Fluid Dynamics of a Terrestrial Magma Ocean - NMSU Geophysics ...

Solomatov: Fluid Dynamics of Terrestrial Magma Ocean 325Temperature, KTemperature, K80007000600050004000300020000 50 100 150Pressure, GPa800070006000500040003000(a)(b)PrvMgOFeO20000 50 100 150Pressure, GPaassumed an ideal mixture of MgSiO 3 and MgO, while Milleret al. (1991b) and Solomatov and Stevenson (1993b) assumeda low melting temperature of perovskite based onthe data by Knittle and Jeanloz (1989). For an ideal eutectic-likesystem, the liquidus T liq approximately follows themelting temperature T prv of pure perovskite MgSiO 3 loweredby the presence of MgO and FeOTprvkB−Tliq≈ nmwTprv(2)∆ Swhere ∆S prv is the entropy change per atom upon meltingof pure perovskite, and n mw is the molar fraction of magnesiowüstite.For n mw ≈ 0.3, ∆S prv ≈ 5 k B [assuming thatthe entropy change is approximately k B per atom (Stishov,1988)] and T prv ≈ 7500 K we obtain T prv – T liq ≈ 450 K.Agee (1990) and Zhang and Herzberg (1994) found thatmagnesiowüstite is the liquidus phase in both chondrites andperidotites just slightly below the upper mantle/lower mantleboundary. A small fraction of Fe in the solid perovskite andthe presence of other components (e.g., Ca-perovskite) inmelt would decrease the liquidus temperature. The estimatesof the entropy change are also not very accurate. As a result,magnesiowüstite can be the first liquidus phase at lowpressures, although at higher pressures it would still besubstituted by perovskite (Fig. 3c).3. FORMATION OF MAGMA OCEANSBY GIANT IMPACTSprv3.1. Mantle Temperature Before ImpactsTemperature, K800070006000500040003000(c)It is quite common to assume that at the latest stages ofEarth’s formation the mantle temperature is close to solidus(Abe and Matsui, 1986; Sasaki and Nakazawa, 1986;Zahnle et al., 1988). However, since the mantle solidus issteeper than the adiabat in the solid mantle, such temperaturedistribution is gravitationally unstable. The effectivedensity contrast between the upper and lower parts of themantle is1−∆ρ ≈ αρH dT sol dTad− ≈ 120 kg m 32 dz dz(3)20000 50 100 150Pressure, GPaFig. 3. (a) Melting curves of perovskite, MgO, and FeO accordingto Boehler (1992) and Zerr and Boehler (1993, 1994). (b) Adiabatsin the convective magma ocean. Liquidus and solidus areshown with a heavy solid line. The beginning of magnesiowüstitecrystallization is shown with a dashed line. (c) An example of calculationswhere magnesiowüstite is the first liquidus phase at thetop of the lower mantle as observed by Agee (1990) and Zhangand Herzberg (1994).where (see Table 1) ρ = 4000 kg m –3 , α ≈ 3 × 10 –5 K –1 , H ≈3 × 10 6 m is the thickness of the mantle, dT sol /dz is the solidusgradient, and dT ad /dz is the adiabatic gradient (Fig. 3).This density contrast causes gravitational instability that issomewhat similar to a classical Rayleigh-Taylor instabilityin a two-layer system. The timescale for the overturn canthen be estimated as (Turcotte and Schubert, 1982)tRT≈ 26 η s∆ρgHwhere η s is the viscosity and g is the gravity.(4)