Joseph G. O’Rourke

Contents lists available atScienceDirect

Earth and Planetary Science Letters

www.elsevier.com/locate/epsl

Prospects for an ancient dynamo and modern crustal remanent magnetism on Venus

Joseph G. O’Rourke

^a,∗

, Cédric Gillmann

^b,c

, Paul Tackley

^d

aSchoolofEarthandSpaceExploration,ArizonaStateUniversity,Tempe,AZ,UnitedStatesofAmerica bRoyalObservatoryofBelgium,Brussels,Belgium

cFreeUniversityofBrussels,DepartmentofGeosciences,G-Time,Brussels,Belgium dDepartmentofEarthSciences,ETHZurich,InstituteofGeophysics,Zurich,Switzerland

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

Articlehistory:

Received29January2018

Receivedinrevisedform16June2018 Accepted28August2018

Availableonlinexxxx Editor:B.Buffett

Keywords:

Venus Venus,interior Venus,surface magneticfields accretion

Venuslacksaninternallygeneratedmagneticfieldtoday.Whetheroneexistedinthepastisunknown, but criticalto atmospheric evolutionand potential habitability. Canonical models assume thecore of Venus iscooling tooslowly forconvection and thusamagneticdynamo to occur today.Core/mantle heatflowissuppressedinthesemodelsafteraputativetransitioninmantledynamicsassociatedwith widespread, volcanicresurfacing.However, recentstudies ofimpactcraters andothersurfacefeatures supportmoresteadyheatlossovergeologictime.PrecipitationofMgOand/orSiO2fromthecorecanalso drivecompositionalconvectionevenwithslowcooling.HerewereevaluatethelikelihoodthatVenushas an“Earth-like”(atleastpartiallyliquidandchemicallyhomogeneous)coreusingnumericalsimulations ofthe coupledatmosphere–surface–mantle–coreevolution. AnEarth-likecoreisonlycompatiblewith the modern lack of a dynamo if the thermal conductivity of core material is towards the higher end of modern estimates (i.e., >100 W m⁻¹K⁻¹). If lower estimates like ∼^{40–50 W m−1K−1 are} actually correct, then wefavor recent proposals that the core hascompletely solidified orpreserved primordial stratification.Any simulationinitialized witha homogeneous,liquidcore predicts aglobal magnetic fieldwith Earth-likesurfacestrengthfor >2–3billionyears after accretion—consistentwith allavailableobservations—andalsosporadicactivitywithinthesurfaceagewhiletemperaturesremain belowtheCuriepointofmagnetite.Therefore,futurespacecraftmissionsshouldprioritizethefirst-ever magnetometermeasurementsbelowtheionospheretosearchforcrustalremanentmagnetism.

©^{2018ElsevierB.V.Allrightsreserved.}

1. Introduction

Venusstandsaloneastheonlymajorplanetwithoutevidence for an internally generated magnetic field either now or in the past.Vigorousconvectionofliquidiron alloyinEarth’souter core hassustainedour geodynamo foratleast3.45 Gyr (e.g.,Tarduno etal., 2010). Venus ispresumablydifferentiated like Earthintoa silicatemantleandmetallic core, butPioneerVenus Orbiter con- strainedthemagneticmomentofVenustolessthan∼^{10−5 times} the modern value forEarth (e.g.,Phillips andRussell, 1987). De- terminingwhetherVenuseverhostedaglobalmagneticfield has myriad implications for its surface habitability (e.g.,Driscoll and Bercovici,2013; FoleyandDriscoll,2016)andtheongoingdebate overthe generalrelationshipbetweenmagneticshielding andat- mosphericerosion(e.g.,Tardunoetal.,2014).

*

Correspondingauthor.

E-mailaddress:jgorourk@asu.edu(J.G. O’Rourke).

Generallyspeaking,therearetwobasicrequirementsforamag- netic dynamo. First, the Coriolis force must strongly affect the fluid flowasindicated bya smallRossbynumberattheequator:

Ro = ^v/(2L),where v is fluidvelocity, L isthe length scaleof the dynamo region, and  is the angular rotation speed. Venus has the longestrotational periodof the major planets,but Ro≈ 10⁻⁵  ^1(versus∼^{10−6 forEarth) should stillsupport dynamo} action(e.g.,Stevenson,2003).Second,themagneticReynoldsnum- ber Rem=^{v L}/λ—where λ is magnetic diffusivity (inversely pro- portional to electrical conductivity)—mustexceed a criticalvalue

∼^{10–100. In the absence of other fluid motions like tidal stir-} ring,thiscriterionmandatesvigorousconvectioninalowviscosity (i.e.,liquid)core.Dynamosconstantlyrequireenergeticinput—any globalmagneticfieldwoulddissipatewithin∼^{104 yrafterconvec-} tionceases(Stevenson,2003).

Canonical models assume Venus has an “Earth-like” core—at least partially liquid and chemically homogeneous—that is cur- rently cooling tooslowly fora dynamo. Thermalconvection only occurs if the heat flow across the core/mantle boundary (CMB) https://doi.org/10.1016/j.epsl.2018.08.055

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

exceeds that which conduction would transport up an adiabatic temperaturegradient.Nimmo(2002) arguedthatatransitionfrom plate tectonics to the stagnant lid regime of mantle convection at ∼^{500 Ma decreased CMB heat flow to nearly zero. Dramatic,} globalchangesinmantledynamicsarecommonlyinvokedtopro- duce “catastrophicresurfacing” andexplain the random distribu- tion of impact craters on the surface (e.g., Strom et al., 1994;

McKinnonetal.,1997).Ongoingdebateoverwhethercatastrophic resurfacingactuallyoccurredhingesonthefractionofcratersthat sufferedpost-impact volcanic modification.Only ∼^{10% ofcraters} were classifiedas obviouslyembayed during the first analysisof radarimagesfromMagellan(e.g.,Strometal.,1994).MonteCarlo models of cratering and non-catastrophic resurfacing can repro- duce this low percentage (Bjonnes et al., 2012), but cannot ex- plain the clustering of obviously embayed craters (O’Rourke et al., 2014). However, radar-dark floors found in ∼^{80% of craters} mayindicate volcanic modification that isnot otherwise obvious inlow-resolution Magellanimagery(Wichman,1999; Herrickand Rumpf,2011).O’Rourkeetal.(2014) showedthatnon-catastrophic resurfacingby thin, localized flows—matching some stratigraphic histories (Guest and Stofan, 1999)—would produce a volcanically modified population withthe samesize andspatial distributions asthedark-flooredcraters.

Even models without catastrophic resurfacing rely on low CMBheat flow to explain the absence of a dynamo today. One- dimensional, parametrized models agree that stagnant lid con- vection suppresses mantle and corecooling ifmelt migration to the surface is relatively inefficient (e.g., Stevenson et al., 1983;

Solomatov andMoresi, 1996; Driscoll andBercovici, 2013, 2014;

O’RourkeandKorenaga,2015;FoleyandDriscoll,2016).However, Armann and Tackley (2012) predicted that magmatic heat pipe dominates in the stagnant lid regime, which leads to unrealisti- cally high rates of crustal production and Earth-like core/mantle heatflow. Anepisodiclidmode,incontrast,suppressescorecool- ingduring quiescentperiods andbetter matchesthe present-day amplitudeofthegeoidandtopography.Recentworkdemonstrates thatatmosphere–surface couplingcausestransitionsbetweendif- ferentmantleconvective regimesthat ultimatelystabilize surface conditions(Noack etal., 2012; GillmannandTackley, 2014). Fac- tors affecting dynamo action have not been fully investigated in thesenewsimulations.

Compositionalbuoyancyproducedbychemicalprocessesispo- tentiallykey todynamoactioninterrestrialplanets.Forexample, the plausible range for the energy sink associated with thermal conduction in Earth’s core (∼^{4–11 TW) overlaps with the es-} timated ∼^{5–15 TW total heat flux across the CMB today (Lay} et al., 2008). There is no problem explaining Earth’s dynamo at present,however,becauseexclusionoflightelementsfromtheso- lidifyinginnercoreprovidescompositionalbuoyancy.Core/mantle heat flow need not exceed the conductive flux along the adi- abat once the inner core nucleates since compositionally dense (butrelatively hot) materialcan sink andcarry heat downwards.

Precipitationof light elements from the core may provide com- positionalbuoyancybefore nucleation of an inner core. O’Rourke andStevenson (2016) first proposed that magnesium could pro- videan earlypowersourceforEarth’sdynamo.Later,Badroetal.

(2016) presentedsupportiveresultsfromdiamond-anvilcellexper- iments.Magnesiumisdeliveredinthe∼10% ofcore-forming iron alloythatchemicallyequilibrateswithmantlesilicateatextremely high-temperatureconditionsintheaftermathofgiantimpacts.The solubilityofmagnesiuminmetaldecreasesrapidlywithtempera- ture, so cooling rates under ∼^{50 K Gyr−1 still provide sufficient} mass flux to drive convection. Hirose et al. (2017) subsequently suggestedthatcrystallizationofsilicon dioxidemayalsooccurat similar rates even if metal/silicate equilibration occurs at more moderatetemperatures near mid-mantle depths. Additional min-

eralphysicsexperimentsarerequiredtoclarifymanydetailsabout these newmechanisms. Regardless, Venus could possibly sustain adynamowithsub-adiabatic heatflowinan Earth-likecoreeven priortoinnercorenucleation.

Tworecentstudiesofferalternativestocanonicalmodelsofthe core. First, Earth-sized planets are expected to form with strati- fied coreswhere theabundances of light elements increase with radius (Jacobsonet al., 2017). Metaladded later tothe coredur- ing accretionchemicallyequilibrateswithsilicatesathighertem- perature/pressure conditions where silicon and oxygen are more solubleinmetal. Earth’sMoon-formingimpactpresumablyelimi- natedthisstratificationthroughmechanicalmixing.Intheabsence ofalateenergeticimpactwithappropriategeometry,thisstratifi- cationmay surviveandcompletelyprevent convection evenwith extremelyrapidcooling.Second,thecoreofVenusmayhavecom- pletely solidified (e.g., Stevenson et al., 1983; Dumoulin et al., 2017). Doppler tracking of Magellan and Pioneer Venus Orbiter measured thetidalLove numberask2=⁰.295±⁰.066 (Konopliv and Yoder, 1996). Elastic deformation models based on a one- dimensional seismological model of Earth’s interior implied that asolid corewouldhavek2≈⁰.17 comparedto0.23<k2 <0.29 fora liquidcore (Konoplivand Yoder,1996).However, Dumoulin etal.(2017) usedaviscoelasticsolutionformantledeformationto arguethatthecoremustbefullysolidiffuturespacecraftfindk2

< 0.27.Verifyingeitherofthesescenarios wouldprofoundlyalter theoriesfortheaccretionofVenusandEarth.

In this paper, we address two fundamental questions. First, doesslow cooling alone explain the modern absence of a global magnetic field? Second, should we prioritize a search forcrustal remanent magnetism on Venus? We run numerical simulations built onrecent models ofEarth’s dynamo (O’Rourke andSteven- son, 2016; O’Rourke etal., 2017) anda previous investigation of coupled atmospheric and mantle dynamics on Venus (Gillmann andTackley,2014).Forsimplicity,wealwaysassumethatthecore lacks significantcompositionalstratification andhasan Earth-like bulkcomposition.Simulationsthatpredictenoughcorecoolingto drive a dynamo at present are taken as evidence for primordial stratification of the core (Jacobson et al., 2017), unless the core hascompletelysolidified.

2. Model

Oursimulations oftheevolution ofVenusincludethree mod- ules tohandletheenergybalance oftheatmosphereandthedy- namicsofthemantleandcore.Weconsidertwo-waycouplingbe- tweentheatmosphereandmantlebasedonhowmeltproduction in the mantle releases greenhouse gases and then surface tem- perature determines theregime of mantleconvection. Thisstudy includes some coupling between the mantle and core because the temperatureof thecore influences mantleconvection, which controls the cooling rateof the core. However, we have not yet formulatedamodelfortheinfluenceofamagneticfieldonatmo- spheric composition. Table 1 defines critical parameters that we useortrackinoursimulations,alongwithvaluesforsomeimpor- tantconstants.

2.1. Evolutionofthemantle

We continue to use the StagYYcode to simulatemantle con- vection (Armann andTackley, 2012; Gillmann andTackley, 2014;

Gillmannetal., 2016). Briefly,we assume acompressible, anelas- tic mantlewithinfinite Prandtl numberin 2D, sphericalannulus geometry with a resolution of 512 azimuthal by 64 radial cells plus1milliontracerstotrackcompositionandmeltfraction.Heat- producingelementsareuniformlydistributedinitiallybutpartition intomelt asinsome casesfromArmannandTackley(2012).We

Table 1

Definitionsanddefaultvaluesforkeyparametersandconstantsusedinoursimulations.Termsdescribingtheradialstruc- tureofthecoreweremodifiedfromLabrosse(2015).

Constant Definition Value Units

Cp Specific heat of the core 750 kJ kg⁻¹K⁻¹

∂TL/∂P Pressure dependence of the core liquidus 9 K Pa⁻¹

Lp Length scale for the density profile in the core 8510 km

Ap Prefactor in the density profile in the core 0.4835

K0 Bulk modulus in the density profile and liquidus 1172 GPa

rc Radius of the core 3110 km

Λm Latent heat of MgO and/or SiO2crystallization 4 MJ kg⁻¹

αm Compositional expansitivity of MgO and/or SiO2 0.8

ρp Density anomaly for the primordial layer 300 kg m⁻³

σy Yield stress at the surface 90 M Pa

∂σy/∂z Increase in yield strength with depth 38.21 Pa m⁻¹

Parameter Definition Units

ρ0 Density of pyroxene–garnet system relative to olivine %

Dp Thickness of the primordial layer in the lower mantle km

Cm Rate of MgO and/or SiO2precipitation K⁻¹

k Thermal conductivity at the center of the core W m⁻¹K⁻¹

kc Minimum conductivity that prevents a modern dynamo W m⁻¹K⁻¹

ri Radius of the inner core km

Tc Initial “super-heat” in the core K

TC M B Temperature at the core/mantle boundary K

QC M B Total core/mantle heat flow TW

QS Secular cooling of the core TW

QR Radiogenic heating in the core TW

QL Latent heat from inner core growth TW

QG Gravitational energy from inner core growth TW

QC Cooling of a perfectly conductive inner core TW

QP Gravitational energy from MgO and/or SiO2precipitation TW

QE Latent heat from MgO and/or SiO2precipitation TW

Φ Total dissipation in the core TW

Φo Dissipation produced at the core/mantle boundary TW

Φi Dissipation produced at the inner core boundary TW

TDM True dipole moment measured at the surface A m²

considerasingleheat-producingcomponentwithaneffectivehalf- lifeof2.43 Gyrandagloballyaveraged,radiogenicproductionrate of5.2×^{10−12W kg−1today(ArmannandTackley,2012).Bound-} aryconditions atthe surface and CMB, respectively, are free slip andisothermal. The atmospheric model sets the surface temper- ature,whereas thecoretemperature evolvesasafunction ofthe core/mantleheatflux asdescribedbelow. Theinitial temperature profilehassmall,randomperturbationsfromanadiabatwithapo- tentialtemperatureof1900 Kandthinboundarylayersatthetop andbottom(GillmannandTackley,2014).

Mantle material is initially a uniform mixture of two end- member components: basalt(20 wt%) and harzburgite (80 wt%).

We assume that basalt consists of a pure pyroxene–garnet sys- tem,whereasharzburgiteiscomposedof25 wt%pyroxene–garnet and75 wt%olivine(ArmannandTackley,2012).Depth-dependent propertieslikedensityandthermalexpansivityandconductivity—

including phase transitions in both systems—are calculated asin XieandTackley(2004) and ArmannandTackley(2012). Wevary the bulk modulus of the pyroxene–garnet system in the lower mantle, which creates a density difference that rises from zero at∼^{800 km depth to a maximum (}

ρ

0 ≈ ^{0–6%) nearthe CMB} (e.g., Nakagawa and Tackley, 2014), to reflect the uncertainty in the density of basalt relative to harzburgite in the lower man- tle (e.g.,Hiroseetal.,2005; Ohtaetal., 2008).At each timestep inthe StagYY code,melt consistingentirelyof basaltis removed fromeach cellifneededtoreturnthetemperaturetothesolidus.

Allmelt generatedabove300 kmrises tothesurface, butwe as- sumethatonly10%ofthetotalmeltproductionendsinextrusive volcanism. We also include a primordial layer of dense material abovetheCMB—presumablyresultingfrominitialmantledifferen- tiation(Nakagawa andTackley,2014)—withdensityequalto 

ρ

p plusthatofbasaltattheCMBtoreducecore/mantleheat flowat earlytimes.

Rheologyisassumedtobeindependentofcompositionandin- cludes Newtonian diffusion creep plus plastic yielding (Gillmann and Tackley,2014; Gillmann etal., 2016). The yield stress is the minimum ofa brittlevalue predictedby Byerlee’s law,withzero cohesionandafrictioncoefficientof0.5,andaductileyieldstress, whichisequalto90 MPaatthesurfacebutincreaseswithdepth to avoid unrealistic yielding in the lower mantle (Nakagawa and Tackley,2015).Inplaceswherethestressishigherthantheyield stress, the effective viscosity is iteratively adjusted to reduce it to the yield stress. In Armann andTackley (2012), plastic yield- ing was favored to produce an episodiclid regime withrealistic crustal thicknesses andvolcanic activityinstead of astagnant lid regime with massive magmatism. No significant changes to the overallevolutionwereobservedwhenductileyieldstresswasvar- ied between80 and 120 MPa in coupled models (Gillmann and Tackley,2014).

2.2. Evolutionoftheatmosphere

Ourtreatmentoftheatmosphereisnearlyidenticaltothesetup in Gillmann and Tackley (2014), which used a one-dimensional (vertical) model adapted from Phillips et al. (2001). Radiative transferisgrey,meaningthatthermalinfraredopacitydependson thealtitudeandabundanceofgreenhousegasesbutnotonwave- length.Self-consistenttemperatureprofilesarecalculatedforara- diativelayerandanunderlyingconductivelayerthatmatchatthe tropopause, assuming hydrostatic equilibrium. Greenhouse gases CO₂ and H₂O are assumed to have constant mixing ratios with altitude. Solar flux increases over time from 70% of its present- dayvalue at 4.5 Gaaccordingto thefaint youngSun hypothesis.

Cloud evolution is not modeled in detail since we have not in- cludedafull,3-Dclimatemodel.Effectivetemperaturesaresimply computedfromthesolarfluxaccordingtotheblackbodylaw,as-

sumingconstantalbedo(Gillmann andTackley,2014;Gillmannet al.,2016).Removingcloudsatanytimewoulddecreasetheplane- taryalbedoandraisethesurfacetemperaturebyseveralhundred degrees.

Greenhousegasesareremovedbyvariousescapeprocesses,but replenished through mantle melting.To set the initial condition, weassume that hydrodynamic escapedriven byextremeUV flux from the Sun caused fast H and O loss in the first ∼^{100 Myr} afteraccretion. Rapidly dryingtheatmospheredecreases thesur- facetemperature, which inturnspeedscrystallization ofthe pri- mordialmagmaoceanandcanexplainthepresent-day²⁰Ne/²²Ne and³⁶Ar/³⁸Ar ratios(Gillmann et al., 2009). Adense, CO₂ atmo- spherewithapartialpressurewithin ∼^{0.5–1%ofthe}present-day value outgasses asthe magma ocean solidifies. Subsequent melt production in our simulations barely increases atmosphericCO2, although some degassing of H andCO may alsooccur in reality atearly times.We assume that all volatiles in extrusive magma are degassed. However, we then use low apparent abundances of volatiles—e.g., a reduced mantle with only 125 ppm of CO2 today—to compensate for the likelihood that high surface pres- surepreventscompletedevolatilizationonVenus(Elkins-Tantonet al., 2007; Gaillard and Scaillet, 2014). Escape of CO₂ is likewise negligible—currentlybelowthedetectionlimitsofmodern instru- ments(GillmannandTackley,2014).Hydrodynamicescapealways remainssignificantforH,butstopsforOafter∼^{500 Myrassolar} EUVfluxdrops.Variousnonthermalescapeprocessesinvolvingso- laremissionandwind,whichdominateatmosphericescapeduring the last ∼^{4 Gyr, are modeled at all times based on an energy-} limited approach that takes into account evolution of solar EUV fluxandpresent-daymeasurements (GillmannandTackley,2014;

Gillmannetal.,2016).

2.3.Evolutionofthecore

Wemakesmallmodificationstoaone-dimensionalparameter- ization built forEarth on a fourth-order expansion of the radial densityandgravity inthe core(e.g.,Labrosse, 2015; O’Rourkeet al.,2017).Internal pressuresarereducedrelativetoEarthby∼^7%

and∼^20%,respectively, attheCMBandcenterofthe core.Fig.1 showstheimpliedliquidustemperaturesinbothplanets.Thecore of Venus completely solidifies after ∼^{420 K of cooling once the} innercorefirstnucleates(comparedto∼^{690 KforEarth).Weas-} sumethattheouter coreischemicallyhomogeneousandexhibits negligibledeviationsfromanisentropicstateexceptinthinbound- arylayers.Inthiscase,thetotalheatflowacrosstheCMBcontrols thethermalandchemicalevolutionofthecore.

2.3.1. Calculatingthetotaldissipation

Standardmodelsfortheevolutionofmetallic coresalwaysin- cludesecularcooling( Q_S),radiogenicheating( Q_R),andtwoterms relatedtothegrowthofaninnercore:latentheatoffreezing( QL) andgravitationalenergyreleasedfromtheexclusionofincompati- blelight elements ( QG). Here we assume that QR resultsexclu- sively from the decay of potassium-40, although recent metal–

silicate partitioning experiments also imply a small contribution from uranium (Chidester et al., 2017; Blanchard et al., 2017). If [K]=^{200 ppm,then Q}R decaysfrom11.2to0.9TWover4.5 Gyr.

Potassium is quite soluble in iron alloy, but experiments imply [K] < 100 ppminEarth’scoresince potassiummostly partitions into silicates during equilibration at the base of a deep magma ocean(e.g.,Hiroseetal.,2013;Blanchardetal.,2017).Weassume thattheinnercorehasinfinitethermalconductivityandmaintains uniformtemperatureequaltotheliquidusatitsboundary.Thisin- troducesanadditionalcoolingterm:

Fig. 1. Basicpropertiesofourmodelforthecore.Radialprofilesof(a)densityand (b)gravity.(c)Temperaturesat theinnercoreandcore/mantleboundariesasso- ciatedwithagiveninnercoreradius.InsimilarmodelsforEarth,theinnercore nucleatesandthecorecompletelysolidifieswhenthetemperatureatthecore/man- tleboundarydropsbelow∼4110 Kand3430 K,respectively(Labrosse,2015).

Q_C

=

^CpM_icK₀

 ∂

TL

∂

P

 

2ri

L²_p

+

^16ri3 5L⁵_p

 

dri

dt



,

(1)

whereM_icisthemassoftheinnercoreandK₀isaneffectivebulk modulus(Labrosse,2015).OtherparametersaredefinedinTable1.

Wecouldalternativelyimplementaninsulatinginnercorewithan adiabatictemperaturegradientbysettingQ_C=^{0 TW.}

PrecipitationofbuoyantcomponentslikeMgOand/orSiO2may begin at a critical temperature and continue thenceforthwith a nearlysteadymassflux(e.g.,O’RourkeandStevenson,2016;Badro etal.,2016;Hiroseetal.,2017).Inthisstudy,wemodelafiducial light component because of uncertainties about the exact parti- tioningbehaviorandrelativeabundancesofeachelement(e.g.,Du etal.,2017).Thisformulationcoversmanyplausiblescenariosthat are energetically equivalent. We calculate the resulting contribu- tiontothetotalheatbudgetfromgravitationalenergyas

QP

=

⁸

3

π

²G

ρ

₀²L⁵_p

α

mCm



f_γ



rc

Lp



−

^fγ



r_i Lp



dTC M B

dt

,

(2) wheretheusefulfunction

f_γ

(

x

) =

^x3



− Γ

3

+ (

1

+ Γ )

x²

5

+ (

Ap

Γ −

1

.

3

)

x⁴ 6

,

(3)

withΓ = (^rc/Lp)²[¹−⁰.3(rc/Lp)²]^{.Wealsoincludealatentheat} Q_E= ΛmC_mM_oc(dT_{C M B}/dt), where M_oc isthe mass ofthe outer core, associated with crystallization of the light components at the CMB. Remarkably, independent experiments indicate that Cm

≈ ² × ^{10−5 K−1 for both MgO and SiO}2 precipitation, so our simulationsare notsensitivetotheexactcomposition ofthepre- cipitate.Energeticimpactsduringaccretionarenecessarytodeliver

∼^{1–3 wt%MgOintothecore(Badroetal.,2016).Ifgiantimpacts} are also required to disruptprimordial stratification (Jacobson et al.,2017),thenanyEarth-sizedcorethatisabletoconvectshould experienceMgOprecipitation.However,SiO2exsolutionispossible

evenifmetal/silicateequilibrationduringcoreformationprimarily occurrednearliquidustemperaturesatmid-mantledepthsandthe initialabundanceofMgisnegligible(Hiroseetal.,2017).

The global energy budget is simply the sum of all the heat sources(e.g.,Labrosse,2015).Thatis,

Q_{C M B}

=

^QS

+

^QR

+

^QP

+

^QE

+

^QL

+

^QC

+

^QG

.

(4)

We assume that thecorebegins witha “hot start,” whereinitial temperatureiselevatedrelativetothebasalmantle(e.g.,Stevenson etal., 1983). Specifically, the initial temperatureequals 4121 K—

slightlyabovethetemperatureatwhichinnercoregrowthbegins (3859 K)—plusan additionalTc. Thecooling rateofthe coreis hence

dT_{C M B}

dt

=

^{QC M B}

−

^QR

Q_S

+

^QP

+

^QE

+

^QL

+

^QC

+

^QG

,

(5)

where Qi=^Qi/(dTC M B/dt) are calculated analytically (Labrosse, 2015). The total dissipation includes all these terms multiplied by appropriateefficiencyfactors, plusa sinktermrelatedtoheat conduction along the adiabatic gradient. We can write the total dissipationasΦ= Φi+ Φo,wherewecalculatecontributionsfrom the inner and outer boundaries, respectively, as in Aubert et al.

(2009):

Φ

i

=

^TD

(

T_L

−

^TC M B

)

T_LT_{C M B}

(

Q_L

+

^QC

) +

^TD

T_{C M B}Q_G (6)

and

Φ

o

=

^TD

(

T_R

−

^TC M B

)

T_RT_{C M B} Q_R

+

^TD

(

T_S

−

^TC M B

)

T_ST_{C M B} Q_S

+

^TD T_{C M B}Q_P

−

TDEK

.

(7)

The effective temperature of dissipation overall is TD, which is calculated along withother terms not definedhereexactly asin Labrosse(2015). All heat sources exceptthose relatedto compo- sitionalbuoyancy ( QG and QP) are hampered by a “Carnot-like”

efficiency term. Those sources with lower effectivetemperatures (e.g.,T_R<T_S<T_L)producerelativelylessdissipation.Weomitted anytermrelatedto QE sinceheat sourcesattheCMBcontribute nothingto the dissipation (i.e., TE−^TC M B=^{0 K). The minimum} requirement for a dynamo is Φ >0 TW. The actual Ohmic dis- sipation produced by Earth’s dynamo is uncertain, but possibly ashighas∼^{3–8 TW(Stelzer andJackson, 2013). We predictthe} true dipole moment (TDM) over time using a scaling law for a strong-field, dipole-dominated dynamo fromAubert etal. (2009) asdetailedinthesupplementarymaterial.

Thethermalconductivityofironalloysunderextremetemper- atureandpressureconditionsiscurrentlyuncertainbyafactorof two to three. Extrapolations of theory andexperiments applica- bletolowerpressureandtemperatureconditionspredictedvalues around^{40–50 W m−1K−1 inEarth’score(e.g.,StaceyandLoper,} 2007). However, recent first-principles calculations (de Koker et al., 2012; Pozzo etal., 2012) anddiamond-anvil cellexperiments (Gomi et al., 2013; Seagle et al., 2013; Ohta et al., 2016) sug- gest that the electrical resistivity at core conditions is several times lower than previously believed. Thermal conductivity was thus calculated as ^{90–130 W m−1K−1 at the CMB, increasing} to ∼150–200 W m⁻¹K⁻¹ at Earth’s inner coreboundary accord- ing to the Wiedemann–Franz “law”. However, direct measure- mentsofthermalconductivityindiamond-anvilcells suggestthat this empirical relation maynot holdat core conditions, andthe earlier estimates of conductivity (∼^{40–50 W m−1K−1) are pos-} sibly still valid (Konôpková et al., 2016). Because thermal con- ductivity does not appear in the global heat budget, we can quicklyre-compute the predicteddissipation fortheentire range

Fig. 2. Minimumtotalcore/mantleheatflowrequiredtodriveadynamoforvarious valuesofthermalconductivityatthecenterofthecoreandtotalOhmicdissipa- tion,assumingVenushasan“Earth-like”corethatischemicallyhomogeneousand fullyconvectiveexceptinthinboundarylayers.Exceptwhereotherwiselabeled, weassumeΦ=^{0 TW,C}m=^{0 K−1,[K]}=^{0 ppm,andk}=^{130 W m−1K−1asfor} theunlabeled,blackcurve.Theygray,shadedregiondenotesheatflowsrequired tomaintainΦ=^{3–8 TW.Theredcurveshowsa}calculationwith[K]=^{200 ppm,} whilethebluecurvehasCm=2×10⁻⁵ K⁻¹.Thelowest,blackcurvehask re- ducedto40 W m⁻¹K⁻¹.(Forinterpretationofthecolorsinthefigure(s),thereader isreferredtothewebversionofthisarticle.)

of plausiblevalues—alwaysassuming thedepth dependencefrom Labrosse (2015) where the average conductivity equals ∼^0.7k—

withoutrepeatingafull simulation.Inparticular,wecalculatethe minimum thermal conductivity (kc) at the center that prevents positivedissipationatpresentdayforeachsimulation.

Fig. 2 shows the minimum QC M B required to produce a dy- namo undervariousconditions.Ifthelowvaluesofthermalcon- ductivity are correct, then the minimum QC M B drops from ∼^2.5 to only 1 TWonce the innercore nucleates.Increasingthe ther- mal conductivity to theupper estimates increasesthe power re- quirementsby ∼^{3.4times.Likewise, raising}Φ toreflectnon-zero Ohmicdissipationcausesanidenticalincreaseintherequiredheat flow. The minimum Q_{C M B} is incrementally boosted for small r_i butskyrocketsforri >2500 kmwithanyradiogenicelementsun- der ourassumption that they areincompatible inthe inner core.

AddingMgOand/orSiO₂ precipitationroughlyhalvestherequired QC M B priortoinnercorenucleationbutnegligiblyreducesitonce ri  ^{750 km. Similarly, inner corenucleation only decreases the} minimum QC M B by∼^50%ifprecipitationisalreadyongoing.

3. Results

Table 2 lists input parameters and key output results from fortysimulations.Numbers1through34usethefullatmosphere–

mantlecouplingdescribedabove(GillmannandTackley,2014).We first consider the possibility that Venus currently hosts a liquid outercorelikeEarth,whichsimplylosesheatbyconductionwith- out rapidfluid motions.Then, wedeterminetheinitialconditions thatpermitthecoretohavecompletelysolidifiedwithsimulations 32–34.Wemostlydiscusssimulations35–40inthesupplementary material.

3.1. Insufficientcoolingofan“Earth-like”core 3.1.1. Examplesimulation

Representativeresultsareobtainedwithslightmodificationsto the preferred atmospheric and mantle evolution from Gillmann and Tackley (2014) in simulation 8 from Table 2. Atmospheric abundances ofCO₂ andH₂Oare initially 84.5 bar and2.7 mbar, respectively. Basalt is set as 3% more dense than harzburgite at

Table 2

Initialconditionsandkeyoutputresultsforeverysimulationperformedforthisstudy.Importantvaluesincludethetimeofinnercorenucleation(ti),thetotalheatflowfrom themantlethroughthesurface( QM),andthethicknessofcrustproducedsince750 Ma(hc).^Simulations35–37haveatmosphere–interiorcouplingbutσy=300 MPaso thatrecentevolutionisinthestagnant(insteadofepisodic)lidregime.^†Simulations38–40havenoatmosphere–interiorcoupling—thesurfacetemperatureisfixedto740 K withσy=^{90 MPasothatVenusalwaysevolvesintheepisodiclidregime.}

# ρ0

(%)

Dp (km)

Tc (K)

[K]

(ppm) Cm (10⁻⁵K⁻¹)

ti (Gyr)

hc (km)

QM(tp) (TW)

QC M B(tp) (TW)

ri(tp) (km)

kc (W m⁻¹K⁻¹)

1 0 0 1000 0 2 1.2 3.7 13.5 8.9 2877 176

2 0 200 1000 0 2 2.3 9.1 15.6 5.6 1851 220

3 0 400 1000 0 2 1.9 3.5 13.9 3.1 1922 120

4 6 0 1000 0 2 1.4 8.5 17.9 3.3 2198 110

5 6 200 1000 0 2 1.6 4.9 15.8 3.0 1974 111

6 6 400 1000 0 2 1.5 8.4 15.1 1.2 2052 44

7 3 0 1000 0 2 1.4 7.3 17.5 4.3 2306 135

8 3 200 1000 0 2 2.0 4.8 15.4 2.7 1654 113

9 3 400 1000 0 2 1.8 6.7 20.1 3.8 1847 150

10 6 200 1000 200 2 1.9 8.9 20.3 5.0 1661 164

11 6 200 1000 400 2 2.6 9.9 17.3 3.9 1233 102

12 6 200 1000 600 2 3.7 12.4 25.1 7.6 889 252

13 0 200 2000 0 2 2.7 12.5 24.8 7.3 1670 304

14 3 200 2000 0 2 2.4 6.7 15.5 4.1 1666 171

15 6 200 2000 0 2 1.8 12.5 15.2 5.4 1946 205

16 0 200 2000 200 2 3.4 10.3 15.8 6.2 1383 236

17 3 200 2000 200 2 3.2 5.9 18.0 5.1 1298 191

18 6 200 2000 200 2 2.2 10.8 17.2 4.8 1701 155

19 0 400 2000 200 2 3.4 9.3 16.5 6.2 1329 240

20 3 400 2000 200 2 2.7 7.9 17.0 5.9 1551 213

21 0 200 1000 200 2 3.4 14.3 16.8 5.7 1340 218

22 0 200 1000 400 2 4.3 5.5 13.5 7.6 619 276

23 0 200 1000 600 2 >4.5 44.3 14.7 7.6 0 154

24 3 200 1000 50 2 2.2 7.7 17.6 4.0 1596 159

25 3 200 1000 100 2 2.4 7.7 17.2 4.5 1480 177

26 3 200 1000 200 2 2.9 5.7 16.6 3.5 1248 122

27 3 200 1000 400 2 4.0 7.6 16.7 5.8 814 202

28 3 200 1000 600 2 >4.5 7.5 19.7 5.5 0 105

29 6 0 1000 0 0 1.3 8.1 16.4 3.4 2295 104

30 6 200 1000 0 0 1.4 7.5 15.4 1.7 2070 59

31 6 400 1000 0 0 1.4 7.2 16.1 1.3 2214 42

32 6 200 0 0 0 0.9 8.9 22.8 1.0 2348 30

33 6 0 0 0 0 0.5 10.0 19.7 1.6 2412 47

34 0 0 0 0 0 0.5 13.0 14.6 6.0 3110 –

35^ 0 200 1000 0 2 1.9 32.3 10.4 4.9 2014 179

36^ 3 200 1000 0 2 1.9 12.5 12.0 1.7 1635 73

37^ 6 200 1000 0 2 1.4 12.8 11.4 1.9 2186 65

38^† 0 200 1000 0 2 2.7 6.2 14.9 5.8 1755 232

39^† 3 200 1000 0 2 2.6 6.8 21.1 6.9 1640 289

40^† 6 200 1000 0 2 1.6 9.4 35.5 3.6 2040 129

the CMB. We include a primordial, dense layer with a thickness of200kmtoinsulatethecoreatearlytimes.Theinitialtempera- tureofthecoreis5092 Kand[K]=^{0 ppm.}PrecipitationofMgO and/or SiO₂ occursatconstant C_m=²×^{10−5 Kat}temperatures below4500 K.Fig.3containssnapshotsofthemantletemperature andcompositionat1 Gyrintervals,whileFig.4illustratesthera- dialtemperatureprofilesinthemantleatthesetime steps.Fig.5 showsthedetailedevolutionofmanycriticalparameters.

Partialmeltingproducesasurfacelayerofbasalticcrustwithin

∼^{20 Myr afterthesimulationstarts. Around}∼^{200 Myr,aneraof} rapidmagmatism begins as plumes upwell fromthe CMB while slab-like downwellingevents occur periodically across the entire mantle(Gillmann andTackley, 2014). Primordial material mostly remains at the CMB, although a small amount is entrained and transportedupwards—butnot abovethe 730 kmphasetransition (equivalent to 660 km in Earth). Horizontal velocities near the surface are typically 0.01–0.1 cm yr⁻¹, butcan reach an orderof magnitudelargerduringresurfacingevents.Sluggishhorizontalve- locities and increasing mantle temperatures are characteristic of thestagnantlidregime (GillmannandTackley,2014).Despitethe rapid melt production and ongoing release of greenhouse gases, surfacetemperaturedecreasesfrom∼^{1070to730 Kat}∼^{960 Myr} assolarEUVfluxreducesatmospheric[H2O]to0.01 mbar.

Lowsurfacetemperaturesfacilitatethetransitionat∼^{960 Myr} toamobilelidregimeresemblingplatetectonics.Increasedviscos-

ityinthe(colder)uppermantlemakesconvectivestressrelatively more likely to exceed the yield stress in viscoplastic rheology.

During this second stage, typical horizontal velocities are a few cm yr⁻¹ withfew quiescentperiods. Mantlemelting causestran- sientjumpsin[H2O]correlatedwithspikesinsurfacetemperature froma∼^{520 Kminimum.Mantle}temperaturesquicklydecreaseat firstassurfaceheatflowrises,butremainroughlyconstantduring thesecondhalfofthisstage.Startingaround∼^{1.7 Gyr,watervapor} begins to accumulate inthe atmosphere asescape processesbe- comelessefficient(GillmannandTackley,2014).Mantledynamics then entera thirdstage wheresurface temperaturesrise andthe mobilelidgraduallystagnates.Meltproductiondropsasconvective velocitiesdecrease.After∼^{2.3 Gyr,thefourthandfinalstagestarts} assurface temperaturelevels off nearthepresent-day 740 K. An episodiclid regime beginsthat features severalprominentspikes insurfaceheatflowassociatedwithlocalizedresurfacingevents—

butnotglobal,catastrophicresurfacing.

Attheendofthesimulation,alayerofbasalt∼^{1000 kmthick} has built up above the CMB—a typical outcome of crustal over- turnsintheepisodiclidregime.Thebasalticlayerismuchthinner in simulations with continuous subduction or perpetual evolu- tioninthestagnantlidregime(NakagawaandTackley,2015).The core/mantletemperaturecontrastcalculatedfroman adiabatpro- jected down fromthe 730 km phase transition in the mantle is similartothe∼^{500–1000 KinferredforEarthtoday.However,ra-}

Fig. 3. Snapshotsoftemperature(left),composition(center),andconcentrationof primordialmaterial(right)inthemantleforourexamplesimulation.Composition rangesfrom0(harzburgite)to1(basalt),withtheprimordialmaterialcoloredas harzburgiteinthecentercolumnandinyellowontheright.Theinnercoreisdrawn toscaleasablackcircleateachtimestep.

diogenic heating in the basaltic layer raises the temperatures in thelower mantle byan additional ∼^{500 K. Therefore,asseen in} Fig.4,thetemperaturecontrastthroughthelowestboundarylayer inthemantle(i.e.,theprimordiallayer)isonly∼^{200 Kattheend} ofthissimulation.

Eachstage ofatmosphere–mantle evolution profoundlyaffects thelikelihoodofdynamoactioninan“Earth-like”core.Corecool- ingdeclinesfrom∼^{47to21 TWduringthefirststageastheCMB} temperaturecontrastrapidlydrops.Thermalconvectionalonesus- tains a magnetic dynamo with Earth-like surface strength even beforeMgO/SiO2precipitationincreasesΦ by∼^50%at∼^{600 Myr.}

Fig.6 showsthat Q_P ≈ ^{0.1 Q}S atthistime, butbothterms con- tributenearly identicalamounts ofdissipation sinceprecipitation isnotpenalizedbya“Carnot-like”efficiency.Duringthemobilelid stage,core/mantleheatflowrisestooscillatearound∼^{30–40 TW.}

Theinnercorenucleatesshortlyafterthisstage,whichmorethan doublesΦbutbarelyaffectsthepredictedTDMsincethedynamo becomesdeeperseated(Aubertetal., 2009;Landeauetal., 2017).

Innercoregrowthcontributesmoretothe heatbudgetthanboth precipitationandsecularcooling—andthusbecomesthedominant

Fig. 4. Radialprofilesoftemperatureinthemantleatthebeginningofourexample simulationandatthetimestepsdepictedinFig.3.

sourceofdissipation.Theeffectivetemperaturesofdissipation(T_D, etal.)slowly decreasetoremainbetweenthetemperaturesatthe core/mantleandinner coreboundaries.Fig.6alsoshowsthatthe sinktermassociatedwiththermalconduction(and,lessobviously, allterms)graduallydecreasesinresponse.

Core/mantleheatflowbeginsasteadydeclineoncethemantle reenters thestagnantlid regime,reachingonly2.7 TW atpresent day.Still,Earth-likefieldstrengthsarepredictedatthesurfacecon- tinuouslyuntil∼^{4 Gyr.Threeregional}resurfacingeventsbetween

∼^{2.3and 3 Gyrdeliver relatively cold materialto the cold/man-} tle boundary. Associated small spikes in QC M B and thus Φ and TDMappear∼^{100 Myr(themantletransittime)aftersurfaceheat} flow reachesa local maximum. Fig. 5 showsthat a burst ofdy- namo activityispredictedtooccurbetween∼^{200–100 Maifthe} thermalconductivityiskc=^{113 W m−1K−1.Thisburstisnotsup-} pressed evenif k is raised to 130 W m⁻¹K⁻¹—the upper endof modern estimates.Incontrast,ifthelowest estimatesforthermal conductivity (∼^{40–50 W m−1K−1) provecorrect, then an “Earth-} like” core is obviously incompatible with the modern lack of a dynamo.Increasingk from40to130 W m⁻¹K⁻¹ bolstersthedis- sipativesinkby ∼^{1 TW.Thisabsolutechangeonly}correspondsto a largeproportional changewhen Φ issmallsince TDM ∝ Φ^0.34 in our scaling. Therefore, k stronglyaffects TDM in recent times butnotbefore∼^{2 Gyr.}

3.1.2. Sensitivitytests

We now test plausible values for parameters that are likely to strongly affect the dynamo while still producing realistic at- mospheric evolution and surface volcanism. Varying the basalt–

harzburgite density contrast at the CMBis the only change that appreciably affects the compositional structure of the mantle in this study. Fig. 7 illustrates that any 

ρ

0  3% will produce a thick, basal layer of basalt. When basalt and harzburgite have similar densities in the lower mantle,slab-like downwellingsare more likely to reach the CMB. A thick layer ofbasalt should re- duce core/mantle heat flow—present-day QC M B =⁸.9, 4.3, and 3.3 TW for [K] = ^{0 ppm, D}p=^{0 km, and} 

ρ

0=^{0, 3, and 6%,} respectively—althoughthisgeneraltrenddoesnotalwaysholdbe- cause of the variable timing and magnitude of regional down- wellingsduringtheepisodiclidphase.

Fig. 8shows how thecritical conductivitythat suppresses re- cent dynamo activitydepends onvarious parameters forsimula- tions 1through 33.Onecorrelation isimmediatelyobvious: sim- ulations that predict Earth-like Q_{C M B} > 5 TW are incompatible withVenushavingan “Earth-like”corebutnodynamotoday,as-