Lunar Cumulate Mantle Overturn: A Model Constrainedby Ilmenite RheologyHaoyuan Li1, Nan Zhang1,2 , Yan Liang3 , Bingcheng Wu4, Nicholas J. Dygert5 ,Jinshui Huang4, and E. M. Parmentier3

1The Key Laboratory of Orogenic Belts and Crustal Evolution, Institution of Earth and Space Sciences, Peking University,Beijing, China, 2Department of Applied Geology, Curtin University, Perth, Western Australian, Australia, 3Department ofEarth, Environmental and Planetary Sciences, Brown University, Providence, RI, USA, 4Department of Geophysics andPlanetary Sciences, University of Science and Technology of China, Hefei, China, 5Department of Earth and PlanetarySciences, University of Tennessee, Knoxville, Knoxville, TN, USA

Abstract Lunar cumulate mantle overturn has been proposed to explain the abundances of TiO2 andheat‐producing elements (U, Th, and K) in the source region of lunar basalts. Ilmenite‐bearing cumulates(IBCs) that were formed near the end of lunar magma ocean solidification are the driving force for overturn.IBCs are enriched with dense TiO2 and FeO contents and have lower viscosity and solidus than those of theunderlying lunar cumulate mantle. We investigate the effects of temperature‐ and ilmenite‐dependentmantle rheology on the dynamic process of lunar cumulate mantle overturn and conditions forlong‐wavelength downwellings of an IBC layer in a 3‐D spherical geometry. Our results show that theilmenite‐induced rheological weakening is necessary to decouple the IBC layer from the top stagnant lid andfacilitate overturn. Models with IBC viscosity derived from the experimental scaling can only produceshort‐wavelength downwellings (spherical harmonic degree >3) and show an overturn timescale more than100 Ma. A viscosity of the IBC layer at least 10−4 lower than that of the ambient mantle can produce thelong‐wavelength (spherical harmonic degree ≤3) downwellings in ~10 Ma and even a hemisphericdownwelling. Such low IBC viscosity requires additional weakening mechanisms, such as remelting or/andwater enrichment. During the overturn, the cold downwellings displace upward the materials from hotlower mantle and produce partial melting in upper mantle, which may serve as a viable mechanism for earlylunar magmatisms. The settling of cold downwellings on the core‐mantle boundary stimulates a transienthigh heat flux, which may contribute to generating an early lunar dynamo event.

1. Introduction

Lunar cumulate mantle overturn (Hess & Parmentier, 1995; Ringwood & Kesson, 1976) has been invoked toexplain major lunar observations such as the asymmetrical concentrations of incompatible trace elements(K, rare earth elements, and P, referred to as KREEP hereafter) and the high‐TiO2 ancient rocks from earlymagmatic events (e.g.,Delano, 1986; Lucey et al., 1998; Nyquist & Shih, 1992; Parmentier et al., 2002; Sheareret al., 2006). Lunar mantle overturn is thought to be a natural consequence of lunar magma ocean (LMO)crystallization. Solidification of LMO starts with olivine, followed by low‐Ca pyroxene, anorthitic plagio-clase, and clinopyroxene. Ilmenite becomes a liquidus phase with anorthite and clinopyroxene near theend of LMO solidification (e.g.,Charlier et al., 2018; Elardo et al., 2011; Elkins‐Tanton et al., 2011; Linet al., 2016; Rapp & Draper, 2018; Snyder et al., 1992). The late cumulate that consists mainly of ilmenite,clinopyroxene, and possible pigeonite are enriched in FeO, TiO2, and KREEP components and hence denserthan the underlying earlier cumulates (e.g.,Charlier et al., 2018; Hess & Parmentier, 1995; Lin et al., 2016;Rapp &Draper, 2018; Snyder et al., 1992). These dense cumulates may have sunk as they crystallized, mixingwith the underlying olivine + low‐Ca pyroxene + clinopyroxene cumulates to form a thicker ilmenite‐bearing cumulate (IBC) layer (e.g.,Dygert et al., 2019; Hess & Parmentier, 1995; Papike et al., 1998;Parmentier et al., 2002; Ringwood & Kesson, 1976). The IBCs being denser than the underlying mafic cumu-lates are hypothesized to have sunk into the deeper mantle by solid‐state mantle flow, a process termedcumulate mantle overturn. Some fraction of KREEP, including the heat‐producing elements (HPEs) K, U,and Th, would have been entrained during overturn and carried into the lunar interior along with theIBCs. Mantle overturn processes also occurred in different terrestrial planets, which have been a focus ofresearch (e.g.,Boukare et al., 2018; Maurice et al., 2017; Scheinberg et al., 2014; Tosi et al., 2013).

©2019. American Geophysical Union.All Rights Reserved.

RESEARCH ARTICLE10.1029/2018JE005905

Key Points:• A small viscosity of ilmenite‐bearing

cumulates beneath the lunar crust isnecessary to facilitate the overturnafter the lunar magma ocean

• To generate a hemisphericdownwelling, additional weakeningmechanisms other than low IBCviscosity are required

• The overturn produces a largeamount of melting and a transienthigh CMB heat flux

Supporting Information:• Supporting Information S1• Data Set S1• Movie S1• Movie S2• Movie S3• Movie S4• Movie S5• Movie S6

Correspondence to:N. Zhang,nan_zhang@pku.edu.cn

Citation:Li, H., Zhang, N., Liang, Y., Wu, B.,Dygert, N. J., Huang, J., & Parmentier,E. M. (2019). Lunar cumulate mantleoverturn: A model constrained byilmenite rheology. Journal ofGeophysical Research: Planets, 124,1357–1378. https://doi.org/10.1029/2018JE005905

Received 15 DEC 2018Accepted 9 APR 2019Accepted article online 16 APR 2019Published online 27 MAY 2019

Author Contributions:Conceptualization: Nan ZhangFormal analysis: Haoyuan Li, NanZhangInvestigation: Haoyuan LiMethodology: Nan ZhangSoftware: Nan Zhang, Bingcheng WuVisualization: Haoyuan Li, JinshuiHuangWriting – review & editing: NanZhang, Yan Liang, E. M. Parmentier

LI ET AL. 1357

The wavelength of lunar cumulate mantle overturn may provide important clues to understanding the ori-gin and asymmetric concentration of mare basalts on the near side of the Moon. Parmentier et al. (2002)investigated whether a spherical harmonic degree‐1 downwelling could lead to the spatial concentrationof a KREEP layer early in lunar evolution through Rayleigh‐Taylor (R‐T) stability analysis and 2‐D finiteamplitude numerical simulations. They found that a weak IBC layer with a viscosity contrast of 10−4 is cap-able of generating a spherical harmonic degree‐1 downwelling. Based on the flow law for olivine, Elkins‐Tanton et al. (2002) concluded that the solidified IBC layer would be too viscous to sink via a R‐T instability.They postulated that impact or radiogenic heating might remelt the IBC layer, which then sank downwardto form a heterogeneous lunar mantle. Several studies have also applied the cumulate overturn to investigatethe lunar geochemical evolution (e.g.,de Vries et al., 2010; Thacker et al., 2009; Yu et al., 2019; Zhaoet al., 2019).

The timescales of lunar cumulate mantle overturn may provide useful information for understanding lunarpaleomagnetic records (Garrick‐Bethell et al., 2016; Lawrence et al., 2008). Cold downwellings during over-turn were hypothesized to create a pulse of heat flux at the core‐mantle boundary (CMB) that might power atransient lunar core dynamo (Weiss & Tikoo, 2014). Thus, the time period and intensity of paleomagneticrecords can offer a geological constraint on the early interior evolution of the Moon. The oldest lunar mag-netic intensity of 20–40 μT dates back to 4.25 Ga (Garrick‐Bethell et al., 2016). Such highmagnitude of paleo-magnetic intensity suggests a robust lunar core dynamo although this high strength is difficult to explainusing our current understanding of small core size and the scaling between core heat flux and surface fieldstrength (see summary by Scheinberg et al., 2018). Lunar paleomagnetism has been explained with differentmechanisms. These mechanisms include the internal convection energy (Evans et al., 2014; Laneuville et al.,2013; Scheinberg et al., 2018; Zhang et al., 2013a), the impact event (Hood & Artemieva, 2008; Le Bars et al.,2011), and the precession movement (Dwyer et al., 2011).

The IBC viscosity has a significant effect on the wavelength and timescales of lunar cumulate mantle over-turn (Parmentier et al., 2002). Dygert et al. (2016) recently measured the viscosity of pure ilmenite and foundit to be more than 3 orders of magnitude lower than dry olivine at mantle temperature and stress. Zhanget al. (2017) found that the IBC viscosity can largely change the plume generation of an overturned lunarcumulate mantle. The present study is built on the laboratory and numerical work of Dygert et al. (2016)and Zhang et al. (2017) by investigating the instability and evolution of a dense IBC layer at the top of thecumulate mantle. Both R‐T analysis and finite amplitude convection modeling are used. We focus on theeffects of IBC thickness and composition‐ and temperature‐dependent viscosity on the wavelength and time-scale of overturn and the early lunar mantle evolution. We track the evolution of mantle structures, likeli-hood of melting, and CMB heat flux during overturn. We show that IBC layer with a viscosity derivedfrom the experimental scaling can only produce short‐wavelength downwellings. A viscosity of a 75‐kmIBC layer at least 10−4 lower than that of the ambient mantle can produce the long‐wavelength (sphericalharmonic degree ≤3) downwellings in ~10 Ma.

2. Model Formulation2.1. Physical and Chemical Properties of the IBC Layer

The thickness, density, viscosity, and HPE content of the IBC layer are fundamental to understanding itsdynamical behavior. The volume (and hence the thickness) and ilmenite fraction of the IBC layer have beenconstrained both petrologically and geophysically (Figure 1A; see also supporting information Figure S1).Based on an assumed initial LMO composition (Buck & Toksöz, 1980; Khan et al., 2007; Longhi, 2006;Taylor, 1982; Warren, 1985, 1986), estimates of an IBC layer thickness prior to overturn have been derivedin previous studies (e.g.,Charlier et al., 2018; Elardo et al., 2011; Elkins‐Tanton et al., 2011; Hess &Parmentier, 1995; Lin et al., 2016; Rapp & Draper, 2018; Snyder et al., 1992). These studies focused onmineral composition and lithological stratification resulting from LMO solidification. Snyder et al. (1992)modeled crystallization of a 400‐km‐deep LMO under a constant pressure of 0.6 GPa and predicted an 18‐km‐thick IBC layer with 16‐vol% ilmenite. For a 1,000‐km‐thick LMO, Elkins‐Tanton et al. (2011) calculateda thicker IBC layer and hence lower ilmenite fraction. Elardo et al. (2011) considered two initial end‐member compositions: “Taylor Whole Moon” (Taylor et al., 2006) and “lunar primitive upper mantle”(Hart & Zindler, 1986; Longhi, 2006). Based on an assumed normative mineralogy, they predicted a

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50‐km‐thick IBC layer with only 2‐ to 4‐vol% ilmenite. Lin et al. (2016) carried out experiments with variouswater contents and predicted a 24‐km‐thick IBC layer with 14‐vol% ilmenite. However, with a lunarprimitive upper mantle bulk composition, Rapp and Draper (2018) suggested more limited ilmeniteproduction; Charlier et al. (2018) found relatively late ilmenite saturation after 97% LMO crystallization.Thus, wide ranges of IBC layer thickness and ilmenite fraction have been reported. Rescaled to an LMOthickness of 1,000 km, IBC layer thickness ranges from 18 to 50 km with the ilmenite volume percentagecorrespondingly varies from 16% to 3.6% (Figures 1A and S1).

The above 18‐ to 50‐km‐thick IBC layer is derived solely from mass balance and phase equilibria considera-tion assuming perfect fractional solidification. However, solidification is a dynamic process that may dis-perse FeO‐TiO2‐rich materials to greater depth than predicted by mass balance. Hess and Parmentier(1995) and Parmentier and Hess (1999) used R‐T instability to explore such redistribution of IBC(Figure 1B) during the early phases of overturn. According to Hess and Parmentier (1995), tiny diapirsdevelop when the layer is still solidifying (Figures 1B and S2). These tiny diapirs migrate downward forminga thicker IBC layer. The thickened IBC layer then form larger diapirs having a sinking velocity by 2 orders ofmagnitude bigger than that for the initial tiny diapir (Text S1 and Figure S2 in the supporting information).Hence, these larger diapirs sweep all previous tiny diapirs before tiny diapirs sink to upper mantle and evolveto a cumulate mantle overturn event. Our initial condition considers such a thicker but diluted solid IBC

Figure 1. Synthesized IBC thickness and fraction (A) from five LMO crystallization models (Figure S1), the thickeningprocess during the IBC crystallization (B), the derived relation between the IBC volume fraction and IBC thickness (C),and two types of initial temperatures (D). The five LMO crystallization models include Snyder et al. (1992), Hess andParmentier (1995), Elkins‐Tanton et al. (2011), Elardo et al. (2011), Lin et al. (2016), and Rapp & Draper (2018; seeFigure S1). Those of Rapp and Draper (2018) and Lin et al. (2016) represent the lower and upper bounds of the ilmenitebudget, respectively. During the crystallization, the IBC layer thickens downward (B; also see Figure S2). The possible IBCthickness‐density range is depicted by the blue shadow zone in (C). The LMO crystallization temperature from Lin et al.(2016) is interpolated into ourmain initial temperature (dashed black curve) in (D) and (E), whichmerges to the peridotitesolidus at the crossing point between Lin et al. (2016) and the peridotite solidus. IBC = ilmenite‐bearing cumulate;LMO = lunar magma ocean; R‐T = Rayleigh‐Taylor.

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layer that could develop due to this process. Parmentier et al. (2002) presented a numerical model illustratingthis process. Our results in the supporting information suggest a >150‐km IBC thickness, which is also con-sistent with estimation from recent study (Zhao et al., 2019). Thus, we consider IBC layer thickness up to150 km, larger than the maximum of 50 km predicted by mass balance alone.

It is important to recognize the likely complexity of the IBC solidification and overturn process. The deposi-tion of progressively denser solid (from an overlying liquid) at the top of a thickening solid layer can lead tooverturn at a range of spatial and temporal scales (Boukare et al., 2018). Dense partially molten diapirs mayalso form before IBC layer solidification is complete, then the segregation of melt from crystallized solidsbecomes a potentially important aspect of IBC instability (Boukare et al., 2019). Resolving these processesis beyond the scope of the present work and part of ongoing studies. The range of values from 50–150 kmthat we have chosen shows the potential importance of processes controlling IBC layer thickness.

We adopt upper and lower bounds on total ilmenite content from previous petrological models (Charlieret al., 2018; Elardo et al., 2011; Elkins‐Tanton et al., 2011; Hess & Parmentier, 1995; Lin et al., 2016; Rapp& Draper, 2018; Snyder et al., 1992) but explore a range of IBC layer thicknesses reflected in the above dis-cussion. In varying the IBC thickness, we correspondingly adjust the density difference while keeping thetotal ilmenite budget fixed. The average between the upper bound (Lin et al., 2016) and the lower bound(Rapp & Draper, 2018) of the ilmenite content (the dark blue line in Figure 1C) is close to the result ofHess and Parmentier (1995).

In addition to its density and thickness, the viscosity contrast between the IBC layer and the underlyingmantle plays a key role in determining the dynamical behavior of an IBC layer (Elkins‐Tanton et al.,2002; Parmentier et al., 2002; Scheinberg et al., 2014). Previous modeling studies either regarded the IBCviscosity to be the same as that of peridotite or arbitrarily assigned a range of values for parameter explora-tions (e.g.,Elkins‐Tanton et al., 2002; Parmentier et al., 2002; Yu et al., 2019; Zhao et al., 2019). Here weexplore a range of IBC viscosities constrained by the results of a recent experimental study which, for the firsttime, measured the viscosity of ilmenite and extrapolated the measurement to the viscosity of harzburgite‐mixed IBC (Figure 2; Dygert et al., 2016). The predicted viscosity of pure ilmenite is ~4 orders of magnitudelower than that of dry harzburgite (approximated by dry olivine).

Because viscosity is temperature dependent, the concentration of HPEs in the IBC layer counteracts theeffect of conductive cooling and hence is fundamental to IBC layer dynamical behavior. HPE content reflectsthe KREEP abundance in the IBC layer (Hess & Parmentier, 1995). The proportion of KREEP entrained inthemantle interior by the IBCs is not well constrained, but values between 20% and 50% have been suggested(Parmentier & Hess, 1999). Here we test a KREEP content ranging from 0% to 50% within the IBC layer, arange that has been used in previous studies (e.g.,Zhang et al., 2013a, 2013b; Zhong et al., 2000). The heatgeneration rate of KREEP is estimated based on the bulk U and Th abundances. The bulk U and Th abun-dances of the present day are taken as 25.7 and 102.8 ppb (Th/U = 4), respectively (e.g., Taylor, 1982). TheMoon is highly depleted of the volatile element K (Albarede et al., 2014; Jones & Palme, 2000; Wang &Jacobsen, 2016). We neglect the K contribution to the thermal budget in most of our calculations. To testthe sensitivity of the results, we present a case with K/Th ratio of 2,500 (section 3.2.5). We refer to the decayrates of heat generation presented in Turcotte and Schubert (2002).

2.2. Governing Equations

To understand the dynamical behavior of the IBC layer, we employ both R‐T instability analysis and finiteamplitude numerical convection models. The R‐T analysis can efficiently predict the dominant wavelengthfor a large range of viscosity contrasts and IBC layer thicknesses. The details of R‐T analysis in the sphericalgeometry are shown in the supporting information.

However, R‐T instability analysis solves only infinitesimal deformation but neglects the evolution. Hence,we introduce a mantle convection model to study the instability of the IBC layer and its subsequent thermo-chemical evolution in a 3‐D spherical shell. Initially, the dense IBC layer is sandwiched between theanorthosite crust and olivine‐pyroxene cumulated formed at earlier stages of LMO solidification(Figure 2A). The thermochemical model is an expansion of those used in our earlier studies (Zhang et al.,2013a, 2013b, 2017), featuring infinite Prandtl number, Boussinesq approximation, composition‐ andtemperature‐dependent rheology, negligible dissipation and adiabatic heat, parameterized treatment of

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core cooling and solidification, and remelting of overturning cumulates. The latter is new in the presentstudy (see equation (3b) below). In nondimensional form, the governing equations for our models are

∇·u¼0; (1)

−∇P þ∇· η_εð Þ¼ RaT−RbCð Þber ; (2)

∂T∂t

þ u·∇ð ÞT ¼ ∇2T þH C; tð Þ; T≤Tsð Þ

1þ Lnonð Þ ∂T∂t

þ u·∇ð ÞT� �

¼ ∇2T þ H C; tð Þ þ LnondTs

drur ; T>Tsð Þ

8>><>>:

3að Þ3bð Þ

∂C∂t

þ u·∇ð ÞC¼0; (4)

Figure 2. Viscosity of ilmenite and IBC. (A) The result of a deformation experiment on ilmenite by Dygert et al. (2016).The figure shows the effects of temperature on viscosity of synthetic ilmenite, where the differential stress is taken as0.3 MPa as an approximation at the base of the lunar crust. Wet and dry olivine viscosities are also shown here for com-parison. At temperatures higher than 1000 °C, the viscosity of ilmenite can be 3 orders of magnitude lower than that of wetolivine and more than 4 orders lower than that of dry olivine. (B) A figure from Zhang et al. (2017), which shows theviscosity for the assemblage of ilmenite and dry olivine with respect to the volume fraction of ilmenite. Prediction of theviscosity of a mixed material is derived from both Tullis (e.g., Tullis et al., 1991) model and isostress model, with thetemperature and differential stress fixed at 1250 °C and 0.3 MPa, respectively. The viscosity of an IBC layer with 7.8‐vol%ilmenite can be 3 orders of magnitude lower than the viscosity of dry olivine according to the isostress estimation. Thethree green squares are selected IBC viscosities, which are used in our viscosity formulation. (C) The converted density(black curve) and relative viscosity (blue area) ranges of the IBC layer. The viscosity of an IBC layer with 12‐vol% ilmenitecan be more than 3 orders of magnitude lower than the viscosity of dry olivine according to the isostress estimation.When a 30‐km‐thick and hence a 3,608‐kg/m3

‐dense IBC layer is adopted, the viscosity range is as shown by theyellow line. (D) The viscosity contrast between a molten and hydrous olivine and dry olivine based on Hirth andKohlstedt (2003). IBC = ilmenite‐bearing cumulate.

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where u is velocity, P is pressure, η is the dynamic viscosity, T is tempera-ture, ber is the unit radial vector, t is time, _ε is the strain rate, H(C, t) is theIBC‐dependent internal heat generation rate, and C is the IBC composi-tion that has a value between 0 (for pure peridotite) and 1 (for pureIBC). The rate of heat production is based on the HPE content of theIBC. Lnon in 3 is the nondimensional latent heat:

Lnon ¼ LCp TL−TSð Þ (5)

where L, Cp, and TL and TS are the latent heat, specific heat, and liquidusand solidus for the peridotite, respectively (Table 1). The reciprocal of Lnonis the Stefan number for phase transformation. The liquidus is assumed tobe that of KLB‐1 peridotite (Hirschmann, 2000; Katz et al., 2003;Laneuville et al., 2013). The largest temperature difference between TLand TS is ~500 K. The thermal (Ra) and chemical (Rb) Rayleigh numbersin equation (2) are defined as follows:

Ra ¼ αρ0gT R0−Rcð Þ3η0κ

; (6)

Rb ¼ αρ R0−Rcð Þ3η0κ

; (7)

where α is the thermal expansivity; ρ0 is the reference density for the mantle, η0 is the reference viscosity; g(=1.63 m/s2) is the lunar surface gravitational acceleration, which decreases with depth (e.g., Zhang et al.,2013a); and κ is the thermal diffusivity. The parameter used to prescribe the IBC density is the buoyancynumber B, which is defined as

B ¼ Rb

Ra¼ ρIBC−ρ0ð Þ

αρ0Tð Þ ; (8)

Equation 3 allows us to investigate the thermal effect caused by the partial melting. Derivation of 3 follows

Laneuville et al. (2013) and the melt generation rate _Γ calculated from the degree of melting (F):

_Γ ¼ ∂F∂t

þ u·∇F: (9)

The degree of melting F is parameterized as a linear function of the dimensionless temperature:

F ¼ T−TS

T l−TS: (10)

We solve the governing equations listed above numerically using CitcomS for the incompressible Stokes flowin a spherical shell (Zhong et al., 2008). The composition field (equation (4)), included to track the advectionof IBC, is solved using the tracer method (e.g., Tackley & King, 2003). In each case, the whole spherical shellis divided into 12 caps, with each cap being divided into 49 × 49 × 49 nodes or 65 × 65 × 65 nodes. The sphe-rical shell includes a 1,400‐km lunar mantle and crust with a core that has a 340‐km radius (Table 1). Nodesare refined radially to increase the resolution in the much thinner IBC layer. Depending on the IBC thick-ness, 5 to 10 elements are assigned to the IBC layer vertically. Free‐slip condition is used as mechanicalboundary condition. The normalized core temperature is initially 1 and evolves by balancing the energy withthe mantle following the approach of Zhang et al. (2013a). The normalized temperature on the upper bound-ary is set to 0.

2.3. Parameter Space and Initial Conditions of Models

Based on studies summarized in section 2.1, we examine a selected range of IBC layer thicknesses (30–150 km, Table 2 and Figure 1B) in our finite element models. Computational convergence limits the

Table 1Model Parameters

Parameter Value

Moon radius, R 1,740 kmOuter core radius, Rc 340–410 kmInner core radius Ri —

Surface gravitational acceleration, g 1.63 m/s2

Gravitational acceleration at CMB, gc 0.86 m/s2

Mantle specific heat, Cp 1,000 J·kg−1·K−1

Thermal diffusivity, κ 10−6 m2/sLatent heat, L 6 × 105 J/molCore thermal conductivity, kc 50 W·m−1·K−1

Thermal expansion of mantle, α 2.3 × 10−5 K−1

Thermal expansion of core, αc 9 × 10−5 K−1

Temperature difference, ΔT 1660 KInitial CMB temperature, Tc 1610 °CMantle density, ρ0 3,400 kg/m3

Density of IBC‐rich material, ρIBC 3,460–3,700 kg/m3

Core density, ρc 7,800 kg/m3

Mantle specific heat, Cp 1,250 J·kg−1·K−1

Core specific heat, Cpc 800 J·kg−1·K−1

Reference viscosity, η0 5 × 1020–5 × 1021 Pa·sCore sulfur content, S 5%

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smallest low‐viscosity layer thickness in our finite element model to ~30 km, and a thickness greater than150 km results from the prediction of a long‐time solidification of LMO (Elkins‐Tanton et al., 2011;Solomon & Longhi, 1977).

In our models, the viscosity of the IBC depends on the ilmenite, water content, and the physical state of IBC.Given the current state of knowledge of IBC rheology, we examine a large range of IBC viscosities withIBC‐to‐peridotite viscosity ratios varying from 10−5 to 10−1. Of these, 10−3 to 10−1 is within the viscosityreduction predicted for the IBC‐harzburgite rheology (Figures 2B and 2C). Reduction of the viscosity to10−5 requires the weakening effect of partial melting of the IBCs (Elkins‐Tanton et al., 2002; alsosection 3.2.1) and water enrichment (Hui et al., 2013). These weakening effects have been experimentallystudied for olivine (Hirth & Kohlstedt, 2003; Mei et al., 2002; Mei & Kohlstedt, 2000; Scott & Kohlstedt,2006), although the weakening for ilmenite and clinopyroxene is unknown and a subject of continuingexperimental rock deformation study (see Tokle et al., 2017). Applying these twoweakening effects of olivineto our IBC layer, we observe a ~3 × 10−3 viscosity reduction (Figure 2D and section 4.1; also see a Matlabscript in the supporting information).

In calculations with the temperature‐dependent viscosity, we used the Arrhenius flow law. The overallrheology formulation is the same as that in Zhang et al. (2017; also see Hu et al., 2018):

η T;Cð Þ ¼ CηIBC þ 1−Cð Þη0½ � exp ERT

−E

RTref

� �; (11)

where E is the effective activation energy, ηIBC is the IBC viscosity, and η0 is the peridotite viscosity at thereference temperature Tref. In this study, we set Tref to 1250 °C. Although experiments on pure ilmenite(Dygert et al., 2016) have shown a smaller activation energy for ilmenite (Figure 2A), we assumed that theilmenite has the same activation energy as that of peridotite for simplicity. (We return to this point insection 4.4.) We test a range of the effective activation energies between 100 and 200 kJ/mol in our simula-tions. These effective activation energies are smaller than the measurement from laboratory experiments toproperly scale our viscosity model for the effect of the power law on viscosity (e.g., Christensen, 1984). Thiseffective viscosity approximation is appropriate and widely applied for investigating lunar mantle dynamics(e.g., Zhang et al., 2013a). As the pressure in the lunar mantle is relatively low, its effect on the effective visc-osity is not explicitly considered in this study. Equation (11) is a simple linear combination between IBC and

Table 2Input Parameters and Key Outputs

Casea KREEPb content (%) TSc Ra B Degreed toverturn (Myr) qmax (mW/m2)

D30v1e‐3V1e21E100 0, 20, 50 × 5.9×105 1.6 4, 11 119 28.0D30v1e‐3V1e21E100_Ts 50 √ 5.9×105 1.6 4, 11 115 29.5D30v1e‐3V1e21KE100 K/Th = 2,500 × 5.9×105 1.6 4, 11(17) 121 25.6D30v1e‐3V1e20E100 50 × 5.9×106 1.6 4, 16 17.6 148D30v1e‐3V1e21E100_m65 50 × 5.9×105 1.6 4, 11 118 30D30v1e‐3V1e21E200 50 × 5.9×105 1.6 4, 10 141 16.2(D30v1e‐4V1e21E100) 50 × 5.9×105 1.6 4, 5, 10 13.1 60.5(D30v1e‐5V1e21E100) 50 × 5.9×105 1.6 3, 7 4.1 71D50v1e0V1e21E100 50 × 5.9×105 0.98 4 (6) — 17.0D50v1e‐2V1e21E100 50 × 5.9×105 0.98 3 (6) >500 37.5D50v1e‐3V1e21E100 50 × 5.9×105 0.98 2, 6 86.4 41(D75v5e‐5V1e21E100) 50 × 5.9×105 0.66 3 4.8 307.8(D100v1e‐4V1e21E100) 50 × 5.9×105 0.51 3 7.2 219.0(D150v5e‐4V1e21E100) 50 × 5.9×105 0.35 3 10.8 160.5(D150v1e‐3V1e21E100) 50 × 5.9×105 0.35 3 19.2 123.6(D150v1e‐4V1e21E100) 50 × 5.9×105 0.35 1 5.1 362.9(D150v1e‐4V1e21Core410E100) 50 × 5.9×105 0.35 1 5.6 300.9

Note. KREEP = K, rare earth elements, and P.aThe case names with the parentheses are cases in which the IBC viscosity and thickness are chosen to explore the long‐wavelength downwellings.bCalculations examine three inclusions of KREEP content, that is, 0%, 20%, or 50%, and the K content inclusion in the IBC layer. cTS denotes the peridotitesolidus as the initial temperature. Otherwise, the initial temperature is assumed to be the crystallization temperature from Lin et al. (2016). dThe numbersin the parentheses are different degrees slightly weaker than the strongest degree.

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harzburgite viscosities and provide an approximation of the viscosity of IBC‐harzburgite mixtures. Thepotential viscosity reduction by melt and water is superimposed on equation (11), which allows the IBC visc-osity to be 10−5 times smaller than the peridotite mantle.

Our model starts from 4.42 Ga, which is the average ferroan anorthosite age (Borg et al., 2011, 2015; Elkins‐Tanton et al., 2011; Snape et al., 2016). Initially, an IBC layer is at the base of a crustal layer which forms aconductive lid. Our thermochemical models introduce no initially preferred wavelength on the IBC bottominterface; instability is caused by the random distributions of tracers (e.g., Scheinberg et al., 2014). The initialtemperature of our models follows the crystallization temperature of LMO and has a 90‐km top thermalboundary layer (e.g.,Konrad & Spohn, 1997; Zhang et al., 2013a). We consider two initial temperatures:One follows the peridotite solidus as in Zhang et al. (2013a), and the other is a smoothed approximationof the crystallization temperature derived from a recent crystallization experiment (Lin et al., 2016; dashedorange curve in Figure 1D). This results in the initial temperature profile shown by the dashed black curve inFigures 1C and 1D). The latter merges to the peridotite solidus at depth.

To determine the extent of overturn, we calculate the fraction of IBC materials in the lower mantle(radius < 1,040 km). We record the time‐dependent IBC fraction IBCl and the final maximum IBC fractionin the lower mantle IBCl

max. The time at which the ratio, IBCr = IBCl/IBClmax, becomes larger than 1% is

regarded as the beginning of overturn process (also shown graphically in section 3.2.1). We define the timeat which IBCr has the largest positive curvature RS of IBCr evolution curve as the overturn completion time,if IBCr is larger than 50%.

toverturn ¼ t max RSð Þ½ �−t IBCr<1%½ � (12)

This definition of toverturn is not sensitive to the choice of IBCr for the overturn starting time. Finally, it isworth noting that toverturn is not accounted for from the starting time of our model.

3. Results

The structure of IBC downwellings was studied by R‐T instability method and by 3‐D finite‐element model-ing of thermochemical convection. R‐T analysis considers the dominant wavelength for small amplitudeinstabilities. The finite amplitude temporal evolution of IBC downwellings is addressed in our numericalthermochemical models.

3.1. Wavelengths of R‐T Instability

The R‐T analysis considers a geometry of three‐layers in a spherical shell, similar to that of Zhong and Zuber(2001) but with one more layer of a 45‐km‐thick lunar crust on top (Wieczorek et al., 2013). The IBC layer issandwiched between the crust and the peridotite mantle. The CMB radius is set to 340 km (Matsuyama et al.,2016). Details of the R‐T analysis are provided in the supporting information.

The dependence of themost unstable downwelling wavelength on the IBC thickness and viscosity contrast isshown in Figure S3A, which is calculated from choices of 100 thicknesses and 50 viscosity contrasts. For a50 km thick IBC layer, an IBC viscosity contrast no smaller 10−2 leads to an unstable downwelling atshort‐wavelengths of harmonic degree 4 or higher. For a 30 km IBC layer, the IBC viscosity must be~1.5 × 10−4 times lower than the peridotite viscosity in order to promote a degree‐1 downwelling(Figure 3A). In general, the thinner the IBC layer, the larger the viscosity contrast is required to formdegree‐1 downwelling (Figure 3A). The predicted pattern of the R‐T diagram (Figure S3A) is different fromthat in Zhong and Zuber (2001) because ourmodel has an overlying crust layer. However, the pattern is simi-lar to the three‐layer model presented by Qin and Zhong (2014), though their model addressed theplume instability.

3.2. Finite Element Modeling of the IBC Layer Evolution

We first compare finite element models using a viscosity that solely depends on IBC content (solid squares InFigure 3A; also see supporting information Table S1) with the R‐T analysis. These models are similar to onesdiscussed below but without temperature‐dependent viscosity. The applied solidus and liquidus are shownin Figure 4A. The wavelength of downwellings in the finite amplitude models are generally consistent withR‐T prediction (Figure 3A). The convective planforms for six representative calculations are shown in

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supporting information Figures S5–S8. The overturn times quantified from these thermochemicalcalculations (Table S1 and Figure S9) confirms the growth rate analysis toverturn ~ ηIBC

1/3 (Hess &Parmentier, 1995; Whitehead, 1988). We will come back to overturn time in section 4.1.

We next consider themore physically relevant scenario with both IBC‐ and temperature‐dependent viscosity(equation (11)) and explore how a stagnant lid influences downwelling dynamics. We calculated 18 cases(Table 2) comprised of 11 combinations of the IBC viscosity and thickness (circles in Figure 3B). These 18cases were chosen to examine a range of IBC layer thicknesses, IBC viscosities, reference viscosities, activa-tion energy, initial temperature profiles, and core radii to investigate the convective structures, evolutions ofthe temperature, possible occurrence of melting, and CMB heat flux. We adopt a labeling scheme for ourmodels based on the IBC layer thickness, IBC viscosity, reference viscosity, KREEP fraction, initial tempera-ture profile, and activation energy (Table 2). For example, case D30v1e‐3V1e21H50E100 has a 30‐km IBClayer, an IBC viscosity contrast of 10−3, a reference viscosity of 1021 Pa·s, 50% KREEP budget, and anactivation energy of 100 kJ/mol. When the default initial temperature profile (Figure 4A) of Lin et al.(2016; without any designation) is changed to peridotite solidus, the case name becomes D50v1e‐2V1e21H50_Ts (designated by TS).3.2.1. Models With the Same IBC Thickness but Different ViscositiesOur first group of models explores the effect of IBC viscosity on the wavelength of downwellings. For an IBCthickness at 30 km, we vary the IBC viscosity contrast from 10−3 to 10−5 (Table 2). The IBC density followsthe thickness‐density relation (Figure 3C). A 30‐km‐thick IBC layer is the thinnest layer for which CitcomScan achieve satisfactory computational convergence.

Our standard or reference case is D30v1e‐3V1e21E100. The 30‐km IBC layer has 7.8‐vol% ilmenite and is208 kg/m3 denser than the underlyingmantle (Figure 2C). This ilmenite content is within the range reportedin the literature (3.6 vol%, Elardo et al., 2011 to 8 vol%, Lin et al., 2016; Figure S1). An IBC consisting ofdunite with 7.8‐vol% ilmenite is expected to have a viscosity reduced by three orders of magnitude relativeto pure dunite (the green square on dashed curve in Figure 2B). The effective activation energy of100 kJ/mol approximates an equivalent non‐Newtonian activation energy of 335–400 kJ/mol, which is thelower bound for the measured activation energy of olivine (Dygert et al., 2016; Hirth & Kohlstedt, 2003;Karato & Wu, 1993). The initial viscosity and density are shown as solid black curves in Figures 4C and 4D.

Due to temperature‐dependent viscosity, case D30v1e‐3V1e21E100 displays a strong stagnant conductive lid.The dense IBC layer becomes unstable after 36 Ma (Figures 5A and 6A), resulting in many cold, small down-wellings (Figure 5B) with strongest wavelengths at spherical harmonic degrees of 4 and 11 (Table 2). Thedownwellings take 20 Myr (model time 60 Ma) to reach the CMB and after 115 Myr (model time 155 Ma),most of the IBC materials reside at the CMB (Figures 5C and 5D). A movie showing the spatial and

Figure 3. The predicted dominant spherical harmonic degree of overturn as a function of the IBC viscosity contrast andthickness for a parameter range relevant to the lunar mantle. The background contours is from Rayleigh‐Taylor analysis,while the square (A) and circle (B) symbols are selected viscosity and IBC thickness for the thermochemical modelswithout and with the temperature‐dependent viscosity (Tables S1 and 2), respectively. IBC = ilmenite‐bearing cumulate.

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temporal evolution of this case is provided in supporting information Movie S1. The low temperature andhigh density of IBCs give rise to inverted temperature profiles (Figures 4A and 4D). Temperature higherthan solidi of peridotites occur in the upper mantle (Figures 4A and 7A), which should lead to a low‐degree melting. The time from overturn initialization to completion is 119 Myr (Table 2; black curve inFigure 6A and dashed lines in Figure 6C). The CMB temperature reduces by ~45 K during the overturnperiod. The resolution test has 12 × 653 grids (case D30v1e‐3V1e21E100_m65) compared to the originalmesh of 12 × 493 shows only slight differences in terms of either the convective structure or overturntimescale (Table 2).

We examine the thermal evolution of IBC layer in the first ~36 Myr before the instability occurs. CaseD30v1e‐3V1e21E100 has 50% KREEP budget (Table 2). We also ran a case that has 20% KREEP budget(Table 2). During the first 10 Myr of our model run, radiogenic heating raises the layer temperature by~17 K higher than the temperature profile of Lin et al. (2016; purple curve in Figure S4), greatly exceedingthe IBC solidus (red curve in Figure S4; van Orman & Grove, 2000; Yao & Liang, 2012). When the KREEPcontent in IBC is reduced to 20% (Table 2), the layer temperature is still ~3 K higher than that from Lin

Figure 4. Initial condition profiles and postoverturn profiles for cases with the temperature‐dependent viscosity. (A) Thetemperature profiles of our models; the solid black curve is the initial solidus from Lin et al. (2016) with a 340‐km‐radiuscore and a 90‐km top boundary layer. Orange solid curve is the liquidus. Dotted curves are temperature profiles ofrepresentative cases at the times of overturn completion. A case with the peridotite initial temperature is presented in (B).(C) The initial viscosity profiles. A low‐viscosity ilmenite‐bearing cumulate layer lies under a 45‐km‐thick crust. (D)Profiles of the density evolution. Solid lines are initial density profiles, where ilmenite‐bearing cumulate layers generallyhave densities 50–250 kg/m3 higher than those of the underlying harzburgite. Dotted curves here are the density profilesafter the overturns.

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et al. (2016) after 10 Ma. After changing the initial temperature to the peridotite solidus (case D30v1e‐3V1e21E100_Ts), the IBC layer temperature is even higher than the IBC solidus (orange curve in FigureS4). Examinations of IBC‐layer temperature above show a clear temperature increase before overturn,although the remelting state is determined by the relative difference between our initial temperature andthe IBC solidus, which is largely dependent on our assumed initial thermal state after LMO solidification.Nevertheless, the IBC‐layer temperature increase before overturn may reasonably induce partialmelting/remelting in the IBC layer, resulting in further reduction of the layer viscosity (Scott & Kohlstedt,2006) below the lower bound of the experimental results (Figures 2B–2D).

Experimentally constrained IBCweakening rheology is not sufficient to promote long‐wavelength downwel-lings. Therefore, we consider models with further reduced IBC layer viscosity. Lower viscosities could resultfrom the presence of melt (Figure S4; e.g., Elkins‐Tanton et al., 2002; Scott & Kohlstedt, 2006) and/or waterenrichment (Hui et al., 2013;Mei &Kohlstedt, 2000). The next two cases (case names with the parentheses inTable 2), consider an IBC viscosity reduction of 1 × 10−4 and 1 × 10−5 (Table 2; circles in Figure 3B). Thesmallest IBC viscosity reduction is limited to 10−5 due to CitcomS computational capability.

Figure 5. The 3‐D thermochemical structures and 2‐D cross sections in the plane whose normal vector is perpendicular tothe page of the corresponding 3‐D structure to compare the downwellings for cases with different ilmenite‐bearingcumulate viscosities, for case D30v1e‐3V1e21E100 before overturn (A and B) and at the end of overturn (C and D), for caseD30v1e‐4V1e21E100 at 16.7 Ma (E and F) and at 58.6 Ma (G and H), and for case D30v1e‐5V1e21E100 at 8.1 Ma (I and J)and 12.3 Ma (K and L). In the 3‐D view, green stands for isosurfaces of the chemical composition at a contour level of 0.8.Transparent blue and red stand for isosurfaces of the residue temperature, with the contour levels set at −130, 130 °C for(A), (E), and (I) and −200, 200 °C for (C), (G), and (K).

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Figure 6. Evolutions of the IBC volume fraction in the lower mantle (A) and the composition/density profiles for cases D30v1e‐4V1e21E100 (B), D30v1e‐3V1e21E100 (C), D50v1e‐3V1e21E100 (D), and D50v1e‐2V1e21E100 (E). The dashed vertical lines stand for the inflexion with the largest curvature before theIBC fractions become relatively flat. The IBC fraction is calculated by dividing the volume of IBC materials in the lower mantle by the total lower mantle volume.IBC = ilmenite‐bearing cumulate.

Figure 7. The evolution of azimuthally averaged temperatures for three cases: (A) D30v1e‐3V1e21E100, (B) D30v1e‐4V1e21E100, and (C) D100v1e‐4V1e21E100,with the initial mantle temperature of Lin et al. (2016).

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Cases D30v1e‐4V1e21E100 and D30v1e‐5V1e21E100 show longer downwelling wavelengths (Figures 5E–5L). The dominant modes are 4, 5, and 10 for D30v1e‐4V1e21E100 and 3 and 7 for D30v1e‐5V1e21E100(Table 2; Figure 5). The overturn times are reduced to 13.1 Myr for case D30v1e‐4V1e21E100 (Figures 6Aand 6B and Figure 7B) and 4.1 Myr (Figure 8) for the lower viscosity case D30v1e‐5V1e21E100 (Table 2).These two cases show that when the viscosity contrast of IBC is smaller than 10−3, the IBC layer decouplesfrom the overlying thermal boundary layer, having an overturn timescale of ~10 Myr (Figure 8). The movieof case D30v1e‐4V1e21E100 is provided in supporting information Movie S2.3.2.2. Models With the Same IBC Viscosity but Different ThicknessesThe R‐T analysis shows that the IBC thickness has a strong effect on the wavelength of downwellings(Figure 3). Our second group of models investigates this effect. The IBC viscosity is fixed to 10−3 and theIBC thickness is increased to 50 and 150 km (Table 2).

Compared to case D30v1e‐3V1e21E100, cases D50v1e‐3V1e21E100, and D150v1e‐5V1e21E100 clearly showlonger wavelengths (Figures 9E–9L). Case D50v1e‐3V1e21E100 does not differ from D30v1e‐3V1e21E100significantly in terms of overturn time (red curve in Figures 6A and 6D). However, D150v1e‐5V1e21E100evolves to a downwelling planform of degree‐3 (Table 2 and Figures 9I–9L) with an overturn time reducedto ~10 Myr, which suggests that an IBC layer thicker than 100 km minimizes the influence of top thermalboundary layer.3.2.3. Models Examining the Effect of Stagnant LidWe consider two cases with the same IBC thickness (50 km) but different relative IBC viscosities of 1 and10−2 (D50v1e0V1e21E100 and D50v1e‐2V1e21E100, Table 2 and Figure 3B). The case with relative IBC visc-osity of 1 (i.e., temperature‐dependent viscosity only) does not develop clear IBC downwellings until after~600 Ma (Figures 10A and 10B) when a weak spherical harmonic degree‐4 structure develops (Figure 10Aand Table 2). Most of the IBC layer is immobilized by increased viscosity due to the thickening of coldthermal boundary.

The case with relative IBC viscosity of 10−2 develops downwellings dominated by degree 3 and degree 6(Table 2). The reduced IBC viscosity weakens the coupling between the IBC layer and the top stagnantlid, facilitating the IBC downwelling (Figure 10C). However, the gradual sinking IBC exhibits differentbehavior from the rapid, long‐wavelength overturn proposed in Hess and Parmentier (1995). Throughthe model evolution of 500 Myr, the IBC layer continues to form small downwellings (Figure 10D, orangecurve in Figure 6A). This case shows that a weak IBC is necessary to decouple the IBC materials from thetop stagnant lid. When the IBC viscosity is not small enough (<10−2), the convection is in a regime of“continuous small‐scale downwellings,” which causes a long‐term preservation of IBCs beneath the over-lying crust. When further reducing the IBC viscosity to 10−3, which is case D50v1e‐3V1e21E100 displayedin Figures 9E–9H, the system returns to its regular behavior and completes overturn in 86 Ma (Figures 6Aand 6D).

Figure 8. Overturn times with respect to the thickness and relative viscosity of the IBC layer for 16 cases (Table S1)without temperature‐dependent viscosity (A) and 11 cases (Table 2) with temperature‐dependent viscosity (B). Theoverturn timescale is determined as the time between IBCr = 1% and the maximum curvature of IBCr (Figure 7A).IBC = ilmenite‐bearing cumulate.

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3.2.4. Models Generating Long‐Wavelength DownwellingsWe next explore the generation of long‐wavelength downwellings and choose the IBC thickness‐viscositycombinations within the long‐wavelength domain in the R‐T parameter space (circles beyond the thicknessof 50 km in Figure 3B). We extend the IBC thicknesses to 75, 100, and 150 km and run the following fourcases: D75v5e‐5V1e21E100, D100v1e‐4V1e21E100 (Figures 10E–10H), D150v5e‐4V1e21E100, andD150v1e‐4V1e21E100 (Figures 10I–10L). The resultant downwelling wavelengths and the overturn time-scales are summarized in Table 2 and Figure 8. All four cases evolve to downwelling wavelengths longerthan degree 3 (Table 2). When the IBC thickness is 100 km and viscosity is 10−4, the overturn time reducesto 7.2 Myr (Figures 6A and 10E–10H).

Our last case has IBC thickness 150 km and the IBC viscosity 10−4 (Table 2), which evolves to a degree‐1downwelling within 8 Ma (Figures 10I and 10J; supporting information Movie S4). This is our only case thatproduces the degree‐1 downwelling (Table 2 and Figures 10I–10L) with the composition‐temperature‐dependent viscosity (equation (11)).

When we compare the influences of IBC thickness and IBC viscosity on the overturn timescale (Figure 8B),we find that the IBC viscosity is more important than the IBC thickness. This is because the IBC viscosityplays the key role in decoupling the IBC layer from the top thermal boundary.

Figure 9. The 3‐D thermochemical structures and 2‐D cross sections to compare the downwellings for cases with differentilmenite‐bearing cumulate thicknesses. Resultant structures for case D30v1e‐3V1e21E100 before overturn (A and B)and at the end of overturn (C and D), for case D50v1e‐3V1e21E100 at 44.6 Ma (E and F) and at 131 Ma (G and H), and forcase D150v1e‐3V1e21E100 at 21.4 Ma (I and J) and 41.1 Ma (K and L). The 3‐D isosurface/contours are constructedfollowing the same procedure as those in Figure 5.

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3.2.5. Sensitivity to Additional ParametersOther factors parameters that affect the dynamics and evolution of our model include the KREEP content,activation energy, and reference viscosity η0 (hence Rayleigh number Ra). We address them through fiveadditional case studies by varying model parameters relative to the reference case D30v1e‐3V1e21E100.We are especially interested in (1) the effects on downwelling and (2) the strength of the top conductivelycooled stagnant lid.

In case D30v1e‐3V1e21H0E100, we reduce the KREEP abundance to the same level as that of the ambientmantle (KREEP content column in Table 2). The calculated mantle structures and dominant wavelengthsare almost identical to those of case D30v1e‐2V1e21H50 despite the lower heat production in the IBC(Table 2). The lower heat production and buoyancy do not cause a distinguishable difference in the overturntime due to the small temperature reduction of the IBC layer during the rapid overturn. Similarly, additionalK budget (K/Th = 2,500) into the KREEP does not change the downwelling structures (case D30v1e‐3V1e21KE100 in Table 2).

Figure 10. The 2‐D thermal/chemical cross sections (A–D) to examine the stagnant lid effect and 3‐D thermochemicalstructures showing the long‐wavelength downwellings (E–L). The thermal (A) and chemical (B) snapshots of caseD50v1e0V1e21E100 at 626 Ma show that no clear downwelling develops within 600 Myr. The chemical evolution of caseD50v1e‐2V1e21E100 is presented at 99 Ma (C) and 316 Ma (D). The initializations of degree‐3 downwellings (E and Fat 13.4 Ma) and settling downs (G and H at 20.6 Ma) for case D100v1e‐4V1e21E100 are shown in the middle row. Modeledinitialization (I and J at 7.9Ma) and formation (K and L at 13Ma) of degree‐1 downwellings for case D150v1e‐4V1e21E100.Color schemes are the same as in Figure 5.

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When we increase the activation energy to 200 kJ/mol, the stagnant lid becomes stronger. The instabilitytakes longer to form and the overturn time increases to ~150 Ma. The CMB heat flux reduces greatly(Table 2).

The Rayleigh number depends on the reference viscosity (equation (6)). When we reduce the reference visc-osity and IBC viscosity by one order of magnitude, the Rayleigh number increases but the viscosity contrastbetween the IBC layer and the mantle beneath does not change. This higher Rayleigh number case (D30v1e‐3V1e20E100) produces smaller downwellings, a shorter overturn timescale, and a larger maximum CMBheat flux of 148 mW/m2 (Table 2 and Figure S10).

4. Discussion4.1. Generation and Timescale of Overturn: Importance of IBC Viscosity

Our modeling shows that the post‐LMO cumulate lunar mantle, without the viscosity reduction due to ilme-nite in IBC (case D501e0V1e21E100), does not produce any downwelling over the timespan of the model,consistent with the result of Elkins‐Tanton et al. (2002). A weak IBC layer, which is implied by recent experi-mental work (Dygert et al., 2016), is necessary to facilitate overturn. For an effective activation energy of100 kJ/mol, an IBC viscosity contrast of 10−2 produces small continuous IBC downwellings instead of arapid single overturn (e.g., case D501e‐2V1e21E100). Only IBC viscosity contrast beyond 10−3 can producean overturn within 100 Ma after LMO solidification. Thicker IBC layers (but with the same ilmenite andKREEP budget) also helps to facilitate overturn. For 10−3 viscosity contrast, an IBC thickness larger than30 km is necessary for producing overturn in 100 Ma (Figure 8). When the IBC thickness is larger than100 km, the overturn time can decrease to less than 10 Ma (Figure 8).

In the cases we considered, the overturn timescale ranges from ~4.1 Ma to more than 500 Ma (Figure 8B).Although the timescale depends on both IBC layer viscosity and thickness, it is more sensitive to the IBCviscosity (Figure 8B). Because of the temperature‐dependent viscosity and hence the growth of a top stag-nant lid, we cannot predict the overturn time by simply using the scaling relationship toverturn ~ ηm

−2/

3ηIBC1/3 suggested by R‐T analysis (Hess & Parmentier, 1995; Whitehead, 1988). Figure 8B suggests that

IBC viscosity contrast of 10−4 is critical for fast decoupling from top stagnant lid and having an overturntimescale of ~10 Ma, when the IBC thickness is no more than 50 km.

We examined the melt‐induced viscosity reduction and explored the parameter space that can produce themelting in the IBC. The solidus of IBC is probably in the range of 1080 to 1200 °C at low pressure, dependingon mineralogy and bulk composition (Thacker et al., 2009; van Orman & Grove, 2000; Wyatt, 1977; Yao &Liang, 2012). With 20–50% of the KREEP budget in the IBC layer (Figure S4), the temperature of IBC layeris always higher than its initial solidus temperature (3–17 °C, Figure S4) and much higher than the IBC soli-dus (Figure S4) after 10 Ma. The melt fraction is at least 7% in the partially molten IBC. Rock deformationexperiments investigating the rheology of partially molten peridotites suggest that strain rate is proportionalto eαφ, where φ is the melt fraction and α depends on the deformation mechanism and is between 21 and 45for melt fractions less than ~25% (Hirth & Kohlstedt, 2003; Mei et al., 2002; Scott & Kohlstedt, 2006). If thisscaling relationship is applicable to the IBC, relative to melt‐free conditions the IBC may be more than anorder of magnitude weaker for a melt fraction of 7%.

An additional weakening mechanism for the IBC layer may be the effect of water. For example, according todislocation creep flow laws, olivine with 10‐ppm H2O deforms at a strain rate approximately a factor of 30faster of dry olivine (at 1 MPa and 1100 °C; Hirth & Kohlstedt, 2003). The water abundance in the earlyLMO is suggested to be a few hundred parts per million, corresponding to concentrations in the last dregsof the magma ocean >10,000 ppm, depending on the initial bulk water content (Elkins‐Tanton, 2011;Hauri et al., 2017; Hui et al., 2013; Lin et al., 2017; Saal et al., 2008, 2013). A fractionally crystalized IBC layerwould be enriched in water (Elkins‐Tanton & Grove, 2011), but ilmenite deformation experiments were runat nominally dry conditions and the effect of water on the rheology of the IBC is unknown. If water and meltweaken the IBC similar to olivine, the viscosity of the wet, partially molten IBC will be more than 2 orders ofmagnitude weaker than at dry subsolidus conditions (Figure 2D), facilitating long‐wavelength cumulatemantle overturn.

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4.2. The Initial Thickness of Our IBC Layer

The initial IBC thickness in our model considers the IBC thickening before the IBC fully solidifies.Thickening of the IBC layer by formation of small downwelling instabilities, while magma ocean solidifica-tion is active (Hess & Parmentier, 1995) is one of the processes that can thicken the IBC layer. Another pro-cess that can be considered is the melt migration. Solidification tends to spread or disperse FeO‐TiO2‐richmaterials over greater distance than that predicted by mass balance calculation. Density stratification ofmelts in the ilmenite + clinopyroxene + pigeonite ± lagioclase ± olivine mush near the end of LMO solidi-fication could result in double‐diffusive convection and reactive infiltration instability in the mush. Thelower part of the mush may include pyroxene + olivine ± plagioclase + melt aggregates formed earlier.Downward infiltration of the heavier, cooler, and reactive FeO‐TiO2‐rich melt would refertilize the earlycumulates. Upward heat loss and solidification would inevitably trap FeO‐TiO2‐rich melts in the solidifiedmush, extending the IBC layer to greater depth. The depth of infiltration, the fraction of trapped melt,and the thickness of the FeO‐TiO2‐rich mush depend on the rates of solidification and sedimentation, bulkviscosity of the late cumulates, fraction of trapped melt, and the density difference between the interstitialmelt and the solid (McKenzie, 2011), none of which is well constrained presently.

Therefore, the range of IBC thickness in our study determined by the diapir percolating does not present thedefinitive evolution model for the last residuum of LMO but only suggests the future effort needed to even-tually understand the IBC formation.

4.3. Structures of Overturn

To aid the interpretation of mantle structures produced by the IBC‐driven overturn in our numerical models,we conducted a simple three‐layer R‐T instability analysis (Figure 3; supporting information). Similar to pre-vious two‐layer R‐T analyses (Parmentier et al., 2002; Scheinberg et al., 2014; Zhong & Zuber, 2001), the gen-eration of a degree‐1 pattern in our three‐layer R‐T analysis is favored by a low IBC viscosity and a thick IBClayer (Figure 3). The broad spectrum of Figure S3A is consistent with a three‐layer R‐T analysis for the lunarplume dynamics (Qin & Zhong, 2014).

With temperature‐dependent viscosity, the wavelengths of IBC downwelling in our numerical modelsbecome shorter because surface conductive cooling produces a top stagnant lid and reduces the amount ofmaterials available for downwelling. This effect also reduces the growth rate of downwellings. With the com-bination of temperature‐ and ilmenite‐dependent viscosities (equation (11)), the generation of long‐wavelength downwellings requires an even lower viscosity in the IBC layer (circles in Figure 3B).

A viscosity contrast of more than 3 orders of magnitude, which is suggested by the rheology of syntheticilmenite (Dygert et al., 2016, Figure 2C), produces downwellings with wavelengths shorter than sphericalharmonic degree 1. A viscosity contrast of 4 orders of magnitude and an IBC layer at least 150 km thick forma degree‐1 downwelling structure (Figures 3B and 10), in agreement with the finding of Parmentier et al.(2002). Furthermore, our model results suggest that IBC viscosity smaller than 10−5 is required for long‐wavelength downwellings if the IBC layer is 50 km thick (Figure 3B). Such low viscosity may be realizedin the presence of remelting during overturn and dissolved water during the last stage of LMO solidification.

4.4. The Timing of the Overturn in the Lunar Evolution

Our model starts from 4.42 Ga, which is the average ferroan anorthosite age (Borg et al., 2011, 2015). Basedon 142Nd isotope geochronology, Boyet et al. (2015) suggested that the LMO crystallization completed before4.44 Ga. Snape et al. (2016) used a two‐stage Pb model to determine the end time of LMO crystallization.They found that the solidification is completed at 4.38 Ga, 50 Ma later than the average FAN age. This 50‐Ma difference in the duration of LMO solidification would not change the decay rate of HPEs much andthe thermal state in the IBC layer. Therefore, the initial time of our model with a 50‐Ma difference shouldnot influence downwelling dynamics.

After overturn, the overturned IBC layer continues to be heated up by KREEP for ~100 Ma (Figures 5 and 9)andmay evolve into the thermal state proposed by Zhang et al. (2017). The overturned IBCs also concentratearound the core. However, the IBC distribution around the CMB will determine whether the IBCs will takeapproximately 300 Myr to rise up and produce the mare basalt from 3.9–3.2 Ga (Zhang, 2014; Zhang et al.,2013a) or flatten into a stable layer around the core (Zhang et al., 2017).

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4.5. Activation Energy of the IBC Layer

Rock deformation experiments revealed an activation energy for viscous deformation of ~300 kJ/mol forpure end‐member ilmenite (Dygert et al., 2016; Figure 2A). The rheology of Fe‐rich pyroxenes is poorlyunderstood but a preliminary study on single‐crystal hedenbergite obtained an activation energy of526 kJ/mol (Kollé & Blacic, 1983). As an approximation, we assumed a uniform harzburgite‐relevant activa-tion energy of ~500 kJ/mol for the entire lunar mantle (Hirth & Kohlstedt, 2003). The activation energy ofthe IBC may be closer to that of olivine than pure ilmenite for two reasons. First, activation energy is knownto scale with the homologous temperature. Fe‐Mg exchange between olivine and ilmenite in the IBC willincrease the geikielite (MgTiO3) component of the ilmenite, raising its homologous temperature and henceits activation energy (c.f. Dygert et al., 2016). Efforts are underway to parameterize an Fe‐Mg‐dependent flowlaw for ilmenite (Tokle et al., 2017), which will reduce this source of uncertainty for the activation energy.Second, for certain matrix topologies, aggregate rheologies have been successfully modeled using flow lawsbased on volume‐averaged terms in the end‐member flow laws (e.g.,Dimanov & Dresen, 2005; Tullis et al.,1991). Because the IBCs are mostly olivine (Figure 1C), an activation energy for olivine may appropriatelyreflect the temperature sensitivity of the IBC rheology.

4.6. Melting Induced by Thermal Evolution and Cumulate Overturn

All the overturn cases evolve toward some kind of “inverted” profiles of our initial peridotite solidus(Figures 4A, 4B, and 7). The intense thermal inversion produced in case D30v1e‐3V1e21E100 takes placeat approximately 45 Myr (Figure 7A) after overturn occurs. During this time, the horizontally averaged tem-perature of the upper mantle can be approximately 30 K higher than the peridotite solidus at a depth of200 km (Figure 4A), while the local temperature maximum can be 80 K higher than the solidus. This“inverted” state slowly evolves afterward, forming a regular convective temperature profile. The high tem-perature in the upper mantle does not drop back down below the solidus until 150 Ma. Because casesD30v1e‐4V1e21E100 and D100v1e‐4V1e21E100 have smaller IBC viscosities and thus faster overturns, theyquickly evolve into an inverted solidus at 8 Ma (Figure 7B) and 4 Ma (Figure 7C), respectively, after over-turns occur. Average temperatures 40 and 30 K higher than the solidus are observed in the upper mantlefor cases D30v1e‐4V1e21E100 and D100v1e‐4V1e21E100 at those times, respectively. In most of our models(except for D50v1e0V1e21E100), we find local temperatures more than 100 K higher than the peridotite soli-dus in some regions between depths of 150 and 600 km after instabilities occur. These anomalous high tem-peratures in the upper mantle last for at least 100 Myr (Figure 7). Regions with temperatures in excess of thesolidus experiences partial melting (Binder, 1982; Grove & Krawczynski, 2009). The extent of melting isexpected to be small because the excess temperatures are far below the liquidus (Figure 4A). A hotter initialtemperature, starting directly at the peridotite solidus, slightly increases the temperature difference from theperidotite solidus beneath the lithosphere (Figure 4B) and results in more melting (D30v1e‐2V1e21E100_Ts). The partial melts are likely enriched in IBC components. Depending on their FeO andTiO2 contents, hence densities, some of the melts may migrate toward the surface whereas others may sinkdeeper into the lunar interior and serve as an agent for convective overturn.

4.7. Distribution and Evolution of IBC in the Lunar Mantle

The distribution and evolution of IBC in the lower mantle depend on the downwelling wavelength. Whenthe IBC viscosity is small and its thickness is large, the resultant downwellings have long wavelengths(Table 2), which brings more IBC into the lower mantle. For example, lower‐mantle concentrations ofIBC in cases D150v1e‐4V1e21E100 and D100v1e‐4V1e21E100 are more than those in cases with short‐wavelength downwellings (Figure 6A). Interestingly, the degree‐1 case D150v5e‐4V1e21E100 shows a sink-ing and reascending trend (pink curve in Figure 6A) because the IBC are pushed toward the opposite hemi-sphere (also see Figures S8Q and S8R). Because the variation of globally averaged composition is beyond 1%after 500 Myr of model calculation, we cannot accurately constrain the evolution of the IBC distributionbeyond that time. The long‐term evolution of the overturned IBC is a subject of future research.

4.8. CMB Heat Flux and Lunar Magnetic Field

A sufficient CMB heat flux could also drive vigorous core convection and an active dynamo (Stegman et al.,2003). While the inferred strength of the lunar magnetic field from paleomagnetism of Apollo samples

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remains difficult to explain with convective cooling of the core (cf.Scheinberg et al., 2016), it is important to understand the CMB fluximplied by our models.

The range of our calculated maximum CMB heat flux is between 20 and350mW/m2. Our results show that smaller IBC viscosity results in a largermagnitude CMB heat flux pulse with shorter duration (Figure 11).Increasing IBC layer thickness does not raise the CMB heat flux signifi-cantly (D30v1e‐3v1e21E100 vs. D50v1e‐3v1e21E100 in Table 2), suggest-ing that overturn‐induced CMB heat flux is more sensitive to the IBCviscosity than the IBC thickness (Figure 11). After overturn, the CMB heatfluxes become negative as KREEP heating develops. Eventually postover-turned IBCs form a mixed pile on the CMB (Figures 5C and 5G; see alsoZhang et al., 2013a). In all our model cases considered, the positiveCMB heat flux lasts less than 80 Myr (Figure 11). The magnetic intensityestimations scaled from our calculated CMB heat flux (e.g., Christensenet al., 2009) can be compared to the observed paleomagnetic record(Figure S11; supporting information). The predicted paleointensities can-not reach the measured paleointensities of 20–40 μT (Evans et al., 2018;Garrick‐Bethell et al., 2016) even with a large core size (D150v1e‐4V1e21E100Core410 in Figure S11). However, the timing of the stimu-lated core dynamo we predict does coincide with an inferred periods oflunar paleomagnetism at 4.25 Ga (Weiss & Tikoo, 2014). The subsequentnegative CMB heat flux of ~250 Ma is not inconsistent with the silent per-iod of paleomagnetism between 4.25 and 4.0 Ga (Weiss & Tikoo, 2014).

5. Conclusions

We use a combination of R‐T stability analysis and finite‐element models to investigate the gravitationalinstability of an IBC layer sandwiched between the anorthosite crust and the peridotite mantle, the styleand dynamics of cumulate overturn, and the subsequent evolution of the lunar mantle. Both the R‐T analysisand finite element 3‐D simulations demonstrate the importance of the thickness and viscosity of the IBC onthe style and planform of convective overturn of a solidified lunar cumulate mantle.

We utilize an IBC viscosity based on recent laboratory study of the rheology of ilmenite (Dygert et al., 2016).This IBC viscosity is formulated into a temperature‐dependent viscosity law for a harzburgite‐mixed IBC,with consideration of viscosity reduction of partial‐melting and water. This formulation accurately estimatesthe viscosity range of IBC before overturn and represents an improvement over the parameter sweep viscos-ity contrasts assumed in the previous studies (e.g., Parmentier et al., 2002).

Our 3‐Dmodel tracks the distribution of ilmenite in the lunar mantle during and after overturn and provid-ing the basis for constraining petrogenesis of the Ti‐rich lunar volcanic glasses. Also, this model assesses thelink between lunar cumulate mantle overturn and the early lunar dynamo.

The main results and conclusions are as follows:

1. With a temperature‐dependent peridotite viscosity but no viscosity reduction due to ilmenite, a very longoverturn time is expected for the unstable IBC layer resulting from LMO fractional crystallization(Figure 10A) due to the viscosity increase resulting from conductive cooling. An IBC viscosity reductionin the layer is necessary to allow overturn on timescales shorter than 100 Myr.

2. With a viscosity reduction due to ilmenite, the IBC layer decouples from the top cold thermal boundaryand can produce short‐wavelength downwellings. An IBC viscosity reduced by a factor of 10−3 due toilmenite is the threshold value for the decoupling. If the IBC layer thickened downward beyond100 km before full crystallization, a viscosity of 10−4 in this layer can form long‐wavelength downwel-lings (longer than degree 3).

3. The IBC layer is likely to be partially molten before overturn due to the abundance of KREEP that it con-tains and its low solidus temperature. Partial melting and water content are thus expected to further

Figure 11. Evolutions of CMB heat fluxes for five representative cases. Theheat fluxes during overturn and for subsequent 200 Ma are shown forD30v1e‐4V1e21E100 (black), D30v1e‐4V1e21E100 (red), D50v1e‐3V1e21E100 (purple), D150v1e‐4V1e21E100 (dark blue), and D150v1e‐4V1e21E100 (light blue). The dotted black straight line depicts theminimum7.1 mW/m2 required for a core dynamo; 0 Ma in these plots denotes over-turn initialization.

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reduce the IBC viscosity. With such viscosity reduction, an IBC layer thicker than 100 km could producea degree‐1 downwelling structure. For an IBC layer thinner than 30 km, an IBC viscosity contrast 10−5 isnot small enough to achieve a degree‐1 downwelling.

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AcknowledgmentsThe authors acknowledge Wei Leng forthe discussion for propagator matrixanalysis of Rayleigh‐Taylor instability.We also thank the M. Laneuville,Sabine Stanley, and an anonymousreviewer for their constructivecomments. This work is supported bythe National Science Foundation ofChina (NSFC) under grant 41674098,China's Thousand Talents Plan (2015;granted to N. Z.). The ComputationalInfrastructure for Geodynamics isacknowledged for distributing theCitcomS software used in this study.Computational work was supported byresources provided by the High‐performance Computing Platform ofPeking University and by the PawseySupercomputing Centre, with fundingfrom the Australian Government andthe Government of Western Australia.The results produced by this modelingwork are available in the supportinginformation.

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