Читать книгу Mantle Convection and Surface Expressions - Группа авторов - Страница 68

Finite‐strain theory provides a compact yet flexible framework for the computation of elastic properties of minerals and rocks at pressures and temperatures of Earth’s mantle (Birch, 1947; Davies, 1974; Stixrude & Lithgow‐Bertelloni, 2005). This framework is constantly being filled with new elasticity data on mantle minerals as obtained from high‐pressure experiments and quantum‐mechanical computations. Experimental measurements of elastic properties at simultaneously high pressures and high temperatures, however, remain challenging. Propagation velocities of ultrasonic waves can now be determined for samples being compressed and heated in multi‐anvil presses to pressures and temperatures corresponding to conditions of the uppermost lower mantle (Chantel et al., 2012; Gréaux et al., 2019, 2016), and laser‐heated diamond anvil cells (DAC) are capable of creating pressures and temperatures as they prevail throughout the entire mantle. Thermal gradients across samples heated in DACs by infrared lasers, however, are not necessarily compatible with requirements imposed by common light or X‐ray scattering techniques used to measure sound wave velocities. First successful measurements of sound wave velocities on samples being simultaneously laser‐heated and compressed in DACs promise future progress in this direction (Murakami et al., 2012; Zhang & Bass, 2016). Full elastic stiffness tensors are needed to assess elastic anisotropy and to constrain rigorous bounds on average elastic moduli. For lower‐mantle minerals, measurements of full elastic stiffness tensors at relevant pressures are limited to ferropericlase (Antonangeli et al., 2011; Crowhurst et al., 2008; Finkelstein et al., 2018; Marquardt et al., 2009b, 2009c; Yang et al., 2016, 2015), bridgmanite (Fu et al., 2019; Kurnosov et al., 2017), and SiO₂ polymorphs (Zhang et al., 2021).

While density functional theory (DFT) computations are more flexible than experiments in terms of addressing extreme pressure–temperature combinations, they can only be as accurate as their underlying approximations such as the local density approximation (LDA) and generalized gradient approximations (GGA) for the electron density distribution and the quasi‐harmonic approximation (QHA) for the vibrational structure. Discrepancies with experimental results have been observed for Fe‐bearing compositions and reflect challenges in treating the localized d electrons of transition metal cations with LDA and GGA. Important developments in the study of mantle minerals with DFT computations include accounting for d electron interactions in terms of the Hubbard parameter U (Stackhouse et al., 2010; Tsuchiya et al., 2006) and bypassing the QHA with ab initio molecular dynamics (Oganov et al., 2001; Stackhouse et al., 2005b) or density functional perturbation theory (Giura et al., 2019; Oganov & Dorogokupets, 2004). As the full potential of these and other improvements is being explored, future progress in reducing discrepancies between the results of experiments and DFT computations can be expected.

A systematic analysis of the sensitivity of computed elastic wave velocities to individual finite‐strain parameters reveals uncertainties on parameters that capture the quasi‐harmonic contribution to elastic properties as a main source of uncertainty. Reported uncertainties on Grüneisen parameters and their strain derivatives propagate to relative uncertainties of several percent on elastic wave velocities for realistic pressures and temperatures of Earth’s mantle. While measurements and computations of elastic properties at combined high pressures and high temperatures will certainly help to reduce this source of uncertainty, consequent and systematic analyses of cross‐correlations between finite‐strain and quasi‐harmonic parameters can avoid overestimating uncertainties by accounting for these correlations when propagating uncertainties. The analysis of parameter correlations, however, requires consistent data sets that include data both at high pressures and at high temperatures and ideally at combinations of both, again pointing out the need to perform experiments at combinations of high pressures and high temperatures. P‐ and S‐wave velocities computed for isotropic polycrystalline aggregates of anisotropic minerals can differ by several percent when using either the Voigt or the Reuss bound. These bounds provide the extreme values for the elastic response of a polycrystalline aggregate of randomly oriented grains and can only be evaluated when full elastic stiffness tensors are available for the respective minerals and at the pressures and temperatures of interest.

Computing elastic wave velocities for realistic bulk rock compositions requires accurate descriptions of elastic properties across solid solutions of mantle minerals. In contrast to the elastic properties of mantle minerals with compositions spanned by binary solid solutions, i.e., olivine and ferropericlase, the change of elastic properties with chemical composition is not well constrained for mantle minerals that form complex solid solutions between several end member compositions, such as pyroxenes, garnets, bridgmanite, and the calcium ferrite‐type aluminous (CF) phase. High‐pressure experimental data on bridgmanite, for example, barely span bridgmanite compositions observed in experiments on peridotitic bulk rock compositions. While DFT computations have addressed wider compositional ranges (Shukla et al., 2016, 2015; Zhang et al., 2016), the results are not always consistent with those of direct measurements or are limited to specific combinations of pressure and temperature, making it difficult to derive finite‐strain parameters. Inter‐ and extrapolations of elastic properties between different mineral compositions and beyond compositional limits imposed by available elasticity data are subject to uncertainties that arise from averaging over elastic properties of different mineral compositions and from assumptions about how elastic properties vary across solid solutions. Experiments on peridotitic and basaltic bulk rock compositions at pressures and temperatures of the lower mantle show substantial spread in observed mineral compositions and, as a consequence, in derived element exchange or partition coefficients. Despite progress in deriving trends for element partitioning with pressure, temperature, and compositional parameters for specific bulk compositions (Hyung et al., 2016; Irifune et al., 2010; Nakajima et al., 2012; Piet et al., 2016; Xu et al., 2017), it remains difficult to assess to which extent experimentally observed mineral compositions are representative of chemical equilibrium. While capable of reproducing pressures and temperatures of the lower mantle, laser‐heating experiments in diamond anvil cells often give rise to strong thermal gradients across the sample that can drive disparate diffusion of chemical elements and bias mineral compositions (Andrault and Fiquet, 2001; Sinmyo and Hirose, 2010). In comparison to typical sample sizes, thermal gradients tend to be less pronounced for experiments in multi‐anvil presses. Recent progress in multi‐anvil press technology has extended achievable pressures to those of the lowermost mantle (Ishii et al., 2019; Yamazaki et al., 2019) and promises to facilitate experiments that better constrain equilibrium mineral compositions at pressures and temperatures of the lower mantle. When modeling seismic properties for a given bulk rock composition, uncertainties on element partitioning and on other compositional variables translate into uncertainties on the chemical compositions of individual mineral phases and hence on computed elastic properties and wave speeds.

Several major mantle minerals go through continuous phase transitions at pressures and temperatures relevant to Earth’s lower mantle. Continuous phase transitions include ferroelastic phase transitions between polymorphs of calcium silicate perovskite (Shim et al., 2002; Stixrude et al., 2007) and between stishovite and CaCl₂‐type SiO₂ (Andrault et al., 1998; Carpenter et al., 2000; Karki et al., 1997b; Lakshtanov et al., 2007) as well as continuous changes in the electronic configurations of ferrous and ferric iron, i.e., spin transitions (Badro, 2014; Lin et al., 2013). Both ferroelastic phase transitions and spin transitions are associated with substantial softening of elastic moduli that is not commonly accounted for in mineral‐physical models. Anomalies in elastic properties that arise from ferroelastic phase transitions can be described in terms of excess properties using Landau theory (Carpenter and Salje, 1998) and integrated into finite‐strain formalisms (Buchen et al., 2018a; Tröster et al., 2014, 2002). To model elastic softening due to spin transitions, a similar approach can be adopted by formulating excess properties that arise from compression‐ and temperature‐induced changes in the distribution of d electrons of iron cations over multi‐electron states. Crystal‐field theory provides the basis for a semi‐empirical formalism that can be used to describe elastic anomalies associated with spin transitions of Fe²⁺ in ferropericlase and of Fe³⁺ in bridgmanite and in the CF phase at ambient temperature. In the absence of direct measurements of elastic softening at combined high pressures and high temperatures, predictions and extrapolations of elastic properties to realistic mantle temperatures based on Landau theory, crystal‐field theory, or DFT computations need to be made with caution.

Integrating elastic anomalies caused by continuous phase transitions into forward models for elastic properties of rocks in Earth’s lower mantle leads to discernible perturbations of P‐ and S‐wave velocities. While the broad spin transition of Fe²⁺ in ferropericlase reduces the computed P‐wave velocities of peridotitic rocks throughout the entire lower mantle, the spin transition of ferric iron in very Fe³⁺‐rich bridgmanite would mostly affect P‐wave velocities in the upper half of the lower mantle. In addition to their direct impact on P‐wave velocities, spin transitions are expected to alter the Fe‐Mg exchange between ferropericlase and bridgmanite (Badro, 2014; Xu et al., 2017) that appears to exert a strong leverage especially on S‐wave velocities. The potential compositional effect of spin transitions on seismic properties could have implications for the interpretations of large low‐velocity provinces (LLVP). A comparison of the overall seismic signature of peridotitic rocks with the preliminary reference Earth model (PREM; Dziewonski & Anderson, 1981) reveals trade‐offs between feasible interpretations in terms of the chemical and thermal structure of the lower mantle. Despite the low volume fraction of free silica phases found in experiments on basaltic bulk rock compositions, the softening of the shear modulus across the ferroelastic phase transition between stishovite and CaCl₂‐type SiO₂ might be strong enough to dominate elastic wave speeds of metabasaltic rocks in the lower mantle. The resulting low elastic wave speeds could give rise to scattering of seismic waves by fragments of recycled oceanic crust and contribute to low elastic wave speeds within LLVPs.