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

332 Origin of the Earth and Moon10 –2 nucleation controlledTcritical for suspensionCrystal size, m10 –310 –4Solidusφ = 60%10 –510 3 10 4 10 5 10 6 10 7ZRheologicalfrontHeat flux, W/m 2Fig. 8. Crystal size controlled by nucleation (solid line) and thecritical crystal size for suspension (dashed line) as functions ofthe heat flux. The crystal size decreases with the heat flux sincethe convective velocities, and thus the cooling rates in the descendingplumes, increase with the heat flux. On the other hand, thecritical crystal size (above which crystals would settle down) increaseswith the heat flux. The early magma ocean is located nearthe intersection of these two curves.SolidusTRheologicalfrontapproximately follow the curve φ = φ cr = const (the dashedline in Fig. 9).This partially crystallized magma ocean has a huge temperaturegradient that follows the curve φ = φ cr and thereforeis gravitationally unstable. The timescale for Rayleigh-Taylor instability can be estimated from equation (4), inwhich the effective density difference across the mantle isnow ∆ρ ≈ ρα∆T superad /2 where ∆T superad is the superadiabatictemperature contrast across the partially crystallized magmaocean (Fig. 3b).An important parameter is the viscosity η s of partiallymolten rocks. Based on recent estimates of the viscosity inthe lower mantle (Karato and Li, 1992; Li et al., 1996; Itaand Cohen, 1998) the viscosity just above the solidus isexpected to be around 10 18 Pa s for the grain size of about10 –3 m. Small amounts of melt reduce the viscosity further.Experimental data suggest that the viscosity is likely to be1–2 orders of magnitude smaller depending on the meltfraction (van der Molen and Paterson, 1979; Cooper andKohlstedt, 1986; Jin et al., 1994; Rutter and Neumann,1995; Hirth and Kohlstedt, 1995a,b; Kohlstedt and Zimmerman,1996). For η s ~ 10 17 Pa s, equation (4) gives t RT ≈20 yr.It is clear that the magma ocean would not “wait” for400 yr until it suddenly “decides” to resolve the instabilityin 20 yr time. The instability starts developing soon afterthe critical crystal fraction is reached at the bottom of themagma ocean and takes the form of solid-state convection,ZZSolidusφ = 60%φ = 60%SolidificationfrontTSolidificationfrontFig. 9. Propagation of the solidification and rheological frontsin the magma ocean.