The influence of mantle internal heating on lithospheric mobility: Implications for super-Earths

Earth and Planetary Science Letters ] (]]]]) ]]]–]]]

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The influence of mantle internal heating on lithospheric mobility: Implications for super-Earths C. Stein a,n, J.P. Lowman b,c, U. Hansen a a

Institut f¨ ur Geophysik, Westf¨ alische Wilhelms-Universit¨ at M¨ unster, Corrensstr. 24, M¨ unster, Germany Department of Physical and Environmental Sciences, University of Toronto Scarborough, 1265 Military Trail, Toronto, ON, Canada M1C 1A4 c Department of Physics, University of Toronto, 60 St. George St., Toronto, ON, Canada M5S 1A7 b

a r t i c l e i n f o

abstract

Article history: Received 20 September 2011 Received in revised form 4 November 2012 Accepted 5 November 2012 Editor: T. Spohn

Super-Earths, a recently discovered class of exoplanets, have been inferred to be of a similar rock and metal composition to the Earth. As a result, the possibility that they are characterised by the presence of plate tectonics has been widely debated. However, as the super-Earths have higher masses than Earth, it is assumed that they will also have higher Rayleigh numbers and non-dimensional heating rates. Accordingly, we conduct a systematic 2D study to investigate the influence of these parameters on the surface behaviour of mantle convection. The main focus of our work considers the response of surface motion to the mantle’s internal heating. However, we also include an analysis of other parameters scaling with planet mass, such as viscosity. In agreement with the findings of Valencia and O’Connell (2009) and van Heck and Tackley (2011) we find plate-like surface mobilisation for increased Rayleigh numbers. But increasing the internal heating leads to the formation of a strong stagnant-lid because the mantle heating effects thermally activated viscosity. Additionally, viscosity is affected by the increased pressures and temperatures of super-Earths. In total, our findings indicate that surface mobility will likely be reduced on super-sized Earths. Our numerical models show that the interior temperature of the convecting system is of vital importance. In planets with a hotter interior plate tectonics is less likely. & 2012 Elsevier B.V. All rights reserved.

Keywords: mantle convection internal heating stagnant-lid convection super-Earths

1. Introduction The Earth is the only planet in our solar system featuring plate tectonics, however, the number of terrestrial planet candidates for plate tectonics appears to be steadily increasing. As a result of advancements in astronomical observations (including transit surveys by the space missions CoRoT, Kepler and Gaia and the implementation of improved techniques for determining radial velocity using ground-based telescopes), the list of recently detected extrasolar planets orbiting their parent star continues to increase. Mostly, these newly discovered exoplanets have masses comparable to Jupiter’s. However, due to improvements in the mass detection limit, the number of planets confirmed to have masses close to that of the Earth is also increasing. Very recently, Doppler spectroscopy was used to detect GJ581e (Mayor et al., 2009), an exoplanet with a mass of only 1.9 Earth masses ðM Þ. Members of this new class of relatively small extrasolar planets, with masses of 1210 M  , are considered to be rocky in

n

Corresponding author. E-mail address: [email protected] (C. Stein).

nature (Rivera et al., 2005; Valencia et al., 2006; Mayor et al., 2009). These ‘super-Earths’ have therefore attracted considerable interest (cf. Haghighipour, 2011) because they might have a surface behaviour resembling the Earth’s (e.g., Valencia et al., 2007b). The interior structure of a super-Earth has been inferred from various mass-radius relationship analyses that assume equations of state (EOS) based on knowledge of the Earth’s structure (Valencia et al., 2006; Sotin et al., 2007; Seager et al., 2007). However, some uncertainties exist when extrapolating the EOS in the range of pressures and temperatures required in super-sized planets (Seager et al., 2007). Moreover, the size of a planet affects the parameters determining the vigour and mode of mantle convection. Hence, when compared with the Earth, the surface expression of mantle convection in a super-Earth might be affected in such a way that a change occurs in lithosphere mobilisation, from plate-like surface behaviour to a stagnant-lid mode. Crucial to the question which tectonic regime prevails is the balance of forces acting on the plates. On the one hand there is resistance of the plates to deformation (e.g., the thickness and viscosity of the plate) and on the other hand there is convective

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Please cite this article as: Stein, C., et al., The influence of mantle internal heating on lithospheric mobility: Implications for superEarths. Earth and Planetary Science Letters (2012), http://dx.doi.org/10.1016/j.epsl.2012.11.011

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traction at the base of the lithosphere (cf. Valencia et al., 2007b; O’Neill and Lenardic, 2007; van Heck and Tackley, 2011). The shear stresses acting on the base of the plate are strongly related to the tectonic regime. Thus internal heating plays an essential role but also other aspects such as variable viscosity or the presence of water affect the tectonic regime and therefore the balance of resistive to convective stresses (Moresi and Solomatov, 1998). Many of the relevant parameters scale directly with the planet’s size but often only single aspects have been considered in previous studies, and these were shown to partly have an opposing effect on lithospheric mobility. Consequently, controversial results have been presented (Valencia et al., 2006, 2007b; O’Neill and Lenardic, 2007; Valencia and O’Connell, 2009; Korenaga, 2010; van Heck and Tackley, 2011). O’Neill and Lenardic (2007) have numerically shown that an increased radius leads to a stagnant-lid mode of convection as the resistive strength of the lithosphere increases more strongly than the convective stresses. Their result is based on scaling of the yield stress. Assuming Earth’s dimensional yield stress, they scale the non-dimensional yield stress to higher masses. As a consequence the convective stresses are no longer sufficient to overcome the high non-dimensional yield stress and thus the resistive strength prevails as dominant. Considering scaling laws, Valencia et al. (2006, 2007a,b) and Valencia and O’Connell (2009) reason that increased masses will result in higher Rayleigh numbers (therefore higher convective stresses) and argue for lithosphere mobilisation on super-Earths. Based on simple relationships for plate thickness and convective stresses Valencia et al. (2007b) and Valencia and O’Connell (2009) scale these to larger mass planets and find that plate thickness (resistive strength) is reduced and thus easily overcome by the increased convective stresses. A potential problem with their result is that their parameterisations are derived from the boundary-layer theory which holds for the plate tectonic regime but differs in the stagnant-lid regime as viscosity changes (Moresi and Solomatov, 1998). Similarly, Korenaga (2010) adopts the scaling laws of the plate tectonic regime in strongly temperature-dependent viscosity with brittle failure. He argues that the resistive force of the plate not only increases with planet size (as assumed by O’Neill and Lenardic, 2007) but also scales with the thickness of the plate. Thus, increasing a planet’s mass (i.e., increasing the Rayleigh number) decreases plate thickness, which had not been considered in O’Neill and Lenardic (2007). Further Korenaga (2010) discusses the effect of water which more strongly controls the likelihood of plate tectonics than the planet’s size and states that the highconvective stresses (as also found by Valencia et al., 2007b) will only lead to plate motion on a wet planet. Finally, van Heck and Tackley (2011) combine analytical scalings and numerical models by directly comparing both their results. Also, they do not scale either Rayleigh number or yield stress with planet size but change both parameters simultaneously. In particular, they considered the scaling of the ratio of convective to yield stress with mass in purely internally heated and purely basally heated convection. For basally heated convection with a friction coefficient (depthdependent yield stress) they emphasise Korenaga’s (2010) result that planet size does not matter for a constant density scaling. Apart from this, their finding that convective stresses outweigh resistive forces agrees with Valencia et al. (2007b) and Valencia and O’Connell (2009). Additionally, van Heck and Tackley (2011) observe that plate tectonics is more likely in purely basally heated convection than in purely internally heated convection. As pointed out by van Heck and Tackley (2011) the likelihood of plate tectonics on more massive planets depends on the heating mode. While the authors compared the two end-member cases of purely internal heating and purely basal heating, a combination of

both heating modes seems more likely. In so-called mixed-heated convection both internal heat sources, owing to the decay of radioactive elements, as well as basal heating from the core play a role. In the past, not only a difference between purely basally heated and purely internally heated convection has been reported by van Heck and Tackley (2011). Also, the prediction of plate tectonics on superEarths by Valencia and O’Connell (2009) and van Heck and Tackley (2011), obtained in purely internally heated systems, seems to contradict the observation of Stein et al. (2004) who investigated a numerical mantle convection model featuring mixed-mode heating and found that stagnant-lid convection occurs for systems with higher internal heating rates. Given this background, we focus on the effect of the mantle internal heating with regard to lithospheric mobility. We compare purely internal heating results (cf. Valencia and O’Connell, 2009; van Heck and Tackley, 2011) with those obtained in mixed-mode heating convection. In addition, we are interested in the combination of different parameters. For example, we are interested in the combination of high Rayleigh numbers and strong internal heating (cf. van Heck and Tackley, 2011). Both increase with mantle depth but have an opposing effect on the surface behaviour. Fig. 1a and b show schematic regime diagrams of the Rayleigh number and heating rates versus yield stress, respectively, showing that high Rayleigh numbers promote surface mobility while strong internal heating leads to a transition from plate mobilisation to stagnant-lid convection (Stein et al., 2004). Thus, while high Rayleigh numbers favour surface mobility, it remains undetermined whether this still applies at high internal heating. Additionally, Fig. 1c and d show schematic regime diagrams of the pressure-dependent and temperature-dependent viscosity contrast versus yield stress, respectively. Like high Rayleigh numbers, a strong pressure dependence of the viscosity enables plate tectonics, whereas the viscous resistance of the plates (i.e., plate viscosity) is higher for a strong temperature dependence hindering plate tectonics (Stein et al., 2004). Thus, all convection and rheological parameters that will likely change with planet mass affect surface mobility. For super-Earths the interplay of these parameters is important. As the mode of heating was shown (Stein et al., 2004; van Heck and Tackley, 2011) to be of great importance we arrange our study in terms of the internal heating. First we analyse the general effect of the individual non-dimensional convection and rheology parameters on the lithospheric mobility. We start by studying the effect of an increasing internal heating rate. Then we systematically add a further complexity to the system and study the surface behaviour resulting from this new parameter with respect to the heating rate. Implementing this approach we first identify the physical mechanisms that cause transitions between mobile-lid and stagnant-lid convection. Finally, we discuss parameter values for super-Earths and present calculations featuring these parameter values.

2. Method To reduce the dependence of our findings on modelling methods we employ two distinct model approaches. Each solves for thermally driven convection in an incompressible Boussinesq fluid with infinite Prandtl number and variable viscosity. The 2D Cartesian geometry experiments were carried out in a square box employing free-slip, impermeable boundaries and reflective side-wall conditions for the temperature. The material is cooled from the top (T¼0) and heated (T¼1) at the bottom. The nondimensional equations describing thermal convection with internal

Please cite this article as: Stein, C., et al., The influence of mantle internal heating on lithospheric mobility: Implications for superEarths. Earth and Planetary Science Letters (2012), http://dx.doi.org/10.1016/j.epsl.2012.11.011

increasing yield stress

C. Stein et al. / Earth and Planetary Science Letters ] (]]]]) ]]]–]]]

Stagnant−Lid Convection

Surface Mobilisation

Stagnant−Lid Convection

Surface Mobilisation

increasing rate of internal heating

increasing yield stress

increasing Rayleigh number

3

Stagnant−Lid Convection

Surface Mobilisation increasing pressure−dep. viscosity

Stagnant−Lid Convection

Surface Mobilisation

increasing temperature−dep. viscosity

Fig. 1. Schematic regime diagrams adapted from Stein et al. (2004) showing that an increase in Rayleigh number and pressure-dependent viscosity leads to a mobilisation of the surface while increasing internal heating rates and temperature-dependent viscosity lead to stagnant-lid convection.

heating rates, H, are @T RaH 2 ¼ H, þ v  =T= T ¼ Ra @t

ð1Þ

=  v ¼ 0,

ð2Þ

=p þ =  r þRaTez ¼ 0,

ð3Þ

where T is temperature, t is time, v is velocity, p is pressure, r ¼ ½Zðrv þ ðrvÞt Þ is the stress tensor and ez is the unity vector in the vertical direction. The (Benard-)Rayleigh number Ra ¼

ag r0 DTd3 kZ

ð4Þ

r0 Qd2 kDT

:

ð7Þ

where DZT is the viscosity contrast due to temperature, DZp is a measure of the pressure dependence and z is height, with z ¼0 at the base of the system. To allow for plate mobilisation, in Model 1 the viscosity is additionally a function of the stress so that

s ZS ¼ Zn þ _Y E

ð8Þ

Zeff ¼ 2=ðZ1 þ Z1 S Þ: ð5Þ

are defined using a, the thermal expansivity; g, the gravitational acceleration; r0 , the reference density; d, the depth of the convecting layer; k ¼ k=r0 cp , the thermal diffusivity; k, the conductivity; cp, the specific heat at constant pressure; DT the superadiabatic temperature difference between the top and bottom; Q, the rate of internal heat generation per unit volume and Z, the surface viscosity. In the case of convection with variable viscosity Rayleigh numbers are no longer unique but vary with the viscosity over the model domain (cf. Schubert et al., 2001). The Rayleigh numbers defined in the interior, for example, are not a priori known due to the lack of knowledge of the resulting temperature field. However, the Rayleigh numbers at the surface or the bottom of the model domain are clearly defined in the case of isothermal boundary conditions. Knowing the viscosity contrast over the model domain, we can compute Rabot and RaHbot . (To improve readability, we omit the index bot in the following when using the Rayleigh number due to internal heating, RaH, defined at the bottom of the model box.) Following from Eqs. (1), (4) and (5) the non-dimensional heating rate reads: H¼

ð1zÞ Z ¼ DZT , T  DZp

and the effective viscosity (cf. Schubert et al., 2001) is

and the Rayleigh number due to internal heating

ag r20 Qd5 RaH ¼ kkZ

The viscosity is variable and depends on temperature and pressure (i.e., on depth due to the application of the Boussinesq approximation, Christensen, 1984) in the form of

ð6Þ

ð9Þ

3

Values of 10 for the plastic viscosity, Zn , and 3 for the yield stress, sY , have been chosen (E_ is the second invariant of the strain-rate tensor) so that the stagnant lid is readily breakable (cf. Stein et al., 2004). For dimensional values see Section 3.3. The equations in Model 1 are solved employing the numerical methods described by Trompert and Hansen (1996): a finite volume method is employed for spatial discretisation and the Crank–Nicolson method is used for temporal discretisation. The algebraic equations are solved iteratively with a multigrid method and SIMPLER as the smoother. In Model 2 dynamic plates are incorporated by specifying time-dependent surface motion as a boundary condition. Applying the force-balance method (cf. Gable et al., 1991), we specify that the shear stress at the base of the plates sums to zero at all times by balancing the stresses associated with viscous resistance against the convective stresses. Thus, we ensure that plates neither drive nor resist the convection as plate velocities are consistent with the buoyancy distribution, meaning that there is no outside force driving the plates. In this study, we have used a rheology-dependent force-balance method, where the base of the plate is determined as the depth where a linear fit of the top temperature profile (associated with the thermal boundary layer) crosses the interior temperature (cf. Moresi and Solomatov, 1995).

Please cite this article as: Stein, C., et al., The influence of mantle internal heating on lithospheric mobility: Implications for superEarths. Earth and Planetary Science Letters (2012), http://dx.doi.org/10.1016/j.epsl.2012.11.011

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Moreover, the plate viscosity is not specified but instead follows the applied viscosity law (Eq. (7)). In contrast to Model 1, Model 2 only features vertical variations in the viscosity (i.e., in Eq. (7) the horizontally averaged temperature /TS is implemented). Model 2 is a hybrid code that uses an explicit finite-difference method to solve the temperature equation and a spectral formulation to solve for the velocities (Gable et al., 1991). In both models we have used an aspect ratio 1 domain with grid sizes of 128  128 or 128  250 for the higher Rayleigh numbers (i.e., results presented from Fig. 6 onwards).

model 1 300 250 0

Δ ηp

4

5 0 −5

200

100

3. Results

0

3.1. Model behaviour comparison

10

20

30

H model 2 300 250 0

Δ ηp

We first provide a comparison of the different plate-modelling methods. Fig. 2 shows regime diagrams spanned by the pressuredependent viscosity DZp and the non-dimensional heating rate H, for Model 1 (Fig. 2a) and Model 2 (Fig. 2b). The parameters for these calculations are a thermal viscosity contrast of DZT ¼ 105 and a top Rayleigh number of Ra¼ 400. For both models we observe three distinct flow regimes distinguished by considering the surface velocity and the mobility (ratio of surface to volume averaged rms velocity; Tackley, 2000). We define stagnant-lid convection as cases featuring surface velocities lower than 1 and mobilities lower than 0.01. For mobilities larger than 0.01 but lower than 0.5 we observe convection with a sluggish lid or transitional behaviour in which the systems show short plate mobilisation phases (cf. Solomatov, 1995; Trompert and Hansen, 1998b; Loddoch et al., 2006) or erratic surface mobilisation with occasional flow reversals (cf. Lowman et al., 2001, 2003; Koglin et al., 2005). For mobilities close to 1 and high surface velocities, which are uniform over the plate area and fall to zero at the plate boundaries (cf. Fig. 2 at H¼3), we find mobile-lid convection. The colour temperature field snapshots show examples of mobile-lid convection (for DZp ¼30 and H¼3) where the cold topmost layer (in blue) sinks and effectively cools the system interior. The temperature field snapshot examples of stagnant-lid convection (for DZp ¼30 and H¼20) show the relatively hot interior in red and the cold undeformed plate in blue. The differences in the interior temperatures of both models (indicated in the colour temperature field plots) are due to the implementation of the geotherm /TS in Eq. (7) of Model 2 and the chosen yield stress in Model 1. The results in Tables 1 and 2 show that these modifications change the mean temperature. The different mean temperatures affect the temperaturedependent viscosity and the resulting surface behaviour. For example, for Model 2 the mobile-lid convection case has a higher interior temperature than Model 1 (cf. also Table 1). Consequently, in the low DZp range where temperature effects dominate, the transition from mobile-lid to stagnant-lid convection occurs at lower H for Model 2. Besides temperature, the choice of the yield stress plays a role in determining the surface behaviour. In Table 2 we compare the results of Model 1 (with sY ¼ 325) and Model 2. While in the mobile-lid regime (parameter set with a high pressure dependence, DZp ) we find that a yield stress between 3 and 4 will give the best agreement to Model 2 results, a yield stress lower than 3 seems more appropriate in the stagnant-lid regime (parameter set with a low pressure dependence). Despite these differences, our findings indicate that the modelling methods both give the same tendencies for the regime transitions in surface mobility with increasing heating rates, H (i.e., a change from mobile-lid to stagnant-lid convection). Fig. 2 also presents the effect of the pressure-dependent viscosity.

0

5 0 −5

200

100

0 0

10

20

30

H Fig. 2. Regime diagrams spanned by the pressure dependence of the viscosity and the internal heating rate for (a) Model 1 and (b) Model 2. Open circles represent the mobile-lid regime and filled circles the stagnant-lid regime. The triangles mark a region of transitional behaviour. The colour temperature fields (with corresponding surface velocity profiles above) show examples of mobile-lid and stagnant-lid convection, where red and blue represent warm and cold material, respectively. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this article.)

Table 1 Comparison of Nusselt number, velocity, mean temperature, mobility M ¼ vsurf =vrms and flow regime for Models 1 and 2. The use of the horizontally averaged temperature /TS in Eq. (7), which we tested in Model 1, affects the system values, explaining deviations between Model 1 and Model 2. Nu

v rms

T mean

M

Regime

0.74 1.05 0.79

0.89 1.6  10  3 0.88

Mobile Stagnant Mobile

1.22 1.26 1.17

0.89 4.6  10  5 1.6  10  3

Stagnant Stagnant Stagnant

DZT ¼ 105 , DZp ¼300, Ra¼ 400, H¼ 10 Model 1 Model 1, /TS Model 2

14.66 8.25 12.80

131.38 296.05 163.60

DZT ¼ 105 , DZp ¼30, Ra¼ 400, H ¼ 20 Model 1 Model 1, /TS Model 2

16.55 16.31 16.81

1639.55 1742.90 1970.42

As expected from a previous study (Stein et al., 2004), the pressure dependence promotes mobility which is manifested in the widened mobile-lid regime for a strong pressure dependence (cf. Fig. 2). However, at high heating rates, the highest pressure dependence examined does not overcome the thermal effect and we find stagnant-lid convection in both models.

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Table 2 Comparison of Nusselt number, velocity, mean temperature, mobility and flow regime for different yield stresses in Model 1. For obtaining similar flow regimes, Model 1 results with sY ¼ 3 provide the best match of Model 2 results. For stagnant-lid convection, the system values are also in good agreement. v rms

T mean

M

12.80 14.66 10.43 9.92

163.60 131.38 180.46 196.48

40

60

model 2

Regime

1e3

DZT ¼ 105 , DZp ¼ 300, Ra ¼400, H ¼10 Model 2 sY ¼ 3 sY ¼ 4 sY ¼ 5

H 20

0

1e4

0.79 0.74 0.96 0.99

0.88 0.89 0.21 0.14

Mobile Mobile Transitional Stagnant

1.17 1.22 1.23

1.6  10  3 5.9  10  4 3.0  10  4

Stagnant Stagnant Stagnant

velocity

Nu

-20 1e5

5

100 10

DZT ¼ 105 , DZp ¼ 30, Ra ¼400, H ¼ 20 Model 2

sY ¼ 3 sY ¼ 4

16.81 16.55 16.50

1970.42 1639.55 1575.43

1 0.1

In the following section we will first provide a qualitative study of the regime transitions using only Model 2. Here, the observed transition values have to be considered as a transition zone rather than a sharp transition point because several aspects affect the value (such as initial conditions, yield stress, form of temperature dependence or aspect ratio). As we only give rough estimates in Section 3.2, we will also give quantitative results in Section 3.3 by applying the parameter ranges for super-Earths to the models. 3.2. Parameter study For super-Earths, a number of convection and rheology parameters are assumed to be higher than for Earth due to the increased size of the planets. (For a more detailed discussion on the parameter range we refer to Section 3.3.) Here, we study the general effect of the single parameters (in non-dimensional units) with respect to the surface behaviour in more detail by considering the velocity (cf. Valencia et al., 2007b). Additionally, we analyse the surface velocity as this is more representative of the surface mobilisation in comparison to the mean velocity. We begin with the heating rate H (or, correspondingly, the Rayleigh number due to internal heating, RaH). In order to find the transition from mobile-lid to stagnant-lid surface behaviour, we model heating rates varying over a wide range. As for some cases we only observe stagnant-lid convection at heating rates slightly higher than zero, we also implement negative heating rates, i.e., cooling rates, to obtain the full range of flow regimes (mobile-lid and stagnant-lid convection) for all cases. In this way we can directly compare how a certain input parameter changes the critical heating rate, Hcrit , at which the transition from mobileto stagnant-lid convection occurs. In Fig. 3, while keeping the Benard–Rayleigh number fixed, with Ra ¼400, we plot the timeaverages of the volume-averaged velocity, vrms (open symbols) and the averaged surface velocity, vsurf (filled symbols) versus the heating rate H and the Rayleigh number due to internal heating, RaH (defined at the bottom of the model domain). Here, we consider the results of Model 2 for the pressure dependence of DZp ¼ 30. We obtain similar behaviour for all pressure dependencies and the same tendencies for Model 1. In agreement with Valencia and O’Connell (2009) we find an increase in volume-averaged rms-velocity with increasing internal heating Rayleigh numbers, suggesting that the more vigorous convection will lead to surface mobilisation. However, the surface velocity decreases upon reaching a critical value (marked with a dashed vertical line). Beyond this value (H¼3) the ratio of vsurf to vrms decreases and the surface becomes immobile. The reason for the observed behaviour is that the convective stress tpZi v (cf. Valencia and O’Connell, 2009; van Heck and Tackley, 2011) not only

0

3.9e7 RaH vrms

7.8e7

vsurf

Fig. 3. Volume-averaged and surface rms-velocity as function of the Rayleigh number due to internal heating (defined at the bottom) for systems with a varying heating rate H. The parameters are Ra ¼ 400 (Rabot ¼ 1.3  106), a temperaturedependent viscosity of DZT ¼ 105 and a pressure-dependent viscosity of DZp ¼ 30. All results are obtained from Model 2. The dashed vertical line marks the transition from plate mobilisation to stagnant-lid convection at H¼ 3.

depends on the velocity but also on the interior viscosity which dramatically decreases. For example, in Model 1 for H¼ 5 we obtain v¼32.23 and Zi ¼ 0:16 giving a convective stress of 5.16 which outweighs the resisting yield stress of 3; for H¼40 the convective stress reduces to 0.043 (v¼4725.51 and Zi ¼ 0:92  105 ) so that the resisting stress dominates. To verify that the observed splitting in velocities is not a result of the mixed-mode heating (which distinguishes our calculations from that of Valencia and O’Connell, 2009), we examine a similar set of calculations where we consider the effect of an insulating bottom boundary (Fig. 4). For these models, which feature purely internal heating, we also observe a splitting of the velocities. However, in this case it occurs at the higher heating rate H of 5.6. Note that for this set of models we have considered purely internal heating, so that unlike the other models, we have no hot bottom boundary layer. Accordingly, the bottom temperature in purely internally heated cases is not fixed but is instead lower than in the mixed-mode case (for H¼5.6 the non-dimensional bottom temperature is about 0.4.). Therefore, the non-dimensional temperature difference between the top and bottom, which is used for scaling, is no longer one and requires a posteriori adjusting. Due to the lower temperature in purely internally heated systems, we find that the transition to stagnant-lid convection occurs at higher heating rates, H (cf. discussion on thermal structure below). Likewise the critical Rayleigh number is higher than in mixed-mode heating, which is indicated in Fig. 4 by the low rms-velocities for the lowest Rayleigh numbers examined. For these slightly overcritical systems we observe a dramatic shift in velocities over a small H range (not shown in other figures). To conclude, as in Valencia and O’Connell (2009) we observe an increase in system velocity for both heating modes. From this, however, it cannot be simply concluded that the convective stresses also increase. In fact, in our cases the viscosity decreases as the temperature increases due to stronger heating. This will reduce the convective stresses and results in stagnant-lid convection, as can be seen in the reduced surface velocity and mobility in our figures. A further difference between our work and the work in previous studies (Valencia and O’Connell, 2009; van Heck and Tackley, 2011) is that we are keeping the Rayleigh

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-20 1e5

Rabot

H 20

0

40

60

3e4 1e3

model 2

3e5

3e6

3e7

3e6

3e7

model 2

1e4 100 velocity

velocity

1e3 100

10

10 1

1 0.1

0

3.9e7 RaH vrms

7.8e7

0.1 3e4

3e5 RaH

vsurf

vrms

Fig. 4. Volume-averaged and surface rms-velocity as function of the Rayleigh number due to internal heating for systems with a varying heating rate H. Note, that these models consider purely internal heating (i.e., a vanishing bottom heat flux) for the model parameters Ra¼ 400, DZT ¼ 105 and DZp ¼ 30. The dashed vertical line marks the transition from plate mobilisation to stagnant-lid convection (at H¼ 5.6).

vsurf Rabot

3e4 1e3

3e5

3e6

3e7

3e6 RaH

3e7

3e8

model 2

number, that enters the momentum equation, constant while only increasing the heating rate H. Increasing the Rayleigh number while keeping the heating rate constant leads to a different result. In Fig. 5 we plot the rms- and surface velocities versus the bottom Benard–Rayleigh number, Rabot, and the Rayleigh number due to internal heating, RaH, for H¼1. As in Figs. 3 and 4, in Fig. 5a and b we show the results for mixed-mode heated systems and purely internally heated systems, respectively. Independent of the heating mode we find that convective stresses increase with the Rayleigh number due to internal heating, RaH, which can be seen in the increasing surface velocity and therefore higher mobility. The increasing Rayleigh number leads to a higher heat loss and a cooling of the system, so that the viscosity contrast is reduced and mobility promoted. These results resemble the results of the previous studies in which the heating rate and Rayleigh number have been increased simultaneously (i.e., where the effect of Rayleigh number outweighs that 2 of the heating rate). As the heating rate H scales to d =DT but the 3 Rayleigh number scales to g DTd , in super-Earths the increase in Rayleigh number will be stronger than that of the heating rate. In Section 3.3 we will use the consistent balance of Rayleigh number and internal heating rate for two example super-Earths. Before discussing the thermal structure further, we will first consider the general effect of increasing the Rayleigh number, Ra. As the Rayleigh number will increase more strongly than the heating rate, we here increase the Rayleigh number by two orders of magnitude to Ra ¼4  104 (Rabot ¼1.3  108) (Fig. 6) compared to the mixed-mode heated models shown in Fig. 3. At this higher Rayleigh number we find that splitting of the surface and global rms velocities is still obtained. However, as expected, the higher convective stresses allow for more mobility and this moves the transition to stagnant-lid convection to higher values of the heating rate H. Here we find that increasing the Rayleigh number by two orders of magnitude increases the critical heating rate (at which the transition to stagnant-lid convection occurs) by about one order of magnitude. For heating rates larger than 18, the higher Rayleigh number of Ra ¼4  104 is not sufficient to overcome the thermal effect and break the stagnant lid. This suggests

velocity

100

10

1

0.1 3e4

3e5 vrms

3e8

vsurf

Fig. 5. Velocities for models with a heating rate of H¼ 1, a temperature-dependent viscosity contrast of DZT ¼ 105 , a pressure-dependent viscosity of DZp ¼ 30 and varying Rayleigh numbers for (a) mixed-mode heating and (b) purely internal heating.

that plate tectonics on super-Earths is more likely, as proposed by Valencia and O’Connell (2009) and van Heck and Tackley (2011). For the case of purely basal heating (H¼0) and a Rayleigh number of Rabot ¼8  108, which van Heck and Tackley (2011) give as a value for a 2 M  planet, we can estimate from our results that plate tectonics indeed occurs. However, for the heated case with H¼40 our results with mixed-mode heating will lead to stagnantlid convection because of the higher interior temperatures which reduce interior viscosity compared to purely internal heating. This indicates that the heating mode has some influence on the surface behaviour (cf. also van Heck and Tackley, 2011). As a next step, we add viscosity changes to the system. Compared to Earth parameters (which are themselves not well constrained), the details of super-Earth rheology are even less constrained and require some supposition. For example, although the gross composition of super-Earths is unknown, it can be inferred from Earth structure models that core temperature and pressure will be higher in super-sized planets (e.g., Sotin et al., 2007). Consequently, we also assume that temperature- and

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C. Stein et al. / Earth and Planetary Science Letters ] (]]]]) ]]]–]]]

-20 1e5

H 20

0

40

60

-20 1e5

1e4

1e4

1e3

1e3 velocity

velocity

model 2

100

10

1

1

0

3.9e9

0.1

7.8e9

H 20

0

0

3.9e9

RaH vrms

40

60

model 2

100

10

0.1

7

7.8e9

RaH vrms

vsurf

Fig. 6. Volume-averaged and surface rms-velocity as function of the Rayleigh number due to internal heating (defined at the bottom) for systems with a varying heating rate H. We consider mixed-mode heating with the parameters as in Fig. 3 (DZT ¼ 105 and DZp ¼ 30) but with a Rayleigh number of Ra ¼4  104 (i.e., Rabot ¼1.3  108). The transition to stagnant-lid convection occurs at H¼ 18 (see dashed vertical line).

H 20

0

40

60

model 2

1e4 1e3 velocity

pressure-dependent viscosity contrasts will be higher. O’Neill and Lenardic (2007) have already discussed the effect of a higher mass on the resistive strength of the lithosphere (stress-dependent viscosity). Their observation is in agreement with the observations of Moresi and Solomatov (1998), Tackley (2000) and Stein et al. (2004) that an increased yield stress leads to stagnant-lid convection as the increased resistive strength outweighs the convective stresses. We will therefore omit this behaviour here and only consider the effect of the temperature- and pressuredependent viscosity contrasts. In Fig. 7a we display the velocities as a function of the heating for the increased temperature-dependence of DZT ¼ 107 . This is an increase of two orders of magnitude compared to the models in Fig. 6 (the pressure dependence of the viscosity and the bottom Rayleigh number remain the same). For higher viscosity contrasts the surface more easily decouples from the interior (Moresi and Solomatov, 1995; Stein et al., 2004) as the viscosity of the convecting interior decreases (cf. Stein and Hansen, 2008). This reduces the convective stresses which can more easily be outbalanced by the resisting stresses. Therefore, we observe that the transition to stagnant-lid convection (i.e., where surface velocity decreases while rms-velocity increases with heating rates) appears for lower H. In fact, in Fig. 7a the transition occurs for a negative heating rate (i.e., cooling rate) of H¼  10. In Fig. 7b we consider the effect of an increase in the pressuredependent viscosity by almost one order of magnitude (to DZp ¼ 200). This change again promotes surface mobility (Stein and Hansen, 2008) so that the transition to stagnant-lid convection occurs at higher H-values (H¼2) than found in Fig. 7a with the lower pressure dependence. A reason for the promotion in mobility will be that viscosity increases with depth (while we find that the system velocity does not change much between Fig. 7a and b), accordingly convective stresses will be higher with a stronger pressure dependence and can more easily overcome the resisting stresses. Finally, in Fig. 8 we present a set of results with further increased viscosity contrasts where DZT ¼ 109 and DZp ¼ 500 (the bottom Rayleigh number remains Rabot ¼1.3  108). In this case, the splitting of the velocities and the transition to stagnant-

-20 1e5

vsurf

100 10 1 0.1

0

3.9e9

7.8e9

RaH vrms

vsurf

Fig. 7. Velocities as function of the internal heating for models with the same bottom Rayleigh number (Rabot ¼1.3  108) as in Fig. 6. In contrast to Fig. 6, in (a) the temperature-dependent viscosity contrast is increased to DZT ¼ 107 and in (b) the pressure-dependent viscosity contrast has also been increased to DZp ¼ 200.

lid convection occurs for negative values of the heating rate (H¼  4.7). For this final set of calculations we analyse the thermal structure to determine an explanation for the observed behaviour. In Fig. 9 we present colour temperature field snapshots, where red to blue represent warm to cold material, respectively. The full range of flow regimes is represented by examining cases with heating rates varying from H¼  10 and H¼5. For H¼  10 (Fig. 9a) we see a typical mobile-lid convection case. The topmost layer strongly deforms and sinks into the interior leading to effective cooling of the interior. For H¼ 5 (Fig. 9b) the deformation of the cold surface layer is diminished and the cold slab no longer reaches the bottom of the model domain. For H¼0 (Fig. 9c) we obtain stagnant-lid convection. The coldest layer is not deformed and no longer cools the interior. Increasing the heating rate further (H¼5 in Fig. 9d) leads to a thinning of the surface plate and a heating-up of the interior. In Fig. 10 we present the temperature-depth profiles for the last set of calculations featuring a high Rayleigh number, strong

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C. Stein et al. / Earth and Planetary Science Letters ] (]]]]) ]]]–]]]

H 20

0

1 40

60

non-dim. height

-20 1e5

model 2 1e4

velocity

1e3

0.8 0.6 0.4 0.2 0

100

0

0.8

1

H = -10 H = -5 H = -3 H=0 H = 10

10 1 0.1

0.2 0.4 0.6 non-dim. temperature

0

3.9e9

7.8e9

RaH vrms

vsurf

Fig. 8. Surface and volume-averaged rms-velocity as function of the heating rate H (and Rayleigh number due to internal heating, RaH, defined at the bottom) for models with the parameters: Rabot ¼ 1.3  108, DZT ¼ 109 and DZp ¼ 500. The transition to stagnant-lid convection is marked by the dashed vertical line at H¼  4.7.

Fig. 10. Temperature–depth profiles of the models featuring Rabot ¼ 1.3  108, a temperature-dependent viscosity of DZT ¼ 109 and a pressure-dependent viscosity of DZp ¼ 500 (cf. Figs. 8 and 9).

Table 3 Critical values of the heating rate, Hcrit , at which a transition from mobile-lid to stagnant-lid convection approximately occurs for the model calculations featuring different Rayleigh numbers and viscosity contrasts (cf. Figs. 3 and 6–8). Case

Rabot

DZT

DZp

Hcrit

1 2 3 4 5

1.3  106 1.3  108 1.3  108 1.3  108 1.3  108

105 105 107 107 109

30 30 30 200 500

3 18  10 2  4.7

0

1.1

Fig. 9. Snapshots of the colour temperature field for models with (a) H¼  10, (b) H ¼  5, (c) H¼ 0 and (d) H ¼5. The other parameters are unchanged from Fig. 8 (Rabot ¼1.3  108, DZT ¼ 109 and DZp ¼ 500). Reds represent warm material and blues cold. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this article.)

viscosity contrasts and varying heating rates. As expected, we observe an increase in mean temperature as the effective heating rate increases (from about 0.4 to 0.85 for H¼ 10 to H¼10, respectively), the lid becomes thinner and we obtain a larger temperature drop over the top boundary layer. According to the temperature-dependence in Eq. (7) this directly converts into a strong viscosity drop over the surface which increases plate resistance but reduces the convective stresses in the low viscosity interior. In contrast to the temperature-dependent viscosity, the pressure dependence acts in the lower part of the model domain and less strongly affects the viscosity drop over the top boundary layer (Stein and Hansen, 2008). Similarly, the effect of the Rayleigh number on the interior temperature was shown to be fairly small in temperature-dependent viscosity convection with basal heating (Trompert and Hansen, 1998a). However, the interior temperature decreases with increasing internal-heating Rayleigh number in purely internally heated systems with H¼1

due to the higher heat loss at the surface (Turcotte and Schubert, 2002). In general, we find that the internal temperature and therefore the heating rate plays a critical role for surface dynamics. Table 3 summarises our findings for Hcrit, the value of H separating the mobile and stagnant-lid regimes, in the suites of calculations shown in Figs. 3, 6–8. Solely increasing the Rayleigh number (case 1 and case 2) enhances mobility so that the transition to stagnantlid convection occurs at higher H. A stronger temperature dependence of the viscosity, however, dramatically reduces the critical H-value (case 2 and case 3). For an increase in pressure dependence, which again promotes mobility, we again observe an increase in the critical H-value (case 4). In combination, we find that high Rayleigh numbers and viscosity contrasts (case 5) yield a critical H-value that is so low that any heating rate (i.e., positive H) leads to stagnant-lid convection.

3.3. Parameter ranges for super-Earths The precise values of the heating rate, Rayleigh number and viscosity contrasts for super-Earths are not known but can be extrapolated from our knowledge of the Earth. In this section we will discuss models with parameter sets relevant to the Earth and two example super-Earths. Following the work of Sotin et al. (2007) and Valencia et al. (2006) we assume that super-Earth material properties are Earth-like. Consequently, the factor by which the non-dimensional heating rate of a super-Earth differs 2 from the Earth is proportional to the change in d =DT, and 3 similarly the Rayleigh number change is proportional to g DTd . The mantle thickness d of a super-Earth can be derived by the mass-radius relationship RpM 0:27 (von Bloh et al., 2007) and the assumption that the proportions of the super-Earths are similar to

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those of Earth, so that for a super-Earth of 5M (or 10M  ) d is 1:54d (or 1:86d , respectively). The temperature difference DT depends on poorly known parameters (e.g., the vigour of convection) and is thus difficult to estimate (Leger et al., 2011). The thermal profile of larger planets is commonly determined using Earth’s thermal evolution models (e.g., Sotin et al., 2007; Papuc and Davies, 2008; Nettelmann et al., 2011; Tachinami et al., 2011). However, these depend on surface and initial temperature. It is assumed that larger masses will lead to a larger accretion energy but the precise value is highly unknown and also the surface temperature of a super-Earth can strongly vary between dayside (2500 K for CoRoT-7b) and nightside (50–75 K for CoRoT-7b) (Leger et al., 2011). In addition, a common assumption for thermal history models of the mantle is that plate tectonics cools the interior (e.g., Papuc and Davies, 2008). If, however, stagnant-lid convection prevails on a planet, the interior temperatures will be higher. Also, Kaltenegger et al. (2011) argue that the effective temperatures will be higher considering the recent results of solar metallicity of some M dwarfs with planets. (For example, the metallicity of Gl581 is [Fe/H]¼  0.1 (Johnson and Apps, 2009) which means that the planet is slightly metal-poor.) We take the estimated temperatures from Papuc and Davies (2008) where the ratio of superadiabatic lower mantle temperature of a 5M  ð10M  Þ super-Earth compared to the Earth gives a temperature difference across the mantle DT (with the same surface temperature as on Earth) of about 1:2DT  ð1:5DT  Þ. As gravitational acceleration follows g  M 0:5 (Valencia and O’Connell, 2009), a heating rate of H  1:9H (  2:3H ) and a Rayleigh number of Ra  9:8Ra (  30:5Ra ) results for a superEarth of 5M  (or 10M ). Additionally, the higher temperatures (as well as pressures) in super-sized planets affect the viscosity. We define the depthdependent viscosity contrast, DZp , and the temperature-dependent viscosity contrast, DZT , as the contrast between top and bottom viscosity:

DZT ¼

DZp ¼

Ztop T

,

ð10Þ

Zbottom p : Ztop p

ð11Þ

Zbottom T

The top viscosity in the models is set to 1 and the bottom viscosity can be derived from the viscosity contrast for Earth (Eq. (7)) as we assume that the mantle material in super-Earths is Earth-like:

Zbottom ¼ DZTDT , T

ð12Þ

Zbottom ¼ DZdp : p

ð13Þ

From Eqs. (10)–(13), we therefore find a thermal viscosity contrast of DZT ¼ DZT1:2 ( ¼ DZT1:5 ) and a depth dependence of   1:86 DZp ¼ DZp1:54 ð ¼ D Z p Þ for the example of a 5M  (10M  ) super Earth. In fact, the viscosity depends on pressure rather than only on the depth which means that the viscosity increases more strongly. Therefore, Eq. (13) changes to

Zbottom ¼ DZgd p p

ð14Þ

as the pressure scales to g  d. This gives a pressure dependence of DZp ¼ DZp3:45 ð ¼ DZp5:89 ) for the example of a 5M  (10M  ) super  Earth. Using a homologous temperature scaling Stamenkovic´ et al. (2011) find the viscosity to increase by more than 15 orders of magnitude. This results in the formation of a sluggish or stagnant lid at the core-mantle boundary which reduces the convective stresses and the likelihood of plate tectonics (Stamenkovic´ et al., 2012). The viscosity structure of super-Earths is, however, highly unclear which

9

is due to the extreme conditions (e.g., Valencia et al., 2006 estimate the pressures in super-Earths 41 TPa) that prevent experimental measurements. Contrary to the assumption of Stamenkovic´ et al. (2011), considering mineral physics Karato (2010) argues that pressures higher than the Earth’s pressures at the core-mantle boundary lead to a pressure-weakening effect which reduces lower mantle viscosity. We here use a combination of both assumptions, i.e., viscosity strongly increases with depth (cf. Eq. (14)) but starts to decrease at a certain point. Following the work of Stein et al. (2011) we assume that the pressure-weakening effect alters the viscosity equation (Eq. (9)) to d zÞ=zd ZSE ¼ Zeff DZðz : d

ð15Þ

The depth at which the viscosity reduction occurs is zd ¼ 1 ð1=M0:27 Þ ¼0.35 and 0.46 for the 5M  and 10M super-Earth, respectively. The decrease in lower mantle viscosity, DZd , is assumed to be between 100 and 1000 (Karato, 2010). Here, we use DZd ¼ 500. In Table 4 we give Earth reference values. With these values we roughly get a Rayleigh number of Rabot  4:8  107 (i.e., a top Rayleigh number of Ra ¼14,400 for our assumed viscosity contrast) and a heating rate of H  15 for the Earth which is similar to the values given by van Heck and Tackley (2011). Furthermore, Schubert et al. (2001) state that a pressure-dependent viscosity contrast of DZp of 30 is reasonable for the Earth. Assuming a surface temperature of T 0 ¼ 300 (Stacey and Davis, 2008) and activation energies of 300–540 kJ/mol for a dry mantle (Karato and Wu, 1993), the temperature-dependent viscosity contrast has the order of 1038 or more. However, modelling Earth-like Rayleigh numbers becomes less feasible at higher viscosity contrasts due to stronger viscosity gradients. Unfeasible extreme parameter values in numerical models is a common problem in geodynamical modelling. A way out of this is to find the right balance of parameters to allow for an Earth-like flow regime. Considering Rayleigh numbers in a range appropriate for the Earth, a contrast of DZT  ¼ 105 produces dynamical similar flow behaviour to the Earth (e.g., Tackley, 2000; Stein et al., 2004). In this case the activation energy of about 32 kJ/mol is one order of magnitude too low. Similarly the yield stress of about 356 MPa is at the lower end of what is found in experiments (Kohlstedt et al., 1995) but their interaction leads to Earth-like behaviour. With the scalings derived above, for a 5M super-Earth the parameter set would thus approximately be Ra  141,120, H  28:5, DZp  1:25  105 , DZT  106 , DZd ¼ 500, zd ¼0.35 and for 10M  : Ra  439,200, H  34:5, DZp  5:0  108 , DZT  3  107 , DZd ¼500, zd ¼0.46. Comparing the parameter sets with our model calculations summarised in Table 3 indicates that the case with the Earth-like parameter set probably resembles the Earth with its surface mobilisation. We find that the Earth-like values lie between our case 1 and case 2 (being closer to case 2). The estimated H  15 Table 4 Earth reference values taken from Stacey and Davis (2008) and Schubert et al. (2001). All values are dimensional and given in SI units. Symbol

a r0 g

DT d

k Z Q k

Dim. value 5

2  10 4000 10 2500 2.9  106 10  6 1021 5.1  10  12 4.6

SI unit 1/K kg/m3 m/s2 K m m2/s Pa s W/kg W/K m

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might be just lower than the Hcrit value, at which a transition to stagnant-lid convection occurs. To be sure that the Earth parameter set is consistent with the Earth having mobile plates, we performed a further model run with Earth-like parameters, i.e., a bottom Rayleigh number of 4.8  107, a heating rate of H¼15, a temperature- and pressure-dependent viscosity contrast of 105 and 30, respectively. Here we used Model 1 to also allow for a comparison between the different modelling methods. Fig. 11a shows the mobility as a function of the time (red solid curve). Clearly, we find mobile-lid convection with an average mobility of approximately 0.99 (red horizontal line). This is in good agreement with the results of Model 2 and consistent with the Earth’s plate motion. The temporal evolution of the mobility in Fig. 11a (green dashed curve) for a 5M  super-Earth model calculation shows that surface mobility is reduced to about 0.76 (green horizontal line) compared to the Earth reference case. The result of a model calculation with parameters representing a 10M  planet can be seen in Fig. 11a (blue dotted line). In comparison to the model runs representing smaller sized planets, the 10M  planet clearly shows further reduced mobility (of about 0.51). Finally, we discuss the results of another set of model runs. As described by O’Neill and Lenardic (2007) the non-dimensional

Constant yield stress σY=3 for all planets

2.5

1ME 5Me 10Me

mobility

2

1.5

1

0.5

0

0

0.002 time Increased yield stress for super-Earths

2.5

1ME 5Me 10Me

2

mobility

0.004

1.5

1

0.5

0 0

0.002 time

0.004

Fig. 11. (a) Mobility versus non-dimensional time for Model 1 simulations with Earth-like parameters (1 Earth mass, Me), parameters representing a 5 Earth mass and a 10 Earth mass planet with a yield stress of 3. (b) Mobility versus nondimensional time for the Earth-like case with a yield stress of 3 and the two superEarth-like cases with increased yield stress (see text). (The resolution of these runs was 256  250.).

2

yield stress sY ¼ d sdim =Z0 k also depends on planet size. Applying this to the 5M  and 10M  planets, the yield stress of 3 we have chosen for the Earth would become 7.1 for the 5M  planet and 10.4 for the 10M  planet. The results of the model runs with super-Earth-like parameters and increased yield stress are summarised in Fig. 11b. As expected from previous work (Moresi and Solomatov, 1998; Tackley, 2000; Stein et al., 2004; O’Neill and Lenardic, 2007) the mobility is lower for the cases with higher yield stresses.

4. Discussion In the study of lithospheric mobilisation on super-Earths it has to be considered that many convection and rheological parameters scale with planet size. We have isolated the effect of some of these parameters to resolve some of the controversies found in previous studies. Our main focus is on the mantle internal heating. Therefore, we mainly analyse the effect of the heating rate, H, which scales with mantle depth squared (cf. Eq. (6)). Investigating systems with both purely internal heating and mixed-mode heating (i.e., internal and basal heat sources), we observe for both heating modes that, for the temperaturedependent viscosities investigated, the surface mobility decreases as the heating rate, H, is increased. However, we also find slight differences between purely internally heated convection and mixed-mode heated systems due to differences in the interior temperatures. In cases with lower temperatures plate tectonics is obtained for a larger parameter space and is thus more likely. Similarly, our results show that in the case of purely basal heating (for H¼0) plate tectonics is more likely than in the mixed-heated cases. The different heating modes used in various works (Stein et al., 2004; Valencia and O’Connell, 2009; van Heck and Tackley, 2011) could thus be one reason for the seemingly controversial results found. Another difference between our work and that of Valencia and O’Connell (2009) and van Heck and Tackley (2011) is the choice of model parameters. The latter increased the heating rate along with the relevant Rayleigh number that enters the momentum equation (cf. Eq. (3)) (i.e., RaH for purely internal heating but Ra for mixed-mode heating). In an initial set of calculations we kept the Benard–Rayleigh number, Ra, constant while increasing H (e.g., Stein et al., 2004). As such the plate mobility found in previous studies can be explained by the increased heat flux through the surface, appearing due to the increased vigour of convection (Rayleigh number), which leads to a cooling of the system (e.g., Turcotte and Schubert, 2002; Grasset and Parmentier, 1998). Increasing the heating rate (and the Rayleigh number due to internal heating, RaH) at a constant Ra we observe an increase in interior temperature which leads to a strong viscosity contrast across the top thermal boundary layer. Though the Rayleigh number increases and the plate thins (as predicted by Valencia and O’Connell, 2009), we find that the high surface viscosity increases plate resistance so that no subduction is obtained. Additionally, the decrease in interior viscosity leads to smaller convective stresses that cannot overcome the resisting forces. As the (Benard-)Rayleigh number scales with mantle depth cubed (cf. Eq. (4)), both Rayleigh number, Ra, and the heating rate H will indeed increase simultaneously in larger sized planets. Performing calculations with higher Rayleigh numbers, we still obtain stagnant-lid convection when the heating rate is increased. However, the transitional value Hcrit at which stagnant-lid convection appears, increases with Ra (cf. Fig. 6). With the values for a 2M  super-Earth taken from the work by van Heck and Tackley (2011) our results (cf. Fig. 6) suggest stagnant-lid convection

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rather than mobilisation of the surface. This is again due to the different modes of heating. van Heck and Tackley (2011) show results for purely internal heating (as Valencia and O’Connell, 2009) and purely basal heating while we compare this with mixed-mode heating. As mentioned above, differences in the interior temperatures occurring between the different heating modes affect the surface mobilisation. In addition, we also assumed that temperatures and pressures are higher in super-Earths, which will also change the viscosity contrast. Considering higher temperatures for super-Earths, as proposed by Kaltenegger et al. (2011), will increase the Rayleigh numbers and reduce the heating rates compared to the ones used by van Heck and Tackley (2011). In Section 3.2 we have shown that this will increase plate mobility. But additionally the temperature-dependent viscosity will increase dramatically with increased temperatures. In our systematic study of nondimensional parameters we have shown that this leads to a dramatic reduction of the critical H-value for the onset of stagnant-lid convection because the reduction in interior viscosity reduces the convective stress. The pressure dependence of the viscosity dampens the viscosity contrast arising from temperature dependence alone. An increased pressure dependence is also of special importance as it counteracts an increased amount of heating (Stein et al., 2004, cf. also Introduction). Our investigation of the interaction between pressure dependence and heating rate showed that (for all pressure dependencies investigated) as heating rates increase stagnant-lid convection results. The effect of the pressure-dependent viscosity can only overcome the thermal effect and weaken the stagnant-lid for low heating rates (cf. Fig. 2). Stamenkovic´ et al. (2012) also suggest that plate tectonics might be less likely on larger planets as the viscosity at the core-mantle boundary (CMB) is so high that it leads to a CMB-lid and reduces the Rayleigh number and convective stresses. However, the viscosity structure is highly unclear which is reflected in the discrepancies between the works of Karato (2010) and Stamenkovic´ et al. (2011). Scaling the viscosity by the melting temperature results in a strong increase in viscosity (Stamenkovic´ et al., 2011), while higher pressures were argued to cause a pressure-weaking effect and a decrease in the lowermost viscosity (Karato, 2010). This was shown to also result in stagnant-lid convection (Stein et al., 2011). Interestingly, both assumptions give the same general result, i.e., a reduced likelihood of plate tectonics. However, in a recent work Tackley et al. (under review) argue against a CMB-lid because of the effect of internal heating. Tackley et al. (under review) find a self-regulating feedback between the temperature and viscosity: as the mantle heats up the viscosity is reduced which in return allows for the loss of radiogenic heat. The numerical simulations result in a superadiabatic temperature profile and a viscosity decrease of no more than 3 orders of magnitude (Tackley et al., under review). This viscosity increase is lower than the one predicted by Stamenkovic´ et al. (2011). Differences in the viscosity structure are due to the ongoing discussions of the physical properties in the lower mantle of higher mass planets. For the lowermost mantle of the Earth a transition from MgSiO3 perovskite to the denser postperovskite has been discovered at pressures of about 120 GPa (Muramaki et al., 2004), leaving perovskite the dominant mineral structure on Earth. As the pressures in larger mass planets are much higher, post-perovskite might be the dominant structure on super-Earths. Therefore, Tackley et al. (under review) have calculated the activation enthalpies of post-perovskite in contrast to Stamenkovic´ et al. (2011) who considered a perovskite rheology. But even the stability of post-perovskite at very high pressures as assumed at the CMB of super-Earths is not certain. A further transition to post–post-perovskite might occur. Umemoto and Wentzcovitch (2011) in fact report a further dissociation of MgSiO3 ,

11

while Grocholski et al. (2010) show that post-perovskite remains stable. In the second part of our work, we merge the increased parameters in parameter sets relevant for super-Earths showing that the thermal effect prevails. Model calculations with parameters estimated for super-Earths with 5 and 10 Earth masses result in a reduced surface mobility. As we are discussing the influence of temperatures in this study, we also have to take into account that recent results obtained from a comparison between spherical and plane-layer convection models show large differences in their mean temperatures (O’Farrell and Lowman, 2010). Cartesian models featuring internal heating have been shown to result in high mean temperatures (e.g., Sotin and Labrosse, 1999) which is a consequence of the core-mantle boundary being as large as the surface layer which is not valid in the sphere. Therefore, we have to consider that our convection calculations are performed in a plane-layer geometry and the rate of heating would need downward adjustment which could shift super-Earths (at least the smallest ones of them) back into the mobile-lid regime. This has to be tested in spherical shell models. Also, further aspects such as the effect of water, which promotes mobility (Korenaga, 2010), in combination with mixed-mode heating has to be investigated in future work. On the whole, we find that the interior temperature of the convective system plays an important role because viscosity is temperature-dependent. As the knowledge of super-Earth values and especially their temperatures is presently still vague, this work is largely a qualitative one. A more quantitative study has to be conducted when values are more precisely known.

5. Conclusion To investigate surface mobility for conditions applicable to super-Earths, we used two numerical mantle convection models capable of yielding plate-like surface motion through different modelling methods. Both models reveal the same system behaviour, namely stagnant-lid convection with increased non-dimensional heating rates. Focusing on internal heating, we investigated systems with both pure internal heating and mixed-mode heating (i.e., heated by internal and basal sources). Our findings indicate that some of the discrepancies in previous works can be explained by the mode of heating. The hotter the interior of the system (and thus the larger the temperature drop over the surface) the less likely plate tectonics becomes. All our model calculations have shown that (due to the temperature dependence of the viscosity) the internal heating rate has a very strong effect on surface behaviour. Calculations with high viscosity contrasts and high Rayleigh numbers have shown that the effect of increased Rayleigh numbers and pressure dependence does not outweigh strong thermal effects. A transition from mobile-lid convection to stagnant-lid convection occurs for increasing heating rates in all cases examined. To date, our knowledge of the parameters and interior structure of super-Earths is still very poor. Consequently, we predominantly discuss parameter regimes and give tendencies for the resulting lithospheric mobility. However, by assuming that superEarth material properties are Earth-like (Sotin et al., 2007; Valencia et al., 2006), the heating rates, Rayleigh number and viscosity contrasts relevant for super-Earths can be estimated. Extrapolating Earth-like model values to super-Earth values results in a change from mobile-lid to sluggish or stagnant-lid convection in our models. Thus, we conclude that plate mobility will likely be lower on Earth-like planets with higher masses. The effect of an increased yield stress further strengthens our

Please cite this article as: Stein, C., et al., The influence of mantle internal heating on lithospheric mobility: Implications for superEarths. Earth and Planetary Science Letters (2012), http://dx.doi.org/10.1016/j.epsl.2012.11.011

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conclusions (cf. Moresi and Solomatov, 1998; O’Neill and Lenardic, 2007). A reduction of plate mobility on super-Earths agrees with the result of O’Neill and Lenardic (2007) and appears to contradict the conclusions of Valencia and O’Connell (2009) and van Heck and Tackley (2011), who both argue for (more) plate mobility on super-Earths. Differences between our work and the last two is the mode of heating and the choice of parameters (i.e., a higher temperature-dependent viscosity in our case). The differences in the interior temperatures resulting from the different heating modes is only minor and does not have much of an effect on the surface behaviour. However, an increased temperaturedependent viscosity strongly reduces surface mobility. Therefore, the most important difference will be the increase in the temperature-dependent viscosity (which we assume as the temperature drop in super-Earths will be higher).

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Please cite this article as: Stein, C., et al., The influence of mantle internal heating on lithospheric mobility: Implications for superEarths. Earth and Planetary Science Letters (2012), http://dx.doi.org/10.1016/j.epsl.2012.11.011