dimensionlessoceanbase

Experimentsonmetal–silicateplumesandcoreformationBYPETEROLSON1,*ANDDAYANTHIEWEERARATNE2,1DepartmentofEarthandPlanetarySciences,JohnsHopkinsUniversity,Baltimore,MD21218,USA2DepartmentofTerrestrialMagnetism,CarnegieInstitution,Washington,DC20015,USAShort-livedisotopesystematics,mantlesiderophileabundancesandthepowerrequirementsofthegeodynamofavouranearlyandhigh-temperaturecore-formationprocess,inwhichmetalsconcentrateandpartiallyequilibratewithsilicatesinadeepmagmaoceanbeforedescendingtothecore.
Wereportresultsoflaboratoryexperimentsonliquidmetaldynamicsinatwo-layerstratiedviscousuid,usingsucrosesolutionstorepresentthemagmaoceanandthecrystalline,moreprimitivemantleandliquidgalliumtorepresentthecore-formingmetals.
SinglegalliumdropexperimentsandexperimentsonRayleigh–Taylorinstabilitieswithgalliumlayersandgalliummixturesproducemetaldiapirsthatentrainthelessviscousupperlayeruidandproducetrailingplumeconduitsinthehigh-viscositylowerlayer.
Calculationsindicatethatviscousdissipationinmetal–silicateplumesintheearlyEarthwouldresultinalargeinitialcoresuperheat.
Ourexperimentssuggestthatmetal–silicatemantleplumesfacilitatehigh-pressuremetal–silicateinteractionandmaylaterevolveintobuoyantthermalplumes,connectingcoreformationtoancienthotspotactivityontheEarthandpossiblyonotherterrestrialplanets.
Keywords:Earth'score;coreformation;magmaocean;metal–silicateplumes1.
IntroductionEvidencefromshort-livedisotopesindicatesthattheEarthaccretedfromsmallerplanetesimalobjectsinarelativelyshortperiodoftime,estimatedat30–40Myr(Woodetal.
2006).
Meteoriteevidencealsoindicatesthatsomeoftheseaccretingobjectsweremassiveandalreadyhaddifferentiatedmetalliccoresoftheirown,whichwereformedwithinafewMyrtime(Kleinetal.
2002).
ThelargegravitationalenergyuxaccompanyingsuchrapidaccretionimpliesthatatransientmagmaoceancouldhavebeenmaintainedtoaconsiderabledepthintheEarth(Solomatov2000).
TheimpactoflargeobjectswouldinjectenormousPhil.
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A(2008)366,4253–4271doi:10.
1098/rsta.
2008.
0194Publishedonline30September2008Onecontributionof14toaDiscussionMeetingIssue'OriginanddifferentiationoftheEarth:pasttopresent'.
*Authorforcorrespondence(olson@jhu.
edu).
Presentaddress:DepartmentofGeologicalSciences,CaliforniaStateUniversity,Northridge,CA91330,USA.
4253Thisjournalisq2008TheRoyalSocietyamountsofkineticandpotentialenergyintothemagmaocean,creatingahighlyturbulentuidenvironmentinwhichthesegregationofthecorematerialfromthemantlematerialtakesplace(Rubieetal.
2003,2007).
IfthecoreofadifferentiatedplanetesimalremainedcoherentfollowingimpactwiththeEarth,itwouldquicklysinkthroughthemagmaocean,andassumingthatthemagmaoceandidnotextendtothecore,themetalwouldthendescendthroughtheunderlyingcrystallinemantlebyoneofseveralpossiblemechanisms,includingfracturepropagation(Stevenson1990)oraslargediapirsderivedfromRayleigh–Taylorinstabilities(Elsasser1963;Idaetal.
1987).
Althoughsuchdirecttransportoflargemetalvolumesintothecoreprobablydidoccur,theevidenceinfavourofmetal–silicatechemicalre-equilibrationsuggeststhatthiswasnotthenorm.
Instead,themantleabundanceofsiderophileelementssuchasNiandCosuggeststhattherewasapost-impactphasecharacterizedbysomeamountofchemicalequilibriumbetweenmetalandsilicates(Chabotetal.
2005).
Elementpartitioningindicatesthisre-equilibrationoccurredatelevatedtemperature(approx.
2500K)andrelativelyhighpressures,intheregionof30–40GPa(Righter&Drake2000).
Equilibrationbetweenmetalsanduppermantlesilicatesathighpressurepriortothenalstageofcoresegregationplacessomeimportantconstraintsonthecoreformationprocess.
Itrequiresthatlargeplanetesimalobjectsdisaggregateatthetimeoforshortlyaftertheirimpact.
Therateofextractionofsiderophileelementsfromliquidmetalsislimitedbytheirmassdiffusivity,whichistypically10K9m2sK1orless.
Evenifthesiderophileextractionprocessisacceleratedbythepresenceofthininterfaciallayers,theirlowmassdiffusivitylimitsthesizeofliquidmetaldropsorlamentsthatcouldequilibratewiththemagmatoafractionofametreorsmaller(Rubieetal.
2003).
Suchsmallmetalvolumescannotdescendasubstantialdistancethroughcrystallinemantleunlessthereisapermeablenetworkalreadyinplace.
Althoughitispossiblethatsuchanetworkmightdevelopinthesolidmantle(Yoshinoetal.
2003)itrequiresacombinationofsmalldihedralanglesbetweenthemoltenmetalandthesolidsilicatesforpermeability,asubstantialmeltfractionorporosity,andalsolargeshearstressesinthesolid(Hondaetal.
1993).
Thisparticularcombinationismostlikelytooccurnearthebaseofthecrystallinemantleratherthannearitstop,becausethedihedralangleatlowerpressuresmaybetoolargetoproducethenecessarypermeability(Rubieetal.
2007).
Otherimportantconstraintsonthesegregationmechanismarethesuper-heatedstateofthecoretodayandthepowerrequirementsofthegeodynamointhepast.
Coresuperheatistheexcesspotentialtemperatureofthecorecomparedwithitsmeltingtemperatureatsurfacepressure.
Basedonthepresent-daygeotherm,thecoresuperheatisnowapproximately1000K.
Accordingtothermalevolutionmodelsofthecore,however,itmayhavebeentwicethislargeatthetimeofthecoresegregation.
Inordertopowerthegeodynamo,thecoremustlosesubstantialheattothemantle,intherangeof4–10TW(Nimmo2007).
ThisimpliesthattheaveragetemperatureofthecorewasfarhigherintheearlyEarth,particularlywhenthesolidinnercorehadnotyetnucleatedandthegeodynamoprobablyreliedonthermalconvectionalone(Labrosseetal.
2001).
Wenotethatalargeinitialcoresuperheatisnoteasilyreconciledwithpercolationastheprimarycoreformationmechanism,becausethecorematerialwouldhavecomeintoapproximatethermalequilibriumwiththecolderprimitivemantle.
P.
OlsonandD.
Weeraratne4254Phil.
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R.
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A(2008)Inthispaper,weexaminesomeuiddynamicalprocessesofcoresegregationusinglaboratoryexperiments.
Weuseconcentratedanddilutesucrosesolutions(cornsyrups)torepresentthecrystallinemantleandthemagmaocean,respectively,andliquidgalliumtorepresentthecore-formingmetals.
Gravitationalsegregationinthisenvironmentoccursthroughtheformationofnegativelybuoyant,mixed-compositionstructuresthatwecallmetal–silicateplumes.
Theseplumesformnaturallyinthethree-uidsystemofourexperiments,andwedemonstratethatmetal–silicateplumesintheearlyEarthcanprovideameansformetalsdispersedinthenear-surfacemagmaoceantodescendrapidlytothecore,eventhroughaveryviscousprimitivemantle.
Weshowthatmetal–silicateplumesarehighlydissipativestructures,andcanresultinlargecoresuperheatwhilesimultaneouslyallowingforsomedegreeofhigh-pressurecore–mantlechemicalre-equilibration.
Finally,wearguethatdescendingmetal–silicateplumestendtoevolveintoascendingsilicateplumeswithtime,suggestingthattheearliestdynamicsofthedeepmantlemayhavebeenplumedominated.
2.
ExperimentalmethodsWeconsiderthedynamicsofliquidgalliuminamoreviscousbutlessdenselayereduid(cornsyrup–watersolutions)inthreedifferentsituationsthatarerelevanttothecoresegregationprocess.
First,weexaminethemotionofasingledropofliquidgalliumasitcrossesthestableinterfacebetweenalow-density,lessviscousdilutedsyrupsolution(representingthemagmaocean)andahigh-density,moreviscousconcentratedsyrupsolution(representingthecrystallinemantle).
Second,weexaminetheinstabilityofagalliumlayerattheinterfacebetweendiluteandconcentratedsyrupsolutions.
Finally,weconsidertheinstabilityofagalliumemulsionintroducedabovealayeredmoreviscousuid.
Theexperimentsweremadeatroomtemperatureinrectangularplastictanks17cmindepthand12!
6cminarea.
Thelowerlayerconsistedofstandardgradecornsyrupthatwasfurtherconcentratedbyheatingandevaporationforseveralhours,thenallowedtocool.
Byvaryingthelengthandtemperatureofheating,wewereabletoproducearangeofconcentrationswithdynamicalviscositiesfrom300to7000Pasanddensitiesfrom1200to1430kgmK3.
Theupperlayerconsistedofthesamesyrupdilutedwithwatertoreduceitsdensityandviscosity.
Typicalupperlayerviscositieswereintherangeof1–10Pas,andthedensityreductionoftheupperlayerrelativetothelowerlayerwastypically10percent.
Variousdyesweremixedintotheupperlayerforvisualizationpurposes,includinguoresceindyeandbluevegetabledye.
Thepresenceofthesedyeshadnodiscernibleeffectsonthedensityorviscosityofeithersolution.
Galliumliquidhasadensityofapproximately6000kgmK3andaviscosityofapproximately2!
10K3Pasnearits298Cmeltingpoint.
Galliumisaconvenientliquidmetalforexperimentationatroomtemperatures,however,becauseitskineticsstronglyinhibitfreezingwhileincontactwithotherliquidssuchascornsyrups.
Galliumreactswithairandwithdissolvedoxygentoformvariousgalliumoxides,althoughtheseoxidationreactionscanbelargelysuppressedforthedurationofanexperimentbyincludingasmallamountofhydrochloricacidinthesyrupsolution.
Liquidgalliumhasalargeinterfacialtensionincornsyrup,sx700mJmK2.
4255Metal–silicateplumesandcoreformationPhil.
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A(2008)Severaldimensionlessparameterscharacterizetheseexperiments.
OneistheReynoldsnumberReZrWR/h,whererisdensity;hisdynamicalviscosityofthehostuid;andWistheterminalvelocityofagalliumdropofradiusRinthatuid.
AnotherparameteristheBondnumberBZDrgR2/s,wheregisgravityandDristhedifferencebetweengalliumandhostuiddensities.
Twoadditionaldimensionlessparametersthatcharacterizethebackgroundstraticationarethedensityratiorm/rpandtheviscosityratiohm/hp.
Throughoutthispaper,thesubscripts'm','p'and'c'denotedensitiesandviscositiesoftheupperlayer(representingthemagma),thelowerlayer(representingthecrystallineor'primitive'mantle)andthemetal(core-formingmaterial),respectively.
TypicalvaluesoftheaboveparametersinourexperimentsareRew10K4,rm/rpx0.
9,hm/hpx0.
01andBx0.
2K20.
InterfacialtensiondominatesintheexperimentswithsmallvaluesofB,whereasbuoyancyeffectsdominateforlargevaluesofB.
Weusetheterms'drops'and'diapirs'todescribeisolatedparcelsofliquidgalliumfallingthroughthesyrupatsmallandlargeBondnumbers,respectively.
3.
Single-dropexperimentsThelow-Reynolds-numbermotionofaliquiddropthroughastratieduidinterfaceisoffundamentalimportanceinavarietyofgeologicalandgeophysicalcontexts,aswellasindustrialsituations(Chi&Leal1989).
ExperimentsonliquidandsoliddropsfallinginastablystratieduidhavebeenmadeforgeophysicalapplicationsbyManga&Stone(1995)andSrdic-Mitrovicetal.
(1999)usingtwo-layeredandcontinuouslystratieduid,respectively.
Thebasicphenomenaareillustratedbythesequenceofimagesingure1.
Asingledropofliquidgalliumisinjectedfromasyringeintotheupper(red)layerofthestratiedtwo-layerhostuid.
Thegalliumdropdeformstheinterfaceasitapproachesthedenserandmoreviscouslower(blue)layer.
Thedropissurroundedbyalmoftheupperlayeruidasitfallsbelowthestaticleveloftheinterface,andatrailingconduit(sometimescalledacaudal)formsbehindthedrop,consistingofuidentrainedfromtheupperlayer.
Asthedropfallsafewofitsowndiametersintothelowerlayer,thelmofupperlayeruidsurroundingpmFigure1.
Timesequenceshowingthemotionofasingledropofliquidgalliumfallingthroughatwo-layerstratiedsucrosesolution,formingatrailingconduit.
Upperlow-viscosityuid(m)appearsred,lowerhigh-viscosityuid(p)appearsblue.
Thelow-viscosityuidappearsgreenwhenpassingthroughthehigh-viscosityuid,illuminatingtheconduit.
Theupper/lowerlayerdensityandviscosityratiosare0.
9and0.
01,respectively.
TheReynoldsandBondnumbersofthegalliumdropare0.
01and0.
3,respectively.
Dropdiameteris4.
3mm.
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A(2008)thedropissqueezedbacktowardsthedrop'strailingedgeandincorporatedintothetrailing(green)conduit.
Becausetheconduituidissolubleinthelowerlayer,surfacetensioninstabilitiesdonotbreaktheconduitorseparateitfromthedrop.
Instead,theconduitremainsconnectedtoboththedropandtheupperlayerasthedropfallsadistancecorrespondingtomanyofitsowndiameters.
Forexample,weobservedinsomeexperimentsthattheconduitremainedcontinuouslyconnectedformorethan40dropdiameters.
Thewidthoftheconduitdependsonthedropsizeandtheviscosityanddensityofthetwohostuids.
Indynamicalterms,therelevantdimensionlessparameteristheratiooftheterminal(downward)speedofthedropWtothecharacteristic(upward)velocityofthebuoyantuidintheconduitw.
Ifw/Wissmall,thenacylindricalconduitwiththesamediameterfollowsthedrop.
Inthissituation,theconduitislledwithupperlayeruidthatisessentiallyinhydrostaticequilibrium,asinastagnantwell.
Alternatively,ifthepropertiesofthelowerlayerpermitthewalloftheconduittoconstrictbyviscousrelaxation,thentheratiow/Wwillnotbesmall.
Inthissituation,theconduitwillprogressivelyconstrictasthedropfalls,asshowningure1.
ThepresenceoftheconduitcausestheterminalvelocityofthedroptodeviatefromthepredictedStokesvelocityofthesamedropinaninnitehomogeneousviscousuid.
Forthedropshowningure1,thedragonitsleadinghalfiswellapproximatedbytheStokesdrag,DCZ2phpRW,butthedragonitstrailinghalfisdifferentbyvirtueoftheconduit.
Theeffectoftheconduitdragontheshapeofthedropisnotevidentingure1,sincetheBondnumberinthiscaseissmallandinterfacialtensiontendstokeepthedropnearlyspherical.
Inothercases,wherebothw/Wandhm/hparesmall,theconduituidisessentiallystaticandonlyitspressuredecitcontributestothedrag,sothatthedragonthetrailinghalfofthedropisDKZp(rpKrm)gR2z,wherezisthedistancethatthedrophasfallenbelowtheinterface.
AtlowReynoldsnumber,thesumofthesedragforcesbalancesthenetweightofthedropFaccordingtoDCCDKZF;3:1whereFZ2pgR3(rcKrp)/3isthenetweightofasemi-sphericalhalf-diapirintheloweruid.
SubstitutingtheexpressionforFandthetwodragfactors,(3.
1)canberewrittenasanequationforthevelocityofthedropasafunctionofitsdepthbelowtheinterface,or,alternatively,itsdepthasafunctionoftimetWzhdzdtZrcKrpgR23hpKrpKrmgR2hpz:3:2Providedtheconduitremainsintact,thesecondtermin(3.
2)producesasteadydecelerationofthedropwithincreasingdepthbelowtheinterface.
Pre-existinglowviscosityconduitsproducedbyaninitialdropcanincreasetheterminalvelocityofsubsequentdropsanddiapirsabovethatgivenby(3.
2).
Figure2showsthemotionofadropalongapre-existingconduit.
Inthiscase,theconduitreducesthedragontheleadinghalfofthedrop,providingalowerresistancedescentpath.
IftheBondnumberissmall(andassumingtheuidthroughwhichitdescendsisnotturbulent)thedropremainsnearlyspherical,asitdoesingure2.
However,iftheBondnumberislarge,thediapirdeformsintoamorestreamlinedsolitarywaveshapeanddescendsmorerapidlywithintheconduit(Olson&Singer1985;Manga&Stone1995).
Theinuenceof4257Metal–silicateplumesandcoreformationPhil.
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A(2008)pre-existingconduitsontheterminalvelocityisseeningure3,whichshowstheincreaseinvelocityforasuccessionforequalvolumedropsdescendingalongthesamepath.
Repeateddropsenlargethelow-resistanceconduit,andthevelocityofsuccessivedropsrstapproachesandthenexceedstheStokesvelocityofaninviscidsphere.
4.
InstabilityofametalpondIfthecore-formingmetalderivedfromalargeplanetesimalobjectremainscoherentfollowingimpact,itwillquicklysinktothebaseofthemagmaoceanandpondatthetopofthehighlyviscousprimitivemantle.
Ifthemetalpondformedthiswayisshallowcomparedwithitslateraldimension,subsequentgravitationallyinducedmotionwillapproximatetheclassicalRayleigh–Taylorinstabilityofathindenselayerabovealessdensebutmoreviscousuid(Chandrasekhar1961;Whitehead&Luther1975).
Inthethree-layersystemenvisonedfortheearlyEarth,theRayleigh–Taylorinstabilityiscontrolledbythematerialpropertiesofthemostviscousregion(i.
e.
theviscosityanddensityoftheprimitivemantle)andthedensityofthemetal(Solomatov2000).
ThecharacteristicgrowthtimeofaRayleigh–TaylorinstabilityinthissituationdependsinverselyonthewavelengthofthedisturbancelandisapproximatelytRTx2phprcKrpgl:4:1(a)(b)(c)(d)Figure2.
(a–d)Sequenceshowingtheplumeevolutionformedbysuccessivedropsofgalliumfallingthroughthelowerlayerofatwo-layerstratiedsucrosesolution.
(a)Initialplumewithgalliumheadandsucrosetail.
(b)Seconddropfallingalongtheinitialconduit.
(c)Subsequentdropfallingalongpre-existingconduit.
(d)Risingstreamlinedsoliton.
Theupper/lowerlayerdensityandviscosityratiosare0.
9and0.
01,respectively.
TheReynoldsandbondnumbersofthegalliumdropsare0.
01and0.
3,respectively.
Dropdiametersare4.
4mm.
P.
OlsonandD.
Weeraratne4258Phil.
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A(2008)Formulaeequivalentto(4.
1)havebeenderivedbyCanright&Morris(1993)forCartesiangeometryandbyRibe&deValpine(1994)fortheRayleigh–Taylorinstabilityofathin,dense,low-viscositylayerinamoreviscous,lessdensesphericalshell.
MetalpondsareexpectedtoformatthebaseofamagmaoceanfollowinglargeimpactsifthesettlingtimeformetalinthemagmaoceanisshortcomparedwithtRT,asitislikelytobe(Reese&Solomatov2006).
Thesituationmaybedifferentforsmallobjects,orlargeobjectsthatdisaggregateonimpact.
Dependingonthesizeofthemetaldrops,thetimeofsettlingandcompactionatthebaseofthemagmaoceanmaynotbeshortwhencomparedwithtRT,inwhichcasetheinstabilityinvolvesametal–silicateemulsion.
Thissituationwillbeconsideredin§5.
Figure4showsasequenceofimagesfromaRayleigh–Taylorinstabilitydevelopinginalayerofliquidgalliumbetweenthickerlayersofdiluteandconcentratedsyrup.
ThegalliumlayerthicknessisinitiallyhZ0.
5cmandtheupperandlowerlayerviscositiesarehmZ10andhpZ6000Pas,respectively.
Thewavelengthoftheinstabilitypredictedfor(4.
1)iscomparabletothetankdimension,andwendtheleaststablemodeiswavenumberoneortwo.
Theclose-upsingure5showthesegmentationofthegalliumlayerbytheniteamplitudeRayleigh–Taylorinstability.
Segmentationoccursatthelocationofupwellinginthelowerviscousuid.
Inourexperiments,theseupwellingstypicallywerefoundatadistanceequaltoone-quarterofthetankwidthfromoneoftheboundaries,andsplitthegalliumlayerintotwopartswithvolumesintheapproximateratioof3:1.
024681012140–2–22468101214dropdistance(cm)droptime(min)C=1/3C=2/9Figure3.
Distanceversustimeplotshowingthemotionofliquidgalliumdropsinthelowerlayerofatwo-layerstratiedsucrosesolution.
Distanceandtimearemeasuredfromtheuid–uidinterface.
Circles,galliumdropinahomogeneousuid;squares,initialdropinastratieduid;triangles,seconddropinthesamestratieduid.
Theupper/lowerdensityandviscosityratiosare0.
9and0.
01,respectively.
TheReynoldsandBondnumbersofthegalliumdropare0.
01and0.
3,respectively.
DashedandsolidlinesshowtheoreticalStokesvelocityforaninviscid(CZ1/3)andarigid(CZ2/9)sphere,respectively.
4259Metal–silicateplumesandcoreformationPhil.
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A(2008)Figure6showshowtheRayleigh–Taylorinstabilityevolvesintime.
Here,theBondnumberislarge,andthedominanceofgravityoversurfacetensioneffectsattheattenedtrailingedgesofthemetalinstabilitiesisclearlyevident.
Inthispaper,weusetheterm'half-diapir'todescribetheshapeoftheleadingmetalingalliumpondlayer,cfluidlayer,pfluidlayer,m(a)(b)Figure4.
(a)OnsetofRayleigh–Taylorinstabilityofaliquidgalliumlayerbetweentwolessdensebutmoreviscoussucrosesolutionlayers.
Theupper/lowerdensityandviscosityratiosare0.
9and0.
01,respectively,andthegalliumlayerthicknessisinitially0.
7cm(tZ0hours).
(b)Laterintime(tZ4hours).
(a)(b)(c)Figure5.
(a–c)TimesequenceshowinggalliumlayersegmentationduringRayleigh–Taylorinstabilityofthethree-layeruidsystemingure4.
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A(2008)largeBondnumbermetal–silicateplumeswithtrailingconduits.
Inthisparticularexperiment,Rex10K4,Bx20andhm/hp/1,thesetofconditionsinwhichtheRayleigh–Taylorinstabilityproduceshalf-diapirswithlargetrailing(a)(b)(c)(d)Figure6.
(a–d)Timesequenceshowingthedevelopmentofhalf-diapirsandtrailingconduitsfollowingRayleigh–Taylorinstabilityofaliquidgalliumlayerbetweentwolessdensebutmoreviscoussucrosesolutionlayers.
Galliumdiapirdiameteron(a,c)is4.
5cm.
Totaldescenttimefrom(a)to(d)is10min.
4261Metal–silicateplumesandcoreformationPhil.
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A(2008)conduits.
Theattrailingedgesofthemetalhalf-diapirsindicatethattheconduitsarelledwithessentiallystaticupperlayeruid.
Asthehalf-diapirsapproachthebaseofthetankandcollapse(thisistheexperimentalequivalenttomergingwiththegrowingcore)thetrailingconduitsconstrictandthenettransportdirectioninthesestructuresreversesfromdownwardtoupward.
Inspiteofthisowreversal,theconduitsremainpartiallyopenlongafterthemetaldiapirhascollapsed,providingapathforupperlayeruidentrainedbytheinstabilitytodrainbackuptotheupperlayer.
Inoneexperiment,weappliedheattothebaseofthetankafterthecollapseofthemetalhalf-diapir,andobservedthattheresidualconduitwasevolvedintoapositivelybuoyantthermalplume,asshowningure7.
5.
InstabilityofadenseemulsionTheexcessabundanceofsomesiderophileelementsinthemantleisevidencethatsomedegreeofmetal–silicateequilibrationatelevatedtemperatureandpressureoccurredpriortonalcoresegregation.
Sincechemicalequilibrationbetweenliquidsisdiffusioncontrolled,itisoftensupposedthatmuchoftheironthatformedthecorewasdispersedasnedropletsinthemagmaocean,andunderwentpartialphaseseparationwherebythemetalsbecameconcentratedatthemagmaoceanbase(Rubieetal.
2003;Woodetal.
2006).
ProcessesinvolvedingravitationallydrivenphaseseparationinimmiscibleuidswithlargeviscositycontrastshavebeeninvestigatedbySato&Sumita(2007).
Here,weareconcernedwiththegravitationalinstabilityofadenselayerinwhichthephaseseparationisincomplete.
ArelatedphenomenonistheRayleigh–Taylorinstabilityofuidscontainingparticlesorbubbles,whichhasbeenexaminedinthecontextofmagmamixing(Thomasetal.
1993;Michioka&Sumita2005)Figure7.
Structureofapositivelybuoyantthermochemicalplumeevolvedfromthetrailingconduitofanegativelybuoyantgalliumplume.
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OlsonandD.
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A(2008)andsedimentingsuspensions(Voltzetal.
2000).
Dependingonthepropertiesofthemixture,eithersedimentplumesorsedimentngersareobservedtoform.
Ifthesizedistributionoftheheavydropsisnotuniform,aswouldbethecaseifthedropsformedfromthedisaggregationofalargeparentbody,thenthelargestoftheheavydropswilldictatethetimingandthetypeofinstabilitythatresults.
Todemonstratethiseffect,wesynthesizeddenseemulsionsbyshakingmixturesofgalliumanddilutedsyrupsolution.
Thegalliumbecamethedispersedphase,formingdrops,whilethesyrupsolutionbecamethecontinuousphase.
Thesizespectrumofthegalliumdropsproducedthiswayisnon-uniformanddependsonthedurationoftheshaking,i.
e.
thetotalkineticenergyinputtotheemulsion.
Inparticular,thereisalwaysoneormorelargerresidualgalliummassespresent.
WethenrepeatedtheRayleigh–Taylorexperimentsdescribedabove,usingthisemulsionastheupperlayeruid.
Figure8showstheinstabilityofalayerofgalliumemulsionabovealessdensebutmoreviscoussyruplayer.
ARayleigh–Taylorinstabilitywiththesamewavenumberisobservedinthiscaseasinthehomogeneousgalliumlayerexperiment,anditoccursapproximatelyonthesametimescale.
Themainreasonforthiscorrespondenceisthatthelargestgalliumvolumescoalesceintohalf-diapirs,whichformtheleadingedgeoftheresultingplumes.
Theprimarydifferenceinthiscaseisthatthetrailingconduitcontainsenoughgalliumdropstogiveitoverallnegativebuoyancyinthelowerlayer.
Aftertheleadinghalf-diapirreachesthebaseofthetank,thesmallergalliumdropsinthetrailingconduitgraduallysedimentout,reducingthebulkdensityoftheconduit.
Inallofouremulsionexperiments,wendthatsmallemulsiondropsintheleadingedgecoalesceduringdescentintoacohesiveuidgalliumfront(gure8)thatbehavessimilartothelargediapirexperimentsingure6.
Ifnoadditionalgalliumisaddedtothesystem,theowintheconduiteventuallyreversesanditdrainsbackintotheupperlayer.
However,ifanotherdoseofemulsionisaddedtothesystematthisstage,itdescendsdownoneoftheestablishedconduits.
Withrepeateddosesofemulsion,theconduitscanbekeptopenforlongperiodsoftime.
Insomecases,theconduitsdevelopagrosslyheterogeneousstructure,inwhichgallium-richemulsiondescendsinonepartoftheconduitwhilebuoyant,gallium-depletedemulsionascendsinanotherpart,while,inothercases,nosuchheterogeneitywasobserved.
Inprinciple,itispossibleforthesestructurestoreachastatisticalsteadystate,inwhichthedownwardtransportofmetaldropsequalstheinputmetaluxandthenettransportoflowerdensitydilutedsyrupsolutioniszero.
Metal–silicateplumesintheearlyEarthequivalenttothesestructureswouldhaveveryhightemperatures,owingtotheconcentratedvolumetricheatingcausedbytheviscousdissipationofkineticenergyofthefallingmetaldrops,aneffectthatisnotsignicantonthescaleoftheseexperiments.
6.
Metal–silicatemantleplumesOurexperimentssuggestthefollowingconceptualmodelformetal–silicatemantleplumesduringcoreformation.
PlumeformationisinitiatedbylongwavelengthRayleigh–Taylorinstabilityofaliquid-richironpondorapartly4263Metal–silicateplumesandcoreformationPhil.
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A(2008)segregatedironemulsionlayeratthebaseofthemagmaocean.
Dependingonthelateralextentofthepondlayer,eachRayleigh–Taylorinstabilityproducesoneormoremetaldiapirsthatsinkintothehighlyviscousprimitivelowermantle.
Trailingconduitslledwithmagmaoceanmaterialarethenentrainedintotheprimitivemantlebehindeachdiapir.
Thematerialintheconduitisassumedtobeamixtureofsilicatemagmaanddropsofcore-formingmetals,andinitiallytheseconduitshavethesameradiusastheirleadingdiapir.
Providedthemagmaoceanviscosityhmisverysmallcomparedwiththeprimitivemantleviscosityhp,theuidinthetrailingconduitremainsinnearlyhydrostaticequilibrium,itscontributiontoviscousdragonthediapirisnegligible,andthetrailingedgeofthediapirisat.
Whilethemetalhalf-diapirsdescendintotheprimitivemantle,theirratesofsinkingaregovernedby(3.
2).
Theevolutionofthetrailingconduitsfluidlayer,mfluidlayer,pgalliumemulsionlayer,c(a)(b)(c)(d)Figure8.
(a,b)TimesequenceshowingthedevelopmentofRayleigh–Taylorinstabilityofaninitialgalliumemulsionlayerbetweentwolessdensebutmoreviscoussucrosesolutionlayers.
(c,d)Thegrowthofagalliumemulsionplume;theupper/lowerdensityandviscosityratiosare0.
9and0.
01,respectively,andtheinitialemulsionlayerthicknessis1.
0cm.
Scaleisvisibleneartheupperuidlayer.
Totalexperimenttimefrom(a)to(d)is52min.
P.
OlsonandD.
Weeraratne4264Phil.
Trans.
R.
Soc.
A(2008)dependsonthepropertiesoftheprimitivemantleandtheconduitmaterial.
Ifthematerialintheconduitislessdenseandmuchlessviscousthantheprimitivemantle,theconduitradiusawillconstrictonatimescalegivenbytconduitZhprpKrmga:6:1Thisistobecomparedwiththecharacteristicdescenttimeofthemetaldiapirthroughtheprimitivemantle.
DenotingtheradialdepthoftheprimitivemantlebyL,thediapirdescenttimeisapproximatelytdescentZL/W,whereWZrcKrpgR23hp6:2istheStokesvelocityofahalf-diapirintheprimitivemantle.
Iftdescent/tconduit,thatis,ifrpKrmrcKrp/aL;6:3thentheconduitradiuswillconformtothehalf-diapirradiusRuntilthehalf-diapirreachesthecore–mantleboundary.
Ontheotherhand,iftdescent[tconduittheconduitwillconstrictbeforethediapirreachesthecore.
Athirdpossibilitycanoccuriftheconduitisdenserthantheprimitivemantleowingtoahighconcentrationofcore-formingmaterialintheconduit.
Inthiscase,themetaldrainsthroughtheconduitintothediapirandthemassofthediapirincreasesduringitsdescent.
PreviousstudiesofcoreformationbyRayleigh–Taylorinstabilityofauniform,globalmetallayerabovethehighviscosityoftheprimitivemantle(Elsasser1963;Stevenson1990)concludedthatthemostunstablemodecorrespondstoeitherasinglediapir(modemZ1instability)or,alternatively,asmallnumberofdiapirs.
Diapirsformedthiswaywouldbetoolargetoequilibratewiththeprimitivemantleduringtheirdescent,becauseonlyasmallfractionofthemetalinasinkingdiapircirculatesthroughthethincontactregionwherethechemicalandthermalexchangeswiththemantleoccurs.
ThecharacteristicdistanceadiapirmusttraveltoequilibratewithitssurroundingsbythemixingactionofitsowninternalcirculationisgivenbyR2W/k,wherekistherelevantdiffusivity(heatormass)andWistheStokesvelocity(6.
2).
TheR4dependenceimpliedbythisexpressionmeansthatonlyrathersmalldiapirsordropswillequilibrateindescentthroughtheprimitivemantle.
AnotherimplicationisthatdiapirssmallenoughtomixbytheirowninternalStokesowwouldsolidifyintherelativelylowtemperatureprimitivemantlelongbeforetheyreachedchemicalequilibration,because,exceptforthelightelementsheliumandhydrogen,thediffusivityofheatisoftheordersofmagnitudelargerthanmassdiffusion.
Chemicalequilibriumduringcoreformationrequiressmallmetaldropsizes,and,inorderforthemetaltoremainliquid,thetemperaturemustbehigh.
Here,wedemonstratethatitispossibletomaintainsuchanenvironmentinanequilibriummetal–silicatemantleplumeconduit.
Figure9isasketchofanidealizedfullydevelopedmetal–silicateplumeconduitwithanassumedcircular4265Metal–silicateplumesandcoreformationPhil.
Trans.
R.
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A(2008)crosssection,containingamixtureofliquidcore-formingmetalandsilicatemelt.
Weusectorepresentthemassfractionofmetalsintheplume,whichweassumearedispersedinthesilicatemeltintheformofindividualdrops.
Inrealitytherewillalwaysbeadistributionofdropsizes,butinthisanalysisweconsiderforpurposesofsimplicityauniformdropsizewithasphericalradiusR0.
WeassumethatthemetalisdrawnfromthemagmaoceanintotheconduitanddescendsthroughtheconduitasacloudofindividualStokesdropsallthewaytothecore.
Theconservationofmassforthecore-formingmaterialintheconduitcanbewrittenrcW0CwcAZFc;6:4whereAZpa2istheplumecross-sectionalarea;Fcisthemassuxofcore-formingmetalintotheplume;W0ZrcKrmgR023hm6:5istheStokesvelocityintheconduitforthemetaldrops;andwistheaveragevelocityofthesilicatemagmaintheplume.
Weareparticularlyinterestedinthelimitingcaseoflowmetalconcentration(c/1)andnegligiblemagmatransport,i.
e.
wx0in(6.
4).
Thislaterconditionismetwhentheconduitmixtureisneutrallybuoyantwithrespecttotheprimitivemantle.
Theadditionofmetalwilltendtoevolveanotherwisepositivelybuoyantsilicatemeltplumetowardsastateofneutralbuoyancy,fordynamicalreasons.
Iftheconduitmaterialisinitiallylessdenseonaveragethanthesurroundingprimitivemantle,itwillconstrict,therebyincreasingitsmetalconcentrationanditsdensity.
NeutralbuoyancyoftheplumemixturerelativetotheprimitivemantlecorrespondstoametalconcentrationofcZrpKrmrcKrm:6:6azTsilicateliq.
plumemetalliq.
adiabatsuperheatL(a)(b)Figure9.
(a)Sketchofasteady-statemetal–silicatemantleplumeconduitusedintheanalysis,showingliquidmetaldropssinkingthroughasilicatemagmamatrix.
(b)Schematicdistributionoftemperaturealongtheplume,includingmetalandsilicatemeltinglines,metaladiabat,plumetemperatureproleandcoresuperheat.
P.
OlsonandD.
Weeraratne4266Phil.
Trans.
R.
Soc.
A(2008)Accordingto(6.
4),(6.
5)and(6.
6),theequilibriumcross-sectionalareaofaneutrallybuoyantplumeconduitisAZ3hmFcrcrpKrmgR02:6:7Therelativemotionofthemetaldropswithrespecttothemagmaprovidesthemajorsourceofheatingintheplume,throughviscousdissipation.
TheamountofdissipativeheatingintheplumecanbeestimatedfromthechangeingravitationalpotentialenergyduetotheStokesvelocityofthemetaldrops.
Forasingledropfallingataspeedgivenby(6.
5),therateofgravitationalenergyreleaseisQZ4prcKrm2g2R05hm:6:8Assumingthatalltheenergyreleasedthiswaygoesintoviscousheat(thisisareasonablygoodassumptionforStokesowaroundisolateddrops),therateofvolumetricheatproductionintheplumefromauniformdistributionofsuchdropsisgivenbyeZ3cQ4pR03Z3rcKrm2g2cR02hm:6:9Thisresultcanbeusedtoestimatethesuperheatacquiredbymetaldropsdescendingfromthemagmaoceantothecorethroughtheconduit.
Ignoringheatconduction,thechangeintemperaturewithtimeforametaldropfallingthroughtheconduitconsistsofanadiabaticincrease(denotedbysubscriptad)plusthesuperadiabatictemperatureincreaseduetotheviscousheating(denotedbysubscriptsa),andisgivenbydTdtZW0dTdzjadCdTdtjsaZgTW0HTCermCp;6:10wheregistheGruneisenparameterandHTZCp=agisthetemperaturescaleheight(hereCpandaarespecicheatandthermalexpansivityofthemagmaintheplume,respectively).
Accordingto(6.
5),(6.
6)and(6.
10)thesuperadiabatictemperatureincreaseduetoviscousheatingexperiencedbyametaldropfallinginaconduitadistanceLataspeedW0isapproximatelyDTsaZeLrmCpW0Z9rpKrmgLrmCp:6:11Althoughtheviscousheatproduction(6.
9)dependsonthemagmaviscosity,whichispoorlyconstrained,theintrinsicsuperadiabatictemperatureincreaseexperiencedbythefallingdrops(6.
11)isindependentofthisviscosity.
Italsohasnoexplicitdependenceonthemetalproperties,aconsequenceofassumingneutralconduitbuoyancy.
Theamountofcoresuperheatthatwouldresultfromthisstyleofmetalsegregationisprobablyquitelarge.
Usingtheparametervaluesintable1,(6.
11)givesDTsaZ3200K.
Thisisfarinexcessoftheincreaseinmetalandsilicate4267Metal–silicateplumesandcoreformationPhil.
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A(2008)liquidustemperaturesacrossthepressureintervaloftheprimitivemantle,whichensuresthatmetal–silicateplumeconduitswouldremainentirelymolten.
Notethatwehaveassumedasmalldensitydifferencebetweentheprimitivemantleandtheconduitmagma;largerdifferencesimplyevenmoredissipativeheatingintheplume.
TheactualsuperheatofthecoreformedthiswaywouldbesmallerthanDTsa,however,forseveralreasons.
First,wehaveignoredtheeffectsofheatdiffusionfromtheplumeintheaboveanalysis.
Second,themetaldiapirsthatproducetheseconduitsmaynotexperienceasmuchsuperheatinginthecourseoftheirdescentthroughtheprimitivemantle.
Andnally,iftheproto-Earthalreadyhadasmallcore,thatmetalwasprobablyatalowertemperaturethantheincomingmetal.
Intable1anominalmetaldropsizeof0.
1misassumed.
Wecanestimatethethresholdradiusforadroptochemicallyre-equilibratewiththemagmaintheconduitbyequatingitscharacteristicmassdiffusiontimeR02=ktoitsdescenttime,tdropZL/W0.
Thiscriterionpredictsthatadropwillnotchemicallyre-equilibratewiththeconduitmagmaifitsradiusfarexceedsR0critZ3khmLrcKrmg1=4:6:12Usingtheparametervaluesintable1,R0critx0:25m,anestimatethatissensitivetotheassumedmassdiffusivityandtheviscosityoftheconduitmaterial.
Ontheotherhand,dropsforwhichR0/R0critcouldlocallyequilibratewiththesurroundingmagmaasafunctionofdepth.
Table1.
Metal–silicatemantleplumemodelproperties.
inputsnotationunitsassumedvaluemetalmassuxFckgsK16!
108magmadensityrmkgmK33.
9!
103primitivemantledensityrpkgmK34.
0!
103core-formingmetaldensityrckgmK39!
103magmaviscosityhmPas1!
107primitivemantleviscosityhpPas1!
1023gravitygmsK29metaldropradiusR0m0.
1massdiffusivitykm2sK11!
10K10primitivemantledepthLm2!
106resultscalculatedvalueplume,metaldiapirradiia,Rkm265metalconcentration(conduit)cn.
d.
0.
02viscousheating(conduit)eWmK30.
125metalsuperheat(conduit)DTsaK3200Rayleigh–TaylorinstabilitytimetRTMyr1diapirdescenttimetdiapirMyr6dropdescenttime(conduit)tdropKyr4conduitconstrictiontimetconduitMyr17.
6massdiffusionradiusR0critm0.
6P.
OlsonandD.
Weeraratne4268Phil.
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A(2008)Otherpropertiesofmetal–silicateplumescanbeestimatedfromthismodelusingtheparametervaluesintable1.
Accordingto(6.
6),theconcentrationofmetalsinaneutrallybuoyantconduitiscx0.
02.
Thesteady-statecross-sectionalareaofaneutrallybuoyantmetal–silicateplumedependsonthemassuxofmetalintotheconduit,accordingto(6.
7).
Coreformationlasting20Myrcorrespondstoatimeaveragemetalmassuxofnearly3!
109kgsK1,assumingtheinitialcoremasswasnegligiblecomparedwithitsnalmass.
ThevalueFcZ6!
108kgsK1intable1correspondstoveequal-sizedplumesactiveoverthe20Myrperiod.
Accordingto(6.
7),theradiusofeachplumewouldbeapproximately265km.
Thetimerequiredtocreatesuchaplumeisgivenbythegrowthtimeoftheinitialinstabilityplusthedescenttimeofahalf-diapirwiththissameradius.
Using(6.
2),theparametervaluesintable1givetdiapirx6Myrforthedescenttimeand,accordingto(4.
1),thegrowthrateoftheinitialinstabilitywouldbeoforder1Myrorless.
Thesearetobecomparedwiththeconstrictiontimeofthetrailingconduit.
Iftheconduitconsistedonlyofsilicatemagmawiththepropertyvaluesintable1,thentconduitx17.
6Myr.
Thismightbeanunderestimate,sinceitislikelythattheconduitwouldalsocontainsomecore-formingmetal,whichwouldreduceitsdensitydecitrelativetotheprimitivemantleandincreasetheconstrictiontime.
Inanycase,themetal–silicateplumewouldbefullydevelopedwithintherequisite20Myrperiod,providedtheprimitivemantleviscositywasnottoomuchlargerthanthe1023Pasassumedinthismodel.
7.
DiscussionOurexperimentsindicatethatlargeirondiapirssinkingthroughamagmaoceanintotheviscousprimitivelowermantleformtrailingconduitslledwiththeuppermantlesilicatemeltsandmetaldrops.
Thelargeirondiapirscouldoriginatefrommajorimpacteventsinvolvingdifferentiatedplanetesimals,orthroughRayleigh–Taylorinstabilitiesofmetalpondsoremulsionsconcentratedatthebaseofthemagmaocean.
Themoltenconduitsprovidealow-resistancepathforadditionalmetaltransportthroughthehigh-viscosityprimitivemantle.
Metal-richsilicateplumesarehighlydissipativeduetoviscousheatingprovidedbythefallingmetaldrops,whichfacilitateshigh-pressuremetal–silicateequilibriumintheconduitandstronglysuperheatsthemetaldropsbeforetheyenterthecore.
Wecanspeculatehowmetal–silicateplumesmightevolvefollowingthemainphaseofcoreformation.
Astheuxofcore-formingmetalthroughtheplumedecreases,thenetbuoyancyoftheplumematerialwouldbecomepositiveintheprimitivemantle.
TheoverallowintheplumewouldthenbeupwardandthesilicatemeltintheconduitwoulddrainbacktowardstheEarth'ssurface.
Whentheowdirectioninthemetal-depletedplumesreversed,theywouldbegintoextractsomeofthesuperheatfromthecore.
Thisprocesscouldservetoinitiatethegeodynamo,byestablishingaregimeofthermalconvectioninthecoreshortlyafterthemainphaseofcoreformation.
Thechangefromneutrallybuoyantmetal–silicateplumesrelativelyrichincore-formingmetalstopositivelybuoyantplumesrelativelydepletedincore-formingmetalsislikelytobeagradualtransition,whichmightbeexpectedtorecyclesignicantamountsof4269Metal–silicateplumesandcoreformationPhil.
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A(2008)metalsbacktowardstheEarth'ssurface.
Reversalinthedirectionofcore-formingplumesoffersanexplanationforwhythereweremajorpulsesofvolcanicactivityandheterogeneouscrustformationintheearlyhistoryofsometerrestrialplanets,suchashasbeenproposedforthecrustaldichotomyonMars(Ke&Solomatov2006).
ItalsosuggeststhatplumesmayhavebeentheinitialphaseofwholemantleconvectionintheEarth.
ThisresearchwassupportedbygrantEAR-0604974fromtheGeophysicsProgramoftheNationalScienceFoundation.
D.
W.
wassupportedbytheCarnegieInstitutionofWashingtonandbytheNASAPlanetaryGeologyandGeophysicsProgramundergrantNNX07AP50G.
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