Earth and Planetary Science Letters 286 (2009) 492–502

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Earth and Planetary Science Letters

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Convection scaling and subduction on Earth and super-Earths

Diana Valencia ⁎, Richard J. O'ConnellDepartment of Earth and Planetary Sciences, Harvard University, 20 Oxford St., Cambridge, MA 02138, USA

⁎ Corresponding author. Now at the Observatoire deNice Cedex 4, France. Tel.: +33 492 00 30 52; fax: +33

E-mail addresses: valencia@oca.eu (D. Valencia), oco(R.J. O'Connell).

0012-821X/$ – see front matter © 2009 Elsevier B.V. Adoi:10.1016/j.epsl.2009.07.015

a b s t r a c t

a r t i c l e i n f o

Article history:Received 5 July 2008Received in revised form 20 June 2009Accepted 7 July 2009Available online 12 August 2009

Editor: T. Spohn

Keywords:plate tectonicssuper-Earthsparameterized convection

Super-Earths are the smallest class of discovered extra-solar planets. Owing to their relatively small mass, somemight resemble Earth and perhaps be habitable. Because of the connection to habitability through thermalevolution, we investigate the tectonic regime of massive terrestrial planets. Two independent studies [O'Neill, C.,Lenardic, A., Oct 2007. Geological consequences of super-sized earths. GRL 34, L19204, Valencia, D., O'Connell, R.J.,Sasselov, D. D., Nov 2007a. Inevitability of plate tectonics on super-earths. ApJL 670, L45–L48.] have reachedopposing conclusions about the likelihood of plate tectonics. Here, we offer possible reasons for the discordantfindings and address threekeyaspects of sustainingplate tectonics: deformation on faults, negative buoyancyandenergy dissipation during subduction. We show that in general, the ratio of driving force to plate resistanceincreaseswith planetarymass. This is a consequence of increasing convective stresses, thinningplates and similarplate structure. We conclude that even though the strength of dry and wet of faults increases for massiveterrestrial planets, the convective stress increases even more allowing deformation to take place. Also, despiteshorter timescales for plate cooling, rocky super-Earths achieve negative buoyancyat subduction zones. Finally, byinvestigating the effects of energy dissipation during subductionwe find thatmassive terrestrial planets dissipateless energy during subduction and hence provide a positive feedback to sustain active-lid tectonics. In conclusion,rocky super-Earths have more favorable conditions than Earth for the subduction of plates, and hence, forsustaining plate tectonics.

© 2009 Elsevier B.V. All rights reserved.

1. Introduction

Despite the formidable observational challenges, fifteen super-Earths have been discovered from the ground in the last four years.Super-Earths are extrasolar planets made primarily of rock and/or icesand voided of a massive hydrogen or helium envelope. There is no hardlimit on planetary mass that separates the gaseous from the (mostly)solid planets, but a reasonable valuemay be in the vicinity of 10M⊕ (Idaand Lin, 2004). Two important characteristics make super-Earthsinteresting objects to study: the ones that are rocky might be similarto Earth, anddependingon their thermal state behabitable; and, thebiasin detection ensures that larger Earth-like planets will be discoveredbefore a true Earth analog. Furthermore, in thenear future several dozensuper-Earths will be discovered with space missions underway—CoRoT(Borde et al., 2003), Kepler (Borucki et al., 2003)—and others underconstruction (Automated Planet Finder, JWST, etc.). To interpret the datathatwill be available, it is timely to set the framework for understandingthe properties of these planets. In this study we address the relevanttopic of tectonics in rocky massive planets.

la Cote d'Azur, BP 4229, 06304492 00 31 21.nnell@geophysics.harvard.edu

ll rights reserved.

The thermal evolution of a planet is intrinsically related to themode of convection, which can be in a statewith plate tectonics or elsewith an immobile stagnant lid. There have been attempts to predictthe mode of convection of rocky super-Earths with oppositeconclusions drawn. Valencia et al. (2007) show that terrestrialsuper-Earths can exhibit plate tectonics, while O'Neill and Lenardic(2007) (hereonafter OL07) determine that super-sized Earths willmost likely be in a stagnant lid regime and that only in some cases willthey experience episodic plate tectonics.

Plate tectonics is a complicated process and one thatwe only evidenceon Earth. Despite the geological data available on Earth, the details ofactive lid tectonics are not completely understood. Nevertheless, thegeneral features of plate tectonics have been laid out over the past30 years and are well recognized. Determining how the onset of platetectonics takes places is a challenge still unresolved. But once subductionhas started, it ismuch easier tomaintain becausedeformation canhappenalong pre-existing faults. Thus, we investigate if plate tectonics can besustained in massive versions of Earth. We approach three questions: isthe increase in fault strength larger than the convective stress as to hindersubduction (as suggested by O'Neill and Lenardic (2007))? Is negativebuoyancy reached at subduction zones in rocky super-Earths to ensurefoundering? Is dissipation during subduction large enough to slow downplates to the point of halting plate tectonics?

The discovery of super-Earths has recently opened an opportunity forcomparative planetology. It promises to widen the current geophysical

493D. Valencia, R.J. O'Connell / Earth and Planetary Science Letters 286 (2009) 492–502

theories that have been developed to explain Earth's and the otherterrestrial planets' properties. This places thinking about the solid planetsin our solar systemwithin awider, more complete planetary context. Weconsider this study to be a step in that direction.

2. Different approaches

The subject of plate tectonics on Earth has been approached in threedifferent ways: with analytical theories, numerical modeling andexperimental work, all complementary to one another. The approach byOL07 to study tectonics on massive Earth-like planets was to adapt thenumerical model by Moresi and Solomatov (1998). This finite elementcode was developed to reproduce plate-like behaviour on Earth. Itconsidered a temperature-dependent viscous mantle heated from belowoverlainbya lithosphere that deformsaccording to its stress regime. In theupper cold part, the lithosphere is brittle and may deform as long as theconvective stress exceeds the yield stress. To circumvent the problem ofhaving plate-like coherent structure while inducing subduction, theyadopted a Non-Newtonian behaviour to a near-surface layer. Thisformalism reinforces zones ofweakness that then can fail under a Byerleecriterion and produce trenches. Due to the well known limitations ofmodeling systems with large Rayleigh numbers (Ra), Moresi andSolomatov (1998) considered values in the range Ra=105–108. Theysuccessfully reproduce plate-like behaviour, episodic foundering andstagnant lid occurring as a function of increasing fault strength. In theregime of plate tectonics they reach an almost exact agreement with thetheoretical prediction between the Nusselt number (Nu) and Ra fromboundary layer theory.

The conclusions drawn by OL07 rest on how the scaling to biggerplanets from this model was done. They consider super-sized Earthsthat have the same density and conclude that larger planets (scalingratio of R/REN1) would require lower yield stresses to achieve at leastepisodic stages of plate tectonism. Thus, they attribute the prevalenceof stagnant lid on rocky super-Earths to the locking of faults caused byhigher pressures from higher gravity values. Furthermore, Fig. 3 intheir paper suggests that smaller planets than Earth can have mobileplates. In a personal communication with A. Lenardic, this puzzlingconclusion was attributed to lower temperatures on smaller planets.The thermal age of the small planets they model is thus, older thantheirmassive counterparts. Lower temperatures increase the viscosity,so that the deviatoric stress increases to aid subduction. This is thefirst clue to the different findings between the two different groups.

On the other hand, thework we presented in Valencia et al. (2007a)used a parameterized analysis of convection in a system heated fromwithin to predict the convective properties on massive terrestrialplanets.We investigated Earth-like planets in the sense that theywouldhave a similar composition: concentration of radioactive elements andFe/Si ratio (expressed as similar core-mass fraction). We accounted forcompression so that massive planets would have larger averagedensities and gravities that scale accordingly. The conclusion was thatdue to thinner plates and larger convective stresses, the conditions forsubduction, would be more favorable on bigger planets.

There are two basic assumptions behind our work, and that of OL07and Moresi and Solomatov (1998): (1) that the stress needed to causedeformation is available from convection. This is justified by relatingEarth's plate motions to the tractions on plates (Bird, 1998; Becker et al.,1999; Rucker andBird, 2007). And (2) that a Byerlee criterion for failure inthe brittle part of the lithosphere adequately captures the behaviour oflarge-scale faults that represent the collective behaviour of randomlyoriented smaller faults. However, a key result that is different in ourfindings is that the pressure–temperature structure of the plates remainsalmost invariant to mass.

Having that themethods employed by the two groups do not differ intheir core reasoning, we first set out to investigate if the “increase in faultstrengthdrastically outweighs the change in convective stress” inmassiveEarth-like planets as suggested by (O'Neill and Lenardic, 2007).

3. Convective parameters

We elaborate on the parameterized convection analysis to showthe implicit dependence of the convective parameters on the mass ofthe planet (M). This is done via the Rayleigh number (Ra). In the caseof a fluid layer of depth D that is internally heated, the Rayleighnumber can be defined in terms of the surface heat flux q:

Ra =ραgD4qκηk

ð1Þ

where g is the gravity and the material properties are: the density ρ,coefficient of thermal expansion α, thermal diffusivity κ, thermal condu-ctivity k, and viscosity η. These parameters are constant and unambigu-ouslydefined in thecaseof ahomogenousfluid layer.On theotherhand, inthe case of a planetary mantle, the material properties as well as gravity,are a functionof temperature (T) andpressure (P) and therefore a functionof depth. Given that we are investigating the conditions the plates aresubject to, the relevantparameters inEq. (1) are those in thevicinityof theplate. For a discussion on the applicability of boundary layer theory andour choice of boundary condition (that of heat flux) see Appendix A.

3.1. Shear stress

In mantle convection, the deviatoric horizontal shear stressunderneath the lithosphere depends on the velocity of the plate andthe depth of the convective layer ℓ,

Δτxz ~ηuplate

ℓ;

with ℓ~D for whole mantle convection. The velocity of a plate isrelated to the Rayleigh number (O'Connell and Hager, 1980; Turcotteand Schubert, 2002a)

uplateea 21

κD

RaRac

� �1=2λ;

where a1 is a coefficient of order unity, λ is the ratio of width to depthof the convective cell (of order unity in the mantle), and Rac~O(1000)is the critical Ra for convection to occur. Thus, the deviatorichorizontal stress has no direct dependence on the depth of the mantle

Δτxz ~καkRac

� �1=2ðηρgqÞ1=2: ð2Þ

Nevertheless, there is still an indirect dependence because anincrease in planetary size due to an increase in mass is accompaniedby an increase in heat content, gravity, average density and possiblyother material properties. Moreover, in a scenario well described as aheated-from-within system, the stress depends on the heat flux,which is a temporal quantity that changes as the planet evolves andcools. In Section 4we show howwe evaluate super-Earths' heat fluxes.

3.2. Boundary layer

In a planet with plate tectonics, the lithosphere is the boundarylayer. For planets with a stagnant lid, the colder and stronger part ofthe plate sits above the boundary layer that participates in theconvection (Davaille and Jaupart, 1993; Solomatov, 1995). Accordingto classical boundary layer theory, the boundary layer thickness iseffectively independent of the depth of the convective layer (depth ofmantle) but depends on the vigour of convection, Ra (O'Connell andHager, 1980; Turcotte and Schubert, 2002a),

δ =1a1

D2

RaRac

� �−s

ð3Þ

494 D. Valencia, R.J. O'Connell / Earth and Planetary Science Letters 286 (2009) 492–502

where s=1/4 when the boundary condition is the heat flux and s=1/3when it is the temperature difference across the mantle. The morevigorous the convection, the thinner the boundary layer. In terms of theheat flux the thickness of the boundary layer or plate thickness is

δ ~α

κkRac

� �−1=4 ρgqη

� �−1=4: ð4Þ

Eqs. (2) and (4) show the dependence of deviatoric shear stressand plate thickness on the material properties of the convective fluidand the heat flux. To obtain the dependence on mass, we obtain thevalues of these fluid properties (density, gravity, viscosity) and theheat flux for super-Earths. We note that α, κ and k might vary amongterrestrial planets, but that the largest uncertainty comes from theviscosity which can change by orders of magnitude. In addition, whileviscosity depends on pressure, temperature and stress (Karato andWu,1993), the dominant effect is temperature and thus, we consider atemperature-dependent viscous mantle.

4. Planetary heat flux

A planet's surface heat flow Q, reflects the amount of heat beingtransported into the mantle from the core Qcore, generated withinfrom radioactive heat sources Qrh and its cooling rate at a given pointin time dT(t)/dt:

QðtÞ = ρCp4πR3

3dTðtÞdt

+ Q rhðtÞ + Q coreðtÞ: ð5Þ

The contribution from radioactive sources to the heat flow is

Q rhðtÞ = ∑iRiγiðtÞ = ∑

iRiγ

0i exp−λiðt−t0Þ;

where Ri is the amount of heat produced per mass by each of themajor radioactive elements 238U, 235U, 232Th, and 40K, γi is the amountof each radioactive element in the planet's mantle at a given time t,and λi is their decay constant.

Provided that these isotopes have very long half-lifes (i.e. t1/2238U=4.4 Gy, t1/2232Th=14 Gy), their concentration in the galaxy is not expected tovary wildly. Thus, it is reasonable to consider planets that have the same

radioactive concentration as Earth, Q rhp ðtÞ = Q rh

⊕ ðtÞ MM⊕

� �. This assump-

tion can subsequently be relaxed to explore its effects on the results.To calculate the other two terms in Eq. (5) and obtain the total heat

flow of a planet Q at any point in time, we note that a thermalevolution model is required. Such a model would yield in principle theexact contribution of radioactive sources to total heat flow (the Ureyratio—U). Instead, we limit our study to planets mostly dominated byradioactive heat production—as in the case of Earth—which areexpected to have mass-scaled versions of total heat flow

4πR2qp = Qp = Q⊕MM⊕

� �: ð6Þ

5. Scaling with mass

To get the dependence of plate thickness and shear stress on mass,we use a detailed internal structure model to obtain the mantle size,average mantle density, average gravity, and viscosity under the platefor rocky super-Earths. Since the details of the model have beenexplained elsewhere (Valencia et al., 2006, 2007b), we only brieflydescribe it here. It solves the differential equations of density, gravity,mass, pressure and temperaturewith depth. It uses a Vinet equation ofstate (Vinet et al., 1989) and a thermal profile that is conductivethroughout the boundary layers and adiabatic in the convectiveinteriors of the mantle and core. Through iteration in the Rayleigh

number and plate thickness convergence is reached to yield aconsistent thermal profile. It also includes all major phase transitionsknown to occur on Earth including post-perovskite. This modelsuccessfully reproduces Earth and the Terrestrial Planets' structure.

The mass dependence of the properties relevant to this study isadequately expressed in a power law relation: ρ~Ma, g~Mb, η~Mc, q~Md,where the exponents a, b, c and d are determined from the resultsof the internal structure model. The expressions for deviatoric hori-zontal shear stress and plate thickness depend on these exponents (fromEqs. (2) and (4))

Δτxz~Mða + b + c + dÞ=2 ð7Þ

δ~Mða + b + c−dÞ=4: ð8Þ

Even without the implementation of an internal structure model,qualitative results can be obtained from the scaling of themost simplifiedscenario: that of a family of isoviscous rocky super-Earths scaled withconstant density. In this simple case, a=c=0, and R~M1/3, so that theheat flux q~M1/3 and gravity scales as g=GM1/3. The stress dependenceon mass is then Δτxz~M1/3 and δ~M−1/6. This elemental scaling showsthat the ratio of shear stress to plate thickness increases with planetarymass.

We investigate if this qualitative conclusion holds true in a generalcase. In particular, it is not known a-priori the effect of viscosity on shearstress. Even amodest increases in temperature, could significantly lowerthe tractions underneath the plates effectively decreasing the drivingforce behind subduction.

Wemodel the viscosity after Davies (1980) by using a power law fit to

the Arrhenius law, so that η = η0TT0

� �−nwhere n=30 and η0 is the

viscosity value at the reference temperature T0. We are interested in theviscosity underneath the lithosphere, so that T is the potentialtemperature. The results from the internal structure model are thatthe planetary radius, average density, gravity, heat flux and viscosityhave a dependence on mass of R~M0.262, ρ~M0.196, g~M0.503, q~M0.476,and η~M−0.64. The viscosity decreases with mass, because the temper-ature underneath the plate slightly increases from 1566 K for Earth to1631 K for a 10M⨁-planet. Thus, the dependence of deviatoric stress andplate thickness is Δτxz~M0.33 and δ~M−0.42. While the exact values forthe exponents may vary depending on the viscosity formalism adopted,the robust result is that the ratio of shear stress to plate thicknessincreases with mass.

We explore the role of n in this qualitative result (Appendix B). Weconclude that the condition for the shear stress to increase with mass

is satisfied as long as n N 2a + b

d+ 1

� �a + b

d−1

, which always holds true. In the

case of constant density scaling a=0, b=d, so that any nN0 satisfies theinequality. For a general case, two conditions are met: 1) the heat fluxroughly scales with gravity q~g, and 2) the density always increaseswithmass (aN0). Therefore, the inequality holds true for any positive n,and even slightly negative values, which are perhaps unrealistic.

The verymodest increase inpotential temperature can be easily seenfrom its relation toviscosity andheatflux (Eq. 65 ofO'Connell andHager,1980)

ðT−TsÞ4η

≈ ðq=KÞ3κ16ρgα

Rac; ð9Þ

where Ts is the surface temperature. By replacing the power law relationbetween viscosity and temperature, it is clear that the internal temper-ature weakly depends on heat flux, gravity and material properties, allquantities that vary with mass:

T≈T0 η0ðq=KÞ3κ16ρgα

Rac

!1= ðn + 4Þ:

Fig. 1. Faults' stress and strength. The shear stress (solid lines) on the fault increases linearlywith mass, while the strength of a dry (long dashed lines) and wet (short dashed lines) faultincreases more slowly. This behaviour is independent of choice of coefficient of friction Weconsidered two friction coefficient values (top): μ=0.2 and (bottom) μ=0.8. The shadedregion shows the range of values for earthquake stress release (Weins, 2001).

495D. Valencia, R.J. O'Connell / Earth and Planetary Science Letters 286 (2009) 492–502

Given that heat flux scales as gravity and both scale as ~M1/2, thepotential temperature (for constant density planets) scales as T~M1/34

given n=30. Including the effects of compression on density wouldmake T more insensitive to mass.

In conclusion, in internally-heated mantles with same radioactiveconcentrations, shear stress underneath the plates increases withplanetary mass. Furthermore, the ratio of shear stress to platethickness increases for terrestrial massive planets undergoing platetectonics. This results provides a possible future benchmark with thework by OL07. It is unclear from their paper if we agree on this matter.

6. Fault strength

We arrive at the discussion of fault strength and investigatewhether or not it outweighs the increase in convective stresses assuggested by OL07. Pressure does increases the frictional strength onfaults and hence, can potentially hinder deformation on large planets.

We determine how the stress and the strength on the faults varieswith planetary mass. For deformation to happen, the shear stress onthe fault (τ) has to overcome its strength (τrock). In the same manneras (Moresi and Solomatov, 1998), we invoke a Byerlee criterion fordeformation on the brittle part of the lithosphere. The Coulombstrength of rocks is expressed using Byerlee's law

τrock = S + μðσ−λσzzÞ

where S is the internal cohesion parameter, σ is the normal stress onthe fault, σzz=ρgz is the lithostatic pressure, and λσzz expresses theeffect of water on reducing the fault's strength. The normal (σ) andshear (τ) components of the stress on the fault are

σ = σzz +12Δσxxð1 + cos2θÞ

τ =12Δσxxsin2θ

where θ is the angle of faulting which is related to the frictioncoefficient μ by tan 2θ=1/μ (Turcotte and Schubert, 2002b), Δσxx=L/δΔτxz is the deviatoric shear stress available from convection, and Lthe length of the plate. Moresi and Solomatov (1998) did not considerthe angle of faulting, however, it does not constitute a major differencein our treatments. In fact, by considering an angle of faulting ourestimates of τrock/τ are more conservative.

We find that despite higher gravity surface values, rocky super-Earthsexperience similar pressures under their lithospheres. This is because theincrease in gravity is offset by a decrease in plate thickness, therebyproducing a somewhat constant average pressure on the plate of differentplanets. Thus, the lithostatic pressure does not increase for bigger planets,and consequently does not increase the fault strength. Furthermore,because the temperature beneath the boundary layer is also somewhatinsensitive to mass as noted before, the P–T plate structure is almostinvariant to planetarymass. This result constitutes amajor differencewithOL07.

We calculate the strength of a fault for the family of rocky super-Earthsand showhowit compares to the fault shear stress in Fig.1.We takeEarth'sstrength and stress values as the reference case, and scale them accordingto the equations described above. The general feature in Fig. 1 is that thestrength of both dry (long-dashed lines) andwet (short-dashed lines) faultsindeed increaseswithmass, aswell as the shear stress experienced on thefault (solid lines). The former happens because of a small contribution ofthe convective stress on the fault's normal stress that hinders sliding. Thisis due to our choice of including the angle of faulting. Hadwe not, as is thecase used in the model of OL07, the fault strength would have remainedconstant (i.e. constant yield stress) independent of planetary mass.

The interesting result is that the increase in the fault's shear stress issteeper than the modest increase in fault strength, so that terrestrial

planets can accommodate deformation and experience subduction. Thereason for this is that the deviatoric horizontal normal stress (Δσxx)increases almost linearly with mass from an increase in deviatorichorizontal shear stress (Δτxz), and a decrease inplate thickness despite amodest increase in plate length.

The following expression relates the ratio of deviatoric shear stressto plate thickness, and the ratio of driving force to plate resistance

Driving ForcePlateResistance

=τ

τrock

� �=

12Δτxz

Lδsin2θ

S + μð1−λÞ σzz +12μΔτxz

Lδð1 + cos2θÞ

:

We find a positive relation between this ratio and planetary mass,which means that subduction and therefore plate tectonics can besustained more easily on terrestrial super-Earths. Fig. 3 shows theinverse of this ratio as being less than 1 and decreasing with increasingplanetary mass (for the case of s=0.25). It is clear that the reasonproposed by OL07 to argue that super-Earths cannot exhibit platetectonics stands indisagreementwithour results.What lies at the core isthat the plate structure they obtain is not invariant with mass. Theyclaim that thepressure is higher in larger planets, and therefore theplatethickness they obtain must be larger than that predicted by classicalboundary layer theory. We explore this in Section 8.1.

We complete our investigation of the behaviour of faults in massiveterrestrial planets by considering the uncertainty in the values of thefriction coefficient and the fluid contribution to pore pressure (i.e. 0bλb0.9—Ranalli, 1995).We compare planets that exhibit the same coefficientof friction justified in thatweconsiderplanetswith the samecomposition.On amicro-scale, the friction coefficient of differentminerals will dependon their structure (Morrow et al., 2000). However, by looking at planetsthat share the same rock composition, we circumvent the problem ofaddressing deformation on individual faults and focus on the collectivebehaviour that is likely to share the same average value of μ .

Fig.1 considers an effect of water/volatiles onpore pressure ofλ=0.6.With a relatively high coefficient of friction of μ=0.8 (bottom), the faultshear stress on Earth required to exceed the strength of a wet fault is

Fig. 2. Negative buoyancy in super-Earths. Top: We show the relative density of the platewith respect to the underlyingmantle for different melt fraction production from 1 to 2.5%melt per kbar. Even at 2.2%melt per kbar the plate for the largest super-Earths is negativelybuoyant. Bottom: We show the fraction of crust to the total lithospheric thickness at thetime of subduction. The crustal thickness decreases with increasing planetary mass, whilethe lithosphere decreases evenmore, so that the fraction of crustal thickness increases formore massive planets.

496 D. Valencia, R.J. O'Connell / Earth and Planetary Science Letters 286 (2009) 492–502

~100 MPa (available if the deviatoric normal stress isΔσxx=300 MPa). Ifwe consider the coefficient of friction to be lower μ=0.2 (top), the shearstress required is ~20 MPa (available ifΔσxx=50MPa). This low value ofμ has been recently proposed as a better value for the effective frictioncoefficient on pre-existing faults (Di Toro et al., 2006), which is alsomoreconsistent with the values of stress release during earthquakes on Earth(shaded area in Fig. 1). We also examine the case where water is lesseffective at increasing pore pressure λ=0.2 (and μ=0.8). This low valueofλ requires shear stresses of ~200 MPa for Earth to exceed the strengthofawet fault (available ifΔσxx=550MPa). The deviatoric shear stress fromconvection has been modeled to be 10–100 MPa (Becker et al., 1999), sosuch a low value of λ seems unlikely.

More importantly, Fig. 1 shows that the bigger terrestrial super-Earths will not require water for sliding to occur. The slope of shearstress on faults (black line), which describes its dependence on mass,is steeper than the curve for strength of wet and dry faults, so that atsome point these curves are crossed. The transition between super-Earths that require water to the ones that do not, will depend on theefficiency of water to increase pore pressure on faults (i.e. λ) and thefriction coefficient. Conversely, smaller planets' tectonic regimewouldbe more affected by the presence or absence of water.

7. Lithospheric density

At convergent margins on Earth, the oldest, coldest and densestplate is expected to subduct under the younger, hotter and buoyantplate. Knowing that super-Earths have faster convective velocities(Valencia et al., 2007a) and hence, younger plates in general, it isappropriate to establish the conditions in which negative buoyancycan still be achieved at subduction zones.

We calculate the mean density of a plate (ρlit) composed of basalticcrust (ρbas=2880 kg/m3) and lithospheric mantle (ρman=3330 kg/m3)at themaximumage of the plate (when its thickness is ~double themeancalculated thickness),

ρlit =1hc

∫0

hcρbas½1 + αðzÞðTm−TðzÞÞ�dz

+1

ð2δ−hcÞ∫

2δ

hcρman½1 + αðzÞðTm−TðzÞÞ�dz

where hc is crustal thickness, the coefficient of thermal expansion isα(z)= a+ bT(z)+ĉ/T(z)2 with the values for â, b and ĉ determined by(Poudjom-Djomani et al., 2001), and Tm is the temperature beneath theplate.Wefind that themean lithosphericdensitywill be denser for everyEarth-like super-Earth provided the crustal thickness does not exceed16% of the total plate thickness during subduction (Fig. 2).

Oceanic crustal thickness depends on the extent of melting, whichdepends on the depth at which the solidus intersects the thermaladiabatic profile of the planet. The expression for crustal thickness hc is

hc =Fmax

2ðz0−zf Þ;

where Fmax=r(P0−Pf) is the maximum fraction of melting, r is the meltproduction per unit of decompression; and z0 and zf are the initial andfinal depths of decompressionmelting corresponding to pressures P0 andPf respectively. For the solidus, we used the expression Tsol=aP2+bP+c with the coefficients experimentally determined by Hirschmann (2000).

The temperature beneath the plate and hence, at ridges varies littleamong rocky super-Earths (for the same surface temperature). Owingto larger gravity values, the intersecting pressure is achieved atshallower depths, so that we anticipate the extent of melting to bereduced with increasing planetary mass. This suggests that the crustwill remain under 16% of the plate's thickness. We perform a simplecalculation to obtain the crustal thickness of each super-Earth byobtaining the depth at which the magma solidus is intersected by an

ascending parcel and the amount of maximum fractional melting.Fig. 2 shows the results. The conclusion is that as long as the change inmelt fraction r with pressure is below 2.4% melt per kbar, buoyancywill be negative at subduction zones for terrestrial super-Earths. Thevalue of 2.4% melt per kbar is larger than the limits consideredappropriate for Earth (1–2%), although unknown for other planets.

Recent studies (Hynes, 2005; Afonso et al., 2007) show that on Earth,the lithosphericmeandensity is not larger than theunderlyingmantle's atsubduction zones. However, Afonso et al. (2007) note that the plate'sdensity is larger than the adiabatic mantle (or ridge column density) andHynes (2005) shows that flooding can easily cause negative buoyancy.Furthermore, Becker et al. (1999) showed that compression could thickenthe cold lithosphere to the point of inducing subduction. These ideassuggest that perhapsnegativebuoyancybefore subductionmight notplayas important a role as previously thought. In any case, we find that super-Earths are able to achieve negative buoyancy at convergent margins.

8. Energy dissipation during subduction

In this section we explore the effects of energy dissipation duringsubduction as a scenario that might halt plate tectonics. Conrad andHager (1999a) recognized that a major source of dissipation can be thesubduction of thick strong plates. If the energy required in subductingthe plate is large, the plate will slow down, thicken even more makingsubductionmore difficult, until eventually it can no longer be sustained.We investigate the effects of this process bymodeling subduction in twoways: bending of the plate—after Conrad and Hager (1999a), andcontinuous shearing of the plate into the mantle.

8.1. Plate bending

Conrad and Hager (1999b, a) modeled the lithosphere as bendinginto the mantle at subduction zones (with some prescribed radius ofcurvature) and calculated the energy dissipated in this process.Through an energy balance they derived a relationship between Nu

Fig. 4. Cartoon of convection and subduction. Subduction is modeled as a continuousshearing of blocks that sink into the mantle at convergent margins. The velocity profileof the core of the shearing mantle is taken to vary linearly (following Turcotte andSchubert, 2002a). The potential energy is equated to balance the energy dissipated atshearing the core of the mantle and at subduction zones.

497D. Valencia, R.J. O'Connell / Earth and Planetary Science Letters 286 (2009) 492–502

and Ra that would capture the effects of plate bending. They arguedthat the exponent s in the Nu–Ra relation would be effectivelydecreased in the presence of strong dissipative subduction zones,whenmost of the energy is dissipated by lithospheric bending insteadof in the shearingmantle. Conrad and Hager (1999b) described Earth'smantle as heated from below and proposed s to be 0.15.

By considering a system with a known surface heat flux andcorrespondingly using an exponent of s=1/4 (see Appendix A), wehave shown that the plates of more massive terrestrial planets arethinner and thus, the effect of dissipation during subduction is small.Thismeans that if the planet avoids having a thick lithosphere to beginwith, it will be easier for the lithosphere to remain thin. The questionthat follows is to determine how thick the plates can be withoutcausing too much energy dissipation during lithospheric bending, asto avoid continuously thickening into a stagnant lid.

Another way to answer this question is to calculate the minimumvalue for s that would still allow for the strength of the fault to be lessthan the shear applied—τrock/τb1—so that deformation can still takeplace. To assess this, we considered different values of s and obtain thevalues for plate thickness, horizontal deviatoric shear and normalstresses on the fault and with these, calculate the fault's shear stress τand strengthτrock as shownabove. The results are shown in Fig. 3 for twodifferent coefficients of friction μ=0.2 (top) and μ=0.8 (bottom). Thenominal case of s=1/4 described in Section 6 shows how the fault'sstrength ismore easily overcomeby the stress on the fault as themass oftheplanet increases. There are three regimes fordeformation dependingon the value of s: i) if s is not too small (s≳0.19) the ratio of strength tostress decreaseswithmass, suggesting that subductionwould take placemore easily in more massive planets, ii) if s is very small (s≲0.13) thisratio increases dramatically to suggest that only Earth would havesubduction, and iii) for intermediate values of s (0.13≲sb0.19), the valueof μ plays an important role.

These results imply that as long as the relation of heat transport tothe vigour of convection is not negligible (sN0.16), planets with largermasses could also experience subduction and hence, plate tectonics.

Fig. 3. Strength–stress ratio for different convection regimes. The values of s in the relationNu~(Ra/Rac)s are shown for different coefficients of friction μ=0.2 (top) and μ=0.8(bottom). s=0.25 corresponds to the nominal case with no dissipation during subduction.Smaller values of s indicate increasingly larger energy dissipation contributions fromlithosphere bending during subduction.

8.2. Subduction shear

Another way to model energy consequences of subduction is toconsider continuous vertical shearing of the plate into the mantle.Fig. 4 shows a cartoon explaining this approach. At subduction zones,an infinitely small block is sheared from the plate into themantle. Thisprocess is continuous so that the work rate of this system per lengthalong the subduction zone (ℓs) is

w = fτyδv0

where τy is the yield stress at which the block slides past the plate intothe mantle, v0 is the vertical velocity of the flow and fδ is the fractionof the plate that experiences this shear. The rest of the plate candeform as a viscous layer. In this formalismwe assume that the stressat which the block yields is independent of depth. This assumption ispartly supported by the fact that the stress release in shallow and deepearthquakes is estimated to be of the same order. It has beenestimated to be of 3–10 MPa for shallow earthquakes and 10–44 MPafor deep earthquakes (N300 km) (Weins, 2001).

In terms of total energy per unit time consumed during subduc-tion, W=fτyδ(u0ℓs)D/L, which amounts to ~1010–1011W fromτy=107–108Pa, δ=100 km, f=1/2, D=3000 km, λ=D/L~2, andwith a plate production rate at mid-ocean ridges u0ℓs=3 km2/s.

We follow the same energy analysis by Turcotte and Schubert(2002a), to balance the potential energy (Φpe) with the dissipation inthe core of the shearing mantle (Φvd

man) and in subduction zones(Φvd

sub). The expression for each of the quantities per strike length is

Φpe = ρgαðT−TsÞDu1 = 20

κLπ

� �1=2

Φmanvd = 4ηu2

0DL

� �λ′

Φsubvd = fτyδ

LDu0

where T is the temperature of the core of themantle, L=u0δ2/(κπ) is thelength of the plate, u0 is the horizontal velocity, and λ′=λ2+λ−2.

Equating the terms and using the definition of Ra in terms of theheat flux (Eq. (1)) yields

Nu =12

Ra4π2λ′

� �1=41 +

f4πλ′

τyκη

δ3

L

" #−1=4

: ð10Þ

It is clear that when τy or f is zero, we recover the classical boundarylayer result. Eq. (10) shows the form inwhichdissipation fromsubduction

498 D. Valencia, R.J. O'Connell / Earth and Planetary Science Letters 286 (2009) 492–502

appears in the Nu–Ra relation (e.g. Conrad and Hager (1999b)). Thecharacterization of the effect as simply changing the exponent (Korenaga,2006) obscures this.

If we define ϕ = f4πλ′

τyκη

δ3L , the result can be used to obtain the

thickness of the plate δ,

δ = δ0½1 + ϕðδÞ�1=4;

where δ0 is the plate thickness of a systemwith no dissipation duringsubduction. This is a quartic equation that expresses a correction toplate thickness from dissipation of energy during subduction. It isclear that the effect of subduction of a strong plate is to increase itssize (ϕN0). This happens because the plate slows down (Nu decreaseswith increasing δ), has more time to cool and thicken. This equationalso captures the fact that there is a positive feedback, the thickeningeffect of subduction is more pronounced when the plates are thick (ϕincreases with δ). The range of the correction term is 0.2bϕb2 onEarth, calculated from the values given above and the range in τy,κ=10−6m2/s, and η=1021Pa s. This yields a corrected plate thicknesson Earth of 1.005δ0bδb1.05δ0—at most 5%.

It is obvious from this result that rocky super-Earthswith their thinnerplates have a very small correction term and hence, their subductionzones are only weakly dissipative.

9. Discussion

Plate tectonics is an evolutionary phenomenon and the conditionsfor it will change as the planet evolves and cools. At present, Earth hasenough heat flux to drive vigorous convection needed for an activeplate state. A planet's heat flux varies with time and while on shorttimescales it might increase (Sleep, 2000; Van Keken et al., 2001),over long timescales it decreases as the planet cools. This means, thatat some stage, when the heat flux falls below a threshold, platetectonics will cease to operate on a planet. The timing for this eventdepends on the thermal evolution. Although, extra sources of heat,such as tidal heating, can aid in achieving and prolonging the lifetimeof active-lid tectonics.

We find that planets of the samemass would have the same potentialto sustain plate tectonics as dictated by their ratio of shear stress to platethickness. Venus is not the counter example because of two importantdifferences with Earth: the lack of water (Mian and Tozer, 1990; Nimmoand McKenzie, 1998) and a higher surface temperature (Lenardic et al.,2008).

Our results are in agreement with the findings of (O'Neill et al.,2007) where they state that Venus and tidally-heated Io are the onlyother objects in the solar system thatmight have had plate tectonics inthe past by being in an episodic regime.

9.1. Differences between the two models

Through a personal communication with A. Lenardic, we learnedthat their choice of scaling yields smaller planets to be colder than thebigger counterparts. Lower interior temperatures translate into higherviscosities and larger driving stresses, which can drive subductionmore easily. Thus, they conclude that small planets can achieve platetectonics while rocky super-Earths cannot.

There are two important differences in our treatment and that ofOL07. We consider (1) scalings that take into account the compressioneffects on radius, gravity and average density; and more importantly,(2) heat flux as the boundary condition to describe the system.

With respect to the first difference, a constant density scaling is notadequate in comparing Earth to massive Earth-like planets. Indepen-dent models (Valencia et al., 2006, 2007c,b; Sotin et al., 2007; Seageret al., 2007) clearly show that the relation between radius and massdeviates from a constant density scaling. The different models, whichvary in sophistication and treatment agree that the radius scales as

R=R⨁(M/M⨁)0.262–0.274. Because of possible unknown phase transi-tions and uncertainties in the equation of state, the exponents are anupper limit. Thus, a convenient approximation for the radius scaling isR~M1/4, which translates to gravity scaling as g~M1/2. We recommendthese scalings for comparing Earth to massive Earth analogs.

As for the more fundamental difference, the main question is howto best apply a (quasi) steady state boundary condition to an evolvingplanet. Heat comes from radioactive sources in the interior; thissuggest scaling for constant heat sources, which turns out to be thesame as that for imposed heat flux. In an internally heated mantle, theinterior temperature adjusts to conform with the imposed fixed heatflux (which is the boundary condition). In a system with a fixedtemperature difference driving convection, the heat flux adjusts.Given the uncertainties in all such models, the most accurate choice isunclear. In a mixed-system, the results might depend onwhich case isdominant (basal to heating ratio).

In a system that is heated-from-below—that is, with a fixedtemperature difference (ΔT) across its convective layer—and constantsurface temperature, the deviatoric stresses are expected to increasewith the size of the planet, due to an increase in gravity and density

Δτbasalxz = ηðTÞ1=3κ1=3 ραgΔTR

ac

� �2=3:

If, however, the viscosity decreases, convection stresses would de-crease as well.

In the caseof internallyheatedplanets,weobtain avery slight increasein temperature drop within the boundary layer for massive planets (seeEq. (9)). This qualitativelyagreeswith the treatmentofOL07, in that largerplanets are hotter. However, the subsequent effect of lower viscosities isfurther accompanied by an increase in heat flux for massive rocky super-Earths, so that the net effect is that the deviatoric stress increases—opposite to what OL07 find. The relevant equation for this case is

Δτinternalxz = ηðTÞ1=2 κραgqkR

ac� �1=2

:

Therefore, the differences in the results may be coming from thedifferent boundary conditions. Our model considers a steady statescenario with heat coming from radioactive sources. It predicts that forthe same concentration, bigger planets will have larger convectivestresses and thinner lithospheres—a favorable scenario for plate tectonics.

In addition, smaller planets are expected to cool more rapidlybecause of their area to volume ratio. Being that the ability to sustainplate tectonics decreases as the planet cools, small planets will ceaseto have mobile lids faster than massive rocky ones.

9.2. Treatment for viscosity

A natural question that arises in the treatment of terrestrialmassive planets, is the effect of pressure in convection and specificallyon viscosity. The pressure at the core–mantle boundary of massiveEarth-like planets increases almost linearly with mass (Valencia et al.,2006), and thus, has a profound effect on viscosity at such depths.

By its temperature and pressure dependent nature, viscosity iseffectively a function of depth, lateral variations are comparativelyminor except in plates, which have a different rheology anyway. Thelimits on convection come from the transfer of heat through theboundary layer. This is the (or at least one) basis for the derivation ofthe parameterized convection models used. These seem to workreasonably quantitatively for the Earth with a viscosity of ~1020Pa sfor the upper mantle, even though the lithosphere is not characterizedwell by such a rheology, and that the lower mantle viscosity is higher(~1022–1023Pa s). Therefore, we argue that if the temperature andpressure regime in the uppermost mantle of a terrestrial super-Earthis similar to the Earth, it should behave similarly. Consequently, the

499D. Valencia, R.J. O'Connell / Earth and Planetary Science Letters 286 (2009) 492–502

pressure effects in the lowermost mantle of rocky super-Earths can beconsidered a secondary effect.

In any case, we think this to be a first step towards understandingconvection and tectonics in massive planets and consider thatpressure effects on viscosity, depth-dependent thermal expansivity,conductivity and diffusivity to be future research.

10. Summary and conclusions

It is important to establish the tectonic regime of a planet whendetermining its thermal state, which in turn is fundamental to thequestion of habitability. Here, we examine from classical boundarylayer theory, how the convective parameters and fault properties scalefor terrestrial super-Earths. We address three key issues of platetectonics: deformation on faults, negative buoyancy, and energydissipation during subduction. Even though the mantle is a compli-cated system (with chemical heterogeneities, chaotic dynamics,complex rheology, etc.), simple models like ours are useful inilluminating relations between important parameters. Moreover, toassess the effect of mass on the characteristics of planets, it is vital toappropriately scale the relevant properties with mass. For this we usea detailed internal structure model that renders simple and robustrelations: heat flux scales with gravity and radius scales as R~M1/4.

One of the main results is that convective stresses increase, platethickness decreases, while the pressure–temperature structure of theplate is almost the same for massive planets with the same radioactiveheat concentration. These qualitative results differ from the findings ofOL07.

Consequently, fault strength modestly increases with planetarymass, while the shear stress that can cause sliding increases evenmore, such that it exceeds the strength of wet and even dry faults ofmassive planets. Deformation is accommodated more easily in biggerEarth-like planets. Conversely, smaller planets require weakeningagents, like water, much more critically if deformation on faults is tobe sustained and subduction to occur.

Furthermore, by determining the crustal thickness inmassive planets,we determine that they reach negative buoyancy at subduction zones.Crustal thickness decreases because, due to increasing gravity values, thesolidus of terrestrial super-Earths is crossed at shallower depths.

Nevertheless andwithout surprise, not all scenarios are conducive toplate tectonics. In particular, we examine the case when the planetdevelops thick plates when the first lithosphere is formed. If the platesare thick enough, the strength of the plate would increase due to asubstantial increase in the lithostatic confining pressure and this wouldprevent any sliding from occurring. On the other hand, if the plates areinitially formed thin—perhaps after the solidification of the magmaocean—the scenario for plate tectonics is favorable.

Through an energy balance, we find that rocky super-Earths thathave achieved plate tectonics, have thinner plates. Hence, they dissipateless energy during subduction, and can maintain plate tectonics easierthan on Earth.

Appendix A. Convection models

Simple models of the average properties of convecting systems havebeen derived on the basis of conservation of momentum and energy;these lead to parameterizations of the heat flow and temperature fromtheparameters that describe the system,primarily theRayleighnumber.These are well known and presented in textbooks (e.g. Turcotte andSchubert, 2002a). Although they are simple, and are sometimes referredto as “box models”, they can accurately capture the interdependence ofparameters and behaviour of convecting systems. In such models theproperties of variables such as temperature should be regarded asappropriate averages over a spatially variable temperaturefield, possiblyaveraged over time as well; as such the models accurately reflectconservation of momentum and energy, on which they are ultimately

based. Suchmodels have long been used to study the thermal evolutionof the Earth (e.g. McKenzie and Weiss, 1975; Davies, 1980; Conrad andHager, 1999b; Korenaga, 2008) and planets.

The governing equations

The Navier–Stokes equation follows from conservation of momen-tum. In dimensionless form it is

κρη

dυi

dt=

1Pr

dυi

dt= υi;jj +

ℓ3

κηð−p;i + fiÞ: ð11Þ

Here the length scale isℓ, the time scale is τ=ℓ2/κ, with κ=k/ρcp thethermal diffusivity, k thermal conductivity, ρ density, cp the heat capacityand η the viscosity. The velocity υi, xi and t are dimensionless, index

notation is used: υi;j≡∂υi

∂xj, and fi and p are the dimensional body force and

pressure. The Prandtl number Pr=η/ρκ is the ratio of momentumdiffusivity to thermal diffusivity, and is ~1024 for the Earth.

The body force is fi=ρgδi3 where gravity is in the x3 direction. Wecan subtract the mean hydrostatic pressure that satisfies p,i= ρg, withρ the horizontally averaged density, Eq. (11) becomes

1Pr

dυi

dt= υi;jj−p;i +

Δρgℓ3

κηδi3

Here p is now the dimensionless nonhydrostatic pressure, scaledusing the viscosity η, and Δρ is the density variation from thehydrostatic state. For thermal density variations Ti, Δρ=ραTi, whereα is the coefficient of thermal expansion. With reference temperatureT0, which we take as the scaling temperature, we then get

1Pr

dυi

dt= υi;jj−p;i +

ρgαT0ℓ3

κηTiT0

δi3

The combination

ρgαT0ℓ3

κη= Ra

is the Rayleigh number; this gives the ratio of buoyancy forces to viscousforces, and is the main parameter governing thermal convection.

For the Earth, with Pr~1024 the equation is then

0 = υi;jj−p;i + RaδT ð12Þ

where δT = TiT0

is the dimensionless temperature variations. Note thatthe Rayleigh number is the only parameter in the equation.

The dimensionless energy equation that governs heat transfer is

∂T∂t + υiT;i−

αgℓcp

TaT0

ð∂p∂t + υip;iÞ = T;ii +gℓcpT0

τ′ij ˙e ′ij +

ℓ2

kT0A ð13Þ

where we scale temperature, length and time as before with T0, ℓ andℓ2/κ, and scale pressure p and deviatoric stress τij′ with ρgℓ.

Using the Gruneisen parameter γ=(αKs)/(ρcp) this becomes

∂T∂t + υiT;i−

ρgℓγKs

TaT0

ð∂p∂t + υip;iÞ = T;ii +ρgℓγKsαT0

τ′ij ˙e ′ij +

ℓ2

kT0A ð14Þ

The Gruneisen ratio is usually of order unity for Earth materials, sothe terms with γ are small if the hydrostatic pressure ρgℓ is smallcompared to the adiabatic bulk modulus Ks. In this case the volumetriccompression of the material will be small and essentially incompres-sible (i.e. υi,i=0). Correspondingly, the effects of adiabatic tempera-ture changes and viscous heating will be small. These terms balanceeach other when integrated over the entire volume of the region

500 D. Valencia, R.J. O'Connell / Earth and Planetary Science Letters 286 (2009) 492–502

(Backus, 1975), i.e. they cannot contribute to the energy of the wholesystem, although they can appear locally.

Hence, for the incompressible case

∂T∂t + υiT;i = T;ii +

ℓ2

kT0A or

∂T∂t = ðT;i−υiTÞ;i +

ℓ2

kT0A: ð15Þ

The term T,i represents conductive heat flow, and the term υiTrepresents advective heat flow. The latter must vanish when thenormal velocity approaches zero at a boundary, near which heattransport will be dominated by conduction. Similarly, in the interior ofa convecting fluid, the temperature gradients will be small and heattransport dominated by convection.

The parameters

The only parameter in the dimensionless equations (12) and (15) isthe Rayleigh number and the heat sources, which also appears in theRayleigh number when they are present. Consequently the solution tothe equations will depend on the Rayleigh number, together withthose that describe the geometry of the boundary–value problem andthe boundary conditions. For the case of a layer with thickness D wetake the length scale ℓ=D. The horizontal scale of convection may beundetermined a priori, or constrained by lateral boundary conditions;it may be characterized as λD where λ is the aspect ratio. Con-sequently, any solution of the problemmust be, in dimensionless form,a function of the Rayleigh number and parameters such as λ describingthe geometry of the system.

We can distinguish three cases with different thermal cases andboundary conditions:

(1) Fixed temperature T=T0 at the bottom and T=0 at the top.In this case the scaling temperature is the temperaturedifference across the layer T0, and the Rayleigh number isRa=ρgαT0D3/κη.

(2) Fixedheatfluxq0 at the topwith afixed temperature at the top, andno heat flux at the top. In this case the scaling temperature can betaken to be T0=q0D/k, which is the temperature difference acrossthe layer for the conductive state. The Rayleigh number isRa=ρgαq0D4/kκη.

(3) Uniform heat sources A distributed in the layer with fixedtemperature at the top. The Rayleigh number can be taken asRa=ρgαAD5/kκη with T0=AD2/k, which is twice the conduc-tive temperature across the layer in the conductive state.Recognizing that the heat flux out of the layer is q0=AD, thisbecomes the same as case 2 above for the heat flux boundarycondition.

Simple convection models

Temperature boundary conditionsIn a uniform fluid layer of depth D the only parameter in the

dimensionless momentum and energy equations is the Rayleighnumber. For the case of fixed temperature boundary condition, theeffect of convection will be to increase the heat transport through thelayer above the value for the conductive state. The dimensionlessmeasure of this is the Nusselt number

Nu =q

qcond=

qkT0 =D

ð16Þ

where q is the heat flux. If there is no convection Nu=1, and convectionincreases its magnitude. For vigorous convection, when the Rayleighnumber is large, a thermal boundary layer forms at the upper and lowerboundaries where heat transport is by conduction. In the interior heattransport is dominated by convection, and the temperature is nearly

isothermal on average. The heat transport can be written q=kT0/δ0where δ0 is the thickness of an equivalent conductive boundary layer.

Nu =q

qcond=

qkT0 =D

=D2δ0

ð17Þ

This must be a function of the Rayleigh number and the aspectratio λ, and if these are separable so that Nu= f(Ra)a0(λ) then

Nu =q

qcond=

qKT0 =D

=D2δ0

= a0ðλÞRaRac

� �1=3ð18Þ

or

δ0 =RacηκραgT0

� �1=3 12a0ðλÞ

: ð19Þ

This states that the equivalent boundary layer thickness δ0 isindependent of the layer thickness D.

A direct consideration of the boundary layer by Howard (1966)came to the same result. He argued that the fluid next to the coldboundary would cool off by conduction and the temperaturewould beTðz; tÞ = Tierf ðz = 2

ffiffiffiffiffiκt

p Þ, where erf(x) is the error function and thetemperature in the interior of the layer Ti=T0/2. The heat flow at thesurface is qðtÞ = kTi =

ffiffiffiffiffiffiffiffiπκt

p, which is the heat flow for a conductive

boundary layer of thickness δðtÞ = ffiffiffiffiffiffiffiffiπκt

p. The cooled layer would

thicken until it became dynamically unstable and it would sink intothe interior to be replaced by hot fluid. The time is estimated as thatfor the boundary layer to become unstable to convection, or when thelayer thickness δ(t)=δc such that the Rayleigh number for theboundary layer Ra=ραgTiδc3/κη=Rac where Rac is the criticalRayleigh number for the onset of convection of a layer.

Assuming that the time for the detachment and replacement of thelayer is short compared to the cooling time, the average heat flow isq=2kTi/δc=2kTi/(D(Rac/Ra)1/3). We have taken Ti=T0/2 since aboundary layer also exists at the bottom of the convecting layer. Thedimensionless heat flow, the Nusselt number, is then

Nu =q

kT0 =D=

kTi = δ0kT0 =D

=D2δ0

=RaRac

� �1=3≈ a0

RaRac

� �1=3ð20Þ

with the equivalent conductive boundary layer thickness

δ0 =1a0

D2

RacRa

� �1=3:

This is the same relation as before, and again the boundary layerthickness is independent of the layer depth. This is appropriate, sinceδc is determined by a local stability criterion which should not beinfluenced by a distant boundary. Of course this arises from theassumption that the boundary layer exists, and by definition is thincompared to the layer depth. The analysis also ignores the manner ofdetachment of the boundary layer, which will be related to the patternand aspect ratio λ of convection.

Heat flux boundary conditionsIf the heat flux q0 through the layer is specified as a boundary

condition, the Rayleigh number is Ra=ρgαq0D4/(kκη). Since the heatflux is specified, the effect of convectionwill be to lower the temperaturedropacross the layerbelow that for conduction alone, anddetermine theinterior temperature Ti of the layer. The scaling relation used for thetemperature boundary condition case, or the boundary layer stabilitycondition (O'Connell and Hager, 1980) yield

Nu =q0D= k2Ti

=D2δ0

= a1RaRac

� �1=4ð21Þ

501D. Valencia, R.J. O'Connell / Earth and Planetary Science Letters 286 (2009) 492–502

where we have used the Nusselt number to represent the ratio of theinternal temperature Ti=q0δ0/k to that for the conductive state, and asbefore a1 should be of order 1. Again, the boundary layer thickness δ0 isindependent of the layer thicknessD, since its stability is determined bya local stability criterion. The Nusselt number reflects this.

If volumetric heat sources A drive convection, the Rayleigh numbercan be taken as Ra=ρgαAD5/(kκη) with T0=AD2/k, which is twice thetemperature difference across the layer in the conductive state. However,recognizing that the heat production per unit area is q0=AD, thisbecomes the same as the case for the heatflux boundary condition. This isconsistentwith the boundary layer stability condition, since the boundarylayerwill respond only to the heat flux through it, and the boundary layerthickness δ0 is independent of the depth of the layer.

It is interesting that the expressions for the Nusselt number havedifferent exponents dependingon theboundary conditions, even thoughthey are derived from the same physical considerations. The scalingreflects the independence of the boundary layer thickness on the layerdepth; the different exponents reflect thatD appears to different powersin the Rayleigh number. Although this resultmay appear anomalous, it isconfirmed by the more rigorous derivation of the Nu–Ra from energyconsiderations. And even if this relation is only approximate, so long asthe exponents are near these values the boundary layer thickness willdepend only weakly on the layer depth.

For models of mantle convection, plate tectonics defines theboundary layer. The uniform surface motion of the plates, as well asthe observed heat flow and bathymetry of the plates reflects aconductively cooling boundary layer that grows thicker with time. Thetime the layer is on the surface is related to the horizontal extent of theplate λD and the plate velocity v0. Using a conductive cooling model ofthe plate gives scaling for the velocities v0 corresponding to the scaledheat flow:

For constant temperature boundary conditions

u0≈a20λκD

RaRac

� �23

and for constant flux boundary conditions

u0≈a21λκD

RaRac

� �12:

These relations are kinematic, however, since they depend on themotion of the surface layer as it cools, and that the layer sinks when itsthickness reaches a critical value.

Boundary layer modelsThe dimensionless equation for conservation of momentum is

Re ρdυi

dt= Re ρð∂υi

∂t + υiυ;iÞ = τij;j + fi ð22Þ

where Re=(ρu0ℓ)/η is the Reynolds' number, u0 is a characteristicvelocity, and the body force fi=Ra(Ti/T0)δi3.

If the Reynolds' number is small, multiplying by vi and integratingover the volume yields

∫Vτ′ij e

′ijdV + ∫

Vυi fidV = 0 ð23Þ

when there is no work done on the boundaries of the region. Thus theviscouswork done on the system is balanced by thework done by sinkingor rising density heterogeneities. This relation is exact, and applies to thevolume averages of dissipation and vertical momentum (i.e. buoyancy)flux for any distribution of density heterogeneities in equilibrium.

When a temperature distribution corresponding to a sinking thermalboundary layer is used (e.g. McKenzie and Weiss, 1975; Turcotte andSchubert, 2002a)Eq. (23)gives the sameresults thatwereobtainedabovefor either the temperature or heat flux boundary conditions.We note that

the buoyancy flux vΔρg can be simply related to the advected heatυTiρcp,which constitutes most of the heat flux away from boundary layers:

υi fi = ðαg = cpÞqadv:

Models based on Eq. (23) allow the consideration of other sourcesof dissipation such as a strong lithosphere, which must be deformedduring subduction (Conrad and Hager, 1999b).

Even though Eq. (23) is based on the integral of buoyancy and fluxover the whole region, the scaling results are the same as those basedonly on the stability of the boundary layer, which leads to results(Eqs. (20) and (21)) that are independent of the overall depth of the layer.This reflects that the limiting process is the conduction of heat across thethermal boundary layer at the surface. In fact, for either flux or temper-ature boundary conditions, the boundary layer is characterized by

T4i

ðq=kÞ3 ≈κηRacραg

ð24Þ

with δ0=Ti/(q/k).If these results are applied to a systemwith an increase in viscosity

at a depth well below the boundary layer thickness, the result will notdepend directly on the deeper viscosity, but rather on the heat fluxdelivered to the top of the layer, or on the temperature maintained atgreat depth, depending on the type of boundary condition specified.

Appendix B

We examine the effect of viscosity in convective stress and adopt apower law fit η=η0(T/T0)−n to the Arrhenius law. Having that thepotential temperature is T = Ts +

qk, (with Ts the surface temperature),

viscosity depends on the surface temperature, heat flux and plate thick-ness, which in turn depends on the viscosity. We obtain a simple expres-sion for viscosity by removing its explicit dependence on plate thickness:

ηη0

=qδkT0

� �−n

1−nTskqδ

+ n2Tskqδ

−…

� �:

Replacing the value for δ from Eq. (4) yields,

η = η0αn=4an1T

n0k

3n=4

2nκn=4

!4= ðn + 4Þρgq3

� �n= ðn + 4Þ1−nTs

kqδ

+ n2 Tskqδ

� �2−…

� �4= ðn + 4Þ:

The first term on the right hand side is practically a constant. Toleading order, the viscosity has a dependence of

η∼ ρgq3

� �n= ðn + 4Þ: ð25Þ

We examinewhether or not there is a case inwhich viscosity effectsmight change the implicit dependence of shear stress with mass, as todecrease in biggerplanets.Wefind that as longas the viscosity decreaseswith temperature, nN0, the dependence of stress on mass is alwayspositive, Δτxz~Mψ, ψN0, where

ψ =ð2n + 4Þða + bÞ + cð4−2nÞ

2ðn + 4Þ :

Thus, the condition for shear stress to increase with mass is

n N 2ða + b

d+ 1Þ

a + bd

−1;

which is always met.

502 D. Valencia, R.J. O'Connell / Earth and Planetary Science Letters 286 (2009) 492–502

References

Afonso, J.C., Ranalli, G., Fernandez, M., 2007. Density structure and buoyancy of theoceanic lithosphere revisited. JRL 34, L10302.

Backus, G.E., 1975. Gross thermodynamics of heat engines in deep interior of Earth. Proc.Nat. Acad. Sci. 72, 1555–1558.

Becker, T.W., Faccena, C., O'Connell, R.J., Giardini, D., 1999. The development of slabs inthe upper mantle: insights from numerical and laboratory experiments. JGR 104,15207–15226.

Bird, P., 1998. Testing hypotheses on plate-driving mechanisms with global lithospheremodels including topography, thermal structure, and faults. JGR 103, 10115–10129.

Borde, P., Rouan, D., Leger, A., 2003. Exoplanet detection capability of the COROT spacemission. Astron. Astrophys. 405, 1137.

Borucki, W.J., Koch, D., Basri, G., Brown, T., Caldwell, D., Devore, E., Dunham, E., Gautier, T.,Geary, J., Gilliland,R., Gould,A., Howell, S., Jenkins, J., 2003. Keplermission: amission tofind earth-size planets in the habitable zone. In: Fridlund, M., Henning, T. (Eds.),Proceedings of the Conference onTowards Other Earths: DARWIN/TPF and the Searchfor Extrasolar Terrestrial Planets, ESA SP-539, pp. 69–81. ESA Publication Division,Noordwijk.

Conrad, c.P., Hager, B.H., 1999a. Effects of plate bending and fault strength at subductionzones on plate dynamics. JGR 104, 17551–17571 aug.

Conrad, c.P., Hager, B.H., 1999b. The thermal evolution of an earth with strong subductionzones. GRL 26, 3041–3044 oct.

Davaille, A., Jaupart, C., 1993. Transient high-Rayleigh-number thermal convection withlarge viscosity variations. J. Fluid Mech. 253, 141–166.

Davies, G., 1980. Thermal histories of convective earth models and constraints onradiogenic heat production in the earth. J. Geophys. Res. 85, 2517–2530.

Di Toro, G., Hirose, T., Nielsen, S., Pennacchioni, G., Shimamoto, T., 2006. Natural andexperimental evidence ofmelt lubrication of faults during earthquakes. Science 311,647–649.

Hirschmann, M.M., 2000. Mantle solidus: experimental constraints in the effects ofperidotite composition. G-cubed 1.

Howard, L., 1966. Convection at high Rayleigh numbers. In: Görtler, H. (Ed.), Proc. 11thInt. Cong. Appl. Mech. Munich, 1964. Springer, pp. 1109–1115.

Hynes, A., 2005. Buoyancy of the oceanic lithosphere and subduction initiation. Int. Geol.Rev. 47, 938–951.

Ida, S., Lin, D., 2004. Toward a deterministic model of planetary formation. I. A desert inthe mass and semimajor axis distributions of extrasolar planets. ApJ 604, 388–413.

Karato, S., Wu, P., 1993. Rheology of the upper mantle: a synthesis. Science 260, 771–778.Korenaga, J., 2006. Geophysical Monograph 164. AGU, Ch. Archean geodynamics and the

thermal evolution of Earths, pp. 7–32.Korenaga, J., 2008.Urey ratio and thestructure andevolutionof earth'smantle. Rev.Geophys.

4, RG2007. doi:10.1029/2007RG000, 241, 2008.Lenardic, A., Jellinek, A.M., Moresi, L.N., 2008. A climate induced transition in the

tectonic style of a terrestrial planet. EPSL 271, 34–42.McKenzie, D., Weiss, N., 1975. Speculations on the thermal and tectonic history of the

earth. Geophys. J. R. Astron. Soc. 42, 131–174.

Mian, Z.U., Tozer, D., 1990. Nowater, no plate tectonics: convective heat transfer and theplanetary surfaces of venus and earth. Terra Nova 2, 455–459.

Moresi, L., Solomatov, V., 1998. Mantle convection with a brittle lithospehre: thoughtson the global tectonics styles of the earth and venus. Geophys. J. Int. 133, 669–682.

Morrow, C., Moore, D., Lockner, D.A., 2000. The effect of mineral bond strength andadsorbed water on fault gouge frictional strength. GRL 27, 815–818.

Nimmo, F., McKenzie, D., 1998. Volcanism and tectonics on venus. Annu. Rev. EarthPlanet. Sci. 26 (1), 23–51.

O'Connell, R.J., Hager, B., 1980. On the thermal state of the earth. In: Dziewonski, A.M.,Boschi, E. (Eds.), Physics of the Earth's Interior. North Holland, pp. 270–317.

O'Neill, C., Lenardic, A., 2007. Geological consequences of super-sized earths. GRL 34,L19204 Oct.

O'Neill, C., Jellinek, A.M., Lenardic, A., 2007. Conditions for the onset of plate tectonics onterrestrial planets and moons. EPSL 261, 20–32.

Poudjom-Djomani, Y., O'Reilly, H., Griffin, S.Y., 2001. The density structure ofsubcontinental lithosphere through time. Earth Planet. Sci. Lett. 184, 605–621.

Ranalli, G., 1995. Rheology of the Earth. Chapman and Hall, Ch. Faults, pp. 90–116.Rucker, W.K., Bird, P., 2007. Basal shear-traction, not slab-pull, may be the dominant

plate-driving force. AGU Fall Meet Supl. (abstract) 88 (52).Seager, S., Kuchner, M., Hier-Majumder, C., Militzer, B., 2007. Mass-radius relationships

for solid exoplanets. ApJ 669, 1279–1297 Nov.Sleep, N.H., 2000. Evolution of the mode of convection within terrestrial planets. JGR

105, 17563–17568.Solomatov, V.S., 1995. Scaling of temperature- and stress-dependent viscosity convection.

Phys. Fluids 7, 266–274.Sotin, C., Grasset, O., Mocquet, A., 2007.Mass-Radius curve for extrasolar Earth-like planets

and ocean planets.Turcotte, D., Schubert, G., 2002a. Geodynamics. Ch. FluidMechanics. Cambridge University

Press, pp. 226–291.Turcotte, D., Schubert, G., 2002b. Geodynamics. Ch. Faulting. Cambridge University Press,

pp. 339–373.Valencia, D., O'Connell, R.J., Sasselov, D.D., 2006. Internal structure of massive terrestrial

planets. Icarus 181, 545–554.Valencia, D., Sasselov, D.D., O'Connell, R.J., 2007c. Detailed models of super-earths: how

well can we infer bulk properties? ApJ 665, 1413 Aug.Valencia, D., Sasselov, D.D., O'Connell, R.J., 2007a. Radius and structure models of the

first super-earth planet. ApJ 656, 545–551 Feb.Valencia, D., O'Connell, R.J., Sasselov, D.D., 2007b. Inevitability of plate tectonics on

super-earths. ApJL 670, L45–L48 Nov.Van Keken, P., Ballentine, C., Porcelli, D., 2001. A dynamical investigation of the heat and

helium imbalance. Earth Planet. Sci. Lett. 188, 421–434.Vinet, P., Rose, J., Ferrante, J., Smith, J., 1989. Universal features of the equation of state of

solids. J. Phys. Cond. Matter 1, 1941–1963.Weins, D.A., 2001. Seismological constraints on the mechanism of deep earthquakes:

temperature dependence of deep earthquake source properties. PEPI 127, 145–163.