Elasticity, Composition and Temperature of the Earth's Lower

Geophys. J. Int. (1998) 134, 291–311

Elasticity, composition and temperature of the Earth’s lower mantle: a reappraisal

Ian Jackson Research School of Earth Sciences, Australian National University, Canberra ACT 0200, Australia. E-mail: [email protected]

Accepted 1998 February 9. Received 1998 January 23; in original form 1997 April 24

SUMMARY The interpretation of seismological models for the Earth’s lower mantle in terms of

chemical composition and temperature is sensitive both to the details of the method- Downloaded from https://academic.oup.com/gji/article/134/1/291/632609 by guest on 01 October 2021 ology employed and also to uncertainties in key thermoelastic parameters, especially for the dominant (Mg, Fe)SiO3 perovskite phase. Here the alternative approaches— adiabatic decompression of the lower mantle for comparison at zero pressure with laboratory data, and the projection of laboratory data to lower-mantle P–T conditions for direct comparison with seismological observations—are assessed, along with the equations of state on which they are based. It is argued that adiabatic decompression of the lower mantle is best effected by a strategy in which consistent third-order finite- strain expressions are required simultaneously to fit the strain dependence of both the seismic parameter and the density. This procedure accords due weight to the most robust seismological observations, namely the wave speeds, and reduces to an acceptable level the otherwise very strong covariance among the fitted coefficients. It is demonstrated that this approach can readily be adapted to include the effects of relaxation, from the Hill average to the Reuss lower bound, of the aggregate bulk modulus which governs the radial variation of density. For the projection of laboratory thermoelastic data to lower-mantle P–T conditions, the preferred equation of state is of the Mie–Gru¨neisen type, involving the addition at constant volume of the pressure along a finite-strain principal (300 K) isotherm and the thermal pressure calculated from the Debye approximation to the lattice vibrational energy. A high degree of consistency is demonstrated between these alternative equations of state. Possible departures of the lower mantle from conditions of adiabaticity and large-scale homogeneity are assessed; = ± these are negligible for the PREM model (Bullen parameter gB 0.99 0.01) but ~ ± significant for ak135 (gB 0.94 0.02), reviving the possibility of a substantially superadiabatic temperature gradient. Experimentally determined thermoelastic properties of the major (Mg, Fe)SiO3 perovskite and (Mg, Fe)O magnesiowu¨stite phases are used to constrain the equation-of-state parameters employed in the analysis of seismological information, although the scarcity of information concerning the pressure and temperature dependence of the elastic moduli for the perovskite phase remains a serious impediment. Nevertheless, it is demonstrated that the combination of a pyrolite composition simplified to the three-component system (SiO2–MgO–FeO) with molar = = XPv 0.67 and XMg 0.89, and a lower-mantle adiabat with a potential temperature of 1600 K—consistent with the preferred geotherm for the upper mantle and transition zone—is compatible with the PREM seismological model, within the residual uncertainties of the thermoelastic parameters for the perovskite phase. In particular, such = ∂ ∂ = consistency requires values for the perovskite phase of KS∞ ( KS/ P)T and q ∂ ∂ ∂ ∂ − −1 ( ln c/ ln V)T of approximately 3.8 and 2, and G/ T near 0.022 GPa K for the lower-mantle assemblage. More silicic models, which have sometimes been advocated, can also be reconciled with the seismological data, but require lower-mantle temperatures which are much higher—by about 700 K for pyroxene stoichiometry. However, the absence of seismological and rheological evidence for the pair of thermal boundary layers separating convection above and below, which is implied by such models,

remains a formidable diﬃculty. Under these circumstances, the simplest possible model, that of grossly uniform chemical composition throughout the mantle, is preferred. Key words: chemical composition, density, elasticity, lower mantle, seismic parameter, temperature.

consistency will be demonstrated between these alternative 1 INTRODUCTION approaches, and the reasons for it will be explored. In marked contrast to the complex seismic structure of the The selection of appropriate values for the equation-of-state overlying transition zone with its generally high wave-speed– parameters for the major lower-mantle phases (Mg, Fe)SiO3 depth gradients and superimposed discontinuities, the lower perovskite and (Mg, Fe)O magnesiowu¨stite will be discussed mantle is distinguished by smooth and relatively subdued in detail. The constraints available from static compression radial (and lateral) variations of seismic wave speeds, especially (P–V –T ) and acoustic studies will be compared and contrasted, for the depth interval ~900–2600 km. That this region is and the key residual uncertainties will be identified. Within plausibly both grossly homogeneous and not too far from an this framework, alternative composition–temperature models adiabatic temperature profile was demonstrated long ago in a will be tested for compatibility with seismological observations. Downloaded from https://academic.oup.com/gji/article/134/1/291/632609 by guest on 01 October 2021 very influential paper by Birch (1952). He also first employed finite-strain equations of state to estimate the density and = 2 − 2 2 EQUATIONS OF STATE seismic parameter w V P (4/3)V S of the (hot) adiabatically decompressed lower mantle, obtaining what have proved to be remarkably durable values of 4.0 g cm−3 and 51 km2 s−2, 2.1 Isothermal compression respectively. These values are distinctly higher than those for There is ample experimental evidence that the 300 K isothermal the common silicate minerals of the crust and upper mantle. compression of geological materials to relatively large strains Noting that only close-packed oxides such as MgO, TiO2 is adequately described by finite-strain equations of state, and Al2O3 display appropriate combinations of high density based on the Taylor expansion of the Helmholz free energy: and incompressibility, Birch boldly concluded that ‘dense = 2+ 3+ 4+ high-pressure modifications of the ferro-magnesian silicates, F ae be ce … ,(1) probably close-packed oxides … are required to explain the in powers of the Eulerian strain, high elasticity of the deeper part of the mantle’. = − 2/3 = − −2/3 That phase transformations are indeed responsible for much e [1 (r/r0) ]/2 [1 (V /V0) ]/2 , (2) of the seismic structure of the transition zone is no longer truncated at third or fourth order (Birch 1952; Mao & Bell seriously questioned (e.g. Ringwood 1975; Ita & Stixrude 1992). 1979; Jeanloz & Knittle 1986; Knittle & Jeanloz 1987). V and However, the extent of any superimposed compositional layer- r are respectively the molar volume and density; the subscript ing in the mantle has yet to be resolved unequivocally. In a 0 identifies their zero-pressure values. Differentiation of eq. (1) companion paper, Jackson & Rigden (1998), a broad review with respect to volume V leads to the fourth-order expression of the elasticity, composition and temperature of the Earth’s for the pressure mantle is presented. Here, it is my intention to explore in more =− ∂ =− − 5/2 + 2 + 3 detail than was appropriate for the more general review a P ( F/dV )T (1 2e) {C1e C2e /2 C3e /6} (3) range of issues bearing specifically upon the analysis of the (e.g. Davies & Dziewonski 1975). The corresponding expressions elastic properties of the lower mantle, and their interpretation for the incompressibility or bulk modulus, in terms of chemical composition and temperature. = − 5/2 + − + − 2 − 3 I will begin with a brief survey of the equations of state best K (1 2e) {C1 (C2 7C1)e (C3 9C2)e /2 11C3e /6}/3, suited to the analysis of mantle elasticity. The approach (4) favoured here and in some other recent studies (Stixrude & Bukowinski 1990; Bina & Helffrich 1992; Stixrude et al. 1992) and its pressure derivatives K∞,K◊ etc., follow by further involves the construction of a third-order Eulerian finite- differentiation. The more familiar third-order expressions are = strain (Birch–Murnaghan) principal isotherm (for 300 K), and retrieved by setting C3 0. The relationships between the Ci addition at constant volume of the thermal pressure given and the bulk modulus and its pressure derivatives evaluated by the simple Mie–Gru¨neisen–Debye prescription. A popular at zero pressure are as follows: alternative, in which finite-strain equations with temperature- C =3K , adjusted coefficients are used to model compression along 1 0 ff = − ∞ high-temperature isentropes (Davies & Dziewonski 1975; Du y C2 9K0(4 K0), & Anderson 1989; Rigden et al. 1991), will also be examined C =27K [K K◊−K∞ (7−K∞ )+143/9] . (5) with particular attention to the need for internal consistency 3 0 0 0 0 0 in modelling both compression and elastic moduli. In fitting Much of the considerable empirical success of the third- such equations of state to seismological models, strategies order Eulerian P(e) equation of state (Birch–Murnaghan) is 2 = − ∞ for treating the covariance among model parameters and attributable to the fact that the ratio C2e /2C1e 3(4 K0)e/2 ∞ ~ according appropriate emphasis to the most robust seismo- is very small, because K0 4 and e%1, justifying truncation logical information (wave speeds, or seismic parameter, rather at third or even second order (Birch 1952). In fitting the than density and pressure) will be presented. Approximate compressional and shear moduli to finite-strain equations of

© 1998 RAS, GJI 134, 291–311 L ower-mantle elasticity, composition and temperature 293 state (see below), there is much less information concerning compression (e.g. Davies & Dziewonski 1975; Duffy& convergence of the respective expansions. Anderson 1989; Rigden et al. 1991). Eqs (2) and (4) then yield for the seismic parameter = = − + − 2.2 The thermal pressure w KS/r (1 2e){C1 (C2 7C1)e ff + − 2 − 3 The e ect of temperature may be incorporated into the (C3 9C2)e /2 11C3e /6)}/3r0 (12a) equation of state in a variety of ways. Of these, the ‘pointwise’ =(1−2e){C (1−7e)+C (e−9e2/2) addition of the thermal pressure required to maintain constant 1 2 + 2 − 3 volume as the temperature is increased, and the ‘global’ C3(e /2 11e /6)}/3r0 . (12b) approach in which the zero-pressure volume V and the finite- 0 The terms within the braces in eq. (12b) are grouped in such strain coefficients C are regarded as temperature-dependent, i a way as to emphasize the fact that the coefficients of the are the most widely used (e.g. Jackson & Rigden 1996). powers of e in eq. 12(a) are not all independent, an important In the former approach, the vibrational behaviour of the consideration in fitting seismological data (see below). Eqs (3) lattice can be simply and effectively modelled through the and (12) provide an internally consistent method for modelling Debye approximation to the vibrational part of the Helmholz (at fourth order) the compression of regions of the Earth’s free energy, given by interior, which are reasonably regarded as grossly homo-

h/T geneous in composition and phase, and subject to an adiabatic Downloaded from https://academic.oup.com/gji/article/134/1/291/632609 by guest on 01 October 2021 = 3 − −t 2 Fth 9nRT (T /H) P ln(1 e )t dt (6) temperature gradient. The more familiar third-order expressions 0 = are retrieved by setting C3 0. (e.g. Stixrude & Bukowinski 1990; see also Jackson & Rigden 1996). The entropy S, internal energy E , and thermal pressure th 2.4 Hot finite-strain isentropes: compressional and shear P follow from the thermodynamic relationships th moduli S=−(∂F/∂T) ,P=−(∂F/∂V) ,E=F+TS.(7) V T Differentiation of Taylor expansions of the strain energy In the above expressions, n and R are respectively the number densities associated with the appropriate elastic deformations of atoms per formula unit and the gas constant, and H is the leads also to finite-strain expressions for the moduli = + characteristic (Debye) temperature with a volume dependence MP K (4/3)G and G which govern the speeds VP and VS of described by compressional and shear waves (Birch 1938; Sammis, Anderson =− ∂ ∂ & Jordan 1970; Davies & Dziewonski 1975): c ( ln H/ ln V)T .(8) rV 2 =(1−2e)5/2{L +L e+L e2/2+L e3/6} , In the quasiharmonic approximation, the Gru¨neisen parameter P 1 2 3 4 2= − 5/2 + + 2 + 3 c depends only upon volume, as rV S (1 2e) {M1 M2e M3e /2 M4e /6} . (13) ∂ ∂ = = ( ln c/ ln V)T d ln c/d ln V q, (9) As a consequence of the truncation of the power-series expansions of the strain energy, the coefficient of the final term which, upon integration with constant q, yields in the braces is inevitably incomplete. An analogous situation = q ffi n c(V ) c0(V /V0) , (10) is exemplified by the form of eq. (4), where the coe cient of e is a linear combination of C and C ; the coefficient of the where c =c(V ). This analysis leads to the widely employed n n+1 0 0 final term is accordingly incomplete. Sammis et al. (1970) and parametrically economical Mie–Gru¨neisen description of showed that the use by Birch (1938) of a version of eq. (13) the thermal pressure: ffi with incomplete coe cients L 2 and M2 led to erroneous = − ∂ ∂ ∂ ∂ DPth(V, T ) P(V, T ) P(V, 300 K) expressions for the derivatives ( VP/ P) and ( VS/ P) evaluated at zero pressure. Accordingly, they recommended that the final =c(V ){E [H(V),T]−E [H(V ), 300 K]}/V . th th term within the braces in eqs (4), (12a) and (13) should be (11) omitted in the interests of self-consistency, and this has generally been the practice ever since (e.g. Davies & Dziewonski Here P(V, T ) is the total pressure and P(V, 300 K) is the 1975; Jeanloz & Knittle 1986; see, however, Ita & Stixrude pressure on the principal (300 K) isotherm, which is often most (1992) for a notable exception). appropriately modelled by the third-order Eulerian expression Combination of eqs (13) to obtain K = [V 2 −(4/3)V 2], (eq. 3, with C =0). S r P S 3 and comparison with eq. (4), now describing the isentropic One of the attractions of eq. (11) as a P–V –T equation of bulk modulus, yields the relationships state is that it is derived within the complete thermodynamic − = framework provided by eqs (1) and (6), as emphasised by L 1 (4/3)M1 C1 , Stixrude & Bukowinski (1990). It is therefore possible to relate, L −(4/3)M =C −7C , in an internally consistent manner, the isothermal conditions 2 2 2 1 − = − of much laboratory experimentation to the isentropic conditions L 3 (4/3)M3 C3 9C2 , prevailing along mantle adiabats (Stixrude & Bukowinski L −(4/3)M =−11C , (14) 1990; Bina & Helffrich 1992; Jackson & Rigden 1996). 4 4 3 which are eqs (2) of Davies & Dziewonski (1975) augmented through the incorporation of the neglected ‘incomplete’ term. 2.3 ‘Hot’ finite-strain isentropes: adiabatic compression This system of four simultaneous equations overdetermines The finite-strain equation of state (eq. 3) with suitably adjusted the three ‘unknowns’ Cn, meaning that the L i and Mi should coefficients has also been widely used to describe adiabatic not all be regarded as independently adjustable. It is evident

1 from eq. (14) that the procedure recommended by Sammis Table 1. Equation-of-state parameters for (Mg, Fe)SiO3 perovskite and magnesiowu¨stite. et al. (1970), whereby L 4 and M4 are set equal to zero = (at fourth order), requires also that C3 0, thus yielding an Phase perovskite magnesiowu¨stite = 2− 2 expression for the bulk modulus KS r[V P (4/3)V S], which Composition (Mg1−x,Fex)SiO3 (Mg1−x,Fex)O is inconsistent with that for P in the sense that K≠r∂P/∂r.If Equation-of-state parameters a finite-strain fit to the M and G models is to be sought, the 3 + + P V0/cm 24.434 1.21x 11.247 1.00x most appropriate strategy would therefore be a simultaneous + KS0/GPa 264.0 162.5 11.5x fit to the P(e), M (e) and G(e), subject to the constraints ∞ = ∂ ∂ P KS0 ( KS/ P)T,0 4.0 4.13 presented as eq. (14). H0/K 1000 673 2 c0 1.31 1.41 q 1 1.3 2.5 Consistency between Mie–Gru¨ neisen–Debye and hot finite-strain isentropes Shear moduli3 − In later sections of this paper, two alternative approaches to G0/GPa 177.3 130.8 75.6x the modelling of the elasticity of the lower mantle will be 1 For alternative equation-of-state fits (with q=2 and/or K∞=3.8) to compared and found to yield consistent inferences concerning the P–V –T data for MgSiO3 perovskite, see Table 3. chemical composition and temperature. The first of these will 2 The slight difference from the value of 1.33 of Jackson & Rigden

= Downloaded from https://academic.oup.com/gji/article/134/1/291/632609 by guest on 01 October 2021 involve the use of finite-strain expressions (4) and (12b) to fit (1996, 1998) results from the use here of KS0 264 GPa, rather than = the variation of pressure and seismic parameter throughout KT0 261 GPa, in fitting the same P–V –T data set. 3 Sources: MgSiO perovskite—Yeganeh-Haeri (1994); (Mg, Fe)O— that part of the lower mantle (~900pressures and temperatures for direct comparison the intersection of the regions of covariance, which provides a with seismological models. The equation of state employed in good fit to both P(e) and w(e). this extrapolation involves addition at constant volume of the The estimates thus obtained for the properties at the thermal pressure obtained from the Mie–Gru¨neisen–Debye high-temperature (T0) foot of the adiabat are then compared = = model (eq. 11) to the pressure on the 300 K isotherm, which is with those calculated at P 0 and T T0 directly from the calculated from the third-order finite-strain expression (eq. 3). Mie–Gru¨neisen–Debye model. The fit thus obtained to the The fact that these two approaches yield consistent results 1600 K isentrope for (Mg0.94Fe0.06)SiO3 perovskite provides is not entirely surprising. The magnitude of the thermal near-perfect recovery of the ‘known’ zero-pressure properties pressure difference between the 300 K isotherm and the mantle (Table 2). The possibility of greater bias has been explored in adiabat may be inferred from calculations, based on the Mie– further tests involving the use of fourth-order rather than third- = Gru¨neisen–Debye equation of state (eq. 11), of the high- order finite-strain fits, higher temperature (T0 2100 K), alternative perovskite equation-of-state parameters (q=2, c =1.39 temperature properties of (Mg, Fe)SiO3 perovskite–(Mg, Fe)O 0 magnesiowu¨stite mixtures (Section 6 below). For example, for —see later discussion), and perovskite–magnesiowu¨stite mix- the simplified pyrolite composition and a potential temperature tures. In none of these calculations does the error in the of 1600 K, the thermal pressure difference is 10±1 GPa over recovered parameters exceed 0.3 per cent in r0, 1.1 per cent in the pressure range of the lower mantle. This amounts to about 40 per cent of the total pressure at the top of the lower mantle, Table 2. Errors in recovery of zero-pressure properties from finite- decreasing to about 8 per cent near the bottom. The thermal strain fits (33

∞ K0 and 3.2 per cent in K0 (Table 2). The fact that much less ∞ extrapolatory bias in r0,K0,K0, etc. is encountered here than in the analysis by Bukowinski & Wolf (1990) is attributed mainly to the additional constraints provided in this study by the w(e) ﬁts.

3 THERMOELASTIC PROPERTIES OF (Mg, Fe)SiO3 PEROVSKITE AND MAGNESIOWU¨ STITE

3.1 MgSiO3 perovskite Recently published synchrotron-based measurements of the variation of unit cell volume with temperature and pressure provide tighter constraints on the thermal expansivity a ∂ ∂ = and its pressure dependence, expressible as ( a/ P)T (∂K /∂T) /K2 , for the (Mg, Fe)SiO perovskite phase than

T P T 3 Downloaded from https://academic.oup.com/gji/article/134/1/291/632609 by guest on 01 October 2021 had previously been available (Wang et al. 1994; Utsumi et al. 1995; Funamori et al. 1996). Jackson & Rigden (1996) have argued that such data are best fitted to an equation of state of the form given by eq. (11) in which the Mie–Gru¨neisen– Debye description of the thermal pressure is combined with a third-order Eulerian (Birch–Murnaghan) finite-strain principal isotherm. Under these circumstances, the complete equation of state ∞ is prescribed by a set of six parameters: V0,K0,K0, H0, c0 and q; of these, it is c0 that is relatively tightly constrained by the new V(P,T) data for MgSiO3 perovskite, combined with the V(T) measurements of Ross & Hazen (1989). Accordingly, KT0 = was fixed at the value corresponding to KS0 264 GPa Figure 1. The sensitivity of selected thermoelastic properties of ∞ = ∂ ∂ (Yeganeh-Haeri 1994) and KT0 ( KT/ P)T0 at 4, consistent (Mg, Fe)SiO3 perovskite to residual uncertainties in q. Two alter- with the static compression study of Knittle & Jeanloz (1987); native Mie–Gru¨neisen–Debye models which fit the thermal pressure H was fixed at 1000 K in accordance with the heat-capacity calculated from the available P–V –T data equally well, namely 0 = = measurements of Akaogi & Ito (1993), and q at 1 in the first (q, c0) (1, 1.31) represented by the solid lines, and (q, c0) (2, 1.39) instance (see discussion below). The least-squares fit of thermal represented by the broken lines, are compared. The perturbations to | ∂ ∂ | | ∂ ∂ | = a, ( KT/ T)P and ( KS/ T)P at the Debye temperature, caused by pressure to eq. (11) for the data identified above (N 237) + + + = ± increasing q from 1 to 2, are respectively 11, 36 and 45 per cent. yields a very well-constrained value of c0 1.31 0.01 with an rms thermal pressure misfit of 0.558 GPa. The slight difference for this latter model is marginally lower at 0.547 GPa. The between these results and those of Jackson & Rigden (1996, insensitivity of misfit to the variation of q is the reason for its Fig. 5 and Table 5) reflects the fact that here K is fixed at S0 exclusion from the formal least-squares analysis. The existing 264 GPa, whereas previously K was fixed at 261 GPa, con- T0 V(P,T) data are unable to distinguish clearly between the sistent with a slightly lower value of K . Very similar values S0 q=1 and q=2 possibilities, although the wider P–T domain for the key thermoelastic parameters of MgSiO perovskite 3 recently explored by Fiquet et al. (1997) should provide addi- have been inferred from recent analyses of spectroscopic tional resolution. These alternative models yield substantially observations: a=(1.8−2.7)×10−5 K−1 for the temperature different temperature and pressure dependence for some of the interval 300–1000 K (cf. Fig. 1) and c =1.43 (Chopelas 1996); 0 key thermoelastic properties (Table 3; see also Funamori et al. a=2.5×10−5 K−1 at 800 K, c =1.3 and q=1 (Gillet, Guyot 0 1996). For example, the impact of choosing the (q=2, c =1.39) & Wang 1996). The consistency among these results suggests 0 model over the (1, 1.31) combination is to increase the values that little further improvement in resolution of c and/or q is 0 of a and especially |(∂K /∂T) | substantially—by about 10 needed or reasonably expected. S P and 40 per cent respectively—at the zero-pressure Debye However, it is worth noting that it is actually the average temperature (Fig. 1). value of c=c (V /V )q for the volume range of the V(P,T) 0 0 Comparison of thermoelastic properties for the (q, c )= measurements that is well constrained, rather than its value at 0 (1, 1.31) and (2, 1.39) models in Table 3 and Fig. 1 reveals the V =V . Since volume varies more with pressure than with 0 following. temperature, the average value of V/V0 for this data set is − ~ approximately given by 1 Pmax/2K0 0.95. Accordingly, it (1) The intrinsic temperature dependence of the bulk modulus ∂ ∂ is expected that alternative (c0,q) combinations which yield given by ( KT/ T)V is particularly sensitive to the variation of = ∂ ∂ similar values of c for V/V0 0.95 will provide equally accept- q.( KT/ T)V is given for the Mie–Gru¨neisen–Debye equation able fits to the existing V(P,T) data set. Thus, increasing q of state by from 1 to 2 should increase c0 by about 5 per cent, consistent ∂ ∂ =− − ∂ ∂ − ∂2 ∂ ∂ = ( KT/ T)V (q 1)c( DEth/ T)V/V c( DEth/ T V ) with the result c0 1.39, which is obtained from least-squares analysis of the same data with q=2 (Table 3). The rms misfit (15)

1 Table 3. Alternative combinations of thermoelastic parameters for MgSiO3 perovskite . ∞ = ∞ = ∞ = ∞ = Parameter (q, KT0) (1, 4.0) (q, KT0) (2, 4.0)(q, KT0) 1, 3.8) (q, KT0) (2, 3.8) DPth rms misﬁt/GPa 0.558 0.547 0.523 0.516 h/K (1000) (1000) (1000) (1000) q (1)(2)(1)(2) c 1.31(1) 1.39(1) 1.33(1) 1.41(1) a/10−5 K−1 1.55 1.64 1.58 1.67 2 2 2 2 KT/GPa 262.4 262.2 262.3 262.1 ∂ ∂ −1 − − − − ( KT/ T)V/GPa K 0.005 0.010 0.005 0.010 ∂ ∂ −1 − − − − ( KT/ T)P/GPa K 0.021 0.027 0.021 0.027 ∞ = ∂ ∂ KT ( KT/ P)T (4.0) (4.0) (3.8) (3.8) ∂2 ∂ ∂ −4 −1 KT/ T P/10 K 1.3 (1.3 at 1600 K) 2.9 (3.9 at 1600 K) 1.2 (1.3 at 1600 K) 2.9 (3.8 at 1600 K) KS/GPa (264) (264) (264) (264) ∂ ∂ − − − − ( KS/ T)P 0.011 0.015 0.010 0.015 ∂ ∂ ( KS/ P)T 3.99 3.97 3.79 3.77 ∞ = ∂ ∂ KSS ( KS/ P)S 3.97 (3.93 at 1600 K) 3.95 (3.90 at 1600 K) 3.77 (3.73 at 1600 K) 3.75 (3.69 at 1600 K) ∂ ∞ ∂ −4 −1 − − − − − − − − Downloaded from https://academic.oup.com/gji/article/134/1/291/632609 by guest on 01 October 2021 ( KSS/ T)P/10 K 0.18 ( 0.41 at 1600 K) 0.12 ( 0.68 at 1600 K) 0.17 ( 0.39 at 1600 K) 0.16 ( 0.76 at 1600 K) 1 Each column represents a separate ﬁt of eq. (11) to DPth calculated from the P–V –T data set discussed in the text; columns are labelled with the ∞ ∞ values chosen for q and KT. The values of KS0, KT0, H0 and q, which are held constant during the regression, are displayed in parentheses; the uncertainty in the value of c0 thus obtained is indicated in parentheses. Other thermoelastic parameters are calculated from the model; all are evaluated at STP unless otherwise indicated. 2 = Chosen for compatibility with KS 264 GPa for MgSiO3 perovskite (Yeganeh-Haeri 1994).

(Jackson & Rigden 1996, eqs B5). For q=1, a small negative values of Watanabe (1982) and it is assumed that q=1. The ∂ ∂ value of ( KT/ T)V results from the slight volume dependence temperature dependence of the bulk modulus and thermal of the thermal energy DEth(H(V),T). For q significantly greater expansion calculated for this initial model are compared with ∂ ∂ than 1, substantial negative contributions to ( KT/ T)V, pro- the data of Sumino, Anderson & Suzuki (1983), Isaak, portional to the specific heat, are available (Fig. 1). It follows Anderson & Goto (1989) and Suzuki (1975) in Fig. 2. It is that accurate (acoustic) measurements of the temperature evident that the initial model overestimates the measured dependence of the bulk modulus will be required if q is to be expansivity (by about 8 per cent at 1300 K) and underestimates | ∂ ∂ | more tightly constrained (see discussion below for MgO). the average value of ( KS/ T)P (by about 14 per cent). An = ∞ = ∂ ∂ (2) The observation that for q 1, KSS ( KS/ P)S is much examination of eqs (B5) of Jackson & Rigden (1996) indicates less temperature-sensitive than its isothermal counterpart that c0 should be decreased while q is increased to reduce the K∞ =(∂K /∂P) , i.e. |(∂K∞ /∂T) | |∂2K /∂T ∂P| (Jackson & T T T SS P % T misfit evident in Fig. 2. Accordingly, c0,qand H0 were varied Rigden 1996), holds also at higher q. Under these circum- in order to effect the reasonable compromise fit to both K (T ) ∞ = ∂ ∂ S stances, the temperature dependence of KSS ( KS/ P)S and a(T ), labelled ‘final’ in Fig. 2. The preferred values of these between 300 K and plausible potential temperatures for the equation-of-state parameters are presented in Table 1. This lower mantle adiabat is clearly negligible (Fig. 1). process highlights the fact that high-temperature acoustic ∂ ∂ The sensitivity of the these q=1 and q=2 descriptions of measurements provide much tighter constraints on K/ T and the available P–V –T data to the variation of the assumed related thermoelastic parameters, especially q, than are likely ∞ to emerge from V(P,T) static compression studies in the value of KT is also demonstrated in Table 3. Together, the lower value of K∞ (3.8) and higher value of q (2) account for forseeable future. a 15 per cent reduction in variance, although the persistently It is well known, of course, that the Debye model greatly large misfit provides clear evidence of non-random errors in oversimplifies the frequency distribution of the normal modes the P–V –T data set (Jackson & Rigden 1996). The alternative of lattice vibration, but also that the Mie–Gru¨neisen–Debye ∞ model nevertheless captures much of the essential physics combinations of K ,qand c0 presented in Table 3 will be employed in Sections 5, 6 and 7 below to assess the robustness responsible for the temperature dependence of specific heat, of inferences concerning the composition and temperature of bulk modulus and thermal expansion (e.g. Jackson & Rigden the lower mantle. 1996, pp. 90–92; Guyot et al. 1996). The willingness shown here to vary H0 and c0 from the values that are reasonably well constrained by the specific heat measurements (at 3.2 MgO 350–700 K) of Watanabe (1982) underscores the fact that the The additional constraints on equation-of-state parameters equation-of-state parameters thus derived should be regarded arising from acoustic measurements are well illustrated by the as effective values which reproduce key features of the situation for MgO which, for reasons of internal consistency thermoelastic behaviour of geological materials. in modelling the properties of lower-mantle assemblages, Finally, it must be stressed that the influence of the Fe/Mg should be fitted to the same six-parameter equation of state. substitution on the properties (other than molar volume) of ∞ Here, ultrasonically measured KS0 and KS0 (Jackson & Niesler silicate perovskite is unknown, but is likely to be small and 1982) are available and are used to constrain the principal is accordingly neglected. Non-stoichiometric wu¨stite Fe1−xO isotherm. Initially, H0 and c0 are fixed at the calorimetric has a lower bulk modulus than MgO, but crystal-field

Figure 2. Refinement of the Mie–Gru¨neisen–Debye equation of state for MgO. The open circles represent the bulk modulus and thermal = expansivity computed from the initial model with KS0 162.5 GPa, ∞ = = = = KS0 4.13, h0 773 K, c0 1.52 and q 1. The filled circles denote the results for the final model obtained after several iterations in which the values of c0,qand H0 are varied (to 1.41, 1.3 and 673 K respectively) to provide a better fit to the data of Suzuki (1975), Sumino et al. (1983) and Isaak et al. (1989), which are indicated by the curves. considerations suggest that the relative magnitudes of the bulk moduli may be reversed for (fictive) stoichiometric FeO (e.g. Jackson et al. 1990). Magnesiowu¨stites containing no more than ~20 mol. per cent FeO are likely to closely approach Figure 3. Finite-strain P(e) fits to the PREM lower mantle of ideal stoichiometry, especially at lower-mantle pressures. K0 Dziewonski & Anderson (1981). The variation of pressure with strain for the (Mg, Fe)O solid solution has accordingly been assumed for the depth interval 870–2670 km is fitted to eq. (3), truncated at to vary linearly with composition between values appropriate third (III) and fourth (IV) order, for values of r0 between 3.95 and 4.04 g cm−3. The curves in panels (b) and (c) display the variation of for the stoichiometric end-members (Table 1). ∞ the model parameters K0 and K0 as r0 is varied. The rms misfit in pressure is plotted in panel (a). Note that the third- and fourth- ffi 4 ADIABATIC DECOMPRESSION OF THE order curves in panel (a) touch, where coe cient C3 in eq. (3) is set LOWER MANTLE equal to zero. The horizontal and vertical tie-lines highlight the values ∞ of r0,K0 and K0 associated with the minimum misfit for the third- order fit. The filled circles labelled ‘3U’ and ‘4U’ represent the 4.1 General approach unconstrained third- and fourth-order P(e) fits of Bukowinski & Wolf Some analyses of the elasticity of the lower mantle (e.g. Jackson (1990); those with roman numerals I–VI correspond to their solutions 1983; Bukowinski & Wolf 1990) have been based solely on ‘thermodynamically’ constrained through eq. (16). the use of eq. (3) to fit the radial covariation of pressure and density prescribed by a suitable gross earth model, e.g. PREM less than 0.05 GPa in pressure (0.2 per cent of the pressure at ∞ (Dziewonski & Anderson 1981). There are two shortcomings 870 km depth) for appropriate combinations r0,K0 and K0, ± ± ± of this approach. First, there is a wide range of covariance of identified in Fig. 3, within the ranges 1, 15 and 25 per ∞ cent respectively. Bukowinski & Wolf (1990) sought to reduce the model parameters r0,K0 and K0 about the global minimum—especially for the more flexible fourth-order fit (Fig. 3). this indeterminacy by imposing upon candidate mineralogical For PREM, the misfit penalty relative to the fourth-order P(e) models the additional constraint = −3 = ∞ = = d minimum at r0 3.992 g cm , K0 212.9 GPa, K0 4.10 is KS(T0) KS(300 K)[r(T0)/r(300 K)] S . (16)

The value of the Anderson–Gru¨neisen parameter, =− ∂ ∂ dS (1/aKS)( KS/ T)P , (17) was estimated from laboratory measurements on relevant oxide and silicate minerals. Nevertheless, their pyrolite-like models I–III and perovskite models IV–VI together sample much of the range of covariance illustrated in Fig. 3. The second difficulty with excessive reliance on P(e) fits to earth models is the fact that neither density nor pressure reflects very directly the seismological information of highest resolution concerning the elasticity of the lower mantle, namely VP(r) and VS(r). Density–depth models constructed from seismic wave speed versus depth models by integration of the Adams–Williamson relationship, dr/r=−g(r) dr/w(r) , (18) for a homogeneous layer subject to adiabatic self-compression, Downloaded from https://academic.oup.com/gji/article/134/1/291/632609 by guest on 01 October 2021 and used as the starting point in inversions of seismological data, have required little modification in order to satisfy the additional constraints available since the 1960s from observations of the Earth’s free oscillations (e.g. Dziewonski & Anderson 1981; see, however, Montagner & Kennett 1996). The fact that density is relatively poorly constrained poses an obvious dilemma for the fitting of finite-strain equations of state to the lower mantle, since the density, and its zero- pressure value, are required in calculation of the strain. Pressure, in turn, is calculated from the density model through the expression r P(r)=−P r(r)g(r) dr , (19) R where R is the radius of the Earth and the gravitational acceleration g, also a functional of density, is given by r g(r)=GM(r)/r2=(4pG/r2) P r2r(r) dr . (20) 0 The parallel fitting of eq. (12b) to the radial covariation of = 2 − 2 seismic parameter w V P (3/3)V S and density r prescribed by an earth model overcomes these two difficulties. Thus, for ffi = ∞ ◊ any given trial value of r0, the coe cients Cn,n 1, 2 (third Figure 4. The covariation of r0,K0,K0 and K0K0 for third- (III) = and fourth- (IV) order fits to PREM P(e) and w(e) for the depth order, ‘IIIw’) or n 1, 2, 3 (fourth order, ‘IVw’), and thus K0, ∞ interval 870–2670 km. The intersections between the (IIIP, IIIw) and K0, etc., may be determined from a linear least-squares fit to the w(e) seismological model. The resulting fits to w(e) for (IVP, IVw) curves define the preferred properties of the decompressed lower mantle at third and fourth order. The solid and broken tie lines the PREM lower mantle are actually very insensitive to the (respectively) highlight the properties associated with the (IIIP, IIIw) variation of r ; the near-invariance of the w(e) fit for arbitrary 0 fit, and with the alternative in which the bulk modulus K0 governing strain e results from appropriate covariation with r0 of the the radial variation of density is relaxed by 2 per cent. In the lowermost ffi ∞ various coe cients (Appendix A). Combinations (r0,K0,K0, panel, plotting symbols rather than lines represent the IIIw and IVw ◊ and K0K0) associated with such IIIw and IVw fits to the covariance curves. PREM lower mantle are compared in Fig. 4 with the corresponding ‘IIIP’ and ‘IVP’ fits to P(e). The range of variation of K ,K∞ , and K K◊ with changing 0 0 0 0 4.2. Relaxation of internal stresses in polycrystalline r is much more limited for the w(e) fits than for the P(e) fits. 0 aggregates and its effect on elastic moduli Consequently, the IIIw and IIIP, and IVw and IVP covariance curves intersect at relatively large angles defining quite precisely The response of (single-phase) polycrystals and multiphase the values of r0 and K0 which allow P(e) and w(e) to be aggregates to applied stress is complicated by the elastic simultaneously well fitted. The compromise which is thus made anisotropy of crystalline material and by differences in average involves a small P(e) misfit penalty (Fig. 3), but almost none bulk and shear moduli between constituent phases. The alter- for the w(e) fit, for the reasons given in Appendix A. Thus this native assumptions of uniform strain and uniform stress procedure appropriately accords greatest weight to the most throughout the stressed aggregate lead to the familiar Voigt ff robust seismological information (i.e. VP,VS and hence w). (upper) and Reuss (lower) bounds on its e ective elastic moduli.

The Hashin–Shtrikman bounds derived from a variational at 25 GPa to 1.001 at 130 GPa. This trend, whereby KH/KR approach (e.g. Watt, Davies & O’Connell 1976) are typically approaches unity with increasing pressure (cf. Stacey 1997, much more closely spaced—carrying the implication that they Fig. 1), is explained by the fact that the bulk moduli increase relate to states intermediate between the Voigt and Reuss markedly with increasing pressure, with pressure derivatives extremes, in which both stress and strain are heterogeneously for the two phases which are more similar than are their bulk distributed. Elastic moduli measured at high frequency and moduli at zero pressure. In such calculations of the elastic room temperature on polycrystals (e.g. Watt 1988) and on properties of multiphase aggregates, it is customary to treat the multiphase aggregates (Watt & O’Connell 1980) of high individual phases as elastically isotropic. An additional contri- acoustic quality compare well with the corresponding averages bution to the internal stress field and associated separation of of the Hashin–Shtrikman bounds, and are also generally well the Voigt and Reuss bounds arises from the elastic anisotropy approximated by the Hill average of the Voigt and Reuss of the crystalline phases. Thus, even if the lower mantle were bounds. composed entirely of orthorhombic (Mg, Fe)SiO3 perovskite, ffi ≠ On su ciently long timescales at high temperature, plastic a value of KH/KR 1 would be expected. The single-crystal deformation will facilitate the relaxation of the internal stress elastic constants for this silicate perovskite are known only at ff field towards uniformity, and the lower modulus given by the room temperature and pressure, where KH/KR di ers from Reuss bound will be appropriate. If the relaxation time appro- unity by only 0.001 (Yeganeh-Haeri 1994). It can be concluded priate for mantle conditions were to exceed the 1–100 s period with some confidence that effects associated with the relaxation of seismic waves, the unrelaxed (Hashin–Shtrikman or Hill of internal stresses in the lower mantle will be dominated by Downloaded from https://academic.oup.com/gji/article/134/1/291/632609 by guest on 01 October 2021 average) modulus KHS or KH would be appropriate for seismic the large contrast in zero-pressure bulk moduli between wave propagation, whereas the relaxed Reuss modulus should the perovskite and magnesiowu¨stite phases (Heinz, Jeanloz certainly be used in describing the density variation with depth & O’Connell 1982). This effect can be allowed for in the in a homogeneous layer (Kumazawa 1969; Stacey 1997). Under decompression of a pyrolite-like lower mantle by requiring these circumstances, the Adams–Williamson relation needs to that the value of K0 associated with the P(e) fit be about 2 per be revised by the incorporation in the right-hand side of cent lower than that for the w(e) fit (see below). eq. (18) of the multiplicative factor KH/KR (Stacey 1997), and the internally consistent fitting of P(e) and w(e) with the 4.3 Decompression of the PREM and ak135 models for same moduli as recommended above would no longer be the lower mantle appropriate. ff An indication of the magnitude of this e ect is provided The IIIP and IIIw (K0, r0) covariance curves for PREM = −3 = by the calculations described in Section 6 below. For the intersect at r0 3.985 g cm and K0 212.5 GPa (Fig. 4, ∞ perovskite–magnesiowu¨stite mixture of simplified pyrolite Table 4). The corresponding IIIP and IIIw values of K0 are composition, KH/KR decreases with increasing pressure along closely consistent at 3.90 and 3.88, respectively; the associated ◊ the 1600 K isentrope from 1.021 at zero pressure through 1.008 value of K0K0 prescribed by the third-order relationship

Table 4. Properties of the adiabatically decompressed lower mantle. ∞ ◊ ∞ ◊ Analysis Fit r0 K0 K0 K0K0 G0 G0 K0G0 gcm−3 GPa – – GPa – – PREM This study III(P, w) 3.985 212.5 3.89(1) [−3.8] 134.1 1.56 [−2.4] 3.90(P) 3.88(w) III(P, w) 3.974 210.3(w) 3.88(w)[−3.8] 132.7 1.55 [−2.4] relaxed 206.2(P) 3.99(P)[−3.8] IV(P, w) 3.984 206.0 4.1(1) −6(1) 130.0 1.75 −4.0 4.33(P) −6.8(P) 4.17(w) −5.3(w) Jeanloz & Knittle (1986) IIIP 4.003 222.5 3.76 [−3.7] − − III(MP,G) – 216.5 3.80 [ 3.1] 135.4 1.52 [ 2.1] IVP 3.994 214.5 4.05 −5.2 –– – − − IV(MP,G) – 204.7 4.38 7.5 130.1 1.77 4.0 ak135 This study III(P, w) 4.007 213.5 4.1(2) [−4.1(2)] 4.28(P)[−4.3](P) 3.96(w)[−3.9](w) IV(P, w) 4.001 211.5 4.1(1) −3.8(3) 4.22(P) −3.5(P) 4.00(w) −4.0(w) ‘III’ and ‘IV’ denote third- and fourth-order finite-strain fits; in combination with ‘III’ or ‘IV’, the label ‘P’ indicates that pressure only was fitted; ‘(P, w)’ and ‘(MP,G)’ indicate simultaneous fits to pressure and seismic parameter, and to compressional and shear moduli, respectively. Elsewhere, the labels P and w in parentheses denote values appropriate for the separate ‘P’ and ‘w’ fits, respectively (see text for details). The label III(P, w) = relaxed refers to the fits for which the requirement K0(IIIw)/K0(IIIP) 1.02 is imposed (see text).

= (eq. 5 with C3 0), ◊= ∞ − ∞ − K0K0 K0(7 K0) 143/9 , (21) is −3.8. It is thus concluded that the PREM P(e) and w(e) can be consistently described by a third-order Eulerian finite-strain model. For the fourth-order (K0, r0) covariance curves, the inter- = −3 = section occurs at r0 3.984 g cm and K0 206.0 GPa (Fig. 4, Table 4). The corresponding IVP and IVw values of ∞ ◊ K0 are respectively 4.34 and 4.17, whilst for K0K0, which = is no longer subject to the constraint C3 0 (eq. 21), the values are −6.8 and −5.3. The greater flexibility of the fourth- order equation of state evidently imposes a more stringent test on the compatibility of P(r) and w(r) as parametrized in PREM. Nevertheless, a fourth-order Eulerian finite-strain = −3 = ∞ = ± model with r0 3.984gcm , K0 206.0 GPa, K0 4.2 0.1 and K K◊=−6±1, provides a viable alternative to the third-

0 0 Downloaded from https://academic.oup.com/gji/article/134/1/291/632609 by guest on 01 October 2021 order fit. The fact that it is possible to fit both P(e) and w(e) consistently, at either third or fourth order, accords with other indications that the Adams–Williamson relationship (eq. 18) is closely obeyed by the PREM model (Dziewonski & Anderson 1981). ff The possibility discussed above that di erent values of K0, pertaining to conditions in which internal stresses are relaxed and unrelaxed, might be appropriate for the P(e) and w(e) fits, respectively, has also been addressed. The impact of the requirement that P(e) and w(e) for PREM be simultaneously = fitted with K0(IIIw)/K0(IIIP) 1.02, shown by the broken horizontal and vertical lines in Fig. 4 and in Table 4, is a slightly lower value of r0 (by 0.3 per cent) and somewhat ∞ higher values of K0, especially for the (relaxed) IIIP fit. This latter inference is at least qualitatively consistent with the fact that the pressure derivative of the Reuss bound for perovskite– magnesiowu¨stite mixtures exceeds the volume average of the pressure derivatives for the individual phases (e.g. Stacey 1998). Internally consistent fits to P(e) and w(e) have also been sought for the Earth model ak135. The velocity model ak135 was designed by Kennett, Engdahl & Buland (1995) to provide an improved fit to global traveltime data. Subsequently, the requirement that this velocity model also fit normal-mode data was used by Montagner & Kennett (1996) to derive associated radial density and attenuation models. Seismic wave speeds Figure 5. Finite-strain P(e) fits to the ak135 lower mantle of Kennett and density are specified at 50 km intervals in depth, across et al. (1995) and Montagner & Kennett (1996). Details as for Fig. 3. each of which the properties vary linearly. In this way, the constraint of low-order polynomial parametrization, common to PREM and other reference models, is avoided. temperature value for the relatively incompressible MgSiO3 ∞ The radial variation of gravitational acceleration g(r) and perovskite, whereas the value of K0 could be reconciled with pressure P(r) for the ak135 model were calculated from the the room-temperature values of 4–5 typical of close-packed density model supplied by Kennett (personal communication, silicate and oxide phases only through an extraordinarily 1996) through numerical integration of eqs (19) and (20) strong temperature sensitivity. Accordingly, the combination above. Third- and fourth-order finite-strain equations of state of parameter values associated with the ak135 IVP minimum were then fitted to P(e) for the ak135 model, with the results is regarded as physically very implausible. Comparison with displayed in Fig. 5. Comparison with Fig. 3 reveals that the the fits to the smoothly parametrized PREM model (Fig. 3) IIIP misfit is about seven times greater than for PREM, thus highlights the risk inherent in fitting the more flexible reflecting the fact that in the absence of low-degree polynomial and therefore less stable fourth-order equation of state to parametrization, the ak135 model is not as smooth. The unsmoothed models. value of r0 associated with the IIIP minimum for ak135 In marked contrast with the experience in fitting P(e), the ∞ is indistinguishable from that for PREM, whereas K0 and K0 IIIw misfit to ak135 is comparable to that for PREM; the gain for ak135 are respectively 5 per cent lower and 15 per cent from smoothing PREM seems to be offset by too restrictive a higher. Interestingly, the ‘IVP’ misfit minimum for ak135 parametrization in the uppermost part of the lower mantle. = −3 = ∞ = occurs at r0 4.095 g cm , K0 303.6 GPa, K0 1.64 and The intersection of the IIIP and IIIw covariance curves for ◊=+ = −3 = K0K0 5.2. This value of K0 greatly exceeds even the room- ak135 (Fig. 6) occurs for r0 4.007 g cm , K0 213.5 GPa.

dependence of both P and w (especially at third order) is consistent with previous conclusions that the density distribution is closely consistent with the Adams–Williamson relation (Dziewonski & Anderson 1981). However, as we have seen, an alternative interpretation in which a relaxed value of the bulk modulus is employed in fitting P(e) is also viable. For the ak135 model, there is greater difficulty in consistently fitting P(e) and w(e). With common r0 and K0 for the IIIP and IIIw fits (Fig. 6), it is evident that a significantly higher ∞ value of K0 is required for the IIIP fit than for the IIIw fit. Comparison of the ak135 and PREM models in Fig. 7 reveals the reason. Under conditions of homogeneity and adiabaticity, the slope of the density–pressure trajectory is given by ∂ ∂ = −1 ( r/ P)S w . (22) It is clear from Fig. 7(a) that the average slope of the r(P) trajectory is substantially lower for ak135 than for PREM (by about 5 per cent), yet the w(P) curves for the two models Downloaded from https://academic.oup.com/gji/article/134/1/291/632609 by guest on 01 October 2021 are essentially indistinguishable (Fig. 7b). The P(e) and w(e) fits for ak135 therefore cannot be entirely compatible; the greater high-pressure incompressibility required of the IIIP fit ∞ is obtained through the larger value of K0. It should be noted that relaxation of the bulk modulus associated with the radial variation of density would have the opposite effect—causing ∂ ∂ −1 ( r/ P)S to exceed w .

4.4 Adiabaticity and homogeneity: discussion The parameter 1−g−1 dw/dr, which can be constructed from seismological wave-speed–depth models (by neglecting the

∞ ◊ Figure 6. The covariation of r0,K0,K0 and K0K0 for third- (III) and fourth- (IV) order ﬁts to ak135 P(e) and w(e) for the depth interval 958–2640 km. Details as for Fig. 4.

∞ K0 is less consistently determined than for PREM, with values of 4.28 and 3.96 for the IIIP and IIIw fits, respectively. These values are compared with those for PREM and for the (IVP, IVw) fit to ak135 in Table 4. Complete consistency between the P(e) and w(e) fits to an earth model is expected only under the very restrictive conditions which underpin the Adams–Williamson relation (eq. 18). The depth interval in question must be: (1) composed of material that is grossly uniform in both composition and phase; (2) subject to an adiabatic temperature gradient; and (3) suitably parametrized. ∞ Thus the fact that a single combination of r0,K0,K0 and Figure 7. Comparison of the PREM and ak135 lower-mantle models. ◊ K0 can be identified for PREM that describes the strain (a) Density versus depth; (b) seismic parameter versus depth.

© 1998 RAS, GJI 134, 291–311 302 I. Jackson depth dependence of the gravitational acceleration g, which is (1975) inference that heterogeneity and superadiabatic tem- ff minor throughout the lower mantle), was used by Bullen and perature gradients should generally have e ects on gB of Birch to assess the likelihood of homogeneity and adiabaticity opposite sign. of various layers of the Earth’s mantle. The basis of this approach is the identity 4.5 The variation of the shear modulus 1−g−1∂w/∂r=(∂K /∂P) +(taw/g){1+(∂K /∂T) /aK }, S S S P S The procedure favoured above of requiring both P(e) and w(e) (23) to be fitted by the appropriate finite-strain equation of state which describes the behaviour of a self-compressed homo- has the advantages of internal consistency and appropriate geneous layer, subject to a superadiabatic temperature gradient weighting of the wave speeds and pressure over previous = − = analyses in which either P(e) only has been fitted (Bukowinski t d(T TS)/dz (Birch 1952). In the event that t 0, the right- hand side of eq. (23) becomes simply dK /dP. Accordingly, & Wolf 1990; Jackson 1983) or P(e) and the compressional S = it is useful to tabulate in conjunction with earth models and shear moduli M(e)(M MP,G) have been separately (Jeanloz & Knittle 1986) or simultaneously (Butler & Anderson (e.g. Dziewonski & Anderson 1981) the parameter g given by B 1978; Davies & Dziewonski 1975) fitted to eq. (3) and = + −1 = gB dKS/dP g dw/dr w dr/dP (24) incomplete versions of eqs (13). Accordingly, the properties (r ,K,K∞ and K K◊) of the adiabatically decompressed

(e.g. Bullen 1975, Chapter 11). For a homogeneous layer, the 0 0 0 0 0 Downloaded from https://academic.oup.com/gji/article/134/1/291/632609 by guest on 01 October 2021 lower mantle (Table 4) are better constrained than in previous close approach of gB to unity is clearly diagnostic of an adiabatic temperature gradient. For PREM, the approximate analyses. −1 = ± However, this procedure in which the V and V information equality of w and dr/dP yields gB 0.99 0.01 for most of P S the lower mantle—consistent with a close approach to adia- is combined to best constrain the compression of the lower ff baticity and homogeneity. More generally, the difference mantle su ers from the disadvantage that it does not provide ∞ between w−1 and dr/dP provides a measure of the superadia- estimates of the zero-pressure shear-mode properties G0,G0 ◊ batic gradient across a homogeneous layer given by and G0. An alternative to the conventional finite-strain analysis of M (e) and G(e), in which the trailing incomplete terms in = −1− P t (g/a)[w dr/dP] (25) eqs (13) have been inappropriately neglected, is provided by (Birch 1952; Butler & Anderson 1978). For ak135, the Bullen the observation that the relationship parameter varies between 0.92 and 0.96—indicative of much G=AK+BP , (28) larger departures from the conditions of homogeneity and adiabaticity. Eq. (25) for the superadiabatic gradient has been with A=0.631(1) and B=−0.899(6), provides an excellent evaluated for ak135 by approximating the derivative with finite description of the variation of the shear modulus in the PREM differences across 100 km depth intervals, and allowing the lower mantle (Stacey 1995). thermal expansivity a to decrease with increasing pressure It follows from eq. (28) that the shear modulus and its first from 3×10−5 K−1 at zero pressure to 1×10−5 K−1 near the and second pressure derivatives, evaluated at zero pressure, base of the lower mantle. The calculated values of t increase are given by −1 −1 from about 0.3 K km to more than 0.7 K km in the deep G =AK , mantle; the superadiabatic temperature increment across the 0 0 ∞ = ∞ + depth interval 960–2640 km is thus almost 900 K, exceeding G0 AK0 B, the temperature increase along the adiabat (Section 6 below). G◊=AK◊ . (29) If the conditions of adiabaticity and homogeneity are both 0 0 relaxed, the variation of density with pressure across the layer Eqs (29) may be used in conjunction with the results of the contains an additional term, as follows: (IIIP, IIIw)or(IVP,IVw) analyses above to infer the values of G ,G∞ and K G◊ (Table 4). These findings are not grossly dr/dP=(∂r/∂P) +[(∂r/∂T) d(T −T )/dz 0 0 0 0 S P,X S inconsistent with those from published III(MP,G) and + ∂ ∂ IV(M ,G) analyses, despite the formal shortcomings of the ( r/ X)P,TdX/dz] dz/dP P latter. = −1− + ∂ ∂ w at/g (j/rg)( r/ X)P,T , (26) where X is a variable describing the changing composition 5 DECOMPRESSED LOWER MANTLE and/or phase of the material and =dX/dz is the gradient in j INTERPRETED AS A PEROVSKITE+ this quantity (Bullen 1975, Chapter 11). If an explanation were MAGNESIOWU¨ STITE MIXTURE sought for the inequality between dr/dP and w−1 in terms of departures from homogeneity rather than from adiabaticity, The properties established above for the adiabatically decom- then the required gradient would be pressed lower mantle may be compared with those expected at zero pressure and the potential temperature T for plausible j=−rg[w−1−dr/dP]/(∂r/∂X) , (27) 0 P,T lower-mantle mineral assemblages. For a wide range of chemical analogous to eq. (25) for the superadiabatic temperature compositions, orthorhombic (Mgx,Fe1−x)SiO3 perovskite and gradient. Heterogeneity alone can be excluded as an explanation (Mgy,Fe1−y)O magnesiowu¨stite will be the dominant mineral for the relatively shallow density–pressure trajectory for phases (e.g. O’Neill & Jeanloz 1990; Irifune 1993). For example ak135 (Fig. 7), at least in steady state, since the associated for the pyrolite model, phase equilibrium studies indicate that stratification, involving material of progressively lower intrinsic the Al2O3 component will be accommodated in dilute solid density with increasing depth, would be unstable with respect solution (6 mol. per cent) in the (Mgx,Fe1−x)SiO3 perovskite to convective overturn. This finding is consistent with Bullen’s phase (Irifune 1994). Under these circumstances, the aluminous

© 1998 RAS, GJI 134, 291–311 L ower-mantle elasticity, composition and temperature 303 ferromagnesian perovskite and magnesiowu¨stite phases together and bulk modulus for the binary mixture are then calculated— account for ~93 per cent of the bulk by mass or volume or the latter as the Hill average of the Voigt and Reuss bounds, 95 mol. per cent. The balance (7 weight per cent) is cubic which differ by about 4 per cent for the ‘pyrolite’ model at = CaSiO3 perovskite. Only very limited mutual solid solubility T0 1600 K and zero pressure. Use of the Hill average, rather is reported between the MgSiO3 and CaSiO3 perovskite phases than the more closely spaced Hashin–Shtrikman bounds, (Irifune et al. 1989; Kesson, Fitz Gerald & Shelley 1994). avoids the need to include the shear modulus at this stage of The possibility of further pressure-induced phase trans- the modelling. This is desirable because of the absence of formations in the deep mantle has been reviewed briefly experimental constraints on ∂G/∂T for the perovskite phase, by Jackson & Rigden (1998, pp. 416–7) and will not be and the possible complicating effects of dispersion associated pursued in detail here. However, it should be noted that the with viscoelastic relaxation. disproportionation of MgSiO3 perovskite into a mixture of its The densities and bulk moduli calculated in this way for component oxides at pressures corresponding to the deeper various choices of the compositional parameters XPv and XMg part of the lower mantle reported by Saxena et al. (1996) was and potential temperature T0 are compared in Fig. 8 with the not observed in recent experiments on the pyrolite composition corresponding properties of the decompressed lower mantle, (Kesson, Fitz Gerald & Shelley 1998). The minor components as inferred from the III(P, w) and IV(P, w) finite-strain fits Al2O3 and CaSiO3, although petrologically very important, presented in Table 4. The properties for the simplified pyrolite are excluded from the following analysis because their impact composition (right-hand diamond in Fig. 8) are calculated at Downloaded from https://academic.oup.com/gji/article/134/1/291/632609 by guest on 01 October 2021 upon the elasticity and density of a pyrolite lower mantle is a potential temperature T0 of 1600 K, which is consistent with much smaller than the effect of residual uncertainties in K∞ the eruption temperatures of mid-ocean-ridge basalts modelled and q for the dominant (Mg, Fe)SiO3 phase (Jackson & as primary magmas, and with the depths of the major seismic Rigden 1998). discontinuities interpreted as phase transformations in an With bulk composition in the system SiO2–MgO–FeO isochemical mantle (Ito & Katsura 1989; Akaogi, Ito & parametrized by the mole fractions XPv of perovskite in Navrotsky 1989; Ita & Stixrude 1992; Jackson & Rigden 1998). = + the two-phase mixture and XMg [MgO]/{[MgO] [FeO]}, and the partitioning of Fe and Mg between the perovskite and magnesiowu¨stite phases represented by the distribution coefficient = Pv Mw k (XFe/XMg) /(XFe/XMg) , (30) the compositions of the respective phases are readily determined. Experimentally determined values of k range widely from about 0.1 to 1 (e.g. Mao, Shen & Hemley 1997, and references therein), reflecting sensitivity to bulk composition, pressure and temperature and also the influence of various technical complications (Kesson, personal communication, 1998). In the companion study of mantle elasticity by Jackson & Rigden (1998), Kesson & Fitz Gerald’s (1991) value Mw Pv= = of (XFe/XMg) /(XFe/XMg) 4 equivalent to k 0.25 was adopted. A somewhat higher value, k=0.45, inferred by Kesson et al. (1997) from studies of the phase chemistry of pyrolite to 135 GPa is employed in the following analysis. Preferential partitioning of Fe into the magnesiowu¨stite phase is accordingly here less pronounced than in the com- Figure 8. A comparison of the density and bulk modulus of the panion study by Jackson & Rigden (1998). However, the adiabatically decompressed lower mantle with the corresponding + aggregate physical properties of perovskite–magnesiowu¨stite properties of (Mg, Fe)SiO3 perovskite (Mg, Fe)O magnesiowu¨stite mixtures remain essentially unchanged, being remarkably mixtures calculated at various potential temperatures T0 and for bulk compositions prescribed by the parameters X and X . The open insensitive to the assumed value of k. For example for the Pv Mg plotting symbols (circles for PREM and squares for ak135) represent pyrolite composition approximated in the three-component the III(P, w) and IV(P, w) solutions of Table 4. The large diamond = = system by XPv 0.67 and XMg 0.89, the density and bulk symbols represent the properties of the simplified pyrolite model modulus calculated as described below vary by less than = with T0 1600 K, calculated with the equation-of-state parameters of 0.1 per cent for variation of k from 0.2 to 1. Once the phase Table 1 (right-hand diamond), and with the alternative perovskite compositions are established for chosen XPv and XMg, the equation of state of column 4, Table 3. The bold vectors radiating ff molecular weights and STP molar volumes V0, and the bulk from the right-hand diamond indicate the e ects of changing each of modulus for magnesiowu¨stite are calculated from molar averages the compositional parameters and the potential temperature. The of the respective end-member properties (Table 1). All other lengths and relative orientations of the vectors correspond to the thermoelastic properties are assumed to be independent of equation-of-state parameters of Table 1. Greater length and a significantly shallower slope for the temperature-change vector would result composition (Section 3 above). from the use of q =2, since |∂K/∂T | is more sensitive to variation of The Mie–Gruneisen–Debye equation of state (eq. 6) with Pv ¨ q than is a (Fig. 1). The trade-off between composition and temperature the total pressure P(V, T ) set equal to zero is then used with is illustrated by the triangle in the upper right which shows that the parameters of Table 1 to calculate the relative volume = the alternative simplified pyrolite (XPv,XMg,T0) (0.67, 0.89, 1600 K) V(T0)/V0 and bulk modulus K(T0) at the chosen potential and perovskite-only (1.0, 0.82, 2900 K) models have indistinguishable temperature T0 and zero pressure for each phase. The density densities and bulk moduli.

It is evident that the calculated high-temperature density V/V0 at fixed pressure was then determined by numerical and bulk modulus for pyrolite provide a reasonable match to integration of the calculated thermal expansivity; V/V0 and S the corresponding properties of the decompressed lower were thus determined on a P–T grid for each phase. However, mantle, although the bulk modulus given by the more robust the values of P at which V/V0 and S were evaluated for each III(P, w) decompressions is overestimated by 2–4 per cent. The phase differ because of the markedly different compressibilities results of a parallel calculation in which the same pyrolite of the two phases. It was therefore necessary to conduct an composition–temperature model is subject to the alternative interpolation at each temperature T to identify values of V/V0 perovskite equation-of-state parameters of column 4, Table 3 and S for the two phases at common values of pressure. The (left-hand diamond in Fig. 8) indicate that this discrepancy is density, entropy and Hill average of the Voigt and Reuss bulk readily accommodated within the remaining uncertainties. moduli of the specified mixture were then calculated on this There is a wide trade-off possible between the compositional common P–T grid. Finally, selected isentropes were located in parameters and the potential temperature (e.g. Davies 1974; (P, T ) space through a procedure in which an isobaric inter- ff Jackson 1983). The nature of this trade-o is evident from the polation was employed to identify the temperature Ti at given relative orientations and lengths of the DXPv, DXMg and DT0 pressure Pi at which the entropy is equal to that at the foot vectors in Fig. 8. Thus, it is possible to choose combinations of an isentrope of specified potential temperature T0, i.e. = of DXPv, DXMg and DT0 such that the vector sum of the S(Pi,Ti) S(0, T0). resulting perturbations in K(T )–r(T ) space is zero. All such The density of the aggregate, obtained in this way from

0 0 Downloaded from https://academic.oup.com/gji/article/134/1/291/632609 by guest on 01 October 2021 combinations of composition and temperature will match those of the individual phases at common P and T, will vary the density and bulk modulus of the decompressed lower with pressure according to the Reuss rather than the Hill mantle equally well. For example, compositions enriched average KSH of the Voigt and Reuss bounds on the adiabatic relative to pyrolite in silica and FeO are allowed, provided bulk modulus which is used in calculating the seismic param- that the potential temperature is increased appropriately. The eter w.KSH describes the behaviour of the aggregate in which combination of silica enrichment to pyroxene stoichiometry the internal stress and temperature fields arising during its = =− (DXPv 0.33), substantial iron enrichment (DXMg 0.07) infinitesimal compression are both unrelaxed (Stacey 1998). =+ ~ 2 ~ and much higher potential temperature (DT0 1300 K) The thermal relaxation has a time constant t d /D 10 s reproduces the density and bulk modulus calculated for (for grain-size d=3mmandthermaldiffusivity D~10−6 m2 s−1) = the pyrolite (T0 1600 K) model. Simple scaling allows the within the seismic band, but a modulus defect DK for the identification of other models with intermediate compositions pyrolite composition at 1600 K and zero pressure of only 0.5 and potential temperatures, and the same high-temperature per cent, which for the present purposes is negligible. values of r and KS. The density and seismic parameter thus calculated as Finally, it should be noted that the composition–temperature functions of pressure along selected high-temperature adiabats trade-off described above is resolvable in principle through the for the perovskite and magnesiowu¨stite phases, and various incorporation into the analysis of another constraint—in the mixtures thereof, are presented, for clarity, as residuals relative form of the shear modulus. However, so much less is known to the Earth model PREM in Fig. 9. The sensitivities of these about the pressure and temperature dependence of the shear properties to reasonable variations of the least well-constrained modulus for (Mg, Fe)SiO3 perovskite that its inclusion is perovskite equation-of-state parameters (Figs 9a and b), deferred until Section 7. temperature (Figs 9c and d) and composition (Figs 9e and f ) are demonstrated. For the simplified pyrolite composition (X =0.67, X =0.89), the coexisting perovskite and mag- 6 PROJECTION OF LABORATORY DATA Pv Mg nesiowu¨stite solid solutions have Mg end-member mole fractions TO LOWER-MANTLE P–T CONDITIONS of 0.92 and 0.83. The two-phase mixture of this composition, with the equation-of-state parameters of Table 1 and potential 6.1 Calculations for perovskite–magnesiowu¨ stite temperature T =1600 K, provides a reasonable match to the mixtures 0 average density for the PREM lower mantle (Fig. 9a), although There is inevitably some sensitivity of the properties of the the density gradient ∂r/∂P is clearly underestimated. It is adiabatically decompressed lower mantle to any departure the density gradient, given for adiabatic compression of from the assumptions of homogeneity and adiabaticity, and homogeneous material by to the choice of equation of state (Fig. 8, Table 4). Included (∂r/∂P) =r/K =w−1=V 2 −(4/3)V 2 , (31) in the latter category is the extrapolatory bias discussed by S S P S Bukowinski & Wolf (1990). It is therefore of interest also to rather than the density itself, which is most tightly constrained examine the alternative approach, in which the properties of by seismological observations. The fact that the calculated the perovskite–magnesiowu¨stite mixture are extrapolated to density gradient given by the slope of the solid line in Fig. 9(a) lower-mantle P–T conditions for direct comparison with the is too low is thus consistent with the observation (Fig. 9b) that seismological model. the calculated seismic parameter for the mixture of simplified Use of the Mie–Gru¨neisen–Debye equation of state (eq. 11), pyrolite composition is consistently higher (by ~4 per cent) with the parameters of Table 1, provides for the straightforward than for PREM throughout the lower mantle. calculation of high-temperature isentropes for perovskite– It is clearly important to explore the sensitivity of this result magnesiowu¨stite mixtures as follows. First, for each phase, the to the residual uncertainties in the equation-of-state parameters pressure P, molar entropy S and various derived thermoelastic for (Mg, Fe)SiO3 perovskite. The critical (most poorly resolved) quantities including the thermal expansivity a were calculated parameters are q and K∞, for which variations from the values + − at each of a series of equally spaced values of V/V0 along the in Table 1 of order 1 and 0.2 respectively are readily principal (300 K) isotherm. The temperature dependence of accommodated by the existing V(P,T) data set for the MgSiO3

Figure 9. The variation relative to the PREM model (Dziewonski & Anderson 1981) of density r and seismic parameter w along selected high- + temperature isentropes for (Mg, Fe)SiO3 perovskite (Mg, Fe)O magnesiowu¨stite mixtures. Panels (a, b), (c, d) and (e, f ) demonstrate the sensitivity ∞ of these profiles to the values chosen for key thermoelastic parameters for the perovskite phase (Tables 1 and 3; with KS0 rounded to two significant figures), and to the variation of temperature and composition, respectively. In panels (e) and (f ), the tie-lines highlighted by the solid circles link the curves for the simplified pyrolite composition ‘py(89)’ with those corresponding to the adiabats for its constituent phases (Mg, Fe)SiO3 = = perovskite (XMg 0.92) and (Mg, Fe)O magnesiowu¨stite (XMg 0.83).

© 1998 RAS, GJI 134, 291–311 306 I. Jackson phase (Section 3 above). These parameters are here allowed to Indeed, with the equation-of-state parameters of Table 1, in ∞ = vary in the combinations identified in Table 3. Reduction of particular KPv 4, there exists a prima facie case for a some- K∞ increases the compressibility along the principal isotherm, what less silicic composition for the lower mantle, as indicated especially at very high pressure, resulting in higher densities in Figs 9(e) and (f ) by the broken curves labelled ‘ol(89)’ for = = = and density–pressure gradients which are more compatible olivine stoichiometry (XPv 0.5, XMg 0.89, T0 1600 K). As with PREM (Fig. 9a). noted above, however, a satisfactory match is obtained with ∞ For the reasons described in Section 3 above, increasing q the simplified pyrolite composition and a lower value of KPv from 1 to 2 is associated with a larger value of c0, resulting in near 3.8, especially if qPv is significantly greater than 1. This somewhat higher thermal expansivity and greatly increased finding is consistent with the fact that a K∞ value near 3.9 is values of |∂K/∂T | at low pressure (Fig. 1). Through the obtained from third-order finite-strain decompression of the ∂ ∂ = ∂ ∂ 2 | ∂ ∂ | identity ( a/ P)T ( KT/ T)P/KT, larger values of ( KT/ T)P PREM lower mantle (Table 4). In order to reconcile these ∞ = ∂ ∂ translate into more strongly pressure-dependent thermal inferences, it is required that KSS ( KS/ P)S does not increase expansivity, which is responsible for the higher densities at significantly with increasing temperature at zero pressure. It very high pressures for the q=2 model in Fig. 9(a). The seismic has been demonstrated that this is a robust characteristic of = ff parameter w KS/r, on the other hand, is most di erent from the Mie–Gru¨neisen–Debye description of the thermal pressure the reference model (calculated with the parameters of Table 1) (eq. 11), as applied to MgSiO3 perovskite (Jackson & Rigden at relatively low pressure, where the bulk modulus is most 1996; Section 3 above). Under these circumstances, thermal ∞ Downloaded from https://academic.oup.com/gji/article/134/1/291/632609 by guest on 01 October 2021 strongly temperature-dependent (Fig. 9b). corrections to KSS for seismological application are indeed The intersection of the various r(P) curves in Fig. 9(a) negligible. reflects the constraint on volume and hence density imposed The trade-off between chemical composition and tem- by the existing MgSiO3 V(P,T) data set; the lack of a similar perature has been explored further through calculations of intersection in Fig. 9(b) indicates that the experimental con- the temperature sensitivity of density and seismic parameter ∂ ∂ straints on K/ T are much weaker. Together, the weaker for (Mg0.88,Fe0.12)SiO3 perovskite—the lower-mantle com- pressure dependence and stronger temperature sensitivity of K position favoured by Stixrude et al. (1992). These calculations, for the q=2, K∞=3.8 model provide a much better match to like those for the best-fitting pyrolite model, are based on the the PREM w(P), and to the density–pressure gradient, although perovskite equation-of-state parameters of column 4, Table 3. the absolute value of density remains marginally too great, by It is evident from Fig. 10 that a satisfactory match to both the about 0.5 per cent. density and seismic parameter profiles for PREM is obtained The sensitivities of the calculated values of r(P) and w(P) to variation of the potential temperature T0 have been explored in calculations in which the values of the equation-of-state parameters (Table 1) and the composition (simplified pyrolite) are held constant. The results, presented in Figs 9(c) and (d), reveal a temperature sensitivity of density at 65 GPa of about 0.5 per cent per 350 K, consistent with a mid-mantle thermal expansivity of about 1.4×10−5 K−1. At the same pressure, the seismic parameter w changes by about 0.7 per cent for a 350 K increase in T ; this value of (dw/w)/dT, namely −2×10−5 K−1 is consistent, through the identity ∂ ∂ = − ( ln w/ T)P a(1 dS), (32) with a value of about 2.4 for the Anderson–Gru¨neisen parameter dS (eq. 17). The sensitivity of the density and seismic parameter to variation of the composition and mineralogy of the lower mantle is demonstrated in Figs 9(e) and (f ). Here, both T0 (1600 K) and the equation-of-state parameters (Table 1) are held constant, while the composition is varied. The properties = = of the perovskite (XMg 0.92) and magnesiowu¨stite (XMg 0.83) phases of the simplified pyrolite model (continuous lines in Figs 9e and f ) indicate that perovskite of this composition by itself is too high in w, and magnesiowu¨stite alone too dense and too low in w to match the seismological data. These curves for the individual phases are displaced to higher density and lower w as XMg is decreased; however, since K is assumed to be independent of composition for the perovskite phase (Table 1), large changes in composition (and/or temperature— see Figs 9c and d) would be needed to eliminate the mismatch Figure 10. The density (a) and seismic parameter (b), calculated in w between the perovskite-only models and PREM (see along selected high-temperature isentropes for (Mg0.88,Fe0.12)SiO3 discussion below). perovskite with the equation-of-state parameters of column 4, Table 3. Admixture of a substantial proportion of the much more These properties are plotted as residuals relative to the PREM model compressible magnesiowu¨stite phase is an attractive alternative. of Dziewonski & Anderson (1981).

© 1998 RAS, GJI 134, 291–311 L ower-mantle elasticity, composition and temperature 307 with this composition for T0 near 2300 K. The variations of relatively well-constrained adiabat for the upper mantle and temperature along the respective adiabats for the best-fitting transition zone presents a major difficulty, as follows. A large = pyrolite (T0 1600 K) and (Mg0.88,Fe0.12)SiO3 perovskite temperature increment (about 800 K for a lower mantle of = (T0 2300 K) models are compared in Fig. 11. Temperature pyroxene stoichiometry) would have to be supported con- increases along the pyrolite 1600 K adiabat from about 1850 K ductively across a pair of thermal boundary layers, presumably at the 660 km discontinuity, consistent with its interpretation several hundred kilometres in total thickness, bounding as the ringwoodite (spinel) perovskite+magnesiowu¨stite separately convecting regions above and below. Given that the phase boundary (Ito & Katsura 1989; Akaogi et al. 1989), to discontinuities and gradients in bulk sound speed through about 2300 K at the core–mantle boundary (the effect of a the transition zone are adequately explained by phase trans- thermal boundary layer within the D◊ layer at the base of the formations in a pyrolite model mantle (Ita & Stixrude 1992; mantle is not included in this analysis). The corresponding Jackson & Rigden 1998), any compositional/thermal boundary temperatures for the (Mg0.88,Fe0.12)SiO3 perovskite 2300 K would have to be located deeper, i.e. within the lower mantle. In adiabat are about 2600 K at 660 km depth and ~3400 K at this scenario, the regions above and below the compositional/ the base of the mantle. thermal boundary layers would exhibit comparable values of It is apparent from the results presented in Figs 8, 9 and 10 density, bulk modulus and seismic parameter, albeit along that consistent results emerge from analyses in which (1) the adiabats substantially offset in temperature. Within the boundary lower mantle is adiabatically decompressed for comparison layers it would be necessary for the thermal and compositional with laboratory thermoelastic data, and (2) the laboratory gradients to be very closely matched if the boundary were Downloaded from https://academic.oup.com/gji/article/134/1/291/632609 by guest on 01 October 2021 data are projected to the P–T conditions of the lower mantle. to be seismologically unobservable as suggested by Jeanloz This encouraging result is in part a consequence of the fact (1991). Moreover, given that at constant composition and that Eulerian isentropes faithfully describe the V(P)S calcu- phase, viscosity is expected to decrease by a factor of about 4 lated from a Eulerian principal isotherm through the Mie– per 100 K increase in temperature, it would be most fortuitous Gru¨neisen prescription for the thermal pressure (Section 3 indeed if a conductively supported temperature increment of above). A satisfactory description of the variation of w and r even a few hundred degrees could be reconciled with the throughout the lower mantle is obtained with the simplified evidence for a modest increase in viscosity (about 30-fold) pyrolite composition, and within the framework provided by across the depth of the mantle (e.g. Lambeck & Johnston the preferred equation of state (eq. 11), with the perovskite 1998). The lack of seismological and rheological evidence for equation-of-state parameters of column 4, Table 3, i.e. K∞=3.8 the thermal boundary layers implied by layered convection is = = ffi q 2 and c0 1.41. Perovskite alone, with the same thermo- just one of many di culties faced by layered models of mantle elastic parameters, and the composition (Mg0.88,Fe0.12)SiO3 convection (Davies & Richards 1992; Davies 1998). favoured by Stixrude et al. (1992), also provides a good match to the radial variation of w and r, but only with the much 6.2 Comparison with previous studies higher temperatures encountered along the 2300 K adiabat. In principle, these alternative composition–temperature models The results of this analysis are broadly consistent with those provide equally viable explanations of the variation of seismic of previous investigations in which relatively low values of a parameter and density within the lower mantle. and normal values of ∂K/∂T for the perovskite phase have However, reconciliation of the high potential temperatures been employed (e.g. Jackson 1983; Bukowinski & Wolf for models substantially more silicic than pyrolite with the 1990; Wang et al. 1994; Zhao & Anderson 1994; Bina & Silver 1997). In marked contrast, analyses based on very high average expansivities (>4×10−5 K−1) and abnormally large |∂K/∂T |~0.05 GPa K−1 for perovskite (Jeanloz & Knittle 1989; Stixrude et al. 1992) have consistently favoured strong silica enrichment of the lower mantle (see also Zhao & Anderson 1994; Bina & Silver 1997). Representative of this group is the work of Stixrude et al. (1992), whose approach has been followed here, with the important difference that their values of the parameters H, c0 and q were constrained by the P–V –T data for (Mg0.9Fe0.1)SiO3 perovskite (Knittle, Jeanloz & Smith 1986; Mao et al. 1991) rather than the superior data which have subsequently become available for the MgSiO3 end-member. The need to reconcile the unusually strongly temperature-dependent thermal expansivity of Knittle et al. (1986; see also Hill & Jackson 1990) with the much lower expansivities measured at high pressure (Mao et al. 1991) led Stixrude et al. (1992) and other analysts (e.g. Bina 1995) to exotic values for many of the thermoelastic parameters. The combination of higher-than-usual values for both c0 (1.96) and q (2.5) ensures high and very strongly temperature-dependent × −5 −1 × −5 −1 Figure 11. The variation of temperature along the 1600 K adiabat expansivity (2.3 10 K at 300 K to 7.0 10 K at for the simplified pyrolite composition, and along the 2300 K adiabat 2000 K!) and unusually strong temperature sensitivity of for (Mg0.88,Fe0.12)SiO3 perovskite, calculated with the perovskite the bulk modulus. With these equation-of-state parameters, ∂ ∂ − −1 equation-of-state parameters of column 4, Table 3. ( KT/ T)P increases in magnitude from 0.043 GPa K at

300 K to −0.077 GPa K−1 at 2000 K. Between 500 and perovskite+magnesiowu¨stite mixtures at STP. For the purpose ∂ ∂ = 2000 K, ( P/ T)V aKT increases by more than 30 per cent, of this calculation, compositions of the coexisting perovskite from 8.3×10−3 to 11.1×10−3 GPa K−1, in marked contrast and magnesiowu¨stite phases appropriate for the simplified with the common observation that aKT is almost independent pyrolite bulk composition are chosen. The elastic moduli K of temperature for T >H/2 (Anderson 1984). and G for magnesiowu¨stite are assumed to vary linearly with The use of this physically implausible combination of values composition between those appropriate for stoichiometric ff for c0 and q inevitably results in substantially di erent con- end-members. In the absence of experimental data, the elastic clusions concerning the composition and temperature of the moduli for perovskite are assumed to be independent of lower mantle. The density and seismic parameter calculated composition. The interesting question is whether, and for = = = for the model (XPv 1, XMg 0.88, T0 1660 K) preferred by which compositions, the properties of the hot decompressed Stixrude et al. (1992) provide a good match to the PREM lower mantle can be reconciled with those of the two-phase model (Jackson & Rigden 1998, Figs 9a and b), with temper- mixture at STP through appropriate combinations of T0 and atures increasing along the adiabat from 2000 K at the 660 km temperature sensitivity of the elastic moduli. discontinuity to about 2750 K near the base of the mantle Both K and G for the hot decompressed lower mantle are without recourse to the much higher temperatures inferred substantially smaller than those for relatively perovskite-rich above. mixtures at 300 K, indicating the need for relatively large thermal corrections. The horizontal and vertical lines indicate the magnitudes of the upward corrections to K and G respectively Downloaded from https://academic.oup.com/gji/article/134/1/291/632609 by guest on 01 October 2021 7 THE SHEAR MODULUS OF A which are required for a downward adjustment to 300 K from PEROVSKITE-RICH LOWER MANTLE a potential temperature of 1600 K. The correction to K has Since little is as yet known about the pressure and temperature been calculated from the equation-of-state parameters of dependence of the shear modulus for (Mg, Fe)SiO3 perovskite, Table 1 as modified in column 4, Table 3. The adjustment to interpretation of the shear-mode properties of the lower mantle G, on the other hand, based on the values of |∂G/∂T | for MgO is necessarily less secure. The state of the art is illustrated in and olivine as plausible upper and lower bounds on the high- Fig. 12, in which the bulk and shear moduli for the adiabatically frequency (anharmonic) derivative for the two-phase mixture, decompressed lower mantle (Table 4) at the unknown potential is largely indicative. It is concluded that an average aggregate ∂ ∂ − −1 temperature T0 are compared with those calculated for value of G/ T near 0.022 GPa K , near the upper end of the plausible range, is required in order to reconcile the experimentally determined shear moduli for perovskite and magnesiowu¨stite with that for a lower mantle of pyrolite composition (containing ~20 volume per cent magnesiowu¨stite) at a potential temperature of 1600 K. For more silicic compositions, higher potential temperatures T0 and/or larger values of |∂G/∂T | are clearly required. For example, Zhao & Anderson (1994) showed that the unusually large (in magnitude) value of ∂G/∂T =−0.035 GPa K−1 admits a wide range of perovskite-rich compositions. Wang et al. (1994) have ∂ ∂ calculated the temperature-averaged values of ( KS/ T)P and ∂ ∂ ( G/ T)P needed to satisfy the radial (PREM) and lateral ∂ ∂ = variation [( ln VS/ ln VP)P 1.8–2.0] of seismic wave speeds, for compositional models of varying silica content. Two distinct trends emerged from the analyses of the radial and lateral variability, and these were found to intersect at a <| ∂ ∂ |< composition near pyrolite with 0.015 ( KS/ T ) 0.020 and Figure 12. A comparison of the elastic moduli of the (hot) adiabatically <| ∂ ∂ |< −1 0.020 ( G/ T)P 0.035 GPa K , in general agreement decompressed lower mantle with those expected for (Mg, Fe)SiO3 with the analysis presented here. perovskite+magnesiowu¨stite mixtures. The compositions of the individual phases are appropriate for the pyrolite bulk composition through eq. (30). The bold line represents the 300 K Hashin–Shtrikman 8 CONCLUSIONS moduli for two-phase aggregates with the indicated proportions of the constituent phases. The properties of the decompressed PREM lower 8.1 Adiabatic decompression of the lower mantle mantle at the unknown potential temperature T0 are indicated by the triangular plotting symbols, whilst the filled circles correspond to (1) In a departure from most previous studies, internally temperature-corrected values, which are to be compared directly with consistent expressions (eqs 3 and 4) have been used to describe the 300 K laboratory data. The thermal correction to the bulk modulus the dependence upon the Eulerian finite strain e of the pressure for the indicative 1300 K change in temperature is calculated as in and bulk modulus. Section 5, for the simplified pyrolite composition and the equation- (2) Reduction of the covariance amongst the fitted of-state parameters of Table 1 as modified in column 4, Table 3 (the ∞ ∂ ∂ − −1 parameters (r0,K0,K0, etc.) and appropriate weighting of the adjustment amounts to an average KS/ T of 0.022 GPa K ). In the absence of direct constraints on ∂G/∂T for the silicate perovskite most robust seismological observations (wave speeds rather than phase, the shear modulus corrections are based on the corresponding density and pressure) are achieved through the simultaneous derivatives for MgO (−0.025 GPa K−1) and olivine (−0.015 GPa K−1), fitting of w(e) and P(e). which are regarded as plausible bounds on the anharmonic derivative (3) Third-order Eulerian finite-strain isotherms and isentropes for the aggregate. appear to be adequate for the range of strains encountered in

© 1998 RAS, GJI 134, 291–311 L ower-mantle elasticity, composition and temperature 309 the lower mantle; caution should be exercised in fitting the perature of 1600 K and perovskite thermoelastic parameters more flexible fourth-order equations of state, especially to K∞=4 and q=1 provides a reasonable match to the lower- unsmoothed models. mantle density, but significantly overestimates its incom- (4) Shear-mode properties for the decompressed lower pressibility and seismic parameter. A wholly satisfactory fit is ∞ ◊ ∞ mantle (G0,G0,G0, etc.) are readily associated with the bulk obtained with a somewhat lower value (3.8) of K and a properties derived by finite-strain adiabatic decompression substantially higher value (2) of q for the perovskite phase. through Stacey’s (1995) observation that G varies linearly with Reconciliation of the shear properties of the lower mantle with K and P (eq. 28). this mineralogy and temperature requires a relatively large (5) The properties thus obtained for the PREM lower mantle value of |∂G/∂T | of 0.022 GPa K−1 for the two-phase mixture. at the high-temperature foot of the adiabat are as follows: (10) There remains, in principle, a significant trade-off = −3 = = ∞ = r0 3.985 g cm , K0 212.5 GPa, G0 134.1 GPa, K0 3.89, between composition and temperature. Thus more silicic com- ∞ = G0 1.56. Allowance for an estimated 2 per cent relaxation positions, sometimes advocated for the lower mantle, can also (from the Hill average KH to the Reuss lower bound KR) provide a satisfactory match to the density and bulk modulus of the bulk modulus which governs the radial variation of or seismic parameter, but only in combination with much higher density yields an alternative fit at third order to w(e) and P(e), temperatures (by about 700 K for pyroxene stoichiometry). In = −3 = = with r0 3.974 g cm , KH0 210.3 GPa, KR0 206.2 GPa, this scenario, the lower mantle would be separated from the G =132.7 GPa, K =3.88, K =3.99 and G∞ =1.55. Unlike overlying upper mantle and transition zone by a prominent

0 H∞0 R∞0 0 Downloaded from https://academic.oup.com/gji/article/134/1/291/632609 by guest on 01 October 2021 PREM, earth model ak135 of Montagner & Kennett (1996) pair of thermal and compositional boundary layers. A major departs significantly from the Adams–Williamson relation, difficulty with such models is that there is no evidence of apparently reopening the possibility of substantial superadiabatic the expected impact of these boundary layers in either the temperature gradients. seismological or the rheological stratification of the mantle. (11) Under these circumstances, the simplest possible model, that of grossly uniform chemical composition throughout 8.2 Preferred equation of state and residual uncertainties and a 1600 K adiabat, continues to provide an adequate in thermoelastic properties of the perovskite and explanation of the observations, within the residual uncertainties magnesiowu¨ stite phases in the thermoelastic properties of the silicate perovskite phase. (6) The preferred P–V –T equation of state combines the Specifically, it is required that K∞~3.8, q~2 for the perovskite Mie–Gru¨neisen–Debye description of the thermal pressure phase and that ∂G/∂T ~−0.022 GPa K−1 for the lower-mantle with the third-order Eulerian finite-strain 300 K isotherm. assemblage. Acoustic studies nearing completion on the close This simple and parametrically economical equation of state structural analogue ScAlO3 (Kung et al. 1996; Kung & Rigden embodies much of the essential physics responsible for the 1997) will soon provide additional insight into the pressure and temperature dependence of specific heat, bulk modulus and temperature derivatives of the elastic moduli for (Mg, Fe)SiO3 thermal expansivity. The completeness of this formalism perovskite. facilitates straightforward and internally consistent conversion (12) For the preferred model of grossly uniform chemical = between isothermal and adiabatic conditions. Another important composition and T0 1600 K, the temperature increases along and apparently rather general feature of this equation of state the adiabat from ~1850 K at the 660 km seismic discontinuity ∞ = ∂ ∂ ~ ◊ is that KSS ( KS/ P)S is much less temperature-dependent to 2300 K near the base of the mantle (above D ). ∞ = ∂ ∂ than KTT ( KT/ P)T. Consequently, thermal corrections to K∞ between 300 K and the potential temperature T for the SS 0 ACKNOWLEDGMENTS mantle adiabat can reasonably be neglected. (7) The moduli K and G are well constrained by acoustic Correspondence with Frank Stacey, discussions with Sue studies for both the silicate perovskite and magnesiowu¨stite Kesson and Brian Kennett, and the suggestions of the reviewers phases, as are the corresponding pressure and temperature are gratefully acknowledged. derivatives for MgO. For MgSiO3 perovskite, recent X-ray P–V –T studies constrain the average value of c for volumes ~ NOTE ADDED IN PROOF V/V0 in the range 0.9–1.0, but not c0 and q separately. ∂K/∂T for the silicate perovskite is accordingly still highly Very recently, Poirier and Tarantola (A logarithmic equation uncertain (~50 per cent); for MgO, acoustic determination of of state, Phys. Earth planet. Inter., in press) have drawn K(T ) tightly constrains q, resolving this ambiguity. Values for attention to an alternative finite-strain equation of state, based the parameters q and K∞ within the ranges 1–2 and 3.8–4.0 ∞ on the natural strain. For the same values of r0, K0 and K0, are readily reconciled with the experimental data for MgSiO3 this alternative ‘logarithmic’ equation of state yields values of perovskite. K at lowermost mantle pressures about 6 per cent lower than for the Birch–Murnaghan equation of state employed in the 8.3 Composition and temperature of the lower mantle present study. This is a timely reminder that there remain considerable uncertainties associated with the choice of the (8) Consistent interpretations emerge from analyses of isothermal equation of state. the properties of the adiabatically decompressed lower mantle and from those in which laboratory data are projected to lower-mantle P–T conditions for direct comparison with REFERENCES seismological models. Akaogi, M. & Ito, E., 1993. Heat capacity of MgSiO3 perovskite, (9) A simplified (SiO2–MgO–FeO) pyrolite model with Geophys. Res. L ett., 20, 105–108. = = molar XPv 0.67 and XMg 0.89 along with a potential tem- Akaogi, M., Ito, E. & Navrotsky, A., 1989. Olivine-modified spinel–

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Inter., 89, 219–245. to the following expression for the perturbation of the fit X(e): Stacey, F.D., 1997. Bullen’s seismological homogeneity parameter, g, 3 applied to a mixture of minerals: the case for the lower mantle, = − n ∑ i − + dX(e) (1/3)(1 2e) G (e /i!)[3dbi 2(n i)bidr0/r0] Phys. Earth planet. Inter., 99, 189–193. i=0 Stacey, F.D., 1998. Thermoelasticity of a mineral composite and a 2 reconsideration of lower mantle properties, Phys. Earth planet. Inter., + ∑ i (dr0/r0) bi+1(e /i!)H . (A3) in press. i=0 Stixrude, L. & Bukowinski, M.S.T., 1990. Fundamental thermodynamic relations and silicate melting with implications for the constitution Regrouping of terms leads to the following expressions for the of D◊, J. geophys. Res., 95, 19 311–19 325. coefficients of the various powers of e within the braces: Stixrude, L., Hemley, R.J., Fei, Y. & Mao, H.K., 1992. Thermoelasticity 0 + − of silicate perovskite and magnesiowu¨stite and stratification of the e :3db0 [b1 2nb0]dr0/r0 , Earth’s mantle, Science, 257, 1099–1101. e1:3db +[b −2(n+1)b ]dr /r , Sumino, Y., Anderson, O.L. & Suzuki, I., 1983. Temperature coefficients 1 2 1 0 0 2 + − + of elastic constants of single crystal MgO between 80 and 1300 K, e :3db2 [b3 2(n 2)b2]dr0/r0 , Phys. Chem. Minerals, 9, 38–47. 3 − + Suzuki, I., 1975. Thermal expansion of periclase and olivine and their e :3db3 2(n 3)b3dr0/r0 . (A4) anharmonic properties, J. Phys. Earth, 23, 145–149. If the b (i=0, 3) are all independent, each of these expressions Utsumi, W., Funamori, N., Yagi, T., Ito, E., Kikegawa, T. & i Shimomura, O., 1995. Thermal expansivity of MgSiO perovskite may be set equal to zero through appropriate choice of the 3 ffi under high pressures up to 20 GPa, Geophys. Res. L ett., 22, perturbations dbi to the fitted coe cients. Thus although the ffi 1005–1008. individual coe cients vary with the perturbation in r0, the fit Wang, Y., Weidner, D.J., Liebermann, R.C. & Zhao, Y., 1994. P–V–T evaluated at arbitrary e is unchanged and so, therefore, is equation of state of (Mg, Fe)SiO3 perovskite: constraints on com- the misfit. position of the lower mantle, Phys. Earth planet. Inter., 83, 13–40. However, this general situation does not apply directly to Watanabe, H., 1982. Thermochemical properties of synthetic high- either the P(e) or the w(e) fit. It is clear from eq. (3) that for pressure compounds relevant to the Earth’s mantle, in High = ffi P(e), b0 0. The coe cient of e accordingly becomes b1dr0/r0, Pressure Research in Geophysics, pp. 441–464, eds Akimoto, S. & so that the fit is unavoidably perturbed significantly by Manghnani, M.H., Center for Academic Publications, Tokyo. variation of r . The rms misfit therefore varies with r as Watt, J.P., 1988. Elastic properties of polycrystalline materials: com- 0 0 parison of theory and experiment, Phys. Chem. Minerals, 15, seen in Fig. 1. For w(e) given by eq. (12), the bi are not all 579–587. independent. The consequence is that the four expressions Watt, J.P. & O’Connell, R.J., 1980. An experimental investigation of given as (A4) above cannot all be simultaneously set equal the Hashin-Shtrikman bounds on two-phase aggregate elastic to zero. Minor perturbations to the fit, and (very limited) properties, Phys. Earth planet. Inter., 21, 359–370. sensitivity of the misfit to variation of r0 are the result.