Suspension in convective layers and style of differentiation of a terrestrial magma ocean

Abstract

Recent physical theories for the formation of the Earth suggest that about 4.5 b.y. ago the mantle of the Earth was partially or completely molten. Fractional crystallization of this hypothetical magma ocean would result in a strong chemical stratification of the Earth's mantle. Such a scenario is controversial from the geochemical point of view. However, it has been noted that the simple scenario of fractional crystallization could be avoidable in a convective magma ocean if crystals remain suspended. In this paper, the problem of suspension is developed with the help of an energetic approach: convection must do some work against gravitational settling. We distinguish three regimes of convective suspensions. Absolute or complete sedimentation occurs when the energy dissipation due to the settling exceeds the heat loss from the convective layer. This is possible only in large-scale systems like magma oceans and implies that cooling can proceed only together with sedimentation, crystallization, and a decrease in the liquidus temperature at a constant pressure. A regime of partial differentiation occurs when the energy dissipation due to the settling is less than the total heat loss but larger than the power which can be spent by convection on the crystal reentrainment process. The differentiation is not complete, and a competition between the rate of cooling, the rate of sedimentation, and the rate of turbulent diffusion determines the degree of differentiation. The third regime is an absolute suspension which could be sustained for an indefinitely long time. In this case, sedimentation starts only when the crystal fraction reaches the maximum packing value: when the viscosity of the magma rapidly increases. The power which can be spent by convection on reentrainment is equal to εαgd/cp of the total energy supply to the convective layer, where ε < 1 is an efficiency factor. This factor is probably about 0.01 and has been estimated from one experiment on convective suspensions and with the help of an analogy with remixing in chemically layered convective layers; we find that both cases are controlled by the energetics of convection. The crystal radius is one of the most crucial and uncertain parameters. If it exceeds about 10^(−2) − 1 cm during crystallization of deep layers (> 15 GPa) or 10^(−3) – 10^(−1) cm during crystallization of shallow layers, the first regime ("fractional crystallization") is unavoidable. The estimates depend on various poorly constrained parameters and processes, such as heat flux, viscosity, thermodynamical disequilibrium and highly variable viscosity convection. For absolute suspension the crystal size must be at least ε^(½) times less, or 10^(−3) – 10^(−1) cm and 10^(−4) – 10^(−2) cm, respectively, if ε ∼ 0.01. The partial differentiation occurs in a narrow (one decade) range between these two regimes. The radius of about 1 cm must be considered as an absolute upper bound above which fractional differentiation is guaranteed. These estimates for the critical crystal size are orders of magnitude lower than suggested previously, and thus the problem of crystal sizes becomes a central one for magma oceans. A necessary condition for reentrainment is the existence of local mechanisms. The absence of such mechanisms to reentrain the particles from the bottom would mean that an absolute suspension is impossible even if the energetics allows it. Turbulence is considered as a possible important factor. A simple model of convection predicts a strong turbulence, provided the viscosity is less than 10^9 – 10^(10) P. Rotation reduces this critical viscosity to 10^5 – 10^8 P but this is still sufficiently large and is reached only near the maximum packing crystal fraction. Power law or Bingham rheology of partial melts can exclude any turbulence already at 20 – 30% of crystal fraction. We also show that the energetic criterion for the absolute suspension with ε ∼ 1 coincides with the condition that the particle concentration gradient suppresses the turbulence.

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