a–b) Time history of the temperature and particle concentration profiles for a case with Ra=108 and Ro=0.3 representing the layer case. For every time step, particle concentration and temperature profiles are shown color coded to illustrate the development of these profiles with time. For example, the time history of the particle concentration profiles shows, at the beginning, a low concentration in the whole box, due to the starting condition of the particles. During the evolution of this case, a dense layer forms at about 0.4 of the box height; and above this layer, the concentration drops to zero. (c–d) The time averaged depth profiles for each case after a statistically steady state is reached.

[52] Details of the layer structure can be inferred from the time‒averaged depth profiles (Figure 11c and 11d). The depth profile of the particle concentration shows a layer of particles, ranging from “slightly above the bottom” to a depth of about z=0.4. Due to the significant extension of the layer, the actual particle concentration is quite low. At the maximum, at a depth of z=0.2, the particle concentration is C=0.025. Due to the Coriolis force, a situation arises where particles stay almost in contact with the bottom, but are mostly kept in suspension. This has a profound influence on the thermal structure.

[53] The temperature profile (Figure 11d) reveals a three‒layered structure. In the statistically stationary state, a hot layer of virtually constant temperature underlies a thin layer with a relatively small temperature gradient. The top layer is characterized by an almost conductive temperature distribution. Strong convection takes place in the lowermost part of the box, leading to a well mixed layer. Since the particles are not, or at least not fully, covering the bottom, they have no strong insulating effect and heat flows into the system. Strong plumes develop (Figure 12), rise from the hot boundary until they reach the region with high particle concentration. The stabilizing effect of the particles, together with the Coriolis force, shears the plume and thus deflecting it from its way upward and prevent it from leaving the particle layer. A series of snapshots, showing the temperature fields at successive time instants, illustrate this behavior. For better visibility, we have marked one emerging hot plume (also visible as dark red area) with a yellow arrow. The plume rises upwards but is also clearly deflected from its path. As explained above, the relatively high concentration of heavy particles slows the plume down, and the last snapshot (Figure 12c) shows that the plume is not able to leave the lowermost layer.

Figure 12.

Time series of the temperature field of a case with Ra=108 and Ro=0.3. The snapshots are made in a temporal interval of 0.05 Stokes' times, beginning at the left. Red colors mean high temperatures, blue colors low temperatures. The arrow marks a plume rising from the bottom upward.

[54] Summarizing, in this configuration with moderate rotation rate, rotation and gravitation form a particle layer close to the bottom, which is, however, not covering it. This particle layer, divides the system into two domains. The lower one is hot and vigorously convecting, and the upper one is hardly convecting with an almost linear temperature gradient. This particular type of layering depends on a delicate balance of the Coriolis and gravitational forces and can thus only be observed in a small parameter range.

3.5 The Ribbon Case

[55] As the rotation is further increased, the particles form a ribbon‒like structure located significantly above the lower boundary. We describe this scenario by the same type of Figures as in the previous cases. The temporal evolution of the depth profile of the particle concentration for a Rossby number of Ro=0.1 is displayed in Figure 13a. The initially homogeneous particle distribution evolves towards the “ribbon‒like” structure in the middle of the box, after already half a Stokes time. The shape of this particle layer, as well as the particle concentration within the layer, fluctuates with time. Interestingly, the maximum particle concentration stays constant at a depth of d=0.4 (Figure 13c). Further, during the entire simulation time, the layer has no contact with the bottom. Due to strong rotation, the system is dominated by a strong shear flow (see also Figure 8). Only at mid depth the flow in y‒direction almost vanishes. Here, the particles sink due to gravitation and once they have entered the shear zone again, they are pushed upwards by the Coriolis force. Complex interaction of the Coriolis force, gravitation, and the fluid flow creates the particle layer at about mid depth and also its internal dynamics.

Figure 13.

(a–c) Time history of temperature and particle density profiles for a case with Ra=108 and Ro=0.1 representing the ribbon case. For every time‒step a color‒coded profile is shown. (e–f): The time‒averaged temperature and particle concentration profiles after the system reaches a statistically stationary state.

[56] Figure 13b displays the time history of the temperature field and shows that, differently from the layer case, no thermal layering in the temperature field can be observed. This is due to the fact that the particles have no contact with the bottom boundary and the shear flow in this case prevents the plumes from rising upward into the particle layer. The time‒averaged temperature profile in Figure 13d shows more details of the vertical temperature distribution in this case. The temperature profile is asymmetric and shows three distinguishable regions of different temperature gradients. At the bottom and the top, the temperature change indicates a mostly conductive heat transport, while in the middle where the particles are located, a more or less constant temperature develops. This behavior can be explained through the interaction of the particles with the fluid and the strong shear flow. The dynamics of the fluid is mostly dominated by the strong shear flow in y‒direction. Rising plumes from the bottom and sinking plumes from the top are deflected by this shear, such that they are unable to reach the particle layer. Heat transport in the top and bottom layer is mainly conductive. If plumes cannot reach the particle layer, then what mechanism does generate an almost constant temperature within the particle region? The particles themselves play the key role here. In the particle region, the influence of the shear flow is minimal. Particle laden fluid is heavy and sinks due to the higher density. Once it reaches the lower shear zone, the Coriolis force will push it upward again. Sinking of the fluid due to particle induced excess of density, and subsequent upward pushing by rotational forces leads to a wave‒like motion of the particle layer (Figure 14). The snapshot in Figure 14a shows the particle concentration. Clearly, the ribbon‒like structure of the high particle concentration can be seen in the middle of the box. This ribbon reflects the wave‒like motion of the particles, where the particles reach their maximum height at the right side of the box and their minimum height in the middle of the box. The distribution of the corresponding vertical velocity is shown in Figure 14b. Upward motion (positive z‒velocity) is visible where the particles reach their lowest position. Downward motion, where the particles are at their maximum height, is found on the right and continued at the left side of the box, due to the periodic boundaries. Comparing the shear flow in Figure 14c, where particles are present, with the shear flow in Figure 8a, where no particles are present in the model, shows that the particles also clearly alter the structure of the positive and negative flow zones. Due to the flows that are created by the presence of the particles, the fluid is effectively mixed and a region of almost constant temperature in the middle of the box develops.

Figure 14.

(a–c) Snapshots of the particle concentration field and the z‒ and y‒velocity of the strong rotational case with Ra=108 and Ta=1010. These three Figures illustrate the wavelike motion of the particles and the mixing of the fluid in the middle of the box due to the up and downward motion of the fluid.

[57] In summary in this rotation dominated case, the developing shear flows have a strong influence on the dynamics of the particles. The opposed flows in the y‒direction at the bottom and the top of the box lead to strong Coriolis forces acting on the particles that follow that flow. The Coriolis force directs the particles to the middle of the box where the y‒flow decreases to nearly zero. Therefore, the particles form a ribbon in the middle of the box with no contact to the bottom boundary.

[58] The previously discussed equatorial cases are summarized in Figure 15. Here, we show snapshots of the temperature field and the particle concentration for different rotation rates, ranging from low rotation (Ro=1, Figures 15a and 15d) over medium (Ro=0.3, Figures 15b and 15e) to high rotational rates (Ro=0.1, Figures 15c and 15f). At a low rotation rate, we observe a vigorously convecting fluid, characterized by many irregular upwellings and downwellings (Figure 15a). The initially evenly distributed particles have settled and form a dense layer at the bottom (Figure 15d).

Figure 15.

Snapshots of the equatorial scenario with Ra=108 and different Rossby numbers. Figures 15a and 15d show the low rotation case, 15b and 15e the layer case, and 15c and 15f the ribbon case. (a–c) Snapshots of the temperature field where red means hot temperatures and blue means cold temperatures. (d–f) Snapshots of the particle concentration field (contour plot).

[59] An increase of the rotation rate lifts the particle layer (Figure 15e), so that it loses its insulating effect. Plumes developing at the lower boundary are able to deform the particle layer. However, they cannot escape through the layer. This creates a hot lower layer with a virtually constant temperature (see also Figure 15b). In the case of strongest rotation (Ro=0.1), we observe the formation of a “ribbon”‒like structure of the particles, close to mid depth of the box (see Figure 15f). The strong Coriolis forces lead to a flow pattern that shows a mostly horizontal dynamic. Plumes that rise from the boundary layers are deflected in y‒direction due to the strong rotation (Figure 15c).

4 Discussion

[60] In order to better understand the iron‒silicate differentiation process in an early Earth's magma ocean, we investigate the dynamics of heavy particles in a strongly convecting fluid being subject to rotation, by means of a numerical model. The magma ocean is simulated as a low Prandtl number (Pr=1) fluid in a 3D Cartesian box, subject to strong convection and rotation. We simulate the fluid at the poles where the rotation axes are aligned with the gravitational force and at the equator where the rotation axis is perpendicular to the gravitational vector. The iron droplets are represented through finite sized, spherical particles that move, due to a combination of the gravitational forces, Coriolis forces and frictional forces between fluid and particles. The particles influence the density of the fluid depending on the local particle concentration. We use a collision algorithm introduced by Verhoeven and Schmalzl [2009] to ensure that the finite size of the particles is conserved throughout the simulation. As boundary conditions, we use a Rayleigh‒Bèrnard setting with stress free boundaries at the top and bottom and periodic boundaries at all sides.

[61] As compared to the “real” magma ocean, the model configuration holds significant simplifications with respect to the parameters, the model geometry and boundary conditions, and further with respect to the particle size. The realistic Rayleigh number in a magma ocean is assumed to be in the range between Ra=1027 and Ra=1029 and the Taylor number in the range between Ta=1025and Ta=1027[Höink et al., 2006; Solomatov, 2000]. Thus, the Rayleigh and Taylor numbers, as adopted here, differ from the realistic numbers by many orders of magnitude. Only the buoyancy number and the Prandtl number are in the range assumed by Solomatov [2000]. Unfortunately, simulations with realistic parameters cannot be expected in the near future. Particularly, the size of the iron droplets is clearly too large. While the realistic size of the particles ranges from 1 cm to 20 cm [Rubie et al., 2003; Ichikawa et al., 2010; Samuel, 2012], our particles are in the range of several meters. A key finding in this study is that particles can be kept in suspension by rotational forces. Smaller particles would be kept easier in suspension, due to their smaller Stokes' velocity. Thus, we feel that our study provides a lower bound with respect to conditions that are required to keep the iron particles in suspension. Certainly, a more detailed parameter study is necessary in order to better estimate the influence of the particle size.

[62] We simulate the large scale dynamics of the magma ocean over its entire scale. Therefore, we are not able to simulate in detail the interaction between two iron droplets. Particularly, our particles are not able to merge and form bigger droplets. In this respect, we employ the findings of Ichikawa et al. [2010] who simulated the particle interaction in great detail, but only in a small area of the magma ocean about a few centimeters in diameter. According to the findings of Ichikawa et al. [2010], the particles have a more or less constant size as long as the fluid flow around them is sufficiently strong. In the bottom boundary layer where velocities can decrease, merging might be more important. This cannot be simulated in our calculations. However, it seems unlikely that for the equatorial cases, this process can prevent the suspension of the particles under the influence of strong rotation, since the droplets are not re‒entrained from the bottom.

[63] The boundary in a magma ocean was probably rough and mushy rather than smooth and solid, as assumed in our model. Complications aiming for such a boundary layer type and also for internal heat sources need to be addressed in future studies.

5 Conclusion

[64] Several conclusions can be drawn from these results. The metal‒silicate differentiation process in a magma ocean does not operate in the same way on a planet‒wide scale, if rotational influences are significant. While in the polar region, the iron would sink to the bottom forming a metal pond, the dynamics is more complex at the equator. At moderate rotation rates, the droplets are kept in a layer, ranging from the bottom to about one third of the magma oceans height. The concentration of particles in this layer is relatively small, but sufficiently high to affect not only the composition but also the thermal structure. In the case of strong rotation (Ro=0.1), our numerical experiments indicate that no uniform, planet‒wide pond of iron‒rich heavy material would form at the boundary between the magma ocean and the solid part of the mantle. Rather the iron particles can stay completely in suspension in the equatorial region.

[65] An accumulation of the iron in a local metal pond in the polar regions can have far‒reaching implications for the growth of the core. The formation of iron‒diapirs from such a pond would thus mainly take place in the polar region resulting in a non‒uniform growth of the core.

[66] Another issue connected to our results is the overabundance of siderophile elements in the Earth's mantle. While the metal rain scenario can basically explain a core formation process, implying a chemical equilibrium between iron and silicates [Rubie et al., 2007], the works of Wade and Wood [2005] and Wood et al. [2006] indicate that the concentration of siderophile elements is by orders of magnitude too high, as compared to the metal‒rain scenario. An early explanation was offered in the work by [Jones and Drake, 1986], proposing that some iron was trapped within the mantle and thus not involved in core‒formation. Up to now, most authors argue that iron would sink too fast in a magma ocean to get trapped [Rubie et al., 2007]. Allowing for rotation to influence the differentiation process offers a plausible explanation for the trapped iron. Iron particles can stay suspended in the equatorial region of the magma ocean. A certain amount of this iron gets trapped in the mantle after its solidification and would explain the overabundance of the siderophile elements.

[67] Further, if a small amount of iron has mixed with the solidifying silicate and stayed in the mantle, that material is heavier than the surrounding mantle. Eventually, this iron‒rich, heavy material sank to the bottom of the solid, convecting mantle and thus contributing to a chemical dense layer at the core‒mantle‒boundary.

Acknowledgments

[68] The authors gratefully acknowledge financial support from the DFG (grant number: HA 1765‒12‒2), and A.M and U.H. thank the Helmholtz Alliance “Planetary evolution and life” for financial support. We also thank the reviewers for their constructive and helpful comments.

Ancillary