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

Solomatov: Fluid Dynamics of Terrestrial Magma Ocean 333which is the primary mechanism of crystallization beyondthe critical crystal fraction (Solomatov and Stevenson, 1993b).Therefore a complete solidification front would follow therheological front toward the surface (Fig. 9).10. DIFFERENTIATION IN THESHALLOW MAGMA OCEANAfter the rheological front reaches the surface, the drivingforce for this type of convection motion disappears andit stops. All important changes occur when the potentialtemperature decreases from liquidus to the critical temperaturefor the rheological transition. The crystal size becomeslarger than the critical one for suspension, the heat fluxdrops, convection slows down, and a stable crust forms atthe surface, reducing further the intensity of convection andthe cooling rate of the magma ocean.The lower bound on the depth H shallow of the remainingshallow magma ocean can be obtained from an adiabat startingat the critical temperature for the rheological transition.Figure 1 suggests that such an adiabat intersects solidusaround 10 GPa or H shallow ~300 km. This estimate is ratheruncertain since even a 100 K error in the estimate of therheological transition increases H shallow by a factor of 2.Besides, small superadiabatic gradients can be preserved fora long time and can increase the estimate of H shallow substantially— note that both solidus and liquidus are quiteflat between 10 and 23 GPa and even small superadiabatictemperatures can extend the depth of the shallow magmaocean all the way through the bottom of the upper mantle.The presence of water would also increase H shallow althoughthe melt fraction in the temperature range between “dry”solidus and “wet” solidus is small so that “dry” solidusmight still be a reasonable basis for defining the depth ofthe shallow magma ocean.Moreover, when the potential temperature drops belowliquidus, the completely molten layer disappears and thesequence “nucleation-growth-dissolution” changes to just“growth.” In this case, the crystals had enough time to reachthe critical size for the cessation of suspension (equations(23) and (34)). Although liquidus is only 300 K abovethe critical temperature for the rheological transition, thebottom of the shallow magma ocean can extend well intothe lower mantle, probably around 40 GPa or so (Fig. 3; seealso Miller et al., 1991b; Solomatov and Stevenson, 1993b;Abe, 1997).Differentiation in the shallow magma ocean takeswhereutpercdiffHshallow~ (35)upercg∆ρdφl=150η1− φl2 2( )l(36)is the percolation velocity according to the Ergun-Orningformula (Soo, 1967; Dullien, 1979) and φ l is the melt fraction.For example, it takestdiff~10822φld ηlyr− 3(37)002 . 10 m 100 Pa sto reduce the average melt fraction to about 2%. Note thathere we used the value of the viscosity of a low-pressure,high-silica, polymerized magma just above the solidus(Kushiro, 1980, 1986). Melt/crystal density inversions (Agee,1998) might result in the formation of more than one moltenlayer. In particular, a gravitationally stable molten layercould be formed at the bottom of the upper mantle in additionto the shallow magma ocean beneath the crust. Coolingand further crystallization of these magmatic layerswould take an even longer time depending on the globalthermal evolution of the mantle. It can be as fast as 10 7 –10 8 yr (Davies, 1990). However, if surface recycling wasinefficient, convection beneath the solid crust would beslower (e.g., Solomatov and Moresi, 1996), and crystallizationtime would be significantly larger.11. DISCUSSION AND CONCLUSIONAlthough the uncertainties in fluid dynamics of magmaoceans are substantial, the following scenario of evolutionof a terrestrial magma ocean can be suggested.A giant impact melts a significant part of one hemisphereof the Earth. Isostatic adjustment quickly redistributes themass to create a more stable spherically symmetric configuration.In the beginning of crystallization of the magmaocean, the temperatures in the deepest parts of the Earthwere probably near the solidus and depended on the poorlyconstrained preimpact thermal state of the mantle. Thisimplies that some portion of the lower mantle could retainsubstantial amounts of primordial volatiles.Crystallization starts from the bottom and in less than1000 yr propagates through the lower mantle. During thisperiod of time, convection can prevent fractionation of crystalsup to the critical crystal fraction around 60% (rheologicaltransition from a low-viscosity suspension to a highviscositypartially molten solid), provided the crystals aresmaller than 10 –3 m. Kinetics of nucleation and crystalgrowth suggest that this is a possibility. Solid-state convectionhelps further cooling and crystallization of the partiallymolten mantle down to the solidus and below. The productof this period of crystallization is essentially undifferentiatedmantle with the remaining shallow partially moltenlayer. This precludes a severe fractionation of minor andtrace elements (Kato et al., 1988a,b; Ringwood, 1990;McFarlane and Drake, 1990; McFarlane et al., 1994). Smallamounts of crystals might settle down and contribute to theformation of an Fe-rich D" layer.The shallow magma ocean is the only part of the mantlethat undergoes any substantial differentiation. This can bereconciled with the observed ratios of major and minor el-