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Olivine flotation and crystallization of a global magma ocean
23
Physics of the Earth and Planetary Interiors, 74 (1992) 23—28 Elsevier Science Publishers By., Amsterdam
Olivine flotation and crystallization of a global magma ocean Siegfried Franck Unieersitaet Potsdam, Telegrafenberg, 0-1561 Potsdam, Germany (Received 30 December 1991; accepted 4 May 1992)
ABSTRACT Franck, 5., 1992. Olivine flotation and crystallization of a global magma ocean. Phys. Earth Planet. Inter., 74: 23—28. According to the so-called olivine flotation hypothesis, basic silicate liquids become denser than coexisting ultramafic crystalline peridotites at high pressures through their higher compressibility. Therefore, the olivine, crystallizing from a global magma ocean, settles to a neutrally buoyant zone at about 250 km depth dividing the molten outer part of the Earth into two oceans separated by a dunite septum. The thermal evolution of such a divided global magma ocean is investigated using simple stability estimates and cooling time calculations. It appears that the cooling times and styles of convection (overturn, penetrative, layered) depend strongly on the surface temperature. Overturn convection leads to a compleate destruction of the dunite septum while in the case of kenetrative convection the septum is only partly passed through. Thus, separate chemical evolution of the two mantle regions operates only at high surface temperatures near the solidus of mantle silicates of about 1500 K and cooling times of about io~years.
1. Introduction Although various models of the accretion and core formation of the Earth have been proposed, there is a general consensus that the outer layer of the Earth was much hotter in the early stages of terrestrial evolution. The possible existence of a primordial terrestrial magma ocean has been discussed by many authors (e.g. Ringwood, 1975) based on calculations of the temperature regime during accretion and core formation. The upper mantle stratigraphy resulting from fractional crystallization of a terrestrial magma ocean has been investigated in detail by Hofmeister (1983). Recently developed high-pressure experimental techniques with the help of a piston-cylinder apparatus (Agee and Walker, 1988a), a multipleanvil high-pressure system (Ohtani, 1985) and shock waves (Ridgen et al., 1988; Miller et al., 1991) have made it possible to test the so-called
Correspondence to: S. Franck, Universitaet Potsdam, Telegrafenberg, 0-1561 Potsdam, Germany. 0031-9201/92/$05.00 © 1992
—
olivine flotation hypothesis (Stolper et al., 1981): basic silicate liquids, because of their higher compressibility become denser than coexisting crystalline peridotitic assemblages at high pressures (~8 GPa). Therefore, the olivine, crystallizing from a magma ocean, settles to a neutrally buoyant zone at about 250 km depth dividing the molten outer part of the Earth into two separated oceans. This concept is shown schematically in Fig. 1. In the present paper I discuss time scales and types of convection for the crystallization of such a molten outer part of the Earth.
2. Cooling times and stability estimations The cooling time of a magma ocean depends mainly on the uppermost layers: the early-formed quench zone, the thermal boundary layer, and the later formed flotation crust, which is important only while the last 30 km of magma solidifies. The top of the quench zone is at surface temperature 1 and the bottom at the solidus I~. Fol-
Elsevier Science Publishers B.V. All rights reserved
24
S. FRANCK PROTOCRUST
-
DENSITY R E
E
0
0 0 0
,o N.
Co 0 0 0
lowing Hofmeister (1983), the quench zone and thermal boundary layer are each about 2 m thick so that 4 m can be used as the non-convecting conductive cap of the magma ocean. Figure 2 shows the model magma ocean including the septum from olivine flotation. Based on the sketch in Fig. 2 the time scales for the crystallization of a magma ocean with septum can be calculated. Following Irvine (1970) the equations for the amount of the heat per unit area after time t can be written: Kcr(TL
1
—
Ts)t/Xr
=Xumo(t)LiPi(1 —P1) +Ksep(TL2~ T~’1)t /[Xumo(t) Kcr(TL1 Fig. 1. Schematic cross-section of the magma ocean, divided into an upper and a lower part (after Agee and Walker, 1988a).
T0
S
—
+Xdun(t) +XUbZ(t)l
Ts)t/Xr +KSep(TL2
/ [Xumo(t)
Ximo(t)L2p2(1
P2)
XI Xumo
250
km
750
~
T~1)t
+ Xdun( t) + XUbZ(t)
TLZ
\\\\\\\\\\\\\\\\~
—
(1)
Xlmo
d
Fig. 2. Sketch of temperature distribution in an upper and lower magma ocean (see also Table 1).
(2)
25
CRYSTALLIZATION OF A GLOBAL MAGMA OCEAN
The left-hand side of eqn. (1) is the heat that flows per unit area of roof in time t. It results from the latent heat of crystallization in the upper magma ocean (first term right-hand side) and heat flow through the septum (second term at right-hand side). Equation (2) describes the latent heat of crystallization from the lower magma ocean (right-hand side) that must be transmitted by heat conduction through the septum (second term at left-hand side) and the roof (first term at left-hand side). The nomenclature of the variables in eqns. (1) and (2) is given in Table 1. The dunite part from the dissolved protocrust grows as the freezing velocity of the crust (Hofmeister, 1983): 2Kt/Xr (3) XdUfl(t) 2A Growth of the olivine upper border zone from the lower magma ocean is also according to the model of Hofmeister (1983): =
XUbZ(t)
=2A(Kt)2
(4)
Table 1 also contains the nomenclature for the variables and the working values for the parame-
ters in eqns. (3) and (4). Numerical estimates (see also Franck and Riedel, 1991) show that Xumo(t) grows much faster than Xdufl(t), XUbZ(t) and Ximo(t). The corresponding velocities of growth are: Xumo/t
=
3 X i0~ km year~
x iO~ km year~
XdUfl/t
=
2.3
XUbZ/t
=
0.12 X i0~ km year1 X
(tin iO~year) Ximo/t ~ 10~4Xumo/t
Nomenclature and working values for the parameters and variables Symbol
Definition
Working value
Kcr Ksep
Crust thermal conductivity
L
Septum thermal conductivity Latent heat of crystallization in umo and Imo, respectively Porosity of umo and imo cumulate, respectively Time Liquidus temperature at the roof Liquidus temperature above the septum Liquidus temperature below the septum Liquidus temperature at 750 km depth Surface temperature Dunite layer thickness Cumulate thickness of Imo Roof thickness Olivine upper border zone Cumulate thickness of umo
6X103 cal cm~ 51 K1 3cal cm~ ~ K’ 5X10~ 100 cal g~
P 1, P2 t
TL1 T~ TL2
T~ 1~ XdUfl Xlmo
Xr XUbZ Xumo
0.2 Variable 1700 K 2150 K 2575 K 3650 K 300 K Variable Variable 4 m Variable Variable
Greek letters
a A
K
Pi, P2
(6)
(7) (8)
These results show that a separated magma ocean will crystallize on a time scale many orders of magnitude longer than for the upper magma ocean because the septum consisting of Xumo, XdUfl and XUbZ will act as a growing roof for the lower magma ocean. The next step is to investigate the stability of the dunite septum in the presence of the thermal convection. It has been found from laboratory
TABLE 1
1, L2
(5)
Thermal expansion coefficient Thermal Dimensionless diffusivity parameter Density of crystal accumulation in umo and lmo, respectively
umo, upper magm a ocean; lmo, lower magma ocean; ubz, upper border zone.
i0~ K~’ 2 s —‘ 8.4 X iO~cm 0.414 2.75 g cm3
26
S. FRANCK
and numerical experiments (e.g. Christensen, 1988; Olsen et al., 1990) that the stability of layered convection depends on a global parameter R, the ratio of the maximum chemical buoyancy ~p~/p to the maximum thermal buoyancy aL~T
R —
(9)
paL~T
where p is the average density, a is the thermal expansion coefficient, and zXT is the temperature contrast across the whole convection layer. Overturn occurs for R < 1; strictly layered convection occurs for R>> 1; and so-called penetrative convection in the neighbourhood of R 1. Using equilibrium crystalline olivine and perovskite-depleted liquid compression curves after Agee and Walker (1988b) given by (10) 3.06 g cm3 + 0.03 g cm3p Gpa’ pliqu 2.74 g cm3 + 0.07 g cm3p Gpa~ (11) pOI
=
=
the density anomaly of the dunite septum L~p~/p can be estimated from the condition of neutral buoyancy. At 8 GPa, equivalent to about 250 km depth, p0l ~liqu 3.3 g cm3. Taking into account the chemical buoyancy variation near to the septum from eqns. (10) and (11) and using a pressure gradient of 1 GPa per 31 km, ~ 3.9 x 10~X(t), where X(t) is the time-dependent half-width of the whole septum, measured in kilometres. The temperature contrast T is taken between the bottom of the quench zone (T ~ and the temperature at the crystallization front. It may be calculated from the temperature drop in the boundary layer and the liquidus gradient of about 3 K km~ (Hofmeister, 1983) in a magma ocean of depth d (in kilometres). =
=
=
=
T= ~
—
~o1 +
3Kd(t)
(12)
The expression for z~p~/p and z~Tare both functions of time and cause the global parameter R to become time-dependent also. With the help of the results of eqns. (5)—(8) it is now possible to investigate the global stability parameter R. For this aim a uniform magma ocean at t 0 is imagined. Later the cumulate at the bottom grows as Xumo(t) and a septum grows with X(t) =
=
XdUfl(t) +XUbZ(t). This means the septum forms from dissolved protocrust and floating olivine, as in the standard model of Ohtani (1985) and Agee and Walker (1988a). The depth d in eqn. (12) is written as d(t) d 0 ~Xumo(t). d0 is the depth of a uniform magma ocean at which the average geotherm at first comes in the ‘olivine + liquid’ stability field. The value of d0 is about 750 km (see Agee and Walker, 1988a, figure 3). Figure 3 shows the functions d(t) and R(t) for the surface temperatures 300, 1500 and 1700 K. It is clearly shown that in the case of low surface temperature (Fig. 3) most of the magma ocean crystallization (from d d0 to d 350 km) proceeds in the regime of complete overturn and that the crystallization front reaches the depth of olivine neutral buoyancy (z~250 km) in the range of penetrative convection (R 1.5). So, according to this result there is no complete separation of the two magma oceans at any time and the floating olivine is distributed by the thermal convection currents. The models with surface temperatures of 1500 and 1700 K correspond to the magma ocean scenarios that take into account the blanketing effect of a proto-atmosphere resulting from degassing of volatiles. Matsui and Abe (1986) have shown that there was a rapid rise in the surface temperature after the Earth had grown to about 0.3 of its present radius due to an increase in the total mass of the atmosphere because of initiation of a complete dehydration reaction of the surface layer. In their ‘standard model’ the surface temperature reaches about 1500 K, the solidus of the silicates. A so-called ‘higher water content model’ gives 1 1700 K which corresponds to the liquidus of silicates. A similar discussion on the feedback between greenhouse effect, surface temperature and magma ocean was published by Stevenson (1988). A detailed investigation of the problem of heat flow and crust formation is given by Miller et al. (1991). According to this, a water-rich atmosphere would form at surface temperatures below 1500 K corresponding 2. If to thea net radiative heat flux of about 150 W m surface temperature is between 1500 and 2000 K the heat flow will be between 150 W m2 and 0.9 MW m~2and no crust will exist. =
=
=
27
CRYSTALLIZATION OF A GLOBAL MAGMA OCEAN
From Fig. 3 it can be seen that in the case of 1500 K the first 300 km of the magma ocean crystallize in the regime of penetrative convection and later there is only layered convection, while at I~ 1700 K whole magma ocean crystallization proceeds in the regime of layered convection. This study has shown that the surface temperature is the main parameter controlling the crystallization of a magma ocean. Variation of other parameters like thermal conductivity, density, latent heat or temperature within reasonable boundaries can not produce similar striking effects. A further interesting effect is the turbulence of convection, which causes the suspension of crystals within the magma. Corresponding investigations by Tonks and Melosh (1990) have also led to significant chemical fractionations. =
=
time (1000 a)
3. Conclusions In the case of low surface temperature, the phenomenon of olivine flotation in a crystallizing magma ocean can not be the reason for the chemical stratification of ~he Earth’s mantle that is manifested in a high Mg/Si ratio in the upper mantle compared with an initially chondritic material. The dunite septum, dividing the molten mantle into an upper and a lower magma ocean is only stable in the case of high surface temperatures of about 1500 K. In this case the formation of a majorite-rich layer enriched in incompatible and volatile elements at a depth between about 400 and 700 km is possible (Ohtani, 1988). In this way there may arise a chemically stratified upper mantle and the seismic wave discontinuities at 400 and 650 km depths may result from chemical boundaries.
time (10 000 a)
Acknowledgements 0
5
10
15 20 25 time (1000 000 a)
30
35
Fig. 3. Time dependence of the global parameter R and magma ocean depth d for surface temperatures ~ = 300 K, 1500 K and T,~= 1700 K. The styles of convection are shown within the pictures.
I thank Professor T.J. Ahrens (Pasadena) for .
sending me the paper of Miller et al. (1991) prior to its publication and Dr. R. Daessler (Potsdam) for help in numerical calculations. Furthermore, I
28
acknowledge the helpful comments of two anonymous referees.
References Agee, C.B. and Walker, D., 1988a. Mass balance and phase density constraints on early differentiation of ehondritie mantle. Earth Planet. Sci. Lett., 90: 144—156. Agee, C.B. and Walker, D., 1988b. Static compression and olivine flotation in ultrabasie silicate liquid. I. Geophys. Res., 93: 3437—3449. Christensen, U., 1988. Is subdueted lithosphere trapped at the 670 km discontinuity? Nature, 336: 462—463. Franek, S. and Riedel, M., 1991. The evolution of a global magma ocean with olivine flotation. Earth, Moon, Planets, 54: 59—65. Hofmeister, A.M., 1983. Effect of a hadeao terrestrial magma ocean on crust and mantle evolution. J. Geophys. Res., 88: 4963—4983. Irvine, T.N., 1970. Heat transfer during solidification of layered intrusions. I. Sheets and sills. Can. I. Earth Sei., 7: 1031—1061. Matsui, T. and Abe, Y., 1986. Evolution of an impact-induced atmosphere and magma ocean on the aeereting Earth. Nature, 319: 303—305.
S. FRANCK
Miller, G.H., Stolper, E.M. and Ahrens, T.J., 1991. The equation of state of a molton komatiite, 2. Application to a komatiite petrogenesis and the hadean mantle. I. Gaophys. Res., 96: 11849—11864. Ohtani, E., 1985. The primordial terrestrial magma ocean and its implication for the stratification of the mantle. Phys. Earth Planet. Inter., 38: 70—80. Ohtani, E., 1988. Chemical stratification of the mantle formed by melting in the early stage of the terrestrial evolution. Teetonophysies, 154: 201—210. Olsen, P., Silver, P.G. and Carlson, R.W., 1990. The large-scale Structure of convection in the Earth’s mantle. Nature, 344: 209—215. Ridgen, S.M., Ahrens, T.J. and Stolper, EM., 1988. Shock compression ofmolten silicate: Results for a model basaltic composition. I. Geophys. Res., 93: 367—382. Riogwood, A.E., 1975. Composition and petrology of the Earth’s mantle. McGraw Hill, New York, 618 pp. Stevenson, D.J., 1988. Greenhouses and magma oceans. Nature, 335: 587—588. Stolper, E.M., Walker, D., Hager, B.H. and Hays, J.F., 1981. Melt segregation from partially molten source region: The importance of melt density and source region size. J. Geophys. Res., 86: 6161—6271. Tonks, W.B. and Melosh, H.J., 1990. The physics of crystal settling and Suspension in a turbulent magma ocean. In: HE. Newson and I.H. Lones (Editors), Origin of the Earth. Oxford University, New York, pp. 151—174.
Physics of the Earth and Planetary Interiors, 74 (1992) 23—28 Elsevier Science Publishers By., Amsterdam
Olivine flotation and crystallization of a global magma ocean Siegfried Franck Unieersitaet Potsdam, Telegrafenberg, 0-1561 Potsdam, Germany (Received 30 December 1991; accepted 4 May 1992)
ABSTRACT Franck, 5., 1992. Olivine flotation and crystallization of a global magma ocean. Phys. Earth Planet. Inter., 74: 23—28. According to the so-called olivine flotation hypothesis, basic silicate liquids become denser than coexisting ultramafic crystalline peridotites at high pressures through their higher compressibility. Therefore, the olivine, crystallizing from a global magma ocean, settles to a neutrally buoyant zone at about 250 km depth dividing the molten outer part of the Earth into two oceans separated by a dunite septum. The thermal evolution of such a divided global magma ocean is investigated using simple stability estimates and cooling time calculations. It appears that the cooling times and styles of convection (overturn, penetrative, layered) depend strongly on the surface temperature. Overturn convection leads to a compleate destruction of the dunite septum while in the case of kenetrative convection the septum is only partly passed through. Thus, separate chemical evolution of the two mantle regions operates only at high surface temperatures near the solidus of mantle silicates of about 1500 K and cooling times of about io~years.
1. Introduction Although various models of the accretion and core formation of the Earth have been proposed, there is a general consensus that the outer layer of the Earth was much hotter in the early stages of terrestrial evolution. The possible existence of a primordial terrestrial magma ocean has been discussed by many authors (e.g. Ringwood, 1975) based on calculations of the temperature regime during accretion and core formation. The upper mantle stratigraphy resulting from fractional crystallization of a terrestrial magma ocean has been investigated in detail by Hofmeister (1983). Recently developed high-pressure experimental techniques with the help of a piston-cylinder apparatus (Agee and Walker, 1988a), a multipleanvil high-pressure system (Ohtani, 1985) and shock waves (Ridgen et al., 1988; Miller et al., 1991) have made it possible to test the so-called
Correspondence to: S. Franck, Universitaet Potsdam, Telegrafenberg, 0-1561 Potsdam, Germany. 0031-9201/92/$05.00 © 1992
—
olivine flotation hypothesis (Stolper et al., 1981): basic silicate liquids, because of their higher compressibility become denser than coexisting crystalline peridotitic assemblages at high pressures (~8 GPa). Therefore, the olivine, crystallizing from a magma ocean, settles to a neutrally buoyant zone at about 250 km depth dividing the molten outer part of the Earth into two separated oceans. This concept is shown schematically in Fig. 1. In the present paper I discuss time scales and types of convection for the crystallization of such a molten outer part of the Earth.
2. Cooling times and stability estimations The cooling time of a magma ocean depends mainly on the uppermost layers: the early-formed quench zone, the thermal boundary layer, and the later formed flotation crust, which is important only while the last 30 km of magma solidifies. The top of the quench zone is at surface temperature 1 and the bottom at the solidus I~. Fol-
Elsevier Science Publishers B.V. All rights reserved
24
S. FRANCK PROTOCRUST
-
DENSITY R E
E
0
0 0 0
,o N.
Co 0 0 0
lowing Hofmeister (1983), the quench zone and thermal boundary layer are each about 2 m thick so that 4 m can be used as the non-convecting conductive cap of the magma ocean. Figure 2 shows the model magma ocean including the septum from olivine flotation. Based on the sketch in Fig. 2 the time scales for the crystallization of a magma ocean with septum can be calculated. Following Irvine (1970) the equations for the amount of the heat per unit area after time t can be written: Kcr(TL
1
—
Ts)t/Xr
=Xumo(t)LiPi(1 —P1) +Ksep(TL2~ T~’1)t /[Xumo(t) Kcr(TL1 Fig. 1. Schematic cross-section of the magma ocean, divided into an upper and a lower part (after Agee and Walker, 1988a).
T0
S
—
+Xdun(t) +XUbZ(t)l
Ts)t/Xr +KSep(TL2
/ [Xumo(t)
Ximo(t)L2p2(1
P2)
XI Xumo
250
km
750
~
T~1)t
+ Xdun( t) + XUbZ(t)
TLZ
\\\\\\\\\\\\\\\\~
—
(1)
Xlmo
d
Fig. 2. Sketch of temperature distribution in an upper and lower magma ocean (see also Table 1).
(2)
25
CRYSTALLIZATION OF A GLOBAL MAGMA OCEAN
The left-hand side of eqn. (1) is the heat that flows per unit area of roof in time t. It results from the latent heat of crystallization in the upper magma ocean (first term right-hand side) and heat flow through the septum (second term at right-hand side). Equation (2) describes the latent heat of crystallization from the lower magma ocean (right-hand side) that must be transmitted by heat conduction through the septum (second term at left-hand side) and the roof (first term at left-hand side). The nomenclature of the variables in eqns. (1) and (2) is given in Table 1. The dunite part from the dissolved protocrust grows as the freezing velocity of the crust (Hofmeister, 1983): 2Kt/Xr (3) XdUfl(t) 2A Growth of the olivine upper border zone from the lower magma ocean is also according to the model of Hofmeister (1983): =
XUbZ(t)
=2A(Kt)2
(4)
Table 1 also contains the nomenclature for the variables and the working values for the parame-
ters in eqns. (3) and (4). Numerical estimates (see also Franck and Riedel, 1991) show that Xumo(t) grows much faster than Xdufl(t), XUbZ(t) and Ximo(t). The corresponding velocities of growth are: Xumo/t
=
3 X i0~ km year~
x iO~ km year~
XdUfl/t
=
2.3
XUbZ/t
=
0.12 X i0~ km year1 X
(tin iO~year) Ximo/t ~ 10~4Xumo/t
Nomenclature and working values for the parameters and variables Symbol
Definition
Working value
Kcr Ksep
Crust thermal conductivity
L
Septum thermal conductivity Latent heat of crystallization in umo and Imo, respectively Porosity of umo and imo cumulate, respectively Time Liquidus temperature at the roof Liquidus temperature above the septum Liquidus temperature below the septum Liquidus temperature at 750 km depth Surface temperature Dunite layer thickness Cumulate thickness of Imo Roof thickness Olivine upper border zone Cumulate thickness of umo
6X103 cal cm~ 51 K1 3cal cm~ ~ K’ 5X10~ 100 cal g~
P 1, P2 t
TL1 T~ TL2
T~ 1~ XdUfl Xlmo
Xr XUbZ Xumo
0.2 Variable 1700 K 2150 K 2575 K 3650 K 300 K Variable Variable 4 m Variable Variable
Greek letters
a A
K
Pi, P2
(6)
(7) (8)
These results show that a separated magma ocean will crystallize on a time scale many orders of magnitude longer than for the upper magma ocean because the septum consisting of Xumo, XdUfl and XUbZ will act as a growing roof for the lower magma ocean. The next step is to investigate the stability of the dunite septum in the presence of the thermal convection. It has been found from laboratory
TABLE 1
1, L2
(5)
Thermal expansion coefficient Thermal Dimensionless diffusivity parameter Density of crystal accumulation in umo and lmo, respectively
umo, upper magm a ocean; lmo, lower magma ocean; ubz, upper border zone.
i0~ K~’ 2 s —‘ 8.4 X iO~cm 0.414 2.75 g cm3
26
S. FRANCK
and numerical experiments (e.g. Christensen, 1988; Olsen et al., 1990) that the stability of layered convection depends on a global parameter R, the ratio of the maximum chemical buoyancy ~p~/p to the maximum thermal buoyancy aL~T
R —
(9)
paL~T
where p is the average density, a is the thermal expansion coefficient, and zXT is the temperature contrast across the whole convection layer. Overturn occurs for R < 1; strictly layered convection occurs for R>> 1; and so-called penetrative convection in the neighbourhood of R 1. Using equilibrium crystalline olivine and perovskite-depleted liquid compression curves after Agee and Walker (1988b) given by (10) 3.06 g cm3 + 0.03 g cm3p Gpa’ pliqu 2.74 g cm3 + 0.07 g cm3p Gpa~ (11) pOI
=
=
the density anomaly of the dunite septum L~p~/p can be estimated from the condition of neutral buoyancy. At 8 GPa, equivalent to about 250 km depth, p0l ~liqu 3.3 g cm3. Taking into account the chemical buoyancy variation near to the septum from eqns. (10) and (11) and using a pressure gradient of 1 GPa per 31 km, ~ 3.9 x 10~X(t), where X(t) is the time-dependent half-width of the whole septum, measured in kilometres. The temperature contrast T is taken between the bottom of the quench zone (T ~ and the temperature at the crystallization front. It may be calculated from the temperature drop in the boundary layer and the liquidus gradient of about 3 K km~ (Hofmeister, 1983) in a magma ocean of depth d (in kilometres). =
=
=
=
T= ~
—
~o1 +
3Kd(t)
(12)
The expression for z~p~/p and z~Tare both functions of time and cause the global parameter R to become time-dependent also. With the help of the results of eqns. (5)—(8) it is now possible to investigate the global stability parameter R. For this aim a uniform magma ocean at t 0 is imagined. Later the cumulate at the bottom grows as Xumo(t) and a septum grows with X(t) =
=
XdUfl(t) +XUbZ(t). This means the septum forms from dissolved protocrust and floating olivine, as in the standard model of Ohtani (1985) and Agee and Walker (1988a). The depth d in eqn. (12) is written as d(t) d 0 ~Xumo(t). d0 is the depth of a uniform magma ocean at which the average geotherm at first comes in the ‘olivine + liquid’ stability field. The value of d0 is about 750 km (see Agee and Walker, 1988a, figure 3). Figure 3 shows the functions d(t) and R(t) for the surface temperatures 300, 1500 and 1700 K. It is clearly shown that in the case of low surface temperature (Fig. 3) most of the magma ocean crystallization (from d d0 to d 350 km) proceeds in the regime of complete overturn and that the crystallization front reaches the depth of olivine neutral buoyancy (z~250 km) in the range of penetrative convection (R 1.5). So, according to this result there is no complete separation of the two magma oceans at any time and the floating olivine is distributed by the thermal convection currents. The models with surface temperatures of 1500 and 1700 K correspond to the magma ocean scenarios that take into account the blanketing effect of a proto-atmosphere resulting from degassing of volatiles. Matsui and Abe (1986) have shown that there was a rapid rise in the surface temperature after the Earth had grown to about 0.3 of its present radius due to an increase in the total mass of the atmosphere because of initiation of a complete dehydration reaction of the surface layer. In their ‘standard model’ the surface temperature reaches about 1500 K, the solidus of the silicates. A so-called ‘higher water content model’ gives 1 1700 K which corresponds to the liquidus of silicates. A similar discussion on the feedback between greenhouse effect, surface temperature and magma ocean was published by Stevenson (1988). A detailed investigation of the problem of heat flow and crust formation is given by Miller et al. (1991). According to this, a water-rich atmosphere would form at surface temperatures below 1500 K corresponding 2. If to thea net radiative heat flux of about 150 W m surface temperature is between 1500 and 2000 K the heat flow will be between 150 W m2 and 0.9 MW m~2and no crust will exist. =
=
=
27
CRYSTALLIZATION OF A GLOBAL MAGMA OCEAN
From Fig. 3 it can be seen that in the case of 1500 K the first 300 km of the magma ocean crystallize in the regime of penetrative convection and later there is only layered convection, while at I~ 1700 K whole magma ocean crystallization proceeds in the regime of layered convection. This study has shown that the surface temperature is the main parameter controlling the crystallization of a magma ocean. Variation of other parameters like thermal conductivity, density, latent heat or temperature within reasonable boundaries can not produce similar striking effects. A further interesting effect is the turbulence of convection, which causes the suspension of crystals within the magma. Corresponding investigations by Tonks and Melosh (1990) have also led to significant chemical fractionations. =
=
time (1000 a)
3. Conclusions In the case of low surface temperature, the phenomenon of olivine flotation in a crystallizing magma ocean can not be the reason for the chemical stratification of ~he Earth’s mantle that is manifested in a high Mg/Si ratio in the upper mantle compared with an initially chondritic material. The dunite septum, dividing the molten mantle into an upper and a lower magma ocean is only stable in the case of high surface temperatures of about 1500 K. In this case the formation of a majorite-rich layer enriched in incompatible and volatile elements at a depth between about 400 and 700 km is possible (Ohtani, 1988). In this way there may arise a chemically stratified upper mantle and the seismic wave discontinuities at 400 and 650 km depths may result from chemical boundaries.
time (10 000 a)
Acknowledgements 0
5
10
15 20 25 time (1000 000 a)
30
35
Fig. 3. Time dependence of the global parameter R and magma ocean depth d for surface temperatures ~ = 300 K, 1500 K and T,~= 1700 K. The styles of convection are shown within the pictures.
I thank Professor T.J. Ahrens (Pasadena) for .
sending me the paper of Miller et al. (1991) prior to its publication and Dr. R. Daessler (Potsdam) for help in numerical calculations. Furthermore, I
28
acknowledge the helpful comments of two anonymous referees.
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