Bullen's parameter as a seismic observable for spin crossovers in the lower mantle
Juan J. Valencia-Cardona, Quentin Williams, Gaurav Shukla, Renata M. Wentzcovitch
aa r X i v : . [ phy s i c s . g e o - ph ] J u l GEOPHYSICAL RESEARCH LETTERS, VOL. ???, XXXX, DOI:10.1002/,
Bullen’s parameter as a seismic observable for spincrossovers in the lower mantle
Juan J. Valencia-Cardona , Quentin Williams , Gaurav Shukla , Renata M.Wentzcovitch D R A F T August 30, 2018, 2:39pm D R A F T - 2
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Corresponding author: Juan J. Valencia-Cardona, Scientific Computing Program, Universityof Minnesota, Minneapolis, Minnesota, USA. ([email protected]) Scientific Computing Program,University of Minnesota, Minneapolis,Minnesota, USA Department of Earth and PlanetarySciences, University of California SantaCruz, Santa Cruz, California, USA Department of Earth, Ocean, andAtmospheric Science, Florida StateUniversity, Tallahassee, Florida, USA Department of Applied Physics andApplied Mathematics, Columbia University,New York City, New York, USA Lamont-Doherty Earth Observatory,Columbia University, Palisades, New York,USA
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Elastic anomalies produced by the spin crossover in ferropericlase have beendocumented by both first principles calculations and high pressure-temperatureexperiments. The predicted signature of this spin crossover in the lower man-tle is, however, subtle and difficult to geophysically observe within the man-tle. Indeed, global seismic anomalies associated with spin transitions havenot yet been recognized in seismologic studies of the deep mantle. A sensi-tive seismic parameter is needed to determine the presence and amplitudeof such a spin crossover signature. The effects of spin crossovers on Bullen’sparameter, η , are assessed here for a range of compositions, thermal profiles,and lateral variations in temperature within the lower mantle. Velocity anoma-lies associated with the spin crossover in ferropericlase span a depth rangenear 1,000 km for typical mantle temperatures. Positive excursions of Bullen’sparameter with a maximum amplitude of ∼ D R A F T August 30, 2018, 2:39pm D R A F T - 4
VALENCIA-CARDONA ET AL.: BULLEN’S PARAMETER AND SPIN CROSSOVERS
1. Introduction
The adiabatic nature of the convecting mantle is a frequently used concept in the geo-physical sciences. For instance, equation of state parameters, which are used to calculatethe elastic and thermodynamic properties of minerals at mantle conditions, are commonlyassumed to be adiabatic within the convecting mantle, e.g., the adiabatic bulk modulusand its derivative. However, various geodynamic simulations and seismological models[
Bunge et al. , 2001;
Dziewonski and Anderson , 1981;
Kennett et al. , 1995;
Mattern et al. ,2005;
Matyska and Yuen , 2000, 2002] suggest that the mantle is regionally nonadiabatic,particularly in the shallow and deep mantle regions, and in some cases, at mid lower mantlepressures. The latter is important because deviations from adiabaticity within the mantleprovide insights into temperature gradients, heat flux, thermal history, thermal bound-ary layers, phase transitions, chemical stratification, and compositional heterogeneities.Therefore, knowledge about the degree of adiabaticity of the mantle helps us to constrainits composition and thermal structures related to mantle convection [
Matyska and Yuen ,2000].A common observable that quantifies the adiabaticity level of the mantle is Bullen’sparameter, η . Introduced and developed by Bullen [1949, 1963], η is a measure of theratio between the actual density increase with pressure within the Earth (as constrained bya combination of seismology, the Earth’s moment of inertia, and mass) with respect to theprofile derived from adiabatic self-compression. As such, it is expected to be unity wherethe mantle is homogeneous, adiabatic, and free of phase transitions. Thus, deviationsof η from unity (generally ∼ ± D R A F T August 30, 2018, 2:39pm D R A F T
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X - 5 consequently, the presence of thermal boundary layers, compositional variations or phasetransitions. Moreover, there is also the possibility that due to internal heating within themantle, the mantle may be systematically subadiabatic [
Bunge et al. , 2001].Evaluations of η in geodynamic simulations are generally done by probing the parameterspace associated with plausible convection models. This includes examining the effects ofpossible variations of the thermal conductivity, thermal expansion coefficient, viscosity,internal heating, and heat flux from the core, each of which directly impact the inferredgeotherms [ Bunge et al. , 2001;
Matyska and Yuen , 2000, 2002]. For instance, if internalheating is relatively significant, subadiabaticity is expected. Additionally, differences inelastic properties between individual phases within an aggregate can also produce varia-tions in Bullen’s parameter, and hence apparent deviations from adiabaticity. This is abulk attenuation effect. Specifically, bulk attenuation phenomena are attributed to inter-nal shear stresses generated from the local mismatch of the elastic moduli of neighboringgrains in a given aggregate. One formulation of bulk attenuation by
Heinz et al. [1982]characterizes it through the ratio of the adiabatic bulk modulus K S and an effective modu-lus (Reuss bound) K E , since the mantle can be assumed to be under hydrostatic pressure.Attenuation is a complicated problem to tackle, because it involves calculating complexmoduli with an associated time dependency [ Heinz et al. , 1982;
Heinz and Jeanloz , 1983;
Budiansky et al. , 1983]. Such bulk attenuation effects are beyond the scope of this study,since the calculations we conduct are not time-dependent, but certainly needs to be ad-dressed to understand systematic deviations of Bullen’s parameter from 1. Here, we study
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VALENCIA-CARDONA ET AL.: BULLEN’S PARAMETER AND SPIN CROSSOVERS how anomalies in bulk modulus induced by spin crossovers affect the Bullen’s parameter,and hence inferred adiabaticity of the lower mantle.Elastic anomalies produced by the spin crossover in ferropericlase (fp) and bridgmanite(bdg), have been documented by both first principles calculations and high pressure-temperature experiments [
Badro et al. , 2003;
Speziale et al. , 2005;
Tsuchiya et al. , 2006;
Wentzcovitch et al. , 2009;
Wu et al. , 2009;
Crowhurst et al. , 2008;
Marquardt et al. , 2009;
Antonangeli et al. , 2011;
Murakami et al. , 2012;
Wu et al. , 2013;
Wu and Wentzcovitch ,2014;
Hsu et al. , 2011, 2012;
Shukla et al. , 2016;
Shukla and Wentzcovitch , 2016]. Thepredicted signatures of this spin crossover in the lower mantle are subtle. Despite thefact that thermally induced velocity heterogeneities associated with this spin crossoverappear to correlate statistically with seismic tomographic patterns observed in deeplyrooted plumes [
Wu and Wentzcovitch , 2014], spherically averaged anomalies have not yetbeen recognized in seismologic studies of the deep mantle. This may be due to difficultiesassociated with resolving gradual changes in the slopes of seismic velocities as a functionof depth, and the trade-offs involved in seismic inversions of depth-dependent velocity anddensity structures. In particular, velocity anomalies associated with the spin crossover infp are anticipated to span a depth range greater than 1000 km at mantle temperatures.Thus, a sensitive seismic parameter is needed to determine the presence or absence ofthis spin crossover signature, which would in turn shed light on the amount of ferroperi-clase in the lower mantle. Bullen’s parameter η is an ideal candidate as it relates seismicwave speeds with density variations, and sensitively records deviations from adiabaticity.Moreover, deviations from Bullen’s parameter can be readily identified because it has a D R A F T August 30, 2018, 2:39pm D R A F T
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X - 7 clear reference value (unity) in an adiabatic mantle that is heated from below. We calcu-lated one dimensional perturbations of η due to changes in composition, temperature, andspin crossover. We achieved this by computing η of different relevant mantle aggregatesalong their own adiabats. We also approximate lateral variations in temperature by mod-eling differing areas and temperature differences between upwellings and downwellings.The mantle phases of the aggregates considered are bridgmanite (bdg: Al- Fe- bearingMgSiO perovskite), CaSiO perovskite (CaPv), and ferropericlase (fp: (Mg,Fe)O). Theaggregates have Mg/Si ratios that range from 0.82 to 1.56 and are harzburgite (Mg/Si ∼ Baker and Beckett , 1999], chondrite (Mg/Si ∼ Hart and Zindler , 1986], py-rolite (Mg/Si ∼ McDonough and Sun , 1995], peridotite (Mg/Si ∼ Hirose andKushiro , 1993], and perovskite only (Mg/Si ∼ Williams and Knittle , 2005]. Thepredicted deviations in η due to the spin crossover are comparable to previously reportedvariations [ Bunge et al. , 2001;
Matyska and Yuen , 2000, 2002;
Mattern et al. , 2005], andmay be sufficiently large to turn up in accurate seismic inversions of this parameter.
2. Method and Calculation Details
We used bdg Mg − x Fe x SiO , (Mg − x Al x )(Si − x Al x )O , (Mg − x Fe x )(Si − x Al x )O ,(Mg − x Fe x )(Si − x Fe x )O ( x = 0 and 0 . − y Fe y O ( y = 0 and 0 . Shukla et al. [2015, 2016] and
Wu et al. [2013]. Results forother x and y values were obtained by linear interpolation. All compositions account forthe spin crossover in fp unless otherwise noted, i.e., bdg’s iron (ferrous and/or ferric) is inthe high spin (HS) state and fp is in a mixed spin (MS) state of HS and low spin (LS) states.For CaPv, we used thermoelastic properties from Kawai and Tsuchiya [2014, 2015], which
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VALENCIA-CARDONA ET AL.: BULLEN’S PARAMETER AND SPIN CROSSOVERS were reproduced within the Mie-Debye-Gr¨uneisen [
Stixrude and Lithgow-Bertelloni , 2005]formalism. The mantle aggregates in this study, namely, harzburgite [
Baker and Beckett ,1999], chondrite [
Hart and Zindler , 1986], pyrolite [
McDonough and Sun , 1995], peri-dotite [
Hirose and Kushiro , 1993], and perovskititic only [
Williams and Knittle , 2005],are mixtures within the SiO - MgO - CaO - FeO - Al O system (ignoring alkalis andTiO is not anticipated to resolvably affect the results). In addition, the Fe-Mg partitioncoefficient K D = x/ (1 − x − z ) y/ (1 − y ) between bdg and fp, which is known to be affected by the spincrossover [ Irifune et al. , 2010;
Piet et al. , 2016], was assumed to be uniform throughoutthe mantle with a value of 0.5. Further details about these compositions can be found in
Valencia-Cardona et al. [2017].The adiabats of the different minerals and aggregates were integrated from their adia-batic gradient, ∂T∂P ! S = αV TC p (1)We denote the molar fraction, molar volume, molar mass, thermal expansion coefficient,and isobaric specific heat of the i th mineral in the mixture as µ i , V i , M i , α i , and Cp i respectively. The aggregate properties such as volume, thermal expansion coefficient,and isobaric specific heat are then V = P i µ i V i , α = P i α i µ i V i /V , and C p = P i µ i C p i .The adiabatic aggregate bulk moduli K S were obtained from the Voigt-Reuss-Hill (VRH)average. Moreover, the aggregate density ρ = P i µ i M i /V and seismic parameter φ = K S /ρ were calculated along the aggregate adiabat, in order to compute adiabatic changesof density with respect to pressure as, D R A F T August 30, 2018, 2:39pm D R A F T
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X - 9 η = φ dρdP (2)where η is the Bullen’s parameter. If η = 1 the mantle is homogeneous and adiabatic,whereas values of η > ρ varies more rapidly with depththan predicted by the adiabat. Furthermore, values of η < D R A F T August 30, 2018, 2:39pm D R A F T - 10
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3. Results and Discussion3.1. Observations of η in the lower mantle Figure 1 shows different η calculations from previous geodynamic [ Bunge et al. , 2001;
Matyska and Yuen , 2000, 2002], seismic [
Dziewonski et al. , 1975;
Dziewonski and Ander-son , 1981;
Kennett et al. , 1995], and seismic plus mineral physics models with a prioristarting conditions [
Mattern et al. , 2005]. Overall, η oscillates between values of ∼ Kennett et al. , 1995;
Montagner and Kennett , 1996], which displays the largest fluctuations. AK135F exhib-ited an average value of η ∼ η above and below those depths were at least of the order of ∼ Dziewonski et al. , 1975] and PREM [
Dziewonski and Anderson , 1981] such large fluctu-ations are not observed, but they could be suppressed by the continuity requirements ofthe polynomial formulations of these models. However, the Bullen’s parameters of theseseismic models do suggest the presence of a thermal boundary layer at the bottom of thelower mantle, as shown by the negative slope of all models in the bottommost hundredto few hundred km of the mantle. Notably, for the mineral physics plus inverse modelcalculation by
Mattern et al. [2005], η values less than one from 800 km to 1300 km wereattributed to iron depletion from their initially pyrolitic compositional model.Two and three dimensional geodynamic calculations of η were first done by Matyska andYuen [2000, 2002], where the effect of varying parameter space properties, such as thermalconductivity, thermal expansion coefficient, and viscosity, lead to different perturbationsin η , but with an average value of ∼ D R A F T August 30, 2018, 2:39pm D R A F T
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X - 11 other geodynamic calculations by
Bunge et al. [2001], which also showed that the presenceof internal heat sources lead to subadiabatic regions. Other thermal contributions, likecore heating, cause superadiabatic temperature gradients at the bottom of the mantle andthus the presence of a thermal boundary layer, as manifested by the negative slopes of η near the base of the mantle. We studied the effect of spin crossovers on lower mantle adiabaticity by examining η excursions for different lower mantle aggregates along their self consistent adiabats. All ofthe adiabats of the different aggregates are listed in Valencia-Cardona et al. [2017]. Figure2 shows the variations of η only due to the spin crossover in fp: only a portion of trivalentiron in bdg is anticipated to undergo a spin transition within the mantle (e.g., Catalli et al. [2010] and
Hsu et al. [2011, 2012]). For compositions with fp, fluctuations in η were ∼ η anomaly whichis nearly twice that of the chondritic composition. The η excursions for the perovskiticcomposition, Pv only, depict the profile of a composition without fp in the lower mantle. We have characterized what Bullen parameter anomalies, due to spin crossovers, mightgenerate for one-dimensional seismic models of an isochemical adiabatic mantle. However,
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VALENCIA-CARDONA ET AL.: BULLEN’S PARAMETER AND SPIN CROSSOVERS the lack of maxima in most Bullen parameter observations (Figure 1) that are at theappropriate depth and have the right breadth to correspond to the spin crossover of fp,led us to probe the effect of lateral temperature variations on deviations of η . Since lateraltemperature variations and their areal distribution at a given depth of hot/upwelling andcold/downwelling material are not well-constrained in the deep mantle (e.g., Houser andWilliams [2009]), we conducted a sensitivity analysis for the effect of thermal variationson η in a pyrolitic mantle. Here, material at each depth is distributed along adiabats withpotential temperatures above (hot) and below (cold) a reference adiabat pinned at 1873K at 23 GPa as in Brown and Shankland [1981] (B&S) (See also [
Akaogi and Akimoto ,1979]). The lateral temperature variations between hot and cold regions that we probedwere ±
250 K, ±
500 K, and ±
750 K in a sequence of 25%:75%, 50%:50% and 75%:25%ratio of the mantle at a given depth being hot:cold (See Figure 3).For all the temperature-average distributions (Figures 3a, 3b, and 3c), we observed thatthe spin crossover anomalies, i.e. deviations from adiabaticity, became more prominent atlower temperatures: this is a natural consequence of the broadening of the spin transitionthat occurs at high temperatures. Conversely, greater amounts of hot material tend tomake spin crossovers more difficult to resolve. Furthermore, we also observed that forlarge temperature variations, ±
750 K, two peaks in η can also be generated at differentdepths in an isochemical thermally heterogeneous mantle (Figure 3c). This phenomenonis attributed to the volume increase with temperature, which increases the pressures thatare required for the spin crossover to occur. Since the amplitude of the perturbationsin η increases also with higher fp content, it is expected that regions with larger cold D R A F T August 30, 2018, 2:39pm D R A F T
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X - 13 harzburgitic chemistry present within the lower mantle, such as subducting slabs, shouldhave substantially greater local fluctuations in the Bullen’s parameter if a local verticalsampling of η over such regions is performed.Beyond lateral temperature variations, we examined the case of coupled compositionaland thermal lateral heterogeneities. The rationale here is that cold, downwelling sub-ducted material is likely to have a larger concentration of harzburgite than ambient man-tle. We utilized a similar temperature averaging scheme, but with cold η values beingharzburgitic. Figure 4 shows different η profiles with the mantle being 75% hot(pyrolite)and 25% cold(harzburgite). For this scenario, perturbations in η due to the spin crossovervary their magnitude and reach a maxima at different depths, depending on the tem-perature difference between the cold downwellings of harzburgitic chemistry and ambientpyrolitic mantle. If the temperature difference is sufficiently large, e.g. ±
750 K, mul-tiple peaks can be observed. Thus, the relative amplitudes and locations of multiplepeaks could, if observed/observable, provide strong constraints on lateral variations inthe geotherm and/or composition of the deep mantle. In particular, the depth at whichthe spin transition-induced peak occurs in Bullen’s parameter is highly sensitive to tem-perature (Figure 3), while the amplitude of its variation is sensitive to composition (Figure2).
4. Geophysical Significance
We have utilized η as an observable for spin crossovers in the lower mantle for the firsttime, in an attempt to reconcile mineral physics with seismic observations and to under-stand how such spin crossovers may affect observations of deviations from adiabaticity D R A F T August 30, 2018, 2:39pm D R A F T - 14
VALENCIA-CARDONA ET AL.: BULLEN’S PARAMETER AND SPIN CROSSOVERS within the mantle. Our results suggest that the spin crossover signatures in η should besufficiently large to turn up in accurate (ca. 1%) seismic inversions for this parameter.Whether such accuracies are achievable is unclear: several decades ago, Masters [1979]concluded that η variations from seismic observations could be resolved with a precisionno better than 2%. Recent results from an inverse Bayesian method, deployed via a neu-ral network technique by De Wit and Trampert [2015], showed that ρ , Vp, and Vs mayeach be resolvable to somewhat better than 1% in the ∼ η might be developed.We also highlight the importance of the chosen temperature profile, as it has a directimpact on η . Elastic moduli, seismic velocities, and aggregate densities strongly depend ontemperature. Hence, super(sub)adiabatic geotherms will lead to different interpretationsof η . As recently showed by Valencia-Cardona et al. [2017], the spin crossover in fp andbdg induces an increment in the adiabat’s temperature of a given aggregate and sucha temperature increment will impact η ’s sensitivity. Because of the potentially complexcoupling of lateral temperature differences with compositional variations, further work onthe effect of spin crossovers on η would likely benefit from an assessment within a threedimensional convective scheme, such as the formulation proposed by Matyska and Yuen [2002].
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5. Conclusions
Apparent deviations from adiabaticity due to spin crossover, as recorded by the Bullen’sparameter, increased in proportion to the aggregate’s ferropericlase content. The magni-tude of these perturbations is generally consistent with the magnitude of variations in η present in previous seismological and geodynamic inversions of η in the lower mantle. Ourresults provide a sense of how much of a perturbation in η , given the spin crossover andlateral temperature variations, might be expected in one dimensional seismic models, withthe net result being of order 1-2%. Accurate characterization of η either globally or locallycould provide constraints on both the lateral temperature distribution and the fp contentat depth, although such determinations hinge critically on achieving sufficient seismic res-olution to resolve spin transitions. Also, the perturbations found in η for different mantletemperature averages highlight the importance of doing vertical seismic velocity profileswith sufficient precision to allow η to be characterized on a regional basis. Our resultsprovide a guide for possible a priori models of η in regionalized inversions of velocity asa function of depth: inversions without spin crossover induced perturbations in η implic-itly assume that spin transitions are absent at depth, and hence that no ferropericlase ispresent in the deep mantle. Acknowledgments.
We thank two reviewers for helpful comments. This work wassupported primarily by grants NSF/EAR 1319368 and 1348066. Q. Williams was sup-ported by NSF/EAR 1620423. Results produced in this study are available in the sup-porting information. The 2016 CIDER-II program (supported by NSF/EAR 1135452) isthanked for providing a portion of the original impetus of this study.
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6. Supplementary Material
The supporting information consists of Text S1, Figure S1, and Table S1. Figure S1shows η for prystine lower mantle minerals, namely, MgSiO , MgO, and CaSiO . TextS1 shows the Bullen’s parameter η derivation. Table S1 the values of η for the differentaggregates. The bulk modulus of a mineral under adiabatic self compression is given by, K S = ρ ∂P∂ρ ! S (3)Hence, K S ρ = ∂P∂ρ ! S = φ (4)where φ is the seismic parameter φ = V P − (cid:16) (cid:17) V S .Furthermore, assuming a homogeneous media under hydrostatic changes in pressureswith respect to depth, dPdr = − ρg (5)where g and r are the acceleration due gravity and depth, respectively.Thus, dPdρ dρdr = − ρg (6) D R A F T August 30, 2018, 2:39pm D R A F T
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X - 17 and using equation (4), 1 = − φρ − g − dρdr (7)Equation (7) is known as the Adams-Williamson equation.If the system is not adiabatic, i.e., equation (7) differs from the unity, we have, η = − φρ − g − dρdr = φ dρdP (8)where η is the Bullen’s parameter [ Bullen , 1963].
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Dziewonski-Anderson, 1981 *(PREM)Dziewonski et al., 1975 * (PEM)Kennett et al., 1995 + Montagner-Kennett, 1996 * (AK135F)Mattern et al., 2005 ** Bunge et al., 2001 *** (no core heating)Bunge et al., 2001 *** (core heating)Matyska-Yuen, 2002 *** ** Mineral physics + Inverse model* Seismic model*** Geodynamic simulation
Figure 1.
Bullen’s parameter η calculations for seismic, geodynamic, and mineralphysics models. D R A F T August 30, 2018, 2:39pm D R A F T - 24
VALENCIA-CARDONA ET AL.: BULLEN’S PARAMETER AND SPIN CROSSOVERS
Pressure (GPa)
Pyrolite Mg/Si: 1.24Chondrite Mg/Si: 1.07Harzburgite Mg/Si: 1.56Pv Only Mg/Si: 0.82Peridotite Mg/Si: 1.30
Depth (km)
Figure 2.
Perturbations of η due spin crossover in fp in lower mantle aggregates. D R A F T August 30, 2018, 2:39pm D R A F T
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Pressure (GPa)
20 40 60 80 100 120
Pressure (GPa)
20 40 60 80 100 120
Pressure (GPa)
Depth (km)
Depth (km)
Depth (km) a) b) c)
Figure 3.
Lateral temperature variations of a) ±
250 K, b) ±
500 K, and c) ±
750 Kfor sequences of 25%:75%, 50%:50% and 75%:25% of the mantle being hot:cold.In accord with the two-state model for temperature that we have assumed, twoisosbestic points are generated near 1250 km and 2000 km depth, respectively.
D R A F T August 30, 2018, 2:39pm D R A F T - 26
VALENCIA-CARDONA ET AL.: BULLEN’S PARAMETER AND SPIN CROSSOVERS
Pressure (GPa) a) +/- 250 Kb) +/- 500 Kc) +/- 750 K
Depth (km)
25% cold (Harzburgite) : 75% hot (Pyrolite)
Figure 4.
Lateral temperature and composition variations of a) ±
250 K, b) ±
500 K,and c) ±
750 K for a mantle being 25% cold(harzburgite): 75 % hot(pyrolite).
D R A F T August 30, 2018, 2:39pm D R A F T
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Pressure (GPa)
MgSiO MgOCaSiO Depth (km)
Figure S1.
Bullen’s parameter η for pristine lower mantle minerals. D R A F T August 30, 2018, 2:39pm D R A F T - VA L E N C I A - C A R D O NA E T A L .: B U LL E N ’ S P A R A M E T E R AN D S P I N C R O SS O V E R S Table 1.
Bullen’s parameter η for aggregates with fp in MS state. Perovskite Only Chondritic Pyrolite Peridotite HarzburgitePressure (GPa) η η η η η
23 0.9983 0.9989 1.0008 0.9990 0.999630 0.9983 0.9981 0.9996 0.9978 0.998140 0.9984 0.9974 0.9985 0.9967 0.996750 0.9985 0.9976 0.9985 0.9968 0.996860 0.9986 0.9987 0.9998 0.9986 0.998770 0.9987 1.0020 1.0041 1.0038 1.004680 0.9987 1.0073 1.0116 1.0125 1.0148
90 0.9990 1.0091 1.0145 1.0157 1.0187100 0.9990 1.0054 1.0092 1.0097 1.0117110 0.9991 1.0016 1.0036 1.0032 1.0038120 0.9992 1.0002 1.0016 1.0007 1.0009125 0.9992 1.0003 1.0016 1.0006 1.000790 0.9990 1.0091 1.0145 1.0157 1.0187100 0.9990 1.0054 1.0092 1.0097 1.0117110 0.9991 1.0016 1.0036 1.0032 1.0038120 0.9992 1.0002 1.0016 1.0007 1.0009125 0.9992 1.0003 1.0016 1.0006 1.0007