Spin state transition and partitioning of iron

Contents lists available atScienceDirect

Earth and Planetary Science Letters

www.elsevier.com/locate/epsl

Spin state transition and partitioning of iron: Effects on mantle dynamics

Kenny Vilella

^a,∗

, Sang-Heon Shim

^b

, Cinzia G. Farnetani

^a

, James Badro

^a,c

aInstitutdePhysiqueduGlobedeParis,SorbonneParisCité,75005Paris,France

bSchoolofEarthandSpaceExploration,ArizonaStateUniversity,781S.TerraceRoad,Tempe,AZ85281,USA cÉcolePolytechniqueFédéraledeLausanne,CH-1015Lausanne,Switzerland

a r t i c l e i n f o a b s t ra c t

Articlehistory:

Received23July2014

Receivedinrevisedform17December2014 Accepted10February2015

Availableonlinexxxx Editor: J.Brodholt

Keywords:

spinstatetransition Earth’slowermantle Fepartitioning convection geodynamics

Experimentalstudies atpressure and temperatureconditionsoftheEarth’s lowermantlehave shown thatironinferropericlase(Fp)andinMg-silicateperovskite(Pv)undergoesaspinstatetransition.This electronic transition changes elasticand transport properties oflower mantle minerals and can play an importantrolein mantle convection.Herewefocusonthe geodynamiceffectofthespin-induced density modifications caused by the volume collapse of Fp and by the variation of Fe partitioning (K^Pv–Fp) betweenFpand Pv.Since K^Pv–Fp behavior strongly dependsonaluminacontent, we explore two end-membercompositions,one Al-bearing(with 4.7 wt%Al2O3inPv) andthe otherAl-free.We use the theoretical model by Sturhahn etal. (2005) to calculatethe spin configuration of Fp overa rangeof pressure–temperatureconditions, and useexperimental resultsto model Fepartitioning.We thenapplytheMie–Grüneisen–Debyeequationofstatetoobtainthedensityofthemineralassemblages.

Thecalculatedamplitudeofthedensitychangeacrossthespinstatetransitionislessthan1%,consistent with experimentsby Maoetal. (2011);our densityprofilesdiffer fromPREMby less than1.5%.The spin-induced density variations are included in a three dimensional convection code (Stag3D) for a compressiblemantle.Wefindsmalltemperaturedifferencesbetweenmodelswithandwithoutspinstate transitions,sinceoverbillionsofyearstherelativetemperaturedifferenceislessthan50 K.Howeverthe relativeRMSverticalvelocitydifferenceis upto15%foranAl-freesystem,butonly lessthan6%foran Al-bearingsystem.

©^{2015ElsevierB.V.}All rights reserved.

1. Introduction

The widely acceptedpyrolitic compositionsconsist of approx- imately 18 vol% ferropericlase (Mg, Fe)O (hereafter called Fp), 75 vol% Mg-silicate perovskite (Mg, Fe)(Al, Si)O3 (hereafter called Pv),and7 vol%Ca-silicateperovskiteCaSiO3(hereaftercalledCaPv) (Ringwood, 1982; Irifune, 1994; Irifune et al., 2010). Evenif the uncertainties inthe compositionof thelower mantle areconsid- ered,currentexperimentsathighpressureandtemperature, cou- pledwithequationsofstate(Jackson,1998; Ricolleauetal.,2009;

Murakamietal.,2012) cannotfullyexplaindensityandseismicve- locitiesinferredbyseismicmodelssuchasPREM(Dziewonskiand Anderson,1981).Thedisagreementrevealsthelargeuncertainties thatstill affectcomposition, temperature, andphysicalproperties inthelowermantle.

*

Correspondingauthor.

E-mailaddress:vilella@ipgp.fr(K. Vilella).

Fyfe (1960) suggestedthat the electronic structure of Fe²⁺ in the octahedral coordination can changeat highpressure. Forex- ample, the3d orbitalsofFe²⁺ inFp, whichis surroundedby six oxygen atoms, split in two different groups with different ener- gies:threeorbitals(t_2g)withalowerenergyandtwoorbitals(e_2g) withahigherenergy(see Lietal., 2004,Fig. 4).Following Hund’s rule,atambientcondition,thestablestatehastwounpairedelec- tronintwot2gorbitals,twounpairedelectronsintwoe2gorbitals, andtwopairedelectrons inat_2g orbital.Thisconfigurationisthe high spin (HS) state. Withcompression, the splitting of the two energylevels canincrease andatsome point theenergygapbe- comeslargeenoughtostabilizethestatewithsixpairedelectrons in the t_2g orbitals. This configuration is the low spin (LS) state.

Sherman (1988) andBurns (1993), witha crystalfield theory,as wellasCohenetal. (1997),withabandtheory,predictedtheoc- currenceofsuchchangeinspinstateatthepressure–temperature conditionsof theEarth’slower mantle.Badro etal. (2003)found aspin statetransitioninFpatapressurerange∼^{60–70 GPaand} atambienttemperature. Athighertemperatures,theoretical mod- els(Sturhahnetal.,2005; Tsuchiyaetal.,2006) predictedthatthe http://dx.doi.org/10.1016/j.epsl.2015.02.009

0012-821X/©^{2015ElsevierB.V.}All rights reserved.

Spinstate transitionsalter theelasticandtransport properties (Jacksonet al., 2006; Lin etal., 2006, 2007b, 2013;Crowhurst et al., 2008;Goncharov et al., 2008, 2009; Antonangelietal., 2011;

Ammann etal., 2011) thereby affecting mantle dynamics.More- overthelowermantledensityismodified bythevolumecollapse due to the lower volume of Fe²⁺ in LS state, and by the spin state induced modificationof Fepartitioning betweenFp andPv.

Boweretal. (2009)andShahnasetal. (2011)calculatedtheprop- ertychanges inducedby theFe²⁺ spinstate transitionin Fp,and conductednumericalsimulations toquantifytheeffectonmantle dynamics. Both studies found increased mantle temperature and enhanced flow velocity. However, Bower etal. (2009) assumeda pureFpcompositionandShahnasetal. (2011)neglectedFeparti- tioning,sothatbothstudiesuseasimplifiedlowermantlecompo- sition.

Here we use a theoretical model (Sturhahn etal., 2005) cou- pledtoanequationofstate (JacksonandRigden,1996) tobuilda densitymodelincludingthe Fe²⁺ spinstate transitionin Fp.The dominant chemical components (e.g., FeO, MgO, MgSiO₃, Fe₂O₃, Al2O3, etc.) are included in order to provide realistic thermody- namic properties of the mineral assemblages (Fp, Pv, and CaPv).

We apply an equation of state to these minerals to obtain their densityasafunction ofpressureandtemperature. Thisapproach enablesusto explore differentcompositionsandto calculatethe correspondingdensityprofile.

Anewaspectofourworkistoconsiderthespinstateinduced Fe partitioning between Pv andFp (K^Pv–Fp). Recent experiments haveshowndifferentbehaviors of K^Pv–Fp foran olivinecomposi- tion (Kobayashi etal., 2005; Sinmyo etal., 2008; Auzende etal., 2008; Sakaietal.,2009) andpyroliticcompositions(Irifune,1994;

Kessonetal.,1998; Wood,2000; Murakamietal.,2005; Irifuneet al.,2010; SinmyoandHirose,2013).Thereforewestudytwoend- membercompositions, anAl-bearingandan Al-freepyrolite,with theircorrespondingFepartitioning.WeassumethatintheAl-free systemFepartitioningfollowsthesamebehavior asintheolivine composition.Thecalculateddensityprofileinthelowermantlefits PREMdensity(DziewonskiandAnderson,1981) within1.5%,using Brownand Shankland (1981)geotherm, andit isconsistent with hightemperatureexperiments(Maoetal.,2011).Thedensitymod- elsarethenincludedintheconvectioncodeStag3D(Tackley,1996) toquantifythelongtermimpactoftheFespinstatetransitionon mantleconvection.

2. Densitymodels

Thisparagraphpresentshowwecalculate:(a)theaveragespin sateofFe²⁺ inFp,(b)theiron contentofFp andPv, considering Fepartitioning,and(c)thedensityvariationsinducedbythespin statetransitionfortwoend-memberlowermantlecompositions.

2.1. Averagespinstateofironinferropericlase

FollowingSturhahnetal. (2005)wecalculatetheaverageFe²⁺ spinconfigurationinFpbyminimizingtheHelmholtzfreeenergy:

F=^U−^{T S.Notethat by} consideringthe Helmholtzfreeenergy, ratherthantheGibbsfreeenergy,Sturhahnetal. (2005)implicitly neglect work variations during the spin state transition. Only LS state Fe²⁺ ionsinteractwitheachother,thus theinternal energy is

S

= −

^kBN

η

LSln

η

LS

g_LS

+ η

HSln

η

HS

g_{H S}

,

(2)

wherek_B isthe Boltzmannconstant, g_LS and g_HS are theenergy degeneracies of the electronic configuration. The free energy is then:

F

=

^N

−

^JLS

η

²_LS

+ η

_HSE_HS

+ η

_LSE_LS

+

kBT

η

LSln

η

LS

gLS

+ η

HSln

η

HS

gHS

.

(3)

Tofindtheequilibriumstateatagivenconditionwesolve

∂

F

∂ η

LS

=

⁰

.

(4)

Byusingthenormalizedequation,weexpressEq.(4)as:

0

= η

_LS

1

+

^gHS

g_LSexp

( −

²

β

J_LS

η

_LS

)

exp

(β(

E_LS

−

^EHS

))

−

¹

,

(5)

with β=^kBT. J_LS dependsonthe iron content andvolume, E_LS and EHS depend on volume (Sturhahn etal., 2005), the remain- ingparametersareassumedtobeconstant.ThesolutionofEq.(5) providesthefractionofLSstateasafunctionofironcontent,vol- ume,andtemperature.Forfurtherdetailsontheparametersvalues pleaserefertoSturhahnetal. (2005).

Thenext stepistoconvertvolumetopressureusingtheMie–

Grüneisen–Debyeequationofstate(JacksonandRigden,1996) and the parameters listed in Tables 1 and 2. At ambienttemperature weusethethirdorderBirch–Murnaghanequationofstate:

P

=

^3KT0 2

V0

V

7/3

−

V0

V

5/3

1

−

³

4

(

4

−

K_T0

)

V0

V

2/3

−

1

+

P_th

,

(6) whiletheeffectoftemperatureisaddedviaathermalpressure:

P_th

= γ (

V

)

V

[

^Eth

(

V

,

T

) −

^Eth

(

V

,

T₀

)],

⁽⁷⁾

wherethesubscriptzeroindicates ambientconditionsforvolume V0,temperatureT0,isothermalbulkmodulusKT0 anditspressure derivative K_T0.TheGrüneisenparameterdependsonvolume:

γ (

V

) = γ

0

V V0

q

,

(8)

where q is assumed to be a constant. The vibrational energy is calculatedfromtheDebyemodel,

E_th

=

^{9nR T4}

θ

³

θ/T

0

x³

e^x

−

^1dx

,

(9)

nisthenumberofatomsperformulaunit, R isthegasconstant, andθ istheDebyetemperature:

θ = θ

0exp

γ

0

− γ (

V

)

q

.

(10)

Fig. 1.FepartitioningbetweenPvandFpversuspressure.(a)FortheAl-freesystemtheexperimentalresults(solidlines)arefrom:Kobayashietal. (2005)(purple);Sinmyo etal. (2008)(orange);Auzendeetal. (2008)(green);Sakaietal. (2009)(black).(b)FortheAl-bearingsystemdataby:Irifuneetal. (2010)(opencircles);Irifune (1994)(open squares);Wood (2000)(squares);Kessonetal. (1998)(circles);Murakamietal. (2005)(opentriangles);SinmyoandHirose (2013)(triangles).Inbothpanelsthebluedashed linerepresentstheconstantpartitioningusedintheReference-modelswithoutspinstatetransition,whereasthereddashedlinerepresentsthevariableK^Pv–Fpusedinour modelswithspinstatetransition(seetext).(Forinterpretationofthereferencestocolorinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)

Experimental results at ambient temperature show that K_T0 for FeOcouldchangewiththespinstate,howeverthepressurerange of the LSstate is too small to determine KT0 precisely (see Lin etal., 2013, formore details). At highertemperature, Mao etal.

(2011)foundaKT0differencewithintheerrorbars.Thechangeof K_T0 forFeOduetospinstatetransitionseemstohaveaminoref- fectonthelowermantledensity,thereforeweusethesamevalue forbothspin states.Thisset ofequationsenables ustocalculate theaverage spin state asa function ofiron content, temperature andpressure.

2.2.Ironcontentinferropericlaseandperovskite

Fe partitioning is measured with the Fe–Mg exchange coeffi- cientbetweenPvandFp,

K^Pv–Fp

=

Fe

Mg

Pv

Fe Mg

Fp

=

1 xFp

−

1

1 xPv

−

1

,

(11)

wherex_Pv andx_Fp arethemolar ironconcentration inPv andFp, respectively.

ExperimentalstudieshaveshownthatFe partitioningdepends stronglyon Al content, perhapsbecause ofthe coupled substitu- tionofAl andFe³⁺ inPvandabsenceofsuch substitutionforFp (Navrotsky,1999).FortheAl-freesystem,thesolidlinesinFig. 1a highlight two different behaviors: Kobayashi et al. (2005) and Sinmyo etal. (2008) found an almost constant partitioningwith pressure, whereas Auzende et al. (2008) and Sakai et al. (2009) showed that Fe partitioning decreases dramatically at ∼^{70 GPa,} the pressure atwhich the spin state transitionoccurs in Fp. For theAl-bearingsystem, Fig. 1b showsthemainexperimental data ascompiledby Irifune etal. (2010).A sharpincrease ofthe par- titioningat∼^{25 GPa(Irifune,}1994; Wood,2000) isfollowedbya sharpdecreaseat∼^{40 GPa.Athigherpressuretheresultsarecon-} flicting:Kessonetal. (1998)andMurakamietal. (2005)measured aconstantpartitioning,whileSinmyoandHirose (2013) foundan increaseofFepartitioningabove90 GPa.

Giventhe strikingdifferencebetweentheAl-free(Fig. 1a)and theAl-bearing(Fig. 1b) cases,we build twoend-member models forlowermantlecomposition.Ourfirstcomposition,calledAl-free, hasa classical lower mantle mineralvolume proportion: 75% Pv, 18%Fpand7% CaPv,with8 wt%FeOinthebulkcomposition.All iron is assumed to be Fe²⁺. Our second composition, called Al- bearing,differs fromthepreviousone by theadditionof4.7 wt%

Al₂O₃andbyassumingthat60%ofironinPvisFe³⁺,theremain- ingisFe²⁺.FerrousironentersintoPvasFeSiO3,ferricironenters

Table 1

Isothermalbulkmodulus(KT0)andvolume(V0)atambientconditionsforseveral compounds.

Compounds KT0

(GPa)

V0 (cm³/mol)

MgO 160^a 11.25^a

FeO (LS) 150^b 10.82^b

FeO (HS) 150 12.18^b

MgSiO3 261^c 24.43^c

0.85MgSiO3–0.15FeSiO3 259^c 24.58^c

0.915MgSiO3–0.085Fe2O3 237^d 24.95^d

0.90MgSiO3–0.10FeAlO3 262^e 24.80^e

0.90MgSiO3–0.10Al2O3 244^e 24.66^e

CaSiO3 236^f 27.45^f

a Spezialeetal. (2001).

b Feietal. (2007).

c Lundinetal. (2008).

d Catallietal. (2010).

e Catallietal. (2011).

f Shimetal. (2000b).

intoPvasFeAlO₃.IfthereisanexcessofFe³⁺,Fe₂O₃ enters into Pv,whereasifthereisanexcessofAl,Al2O3 entersintoPv.

These compositions are used in the models that include the spin state transition and a variable partitioning coefficient, as showninFig. 1 (reddashed lines), aswell asinthe correspond- ing reference models (called Reference Al-free andReference Al- bearing), which are calculated without spin state transition and witha constant partitioningcoefficient, asshown in Fig. 1(blue dashedlines).Theunderlyingassumptionisthatthechangeofpar- titioningiscausedbythespinstatetransition.Badroetal. (2005) provide a thermodynamic argument for the Al-free system, and findthat

K^Pv–Fp

=

^K0exp

−

V

(

P

−

^Ptr

)

R T

,

(12)

whereV =¹.36 cm³/mol istheFeOvolumedifferencebetween HS andLSstates(Feiet al.,2007) and Ptr isthe pressureof the spin state transition. The red dashed linein Fig. 1a is calculated usingEq.(12),whichisinagreementwithexperimentalresultsby Auzendeetal. (2008)andbySakaietal. (2009).FortheAl-bearing system,weuseafitofexperimentalresultsbyIrifuneetal. (2010).

2.3. Densityasafunctionoftemperatureandpressure

Wecalculatethedensityofeachmineralusingtheaveragespin configurationandtheMie–Grüneisen–Debyeequationofstatede- tailedinEq.(6)(standardvaluesarelistedinTables 1 and2).The composition of each mineral species changes with pressure and

q 1.4 1.3 0.6 a Fiquetetal. (2000).

b JacksonandNiesler (1982).

c Shimetal. (2000a).

temperature because of variations in Fe partitioning. The initial volumeandbulkmodulus(V₀andK_T0)areobtainedbylinearin- terpolationbetweentheMg- andFe-end-members.Wethencalcu- latethedensityoftherockastheweightedaverageofconstituent mineraldensities.

The relative densitydifference (

ρ

=¹⁰⁰(

ρ

Spin−

ρ

Ref)/

ρ

Spin) betweenthemodelwithspinstate transitionandvariable K^Pv–Fp and the corresponding reference model is shown for Al free (Fig. 2a)andAl bearing(Fig. 2b)compositions.Asfoundby theo- reticalandexperimentalstudies,thespin statetransitionissharp at low temperature and broad at high temperature. At 1200<

T <1800 K, our transition is sharper than experimental results (Komabayashi et al., 2010; Mao et al., 2011), but the difference vanishes athigher temperatures(T >2000 K) appropriate to the lowermantle.Theamplitudeofthedensitychangeis∼^0.8%forAl- freesystemandonly∼^{0.5%forAl-bearing system,thisdifference} isduetotheFepartitioning.InFig. 2bthepeakof

ρ

at35 GPais duetoFepartitioningandclearly reflectsthe K^Pv–Fp trendshown isFig. 1b.Finally,wenotethat thecalculateddensitychangesare comparablewiththose found inexperimental results,although a slightmismatchstillexists.Forexample,atT=^{2000 K wefindthe} samedensitychange asMao etal. (2011),butour transitionoc- cursatapressure10 GPalower.Thisseemsacceptable,giventhat asimilarmismatchisobservedbetweenexperimentalstudies(for areviewseeLinandTsuchiya,2008).Atagivenpressure,theden- sitychangevarieswithtemperature,sothetemperaturederivative of densityis affected by the spin state transition. Therefore, the thermal expansivity, definedas

α

=(−¹/

ρ

)(∂

ρ

/∂T)p, isalso af- fectedbythespin statetransition. Atlowermantletemperatures,

α

islocallyincreasedupto∼^{40%.Wealsonotethatthemismatch} betweenour densityprofile andPREM density,using Brown and Shankland (1981)geotherm,islessthan0.6%forreferencemodels.

Furthermore we conducted a number ofsimulations (not shown

mensional geodynamicmodel.ThecodeStag3D(Tackley,1996) in cartesiangeometrysolves the non-dimensionalequationsgovern- ing mantleconvection, assuming thetruncatedanelastic approxi- mation.Theequationsare:conservationofmass

∇ • ( ρ

v

) =

⁰

,

(13)

conservationofmomentum

∇ • σ − ∇

^p

=

^Ra

ρα

T

^z

,

(14) conservationofenergy

ρ

Cp

D T

Dt

= −

Dis

αρ

Tvz

+ ∇ • (

k

∇

T

) + ρ

H

+

^Dis

Ra

σ : ˙ .

(15) k is the depth-dependent thermal conductivity,

α

the depth- dependent thermal expansivity and

ρ

the depth-dependent den- sity.T istemperature,T=^T−^Tref thetemperatureanomalywith respecttotheadiabatictemperatureTref,pthedynamicpressure, v the velocity vector, H the internal heating rate, C_p the spe- cific heatcapacity,

σ

thedeviatoricstresstensor,

˙ thestrainrate tensor,^{z isa unit vector in the vertical direction. The truncated} anelasticapproximationneglectsthevariationsoftemperaturedue to the dynamic pressure, thereforewe use the dynamic pressure rather than the total pressure. The two non-dimensional num- bers are the surface dissipation number Di_s=

α

sg D/C_p and the RayleighnumberRa=

ρ

g

α

T D³/

ηκ

(seeTable 3).Toincludeour lower mantle densitymodels (

ρ

model) we mustmodify the equa- tion ofconservationofmomentum,since thethermalexpansivity dependsondepth andspin state transition(i.e.,ontemperature).

Wemodifytheequationofconservationofmomentumasfollows

∇ • σ − ∇

^p

=

^Ra

ρ

_model

ρ

th

^z

,

(16)

where

ρ

th=

α

T and

ρ

model=

ρ

model(T_ref,p)−

ρ

model(T,p). With this formulation the thermal expansivity does not appear explicitly.Weapproximatemantleviscosityviathefollowingequa- tion,

Fig. 2.Relativedensitydifference(%)betweenthemodelwithspinstatetransitionandthereferencemodel(withoutspinstatetransition),asafunctionofpressureand temperature,for(a)Al-free,and(b)Al-bearingcompositions.Thegeotherm(blackline)isfromournumericalsimulations,andthedottedareacorrespondstotherangeof possiblelowermantletemperaturesbyDeschampsandTrampert (2004),inferredfromseismicmodelscombinedwithexperimentalmineralogydata.

Table 3

Mantleparametersusedinthenumericalsimulations.

Symbol Parameter Value Non-dimensional

value Ra0a SurfaceRayleigh

number

10⁷ N/A

Disb Surfacedissipation number

1.18 N/A

ρ0 Surfacedensity 3300 kg/m³ 1.0

g Gravity 9.81 m/s² 1.0

α0 Surface

expansivity

5×10⁻⁵K⁻¹ 1.0 T Superadiabatic

temperaturescale

2500 K 1.0

d Mantlethickness 2890 km 1.0

κ0 Surfacethermal diffusivity

7×^10−7m2/s 1.0 CP,0 Surfaceheat

capacity

1200 J/(kg K) 1.0

η0 Referenceviscosity 1.4×^{1022Pa s 1.0}

H Internalheating

rate

7.38×^10−12W/kg 15.5^c ρ410 Densitychangeat

410km

198 kg/m³ 0.06

ρ660 Densitychangeat 660km

462 kg/m³ 0.14

γ660 Clapeyronslopeat 410km

−^{2.5 MPa/K} −^0.066

γ660 Clapeyronslopeat 660km

2.5 MPa/K 0.066

a Ra0=ρgαT D³/ηκ. b Dis=αsg D/Cp.

c CorrectedforCartesiangeometry,seeTackley (1996)forfurtherdetails.

η (

T

,

d

) = η

0exp

(

d

η

_d

)

exp

13

.

8

T

+

0

.

88

,

(17)

providinga four orders ofmagnitudevariation withtemperature andatenfoldincreasefromthesurfacetothecoremantlebound- ary(CMB),for

η

d=².3.At660 kmdepththeviscosityincreases byafactor30duetothephasetransition.Numericalvaluesofthe Clapeyronslopeandof thedensitychange induced bythe phase transitionsat660 kmand410 kmdeptharegiveninTable 3.

The modeled domain is divided in 1024 × ¹⁰²⁴ × ^{128 ele-} ments,providing aspatial resolutionof ∼^{22.6 km.The boundary} conditionsat top and bottom are free slip velocity and constant temperature(300 Kand4000 K,respectively),thesideboundaries are periodic. The temperature field at equilibrium (i.e., after the equivalent of15 Ga) isused asinitial condition for thedifferent numericalsimulations. Thefourmodelsarerun forfurther20 Ga andwe avoidanyinfluenceoftheinitialcondition byconducting timeaveragesonlyoverthelast7 Ga.

4. Results

Fig. 3showssnapshotsoftheanomalyofpotentialtemperature fortheAl-bearingcaseanditscorrespondingreferencecase.Con- vectioninbothsnapshotsseemstobeidentical.Somecolddown- wellingsarestopped at660 kmdepthandcreateavalanchesthat reachthelowermantle(Fig. 4).Hotupwellingsarealsostoppedat 660 kmdepth andformsecondary plumesthatreachthe surface (Fig. 4).

Thetemperature field is averagedhorizontally andtemporally inorder tocalculate therelative temperature differencebetween models with and without spin state transition. The temperature difference (Fig. 5a) is almost constant in the whole mantle and varies sharply close to the surface and to the CMB. For the Al- bearingcomposition, the temperatureincreases by ∼^0.6%,equiv- alentto a ∼^{20 K difference at}∼^{2500 km depth.The temporally} averaged surface heat loss for the referencecase is 42.7 TW, in therangeofvalueinferred byJaupartetal. (2007),andincreases

Fig. 3.Potentialtemperatureanomaly(T–Tm, whereTm isthe averagetempera- ture)fortheAl-bearingcasewiththespinstatetransition(a)andwithout(b),the twoisosurfaces(−^{450 Kand200 K)highlight}downwellingsandupwellings. For graphicalreasons thecoldisosurface(blue)istransparentatshallowdepthsand thetemperaturescaleistruncated.(Forinterpretationofthereferencestocolorin thisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)

to43.2 TWwiththespinstate transition.Thereforethespinstate transitionslightlyincreasestheefficiencyofheat transfer.Forthe Al-free composition, the effects are similar but more significant withatemperatureincrease of∼^{1.6%,equivalent toa} ∼^{50 Kdif-} ferenceat∼^{2500 kmdepth.Thetemporallyaveragedsurfaceheat} lossforthereferencecaseis42.7 TWand43.8 TWwiththespin statetransition,this∼^{2.5%differenceismuchsmallerthantheun-} certaintyfortheEarth(therangeofplausiblevaluesby(Jaupartet al., 2007) is 43–49 TW).Since thetemperatureprofileis closein bothmodels,thecorrespondingrelativedensitydifference(Fig. 5b) reflectsthedensitychangeduetothespinstatetransitionfoundin Fig. 2.IntheAl-freecasetherelative densitydifference increases at∼^{100 GPaanddeeper,butdecreasesinthelowermost}∼^{150 km} (125<P <135 GPa) because the sharp temperature rise in the thermalboundary layerreducesthe extentof LSstate (asshown bythegeotherminFig. 2).IntheAl-bearingcasetherelativeden- sity increases at ∼^{40 GPa, is almost constant to} ∼^{100 GPa and} thenincreasestotheCMB.

Fig. 4 showsa slice of the temperature field and the density changeinducedbythespinstatetransition.InplumestheLStoHS transitionoccursatgreaterdepth,sothatthesurroundingmantle is in LSstate while the plume center is in HS state. This lateral densitydifference increasesplume buoyancyandshouldenhance the upwelling velocity. Inslabs theHS to LStransition occursat shallowerdepth,thusthesurroundingmantleisinHSstatewhile the slab center is in LS state. The increased lateral density dif- ference should enhance the downwelling velocity. We first study theRMSverticalflowvelocitybyseparatingupwellinganddown- wellingmaterialwithacriterionsolelybasedontheverticalveloc- ity.Notethatonlyinthenextparagraphwewillfocusonplumes and slabs by introducing a further criterion basedon the excess oftemperature.Fig. 6ashowsthat fortheAl-freesystemthespin statetransitionmodifiestheverticalvelocityofbothupwellingand downwellingmaterialthroughoutthe wholemantle.Inparticular, theRMS verticalvelocity difference forupwellingis5–15% faster

Fig. 4.Verticalcross-sectionoftheanomalyofpotentialtemperature(top)andthedensitychangeinducedbythespinstatetransition(bottom)foratypicalslab(left)anda typicalplume(right).TemperatureanomalyisdefinedasT−^Tm,whereTmistheaveragetemperature.Thedensitychangeisequivalenttothespinstate:blueisHSstate, redisLSstate.TheseresultsareextractedfromtheAl-freecase.(Forinterpretationofthereferencestocolorinthisfigurelegend,thereaderisreferredtothewebversion ofthisarticle.)

Fig. 5.(a)Horizontallyandtemporallyaveragedtemperaturedifferencerelativeto reference(%)fortheAl-freecase(orangeline)andfor theAl-bearingcase(pur- pleline).(b)Atypicalrelativedensitydifferencehorizontallyaveraged(%)forthe Al-freecase(orange line)and fortheAl-bearingcase(purpleline).Thepressure scaleisextractedfromnumericalsimulations.(Forinterpretationofthereferences tocolorinthisfigurelegend,thereaderisreferredtothewebversionofthisarti- cle.)

inthelowermostmantle (i.e., D>2300 km),itremains constant (∼^{5% faster) throughout thelower mantleand decreases onlyin} the upper mantle. For downwellings the relative velocity differ- enceis4–17%fasterthroughoutthelowermantle,reachingapeak value of ∼^{17% at} ∼^{2400 km. For the sum of the two contribu-} tionstherelativevelocitydifferenceincreaseswithdepth,reaching

Fig. 6.HorizontallyandtemporallyaveragedrelativeRMSverticalvelocitydifference (%)versusdepth(km).ForbothAl-free(a)andAl-bearing(b)system,werepresent the downwellingvelocity(bluelines),theupwellingvelocity(redlines),selected as negativeorpositive flowvelocity, respectively,and thevelocityofthe whole convectivefluid(blacklines).Thepressurescaleisextractedfromnumericalsimu- lations.(Forinterpretationofthereferencestocolorinthisfigurelegend,thereader isreferredtothewebversionofthisarticle.)

∼^{15%attheCMB. FortheAl-bearingsystem(Fig. 6b), theeffects} ofthe spinstate transitionare similar.The maindifference bears on the amplitude ofthe velocitydifference, which islesssignifi- cant forthis composition (upto ∼^{6% for} upwellings, up to ∼^8%

fordownwellingsandupto ∼^{6% forthewholeconvective fluids)}

Fig. 7.HorizontallyandtemporallyaveragedrelativeRMSverticalvelocitydifference (%)versusdepth(km).ForbothAl-free(a)andAl-bearing(b)system,werepresent theimpactofspinstatetransitioninthelowermantleonslabs(bluelines)and onplumes(redlines),selectedviathemethodexplainedinthetext.Thepressure scaleisextractedfromnumericalsimulations.(Forinterpretationofthereferences tocolorinthisfigurelegend,thereaderisreferredtothewebversionofthisarti- cle.)

inagreementwiththelowerdensitychange(Fig. 5b).Surprisingly therelativevelocitydifferencedoesnotseemtobeaffectedbythe changeofFepartitioningat30 GPa.

Notethat averages for downwellings andupwellings material, sodefinedby the vertical velocity, cannot providedetailed infor- mationonslabsandplumes,whichrepresentonlyasmallfraction ofthewholeconvectivematerial.Tofocusonplumesandslabswe needtouseatemperaturebasedcriterion(Labrosse,2002).A point (x,y,z)belongstoaplumeif:

T

(

x

,

y

,

z

) >

T_mean

(

z

) +

^ph,T

[

^Tmax

(

z

) −

^Tmean

(

z

) ]

⁽¹⁸⁾

ortoaslabif:

T

(

x

,

y

,

z

) <

T_mean

(

z

) +

^pc,T

[

^Tmin

(

z

) −

^Tmean

(

z

)],

⁽¹⁹⁾

where T_mean, T_min andT_max arethe averaged, theminimumand themaximum temperatureat a givendepth, respectively, pc,T = p_h,T =⁰.45 are constants, the chosen value is from Galsa and Lenkey (2007).Weapply thisalgorithmacrossallthelowerman- tle,exceptforthelast50 kmclosetothebottomboundary,since slabs do not penetrate the thermal boundary layer. For the Al- free system (Fig. 7a), with the spin state transition plumes are faster (up to 20%) in the lowermost mantle (D<2100 km) but become slower (up to −^{10%) once they reach shallow depths} (700<D<2100 km),whereas slabsarefaster(upto30%)inthe wholelowermantle.FortheAl-bearingsystem(Fig. 7b)thetrends aresimilar,buttherelativevelocitydifferenceisreducedto∼^15%, showinghowimportantis toincludeAl in lowermantle compo- sitionmodels.We findthat theeffectofthe spin statetransition istoincreaseslabsandplumesverticalvelocitiesatdepthsgreater than2100 km(i.e.,P>100 GPa).Theseresultsshowthatthesim- plereasoningpresentedabove,whichpredictstheverticalvelocity solelyonthelocallateraldensitydifference,isvalidatfirstorder butsomedisagreementsremain.Forinstanceplumesandslabsare affectedbythespin statetransitionatthetopofthelower man- tle, whereas forthis pressure–temperature condition there is no

lateraldensitydifference.Thisresultisnotsurprising,sinceatin- finite Prandtl number the whole convective fluid is immediately affectedbyalocalchange.Slabscouldbefasterinthewholeman- tlebecausetheir deepestpart,whichisundergoingthespinstate transition,isabletopulldownthewholeslab.

We also calculated the averaged time required to travel the lower mantle depth. We found that the spin state transition slightlyslowsdown plumes(2.8%forAl-bearing systemand0.3%

forAl-free),butacceleratesslabs (8.3%forAl-bearingsystemand 14.1%forAl-free).Overall,foranAl-freecomposition,withthespin state transition the RMS vertical velocity is faster in the lower mantle,which implies an enhanced convection.However, theef- fectsof spin state transition ontemperature are smallcompared with our uncertainties. For example the plausible geotherm in- ferred by Deschamps andTrampert (2004) has an uncertainty of

∼^{500 K,whichislargecomparedwiththe50 Kdifferencecaused} bythespinstatetransition.ForanAl-bearingcompositiontheam- plitudeoftherelativechangeofaveragetemperatureandvelocity islowerandbecomesnotsignificant.Whenwefocusonslabsand plumes, rather than study the average vertical flow velocity, we findthattheeffectofthespinstate transitionistoenhancetheir verticalvelocity.

5. Discussion

Weconductednumericalsimulationsofmantleconvectionwith andwithoutthedensitychangeinduced bythe spinstate transi- tioninFp.Anovelaspectofourapproachistoconsiderthatiron content inFp varieswithpressure andtemperature, asindicated by recent experiments on iron partition coefficient (K^Pv–Fp) be- tweenFpandPv.Sincetheexperimentsshowadifferentbehavior of K^Pv–Fp depending on alumina content, we explored two end- membermantlecompositions,oneAl-free(i.e.,75%Pv,18%Fp,and 7% CaPv)andone Al-bearing,with4.7 wt% Al2O3.The advantage of calculatinglower mantle densityusingexperimental measure- ments of single phase minerals is that we can consider several cases, each with a plausible mantle mineralogy. Our calculated density profiles fit PREM densitywithin 1.5%, andare consistent withrecenthightemperatureexperimental data(Komabayashiet al., 2010; Mao et al., 2011). We have shown that for plausible lower mantle compositions, the global densityincrease of ∼^0.7%

induced by the spin state transitionin Fp hasa minoreffect on mantletemperature.Moreprecisely,spatiallyandtemporallyaver- agedmantletemperaturesdifferbylessthan50 Kbetweenmodels withand withoutspin state transition, whereas thetotal surface heat loss differs by lessthan 2.5%. For Al-freecomposition, flow velocity issignificantlyaffected bythespin state transition,since RMS vertical velocity of downwellings and upwellings differ by

∼^{17% and} ∼^15%, respectively. However, for Al-bearing composi- tion,theaverageflowvelocityismoderatelyaffected(differbyless than∼^6%).

Boweretal. (2009) founda moreimportanteffectofthespin state transitionon mantletemperatures (up to 10%) andon the verticalflow velocity (upto25% whilewe findupto 15%forAl- freesystem).Therearetworeasonsforthisdifference.First,Bower etal. (2009) explored the effectof a 2–4% densityincrease. This value isappropriate forpure Fp (Linand Tsuchiya, 2008), butit is certainly excessivefora lower mantle composition with∼20%

Fp. The second reason concerns the range of acceptable values formantlepotentialtemperature.Petrologicalstudies(e.g.,Putirka, 2005and referencestherein) indicate that mantle potential tem- peratureisaround1550 K,whereasBoweretal. (2009)geotherm isonlyat∼^{1050 K.Theroleofthespinstatetransitionisthereby} enhanced,sinceatlowtemperaturesthetransitionissharper.

Numerical simulations by Shahnasetal. (2011) include depth dependentpropertiesandconsideraplausiblelowermantlecom-

can be compared to our results and indeed show a minor ef- fect on mantletemperature(with a maximumof 3% variation at 1000 km depth) and on the radial mass flux. Onlyfor more ex- treme models, which reduce to a pure Fp lower mantle, and in presence ofthe PPv transition, theauthors conclude that mantle mixing is enhanced. Shahnas et al. (2011) conclusions are simi- lartoours,howevertheydidnotshowtheeffectonverticalflow velocity, since velocity variations were only inferred from lateral densitychanges.

Inthe following wediscuss underwhich conditionsit ispos- sible to infer fluid velocity uniquely from local lateral density changes.Thesimplerelationbetweenascentvelocity(Vz)andlat- eral density change (

ρ

) was shown by Batchelor (1954) for a steadylaminarplume:

V_z

∝

^r

2p

ρ

g

η ,

⁽²⁰⁾

where rp is the plume radius. However, this expression is valid underanumberofrestrictiveconditions:First,itisvalidonlyfar fromthe source andtheinterfaces; Kaminski andJaupart (2003) haveshownthat the velocity isconstant onlyat adistance from theinterfacecorrespondingtofivetimestheplumeradius.Forex- ample,ifr_p=^{100 km,Eq.(20)holds500 kmabove theCMBand} 500 kmbelowthe660 kmdiscontinuity.Second,itisvalidonlyif thesourceofbuoyantmaterialprovidesaconstantfluxovertime.

Clearlythisconditionisnotsatisfiedformantleconvection,where timevaryingplatevelocitiesinduceavaryinginfluxatsubduction zones. Third, to obtain this constant ascent velocity, we need to makethe assumptionofzeropressure onthe sidesoftheplume conduit.This impliesthat the conduit mustbe straight andcon- tinuous,becausebendingoftheconduitinducescompressionand decompression(Gareletal., 2014).It isclearthat mantleplumes andslabsarenotstraightandcontinuous.Thusthesimplerelation between lateral

ρ

and vertical flow velocity may have limited applicationsintheEarth’smantle.

Since the presence of alumina inthe lower mantle is widely accepted,theAl-bearingsystemisthemostplausiblecomposition.

Forthiscomposition,thedensitychangeinducedbythespinstate transitioninFp issmall(<0.5%)aswell asits impactondynam- ics.Thisiscoherentwithseismicobservationsthatdonotexhibit anymantlelayering atthedepth rangecorresponding tothespin statetransition.Ourresultsconfirmthatthereisnodisagreement between seismic observations and experimental results at high pressure–temperatureconditions.The enhancedconvectioninthe lowermostmantleisarobust consequenceofthespinstate tran- sition,that could affectthe stability of thetwo Large LowShear VelocityProvinces(LLSVP)beneathAfricaandthePacific(Garnero and McNamara, 2008). The nature of LLSVP is subject of de- bates,someauthorsclaimathermalorigin(Schuberthetal.,2009;

Daviesetal., 2012), whilemostofstudiesintroduce achemically distinct reservoir (Deschamps and Tackley, 2008, 2009; Li et al., 2014) relatedtorecycledbasalticcrust oriron richprimitivema- terial.Thesignificantincrease ofslabs andplumesvelocityinthe deepest part of the mantle may imply a destabilization of such reservoirs. However, thissuggestionis speculativebecause ofthe poorknowledgeofthelowermostmantlecoupledtothecomplex- ityofmantledynamics.

Finally, we note that we did not explore all the implications ofspin statetransitions,henceourconclusionscouldchangewith

of Fp, Pv and PPv with their corresponding Fe partitioning (see Lin et al., 2013; Badro, 2014, and reference therein) would add morecomplexities.Second,Fepartitioningcouldhavesomemajor consequences on the Earth’s mantle, since iron content changes dramatically theradiative thermal conductivity(Goncharov etal., 2010),theelectricalconductivity(DobsonandBrodholt,2000),and maychangethe viscosity,whichisa keyparameterfordynamics (NaliboffandKellogg,2007).Ammannetal. (2011)findanegligi- bleeffectofspinstatetransitionsonviscosity,buttheeffectofFe partitioningonviscosityisunknown.Fig. 1showedthatimportant variations ofFepartitioningarelikelytooccurinthelowerman- tle,thereforeinafuturestudywewillexploretheimplicationsof aviscositychangeassociatedtoaFepartitioningchange.

Fe partitioning mayalso be important to understand yet un- explained observations, for exampleRicolleau etal. (2009) high- lighted that a typical pyrolitic composition has a density profile with a steeper slope than PREM density;this disagreement can- not be resolved by considering uncertainties of both PREM and experimental resultsandrequires anotherexplanation.Cobdenet al. (2009)usedmineralphysicsconstraintswiththeir correspond- ing uncertainties to interpret seismic data. They concluded that the seismic observations can be explained with a superadiabatic geotherm combined to a gradual change of the bulk chemistry.

They suggestedan increase of the MORB proportion withdepth, and they indicated that a Fe-enrichment is also possible. One can speculate that a higher superadiabatic geotherm coupled to an increase of Fe partitioning due to temperature could provide an explanation to seismic observations. Understanding to which extent Fe partitioning affects the lower mantle requires further work and experimental results on effects of temperature, alu- mina contentandbulk iron (Maoetal., 1997; Sakai etal., 2009;

Irifuneetal.,2010).

Acknowledgements

We thankRhodriDaviesandan anonymousreviewer fortheir thoroughandconstructivecommentsthatimprovedthepaper.We alsothankJohnBrodholtforhiseditorialhandling.Numericalcom- putations were performedonthe S-CAPADplatform,IPGP, France andusing HPCresources fromGENCI-IDRIS (Grant 2013-047033).

ThisisIPGPcontribution3612.

Appendix A. Supplementarymaterial

Supplementarymaterialrelatedtothisarticlecanbefoundon- lineathttp://dx.doi.org/10.1016/j.epsl.2015.02.009.

References

Ammann,M.W.,Brodholt,J.P.,Dobson, D.P.,2011.Ferrousirondiffusioninferro- periclaseacrossthespintransition.EarthPlanet.Sci.Lett. 302,393–402.http://

dx.doi.org/10.1016/j.epsl.2010.12.031.

Antonangeli,D.,Siebert,J.,Aracne,C.M.,Farber,D.L.,Bosak,A.,Hoesch,M.,Krisch, M., Ryerson, F.J., Fiquet, G., Badro, J., 2011. Spin crossover in ferropericlase athighpressure:aseismologicallytransparenttransition?Science 331,64–67.

http://dx.doi.org/10.1126/science.1198429.

Auzende,A.L.,Badro,J.,Ryerson,F.J.,Weber,P.K.,Fallon,S.J.,Addad,A.,Siebert,J.,Fi- quet,G.,2008.Elementpartitioningbetweenmagnesiumsilicateperovskiteand ferropericlase:newinsightsintobulklower-mantlegeochemistry.EarthPlanet.

Sci.Lett. 269,164–174.http://dx.doi.org/10.1016/j.epsl.2008.02.001.

Badro,J.,2014.Spintransitionsinmantleminerals.Annu.Rev.EarthPlanet.Sci. 42, 231–248.http://dx.doi.org/10.1146/annurev-earth-042711-105304.

Badro,J.,Fiquet,G.,Guyot,F.,Rueff,J.P.,Struzhkin,V.V.,Vankó,G.,Monaco,G.,2003.

IronpartitioninginEarth’smantle:towardadeeplowermantlediscontinuity.

Science 300,789–791.http://dx.doi.org/10.1126/science.1081311.

Badro,J.,Rueff,J.P., Vankó,G.,Monaco, G.,Fiquet,G., Guyot,F.,2004. Electronic transitionsinperovskite:possiblenonconvectinglayersinthe lowermantle.

Science 305,383–386.http://dx.doi.org/10.1126/science.1098840.

Badro,J., Fiquet,G.,Guyot,F.,2005.ThermochemicalStateoftheLowerMantle:

NewInsights fromMineral Physics.Geophysical MonographSeries, vol. 160.

AGU,pp. 241–260.http://dx.doi.org/10.1029/160GM15.

Batchelor,G.K.,1954.Heatconvectionandbuoyancyeffectsinfluids.Q.J.R.Mete- orol.Soc. 80,339–358.http://dx.doi.org/10.1002/qj.49708034504.

Bower,D.J.,Gurnis,M.,Jackson,J.M.,Sturhahn,W.,2009.Enhancedconvectionand fastplumesinthelowermantleinducedbythespintransitioninferropericlase.

Geophys.Res.Lett. 36,L10306.http://dx.doi.org/10.1029/2009GL037706.

Brown,J.M.,Shankland,T.J.,1981.ThermodynamicparametersintheEarthasde- terminedfromseismicprofiles.Geophys.J.R.Astron.Soc. 66,579–596.http://

dx.doi.org/10.1111/j.1365-246X.1981.tb04891.x.

Burns,R.G.,1993.MineralogicalApplicationsofCrystalFieldTheory.CambridgeUni- versityPress,Cambridge,UK.

Catalli,K.,Shim,S.H.,Prakapenka,V.B.,Zhao,J.,Sturhahn,W.,Chow,P.,Xiao,Y.,Liu, H.,Cynn,H.,Evans,W.J.,2010.SpinstateofferricironinMgSiO3perovskiteand itseffectonelasticproperties.EarthPlanet.Sci.Lett. 289,68–75.http://dx.doi.

org/10.1016/j.epsl.2009.10.029.

Catalli,K.,Shim,S.H.,Dera,P.,Prakapenka,V.B.,Zhao,J.,Sturhahn, W.,Chow,P., Xiao,Y.,Cynn,H.,Evans,W.J.,2011.EffectsoftheFe³⁺spintransitiononthe propertiesofaluminous perovskite–new insights for lower-mantle seismic heterogeneities.EarthPlanet.Sci.Lett. 310,293–302.http://dx.doi.org/10.1016/

j.epsl.2011.08.018.

Cobden, L., Goes,S., Ravenna, M., Styles,E., Cammarano, F., Gallagher,K., Con- nolly,J.A.D.,2009.Thermochemicalinterpretationof1-Dseismicdataforthe lowermantle:thesignificanceofnonadiabaticthermalgradientsandcomposi- tionalheterogeneity.J.Geophys.Res.,SolidEarth 114,B11309.http://dx.doi.org/

10.1029/2008JB006262.

Cohen,R.E.,Mazin,I.I.,Isaak,D.G.,1997.Magneticcollapseintransitionmetalox- idesathigherpressure:implicationsfortheEarth.Science 275,654–657.http://

dx.doi.org/10.1126/science.275.5300.654.

Crowhurst,J.C.,Brown,J.M.,Goncharov,A.F.,Jacobsen,S.D.,2008.Elasticityof(Mg, Fe)Othrough the spin transitionofiron inthe lower mantle. Science 319, 451–453.http://dx.doi.org/10.1126/science.1149606.

Davies,D.R.,Goes,S.,Davies,J.H.,Schuberth,B.S.A.,Bunge,H.P.,Ristema,J.,2012.

ReconcilingdynamicandseismicmodelsofEarthslowermantle:thedominant roleofthermalheterogeneity.EarthPlanet.Sci.Lett. 353–354,253–269.http://

dx.doi.org/10.1016/j.epsl.2012.08.016.

Deschamps,F.,Tackley,P.J.,2008.Searchingformodelsofthermo-chemicalconvec- tionthatexplainprobabilistictomographyI. Principlesandinfluenceofrhe- ologicalparameters.Phys. EarthPlanet.Inter. 171,357–373.http://dx.doi.org/

10.1016/j.pepi.2008.04.016.

Deschamps,F.,Tackley,P.J., 2009.Searchingfor modelsofthermo-chemicalcon- vectionthat explain probabilistictomography II–influenceof physicaland compositionalparameters.Phys.EarthPlanet.Inter. 176,1–18.http://dx.doi.org/

10.1016/j.pepi.2009.03.012.

Deschamps,F.,Trampert,J.,2004.Towardsalowermantlereferencetemperature andcomposition.