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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 eVected 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 coeYcients. It is demon- strated that this approach can readily be adapted to include the eVects 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 seismo- logical 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 uncertain- ties 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 tempera- tures 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, © 1998 RAS 291 292 I. Jackson remains a formidable diYculty. 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. DiVerentiation 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 & HelVrich 1992; Stixrude et al. 1992) and its pressure derivatives K∞,K◊ etc., follow by further involves the construction of a third-order Eulerian finite- diVerentiation. 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 coeYcients are used to model compression along 1 0 V = − ∞ 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; DuVy& convergence of the respective expansions. Anderson 1989; Rigden et al.
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