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
104
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-
108
C.R.S. da SilÕa et al.r Physics of the Earth and Planetary Interiors 118 (2000) 103–109
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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