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Earth and Planetary Science Letters

www.elsevier.com/locate/epsl

Subduction and atmospheric escape of Earth’s seawater constrained by hydrogen isotopes

Hiroyuki Kurokawa

^∗

, Julien Foriel, Matthieu Laneuville, Christine Houser, Tomohiro Usui

Earth-LifeScienceInstitute,TokyoInstituteofTechnology,2-12-1Ookayama,Meguro,Tokyo152-8550,Japan

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

Articlehistory:

Received28March2018 Accepted10June2018 Availableonline22June2018 Editor: B.Buffett

Keywords:

globalwatercycle hydrogenisotopes subduction atmosphericescape earlyEarth seawater

The hydrogen isotopic (D/H) ratio reflects the global cycling and evolution of water on Earth as it fractionates through planetary processes. We model the water cycle taking seafloor hydrothermal alteration, chemical alteration of continental crust, slab subduction, hydrogen escape from the early Earth,anddegassingatmid-oceanridges,hotspots,andarcsintoaccount.ThedifferencesinD/Hratios betweenpresent-day oceans,oceanic andcontinental crust,and mantleare thoughttoreflect isotopic fractionationthroughseaflooralteration,chemicalalteration,andslabdehydration.However,ifthespeed ofplatetectonicshasbeennearlyconstantthroughoutEarth’shistory,thedegassingandregassingrates aretoosmalltoreachthepresent-dayD/Hratios.Weshowthat(a)hydrogenescapefromreducedearly atmosphere,(b)secularnetregassing,or(c)fasterplatetectonicsonearlyEarthisneededtoreproduce thepresent-dayD/Hratiosofthewaterreservoirs.ThelowD/HratioofArcheanseawaterat3.8Gahas previouslybeeninterpretedasasignatureof(a)hydrogenescape,butwefinditcanalsobeexplained eitherby(b)secularnetdegassingorby(c)fasterplatetectonicsonearlyEarth.Theratesofhydrogen escape fromearly Earthand secularregassing onpresent-day Earthare constrained tobelower than 2.1×^{1011 kg/yrand 3}.9×^{1011 kg/yr.}Consequently, the volumeof waterinthe present-day mantle couldresultentirelyfromtheregassingthroughEarth’shistory.Inthatcase,thevolumeofinitialoceans could be2to3timeslargerthanthatofcurrentEarth.We suggestthat,inaddition totheD/Hratio ofArcheanseawater,identifyingtheD/HratiosofbothseawaterandmantlethroughoutEarth’shistory wouldallowtodistinguishtheseevolutionaryscenarios.

©^{2018ElsevierB.V.Allrightsreserved.}

1. Introduction

Waterplaysacriticalroleincontrollingthephysicalandchem- ical evolution on Earth through atmosphere–ocean–continent in- teractionswhichcontroltheatmosphericcomposition,theclimate through the carbon cycle, the subduction of water, and possibly eventheemergenceandevolutionoflife(e.g.,Gaillardetal.,2011;

Walker,1977;Höningetal.,2014;HöningandSpohn,2016;Dohm and Maruyama, 2015). Furthermore, the abundance of water in Earth’sinteriorinfluences themantlemelting, rheology,andstyle ofconvection(e.g.,Hirschmann,2006;KaratoandJung,2003;Mei andKohlstedt,2000).

The abundanceof water onthe surface andin the interioris controlledbythedeepwatercyclebetweentheoceansandman- tle, and loss caused by hydrogen escape throughout Earth’s his- tory.Slabsubductiontransportswaterasboundandporewaterin metamorphicrocks andsedimentsthat originate fromhydrother-

*

Correspondingauthor.

E-mailaddress:[email protected](H. Kurokawa).

malalterationoftheseafloorandchemicalalteration ofcontinen- tal crust (e.g., Jarrard, 2003; Bodnar et al., 2013; Höning et al., 2014; Höning and Spohn, 2016). While the majority of the sub- ducted water returns to the oceans directly by updip transport andindirectlybyarcvolcanism,sometraceamountsofwatermay remain in the mantle. Water can return fromthe mantle to the exosphere (here defined asthe atmosphereand hydrosphere) by degassing at mid-ocean ridges and ocean islands. Although the photolysis of water has a negligible effect on the loss of hydro- genfromtheatmospheretoday,thehydrogenescapethrough the photolysisofmethaneinthereducedearlyatmospherebeforethe greatoxidationevent(GOE)at2.5 Gacouldhaveamoresignificant impact(Catlingetal.,2001).

Despite its importance to control the water budget, the bal- ance between the degassing and regassing as well as the early hydrogenescapefluxispoorlyunderstood.Wenotethattheterm

“regassing”meansthewatertransporttothemantleasboundand pore water in metamorphic rocks and sediments. Net regassing fromtheoceanstothemantlehasbeensuggestedfromthegeo- chemical estimates(Ito etal., 1983). The continentalfreeboard is https://doi.org/10.1016/j.epsl.2018.06.016

0012-821X/©^{2018ElsevierB.V.Allrightsreserved.}

proposedtobenearly constantfromtheendoftheArchean(e.g., Schubert and Reymer, 1985) and is interpreted asthe degassing andregassingrates almost achievingbalance (e.g.,Lécuyer etal., 1998; Parai and Mukhopadhyay, 2012). However, Korenaga et al.

(2017) recently argued that the relative buoyancy of continental lithospherewithrespecttooceaniclithospherewashigherinthe past,whichrequiresalargervolume ofoceanicwateratthetime tokeep continentalfreeboard constant.Hydrogenescapeonearly Earthisevenmorepoorlyconstrainedastheatmosphericcompo- sitionatthattimeisnotwellknown.

Hydrogen isotope (D/H) ratio has been used to constrain the globalcycleandlossofwateronEarth,asitfractionatesthrough planetaryprocesses.Earth’smantleisknowntohaveδD= −⁸⁰h to −⁶⁰h ⁽δD= [(D/H)_sample/(D/H)_reference−¹]×^103,wherethe reference is the standard mean ocean water, hereafter SMOW), which is lower than that of today’s oceans (defined hereas the totalhydrosphere)(Kyser andO’Neil,1984;Clogetal.,2013).The lowmantleδD valuehasbeenconsideredtosuggestthattheman- tlebecameisolatedfromtheoceansthroughgeologictime(Kyser andO’Neil,1984),orthatwaterinthemantlehasbeenisotopically fractionated from the source seawater because of seafloor alter- ation andslab dehydrationprocesses (Lécuyer etal., 1998; Shaw etal.,2008,2012).IsotopicanalysisofArcheanmineralsandrocks hasfoundthatArcheanseawaterhasaδD valuelowerthanthatof present-dayoceans, whichhasbeen interpretedasa signature of waterlosscausedbythehydrogenescape(Hrenetal.,2009;Pope etal.,2012).

While D/H ratio has been widely utilized to constrain those processesof thewatercycle andloss,there isno comprehensive modeloftheD/Hevolutionwhichinvolvesallrelevantprocesses.

Previousstudies considered the degassingandregassing(Lécuyer etal.,1998;Shawetal.,2008)orthehydrogenescape(Popeetal., 2012) only.Inaddition,theD/HratiosofthemantleandArchean seawater havebeen considered separately to constrain these dif- ferentprocesses.Becauseall waterreservoirsare coupledtoeach other,alltheseprocessesandD/Hconstraintsshouldbeconsidered simultaneously,whichistheaimofthisstudy.

Wemodeltheglobalwatercycletakingseafloorhydrothermal alteration,chemicalalterationofcontinentalcrust,slabsubduction, atmosphericescape,anddegassingatmid-oceanridges,hotspots, andarcsintoaccount.The modelcalculationsarecompared with theD/Hratiosofwaterindifferentreservoirsonpresent-dayEarth andofArcheanseawatertoconstraintheratesofhydrogenescape from early Earth and of secular regassing on present-day Earth.

Section 2 presents the model. Section 3 shows the results. The implications fortheevolution ofwateron Earthare discussed in Section4.WeconcludeinSection5.

2. Methods 2.1. Model

We constructed a global water cycle model taking the D/H compositionsintoaccount.Fourreservoirs wereconsideredinour model: the oceans, continental crust, oceanic crust, and mantle.

Thesereservoirsexchangewaterthroughseafloorhydrothermalal- teration, chemical alteration of continents, slab subduction, and degassingatmid-oceanridges,hotspots,andarcs(Fig. 1).Hydro- gen lossto space induced by photolysis of methanewas consid- ered to have occurred on early Earth before the GOE at 2.5 Ga.

The oceans in our model include water in small reservoirs that exchange water with oceans over short timescales: atmosphere, biosphere,surface water,groundwater,andglaciers/polarice,see Table2.Hereaftertheoceanswiththesmallreservoirsarereferred toas“bulkoceans”whenwewouldliketodistinguishthemfrom seawaterwithoutsmallreservoirs.Exploringthepossiblerangesof

Fig. 1.Schematic view of Earth’s water cycle in our model.

fluxes, we constrain thewater cycleanddiscusstheimplications fortheevolutionofwateronEarth.

EvolutionofthemassesandD/Hratiosofwaterineachreser- voirwascalculatedbyusingthefollowingequations,

dM_i

dt

=

sources

F_k

−

sinks

F_k (1)

d

dt

(

M_iI_i

) =

sources

F_kf_kI_i

−

sinks

F_kf_kI_i (2)

fre

far

=

fdehy (3)

F_ar F_ar

+

^Fre

f_ar

+

^Fre F_ar

+

^Fre

f_re

=

^{1 (4)}

whereM_iandI_iarethemassandD/Hratioofwaterinthereser- voiri,andFkand fk arethefluxoftheprocesskanditsfraction- ationfactor.Thesourcesandsinksforeachreservoiraredescribed in Fig. 1. Hereafter subscripts i = ^{o, cc, oc, and m denote the} oceans, continentalcrust, oceanic crust, andmantle, respectively.

The subscript i inEquation (2) denotes a reservoir other thani.

Subscriptsk=^{ch,ar,se,de,es,we,andredenotethechemicalal-} teration, arcvolcanism,seaflooralteration,degassing,atmospheric escape,weathering,andregassing, respectively. Equations(3) and (4) give far and f_de by consideringthedehydration-inducedfrac- tionation and mass balance. Assuming d/dt=^{0 in} Equation (2) givesasteadystateinD/H,whichisusefultounderstandthenu- mericalresults(AppendixA).

Our modelassumed that the fluxesdepend on the massesof waterinthereservoirsandtimeasfollows:

F_ch

=

^F_ch^{0 Ac}

(

t

)

A⁰_c (5)

F_se

=

^F_se⁰

×

^f

(

t

)

(6)

Far

=

^Far⁰

M_oc

(

t

)

M⁰_oc

×

^f

(

t

)

(7)

F_de

=

^F_de^{0 Mm}

(

t

)

M⁰_m

×

^f

(

t

)

¹² (8)

Fwe

=

^F0we

Mcc

(

t

)

M⁰_cc (9)

F_re

=

^F_re^0Moc

(

t

)

M⁰_oc

×

^f

(

t

)

(10)

wheresuperscript0denotesthereference(present-day)values,Ac is the continental area, and f(t) is the speed of plate tectonics scaledtothatonpresent-dayEarth,whichaccountsforthechange ofmantle-convectionspeed duetothecooling suggestedbycon- ventional thermal-evolution models (e.g., Stevenson et al., 1983;

Höning and Spohn, 2016). The difference in the dependence on f(t) betweenEquations (6), (7), (10) and(8) originatesfromthe boundary-layermodel(HöningandSpohn,2016).

The continental area Ac as a function of time is given by (McLennanandTaylor,1982),

Ac

A⁰_c

=

⎧ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎩

0

.

1875

×

t Gyr

(

t

<

0

.

8 Gyr

)

0

.

15

+

0

.

929

×

t Gyr

−

0

.

8

(

0

.

8 Gyr

<

t

<

1

.

5 Gyr

)

0

.

8

+

0

.

0667

×

t Gyr

−

1

.

5

(

1

.

5 Gyr

<

t

)

(11) Wenotethat,though theevolutionofthecontinentalcoverageis controversial,ourresultsareshowntodependonlyweaklyonthe choiceofthecontinentalgrowthmodel(subsection4.7).

Theplatespeed scaled by thepresent-day value f(t) isgiven by,

f

(

t

) =

10^{−(t/4.5 Gyr−1)} (12)

Equation (12) reflects the model of Höning and Spohn (2016), whicharguedthatthespeedofsubductionwas∼^{2–10timesfaster} when the potential temperatureof the mantle was ∼^{100–300 K} hotter than today (Herzberg et al., 2010). In contrast, a recent model based on the energetics of plate-tectonic mantle convec- tionproposed that the tectonicspeed andsurface heat fluxhave beennearly constant throughoutEarth’shistory(Korenaga, 2003;

Korenaga et al., 2017), which implies f(t)=^{1. We considered} boththeformerandlattercases.Hereafterthetwothermalevolu- tionmodelsare referredtoasthefasterandslowerplatetecton- ics(PT) models, respectively. Models assuming time-independent (constant)fluxeswerealsoexplored toshow thebasicproperties oftheevolutionofD/H(AppendixB).

2.2.Parameters

Our assumptions on parameters are summarized in Table 1 (Supplementary text S1). We assumed M⁰_o=^{1 ocean, M}cc⁰ =⁰.2 oceans,andM⁰_oc=⁰.1 oceans(1ocean= ^1.4× ^{1021kg).Mostof} ourmodelsassumedM⁰_m=^{1 ocean,butsomemodelsshowedthe} integratedregassingtobe largerthan1ocean.Inthesecases,we increasedM⁰_m iterativelyuntilthemodelresultsinM_m atpresent whichagreeswiththeassumedM_m⁰.

The present-day net-regassing flux F_re,net (=^Fre⁰ − ^F_de⁰) and escapeflux before 2.5 Ga Fes are treatedasindependent param- eters.Weassumed F_de⁰ =¹.0×^{1011kg/yr, F}se⁰ =¹⁰×^1011kg/yr, F_ch⁰ =¹.5×^{1011 kg/yr,F}we⁰ =¹.0×^{1011kg/yr, F}re⁰ =^F0_de+^Fre⁰,net, and F_ar⁰ =^Fse⁰ +^Fwe⁰ −^Fre⁰, respectively (Supplementary text S2).

Modelsassumingdifferentvaluesofthefluxeswerealsoexplored toshowthebasicbehaviorofthesystem(AppendixB).

Ourstandardmodelassumedthevaluesoffractionationfactors assummarizedinTable1.Consideringtheuncertaintyoffraction- ationfactors(Supplementarytext S3),wealsoexaminedthecases wheredifferentvalueswereassumed.

While our model assumed that the majority of Earth’s wa- terhadbeendeliveredbeforethesolidification ofmagmaoceans, partofwater mighthavebeen delivered by thelate accretionof comets. In order to investigatethe influence of the possible late accretionofcomets,wealsoexploredmodelswherethecometary delivery was implementedby an input of0.01 oceanwater with δD=¹⁰⁰⁰h^at4.1 Ga(SupplementarytextS5).

Table 1

Summaryofparametersandinitialconditions(seeSupplementarytextS1–4forref- erences).1ocean=1.4×10²¹kg.Anapostrophedenotesthefractionationfrom thewaterinoceansbeforethecorrectionbyaddingsmallreservoirs(Supplemen- tarytextS3).

Present-day water fluxes [10¹¹kg/yr]

F_de⁰ 1.0

Fse⁰ 10

F_ch⁰ 1.5

F_we⁰ 1.0

Fre⁰ F_de⁰ +Fre,net

F_ar⁰ F⁰_se+^Fwe⁰ −^Fre⁰

Fractionation factors

10³lnfde 0 (standard), 10

10³lnf_se −³⁰

10³lnf_ch −80

10³lnfwe 0

10³lnfre Equations (3) and (4)

10³lnfar Equations (3) and (4)

10³lnfdehy −40 (standard),−²³

10³lnfes −150

Present-day water masses [ocean]

M⁰_o 1

M⁰cc 0.2

M⁰_oc 0.1

M⁰m 1 or>1

Initial water masses

Mⁱ_o M_o⁰+^Mcc⁰+^Mes+^Mre

Mⁱ_cc 0 ocean

Mⁱoc M⁰oc

Mⁱ_m M⁰_m−^Mre

2.3. Initialconditions

The initial conditions ofwater volumeswere assumedasfol- lows(Table 1 andSupplementarytext S4). Because there was no continentalcrustinitially (Equation (11)), Mⁱ_cc=^{0 oceanswasas-} sumed.Thewaterinsedimentsonpresent-dayEarthwasassumed to be initially partitioned into the oceans. We further assumed that theintegratedwater lostby escape M_es originatedfromthe oceans. Therefore,theinitial waterinoceanswas givenby Mⁱ_o= M⁰_o+^M0cc+^Mes+^Mre,whereMre istheintegratednetregassing.

The regassedwater Mre isnot givenapriori,butitwas givenby iteration.Theinitialmantlewaterwasgivenby Mⁱ_m=^M0m−^Mre.

WeassumedthatallthereservoirshavethesameδD anditera- tivelychangethevaluetoobtainthepresent-dayoceanicδD value which equals the SMOW (Supplementary text S4). We note that the initial δD values obtainedby thisprocedure were within the rangeofcarbonaceouschondrites:δD= −²⁰⁰h^{to300‰(Lécuyer} etal.,1998).

2.4. ConstraintsfromD/H

Modelsareconsideredsuccessfuliftheysatisfy theconstraints on the D/H ratios of present-day reservoirs (Table 2) and of the Archean seawater. Present-day Earth’s seawater have δD=⁰h by definition. Summing up water in oceans andsmall reservoirs (atmosphere,biosphere, surface water,groundwater, andglaciers) leads to the bulk oceanic δD= −¹²h ^to −⁷.1h^{. The gap from} SMOWmostlyoriginatedfromthecontributionoflowδD glaciers andpolar ice. Sedimentary rocks on continentalcrust and meta- morphic rocks on oceanic crust have δD= −¹⁰⁰h ^to −⁶⁰h

Table 2

SizesandD/H ratiosofwater reservoirson present-dayEarth.1ocean=1.4× 10²¹kg.

Reservoir Amount of water (ocean) δD (‰)

Oceans (including small reservoirs) 1.0 −12 to−7.1

Oceans 0.98^a 0^a

Atmosphere 9.3×^10−6a −^{70 to}+^10b

Biosphere 3.4×^10−5a −^{130 to}−^70c

Surface water 1.5×^10−4a −^{300 to}+^10d

Ground water 7.5×10^−3a −300 to+10^d

Glaciers/Polar ice 2.4×^10−2a −^{400 to}−^300c

Oceanic crust 0.10^a −^{50 to}−^30f

Continental crust 0.20^a −^{100 to}−^60c

Mantle 1.0^e −^{80 to}−^60g

a Bodnaretal.(2013) andreferencestherein.

b “Volumetricallymostimportantmeteoricwaters”fromSheppard(1986).

c Lécuyeretal.(1998) andreferencestherein.

d Popeetal.(2012) andreferencestherein.

e Korenagaetal.(2017) andreferencestherein.

f Lécuyeretal.(1998);Shawetal.(2008) andreferencestherein.

g KyserandO’Neil(1984);Clogetal.(2013).

and δD= −⁵⁰h ^to −³⁰h^, respectively (Lécuyer et al., 1998;

Shaw et al., 2008). Earth’s mantle has δD= −⁸⁰h ^to −⁶⁰h (Kyser and O’Neil,1984;Clog et al.,2013).

TheD/H ratioofpaleo-seawaterhasbeen estimatedfromiso- topic analysis of minerals and rocks which interacted with sea- water (Wenner and Taylor, 1974; Lécuyer et al., 1996; Kyser et al., 1999; Hrenetal., 2009;Pope etal., 2012). Weadopted δD=

−²⁵±⁵h ^{proposed by Pope et al. (2012) as the constraint on} 3.8 Ga seawater, because they have derived this value by com- bininghydrogen andoxygen isotopemeasurements of serpentine samples, in which primitive isotopic signatures have been well preserved. We will discussother datasets in subsection 4.4. We assumedthat the rangeof δD values betweenthose of the bulk ocean and the ocean (seawater) should match the evaluated δD ofthe Archean seawater. Assuming constant lnf corresponds to thecasewherethemassratioofglaciers/polaricetobulk oceans hasbeen constantthrough time. Becausethere isno evidence of glaciationbefore approximately 2.9 Ga (Younget al., 1998), con- sideringtherangeofδD isaconservativeassumption.

3. Results

TherangesofFre,netandFeswheretheconstraintsonpresent- day and Archean D/H were satisfied are shown for the slower and faster PT models in Figs. 2a and 2b, respectively. The stan- dardmodelwasassumedforfractionationfactors.Theinitialwater mass in the oceans M_oⁱ is alsoshown. We calculated the evolu- tionofmassesandD/Hratiosofthewaterreservoirs foreachset of Fre,net and Fes. Examplesof evolutionary tracks are shownin Figs.3and4.The resultsfordifferentvaluesoffractionation fac- torsareshowninFig.5.Theinfluenceofthepossiblelateaccretion of comets are considered in Fig. 6. As explained below, the re- sultsshowed that theslower PT model assuming F_re,net=^{0 and} Fes=^{0 doesnotsatisfythe}constraintsonD/H(Fig.2a).Therefore, hydrogenescape fromthe reducedearly atmosphere, secular net regassing(Fig.2a),orfasterplatetectonicsonearlyEarth(Fig.2b) isrequired.

3.1. TheslowerPTmodel

The evolution of δDi and Mi (where i is an arbitrary reser- voir)in theslowerPT model assuming F_re,net=^{0 and F}es=^{0 is} showninFig.3a.TheoceanicδDo increasedthroughtimebecause oftheisotopicfractionationfromseaflooralteration,slabdehydra- tion,andchemical alterationofcontinents. Alloftheseprocesses

Fig. 2.RangeofFres,net and Fes wherethe constraintson D/Haresatisfied(the hatchedareas).Resultsfor(a)slowerplatetectonics(PT)modeland(b)fasterPT modelareshown.Thepresent-dayD/Hratiosofthewaterreservoirswererepro- ducedaboveandbelowthesky-bluelinefor(a)and(b),respectively.TheD/Hof the Archeanseawaterwasreproduced belowthe redline.Colorcontourdenotes Mⁱ_o.MarksinfigurescorrespondtotheparametersetsshowninFigs.3and4.(For interpretationofthecolorsinthefigure(s),thereaderisreferredtothewebversion ofthisarticle.)

led toD-enrichment inliquidwater.The waterintheoceans Mo decreased in response to the increase of the waterin sediments Mccbecausecontinentalgrowthpromotedchemicalalteration.The mantleδDm decreasedthroughtimebecauseofthesubductionof deuterium-poor water ashydrousminerals.The continentalcrust δDcc increasedthroughtime followingtheincrease oftheoceanic δDo.TheoceaniccrustδDocinitiallydecreasedbecauseoftheiso- topicfractionationbyseaflooralterationandthenincreasedinthe latter periodtime following theincrease ofthe oceanicδDo. The integratedincreaseinδDocausedbythedeepwatercycleandcon- tinental growthreached ∼ +²⁰h^{,which isenough to reproduce} theincreaseofδDofromthelowδD Archeanseawater(Popeetal., 2012).However,thepresent-dayδDm inthemodeldisagreedwith the value inferredfromsample analyses (Kyser andO’Neil, 1984;

Clog etal., 2013).Thedeepwatercycleinourmodelevolvedto- ward asteadystate givenby Do−^m≡δDo−δDm∼⁷⁰h^,which is consistent with Do−^m of the present-day Earth(Appendix A andAppendixB),buttheratesofdegassingandregassingaretoo smalltoreachthesteadystatewithin 4.5Gyr. Theconstraintson thecontinentalcrustδDccandoceaniccrustδDocweresatisfiedin allcasesbecausebothδD_ccandδD_ocareinasteadystatewiththe oceanicδDo (AppendixA).

SecularnetregassingandhydrogenescapecanincreaseDo−^m (Figs. 3b and 3c). The secular regassing transported deuterium- poor water fromthe oceans to mantle andthe hydrogen escape removed deuterium-poor water fromthe oceans(more precisely, from the atmosphere, which exchanges water with the oceans).

The net regassing on present-day Earth in the slower PT model

Fig. 3.Timeevolutionofthepointslabeled 3a–dfromtheslowerPTmodelsinFig.2a.Left:δD ofoceans(thincyanlines),bulkoceans(thickcyanlines),oceaniccrust (greenlines),continentalcrust(yellowlines),andmantle(purplelines)asafunctionoftime.DatapointsareδD valuesofreservoirsonpresent-dayEarth(subsection2.4 andTable2)and3.8 Gaseawater(Popeetal.,2012).TheshadedrangedenotesthepossiblerangeofoceanicδD (seesubsection2.4).Right:massesofwaterinbulkoceans (cyanlines),oceaniccrust(greenlines),continentalcrust(yellowlines),andmantle(purplelines)asafunctionoftime.

led to steadydecrease and increase in Mo and Mm,respectively (Fig.3b). The mantleδDm startedto increase at∼³.5 Ga follow- ingtheincreaseoftheoceanicδDo.Thehydrogenescaperemoved waterfromtheoceans Mo before theGOEat 2.5 Ga(Fig. 3c).In contrasttothesecularregassing(Fig.3b),thehydrogenescapeand itscessationat2.5 GaresultedinthekinkintheevolutionofδDo (and consequently, in that of δDcc and δDoc) at 2.5 Ga. The co- existence ofthe secular regassingdiminishes the kink caused by the cessation of hydrogen escape, depending on F_re,net and F_es (Fig.3d).

AcomparisonoftheresultsintheslowerPT modelswiththe D/Hconstraints allowed us to constrain Fre,net and Fes (Fig. 2a).

Because the change in δD values was too small to reproduce present-dayDo−^m inthemodelassuming Fre,net=^{0 andF}es=⁰ (Fig. 3a), this provided a lower limit on Fre,net and Fes that can satisfy the constraint on present-day D/H (the sky-blue line in Fig.2a). We note that there was alsoa upper limit intheseval- uesabove which D_o−m was too large asseenin Fig. 2b,butit wasoutsidetherangeofFig. 2a. Ontheother hand,the increase intheoceanicδDo inferred fromthelow δD of theArcheansea- water(Pope etal.,2012) was reproduced inthemodel assuming

Fre,net=^{0 and F}es=^{0 (Fig. 3a). Because both the secular re-} gassingand hydrogen escapepromote the increase in δDo, there was anupperlimit on Fre,net and Fes tosatisfy theconstrainton theArchean seawater(the red lineinFig.2a). Allthe constraints onD/Hwere satisfiedinthelimitedrangeof Fre,net andFes (the hatchedareainFig.2a).

Therangeof Fre,netandFesdependsontheassumedvaluesof fractionation factors. Withinthe range of uncertainties (Table 1), assumingsmallerfractionationfordehydrationledtothehatched area moving to theright (Fig. 5a), whereas assuming larger frac- tionationfordegassingledtothehatched areamovingtotheleft (Fig.5c).

Thepossiblelateaccretionofcometsmoderatelyinfluencesthe resultingrange of Fre,net and Fes,butthe D/Hconstrains still re- quireat leasteitherone of thetwo mechanisms: secular net re- gassingorhydrogenescape(Fig.6a). Becausecomets haveahigh D/Hratio,theinputresultedinanincreaseinδDo (Fig.6b),while itscontributiontoEarth’swaterbudget(∼⁰.01 oceanwater,Sup- plementary text S5) is negligible. The input of D-enriched water decreased Fre,net and Fes required to reproduce the present-day

Fig. 4.The same as Fig.3, but the time evolution for the points labeled 4a–d from the faster PT models in Fig.2b are shown.

D/H ratios and increased these fluxes required to reproduce the D/HratiooftheArcheanseawater.

3.2. ThefasterPTmodel

In contrast to the slower PT model assuming F_re,net=^{0 and} Fes=^{0 (Fig. 3a), the efficient water cycle on early Earth in the} fasterPT modelledtothe δD valuesnearlyreaching steadystate in Do−^m (Appendix A) even assuming Fre,net=^{0 and F}es=⁰ (Fig. 4a). As withtheslower PT model,the isotopicfractionation resultedfromtheseaflooralteration,slabdehydration,andchem- ical alteration of continents. The faster PT model assuming the secularregassingshowedacontinuousincreaseinboththeoceanic δDo andmantleδDm (Fig.4b),which isqualitativelythe sameas theslowerPT model(Fig.3b),butthechangeislargerbecauseof thehigherregassingrateinthepast.Thoughthehydrogenescape hadaminorinfluenceontheevolutionofthewatermassesinthe reservoirs,itseffectontheevolutionofD/Hwasmorepronounced (Figs.4cand4d)becauseoftheefficientfractionationbyhydrogen escapecomparedtotheotherprocesses(Table1).

ThefasterPTmodelsassuming Fre,net=^{0 and F}es=^{0 satisfied} boththeconstraintsfromD/Hofthepresent-daywaterreservoirs

andoftheArcheanseawater(Fig.2b).Becauseassumingthesecu- larregassingorhydrogenescapeincreasedthechangeinδDoand the present-dayDo−^m (Fig. 4), boththe constraintsonpresent- dayandArchean D/Hgaveanupperlimit on F_re,net and F_es (the purple and red linesin Fig. 2b, respectively). Neither the secular regassingnorhydrogenescape isnecessarilyrequiredandtheal- lowedrangeofFre,netandFes wassmallerthanthatintheslower PTmodel.

Changing the values of fractionation factors in the faster PT modelshowedasimilarbehaviortotheslowerPTmodel(Figs.5b and5d),butthekey result—neitherthesecular regassingnorhy- drogenescapeisnecessarilyrequiredinthefasterPTmodel—does notchange.

Assuming cometaryinput slightlychangedtherange of Fre,net and Fes allowed toreproduceD/Hconstraints(Fig. 6c).The input ofD-enrichedwater(Fig.6d)decreased Fre,net andFes allowedto reproduce the present-day D/Hratios and increased thesefluxes allowedtoreproducetheD/HratioofArcheanseawater.Again,the resultsshowedthatneitherthesecularregassingnorhydrogenes- capeisnecessarilyrequiredinthefasterPTmodel.

Comparedtothe slowerPTmodels (Figs.2a and3), thefaster PT models resulted in a large decrease in the oceanic volume

Fig. 5.The same as Fig.2, but different values of fractionation factors are assumed (Table1). The cases for (a, b) 10³lnfdehy= −23 and (c, d) 10³lnfde=10 are shown.

Fig. 6.The same as Fig.2, but cometary input was assumed at 4.1 Ga (see text).

(Figs.2band4).InthefasterPTmodels,differentdependenceon thethermalevolutionisassumedfortheregassing(Equation (10)) anddegassing (Equation (8)) as predictedby the boundary-layer model of the thermal evolution (Höning and Spohn, 2016). The differenceresultedinthenetregassingbeinglargerintheearlier period.Eveninthe casewherethe balancewas assumedforthe present-dayEarth(F_re,net=^0),∼^{0.7oceansofwatersubductedin} 4.5 Gyr(Fig.4a).ThefasterPTmodelsassumingthenetregassing todayshowedmuchhigherregassinginthepast,leadingto∼⁴.3 oceansofwatersubductedthroughoutEarth’shistory(Fig.4b).

4. Discussion

4.1. EvolutionofwateronEarthconstrainedbyD/H

The evolution of water on Earth can be constrainedby com- paringourresultswiththeD/Hratiosofpresent-daywaterreser- voirs(subsection2.4andTable2).The differencesinδD between the present-day oceans, continentaland oceaniccrust, and man- tle were shown to resultfrom isotopic fractionationthrough the seaflooralteration,slabdehydration,andchemicalalteration (Sec- tion 3). The D/H ratios of present-day reservoirs can be under- stoodbyusingthesteadystate(Appendix AandAppendixB).We note that a steadystate inhydrogen isotopecompositions ofthe oceansandmantlehasbeenproposedbypreviousstudies(Taylor, 1974; Javoy, 2005),though dehydration-induced fractionationhas not been considered in them. However, the model also showed that the rates of present-day regassing and degassing are small sothatthesystemdoesnotreachthepresent-dayD_o−m within 4.5 Gyr(Fig.3a).Therefore,thehydrogenescapefromthereduced early atmosphere, secular regassing, or faster plate tectonics on early Earth was needed to explain the present-day state of D/H ratiosinthereservoirs(Fig.2).

ThelowδD ofArcheanseawater(Popeetal.,2012)furthercon- strainstheevolution ofwateronEarth(subsection2.4).Both the slowerandfasterPTmodels assuming F_re,net=^{0 and F}es=^{0 re-} sulted in the secular increase in the oceanic δDo as a result of thedeep watercycleandcontinentalgrowth, whichagreed with theconstraintontheArchean seawater(Figs.3aand4a).Because thesecularregassingandhydrogenescapepromotetheincreasein δDo,theconstrainton δD oftheArcheanseawatergavean upper limitonF_re,netandF_es(Fig.2).

Thesethreepossibilities—thehydrogenescape,fasterplatetec- tonics on early Earth, andsecular regassing—are mutually exclu- sive. For instance, in our standard model, the slower PT model needed F_re,net=⁰.6–2.1×^{1011 kg/yror F}es=^0–1.2×^{1011 kg/yr} (Fig.2a).Assuminglarger Fre,netleadstolower Fes,andviceversa.

Ontheotherhand,thefasterPTmodelallowedmuchsmallerval- ues:F_re,net=^0–0.2×^{1011kg/yrandF}es=^0–1.0×^1011kg/yr.

Consideringthepossiblerangesoffractionationfactorsandlate accretion of comets (Figs. 2, 5, and 6) yielded Fre,net<3.9× 10¹¹ kg/yr and F_es<2.1×^{1011 kg/yr. The upper limit of the} present-day regassing rate is higher than the upper limit pro- posedbyParai andMukhopadhyay(2012) fromthe constrainton thesea-levelchange(F_re,net=¹.0×^{1011kg/yr),butiscomparable} withtherangerecently arguedby Korenaga etal.(2017) consid- eringthechangeinbuoyancyofcontinentallithosphererelativeto oceanic lithosphere through time: Fre,net=3.0–4.5×10¹¹ kg/yr.

Theupperlimit oftherateofwaterlossduetohydrogenescape Fes=².1×^{1011 kg/yr}corresponds to ∼^{900ppmv ofCH}4 (Equa- tion S1).ThehighCH4 concentrationmightbepossibleduringthe earlyArchean(Kharechaetal.,2005).

Thesecularregassing,hydrogenescape,andchemicalalteration promoted by continental growth contributed to the secular de- crease of oceanic water in our model. In the slower PT model, thepossiblemaximumvalue of Mⁱ_o is givenwhen Fre,net=³.9×

10¹¹ kg/yr was assumed (Fig. 5a). The initial mass of the bulk oceanswas2.5oceans(considering1.3oceanslatersubductedand 0.2oceansformedsediments).InthefasterPTmodel,thepossible maximum value of M_oⁱ is givenwhen Fre,net=⁰.83×^{1011 kg/yr} wasassumed(Fig.5b),theinitialmasswas2.9oceans(1.7oceans subducted and 0.2 oceans formed sediments). These values are comparablewiththeestimate ofKorenagaetal.(2017) torecon- cilethecontinentalfreeboardwiththegeologicconstraints.

We suggest that the D/H constraints are consistent with the secular regassingscenario where water in the present-day man- tleentirelyresultedfromtheregassingthroughoutEarth’shistory.

The scenario is consistent not only with thegeologic constraints on continental freeboard when the change in relative buoyancy of continental lithosphereis taken into account (Korenaga etal., 2017), but alsowith the theoretical predictions of crystallization of magma oceans, where the majority of water was partitioned into theatmosphereand oceans(Hamanoetal., 2013).Such ini- tialconditionshavebeenarguedtobeidealforinitiatingtheplate tectonics(Korenaga,2013).

4.2. ImplicationsforfutureanalysisofD/HonearlyEarth

The threescenarios thatexplain theδD valuesofthe present- day and Archean reservoirs—the hydrogen escape, secular re- gassing, and faster platetectonics on early Earth—can be distin- guished by futureanalyses of samplesthat record theD/H ratios of the seawater andmantle on early Earth. The evolution of the oceanicandmantleδD forvarioussets of Fre,net andFes andfor theslowerandfasterPTmodelsisshowninFig.7.Thoughthein- crease inthe oceanicδDo hasbeen interpreted asa signature of hydrogen escape(Popeetal., 2012),Figs. 7aand7b showed that the increase is possible without the escape. Instead, the kink at the time of the GOE, if confirmed, would be a signature of hy- drogen escape from reduced early atmosphere. Constraining the oceanicδDo atthetime oftheGOEwouldhelp usto distinguish thescenarios.

While the difference in the oceanic δD_o between the slower and fasterPT models is small(Figs. 7aand 7b),the mantle δDm signal candiscriminatebetweenthesescenarios(Figs.7cand7d).

In contrast to the slower PT model where the mantle δDm de- creasedcontinuously,the fasterPTmodelshoweda rapidchange in the earlierperiod.Identifying the pastmantle δD_m aswell as the oceanic δDo wouldallow usto distinguish the three scenar- ios. InadditiontotheD/Hratiosoftheoceansandmantleinthe Archean, those fromtheHadean to thePhanerozoic, ifcombined withourmodel,wouldbeusefultoconstraintheevolutionarysce- narios.

4.3. Comparisonwithpreviousstudies

Earth’s deep water cycle and loss by hydrogen escape have beeninvestigatedinpreviousstudiesbyusingconstraintsonD/H.

Lécuyer et al. (1998) modeled the deep water cycle considering two reservoirs:the oceansand mantle.They argued that hydro- genisotopevariationsoftheoceansintimemayhaveoccurredin responsetotheimbalancebetweentheratesofregassingandde- gassing. Theirmodel showedthat thesystem eventually reached a steady state in δD_o. In contrast, our model showed that δD_o keptchanging(increasing)inresponsetotheimbalance(regassing) (Section 3). Thedifference originated fromthe assumptionabout the isotopic fractionation:they assumed constant δD values both for regassed and degassed water, whereas we calculated the δD values from the fractionationfactors (subsection 2.1). Inthat re- gard, the results of our model is more realistic than those of Lécuyeretal.(1998).

Fig. 7.δD of(a, b)oceansand(c, d)mantleasafunctionoftime.Theresultsfor(a, c)theslowerand(b, d)fasterPTmodelsareshown.Shadedareasshowthepossible rangesofevolutionarytracksthatcorrespondtotheparametersetsinhatchedareasin(a, c)Fig.2aand(b, d)Fig.2balongthesky-blueandredlines.Colorsoftheshaded rangescorrespondtothoseofboundarylinesinFig.2.DatapointsareδD ofArcheanseawaterandofpresent-daymantle(thesameasFig.3).

Shaw et al. (2008) also modeled the long-term water cycle betweenthe oceansandmantle by takingthe isotopicfractiona- tion caused by the seafloor alteration and slab dehydration into account.Assumingthenetbalancebetweentheregassingandde- gassing,theyshowedthecontinuousincreaseanddecreaseinδDo andδDm, respectively. Theirmodel even resulted in all the deu- terium inthe system beingpartitioned intothe oceans afterthe modelwas integratedfor a long enough time. Contrary to Shaw et al. (2008), our model showed that assuming the balance in the fluxes resulted in δD_o and δD_m reaching a steady state de- terminedby thefractionationfactors(Appendix B).Thedifference originatedfromthetreatmentoffractionationfactoroftheseafloor alteration:theydefinedthefractionationfactorastheratioofthe alteredMORBD/HtounalteredMORBD/H,whilewedefineditas theratioofalteredMORBD/HtoseawaterD/H.Becausethewater inthehydrousmineralsproducedbytheseaflooralterationorigi- natedfromseawater,ourassumptionismorerealisticthanthatof Shawetal.(2008).

Popeetal.(2012) usedmass-balancecalculationstoderivethe amount of water lost by the hydrogen escape based on δD of Archeanseawater.Theyconcludedthat ∼^{0.1oceansofwaterwas} lostduetotheescape.Incontrast,we showedthatthelower δD ofArcheanseawatercanbeexplainedbytheisotopicfractionation causedby thedeep water cycle(seafloor alteration and slab de- hydration)andthatthehydrogenescapeisnotnecessarilyneeded (Section2).The conclusionsdifferbecause theyneglected thein- fluence of the regassingand degassing on the δD change in the

mass-balance calculations. Theyjustified neglecting the contribu- tionofthemantletotheevolutionofδD_obyassumingthenetbal- ance betweentheregassinganddegassing.However,δDo changes becauseof thedeep watercycle, evenin thecasewherethe net balancewasassumed(AppendixB).

4.4. OtherdatasetsofPrecambrianseawaterD/H

Weadopted theD/HratioofArchean seawaterat3.8 Gacon- strained by Pope et al. (2012), butother studies have also esti- matedthepaleo-seawaterD/Hratio(Lécuyeretal.,1996;Kyseret al., 1999; Hren etal.,2009). WhereasPope etal.(2012) have re- portedδD= −²⁵±⁵h^andδ¹⁸O=².3h^{fromserpentinesamples} from3.8 GaIsua SupracrustalBeltinWestGreenland,Hrenetal.

(2009) have constrainedδD= −⁷⁰h ^to−⁵h^andδ¹⁸O= −¹⁸h to−⁸h^{from3.4 GaBuckReefChertrocksinSouthAfrica.Also,by} assumingδ¹⁸O= −¹⁰h^{assuggestedbya}theoreticalmodel(Kast- ingetal.,2006),Hrenetal.(2009) have proposedthattheArchean seawater had δD= −⁶⁰h^. Reproducing both δD= −²⁵±⁵h ^at 3.8 Ga and δD= −⁶⁰h ^{at 3.4 Ga in our model is difficult and} mightrequireasingular eventbetweentheinterval.The discrep- ancy in δ¹⁸O values in the two studies suggeststhat these data mightbeincompatible.Becausecherthaslowerconcentrationsof water than serpentine and the temperature dependence of D/H fractionationbetweenchertandwaterisnotstraightforward,Pope etal.(2012) arguedthat chertisalesswell-suitedproxy,sothat we chose to usePopeet al.(2012)’sdata asareliable anchor in theArcheanfortheevolutionofseawaterD/H.

Lécuyeretal.(1996) have proposedδD=⁰±²⁰h^{frommafic–}

ultramaficsamplesfromtheChukotatGroupoftheLowerProtero- zoic(2.0–1.9 Ga)CapeSmithfoldbelt.TheyhavereportedtheδD valuesmatchingcloselythosefoundinmodernmetavolcanicrocks.

Kyser etal. (1999) have constrained the D/Hratio of 2.8–2.6 Ga seawater asδD>−²⁰h ^{fromserpentine minerals fromArchean} Abitibigreenstonebelt inOntario.Because theestimatesofthese earlierstudieswerelesssystematiccomparedtoPopeetal.(2012), wedidnotplotthemexplicitly.However,theseestimatesareinac- cordancewiththetrendfromtheδD valueofPopeetal.(2012) to thatofpresent-dayseawaterso thatadoptingthemwouldhardly changeourresultsquantitatively.

4.5. Subductionregimes

Thisstudyinvestigatedthesecular evolutionofthe globalwa- ter cycle with the numerical modeling under a wide range of parameterspaces(Fig. 2),yetassumedalimitedcaseofcoldsub- duction(SupplementarytextS3,10³lnf_dehy= −^{40 to}−^23)where themajor waterreleasefromtheslab tothesurface (Far) occurs mainlydueto dehydrationofthe hydrousmineralsofamphibole andlawsonite (SchmidtandPoli, 1998;Maruyama andOkamoto, 2007).However, ourmodelalsoexploredtwo extremesubduction regimes of faster and slower PT. Subduction velocity along with plate thermal structure and wedge mantle viscosity strongly in- fluences slab surface temperature, which defines the stability of hydrousminerals (KincaidandSacks, 1997;Peacock, 1993).Each hydrousmineralmay havea differentfractionation factorofD/H duetodecomposition.Inthecaseofahotslab,waterreleasedby major dehydration events would be in isotopic equilibrium with amphibole(∼^{350◦C),epidoteandamphibole(}∼^{400◦C),andchlo-} rite, epidote, and amphibole (∼^550◦C), respectively (Maruyama andOkamoto, 2007). The fractionation factors fdehy of thesehy- drousmineralsrangefrom10³lnf_dehy= −^{23 to}−48 (Suzuokiand Epstein, 1976; Graham etal., 1984; Chacko etal., 1999). As this rangeissimilartothatinthecaseofacoldslab,ourresultswould alsoapply to the caseofa hot slab, though theregassing would mainlybecontributedtobythecold-slabsubduction.ThefasterPT mighthaveasmallernetregassingflux F_re becauseofthedomi- nanceofhotslabs,butourmodelacknowledgedthisandinvolved suchuncertaintiesofFre.

4.6. Onsettimeofplatetectonics

Our model assumed that plate tectonics has operated since 4.5 Ga, soon afterthe solidification of magma oceans. The onset timeofplatetectonicsiscontroversial,butsomestudieshavesug- gestedthat, on the basis of thegeochemistry ofHadean zircons, plate tectonics may have already been operating in the Hadean (Harrison,2009;Korenaga,2013,andreferencestherein).Korenaga (2013) arguedthattheidealinitialwaterdistributiontodriveplate tectonics is voluminiousoceans underlain by a dry mantle.Such conditionswere showntobeconsistentwiththeconstraintsfrom hydrogenisotopes(subsection4.1),thoughourmodelcanapplyto otheronsettimesofplatetectonics.

4.7. Uncertaintyinthecontinentalgrowthmodel

The behavior of our modeldepended only weakly on the as- sumed continentalgrowth model.Thus we demonstratethat the evolution ofD/Handsurface watermassis a poorconstrainton continentformation. Assensitivityanalysis, we compared there- sultsusingthecontinentalgrowthmodelsofMcLennanandTaylor (1982) (Fig. 2) and Armstrong (1981) (Fig. S1) where the conti- nental coverage increased linearly in the first 1 Gyr and kept a constant value. We found that the dependence ofthe results on

theuncertaintyinthecontinentalgrowthmodelissmallcompared tothatontheuncertaintyinthefractionationfactors(Fig.5).

5. Conclusions

WemodeledtheevolutionofthemassesandD/Hratiosofwa- terin theoceans, continentalandoceaniccrust,andmantle.The modelconsidered watertransport andhydrogenisotopicfraction- ation by seafloor hydrothermal alteration, chemical alteration of continentalcrust,slabsubduction,hydrogenescape,anddegassing at mid-ocean ridges, hot spots, and arcs. The differences in D/H ratios between the present-day oceans, oceanic and continental crust,andmantlewereshowntoresultfromisotopicfractionation by seaflooralteration,slabdehydration,andchemicalweathering.

Thecurrentdegassingandregassingratesweretoosmalltoreach the present-day D/H, so an additional mechanism was required.

We showedthreeevolutionaryscenarios thatcan accountforthe present-dayD/Hratios:(a)hydrogenescapefromareducedearly atmosphere,(b)secularnetregassing,and(c)fasterplatetectonics onearlyEarthexpectedfromconventionalthermalevolutionmod- els.AlowD/HratioofArcheanseawaterat3.8 Gahasbeeninter- pretedasasignatureofthehydrogenescapefromareducedearly atmosphere. However, ourmodelshowedthatthesecular netre- gassingthroughoutEarth’shistoryorfasterplatetectonicsonearly EarthcanalsoreproducetheconstraintsonD/H. Thesethreesce- narios are mutuallyexclusive.The ratesofhydrogen escapefrom early Earth and secular regassing on present-day Earth are con- strained to be lower than 2.1×^{1011 kg/yrand 3}.9×^{1011 kg/yr,} respectively. Theinitialoceanscould be2–3timesasvoluminous asthatoncurrentEarthinthesecularregassingscenario.Asigna- tureofhydrogenescapebeforetheGOEisvisibleasakinkinthe slopeoftheevolutionoftheoceanicD/Hratio.ThemantleD/Hra- tio inthe fasterplate tectonics modeldecreased onlyduring the earlierperiod,whereastheslowerplatetectonicsmodelpredicted the mantle D/Hratio has beencontinuously decreasing through- out Earth’s history. Therefore, we emphasize the importance of measurements to constrain mantle D/H ratio throughout Earth’s history, in addition to that of seawater at the time of the GOE, todistinguishthesethreeevolutionaryscenarios.

Acknowledgements

We thank two anonymous reviewers for comments and sug- gestions. HK was supported by JSPS KAKENHI Grant (15J09448, 17H06457, 18K13602) and JSPS Core-to-Core Program “Interna- tional Network of Planetary Sciences.” JF was supported by JSPS KAKENHI Grant (16K05619). CH was supported by MEXT KAK- ENHIgrant(15H05832).TUwassupportedbyJSPSKAKENHIGrant (17H06454, 17H06459). This study was supported by the WPI- fundedEarth-LifeScienceInstituteatTokyoInstituteofTechnology.

Appendix A. DerivationofasteadystateinD/H

The evolutionof δD valuesofreservoirs canbe understoodas the change toward a steady state. From Equation (2), the steady stateisgivenby,

Io

I_m

=

^Fdefde F_ref_re

Frefre

+

Farfar

F_sef_se

+

^Fchf_ch (A.1)

I_cc

I_o

=

^Fchfch

F_wef_we (A.2)

I_oc

Io

=

^Fsefse

+

^Fchf_ch Frefre

+

^Farfar

.

(A.3)

Fortheparametervaluesusedinourstudy(Table1)wherewater inoceaniccrustmostlydegassedthrougharcvolcanism,Far^Fre,

Fig. 8.δD ofoceans(thincyanlines),bulkoceans(thickcyanlines),oceaniccrust(greenlines),continentalcrust(yellowlines),andmantle(purplelines)asafunction oftime inthe constant-fluxmodel(seetext).Theshaded rangedenotes thepossiblerangeofoceanic δD (seesubsection2.4).Results forthe dehydrationmodel(a:

Fde=¹×^{1011kg/yr,b:F}de=⁵×^{1011kg/yr,andc:F}de=¹⁰×^{1011kg/yr)andthe}non-dehydrationmodel(d:Fde=⁵×^{1011kg/yr)areshown.Datapointsare}δD values ofreservoirsonpresent-dayEarth(Table2)and3.8 Gaseawater(Popeetal.,2012).

F_se^Fch,andF_ch∼^Fwe.HerewefurtherassumeF_de=^Fre.Equa- tions (3) and (4) are approximated as fre∼ ^fdehy and far∼^1.

SubstitutingtheserelationsforEquations(A.1)–(A.3) gave, I_o

Im

∼

^fde

fsef_dehy (A.4)

I_cc Io

∼

^fch

fwe

(A.5) Ioc

I_o

∼

^fse

.

(A.6)

Therefore,wecanderive,

D_o−m

∼

^103lnfde

−

^103lnfse

−

^103lnfdehy (A.7)

Dcc−^o

∼

^103lnfch

−

^103lnfwe (A.8)

D_oc−o

∼

^103lnfse (A.9)

Substituting the fractionation factors (Table 1) gave D_o−m ∼ 70h^,Dcc−^o∼ −⁸⁰h^,andDoc−^o∼ −³⁰h^{,whichisconsistent} withtheδD valuesobtainedfromfieldsamples(Table2).Herean apostrophedenotesthefractionationfromseawaterbeforethecor- rectionbyaddingsmallreservoirs.

Ontheother hand,consideringawatercyclewithoutslabde- hydration(Fre^Far) leadstoadifferentsteadystate.Inthiscase Equation (A.1) gave,

I_o I_m

∼

^fde

f_se (A.10)

D_o−m

∼

^103lnfde

−

^103lnf<