The Persistence of Oceans on Earth-Like Planets: Insights from The

arXiv:1501.00735v1 [astro-ph.EP] 5 Jan 2015 ufc cas(.. al fBuaae l 2001), val- larger al. discount et to Bounama seems the of work in recent found 1 is more Table as although water (e.g., much as oceans times any- surface 10 hold to may 1 it from that where suggested long have mantle Earth’s 1996; 2005). al. al. et silicate et (Kohlstedt in Hirschmann pressure chemical water with of their increases solubility in minerals the hydrogen Furthermore, have minerals not formula. anhydrous do nominally which so-called (NAMs), the of even true OH l.Wtr ihri oeua omo sa H an as or man- form silicate molecular the in (2) either and Water, oceans surface the tle. (1) voirs: the surface. controlling the in at be water may mass of presently of is supply the effect it to the super-Earths, look what must to unclear we this that, extending sup- For In the surface. control mantle. the not on permits water does of it liquid, ply remain although to compositions. water atmosphere, atmospheric surface Earth’s of variety the on water a However, The liquid given maintain in surface stars. can region its orbital their planet the Earth-like of as an (HZ) defined which is zones which zone habitable of habitable the classical some in able tele- super-Earths, future be be more With will may we even 2012). JWST, characterize and al. GMT to as et such 2014; Berta resources 2013; scope al. al. et et Wilson Fraine (e.g., made been have measurements iiNpueG 24i h nypae ntesuper- the the in Currently, (1 planet range only Kepler. the mass by is Earth detected 1214 GJ size mini-Neptune similar their plan- of the ground-based of than with ets brightness techniques up planets, spectroscopic follow the to and TESS easier and photometric much The closeness be will their 2015). of stars, of thousands detect virtue al. bright- to et by the (Ricker expected The of planets is survey stars, small grow. planned closest to a and continues is est which size mission, super-Earth TESS and Earth of [email protected] eceia tde ftemtrasta aeup make that materials the of studies Geochemical ae nteErhi on ntopiayreser- primary two in found is Earth the on Water nteps-elrea h ubro nw planets known of number the era, post-Kepler the In rpittpstuigL using 2015 typeset 6, Preprint January version Draft − H ESSEC FOEN NERHLK LNT:ISGT F INSIGHTS PLANETS: EARTH-LIKE ON OCEANS OF PERSISTENCE THE ru,cndsov nslct ieas hsis This minerals. silicate in dissolve can group, agroen u lortr htwtrt h atemr rapidly more longer. sma much mantle than persist years the early will to their conditions), in water surface habitable stable that less (assuming be return therefore insta also may layer Earths but boundary by oceans convection larger para a than oceans rather through ma Surface convection, silicate layer convection. modeled the of differe be parameterizations of Important different convection viscosity. can for The water-dependent cycle the model. through water recycling cycle volatile deep a with The coupled tectonics. plate nti ae epeetasre fmdl o h epwtrcycle water deep the for models of series a present we paper this In 1. A INTRODUCTION T − avr-mtsna etrfrAtohsc,6 adnSt Garden 60 Astrophysics, for Center Harvard-Smithsonian E tl mltajv 5/2/11 v. emulateapj style X 10 M ⊕ o hc atmospheric which for ) ar cafradDmtrSasselov Dimitar and Schaefer Laura rf eso aur ,2015 6, January version Draft + ABSTRACT or CYCLE alihnme eed ntevsoiyo h system, the of viscosity the The on flows. which depends heat number number, conductive Rayleigh Nusselt is and convective the system the and a compares convection, whether the to mod- describes parameters: unstable parameterized which dimensionless heated The two number, plane-parallel, on Rayleigh etc.). differ- primarily below, rely for vs. els convection vs. spherical of within from (e.g. models systems 3D Parameter- and ent using and complicated more models 2011). 2D 2001; from 1D expensive derived simplified al. parameterization et are al. a Crowley (e.g. et models Bounama 2011; convection viscosity ized 1989; al. water-dependent et Sandu mod- a convection Schubert & parameterized McGovern incorporating of use els the through ied mecha- Hirschmann exchange see the mantle, of and the (2006). surface of review the levels detailed between deeper a nisms subducting to For the transported in be mantle. contained can signif- water slabs but the oceanic small of a fraction re- shallow, However, icant through is surface water volcanism. the this water-induced to of back Much immediately subducting, leased seafloor. or oceanic sinking, the water-rich the to by of mantle of loss deep return of the the into is rate water which the ingassing, so-called and through between (MORs) mantle mantle balance the ridges from a The mid-ocean outgassing volcanic is at via tectonics. surface return wa- of the plate rate deep at the to the water tied by of is controlled abundance which is cycle, surface Earth’s ter/silicate mantle. the the on have of ter can flow This material the 1993). on or- effect Wu several significant & by a (Karato viscosity water magnitude the of of 1990; minor reduce even ders effect can of the water Karato presence of the of 1996; amounts that Studies show rheology Kohlstedt 2007). silicate on & Hirth Ohtani & e.g., Litasov conductivity, (see electrical temper- influ- changes, ocean etc. melting strongly phase as 2.5 rheology, minerals such to atures, properties, these material in 0.5 their water ences from 2012). of ranging Kohlstedt presence & values The (Hirschmann water of of (OMs) favor masses in ues h epwtrcceo at a enstud- been has Earth on cycle water deep The wa- of abundance the that shown been has it fact, In esalradmr essetfrsingle for persistent more and smaller re csi ufc ae otn r found are content water surface in nces iiy mle lnt aeinitially have planets Smaller bility. ,Cmrde A02138 MA Cambridge, ., lrpaes u hi habitability their but planets, ller nsprErh experiencing super-Earths on eeie ovcinmodels convection meterized hnlre lnt.Super- planets. larger than tei ikdt h volatile the to linked is ntle O H DEEP-WATER THE ROM 2 Schaefer & Sasselov which itself is dependent on temperature, pressure and water fugacity. The abundance of water in the mantle, R p δ which helps determine the viscosity, evolves along with u the mantle temperature. Here we will use a parameterized convection model to solidus study the deep water cycles of super-Earths. In the current era of exoplanet studies, we are still search- ing for Earth-like planets in Earth-like orbits around Sun-like stars, but what we have found and what we can soon characterize, are super-Earths (∼ 1 − 2R⊕, ∼ Radius 1−10M⊕). Although super-Earths may have cool enough surfaces to retain liquid water, tectonics and mantle convection plays an important role in controlling its abundance at the surface. The question of whether these planets will have plate tectonics has been discussed previously in the literature (e.g. Valencia et al. δ 2007; O’Neill & Lenardic 2007; Korenaga 2010b; c Noack & Breuer 2013; Stamenković& Breuer 2014), R T T T T T c s p u l c although the issue has not been settled. Here, we as- Temperature sume plate tectonics as a starting point in order to apply Figure 1. Schematic of temperature-depth profile. See text for a similar water cycle model. The question this work then details. addresses is whether the different pressure regimes inside super-Earths affect the deep water cycle, and whether cally the characteristic viscosity. Discussion in the lit- surface oceans are tectonically sustainable on these plan- erature is often contradictory on where to define the ets. This has important implications for the potential characteristic viscosity: as an average value (see e.g. habitability of super-Earths. McGovern & Schubert 1989; Tajika & Matsui 1992; In this paper, we address these questions using a pa- Sandu et al. 2011), or at the base of a boundary layer rameterized thermal evolution model coupled with a wa- (see e.g. Deschamps & Sotin 2000; Dumoulin et al. ter cycle model. We give a full description of the model 2005; Stamenkovićet al. 2012). Here we will try both in Section 2. In Section 3, we describe results for mod- types of models, starting with the single-layer models els which have either single layer convection or bound- typically used with volatile evolution models, which use ary layer convection for a number of super-Earth mod- average viscosities. We will then look at a boundary layer els. We also describe the response of the models to rea- model. We explore both models here to see the effect on sonable variations in the parameter values. Section 4 the behavior of water, and note that we are not attempt- discusses implications for the persistence of oceans on ing to reproduce the Earth, but explore two valid, but super-Earths. different, models. 2. METHODS The water cycle model is coupled to the thermal model through the viscosity, which is dependent on the water Mantle convection in the terrestrial planets can be abundance in the mantle (see e.g. McGovern & Schubert controlled either by the stability of the whole man- 1989; Sandu et al. 2011). Models of the Earth have tle layer (single layer convection, e.g. Schubert et al. shown that the viscosity and mantle temperature create (2001)) or by the stability of two boundary layers: a a feedback loop (Schubert et al. 2001; Crowley et al. cold boundary layer at the surface and a hot bound- 2011). When the mantle is warm, convection is vigor- ary layer at the interface of the silicate mantle with the ous and the mantle cools quickly. As the temperature metallic core (see e.g. Turcotte & Schubert 2002). Heat drops, the viscosity increases, causing the convection to from either secular cooling or decay of radioactive ma- become sluggish. Sluggish convection means that less terials (or both) is transferred by conduction through heat is removed from the mantle, causing it to heat up. the boundary layers. The interior region is convective The viscosity increases, and the cycle repeats. This cycle and thus approximately adiabatic. See Figure 1 for a has been shown to be enhanced by the presence of wa- schematic representation of the thermal profile. Single ter (McGovern & Schubert 1989; Crowley et al. 2011). layer convection models typically neglect the heat flux Crowley et al. (2011) describe how the water and tem- from the core, which is small, for simplicity and are perature feedbacks interact for temperature and water- entirely heated from within by radioactive decay. Nu- dependent viscosities. Cowan & Abbot (2014) studied merical simulations are used to determine scaling laws the effect of sea floor pressure on a steady state model relating the heat flux and mean temperature to the de- of the deep water cycle, without considering the effect of gree of convection. These simulations have been done planetary thermal evolution. for a variety of different boundary conditions (free slip, In the following sections, we first describe selection of no slip, etc.), and for different material properties (iso- the super-Earth model parameters. We then describe viscous, highly temperature-dependent viscosity, etc.) the parameterized thermal model, followed by the water (e.g., Honda & Iwase 1996; Deschamps & Sotin 2000; cycle model. Section 3 will discuss the results for these Dumoulin et al. 2005). Choosing the proper scaling re- models. lations is therefore important for a successful model. The parameter which is of the most impor- 2.1. Super-Earth Models tance in determining the convective behavior is typi- Oceans on Super-Earths 3

the mantle potential temperature Tp) through the equa- Table 1 tion: Super-Earth Model Parameters. Rp 3 2 Mass Rp Rcore hρmi hgi hTmi = ǫmTp = 3 3 r T (r)dr (2) −3 −2 R − R Z (M⊕) (R⊕) (R⊕) (kg m ) (m s ) p c Rc 1 1 0.547 4480 9.8 See Tajika & Matsui (1992) for a derivation of this equa- 2 1.21 0.649 4960 13.4 3 1.34 0.717 5470 16.4 tion. We approximate the adiabat as: 5 1.54 0.814 5970 20.7 αg T (r)= Tp + Tp ∆r (3) Note. — Core mass fraction is held fixed at 0.3259, and cp scaling relations from Valencia et al. (2006) are used to determine R and R . See text for details. p core Values of ǫm for the different planet masses are given in Table 5. We use the scaling relations of Valencia et al. (2006) In the mantle, heat is generated by decay of radioactive to calculate the planetary parameters for planets of 1, 2, elements. The heat flux from radioactive decay is domi- 24 238 235 232 40 3, and 5 Earth masses (1M⊕ = 5.97 × 10 kg). Larger nated by U, U, Th, and K. The heat produced planets are not considered here because Noack & Breuer is calculated from the equation: (2013) find that the peak likelihood for plate tectonics 9 occurs for planets between 1 − 5M⊕. The scaling laws Q(t)= ρm CiHiexp[λi(4.6 × 10 − t)] (4) of Valencia et al. (2006) assume a constant core mass X fraction of 0.3259. The planetary and core radii then where Ci is the mantle concentration of the element by ai scale by Ri ∼ Ri,⊕(Mp/M⊕) , where i = c (core) or mass, Hi is the heat production per unit mass, and λi is p (planet), ac = 0.247, ap = 0.27, and values for the the decay constant. The decay constants, heat produc- Earth are indicated by ⊕. The average mantle density tion rates, and abundances relative to total uranium are hρmi is calculated from the mantle mass and volume. taken from Turcotte & Schubert (2002). In this paper, The average gravitational acceleration hgi is found from we assume that all super-Earths have the same ratios of 2 radioactive elements as the present day Earth. The nom- GMp/Rp. Values for these parameters are given in Table 1. We take a constant water mantle mass fraction of inal bulk silicate Earth (i.e., primitive mantle) contains 1.4 × 10−3 for the nominal models. This is equivalent ∼ 21 ppb U (McDonough & Sun 1995). to 4 ocean-masses (OM) of water for the Earth, where 1 Using boundary layer theory, the heat flux out of the 21 mantle is given by: OM is equal to 1.39 × 10 kg H2O. We will later explore the effect of variable water on the results. (T − T ) q = k u s (5) m δ 2.2. Thermal Evolution Model u where k is the mantle conductivity, δ is the boundary Following models of Earth’s deep water cycle u layer thickness, and the T is the temperature at the base (e.g. McGovern & Schubert 1989; Schubert et al. 2001; u of the boundary layer (see Figure 1). T is calculated Sandu et al. 2011), we will first consider models heated u from hT i using the adiabatic temperature profile of the only from within (i.e., heat flux q = 0). These models m c mantle. are considered single-layer convection because the whole The thickness of the upper boundary layer is given by mantle convects. There is a conductive upper thermal the global Rayleigh number: boundary layer that governs heat loss from the surface. For most of the lifetime of the Earth, such models have β β Racr κη(T, P )Racrit been shown to give good fits to the observed mantle vis- δu = Z = (6) cosity and heat flux (Schubert et al. 2001). The ther- Ra gαρm∆T mal evolution model requires solution of the mantle heat where Z is the thickness of the mantle, Ra is the transfer equation: Rayleigh number of the whole mantle, Racrit is the critical Rayleigh number for convective instability, κ is the dhTmi ρmCpVm = −Asqs + Acqc + VmQ(t) (1) mantle thermal diffusivity, η(T, P ) is the characteristic dt mantle viscosity, and α is the thermal expansivity. ∆T where ρm is the mantle density, Cp is the mantle heat is the temperature drop across the mantle minus the adi- capacity, Vm is the mantle volume, As, Ac are the sur- abatic temperature change. We use the mantle viscosity face area’s of the planet and core, qs and qc are the con- calculated with hTmi and hP i, which is characteristic of ducted heat fluxes through the surface and core-mantle the whole mantle layer. This choice dictates the behav- boundary (CMB), respectively, and Q(t) is the heat pro- ior of our model, as will be discussed later. The viscosity duced by radioactive decay within the mantle. In this parameterization is described in Section 2.4. model, there is no heat flux from the core, so the sec- Two other parameters are derived from the thermal ond term on the right side vanishes. The temperature model. The areal spreading rate is the rate at which modeled in any parameterized convection model is some- new ocean crust is being created. McGovern & Schubert what ambiguous. Here, we will follow the convention of (1989) parameterized the areal spreading rate using the McGovern & Schubert (1989) and take the temperature current volume of the oceans and the present day heat to be the spherically-averaged mantle temperature. We flux, which are unconstrained for exoplanets. Instead, we can relate this averaged temperature to the temperature follow Sandu et al. (2011), who relate the areal spread- of the mantle adiabat extrapolated to the surface (i.e. ing rate to the convective velocity uc) and the length of 4 Schaefer & Sasselov

rameterized as: Table 2 Nominal Thermal Model Parameters. routgas = ρm,vdmS (10)

param. units Upper Mantle Lower Mantle Core where ρm,v is the density of volatiles in the mantle,, dm is 0.21 the depth of melting, and S is the areal spreading rate of Racrit 1100 0.28Ra — α ×10−5 K−1 2 1 — the mid-ocean ridges. In McGovern & Schubert (1989), k W m−1 K−1 4.2 4.2 — dm is kept at a constant value of 100 km. The ingassing κ m2 s−1 10−6 10−6 — rate is parameterized as: −1 −1 Cp J kg K 1200 1200 840 −3 a ρ kg m 3300 hρmi 8400 ringas = fbasρbasdbasSχr (11) a See Table 1 where fbas is the mass fraction of volatiles in the hydrated basalt layer, ρbas is the density of basalt, dbas is the average thickness of the basalt (held constant at 5 Table 3 km) and χr is an efficiency factor reflecting the incom- Water Cycle Parameters. plete transport of the water in the hydrated basalt layer into the deep mantle. With all parameters except S and param. name value ρm,v held constant in equations (10) and (11), we found χr regassing efficiency 0.03 that all planets necessarily reached a steady state (i.e., χ degassing efficiency 0.02 d dMH2O,m fbas hydrated basalt fraction 0.03 = 0), where the mantle water abundance is −3 dt ρbas basalt density (kg m ) 3000 given by setting ringas equal to routgas, and solving: Lridge Mid-ocean ridge length 1.5×(2πRp) −γ K Solidus depression constant (K wt% ) 43 MH2O,m fbasρbasdbasSχr γ Solidus depression coeffiient 0.75 ρm,v = = (12) θ Melt fraction exponent 1.5 Vm dmS DH2O Silicate/melt partition coefficient 0.01 fbasρbasdbasχrVm MH2O,m = (13) dm the spreading centers (Lridge) where ocean crust is created: While a case may be made that the Earth is in a steady S =2L u (7) state, there is little reason to suppose that this is a ridge c necessary condition for all exoplanets experiencing plate The convective velocity is determined by the convec- tectonics. We therefore follow here the volatile evolu- tive layer overturn time from boundary layer theory tion model of Sandu et al. (2011), described briefly be- (Schubert et al. 2001, ch. 8) using the equation: low. In this model, the depths of melting and hydration are not held constant, but vary based on local tempera- 5.38κ(Rp − Rc) tures. We found that in these models, steady-state was uc = 2 (8) δu rarely achieved. Parameters used in the volatile evolution model are given in Table 3. The outgassing rate is We parameterize the length of the mid-ocean ridges as given by: 1.5 times the planetary circumference. This parameterization is chosen to give the present day mid-ocean ridge routgas = ρmhFmeltihXmeltiDmeltSχd (14) (MOR) length on the Earth of ∼60,000 km. We describe where hFmelti and hXmelti are the average fraction of results using smaller Lridge values in Section 4. Values for the parameters used in the thermal evolu- melting and average abundance of water in the melt over the melt layer thickness, Dmelt, S is the areal spreading tion model are given in Table 2. The value of Racrit for the mantle is taken from Schubert et al. (2001). We use rate and χd is the degassing efficiency, which accounts for constant values for the heat capacity, thermal conductiv- incomplete transport of water to the surface. Whereas ity, thermal expansivity and thermal diffusivity, although McGovern & Schubert (1989) used a constant value for these parameters are all known to be pressure-dependent. the melt layer thickness, Sandu et al. (2011) used the mantle thermal profile and the peridotite solidus to deter- 2.3. Volatile Evolution Model mine the melt layer thickness. The thermal profile used is composed of the conductive upper thermal boundary Volatile evolution models harken back to layer and the upper mantle adiabat, and the intersection McGovern & Schubert (1989), repeated with vari- of this profile with the hydrated solidus curve for peri- ations by many others. The volatile evolution model dotite determines where melt forms (see Fig. 1). Water involves calculation of outgassing and ingassing rates for dissolved in silicates lowers their solidus (the temperature water based on mantle melting and surface hydration. at which partial melting begins), and the water partitions The rate of change of the water abundance in the mantle preferentially into the melt. Using the parameterization is given by combining the ingassing and outgassing of Katz et al. (2003) for wet melting, the solidus depres- rates: sion is given by: dMH2O,m = ringas − routgas (9) γ dt Tsol,wet = Tsol,dry − ∆TH2O = Tsol,dry − KXmelt (15) This equation is solved simultaneously with the heat where K and γ are empirically determined constants for transfer equation at each time. In the simplest form of peridotite (see Table 3). The melt fraction and water McGovern & Schubert (1989), the outgassing rate is pa- abundance in the melt are determined where the mantle Oceans on Super-Earths 5 thermal profile is above the wet solidus temperature and are given by: Table 4 Viscosity Parameters. θ T − Tsol,wet Fmelt = (16) parameter Olivine (wet) Perovskite(dry) Tliq,dry − Tsol,dry 25 21 η0 (Pa s) 1.08 × 10 1 × 10 r 1.0 ... XH2O,m −1 Xmelt = (17) Ea (kJ mole ) 335 300 3 −1 DH2O + Fmelt(1 − DH2O) Va (cm mole ) 4.0 2.5 −1 −1 Rgas (J mole K ) 8.314 where XH2O,m is the water mass fraction in the mantle, DH2O is the silicate/melt partition coefficient, and θ is then given by: an empirically determined exponent. The dry liquidus and solidus equations are taken from Zhang & Herzberg n −p r Ea + P Va ǫ˙ = Acrpτ d lnfH2Oexp − (20) (1994) and Hirschmann et al. (2009), respectively. The RgasT melt fraction and water fraction are averaged over the melt zone thickness at each timestep for use in equation where n = 1 for diffusion creep, Acrp is a material param- (9). eter, d is the grain size in microns, Ea is the activation The return of water from the surface back into the energy, Va is the activation volume, and Rgas is the ideal mantle through ocean plate subduction is described by: gas constant. This equation shows that the response to stress depends on the pressure and temperature, as well

ringas = fbasρbasDhydrSχr (18) as the water fugacity (fH2O) in a non-linear way. The viscosity is then derived from the constitutive law: where f is the mass fraction of water in a hydrated bas τ basalt layer of thickness Dhydr, ρbas is the density of ηeff = (21) basalt, and χr is the regassing efficiency, which accounts 2˙ǫ for imperfect return of water to the mantle. The hy- We combine the constant parameters Acrp and the grain drated layer thickness is measured from the surface, down size into a normalization factor eta0 to arrive at the ef- to the depth at which the temperature reaches the stabil- fective viscosity. The form of the effective viscosity is ity boundary of serpentinite (i.e., serpentine is not stable then: at lower depths). The upper thermal boundary layer has a linear conductive temperature profile (see eq. 5), where −r Ea 1 1 ηeff = η0fH2Oexp ( − )+ the temperature change with depth is (Tu − Ts)/δu. The Rgas T Tref depth to the serpentine stability temperature is therefore (22) 1 P Va Pref Va given by: ( − ) Rgas T Tref Tserp − Ts Tserp − Ts Dhydr = δu = k (19) We normalize so that η(Tref = 1600, XH2O= 500 ppm, Tu − Ts qm 21 Pref = 0)= 10 Pa s, which gives reasonable viscosities where Tserp is the highest temperature of serpentine sta- for the Earth. Parameters are given in Table 4, and are bility at pressures below ∼ 3 GPa and is ∼ 973 K taken primarily from Hirth & Kohlstedt (2003) for wet (Ulmer & Trommsdorff 1995). The hydrated layer is diffusion in olivine. These authors measured values of r necessarily smaller than the thermal boundary layer, and of ∼ 0.7 − 1.2 on polycrystalline olivine samples. Recent is further restricted to hold no more water than is present work on Si diffusion in olivine and single-crystal deforma- at the surface in a given instant in order to maintain wa- tion experiments suggests that the dependence on water ter mass-balance. We use the values for χr and χd de- abundance may be much lower (r =0 − 0.33) (Fei et al. termined by Sandu et al. (2011). 2013; Girard et al. 2013). However, more work needs A final word about parameterized volatile evolution to be done to reconcile these experiments with the poly- models: As noted by a reviewer, the equations described crystalline experiments. In this paper we use results from above assume that water transported into the mantle is the earlier studies for the nominal models, but we also instantaneously transported throughout the mantle and explore the effect of different r values on our findings. available for outgassing. In the real world, of course, the To convert from mass fraction water in the mantle to mantle is not homogeneous and the spreading zone melt fugacity fH2O, we use the formulation of Li et al. (2008), centers will likely become dehydrated before they can be which is an empirical relationship between the water fu- replenished by advection from subducted water. There- gacity and the concentration of water in olivine (COH , fore outgassing rates calculated here are upper limits on atomic H/106 Si): the true values. 2 3 lnfH2O = c0 + c1lnCOH + c2ln COH + c3ln COH (23) 2.4. Viscosity where c0 = −7.9859,c1 = 4.3559,c2 = −0.5742, and Water dissolved in silicate minerals such as olivine re- c3 = 0.0227. This relationship is derived from olivine duces the mineral’s strength (e.g., Chopra & Paterson solubility data between ∼ 1373 − 1600 K and is only 1981), and therefore its viscosity. In experimental liter- strictly valid within that temperature range. However, ature, the water dependence of the viscosity is parame- the relationship varies only marginally with temperature terized via the water fugacity fH2O, which depends on for a wide range of COH and fH2O, so we apply it to temperature and pressure. The rheological or constitu- the whole temperature range considered here for lack tive law relating stress τ and strain rateǫ ˙ for olivine is of other available data. It is straightforward to convert 6 Schaefer & Sasselov

3000 Table 5 Initial Temperature Parameters.

Mass ǫm hTm,ii Tc,i 2500 (M⊕) (K) (K) 1 1.19 3000 3810 2 1.32 3330 4630 3 1.44 3630 5350 2000 5 1.64 4130 6640 (K) p from the mantle water mass fraction to the concentration T in COH using the molecular weights of water and olivine. 1500

2.5. Initial parameters 1000 The initial temperature parameters for each of the super-Earth models are given in Table 5. Previ- ous parameterized convection models have shown that 500 initial temperatures do not significantly affect the 0 2000 4000 6000 8000 10000 thermal evolution beyond a few hundred Myr (see time (Myrs) e.g. Schubert et al. 1980; McGovern & Schubert 1989). 24 Therefore, the present models are all started with the same initial mantle potential temperature Tp, which is the temperature of the mantle adiabat extrapolated to the surface. The initial mantle potential temperature is set to 2520 K for all models, which is equivalent to an average mantle temperature of ∼ 3000 K for the Earth. 22 The average mantle temperatures are calculated accord- ingly for all super-Earth models. In models described in a later section, which include core evolution, we assume an initial temperature contrast across the CMB of 100 K. Therefore, the initial core temperature is set equal to the temperature extrapolated along the mantle adia- viscosity (Pa s) 20 bat to the CMB plus 100 K. We describe the effect of different initial temperatures in Section 4.

3. RESULTS For our nominal model, we focus on the evolution of 18 temperature and water abundances in both the mantle 0 2000 4000 6000 8000 10000 and on the surface. In the following sub-section, we in- time (Myrs) troduce a second model which includes core-cooling and −3 x 10 uses a different characteristic mantle viscosity for the up- 1.4 per mantle that produces significantly different results. 3.1. Nominal Model - Single Layer Convection 1.2 Figures 2 and 3 show results for the nominal model using parameters from Table 3 and a mantle water 1 abundance of ∼1400 ppm (4 OM for 1 M⊕). Figure 2 shows the evolution of the mantle potential temperature, 0.8 viscosity and water abundance for the nominal model water

using two different viscosity parameterizations. For the X curves shown in red, the activation volume is set to 0, so 0.6 the viscosity is pressure-independent. The blue curves include a non-zero activation volume (see Table 4), so 0.4 the viscosity depends on pressure as well as temperature 1 M⊕ and water fugacity. The viscosity is calculated with the 2 M⊕ 0.2 3 M average mantle temperature hTmi and the mid-mantle ⊕ pressure. 5 M⊕ For the pressure-independent viscosity, Fig. 2a seems 0 0 2000 4000 6000 8000 10000 to indicate that the large planets cool off more rapidly time (Myrs) than the smaller planets. The 5 M⊕ planet has a final potential temperature ∼240 K lower than that of the Figure 2. Nominal model with single layer convection. (a) mantle potential temperature, (b) mantle viscosity, and (c) mantle wa- 1 M⊕ planet. This is counter to standard intuition: ter mass fraction. Red lines represent calculations done with a large planets should cool off more slowly than small pressure-independent viscosity, blue lines represent the use of a pressure-dependent viscosity. Line styles indicate planet mass according to the legend. Oceans on Super-Earths 7

1.8 has the lowest mantle viscosity, but this outgassing is 1 M⊕ much less complete than for the pressure-independent 1.6 case. The larger planets show steady but very gradual 2 M⊕ outgassing, as their temperatures increase and their 1.4 3 M⊕ viscosities drop. In fact, the 5 M⊕ planet has delayed 5 M 1.2 ⊕ onset of outgassing by ∼1 Gyr, due to a very large thermal boundary and low surface heat flux, both of 1 which can be attributed to the large initial mantle viscosity. 0.8 The activation volume that we use here for the water (OM) olivine viscosity is 4 cm3 mol−1, which is derived from 0.6 experimental data. However, the activation volume for other silicates has been shown to decrease with 0.4 pressure (Stamenkovićet al. 2011). In their thermal 0.2 models, Stamenkovićet al. (2012) use activation volumes for pressures at the planet’s CMB (2.5 cm−3 0 mol−1 for Earth). Therefore, a slightly lower activation 0 2000 4000 6000 8000 10000 time (Myrs) volume may be more appropriate, particularly for the larger planets. Intermediate values of Va give Figure 3. Surface water abundance for the nominal models. The water abundance is equivalent to 4 ocean masses of water in the mantle temperatures intermediate to those shown in Earth-sized planet. Fig. 2a for the pressure-dependent viscosities. The mantle water abundance drops less precipitously in planets due to their smaller surface-area/volume ratios. early times, and ingassing is much slower than for the However, note that these are potential temperatures pressure-independent case, but more complete than for (i.e, the mantle temperature extrapolated to the surface the pressure-dependent model shown in Fig. 2b. The along an adiabat). The average mantle temperature of pressure-dependence of the viscosity is therefore an the 5 M⊕ planet is in fact hotter than the 1 M⊕, but by important parameter to include in the models. The only ∼80 K. The adiabatic gradient is much steeper for value of activation volume will affect the final volumes the large planets due to dependence on gravity, so the of the surface oceans and the mantle temperature. near-surface temperatures are therefore much lower. As Figure 3 shows the surface abundance of water for the seen in Fig. 2b, the hotter average mantle temperatures nominal model for each of the super-Earths. This figure of the large planets results in lower mantle viscosities is the surface corollary to Fig. 2c. We show only the and therefore more rapid cooling. pressure-dependent models for clarity. Note that while The lower near-surface temperatures of larger planets the relative abundance of water is the same for each reduces the degree of melting, which strongly affects planet, the total planetary water mass is 4 (8, 12, 20) their water cycles, shown in Fig. 2c. There is a sharp ocean masses for the 1 (2, 3, 5 M⊕) planet. The time- decrease in mantle water abundance in the first 2 Gyr dependent behaviors of the planets do not scale simply due to rapid early outgassing for all planets. As mantle with planet mass. The 1 M⊕ planet has a much more temperature drops, near-surface temperatures drop significant outgassing phase at early times, followed by below the solidus temperature, which halts melting and ingassing. The larger planets show much more gradual outgassing. This is followed by a rapid ingassing period. outgassing, with delayed onset of outgassing for the 5 Ingassing is limited by the mass of the water at the M⊕ planet. Due to the delayed outgassing, the 5 M⊕ surface and the thickness of the hydrated surface layer, planet has less surface water than the 3 M⊕ planet which is regulated by the surface heat flux. All but the for most of their lifetimes. However, the 3 and 5 M⊕ smallest planet have ingassed nearly all of their water planets have roughly equal surface water abundances at by ∼2 Gyr. The deep water cycles of these planets have 10 Gyr. For smaller values of the activation volume, the effectively ceased. However, it should be noted that for surface oceans will be substantially larger. all planets, this leaves a small residual surface reservoir of water that effectively cannot be lost to the mantle. For the pressure-dependent viscosity calculations (blue 3.2. curves) both the average and potential temperatures of Boundary Layer Convection the smaller planets cool more quickly than the larger Volatile evolution models have typically assumed planets (see Fig. 2a). In fact, the largest planets single-layer convection, which has been shown to work initially heat up substantially before beginning to cool, well for the Earth. However, thermal evolution models so near-surface melting persists for much longer than in that neglect volatile evolution often assume that mantle the pressure-independent case. The final temperature convection is controlled by boundary layer instability, in of the 5 M⊕ planet is ∼1000 K higher than that of the which convective instability is determined by the local 1 M⊕ planet. The slow cooling of the planets can be conditions at the boundary layers rather than the man- attributed to the substantially larger viscosities in this tle as a whole. We will now describe how these mod- model, due primarily to the pressure dependence of the els differ from the nominal model described above. For viscosity (see Fig. 2b). The changes in water abundance the boundary layer models, we assume mixed heating, shown in Fig. 2c are much more gradual than with with a non-zero heat flux from the core (taken here to be pressure-independent viscosities. There is a rapid early both solid and isothermal) and a conductive lower bound- outgassing phase only for the smallest planet, which ary layer. Here we are less interested in reproducing the 8 Schaefer & Sasselov

Earth than in exploring the behavior of the models for little variability in the results for surface oceans when different planet masses. When using the same physically- holding α constant. plausible parameters here, we show that the two types of Results for the boundary layer convection model are models give fundamentally different results, due to the discussed in the following subsection. Afterwards, we choice of characteristic mantle viscosity. discuss how parameters affect the two different models. For the boundary layer models, a second heat transfer equation is needed for the core: 3.2.1. Results - Boundary Layer Convection Model dT Results are shown in Figure 4. Core temperatures are ρ C V c = −A q (24) c p,c c dt c c normalized to their initial values, because they vary by ∼3000 K. The upper boundary layer is calculated using where variables are as in eq. (1), but defined for the core the pressure-dependent olivine viscosity using temper- rather than the mantle. The core is isothermal, and so ature and pressure at the base of the boundary layer. is characterized by a single temperature Tc. We neglect The lower boundary is calculated using the pressure- radioactive decay in the core, as well as latent heat due dependent perovskite viscosity from Stamenkovićet al. to possible core freeze-out. The heat flux out of the core (2011) using temperatures and pressures at the CMB. is given by: This viscosity does not depend on the mantle water (Tc − Tl) abundance. qc = k (25) δc Mantle potential temperatures decrease more rapidly and significantly here than for the nominal model. All where δc is the lower thermal boundary layer, Tc is the planets have nearly identical potential temperature core temperature, and Tl is the mantle temperature at evolution, in contrast to the pressure-dependent results the top of the lower thermal boundary layer (see Fig. 1). shown in Fig. 2a. The upper thermal boundary layer We use a local Rayleigh number to define the thickness of thickness is comparable for all planets, so the upper the lower boundary layer. The boundary layer thickness mantle viscosity used here is only weakly dependent on is determined by setting it equal to its critical thickness, pressure (blue curves, Fig. 4c). In contrast, the viscosity the point at which it becomes unstable to convection. of the lower thermal boundary layer is strongly depen- 1 3 dent on pressure, and therefore the core temperature κη(Tc, Pcmb)Racrit,l δc = (26) (Fig. 4b) is highly dependent on planet mass. The 1 gαρm(Tc − Tl) M⊕ planet’s core cools by ∼35%, whereas the 5 M⊕ planet’s core cools only ∼5%. where η(T , P ) is the viscosity of the lower bound- c cmb Figure 4d shows the surface water inventories for ary layer. For the lower boundary layer, we use a per- the boundary layer model, with 4 (8, 12, 20) OM of ovskite rheology. The choice of rheologies will be dis- initial water. In comparison to Figure 3, it is obvious cussed more below. The critical Rayleigh number of the 0.21 that significantly more water is outgassed from the lower boundary layer is given by Racrit,l = 0.28Ra mantle here. The 5 M⊕ planet has a peak surface water (Deschamps & Sotin 2000), where Ra is the global abundance of ∼8 OM, in comparison to ∼1.6 in Fig. 3. Rayleigh number used in the definition of δ . The global u However, the residence time is much shorter. All of the Rayleigh number is defined here as in eq. (6), except for planet’s lose most of their surface water inventory back the characteristic viscosity. For the characteristic vis- into the mantle by ∼4.5 Gyr. After complete ingassing, cosity, we use the value defined by the temperature and the 1 M⊕ planet has the largest remaining surface water pressure at the base of the upper thermal boundary layer. abundance, with ∼0.2 OM at 10 Gyr. All planets have This has a signifcant effect on the outcome of the models significantly lower surface water abundances at 10 Gyr as shown below. for the boundary layer models than for the nominal The viscosity of olivine is used for the upper thermal models shown in Figure 3. For the boundary layer boundary layer, but olivine is not a stable phase at the models, we find little difference between the pressure- pressures and temperatures of the lower thermal bound- dependent and pressure-independent viscosities, except aries. We therefore use the viscosity law for perovskite for the cooling of the core. The peak surface water derived by Stamenkovićet al. (2011) for the lower ther- abundances remain essentially unchanged, but the pres- mal boundary. Parameter values are given in Table 4. ence of surface water does persist for ∼0.5 Gyr longer The lower mantle of super-Earths likely consists of per- in the pressure-independent case. This is not surprising, ovskite transitioning to post-perovskite for larger planets since the pressures of the upper thermal boundary layer (see e.g. Valencia et al. 2006; Tackley et al. 2013). No are very low and do not signficantly affect the viscosities. experimental data exists on the water-dependence of the viscosity of perovskite, so the value of r is set to zero. Stamenkovićet al. (2011) also give the dependence of 3.3. the activation volume as a function of pressure. In the Dependence on parameters nominal models, we use a value of 2.5 cm3 mol−1, which Many of the parameters used in the thermal and is the value at the Earth’s CMB. volatile evolution models are poorly constrained for the We use a slightly smaller value of α for the lower Earth, much less super-Earths. In the following section, thermal boundary layer (see Table 2), which is taken we exam the effect of varying several of these parame- from Tackley et al. (2013) for the Earth’s CMB. ters on the results of both the nominal and the bound- Stamenković& Breuer (2014) found that scaling α with ary layer convection models. We refer the reader to planetary mass was as important as scaling ρm to the Stamenković& Breuer (2014) for an analysis of the ef- calculation of planetary temperatures. However, we find fect of β, αm, κm, and ρm on thermal evolution models. Oceans on Super-Earths 9

2600 1 1 M⊕ 2 M 0.95 2400 ⊕ 3 M ⊕ 0.9 5 M 2200 ⊕ 0.85 (K)

(K) 0.8

2000 c,init p T /T c T 0.75 1800 0.7 1600 0.65

1400 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 time (Myrs) time (Myrs) 32 8

1 M⊕ 30 2 M⊕ 3 M 28 6 ⊕ 5 M⊕ 26

24 4

22 water (OM) viscosity (Pa s)

20 2

16 0 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 time (Myrs) time (Myrs) Figure 4. Results for nominal models with boundary layer convection heating. (a) mantle potential temperature, (b) core temperature normalized to initial core temperature, (c) thermal boundary layer viscosities (upper = blue, lower = red), (d) surface water abundance in ocean masses. Line styles indicate planet mass according to the legend. 3.3.1. Fugacity coefficient outgassed significantly. For the boundary layer convection model (Fig. 5b), Figure 5 explores the effect of the fugacity coefficient r the higher r value increases the amount of outgassing on the results of the previous models for planets of 1 and for both the 1 and 5 M⊕ planets and shifts the peak of 5 M⊕. Other planet masses are not shown for clarity. outgassing to slightly earlier times. However, ingassing This figure shows r values of 0.7, 1.0 (nominal), and also occurs more rapidly, so the oceans persist only 1.2. Hirth & Kohlstedt (2003) give values of 0.7 − 1.0 until ∼4 Gyr. The lower r value reduces the amount for wet diffusion creep of olivine and r = 1.2 for wet of outgassing for both planets, and causes the surface dislocation creep of olivine, which is why we chose these water to persist for longer. For the 1 M⊕ planet and r values. It should be noted that the viscosities were not = 0.7, the water abundance is relatively constant over renormalized with the change of r values. Therefore the the planet’s lifetime. There is about 0.4 OM of water variances reflect the absolute changes in viscosity. remaining on the surface after 10 Gyr. For the single layer convection model (Fig. 5a), the initial outgassing phase is significantly stronger for 3.3.2. Total water abundance r=1.2 for the 1 M⊕ planet, but the final abundance is nearly the same as the nominal value. In contrast, Figure 6 shows the surface water abundances using outgassing is slightly delayed for r = 0.7 but the final different initial mantle water abundances. The surface water abundance is ∼0.2 OM larger. For the 5 M⊕ water abundances appear to be a fairly straightforward planet, the larger r value increases the amount of water function of water abundance. Models with larger outgassed, whereas the lower r value both further water abundance have larger surface inventories. The delays outgassing and limits the total amount of water outgassing occurs earlier for the single layer convection 10 Schaefer & Sasselov

2.0 r = 1.0 12 r = 1.2 r = 0.7 1.6 10

8 1.2

6 water (OM)

water (OM) 0.8 4

0.4 2

0 0 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 time (Myrs) time (Myrs) Figure 5. Surface water abundances for (a) the nominal single layer convection model and (b) boundary layer convection models using diﬀerent values of the fugacity exponent r. Blue lines are for 1 M⊕, red lines for 5 M⊕.

3.5 18 2 OM 16 4 OM 3 6 OM 14 2.5 12

2 10

1.5 8 water (OM) water (OM) 6 1 4 0.5 2

0 0 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 time (Myrs) time (Myrs) Figure 6. Surface water abundances for (a) the nominal single layer convection model and (b) boundary layer convection models for different initial water abundances. Total water abundances are equivalent to 2, 4 and 6 ocean masses of water for the Earth-mass planet. Abundances for the larger planets are the same in terms of mantle mass fraction of water. Colors indicate planet mass (blue 1 M⊕, red 5M⊕), and line styles indicate water abundance. models (Fig. 6a), and oceans persist later for the Although models for the Earth show limited sensitiv- boundary layer convection models (Fig. 6b). The ity to initial temperature (e.g. McGovern & Schubert effect of varying initial water abundances produces (1989), Tajika & Matsui (1992)), this appears not results similar to changes in the fugacity coefficient to be the case for the super-Earth models. Figure shown in Fig. 5. However, one major difference to 7 compares results for the single layer model for the note is that although both larger water abundances nominal starting mantle temperature of 2520 K, 2000 and larger fugacity coefficients increase outgassing, K, and 3000 K for the 1 (blue) and 5 (red) M⊕ planets. the persistence of the oceans differs. Surface oceans Mantle temperatures for the smaller planets converge persist longer (til ∼8 Gyr for the 5 M⊕ planet) for en- within ∼ 4 Gyr on the nominal results (Fig 2a) but hanced water, whereas increasing the fugacity coefficient for the hotter starting temperature the 5 M⊕ planet shortened the ocean lifetime. Another thing to note remains persistently hotter throughout its lifetime. The is that the surface water abundances for the boundary hotter initial temperatures effect the water cycle for all layer model with 2 (10) OM of water are nearly identical. of the planets. Although the 1 M⊕ planet converges to nearly the same temperatures, the initially hotter planet outgasses 3× more water within the first 500 Myr. The 3.3.3. Initial mantle temperature water is gradually ingassed over the planet’s lifetime, but the final abundance of water remains slightly larger Oceans on Super-Earths 11

2 8 150 % 100 % 50 % 1.6 6

1.2 2520 K 3000 K 4 2000 K

water (OM) 0.8 water (OM)

2 0.4

0 0 2000 4000 6000 8000 10000 0 time (Myrs) 0 2000 4000 6000 8000 10000 time (Myrs) Figure 7. Surface water for the single layer convection model. Solid lines show the nominal model (also shown in Fig. 2 and Figure 8. Surface water abundances for the boundary layer con- 3), compared with models with either a higher (3000 K) or lower vection model. Solid lines show the nominal model (also shown (2000 K) starting mantle potential temperature. Line styles indi- in Fig. 4), compared with models using either Lridge = 100% cate temperature according to the legend. Colors indicate planet (dash) or 50% (dash-dot), respectively of the planetary circum- mass (blue 1 M⊕, red 5M⊕). ference. Colors indicate planet mass (blue 1 M⊕, red 5M⊕). The nominal model uses Lridge = 150% of the planetary circumference. than for the colder starting planet. The 5 M⊕ planet temperature is minor for both the 1 and 5 M⊕ planet, begins outgassing much more rapidly, and continues so we do not show it here. For the surface water steadily outgassing for its lifetime. It ends with ∼2 OM abundances, the lower Lridge values reduce the peak of water on the surface. Lower initial temperatures, surface water abundance and significantly prolong the which are more widely used in the literature (e.g. ingassing of the surface water after the initial outgassing Stamenkovićet al. 2012; Noack & Breuer 2013), delay phase. The 5 M⊕ planet has ∼0.5 OM of surface water and reduced the degree of outgassing, particularly for remaining at 10 Gyr, whereas the 1 M⊕ planet has larger planets. For an initial potential temperature of ∼0.8 OM of surface water remaining for an Lridge = 50%. 2000 K, the 5 M⊕ planet does not begin degassing until ∼4 Gyr. Given that surface water is likely necessary for plate tectonics (Korenaga 2010b), these planets may 3.3.5. Other parameters not experience plate tectonics at all, until very late in Another parameter that can significantly affect the their lifetimes. These planets will likely start in a stag- model results is the viscosity parameterization. We nant lid mode, which we have not attempted to model have chosen here to normalize the viscosity to 1022 Pa here. However, it is likely that the stagnant lid would s for a reference state of 1600 K, 0 Pa, and 500 ppm allow the planet to heat up earlier so that formation of water. This gives a reasonable viscosity for the Earth’s the oceans may not be as delayed as shown here. The mantle. However, there are other choices that could boundary layer models show very limited dependence on be made. Sandu et al. (2011) tuned their model to the initial temperature and so are not shown here. For match the present day Earth’s viscosity and heat flux, an initial starting temperature of 3000 K, the planets by normalizing their viscosity to 2.3 × 1021 Pa s at 2300 evolve at virtually the same temperatures, and the K and 500 ppm of water. Using this reference value surface water abundances are only slightly enhanced. 25 for the mid-mantle pressure, we get η0 = 2.5 × 10 The different initial water abundances shown in Fig. 6 Pa s, roughly an order of magnitude lower than the had a far larger impact on that model’s results. value used here. For the single layer model, using this normalization factor results in mantle temperatures 3.3.4. cooler by ∼100-150 K, and surface water abundances Mid-ocean ridge length that resemble the hotter starting temperature in Fig. 7. While we use a parameterization that produces The abundance of radioactive elements affects planets the Earth’s present day mid-ocean ridge length, we in both models. A recent paper shows that older planets, note that the length of the ridges on Earth may have those formed early in the galaxy’s lifetime, will have changed throughout time. However, results for both much lower abundances of radioactive elements, whereas models change only slighlty with different values for younger planets may have up to 7 times more radioactive Lridge. For the single layer model, the temperatures heat production (Frank et al. 2014). For the single with variable Lridge do not change, and surface water layer models, increasing the uranium abundance by abundances only slightly decrease. The effect on the a factor of two increases the mantle temperatures by water abundance for the boundary layer models is 100-200 K. In particular, all but the 1 M⊕ planet heat moderate, as shown in Figure 8. We show results up within the first 2 Gyr, rather than cooling. The 5 for models using lower values of Lridge, which would M⊕ planet reaches a peak temperature of ∼3200 K, correspond with slower plate growth. The effect on compared to ∼3000 K for the nominal results. The 12 Schaefer & Sasselov surface water abundances are increased slightly, but to a We have focused in the discussion of results on lesser extent than seen for a raise of initial temperature the surface water abundances of the different models (Fig. 7). The mantle temperatures of the models heated because this is a potentially observable parameter, and from below increase by ∼100 K. The peak surface water one that has implications for the possible habitability abundances do not change from the nominal results, of super-Earths. Concerns have been raised about the but the ingassing of the water back to the mantle takes habitability of, or more importantly the ability of life a longer amount of time. The 1 M⊕ has substantial to begin on, planets with global ocean coverage. Some surface water until ∼ 4 Gyr, whereas the 5 M⊕ planet’s fraction of continental surface, which provides a shallow surface water persists to ∼ 6 Gyr, compared to ∼4.5 water environment, may be necessary for life to begin Gyr for the nominal models. and for the evolution of complex life. Cowan & Abbot (2014) derive a maximum ocean depth that separates planets with continents from totally water-covered 4. DISCUSSION OF RESULTS planets using a crustal buoyancy model. They find that 4.1. Stagnant lid regime maximum ocean depth scales with surface gravity as −1 dmax ∼ 11.4 (g/g⊕) km. Applying this to the results Many terrestrial planets, such as Mars and Venus, do for the models above gives us the minimum surface not experience plate tectonics, but are in a stagnant- area coverage on each planet. We show these results lid regime. For these planets, a thick lithosphere insu- for both the single layer and boundary layer models lates the convecting portion of the mantle from the sur- in Figure 9. The single layer convection planets have face. Stagnant lids develop when planets are too cool to slightly less than 50% areal coverage by oceans. The 5 maintain convection across the entire silicate layer. It M⊕ planet has minimal ocean areal fraction until ∼2 is likely that most planets can transition between plate Gyr, which would suggest that life would be difficult tectonics and the stagnant or sluggish lid regimes (Sleep to begin on such a planet in its infancy. For boundary 2000). However, the mechanisms by which this transi- layer convection, the 3 and 5 M⊕ planets have areal tion occurs are poorly understood. We can speculate, coverages greater than 1, which indicates that they will however, on how the transition would effect the models have global oceans. These planets are not likely to have described. The formation of a stagnant lid insulates the exposed continents. However, the 1 and 2 M⊕ planets mantle and allows the temperature to increase, which have significant continental area for both convection would lower the viscosity and increase convective vigor. modes, which indicates that these planets may be best Regassing would halt in this regime, as there is no trans- places to search for life. port mechanism for water to return to the mantle (see e.g. Morschhauser et al. 2011). For all of the planets in the boundary layer model, the formation of a stagnant lid would likely extend the lifetime of the surface oceans, 4.4. Additional planetary processes by halting regassing. The planets could also heat up The models presented here are not comprehensive pa- enough to re-initiate mantle melting and degassing. Fu- rameterizations of processes that can affect the water ture work will look at the effect of such a transition on budget of a planet’s surface. One particularly impor- the persistence of surface oceans. tant process is the loss of water from a planet’s atmosphere (Wordsworth & Pierrehumbert 2013). Our 4.2. Steady state versus thermal evolution model is agnostic to the form that water takes at the As discussed in section 2.3, using the volatile evolu- surface (i.e., water, ice, steam, etc.), other than by tion parameterization of McGovern & Schubert (1989), constraining the surface temperature. However, the planets were found to achieve a volatile steady state. surface temperature will vary with stellar type, or- For a planet with a large global water abundance, most bital distance, atmospheric composition, and age. As of the water will be found at the surface of the planet Wordsworth & Pierrehumbert (2013) show, loss of wa- for the planet’s lifetime. For planets with global water ter vapor depends not only on the surface tempera- abundances smaller than the steady state value, all ture, but also on the CO2 abundance of the atmo- water will be trapped within the mantle. Note that sphere. Significant loss of water vapor on hot CO2-rich Cowan & Abbot (2014) considered a steady-state solu- planets would reduce the amount of regassing, so the tion to determine the water mass fraction necessary for mantle would become more dehydrated over time. We a planet to be considered a water planet (>90% surface also neglect continental crust formation and weathering ocean coverage). However, in the models considered (Honing et al. 2014). Continental crust effectively acts here, steady state was never reached. In fact, the results as an insulating barrier, but also serves as a sink for discussed above show widely divergent outcomes for the radioactive elements which are extracted from the man- same planet based upon the characteristic upper mantle tle. Weathering of continents and sedimentation rates, viscosity chosen. Once the upper mantle temperature which are enhanced by living organisms, can affect the drops below the peridotite solidus, the melt layer disap- regassing rate into the oceans. Honing et al. (2014) sug- pears and the volatile cycle effectively ceases, although gest that planets without life will evolve to have more slow ingassing may continue until the surface becomes surface water (therefore lower continental coverage), and depleted of all water. dryer mantles than planets with life. Weathering also acts as climate control by stabilizing atmospheric CO2, which affects the retention of water in the atmosphere 4.3. Ocean depths on Super-Earths as described above (Abbot et al. 2012). Much further work needs to be done to truly understand the feedbacks Oceans on Super-Earths 13

0.4 1.8 1 M⊕ 1.6 0.35 2 M⊕

1.4 3 M⊕ 0.3 5 M⊕ 1.2 0.25 1 0.2 0.8 0.15 0.6 surface area fraction surface area fraction 0.1 0.4

0.05 0.2

0 0 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 time (Myrs) time (Myrs) Figure 9. Minimum surface area coverage for the nominal models with a) single layer convection and b) boundary layer convection. The minimum surface area coverage is determined assuming the maximum ocean depth based on the scaling of Cowan & Abbot (2014). Planets with greater than 90% ocean coverage are considered water planets. that will contribute to the persistence of oceans on super- the quality of this paper. Earths. 5. SUMMARY REFERENCES We explored two different scaling parameterizations for plate tectonics planets using either single layer Abbot, D.S., Cowan, N.B., & Ciesla, F.J. 2012, ApJ, 756, 178. Berta, Z. K., and 9 coauthors. 2012. ApJ, 747, 35. convection or boundary layer convection. Mantle tem- Bounama, C., Franck, S., von Bloh, W. 2001, HESS, 5, 569. peratures are significantly hotter for the first model, and Breuer, D., Labrosse, S., Spohn, T. 2010, SSRv 152, 449. the surface water abundance lower, but more persistent. Chopra, P.N., Paterson, M.S. 1981, Tectp. 78, 453. For many different parameters, smaller planets will Cowan, N., Abbot, D. 2014, ApJ, 781, 27. have initially larger surface oceans, but lose them more Crowley, J. W., Gerault, M., O’Connell, R. J. 2011, E&PSL, 310, 380. rapidly than larger planets. Larger planets show delayed Deming, D., and 11 coauthors. 2009, PASP 121, 952. outgassing, which may compromise them as locations on Deschamps, F., Sotin, C. 2000, Geophys. J. Int. 143, 204. which life can originate. The boundary layer convection Dressing, C., et al. 2014, in preparation. model cools very rapidly due to vigorous convection Dumoulin, C., Doin, M.P., Arcay, D., Fleitout, L. 2005, Geophys. driven by both an upper and a lower thermal boundary J. Int. 160, 344. Fei, H., Wiedenbeck, M., Yamazaki, D., Katsura, T. 2013, Nature layer. The surface water abundances on the massive 498, 213. planets in the boundary layer model are extremely Fraine, J.D. and 9 coauthors. 2013, ApJ, 765, 127. high and suggest that these planets will have limited, Frank, E.A., Meyer, B.S., Mojzsis, S.J. 2014. Icarus, 243, 274. if any, continental coverage. However, these oceans Girard, J., Chen, J., Raterron, P., Holyoke, C. W. III. 2013, persist for less than half of the planet’s lifetime. Upon PEPI, 216, 12. Hirschmann, M.M. 2006, AREPS, 34, 629. complete ingassing of the oceans, the massive planets Hirschmann, M.M., Kohlstedt, D. 2012, PhT, 65, 40. are effectively tectonically dead, and therefore unlikely Hirschmann, M.M., Aubaud, C., Withers, A.C. 2005, E&PSL to be habitable. 236, 167. Observations of rocky exoplanets in the 1 - 5 M⊕ Hirschmann, M. M., Tenner, T., Aubaud, C., Withers, A. C. range are already producing very accurate planet 2009, PEPI, 176, 54. Hirth, G., Kohlstedt, D.L. 1996, E&PSL 144, 93. radii and mass determinations (Dressing et al. 2014). Hirth, G., Kohlstedt, D.L. 2003, Many of these exoplanets have ages, determined from Honda, S., Iwase, Y. 1996, E&PSL, 139, 133. asteroseismic ages of their host stars. In the era of Honing, D., Hansen-Goos, H., Airo, A., Spohn, T. 2014, P&SS, JWST and large ground-based telescopes, some of the 98, 5. nearest exoplanets will have atmospheric spectroscopy Karato, S. 1990, Nature. 347, 272. Karato, S. 2012, Icarus, 212, 14. capable of distinguishing different geochemical regimes, Karato, S., Wu, P. 1993, Sci, 260, 771. like some of the extremes described here. Understanding Katz, R.F., Spiegelman, M., Langmuir, C.H. 2003, GGG, 4, 1073 the general features of the deep water cycle across Kohlstedt, D. L., Keppler, H., Rubie, D.C. 1996, CoMP, 123, 345. rocky planets of different mass and structure might give Korenaga, J. 2010a, JGR, 115, B11405. Korenaga, J. 2010b, ApJ, 725, L43. us unique windows into their interior through its sin- Li, Z. A., Lee, C.A., Peslier, A.H., Lenardic, A., Mackwell, S.J. gular effect on atmospheric and surface water abundance. 2008, JGR, 113, B09210. Litasov, K.D., Ohtani, E. 2007, GSA Special paper. 421, 115. McDonough, W.F., Sun, S.S. 1995, ChGeo, 120, 223. McGovern, P. J., Schubert, G. 1989, E&PSL, 96, 27. We thank Li Zeng for helpful discussions and an anony- McNamara, A.K., van Keken, P. E. 2000, GGG, 1, 1027. mous referee for a detailed review that greatly enhanced Morschhauser, A., Grott, M., Breuer, D. 2011. Icarus, 212, 541. 14 Schaefer & Sasselov

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