## 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 *k**m* 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 ^{142}Nd/ ^{144}Nd and ^{182}Hf/ ^{182}W. Indications for a nearly molten state of the Earth are given especially through the measurement of the ^{142}Nd/ ^{144}Nd 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 ^{182}Hf‒ ^{182}W ratio analysis. Hafnium (^{182}Hf) 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 (^{182}W). 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 ^{182}Hf‒ ^{182}W 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]:

[7] Here, *ρ*_{m} stands for the density of the metal, *ρ*_{s} for the density of the silicate, *d*_{m} is the diameter of a metal droplet, *v*_{s}the 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 *d*_{m}=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 *d*_{m}=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}−10^{4}) and Reynolds numbers (10^{−2}−10^{3}) 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 *d*_{m}<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].

[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 *R**a*=10^{26} to *R**a*=10^{29} [*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 10^{25} and 10^{27}[*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 *R**o*=1. The Rossby number of the magma ocean following the estimations of the Rayleigh and Taylor numbers above is in a range of *R**o* = 0.03 up to *R**o* = 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 *R**o*=0.1 and *R**o*=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.