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The composition and geotherm of the lower mantle: constraints from the elasticity of silicate perovskite
Physics of the Earth and Planetary Interiors 118 Ž2000. 103–109 www.elsevier.comrlocaterpepi
The composition and geotherm of the lower mantle: constraints from the elasticity of silicate perovskite Cesar R.S. da Silva a
a,b,)
, R.M. Wentzcovitch
a,b
, A. Patel c , G.D. Price c , S.I. Karato
d
Department of Chemical Engineering and Materials Science, UniÕersity of Minnesota, Minneapolis, MN 55455, USA b Minnesota Supercomputer Institute, UniÕersity of Minnesota, Minneapolis, MN 55455, USA c Department of Geological Sciences, UniÕersity College London, London, UK d Department of Geology and Geophysics, UniÕersity of Minnesota, Minneapolis, MN 55455 USA Received 3 June 1999; received in revised form 14 September 1999; accepted 14 September 1999
Abstract A newly developed parameterization of the third-order isentropic finite strain equation of states ŽEOS. is used in conjunction with experimental data and theoretical results on MgSiO 3 perovskite. New geotherms for the lower mantle are derived by comparison with preliminary reference earth model ŽPREM.. The geotherms are adiabatic up to 1500 km depth and super-adiabatic thereafter. A description of the critical steps in obtaining the new parameterization is also given. q 2000 Elsevier Science B.V. All rights reserved. Keywords: Geotherm; Lower mantle; Silicate perovskite
The seismologically derived preliminary reference earth model ŽPREM. ŽDziewonski and Anderson, 1981. and other whole Earth models ŽKennett and Engdahl, 1991; Morelli and Dziewonski, 1993. have provided precise limits on the density, pressure and elastic properties of the earth’s interior as a function of depth, z. Despite this, the earth’s composition and thermal structure are still relatively unconstrained. In order to obtain a better description of these two vital aspects of the earth’s interior, seismic data must be compared with mineralogical data and thermodynamic models ŽWang, 1972; Brown and Shankland,
)
Corresponding author. Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455, USA
1981; Anderson, 1982; Ito and Katsura, 1989; Stacey, 1992; Anderson, 1998., however, because of uncertainties and limitations in previously used models, there are considerable discrepancies between the various proposed descriptions of the earth’s thermal and compositional structure ŽWang, 1972; Brown and Shankland, 1981.. Thus, for example, there are differences of up to 2000 K for the proposed temperatures of the core–mantle boundary ŽCMB. ŽAnderson, 1982; Boehler, 1993., and the chemical composition of the lower mantle ŽLM. is claimed to be adequately described by both silica rich Ž‘‘perovskite’’. and or silica depleted Ž‘‘pyrolite’’. mineralogical models ŽZhao and Anderson, 1994.. In this study, we calculate geotherms for these two extreme compositional models, based on the high pressure bulk modulus of MgSiO 3 perovskite calculated using
0031-9201r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 1 - 9 2 0 1 Ž 9 9 . 0 0 1 3 3 - 8
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C.R.S. da SilÕa et al.r Physics of the Earth and Planetary Interiors 118 (2000) 103–109
first principles techniques, experimental data, and a newly developed parameterization of the third-order finite strain isentropic equation of state ŽEOS. suitable to extrapolate the volume Ž V . and the adiabatic bulk modulus Ž K S . in a consistent manner to lower mantle pressure, P, and temperature, T, conditions. Our results suggest that irrespective of the compositional model, the temperature profile in the lower mantle is characterized by nearly adiabatic gradient in the shallow portions Žf 700 to f 1500 km. followed by significantly super-adiabatic gradient in the deeper portions leading to the CMB temperature of about 4000 " 500 K for ‘‘pyrolite’’ and 4600 " 500 K for ‘‘perovskite’’. Despite the large uncertainties, the lower bound limit of these geotherms still imply in super-adiabatic behavior for depths greater than 1500 km. We also find from materials considerations alone that if the lower mantle were silica rich then a thermal boundary layer Žwith DT f 500 K. at the transition zone would be required, while no such thermal boundary would be predicted for a ‘‘pyrolitic’’ lower mantle. In order to constrain the thermal and compositional state of the mantle it is essential to have high quality elasticity data for the component mantle phases, as both a function of P and T. To address the first of these, we have calculated the bulk modulus Ž K S . of MgSiO 3 perovskite by using a firstprinciples method based on density functional theory and the plane wave pseudopotential approach ŽHohenberg and Kohn, 1964; Kohn and Sham, 1965; Singh, 1994.. Equilibrium structures at arbitrary pressures were first obtained using a structural optimization technique ŽWentzcovitch et al., 1993; Wentzcovitch, 1995. based on first-principles damped molecular dynamics ŽMD. with variable cell shape ŽVCS.. This is, in many respects, identical to our previous calculation of the elastic constants of MgSiO 3 perovskite at P s 0 GPa ŽWentzcovitch et al., 1995., where we used Troullier–Martins pseudopotentials ŽTroullier and Martins, 1991., the local density approximation ŽLDA. for the exchange correlation functional ŽCeperley and Alder, 1981; Perdew and Zunger, 1981., and a plane-wave cut-off of 70 Ry. However, in the present study, we used four k-points in the irreducible Brillouin Zone ŽIBZ. of the unstrained Pbnm structure and up to eight k-points in the strained configurations.
The P = V curve obtained from our first principles calculations enable us to obtain zero Kelvin K S ŽT s 0, P s 0. s 262 " 2 GPa and K X Ž0,0., and its pressure derivative K SX ŽT s 0, P s 0. s 3.86 " 0.06. These values are very close to the accepted experimental value at ambient temperature of K Ž300,0. s 261 GPa and K X Ž300,0. s 4. However, in order to constrain the thermal behavior of the LM, it is necessary to establish the K S Ž P,T . surface for the major lower mantle forming phases, which can then be compared to the known K PREM Ž z . to define T Ž z .. We have calculated K S Ž P,T . by using our new parameterization of the third-order finite strain isentropic EOS ŽZhao and Anderson, 1994., which assumes that K S is simply a function of volume, V, as suggested by Birch’s Law ŽBirch, 1961.. The development of the parameterization is summarized in Appendix A. Figs. 1 and 2 display the calculated V Ž P,T . and K S Ž P,T ., respectively, for MgSiO 3 perovskite, which we have obtained by fitting this EOS to an experimental database ŽWang et al., 1994; Utsumi et al., 1995; Funamori et al., 1996. restricted to the range of 0–30 GPa and 293–2000 K. By comparing the calculated K S Ž P,T . surface with K PREM , we inferred the temperature of the lower mantle as a function of pressure or equivalently depth. Note that it seems unreasonable to calculate a geotherm by comparing the shear modulus surface, G S Ž P,T ., as the effects of anelasticity would tend to significantly overestimate the temperature ŽKarato, 1993.. Similarly, the uncertainty in Fe content prevents us from obtaining precise values of T Ž z . by directly comparing r Ž P,T . with r PREM . On the other hand, the effect of Fe on the compressibil-
Fig. 1. V Ž P,T . obtained by the procedure outlined in Appendix A. Pressure in GPa.
C.R.S. da SilÕa et al.r Physics of the Earth and Planetary Interiors 118 (2000) 103–109
Fig. 2. K S Ž P,T . obtained by the procedure outlined in Appendix A. Plotted lines correspond to temperatures ranging from 300 to 5100 K separated by intervals of 800 K. Black dots correspond to PREM values.
ity of perovskite is insignificant ŽMao et al., 1991. and the effect of inelasticity is small. Since thermoelastic data on magnesiowustite is quite well con¨ strained, we simply used available experimental data ŽRichet et al., 1989; Sumino and Anderson. to determine the effective K S for a perovskiteq magnesiowustite assemblage characteristic of a ‘‘pyrolitic’’ ¨ LM. In Fig. 3, we show our calculated geotherms for a ‘‘perovskite’’ and a ‘‘pyrolite’’ LM and compare with others ŽBrown and Shankland, 1981; Anderson, 1982; Ito and Katsura, 1989; Stacey, 1992.. The error bars in ours reflect uncertainties related to the calculated K S Ž P,T . surface Žsee Appendix A.. For the pyrolite model, the estimated temperature at 670 km of 1700 " 110 K is in excellent agreement with that recently inferred for the base of the transition zone from Fe–Mg partitioning experiments ŽKatsura and Ito, 1996.. In contrast, we calculate from purely materials considerations that for a pure ŽMg,Fe.-perovskite lower mantle T670 f 2060 " 110 K, suggesting that if the mantle were to have this silica rich composition, a thermal boundary layer would have to exist at z s 670 km, with DT f 350 K, this conclusion is in keeping with fluid dynamical models of layered mantle convection ŽPeltier, 1989; Peltier et al., 1997.. Comparing our geotherms with the nearly adiabatic geotherm of Brown and Shankland Ž1981., we see that both compositional models predict nearly
105
adiabatic gradient in the shallow portions of the LM but the temperature gradient significantly exceeds the adiabatic gradient for depth over f 1500 km and leads to the CMB temperature of f 4000 K. The temperatures at CMB estimated in our models exceed the estimated melting temperature of core materials ŽJeanloz, 1989; Boehler, 1993. at CMB conditions without invoking additional thermal boundary layerŽs.. It is also close to the solidus temperature of lower mantle materials measured by recent diamond anvil experiments Žf 4000 K. ŽZerr et al., 1998.. The steeper temperature gradients in the deep LM inferred from our analysis suggest that convective heat transport is not very efficient in this region. This could be caused by high viscosities ŽNakada and Lambeck, 1989; Mitrovica and Forte, 1997.. Mitrovica and Forte Ž1997. indicated that the viscosity changes only by a factor of hundred or less in the lower mantle. This observation is difficult to reconcile with an adiabatic gradient because the melting temperature of lower mantle minerals increases significantly with depth ŽBoehler, 1993.. With the superadiabatic gradients found here, it is possible to reproduce the nearly constant viscosity inferred from geodynamic modelling ŽKarato, 1998..
Fig. 3. Lower mantle geotherms. The effects of mantle mineralogical composition on the bulk modulus were modeled using an averaging technique for a two-component aggregate ŽWatt et al., 1976.. Pyrolitic composition was assumed to be Ž84 wt.% Mg 0.9 Fe 0.1 SiO 3 q16 wt.% Mg 0.88 Fe 0.12 O.. Thermoelastic parameters for magnesiowustite were assumed to be ŽSpetzler, 1970; ¨ Richet et al., 1989; Mao et al., 1991.: K S Ž0,0. s160y7.5x Fe GPa, ŽEK S rEP .< Ts 0 s 4.2 and ŽEK S rET .< Ts0 sy00016 GPa Ky1 , and E 2 K S rEPET s 0.0001 Ky1 . The latter was determined for periclase.
106
C.R.S. da SilÕa et al.r Physics of the Earth and Planetary Interiors 118 (2000) 103–109
We also note that the lower mantle has been considered to have nearly adiabatic temperature gradient because Bullen’s inhomogeneity parameter is nearly one ŽBirch, 1952.. The superadiabatic gradients obtained in our calculations ŽdTrd z f 0.7–1.0 Krkm. correspond to the Bullen’s parameters of around 0.90 " 0.02 which is within the acceptable range of uncertainty of seismological models. In addition, we also note that the density–depth relation is not well constrained by PREM. Kennett Ž1998. recently analyzed the density–depth relation in the LM based primarily on normal mode data and found density gradients significantly smaller than PREM values for depths greater than f 1500 km, implying that temperature gradients are likely to be superadiabatic. This steeper temperature gradient beyond 1500 km depth seems to be a very robust feature that stands out in all temperature profiles we derived by comparing PREM data with our derived K S Ž P,T . using somewhat different parameters. In our approach, the superadibatic gradient seems to be a unavoidable conclusion if the lower mantle is assumed to be homogeneous. This assumption has lately been questioned by Kellogg et al. Ž1999. and Ž1999.. These authors van der Hilst and Karason ´ propose the existence of a boundary between compositionally distinct regions at a depth around 1600 km. Beyond this depth, a lower mantle that is about 4% denser and hotter could be dynamically stable. We simply note that the direct comparison between PREM and our derived K S Ž P,T . reveals a region with distinct properties beyond f 1500 km. Finally, seismic tomography Žvan der Hilst et al., 1997. suggests that the 670 km discontinuity does not act as a global barrier to mantle convection at the present time and in the recent past. Therefore, some mixing must occur between UM and LM and a model of LM such as pure perovskite composition seems inappropriate. Our geothermal profiles are, however, compatible with both pyrolitic and some chemically layered model, the latter being associated with a minor thermal boundary layer near the 670 km discontinuity. In summary, this study has determined accurate first principles bulk modulus which were used in conjunction with experimental data on V Ž P,T . of MgSiO 3 perovskite phase at moderately high tem-
peratures and pressures to fit a K S Ž P,T . surface based on a new parameterization of the isentropic EOS. By comparing the K S Ž P,T . surface with PREM data we have inferred a geotherm for a ‘‘perovskite’’ and a ‘‘pyrolite’’ LM. The basic assumption involved in our procedure is that K S is essentially a function of density only ŽBirch’s law of corresponding states., rather than a specific function of pressure and temperature separately. A possible contribution from intrinsic temperature dependence of K S ŽAnderson, 1987. will reduce the inferred temperatures to some extent. In contrast to most of the previous models ŽBrown and Shankland, 1981; Anderson, 1982; Ito and Katsura, 1989; Stacey, 1992., ours do not require an additional thermal boundary layer at the CMB to satisfy the condition that the temperature at CMB exceeds the melting temperature of core materials ŽBoehler, 1993.. A thermal boundary layer at the bottom of the mantle, if any, seems very broad as has been shown in some numerical modeling assuming high viscosity in the LM Žvan den Berg and Yuen, 1998.. Our geotherm for pyrolitic composition is the first materials based geotherm that independently satisfies current estimates of the temperatures of the shallow and deep portions of the lower mantle and is also consistent with current estimates of its viscosity profile.
Acknowledgements We wish to acknowledge computational resources from the Minnesota Supercomputer Institute, the National Science Foundation Žaward No. EAR 9628042 to RMW and xxxx to S.-i Karato., the Brazilian National Research Council ŽCNPq award to CRSS., and the British NERC ŽAP and GDP..
Appendix A Here we outline the parameterization of the thirdorder finite strain isentropic EOS we have used to extrapolate volume and bulk modulus obtained at low and moderately high temperatures to the temperatures and pressures prevailing in the lower mantle. Our starting point is a well accepted EOS for the
C.R.S. da SilÕa et al.r Physics of the Earth and Planetary Interiors 118 (2000) 103–109
temperature dependent adiabatic bulk modulus K S Ž P,T . ŽZhao and Anderson, 1994.: K S Ž P ,T . s K S Ž 0,Tf . Ž 1 q 2 f . = 1q Ž
3 K SX
5
5 2
K S Ž P ,0 . s K S Ž 0,0 . Ž 1 q 2 f 0 .
In this equation f s 1r2wŽ V Ž0,Tf .rV Ž P,T y 1x is the Eulerian finite strain, K SX Ž0,Tf . s ŽEK SrEP . SŽ0,Tf . is the adiabatic pressure derivative of the adiabatic bulk modulus, and Tf is the temperature at the intersection of the adiabat passing through Ž P,T . with the zero pressure axis. Tf is related to T by:
Ž 0,Tf .
g Ž P ,T . s g Ž 0,0 .
ž
V Ž P ,T . V Ž 0,0 .
q
/
K S Ž P ,T . ' K S Ž r Ž P ,T . .
ž
2
= Ž 1 q Ž 3 K SX Ž 0,0 . y 5 . f f
Ž 3.
2
K SX Ž 0,Tf . y 6 f
/
Ž 4.
which we have fit to the database. K S Ž0,Tf . and K SX Ž0,Tf . are then used in Eq. Ž1. to obtain K S Ž P,T ..
1 The approximation that G r K S in Eq. Ž2. is independent of pressure is reasonable. Some of the dependence can be absorbed by the parameters G 0 and q.
Ž 6.
where f f s 1r2wŽ V Ž0,0.rV Ž0,Tf .. 2r3 y 1x. Generally, V Ž0,Tf . is expressed in term of the thermal expansion coefficient a Ž0,T .:
H0Ta Ž0,T .dT
5
P s 3 K S Ž 0,Tf . f Ž 1 q 2 f .
2
V Ž 0,T . s V Ž 0,0 . e
This assumption leads to unique analytical forms for K S Ž0,Tf . and K SX Ž0,Tf ., which are then used in the adiabatic third order finite strain EOS for P ŽBoehler, 1993.:
3
5
.
Additionally, we require Eq. Ž1. to be consistent with Birch’s law ŽBirch, 1961.. This empirical law derived from velocity–density systematic in minerals, recognizes that the adiabatic bulk modulus of similar solids with the same molecular weight depends essentially on the density. Mathematically we have:
= 1q
where f 0 s 1r2wŽ V Ž0,0.rV Ž P,0 y 1x. Furthermore, r Ž0,Tf . s r Ž P,0. must hold for some negative P. Therefore, consistency with Birch’s law requirement allows us simply to substitute r Ž0,Tf . for r Ž P,0. in Eq. Ž5. and obtain:
Ž 2.
where g , the thermodynamical Gruneisen parameter, ¨ is approximated by ŽPoirier, 1991. 1
Ž 5.
.. 2r3
K S Ž 0,Tf . s K S Ž 0,0 . Ž 1 q 2 f f .
gP KS
= 1 q Ž 3 K SX Ž 0,0 . y 5 . f 0
Ž 1. .. 2r3
T s Tf e
We now derive expressions for K S Ž0,Tf . and K SX Ž0,Tf .. Let us recall that:
2
Ž 0,Tf . y 5 . f
107
Ž 7.
where a Ž0,T . is parameterized as:
a Ž 0,T . s a q bT y cT 2
Ž 8.
Similarly, we are able to map K S Ž P,T . into the Ž K S P,0. line: 5 U
K S Ž P ,T . s K S Ž 0,0 . Ž 1 q 2 f .
2
= 1 q Ž 3 K SX Ž 0,0 . y 5 . f U
Ž 9.
where f U s 1r2wŽ V Ž0,0.rV Ž P,T .. 2r3 y 1x. Internal consistency requires K S Ž P,T .’s calculated from Eqs. Ž1. and Ž9. to be mathematically identical. This leads to: K SX Ž 0,Tf . s
2r3
5 q
2 3
Tf
1 q ye H0
a Ž0,T . dT
3
Ž 10 .
where y s 2rŽ3 K X Ž0,0. y 5. y 1. Eq. Ž4. and the auxiliaries Ž2., Ž6., and Ž10. form a coupled set of equations parameterized in terms of V Ž0,0., K Ž0,0., K X Ž0,0., a, b, and c. Eq. Ž4., which relates P, T, and V, is then fit to the experimental database collected by Funamori et al. Ž1996.. The fitting is accomplished by using a standard multidimensional nonlinear regression algorithm ŽBeving-
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ton, 1969.. We chose to constrain K Ž0,0. to 262 " 2 GPa and K X Ž0,0. to 3.86 " 0.06. These values were obtained by fitting Eq. Ž5. to the calculated V vs. P relation. We also constrained a Ž300. to a safe interval Ž2.0 " 0.5. = 10y5 as indicated by experiments ŽKnittle et al., 1986; Ross and Hazen, 1989; Mao et al., 1991; Funamori and Yagi, 1993; Wang et al., 1994; Chopelas, 1996; Jackson and Rigden, 1996; Gillet et al., 1996; Jackson, 1998.. We have found that by varying q in the range 0.6–10 does not alter significantly the results, therefore q was set to unity. The value of g 0 was also constrained to a safe interval Ž1.3 " 0.3.. The uncertainties in the values of g 0 and a Ž300. are the very predominant contributions to the uncertainties in K S Ž P,T .. For a Ž300. and g 0 in the middle of the interval, we obtained ˚ 3, b s 1.37 " 0.05 = 10y8 Ky1, V0 s 162.0 " 0.2 A and c s 0.43 " 0.02 K 2 , which produces a s 2.06 " 0.4 = 10 -5 and a normalized x 2 s 0.88.
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The composition and geotherm of the lower mantle: constraints from the elasticity of silicate perovskite Cesar R.S. da Silva a
a,b,)
, R.M. Wentzcovitch
a,b
, A. Patel c , G.D. Price c , S.I. Karato
d
Department of Chemical Engineering and Materials Science, UniÕersity of Minnesota, Minneapolis, MN 55455, USA b Minnesota Supercomputer Institute, UniÕersity of Minnesota, Minneapolis, MN 55455, USA c Department of Geological Sciences, UniÕersity College London, London, UK d Department of Geology and Geophysics, UniÕersity of Minnesota, Minneapolis, MN 55455 USA Received 3 June 1999; received in revised form 14 September 1999; accepted 14 September 1999
Abstract A newly developed parameterization of the third-order isentropic finite strain equation of states ŽEOS. is used in conjunction with experimental data and theoretical results on MgSiO 3 perovskite. New geotherms for the lower mantle are derived by comparison with preliminary reference earth model ŽPREM.. The geotherms are adiabatic up to 1500 km depth and super-adiabatic thereafter. A description of the critical steps in obtaining the new parameterization is also given. q 2000 Elsevier Science B.V. All rights reserved. Keywords: Geotherm; Lower mantle; Silicate perovskite
The seismologically derived preliminary reference earth model ŽPREM. ŽDziewonski and Anderson, 1981. and other whole Earth models ŽKennett and Engdahl, 1991; Morelli and Dziewonski, 1993. have provided precise limits on the density, pressure and elastic properties of the earth’s interior as a function of depth, z. Despite this, the earth’s composition and thermal structure are still relatively unconstrained. In order to obtain a better description of these two vital aspects of the earth’s interior, seismic data must be compared with mineralogical data and thermodynamic models ŽWang, 1972; Brown and Shankland,
)
Corresponding author. Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455, USA
1981; Anderson, 1982; Ito and Katsura, 1989; Stacey, 1992; Anderson, 1998., however, because of uncertainties and limitations in previously used models, there are considerable discrepancies between the various proposed descriptions of the earth’s thermal and compositional structure ŽWang, 1972; Brown and Shankland, 1981.. Thus, for example, there are differences of up to 2000 K for the proposed temperatures of the core–mantle boundary ŽCMB. ŽAnderson, 1982; Boehler, 1993., and the chemical composition of the lower mantle ŽLM. is claimed to be adequately described by both silica rich Ž‘‘perovskite’’. and or silica depleted Ž‘‘pyrolite’’. mineralogical models ŽZhao and Anderson, 1994.. In this study, we calculate geotherms for these two extreme compositional models, based on the high pressure bulk modulus of MgSiO 3 perovskite calculated using
0031-9201r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 1 - 9 2 0 1 Ž 9 9 . 0 0 1 3 3 - 8
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C.R.S. da SilÕa et al.r Physics of the Earth and Planetary Interiors 118 (2000) 103–109
first principles techniques, experimental data, and a newly developed parameterization of the third-order finite strain isentropic equation of state ŽEOS. suitable to extrapolate the volume Ž V . and the adiabatic bulk modulus Ž K S . in a consistent manner to lower mantle pressure, P, and temperature, T, conditions. Our results suggest that irrespective of the compositional model, the temperature profile in the lower mantle is characterized by nearly adiabatic gradient in the shallow portions Žf 700 to f 1500 km. followed by significantly super-adiabatic gradient in the deeper portions leading to the CMB temperature of about 4000 " 500 K for ‘‘pyrolite’’ and 4600 " 500 K for ‘‘perovskite’’. Despite the large uncertainties, the lower bound limit of these geotherms still imply in super-adiabatic behavior for depths greater than 1500 km. We also find from materials considerations alone that if the lower mantle were silica rich then a thermal boundary layer Žwith DT f 500 K. at the transition zone would be required, while no such thermal boundary would be predicted for a ‘‘pyrolitic’’ lower mantle. In order to constrain the thermal and compositional state of the mantle it is essential to have high quality elasticity data for the component mantle phases, as both a function of P and T. To address the first of these, we have calculated the bulk modulus Ž K S . of MgSiO 3 perovskite by using a firstprinciples method based on density functional theory and the plane wave pseudopotential approach ŽHohenberg and Kohn, 1964; Kohn and Sham, 1965; Singh, 1994.. Equilibrium structures at arbitrary pressures were first obtained using a structural optimization technique ŽWentzcovitch et al., 1993; Wentzcovitch, 1995. based on first-principles damped molecular dynamics ŽMD. with variable cell shape ŽVCS.. This is, in many respects, identical to our previous calculation of the elastic constants of MgSiO 3 perovskite at P s 0 GPa ŽWentzcovitch et al., 1995., where we used Troullier–Martins pseudopotentials ŽTroullier and Martins, 1991., the local density approximation ŽLDA. for the exchange correlation functional ŽCeperley and Alder, 1981; Perdew and Zunger, 1981., and a plane-wave cut-off of 70 Ry. However, in the present study, we used four k-points in the irreducible Brillouin Zone ŽIBZ. of the unstrained Pbnm structure and up to eight k-points in the strained configurations.
The P = V curve obtained from our first principles calculations enable us to obtain zero Kelvin K S ŽT s 0, P s 0. s 262 " 2 GPa and K X Ž0,0., and its pressure derivative K SX ŽT s 0, P s 0. s 3.86 " 0.06. These values are very close to the accepted experimental value at ambient temperature of K Ž300,0. s 261 GPa and K X Ž300,0. s 4. However, in order to constrain the thermal behavior of the LM, it is necessary to establish the K S Ž P,T . surface for the major lower mantle forming phases, which can then be compared to the known K PREM Ž z . to define T Ž z .. We have calculated K S Ž P,T . by using our new parameterization of the third-order finite strain isentropic EOS ŽZhao and Anderson, 1994., which assumes that K S is simply a function of volume, V, as suggested by Birch’s Law ŽBirch, 1961.. The development of the parameterization is summarized in Appendix A. Figs. 1 and 2 display the calculated V Ž P,T . and K S Ž P,T ., respectively, for MgSiO 3 perovskite, which we have obtained by fitting this EOS to an experimental database ŽWang et al., 1994; Utsumi et al., 1995; Funamori et al., 1996. restricted to the range of 0–30 GPa and 293–2000 K. By comparing the calculated K S Ž P,T . surface with K PREM , we inferred the temperature of the lower mantle as a function of pressure or equivalently depth. Note that it seems unreasonable to calculate a geotherm by comparing the shear modulus surface, G S Ž P,T ., as the effects of anelasticity would tend to significantly overestimate the temperature ŽKarato, 1993.. Similarly, the uncertainty in Fe content prevents us from obtaining precise values of T Ž z . by directly comparing r Ž P,T . with r PREM . On the other hand, the effect of Fe on the compressibil-
Fig. 1. V Ž P,T . obtained by the procedure outlined in Appendix A. Pressure in GPa.
C.R.S. da SilÕa et al.r Physics of the Earth and Planetary Interiors 118 (2000) 103–109
Fig. 2. K S Ž P,T . obtained by the procedure outlined in Appendix A. Plotted lines correspond to temperatures ranging from 300 to 5100 K separated by intervals of 800 K. Black dots correspond to PREM values.
ity of perovskite is insignificant ŽMao et al., 1991. and the effect of inelasticity is small. Since thermoelastic data on magnesiowustite is quite well con¨ strained, we simply used available experimental data ŽRichet et al., 1989; Sumino and Anderson. to determine the effective K S for a perovskiteq magnesiowustite assemblage characteristic of a ‘‘pyrolitic’’ ¨ LM. In Fig. 3, we show our calculated geotherms for a ‘‘perovskite’’ and a ‘‘pyrolite’’ LM and compare with others ŽBrown and Shankland, 1981; Anderson, 1982; Ito and Katsura, 1989; Stacey, 1992.. The error bars in ours reflect uncertainties related to the calculated K S Ž P,T . surface Žsee Appendix A.. For the pyrolite model, the estimated temperature at 670 km of 1700 " 110 K is in excellent agreement with that recently inferred for the base of the transition zone from Fe–Mg partitioning experiments ŽKatsura and Ito, 1996.. In contrast, we calculate from purely materials considerations that for a pure ŽMg,Fe.-perovskite lower mantle T670 f 2060 " 110 K, suggesting that if the mantle were to have this silica rich composition, a thermal boundary layer would have to exist at z s 670 km, with DT f 350 K, this conclusion is in keeping with fluid dynamical models of layered mantle convection ŽPeltier, 1989; Peltier et al., 1997.. Comparing our geotherms with the nearly adiabatic geotherm of Brown and Shankland Ž1981., we see that both compositional models predict nearly
105
adiabatic gradient in the shallow portions of the LM but the temperature gradient significantly exceeds the adiabatic gradient for depth over f 1500 km and leads to the CMB temperature of f 4000 K. The temperatures at CMB estimated in our models exceed the estimated melting temperature of core materials ŽJeanloz, 1989; Boehler, 1993. at CMB conditions without invoking additional thermal boundary layerŽs.. It is also close to the solidus temperature of lower mantle materials measured by recent diamond anvil experiments Žf 4000 K. ŽZerr et al., 1998.. The steeper temperature gradients in the deep LM inferred from our analysis suggest that convective heat transport is not very efficient in this region. This could be caused by high viscosities ŽNakada and Lambeck, 1989; Mitrovica and Forte, 1997.. Mitrovica and Forte Ž1997. indicated that the viscosity changes only by a factor of hundred or less in the lower mantle. This observation is difficult to reconcile with an adiabatic gradient because the melting temperature of lower mantle minerals increases significantly with depth ŽBoehler, 1993.. With the superadiabatic gradients found here, it is possible to reproduce the nearly constant viscosity inferred from geodynamic modelling ŽKarato, 1998..
Fig. 3. Lower mantle geotherms. The effects of mantle mineralogical composition on the bulk modulus were modeled using an averaging technique for a two-component aggregate ŽWatt et al., 1976.. Pyrolitic composition was assumed to be Ž84 wt.% Mg 0.9 Fe 0.1 SiO 3 q16 wt.% Mg 0.88 Fe 0.12 O.. Thermoelastic parameters for magnesiowustite were assumed to be ŽSpetzler, 1970; ¨ Richet et al., 1989; Mao et al., 1991.: K S Ž0,0. s160y7.5x Fe GPa, ŽEK S rEP .< Ts 0 s 4.2 and ŽEK S rET .< Ts0 sy00016 GPa Ky1 , and E 2 K S rEPET s 0.0001 Ky1 . The latter was determined for periclase.
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We also note that the lower mantle has been considered to have nearly adiabatic temperature gradient because Bullen’s inhomogeneity parameter is nearly one ŽBirch, 1952.. The superadiabatic gradients obtained in our calculations ŽdTrd z f 0.7–1.0 Krkm. correspond to the Bullen’s parameters of around 0.90 " 0.02 which is within the acceptable range of uncertainty of seismological models. In addition, we also note that the density–depth relation is not well constrained by PREM. Kennett Ž1998. recently analyzed the density–depth relation in the LM based primarily on normal mode data and found density gradients significantly smaller than PREM values for depths greater than f 1500 km, implying that temperature gradients are likely to be superadiabatic. This steeper temperature gradient beyond 1500 km depth seems to be a very robust feature that stands out in all temperature profiles we derived by comparing PREM data with our derived K S Ž P,T . using somewhat different parameters. In our approach, the superadibatic gradient seems to be a unavoidable conclusion if the lower mantle is assumed to be homogeneous. This assumption has lately been questioned by Kellogg et al. Ž1999. and Ž1999.. These authors van der Hilst and Karason ´ propose the existence of a boundary between compositionally distinct regions at a depth around 1600 km. Beyond this depth, a lower mantle that is about 4% denser and hotter could be dynamically stable. We simply note that the direct comparison between PREM and our derived K S Ž P,T . reveals a region with distinct properties beyond f 1500 km. Finally, seismic tomography Žvan der Hilst et al., 1997. suggests that the 670 km discontinuity does not act as a global barrier to mantle convection at the present time and in the recent past. Therefore, some mixing must occur between UM and LM and a model of LM such as pure perovskite composition seems inappropriate. Our geothermal profiles are, however, compatible with both pyrolitic and some chemically layered model, the latter being associated with a minor thermal boundary layer near the 670 km discontinuity. In summary, this study has determined accurate first principles bulk modulus which were used in conjunction with experimental data on V Ž P,T . of MgSiO 3 perovskite phase at moderately high tem-
peratures and pressures to fit a K S Ž P,T . surface based on a new parameterization of the isentropic EOS. By comparing the K S Ž P,T . surface with PREM data we have inferred a geotherm for a ‘‘perovskite’’ and a ‘‘pyrolite’’ LM. The basic assumption involved in our procedure is that K S is essentially a function of density only ŽBirch’s law of corresponding states., rather than a specific function of pressure and temperature separately. A possible contribution from intrinsic temperature dependence of K S ŽAnderson, 1987. will reduce the inferred temperatures to some extent. In contrast to most of the previous models ŽBrown and Shankland, 1981; Anderson, 1982; Ito and Katsura, 1989; Stacey, 1992., ours do not require an additional thermal boundary layer at the CMB to satisfy the condition that the temperature at CMB exceeds the melting temperature of core materials ŽBoehler, 1993.. A thermal boundary layer at the bottom of the mantle, if any, seems very broad as has been shown in some numerical modeling assuming high viscosity in the LM Žvan den Berg and Yuen, 1998.. Our geotherm for pyrolitic composition is the first materials based geotherm that independently satisfies current estimates of the temperatures of the shallow and deep portions of the lower mantle and is also consistent with current estimates of its viscosity profile.
Acknowledgements We wish to acknowledge computational resources from the Minnesota Supercomputer Institute, the National Science Foundation Žaward No. EAR 9628042 to RMW and xxxx to S.-i Karato., the Brazilian National Research Council ŽCNPq award to CRSS., and the British NERC ŽAP and GDP..
Appendix A Here we outline the parameterization of the thirdorder finite strain isentropic EOS we have used to extrapolate volume and bulk modulus obtained at low and moderately high temperatures to the temperatures and pressures prevailing in the lower mantle. Our starting point is a well accepted EOS for the
C.R.S. da SilÕa et al.r Physics of the Earth and Planetary Interiors 118 (2000) 103–109
temperature dependent adiabatic bulk modulus K S Ž P,T . ŽZhao and Anderson, 1994.: K S Ž P ,T . s K S Ž 0,Tf . Ž 1 q 2 f . = 1q Ž
3 K SX
5
5 2
K S Ž P ,0 . s K S Ž 0,0 . Ž 1 q 2 f 0 .
In this equation f s 1r2wŽ V Ž0,Tf .rV Ž P,T y 1x is the Eulerian finite strain, K SX Ž0,Tf . s ŽEK SrEP . SŽ0,Tf . is the adiabatic pressure derivative of the adiabatic bulk modulus, and Tf is the temperature at the intersection of the adiabat passing through Ž P,T . with the zero pressure axis. Tf is related to T by:
Ž 0,Tf .
g Ž P ,T . s g Ž 0,0 .
ž
V Ž P ,T . V Ž 0,0 .
q
/
K S Ž P ,T . ' K S Ž r Ž P ,T . .
ž
2
= Ž 1 q Ž 3 K SX Ž 0,0 . y 5 . f f
Ž 3.
2
K SX Ž 0,Tf . y 6 f
/
Ž 4.
which we have fit to the database. K S Ž0,Tf . and K SX Ž0,Tf . are then used in Eq. Ž1. to obtain K S Ž P,T ..
1 The approximation that G r K S in Eq. Ž2. is independent of pressure is reasonable. Some of the dependence can be absorbed by the parameters G 0 and q.
Ž 6.
where f f s 1r2wŽ V Ž0,0.rV Ž0,Tf .. 2r3 y 1x. Generally, V Ž0,Tf . is expressed in term of the thermal expansion coefficient a Ž0,T .:
H0Ta Ž0,T .dT
5
P s 3 K S Ž 0,Tf . f Ž 1 q 2 f .
2
V Ž 0,T . s V Ž 0,0 . e
This assumption leads to unique analytical forms for K S Ž0,Tf . and K SX Ž0,Tf ., which are then used in the adiabatic third order finite strain EOS for P ŽBoehler, 1993.:
3
5
.
Additionally, we require Eq. Ž1. to be consistent with Birch’s law ŽBirch, 1961.. This empirical law derived from velocity–density systematic in minerals, recognizes that the adiabatic bulk modulus of similar solids with the same molecular weight depends essentially on the density. Mathematically we have:
= 1q
where f 0 s 1r2wŽ V Ž0,0.rV Ž P,0 y 1x. Furthermore, r Ž0,Tf . s r Ž P,0. must hold for some negative P. Therefore, consistency with Birch’s law requirement allows us simply to substitute r Ž0,Tf . for r Ž P,0. in Eq. Ž5. and obtain:
Ž 2.
where g , the thermodynamical Gruneisen parameter, ¨ is approximated by ŽPoirier, 1991. 1
Ž 5.
.. 2r3
K S Ž 0,Tf . s K S Ž 0,0 . Ž 1 q 2 f f .
gP KS
= 1 q Ž 3 K SX Ž 0,0 . y 5 . f 0
Ž 1. .. 2r3
T s Tf e
We now derive expressions for K S Ž0,Tf . and K SX Ž0,Tf .. Let us recall that:
2
Ž 0,Tf . y 5 . f
107
Ž 7.
where a Ž0,T . is parameterized as:
a Ž 0,T . s a q bT y cT 2
Ž 8.
Similarly, we are able to map K S Ž P,T . into the Ž K S P,0. line: 5 U
K S Ž P ,T . s K S Ž 0,0 . Ž 1 q 2 f .
2
= 1 q Ž 3 K SX Ž 0,0 . y 5 . f U
Ž 9.
where f U s 1r2wŽ V Ž0,0.rV Ž P,T .. 2r3 y 1x. Internal consistency requires K S Ž P,T .’s calculated from Eqs. Ž1. and Ž9. to be mathematically identical. This leads to: K SX Ž 0,Tf . s
2r3
5 q
2 3
Tf
1 q ye H0
a Ž0,T . dT
3
Ž 10 .
where y s 2rŽ3 K X Ž0,0. y 5. y 1. Eq. Ž4. and the auxiliaries Ž2., Ž6., and Ž10. form a coupled set of equations parameterized in terms of V Ž0,0., K Ž0,0., K X Ž0,0., a, b, and c. Eq. Ž4., which relates P, T, and V, is then fit to the experimental database collected by Funamori et al. Ž1996.. The fitting is accomplished by using a standard multidimensional nonlinear regression algorithm ŽBeving-
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ton, 1969.. We chose to constrain K Ž0,0. to 262 " 2 GPa and K X Ž0,0. to 3.86 " 0.06. These values were obtained by fitting Eq. Ž5. to the calculated V vs. P relation. We also constrained a Ž300. to a safe interval Ž2.0 " 0.5. = 10y5 as indicated by experiments ŽKnittle et al., 1986; Ross and Hazen, 1989; Mao et al., 1991; Funamori and Yagi, 1993; Wang et al., 1994; Chopelas, 1996; Jackson and Rigden, 1996; Gillet et al., 1996; Jackson, 1998.. We have found that by varying q in the range 0.6–10 does not alter significantly the results, therefore q was set to unity. The value of g 0 was also constrained to a safe interval Ž1.3 " 0.3.. The uncertainties in the values of g 0 and a Ž300. are the very predominant contributions to the uncertainties in K S Ž P,T .. For a Ž300. and g 0 in the middle of the interval, we obtained ˚ 3, b s 1.37 " 0.05 = 10y8 Ky1, V0 s 162.0 " 0.2 A and c s 0.43 " 0.02 K 2 , which produces a s 2.06 " 0.4 = 10 -5 and a normalized x 2 s 0.88.
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