Earth and Planetary Science Letters

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1 Earth and Planetary Science Letters 286 (2009) Contents lists available at ScienceDirect Earth and Planetary Science Letters journal homepage: Convection scaling and subduction on Earth and super-earths Diana Valencia, Richard J. O'Connell Department of Earth and Planetary Sciences, Harvard University, 20 Oxford St., Cambridge, MA 02138, USA article info abstract Article history: Received 5 July 2008 Received in revised form 20 June 2009 Accepted 7 July 2009 Available online 12 August 2009 Editor: T. Spohn Keywords: plate tectonics super-earths parameterized convection Super-Earths are the smallest class of discovered extra-solar planets. Owing to their relatively small mass, some might resemble Earth and perhaps be habitable. Because of the connection to habitability through thermal evolution, we investigate the tectonic regime of massive terrestrial planets. Two independent studies [O'Neill, C., Lenardic, A., Oct 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 reached opposing conclusions about the likelihood of plate tectonics. Here, we offer possible reasons for the discordant findings and address three key aspects of sustaining plate tectonics: deformation on faults, negative buoyancyand energy dissipation during subduction. We show that in general, the ratio of driving force to plate resistance increases with planetary mass. This is a consequence of increasing convective stresses, thinning plates and similar plate structure. We conclude that even though the strength of dry and wet of faults increases for massive terrestrial planets, the convective stress increases even more allowing deformation to take place. Also, despite shorter timescales for plate cooling, rocky super-earths achieve negative buoyancyat subduction zones. Finally, by investigating the effects of energy dissipation during subduction we find that massive terrestrial planets dissipate less 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, for sustaining plate tectonics 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 ices and voided of a massive hydrogen or helium envelope. There is no hard limit on planetary mass that separates the gaseous from the (mostly) solid planets, but a reasonable value may be in the vicinity of 10M (Ida and Lin, 2004). Two important characteristics make super-earths interesting objects to study: the ones that are rocky might be similar to Earth, and depending on their thermal state be habitable; and, the bias in detection ensures that larger Earth-like planets will be discovered before a true Earth analog. Furthermore, in the near future several dozen super-earths will be discovered with space missions underway CoRoT (Borde et al., 2003), Kepler (Borucki et al., 2003) and others under construction (Automated Planet Finder, JWST, etc.). To interpret the data that will be available, it is timely to set the framework for understanding the properties of these planets. In this study we address the relevant topic of tectonics in rocky massive planets. Corresponding author. Now at the Observatoire de la Cote d'azur, BP 4229, Nice Cedex 4, France. Tel.: ; fax: addresses: valencia@oca.eu (D. Valencia), oconnell@geophysics.harvard.edu (R.J. O'Connell). The thermal evolution of a planet is intrinsically related to the mode of convection, which can be in a state with plate tectonics or else with an immobile stagnant lid. There have been attempts to predict the mode of convection of rocky super-earths with opposite conclusions drawn. Valencia et al. (2007) show that terrestrial super-earths can exhibit plate tectonics, while O'Neill and Lenardic (2007) (hereonafter OL07) determine that super-sized Earths will most likely be in a stagnant lid regime and that only in some cases will they experience episodic plate tectonics. Plate tectonics is a complicated process and one that we only evidence on Earth. Despite the geological data available on Earth, the details of active lid tectonics are not completely understood. Nevertheless, the general features of plate tectonics have been laid out over the past 30 years and are well recognized. Determining how the onset of plate tectonics takes places is a challenge still unresolved. But once subduction has started, it is much easier to maintain because deformation can happen along pre-existing faults. Thus, we investigate if plate tectonics can be sustained in massive versions of Earth. We approach three questions: is the increase in fault strength larger than the convective stress as to hinder subduction (as suggested by O'Neill and Lenardic (2007))? Is negative buoyancy reached at subduction zones in rocky super-earths to ensure foundering? Is dissipation during subduction large enough to slow down plates to the point of halting plate tectonics? The discovery of super-earths has recently opened an opportunity for comparative planetology. It promises to widen the current geophysical X/$ see front matter 2009 Elsevier B.V. All rights reserved. doi: /j.epsl

2 D. Valencia, R.J. O'Connell / Earth and Planetary Science Letters 286 (2009) theories that have been developed to explain Earth's and the other terrestrial planets' properties. This places thinking about the solid planets in our solar system within a wider, more complete planetary context. We consider this study to be a step in that direction. 2. Different approaches The subject of plate tectonics on Earth has been approached in three different ways: with analytical theories, numerical modeling and experimental work, all complementary to one another. The approach by OL07 to study tectonics on massive Earth-like planets was to adapt the numerical model by Moresi and Solomatov (1998). This finite element code was developed to reproduce plate-like behaviour on Earth. It considered a temperature-dependent viscous mantle heated from below overlain by a lithosphere that deforms according to its stress regime. In the upper cold part, the lithosphere is brittle and may deform as long as the convective stress exceeds the yield stress. To circumvent the problem of having plate-like coherent structure while inducing subduction, they adopted a Non-Newtonian behaviour to a near-surface layer. This formalism reinforces zones of weakness that then can fail under a Byerlee criterion and produce trenches. Due to the well known limitations of modeling systems with large Rayleigh numbers (Ra), Moresi and Solomatov (1998) considered values in the range Ra= They successfully reproduce plate-like behaviour, episodic foundering and stagnant lid occurring as a function of increasing fault strength. In the regime of plate tectonics they reach an almost exact agreement with the theoretical prediction between the Nusselt number (Nu) and Ra from boundary layer theory. The conclusions drawn by OL07 rest on how the scaling to bigger planets from this model was done. They consider super-sized Earths that have the same density and conclude that larger planets (scaling ratio of R/R E N1) would require lower yield stresses to achieve at least episodic stages of plate tectonism. Thus, they attribute the prevalence of stagnant lid on rocky super-earths to the locking of faults caused by higher pressures from higher gravity values. Furthermore, Fig. 3 in their paper suggests that smaller planets than Earth can have mobile plates. In a personal communication with A. Lenardic, this puzzling conclusion was attributed to lower temperatures on smaller planets. The thermal age of the small planets they model is thus, older than their massive counterparts. Lower temperatures increase the viscosity, so that the deviatoric stress increases to aid subduction. This is the first clue to the different findings between the two different groups. On the other hand, the work we presented in Valencia et al. (2007a) used a parameterized analysis of convection in a system heated from within to predict the convective properties on massive terrestrial planets. We investigated Earth-like planets in the sense that they would have a similar composition: concentration of radioactive elements and Fe/Si ratio (expressed as similar core-mass fraction). We accounted for compression so that massive planets would have larger average densities and gravities that scale accordingly. The conclusion was that due to thinner plates and larger convective stresses, the conditions for subduction, would be more favorable on bigger planets. There are two basic assumptions behind our work, and that of OL07 and Moresi and Solomatov (1998): (1) that the stress needed to cause deformation is available from convection. This is justified by relating Earth'splatemotionstothetractionsonplates(Bird, 1998; Becker et al., 1999; Rucker and Bird, 2007). And (2) that a Byerlee criterion for failure in the brittle part of the lithosphere adequately captures the behaviour of large-scale faults that represent the collective behaviour of randomly oriented smaller faults. However, a key result that is different in our findings is that the pressure temperature structure of the plates remains almost invariant to mass. Having that the methods employed by the two groups do not differ in their core reasoning, we first setout to investigate if the increase in fault strength drastically outweighs the change in convective stress in massive Earth-like planets as suggested by (O'Neill and Lenardic, 2007). 3. Convective parameters We elaborate on the parameterized convection analysis to show the implicit dependence of the convective parameters on the mass of the planet (M). This is done via the Rayleigh number (Ra). In the case of a fluid layer of depth D that is internally heated, the Rayleigh number can be defined in terms of the surface heat flux q: Ra = ραgd4 q κηk where g is the gravity and the material properties are: the density ρ, coefficient of thermal expansion α, thermal diffusivity κ, thermalconductivity k, and viscosity η. These parameters are constant and unambiguously defined in the case of a homogenous fluid layer. On the other hand, in the case of a planetary mantle, the materialpropertiesaswellasgravity, are a function of temperature (T) and pressure (P) and therefore a function of depth. Given that we are investigating the conditions the plates are subject to, the relevant parameters in Eq. (1) are those in the vicinity of the plate. For a discussion on the applicability of boundary layer theory and our choice of boundary condition (that of heat flux) see Appendix A Shear stress In mantle convection, the deviatoric horizontal shear stress underneath the lithosphere depends on the velocity of the plate and the depth of the convective layer l, Δτ xz ~ η u plate l ; with l~d for whole mantle convection. The velocity of a plate is related to the Rayleigh number (O'Connell and Hager, 1980; Turcotte and Schubert, 2002a) u plate ea 2 1 κ Ra 1 = 2 λ; D Ra c where a 1 is a coefficient of order unity, λ is the ratio of width to depth of the convective cell (of order unity in the mantle), and Ra c ~O(1000) is the critical Ra for convection to occur. Thus, the deviatoric horizontal stress has no direct dependence on the depth of the mantle Δτ xz ~ κα 1 = 2 ðηρgqþ 1 = 2 : ð2þ krac Nevertheless, there is still an indirect dependence because an increase in planetary size due to an increase in mass is accompanied by an increase in heat content, gravity, average density and possibly other material properties. Moreover, in a scenario well described as a heated-from-within system, the stress depends on the heat flux, which is a temporal quantity that changes as the planet evolves and cools. In Section 4 we show how we evaluate super-earths' heat fluxes Boundary layer In a planet with plate tectonics, the lithosphere is the boundary layer. For planets with a stagnant lid, the colder and stronger part of the plate sits above the boundary layer that participates in the convection (Davaille and Jaupart, 1993; Solomatov, 1995). According to classical boundary layer theory, the boundary layer thickness is effectively independent of the depth of the convective layer (depth of mantle) but depends on the vigour of convection, Ra (O'Connell and Hager, 1980; Turcotte and Schubert, 2002a), δ = 1 D Ra s ð3þ a 1 2 Ra c ð1þ

3 494 D. Valencia, R.J. O'Connell / Earth and Planetary Science Letters 286 (2009) where s=1/4 when the boundary condition is the heat flux and s=1/3 when it is the temperature difference across the mantle. The more vigorous the convection, the thinner the boundary layer. In terms of the heat flux the thickness of the boundary layer or plate thickness is α 1 = 4 ρgq 1 = 4 δ ~ : ð4þ κkra c η Eqs. (2) and (4) show the dependence of deviatoric shear stress and plate thickness on the material properties of the convective fluid and the heat flux. To obtain the dependence on mass, we obtain the values of these fluid properties (density, gravity, viscosity) and the heat flux for super-earths. We note that α, κ and k might vary among terrestrial planets, but that the largest uncertainty comes from the viscosity which can change by orders of magnitude. In addition, while viscosity depends on pressure, temperature and stress (Karato and Wu, 1993), the dominant effect is temperature and thus, we consider a temperature-dependent viscous mantle. 4. Planetary heat flux A planet's surface heat flow Q, reflects the amount of heat being transported into the mantle from the core Q core, generated within from radioactive heat sources Q rh and its cooling rate at a given point in time dt(t)/dt: QðtÞ = ρcp 4πR3 dtðtþ + Q rh ðtþ + Q core ðtþ: ð5þ 3 dt The contribution from radioactive sources to the heat flow is Q rh ðtþ = R i γ i ðtþ = R i γ 0 i exp λ i ðt t 0 Þ; i i where R i is the amount of heat produced per mass by each of the major radioactive elements 238 U, 235 U, 232 Th, and 40 K, γ i is the amount of each radioactive element in the planet's mantle at a given time t, and λ i is their decay constant. 238U Provided that these isotopes have very long half-lifes (i.e. t 1/2 = Gy, t Th 1/2 =14 Gy), their concentration in the galaxy is not expected to vary wildly. Thus, it is reasonable to consider planets that have the same radioactive concentration as Earth, Q rh p ðtþ = Q rhðtþ M. This assumption can subsequently be relaxed to explore its effects on the results. M 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 thermal evolution model is required. Such a model would yield in principle the exact contribution of radioactive sources to total heat flow (the Urey ratio U). Instead, we limit our study to planets mostly dominated by radioactive heat production as in the case of Earth which are expected to have mass-scaled versions of total heat flow 4πR 2 M q p = Q p = Q : ð6þ 5. Scaling with mass M 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 plate for rocky super-earths. Since the details of the model have been explained elsewhere (Valencia et al., 2006, 2007b), we only briefly describe it here. It solves the differential equations of density, gravity, mass, pressure and temperature with depth. It uses a Vinet equation of state (Vinet et al., 1989) and a thermal profile that is conductive throughout the boundary layers and adiabatic in the convective interiors of the mantle and core. Through iteration in the Rayleigh number and plate thickness convergence is reached to yield a consistent thermal profile. It also includes all major phase transitions known to occur on Earth including post-perovskite. This model successfully reproduces Earth and the Terrestrial Planets' structure. The mass dependence of the properties relevant to this study is adequately expressed in a power law relation: ρ~m a, g~m b, η~m c, q~m d, where the exponents a, b, c and d are determined from the results of the internal structure model. The expressions for deviatoric horizontal shear stress and plate thickness depend on these exponents (from Eqs. (2) and (4)) ða + b + c + dþ = 2 Δτ xz ~ M ð7þ δ~ M ða + b + c dþ = 4 : ð8þ Even without the implementation of an internal structure model, qualitative results can be obtained from the scaling of the most simplified scenario: that of a family of isoviscous rocky super-earths scaled with constant density. In this simple case, a=c=0, and R~M 1/3, so that the heat flux q~m 1/3 and gravity scales as g=gm 1/3. The stress dependence on mass is then Δτ xz ~M 1/3 and δ~m 1/6. This elemental scaling shows that the ratio of shear stress to plate thickness increases with planetary mass. We investigate if this qualitative conclusion holds true in a general case. In particular, it is not known a-priori the effect of viscosity on shear stress. Even a modest increases in temperature, could significantly lower the tractions underneath the plates effectively decreasing the driving force behind subduction. We model the viscosity after Davies (1980) by using a power law fitto T n the Arrhenius law, so that η = η 0 where n=30 and η0 is the T 0 viscosity value at the reference temperature T 0. We are interested in the viscosity underneath the lithosphere, so that T is the potential temperature. The results from the internal structure model are that the planetary radius, average density, gravity, heat flux and viscosity have a dependence on mass of R~M 0.262, ρ~m 0.196, g~m 0.503, q~m 0.476, and η~m The viscosity decreases with mass, because the temperature underneath the plate slightly increases from 1566 K for Earth to 1631 K for a 10M -planet. Thus, the dependence of deviatoric stress and plate thickness is Δτ xz ~M 0.33 and δ~m While the exact values for the exponents may vary depending on the viscosity formalism adopted, the robust result is that the ratio of shear stress to plate thickness increases with mass. We explore the role of n in this qualitative result (Appendix B). We conclude that the condition for the shear stress to increase with mass a + b +1 d is satisfied as long as n N 2, which always holds true. In the a + b d 1 case of constant density scaling a=0,b=d,sothatanynn0 satisfies the inequality. For a general case, two conditions are met: 1) the heat flux roughly scales with gravity q~g, and 2) the density always increases with mass (an0). Therefore, the inequality holds true for any positive n, and even slightly negative values, which are perhaps unrealistic. The very modest increase in potential temperature can be easily seen from its relation to viscosity and heat flux (Eq. 65 of O'Connell and Hager, 1980) ðt T s Þ 4 η ðq=kþ3 κ 16ρgα Ra c; where T s is the surface temperature. By replacing the power law relation between viscosity and temperature, it is clear that the internal temperature weakly depends on heat flux, gravity and material properties, all quantities that vary with mass: T T 0 η 0 ðq=kþ 3 κ 16ρgα Ra c! 1 = ðn +4Þ : ð9þ

4 D. Valencia, R.J. O'Connell / Earth and Planetary Science Letters 286 (2009) Given that heat flux scales as gravity and both scale as ~M 1/2,the potential temperature (for constant density planets) scales as T~M 1/34 given n=30. Including the effects of compression on density would make T more insensitive to mass. In conclusion, in internally-heated mantles with same radioactive concentrations, shear stress underneath the plates increases with planetary mass. Furthermore, the ratio of shear stress to plate thickness increases for terrestrial massive planets undergoing plate tectonics. This results provides a possible future benchmark with the work 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 investigate whether or not it outweighs the increase in convective stresses as suggested by OL07. Pressure does increases the frictional strength on faults and hence, can potentially hinder deformation on large planets. We determine how the stress and the strength on the faults varies with planetary mass. For deformation to happen, the shear stress on the fault (τ) has to overcome its strength (τ rock ). In the same manner as (Moresi and Solomatov, 1998), we invoke a Byerlee criterion for deformation on the brittle part of the lithosphere. The Coulomb strength of rocks is expressed using Byerlee's law τ rock = S + μðσ λσ zz Þ where S is the internal cohesion parameter, σ is the normal stress on the fault, σ zz =ρgz is the lithostatic pressure, and λσ zz expresses the effect of water on reducing the fault's strength. The normal (σ) and shear (τ) components of the stress on the fault are σ = σ zz Δσ xxð1 +cos2θþ τ = 1 2 Δσ xx sin2θ where θ is the angle of faulting which is related to the friction coefficient μ by tan 2θ=1/μ (Turcotte and Schubert, 2002b), Δσ xx =L/ δδτ xz is the deviatoric shear stress available from convection, and L the length of the plate. Moresi and Solomatov (1998) did not consider the angle of faulting, however, it does not constitute a major difference in our treatments. In fact, by considering an angle of faulting our estimates of τ rock /τ are more conservative. We find that despite higher gravity surface values, rocky super-earths experience similar pressures under their lithospheres. This is because the increase in gravity is offset by a decrease in plate thickness, thereby producing a somewhat constant average pressure on the plate of different planets. 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 somewhat insensitive to mass as noted before, the P T plate structure is almost invariant to planetary mass. This result constitutes a major difference with OL07. We calculate the strength of a fault for the family of rocky super-earths and show how it compares to the fault shear stress in Fig.1. We take Earth's strength and stress values as the reference case, and scale them according to the equations described above. The general feature in Fig. 1 is that the strength of both dry (long-dashed lines)and wet(short-dashed lines)faults indeed increases with mass, as well as the shear stress experienced on the fault (solid lines). The former happens because of a small contribution of the convective stress on the fault's normal stress that hinders sliding. This is due to our choice of including the angle of faulting. Had we not, as is the case used in the model of OL07, the fault strength would have remained constant (i.e. constant yield stress) independent of planetary mass. The interesting result is that the increase in the fault's shear stress is steeper than the modest increase in fault strength, so that terrestrial Fig. 1. Faults' stress and strength. The shear stress (solid lines) on the fault increases linearly with mass, while the strength of a dry (long dashed lines) andwet(short dashed lines) fault increases more slowly. This behaviour is independent of choice of coefficient of friction We considered two friction coefficient values (top): μ=0.2 and (bottom) μ=0.8. The shaded region shows the range of values for earthquake stress release (Weins, 2001). planets can accommodate deformation and experience subduction. The reason for this is that the deviatoric horizontal normal stress (Δσ xx ) increases almost linearly with mass from an increase in deviatoric horizontal shear stress (Δτ xz ), and a decrease in plate thickness despite a modest increase in plate length. The following expression relates the ratio of deviatoric shear stress to plate thickness, and the ratio of driving force to plate resistance Driving Force PlateResistance = τ τ rock 1 = 2 Δτ L xzδ sin2θ S + μð1 λþ σ zz + 1 : 2 μδτ L xzδ ð1 + cos2θþ We find a positive relation between this ratio and planetary mass, which means that subduction and therefore plate tectonics can be sustained more easily on terrestrial super-earths. Fig. 3 shows the inverse of this ratio as being less than 1 and decreasing with increasing planetary mass (for the case of s=0.25). It is clear that the reason proposed by OL07 to argue that super-earths cannot exhibit plate tectonics stands in disagreement with our results. What lies at the core is that the plate structure they obtain is not invariant with mass. They claim that the pressure is higher in larger planets, and therefore the plate thickness they obtain must be larger than that predicted by classical boundary layer theory. We explore this in Section 8.1. We complete our investigation of the behaviour of faults in massive terrestrial planets by considering the uncertainty in the values of the friction coefficient and the fluid contribution to pore pressure (i.e. 0bλb 0.9 Ranalli, 1995). We compare planets that exhibit the same coefficient of friction justified in that we consider planets with the same composition. On a micro-scale, the friction coefficient of different minerals will depend on their structure (Morrow et al., 2000). However, by looking at planets that share the same rock composition, we circumvent the problem of addressing deformation on individual faults and focus on the collective behaviour that is likely to share the same average value of μ. Fig. 1 considers an effect of water/volatiles on pore pressure of λ=0.6. With a relatively high coefficient of friction of μ=0.8 (bottom), the fault shear stress on Earth required to exceed the strength of a wet fault is

5 496 D. Valencia, R.J. O'Connell / Earth and Planetary Science Letters 286 (2009) ~100 MPa (available if the deviatoric normal stress is Δσ xx =300 MPa). If we consider the coefficient of friction to be lower μ=0.2 (top), the shear stress required is ~20 MPa (available if Δσ xx =50 MPa). This low value of μ has been recently proposed as a better value for the effective friction coefficient on pre-existing faults (Di Toro et al., 2006), which is also more consistent with the values of stress release during earthquakes on Earth (shaded area in Fig. 1). We also examine the case where water is less effective at increasing pore pressure λ=0.2(andμ=0.8). This low value of λ requires shear stresses of ~200 MPa for Earth to exceed the strength of awetfault(availableifδσ xx =550 MPa). The deviatoric shear stress from convection has been modeled to be MPa (Becker et al., 1999), so such 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 shear stress on faults (black line), which describes its dependence on mass, is steeper than the curve for strength of wet and dry faults, so that at some point these curves are crossed. The transition between super- Earths that require water to the ones that do not, will depend on the efficiency of water to increase pore pressure on faults (i.e. λ) and the friction coefficient. Conversely, smaller planets' tectonic regime would be more affected by the presence or absence of water. 7. Lithospheric density At convergent margins on Earth, the oldest, coldest and densest plate is expected to subduct under the younger, hotter and buoyant plate. Knowing that super-earths have faster convective velocities (Valencia et al., 2007a) and hence, younger plates in general, it is appropriate to establish the conditions in which negative buoyancy can still be achieved at subduction zones. We calculate the mean density of a plate (ρ lit ) composed of basaltic crust (ρ bas =2880 kg/m 3 ) and lithospheric mantle (ρ man =3330 kg/m 3 ) at the maximum age of the plate (when its thickness is ~double the mean calculated thickness), ρ lit = 1 h c 0 h c ρ bas ½1+αðzÞðT m TðzÞÞŠdz 1 + ð2δ h c Þ 2δ ρ hc man ½1+αðzÞðT m TðzÞÞŠdz where h c is crustal thickness, the coefficient of thermal expansion is α(z)=a +b T(z)+ĉ/T(z) 2 with the values for â, b and ĉ determined by (Poudjom-Djomani et al., 2001), and T m is the temperature beneath the plate. We find that the mean lithospheric density will be denser for every Earth-like super-earth provided the crustal thickness does not exceed 16% of the total plate thickness during subduction (Fig. 2). Oceanic crustal thickness depends on the extent of melting, which depends on the depth at which the solidus intersects the thermal adiabatic profile of the planet. The expression for crustal thickness h c is h c = F max 2 ðz 0 z f Þ; where F max =r(p 0 P f )isthemaximumfractionofmelting,r is the melt production per unit of decompression; and z 0 and z f are the initial and final depths of decompression melting corresponding to pressures P 0 and P f respectively. For the solidus, we used the expression T sol =a P 2 +b P+c with the coefficients experimentally determined by Hirschmann (2000). The temperature beneath the plate and hence, at ridges varies little among rocky super-earths (for the same surface temperature). Owing to larger gravity values, the intersecting pressure is achieved at shallower depths, so that we anticipate the extent of melting to be reduced with increasing planetary mass. This suggests that the crust will remain under 16% of the plate's thickness. We perform a simple calculation to obtain the crustal thickness of each super-earth by obtaining the depth at which the magma solidus is intersected by an Fig. 2. Negative buoyancy in super-earths. Top: We show the relative density of the plate with respect to the underlying mantle 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 negatively buoyant. Bottom: We show the fraction of crust to the total lithospheric thickness at the time of subduction. The crustal thickness decreases with increasing planetary mass, while the lithosphere decreases even more, so that the fraction of crustal thickness increases for more massive planets. ascending parcel and the amount of maximum fractional melting. Fig. 2 shows the results. The conclusion is that as long as the change in melt fraction r with pressure is below 2.4% melt per kbar, buoyancy will be negative at subduction zones for terrestrial super-earths. The value of 2.4% melt per kbar is larger than the limits considered appropriate for Earth (1 2%), although unknown for other planets. Recent studies (Hynes, 2005; Afonso et al., 2007) show that on Earth, the lithospheric mean density is not larger than the underlying mantle's at subduction zones. However, Afonso et al. (2007) note that the plate's density is larger than the adiabatic mantle (or ridge column density) and Hynes (2005) shows that flooding can easily cause negative buoyancy. Furthermore, Becker et al. (1999) showed that compression could thicken the cold lithosphere to the point of inducing subduction. These ideas suggest that perhaps negative buoyancy before subduction might not play as 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 during subduction as a scenario that might halt plate tectonics. Conrad and Hager (1999a) recognized that a major source of dissipation can be the subduction of thick strong plates. If the energy required in subducting the plate is large, the plate will slow down, thicken even more making subduction more difficult, until eventually it can no longer be sustained. We investigate the effects of this process by modeling subduction in two ways: bending of the plate after Conrad and Hager (1999a), and continuous shearing of the plate into the mantle Plate bending Conrad and Hager (1999b, a) modeled the lithosphere as bending into the mantle at subduction zones (with some prescribed radius of curvature) and calculated the energy dissipated in this process. Through an energy balance they derived a relationship between Nu

6 D. Valencia, R.J. O'Connell / Earth and Planetary Science Letters 286 (2009) and Ra that would capture the effects of plate bending. They argued that the exponent s in the Nu Ra relation would be effectively decreased in the presence of strong dissipative subduction zones, when most of the energy is dissipated by lithospheric bending instead of in the shearing mantle. Conrad and Hager (1999b) described Earth's mantle as heated from below and proposed s to be By considering a system with a known surface heat flux and correspondingly using an exponent of s=1/4 (see Appendix A), we have shown that the plates of more massive terrestrial planets are thinner and thus, the effect of dissipation during subduction is small. This means that if the planet avoids having a thick lithosphere to begin with, it will be easier for the lithosphere to remain thin. The question that follows is to determine how thick the plates can be without causing too much energy dissipation during lithospheric bending, as to avoid continuously thickening into a stagnant lid. Another way to answer this question is to calculate the minimum value for s that would still allow for the strength of the fault to be less than the shear applied τ rock /τb1 so that deformation can still take place. To assess this, we considered different values of s and obtain the values for plate thickness, horizontal deviatoric shear and normal stresses on the fault and with these, calculate the fault's shear stress τ and strength τ rock as shown above. The results are shown in Fig. 3 for two different coefficients of friction μ=0.2 (top) and μ=0.8 (bottom). The nominal case of s=1/4 described in Section 6 shows how the fault's strength is more easily overcome by the stress on the fault as the mass of the planet increases. There are three regimes for deformation depending on the value of s:i)ifs is not too small (s 0.19) the ratio of strength to stress decreases with mass, suggesting that subduction would take place more easily in more massive planets, ii) if s is very small (s 0.13) this ratio increases dramatically to suggest that only Earth would have subduction, and iii) for intermediate values of s (0.13 sb0.19), the value of μ plays an important role. These results imply that as long as the relation of heat transport to the vigour of convection is not negligible (sn0.16), planets with larger masses could also experience subduction and hence, plate tectonics. Fig. 4. Cartoon of convection and subduction. Subduction is modeled as a continuous shearing of blocks that sink into the mantle at convergent margins. The velocity profile of the core of the shearing mantle is taken to vary linearly (following Turcotte and Schubert, 2002a). The potential energy is equated to balance the energy dissipated at shearing the core of the mantle and at subduction zones Subduction shear Another way to model energy consequences of subduction is to consider 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 the mantle. This process is continuous so that the work rate of this system per length along the subduction zone (l s )is w = f τ y δv 0 where τ y is the yield stress at which the block slides past the plate into the mantle, v 0 is the vertical velocity of the flow and fδ is the fraction of the plate that experiences this shear. The rest of the plate can deform as a viscous layer. In this formalism we assume that the stress at which the block yields is independent of depth. This assumption is partly supported by the fact that the stress release in shallow and deep earthquakes is estimated to be of the same order. It has been estimated to be of 3 10 MPa for shallow earthquakes and MPa for deep earthquakes (N300 km) (Weins, 2001). In terms of total energy per unit time consumed during subduction, Ẇ=fτ y δ(u 0 l s )D/L, which amounts to ~ W from τ y = Pa, δ=100 km, f =1/2, D=3000 km, λ=d/l~2, and with a plate production rate at mid-ocean ridges u 0 l s =3 km 2 /s. We follow the same energy analysis by Turcotte and Schubert (2002a), to balance the potential energy (Φ pe ) with the dissipation in the core of the shearing mantle (Φ man vd ) and in subduction zones sub ). The expression for each of the quantities per strike length is (Φ vd Φ pe = ρgαðt T s ÞDu 1 = 2 0 Φ man vd =4ηu 2 D 0 L Φ sub vd = f τ y δ L D u 0 λ κl 1 = 2 π where T is the temperature of the core of the mantle, L=u 0 δ 2 /(κπ)isthe length of the plate, u 0 is the horizontal velocity, and λ =λ 2 +λ 2. Equating the terms and using the definition of Ra in terms of the heat flux (Eq. (1)) yields Fig. 3. Strength stress ratio for different convection regimes. The values of s in the relation Nu ~ (Ra/Ra c ) s are shown for different coefficients of friction μ=0.2 (top) andμ=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 from lithosphere bending during subduction. Nu = 1 " # Ra 1 = 4 1+ f τ y δ 3 1 = 4 2 4π 2 λ 4πλ : ð10þ κη L It is clear that when τ y or f is zero, we recover the classical boundary layer result. Eq. (10) shows the form in which dissipation from subduction

7 498 D. Valencia, R.J. O'Connell / Earth and Planetary Science Letters 286 (2009) appears in the Nu Ra relation (e.g. Conrad and Hager (1999b)). The characterization of the effect as simply changing the exponent (Korenaga, 2006) obscures this. If we define ϕ = f τy δ 3 4πλ κη L, the result can be used to obtain the thickness of the plate δ, δ = δ 0 ½1+ϕðδÞŠ 1 = 4 ; where δ 0 is the plate thickness of a system with no dissipation during subduction. This is a quartic equation that expresses a correction to plate thickness from dissipation of energy during subduction. It is clear that the effect of subduction of a strong plate is to increase its size (ϕn0). This happens because the plate slows down (Nu decreases with increasing δ), has more time to cool and thicken. This equation also captures the fact that there is a positive feedback, the thickening effect of subduction is more pronounced when the plates are thick (ϕ increases with δ). The range of the correction term is 0.2bϕb2 on Earth, calculated from the values given above and the range in τ y, κ=10 6 m 2 /s, and η=10 21 Pa s. This yields a corrected plate thickness on Earth of 1.005δ 0 bδb1.05δ 0 at most 5%. It is obvious from this result that rocky super-earths with their thinner plates have a very small correction term and hence, their subduction zones are only weakly dissipative. 9. Discussion Plate tectonics is an evolutionary phenomenon and the conditions for it will change as the planet evolves and cools. At present, Earth has enough heat flux to drive vigorous convection needed for an active plate state. A planet's heat flux varies with time and while on short timescales it might increase (Sleep, 2000; Van Keken et al., 2001), over long timescales it decreases as the planet cools. This means, that at some stage, when the heat flux falls below a threshold, plate tectonics will cease to operate on a planet. The timing for this event depends on the thermal evolution. Although, extra sources of heat, such as tidal heating, can aid in achieving and prolonging the lifetime of active-lid tectonics. We find that planets of the same mass would have the same potential to sustain plate tectonics as dictated by their ratio of shear stress to plate thickness. Venus is not the counter example because of two important differences with Earth: the lack of water (Mian and Tozer, 1990; Nimmo and 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 only other objects in the solar system that might have had plate tectonics in the past by being in an episodic regime Differences between the two models Through a personal communication with A. Lenardic, we learned that their choice of scaling yields smaller planets to be colder than the bigger counterparts. Lower interior temperatures translate into higher viscosities and larger driving stresses, which can drive subduction more easily. Thus, they conclude that small planets can achieve plate tectonics while rocky super-earths cannot. There are two important differences in our treatment and that of OL07. We consider (1) scalings that take into account the compression effects 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 not adequate in comparing Earth to massive Earth-like planets. Independent models (Valencia et al., 2006, 2007c,b; Sotin et al., 2007; Seager et al., 2007) clearly show that the relation between radius and mass deviates from a constant density scaling. The different models, which vary in sophistication and treatment agree that the radius scales as R=R (M/M ) Because of possible unknown phase transitions and uncertainties in the equation of state, the exponents are an upper limit. Thus, a convenient approximation for the radius scaling is R~M 1/4, which translates to gravity scaling as g~m 1/2.Werecommend these scalings for comparing Earth to massive Earth analogs. As for the more fundamental difference, the main question is how to best apply a (quasi) steady state boundary condition to an evolving planet. Heat comes from radioactive sources in the interior; this suggest scaling for constant heat sources, which turns out to be the same as that for imposed heat flux. In an internally heated mantle, the interior temperature adjusts to conform with the imposed fixed heat flux (which is the boundary condition). In a system with a fixed temperature difference driving convection, the heat flux adjusts. Given the uncertainties in all such models, the most accurate choice is unclear. In a mixed-system, the results might depend on which case is dominant (basal to heating ratio). In a system that is heated-from-below that is, with a fixed temperature difference (ΔT) across its convective layer and constant surface temperature, the deviatoric stresses are expected to increase with the size of the planet, due to an increase in gravity and density xz = ηðtþ 1 = 3 κ 1 = 3 ραgδt 2 = 3 a R c : Δτ basal If, however, the viscosity decreases, convection stresses would decrease as well. In the case of internally heated planets, we obtain a very slight increase in temperature drop within the boundary layer for massive planets (see Eq. (9)). This qualitatively agrees with the treatment of OL07, in that larger planets are hotter. However, the subsequent effect of lower viscosities is further 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 xz = ηðtþ 1 = 2 κραgq 1 = 2: kr a c Δτ internal Therefore, the differences in the results may be coming from the different boundary conditions. Our model considers a steady state scenario with heat coming from radioactive sources. It predicts that for the same concentration, bigger planets will have larger convective stresses and thinner lithospheres a favorable scenario for plate tectonics. In addition, smaller planets are expected to cool more rapidly because of their area to volume ratio. Being that the ability to sustain plate tectonics decreases as the planet cools, small planets will cease to have mobile lids faster than massive rocky ones Treatment for viscosity A natural question that arises in the treatment of terrestrial massive planets, is the effect of pressure in convection and specifically on viscosity. The pressure at the core mantle boundary of massive Earth-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 is effectively a function of depth, lateral variations are comparatively minor except in plates, which have a different rheology anyway. The limits on convection come from the transfer of heat through the boundary layer. This is the (or at least one) basis for the derivation of the parameterized convection models used. These seem to work reasonably quantitatively for the Earth with a viscosity of ~10 20 Pa s for the upper mantle, even though the lithosphere is not characterized well by such a rheology, and that the lower mantle viscosity is higher (~ Pa s). Therefore, we argue that if the temperature and pressure regime in the uppermost mantle of a terrestrial super-earth is similar to the Earth, it should behave similarly. Consequently, the

8 D. Valencia, R.J. O'Connell / Earth and Planetary Science Letters 286 (2009) pressure effects in the lowermost mantle of rocky super-earths can be considered a secondary effect. In any case, we think this to be a first step towards understanding convection and tectonics in massive planets and consider that pressure 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 when determining its thermal state, which in turn is fundamental to the question of habitability. Here, we examine from classical boundary layer theory, how the convective parameters and fault properties scale for terrestrial super-earths. We address three key issues of plate tectonics: deformation on faults, negative buoyancy, and energy dissipation during subduction. Even though the mantle is a complicated system (with chemical heterogeneities, chaotic dynamics, complex rheology, etc.), simple models like ours are useful in illuminating relations between important parameters. Moreover, to assess the effect of mass on the characteristics of planets, it is vital to appropriately scale the relevant properties with mass. For this we use a detailed internal structure model that renders simple and robust relations: heat flux scales with gravity and radius scales as R~M 1/4. One of the main results is that convective stresses increase, plate thickness decreases, while the pressure temperature structure of the plate is almost the same for massive planets with the same radioactive heat concentration. These qualitative results differ from the findings of OL07. Consequently, fault strength modestly increases with planetary mass, while the shear stress that can cause sliding increases even more, such that it exceeds the strength of wet and even dry faults of massive planets. Deformation is accommodated more easily in bigger Earth-like planets. Conversely, smaller planets require weakening agents, like water, much more critically if deformation on faults is to be sustained and subduction to occur. Furthermore, by determining the crustal thickness in massive planets, we determine that they reach negative buoyancy at subduction zones. Crustal thickness decreases because, due to increasing gravity values, the solidus of terrestrial super-earths is crossed at shallower depths. Nevertheless and without surprise, not all scenarios are conducive to plate tectonics. In particular, we examine the case when the planet develops thick plates when the first lithosphere is formed. If the plates are thick enough, the strength of the plate would increase due to a substantial increase in the lithostatic confining pressure and this would prevent any sliding from occurring. On the other hand, if the plates are initially formed thin perhaps after the solidification of the magma ocean the scenario for plate tectonics is favorable. Through an energy balance, we find that rocky super-earths that have achieved plate tectonics, have thinner plates. Hence, they dissipate less energy during subduction, and can maintain plate tectonics easier than on Earth. Appendix A. Convection models Simple models of the average properties of convecting systems have been derived on the basis of conservation of momentum and energy; these lead to parameterizations of the heat flow and temperature from the parameters that describe the system, primarily the Rayleigh number. These are well known and presented in textbooks (e.g. Turcotte and Schubert, 2002a). Although they are simple, and are sometimes referred to as box models, they can accurately capture the interdependence of parameters and behaviour of convecting systems. In such models the properties of variables such as temperature should be regarded as appropriate averages over a spatially variable temperature field, possibly averaged over time as well; as such the models accurately reflect conservation of momentum and energy, on which they are ultimately based. Such models have long been used to study the thermal evolution of the Earth (e.g. McKenzie and Weiss, 1975; Davies, 1980; Conrad and Hager, 1999b; Korenaga, 2008) and planets. The governing equations The Navier Stokes equation follows from conservation of momentum. In dimensionless form it is κρ dυ i = 1 dυ i = υ η dt Pr dt i;jj + l3 κη ð p ;i + f i Þ: ð11þ Here the length scale is l,thetimescaleisτ=l 2 /κ,withκ=k/ρc p the thermal diffusivity, k thermal conductivity, ρ density, c p the heat capacity and η the viscosity. The velocity υ i, x i and t are dimensionless, index notation is used: υ i;j υ i,andf i and p are the dimensional body force and x j pressure. The Prandtl number Pr=η/ρκ is the ratio of momentum diffusivity to thermal diffusivity, and is ~10 24 for the Earth. The body force is f i =ρgδ i3 where gravity is in the x 3 direction. We can subtract the mean hydrostatic pressure that satisfies p,i =ρ g, with ρ the horizontally averaged density, Eq. (11) becomes 1 dυ i Pr dt = υ i;jj p ;i + Δρgl3 κη δ i3 Here p is now the dimensionless nonhydrostatic pressure, scaled using the viscosity η, and Δρ is the density variation from the hydrostatic state. For thermal density variations T i, Δρ=ραT i, where α is the coefficient of thermal expansion. With reference temperature T 0, which we take as the scaling temperature, we then get 1 dυ i Pr dt = υ i;jj p ;i + ρgαt 0l 3 The combination ρgαt 0 l 3 = Ra κη κη T i T 0 δ i3 is the Rayleigh number; this gives the ratio of buoyancy forces to viscous forces, and is the main parameter governing thermal convection. For the Earth, with Pr~10 24 the equation is then 0=υ i;jj p ;i + RaδT ð12þ where δt = T i T 0 is the dimensionless temperature variations. Note that the Rayleigh number is the only parameter in the equation. The dimensionless energy equation that governs heat transfer is T t + υ it ;i αgl T a ð p T 0 t + υ ip ;i Þ = T ;ii + gl τ c p T ij 0 c p e ij + l2 kt 0 A ð13þ where we scale temperature, length and time as before with T 0, l and l 2 /κ, and scale pressure p and deviatoric stress τ ij with ρgl. Using the Gruneisen parameter γ=(αk s )/(ρc p ) this becomes T t + υ it ;i ρglγ T a ð p T 0 t + υ ip ;i Þ = T ;ii + ρglγ τ K s αt ij 0 K s e ij + l2 kt 0 A ð14þ The Gruneisen ratio is usually of order unity for Earth materials, so the terms with γ are small if the hydrostatic pressure ρgl is small compared to the adiabatic bulk modulus K s. In this case the volumetric compression of the material will be small and essentially incompressible (i.e. υ i,i =0). Correspondingly, the effects of adiabatic temperature changes and viscous heating will be small. These terms balance each other when integrated over the entire volume of the region