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Keywords:

  • mantle convection;
  • numerical modeling;
  • plate tectonics;
  • super-Earths;
  • viscosity structure

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Models
  5. 3. Results
  6. 4. Conclusion
  7. Acknowledgments
  8. References

[1] The quest to identify habitable planets has raised interest in the surface dynamics of terrestrial bodies. In this context super-Earths (a new class of exoplanets) have become of special interest in the past decade. Scalings to super-Earth sizes, when compared to the Earth, suggest changes to convective stresses and mantle temperatures which can cause either an increase in surface mobility or in plate resistance. Mantle viscosity, which depends on temperature, stress and pressure, plays a critical role in both cases. New mineralogical assumptions suggest that the viscosity in super-Earths acts differently than in the Earth, and what had been assumed for super-Earths. In planets larger than the Earth, pressure will become so high that pressure-weakening and a decrease of viscosity in the lowermost mantle results. We present a numerical convection study featuring this viscosity decrease and find that this leads to a reduction in surface mobility.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Models
  5. 3. Results
  6. 4. Conclusion
  7. Acknowledgments
  8. References

[2] Since their discovery, extra-solar planets have become an active research topic [e.g., Mayor and Queloz, 1995; Burrows et al., 2000; Rivera et al., 2005]. Especially the smaller planets with masses close to but exceeding the Earth's, the super-Earths (<10 M [Ida and Lin, 2004]). These planets have attracted substantial interest because they might feature Earth-like surface motion [Valencia et al., 2007b] due to their assumed Earth-like structure [Seager et al., 2007; Sotin et al., 2007; Valencia et al., 2007a]. This may increase the possibility of habitability [von Bloh et al., 2007].

[3] Numerical and theoretical convection studies have considered the scalings and parameters applicable for super-Earths and have analysed the surface behaviour. Valencia and O'Connell [2009] find that the increased mass of super-Earths leads to higher driving forces for mantle convection, raising the likelihood of mobilisation of the surface. However, in addition to convective vigour, rheological aspects are a necessary ingredient for plate modeling [e.g., Solomatov, 1995]. The strong temperature dependence of the mantle's viscosity is most important but stress and pressure dependence are also a prerequisite for steady plate mobilisation as known from the Earth [Tackley, 2000; Stein et al., 2004]. O'Neill and Lenardic [2007], who scale the stresses with the planet size, argue that stagnant-lid convection with no actively moving plates is most likely for super-Earths.

[4] The pressure dependence of mantle viscosity, though shown to have a strong impact on surface mobility [Stein et al., 2004], has often been neglected in super-Earth simulations [Valencia and O'Connell, 2009; O'Neill and Lenardic, 2007]. Recently, however, new mineralogical assumptions on super-Earths that might have an additional profound impact on surface mobility have been provided by Karato [2010]. Karato [2010] argues that in super-Earths, pressures become so high that pressure-weakening starts when exceeding 0.1–0.3 TPa, which occurs for masses greater than the Earth's. This means that for terrestrial planets larger than the Earth the viscosity in the lower mantle decreases relative to the upper mantle.

[5] Motivated by this scenario, we present mantle convection models featuring plates and a decrease in lower mantle viscosity to account for the sizes of super-Earths. We systematically study the effect of the decrease in the lower mantle viscosity as well as different depths at which this viscosity transition occurs. We compare the results to models not featuring a decrease in the lower mantle viscosity (Earth models) and analyse the surface mobility.

2. Models

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Models
  5. 3. Results
  6. 4. Conclusion
  7. Acknowledgments
  8. References

[6] We employ a mantle convection model with plates (MC3D) that satisfies the equations for the conservation of energy, mass and momentum for an incompressible fluid. In the non-dimensional form, we solve the set of Boussinesq equations

  • equation image
  • equation image

and

  • equation image

where T is temperature, t is time, v is velocity, H is the internal heating rate, p is the dynamic pressure, η is the dynamic viscosity, ez is the vertical unit vector, and Ra is the Rayleigh number,

  • equation image

Here, α denotes the thermal expansivity; g, the gravitational acceleration; ρ, the density; ΔT, the temperature drop across the convecting system of depth d; κ, the thermal diffusivity; and η0 is the surface viscosity. Accordingly, the Rayleigh number, Ra, represents the top Rayleigh number.

[7] Plate-like surface movement (i.e., a spatially uniform surface velocity) is achieved by specifying time-dependent surface motion as a boundary condition. Plate velocities are determined dynamically by implementing a force-balance method [cf. Gable et al., 1991] that ensures that the shear stress on the base of the plates sums to zero at all times. As a result, plates neither drive nor resist the convection. Plates are modelled as high viscosity regions with the model viscosity, ηpT, being a function of pressure and geotherm (cf. C. Stein et al., A comparison of mantle convection models featuring plates, submitted to Geochemistry, Geophysics, Geosystems, 2011a):

  • equation image

where ΔηT and Δηp are the temperature- and pressure-dependent viscosity contrasts, respectively. 〈T〉 is the horizontally averaged temperature and z the vertical coordinate with a value of 0 at the bottom and 1 at the surface of the model domain. This rheology-dependent force-balance method has been compared to a fully rheological model (temperature-, pressure- and stress-dependent viscosity) and shows the same system behaviour (Stein et al., submitted manuscript, 2011a).

[8] For the investigation of a decreasing lower mantle viscosity [Karato, 2010] we modified the viscosity equation (equation (5)) for z < zd so that

  • equation image

where Δηd specifies the decrease in lower mantle viscosity relative to ηpT, and zd defines the height above the core-mantle boundary at which the viscosity starts to decrease with pressure.

[9] As an example we consider GJ876d, a 7.5 M planet [Rivera et al., 2005]. According to von Bloh et al. [2007], Sotin et al. [2007] and Valencia et al. [2006] the mass-radius relation of super-Earths scales according to R = M0.27. This gives a radius of approximately 1.72 R for GJ876d. Assuming that super-Earths have a similar structure to Earth, we consider the mantle thickness dSE to also be 1.72 times that of the Earth's mantle, d. Following Karato's [2010] finding that pressures in planets just slightly bigger than the Earth are high enough that there is a reduction in lower mantle viscosity in comparison with the mantle above, we take the Earth's mantle thickness d as the critical depth where a decrease in viscosity starts. In the example of the 7.5 M super-Earth, the critical depth would thus be d = dSE/1.72. Scaling the model thickness with the mantle thickness dSE leads to the non-dimensional critical depth dd = d/dSE = 0.58. As we define the non-dimensional height in our model to be zero at the base and one at the top, we find GJ876d's viscosity transition occurs at zd = 1 – dd = 0.42.

3. Results

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Models
  5. 3. Results
  6. 4. Conclusion
  7. Acknowledgments
  8. References

[10] We begin with models featuring the viscosity structure of equation (5) (i.e., without a decrease in lower mantle viscosity). Figure 1 presents the surface mobility (= surface velocity/rms-velocity [Tackley, 2000]) for two sets of parameters. Model A uses Ra = 65, ΔηT = 109 and Δηp = 500 (consequently the bottom Rayleigh number in this case is 65ΔηTηp = 1.3 × 108); Model B uses more moderate parameters: Ra = 400, ΔηT = 105 and Δηp = 30 (bottom Rayleigh number of 1.3 × 106). In order to find the transition from mobile-lid convection to stagnant-lid convection we vary the non-dimensional heating rate, H, from negative heating rates (cooling rates), H = −20, to positive heating rates, H = 60. The implementation of negative values for H is motivated by our use of Cartesian geometry models and yields temperatures consistent with a spherical shell system of comparable vigour featuring a small amount of internal heating [O'Farrell and Lowman, 2010]. Both models show the same system behaviour: mobility decreases as the non-dimensional rate of internal heating increases. For low heating rates the mobility is close to 1, meaning that the surface moves about as fast as the interior. This is a characteristic of mobile-lid convection. For higher H the mobilities are close to 0, which characterizes stagnant-lid convection. A reduction in surface mobility for hotter mantle temperatures has also been observed by O'Neill et al. [2007]. In this work, the motivation for examining the effect of increased non-dimensional internal heating rate on surface mobility is that the non-dimensional heating rate, H, scales with the square of the mantle depth and thus will likely be higher in a super-Earth in comparison to Earth. C. Stein et al. (The influence of mantle internal heating on lithospheric mobility: Implications for Super-Earths, submitted to Earth and Planetary Science Letters, 2011b) analyze this effect in detail.

image

Figure 1. Mobility as a function of the rate of internal heating. The model parameters for Model A are: Ra = 65, ΔηT = 109 and Δηp = 500, and for Models B: Ra = 400, ΔηT = 105 and Δηp = 30.

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[11] We also show the temperature fields from four calculations. To illustrate the full range of flow regimes, the lower frame of Figure 1 shows examples from the mobile-lid, sluggish-lid and stagnant-lid regimes using the moderate parameter set calculations (Model B). One example of the mobile-lid regime from the more computationally demanding case (Model A) is also shown. For Model A we find that stagnant-lid convection occurs for H exceeding about −2. To obtain surface mobility we have chosen H = −10 (labeled Model A-10). As can be inferred from the surface mobility plot, in Model A-10 the top layer takes part in the convection and subduction of cold material (here in blue) is observed. Model B1 (moderate model parameters with H = 1) is also an example of mobile-lid convection but has a hotter interior than in Model A-10 due to the higher H. The temperature field shows a well-mixed interior with a hot plume on the left (in red) and downwelling of cold material (in blue) on the right. When H is increased to 5 (model B5) we observe strongly reduced mobility. The temperature field shows hardly any cold material sinking into the interior (which contributes to heating up the system) and the surface becomes sluggish. Finally, for Model B20 (i.e., Model B with H = 20) we find stagnant-lid convection, where the surface does not move. In fact, in this case the interior is so hot that heat flows from the mantle into the core. Together, these four models provide examples of very cold and hot mantles and also different flow regimes. In particular, we identify two examples with mobile-lid convection to determine if the surface mobilization that occurs on the Earth is maintained when using a super-Earth-like viscosity structure. Motivated by Karato's [2010] proposal, we now investigate the effect of a decreasing lower mantle viscosity on these four reference cases. Specifically, we study the influence of the viscosity contrast Δηd and the effect of the height zd below which the viscosity reduction is applied (cf. equation (6)). The thick line in Figure 2 shows the viscosity-depth profile that results once Model B reaches a steady state (cf. Figure 1). In addition, we present profiles for two cases in which the viscosity decreases below a height of zd = 0.42. In these cases, the basal viscosity is decreased by a factor of 10 and 100 compared to the reference case Δηd = 1 (top figure). In the right-hand figure Δηd is 10 but the height at which the viscosity reduction starts varies.

image

Figure 2. Viscosity-depth profile for the steady state Model B20 (thick solid line). In addition, we show example viscosity profiles with different viscosity contrasts Δηd and varying heights zd.

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[12] First we determine how the system behaviour (particularly the surface motion) changes if we decrease the lower mantle viscosity by a factor of Δηd. Figure 3 displays the mobilities as a function of the viscosity reduction for the four reference models that represent the three different flow regimes. Despite exemplifying mobile-lid convection when Δηd = 1 (i.e., if there is no viscosity decrease in the lower part of the mantle) Model A-10 and Model B1 show a clear transition to stagnant-lid behaviour and a strongly reduced mobility for higher viscosity contrasts, Δηd. For Model A-10 a viscosity decrease of 10 is sufficient to reduce the mobility dramatically. For the calculation with more moderate parameters (Model B1) an even lower value of Δηd is sufficient to halt surface mobility. Models B5 and B20 which are in the sluggish or stagnant-lid regime for Δηd = 1 essentially show the same behaviour. With decreasing bottom viscosity (increasing Δηd) the surface mobility diminishes further and the models remain in stagnant-lid convection.

image

Figure 3. Mobility as a function of the viscosity decrease below the height zd = 0.42. The parameters of Model A and Model B are the same as in Figure 1 with A-10 implying that H = −10 has been chosen. For Model B we have chosen rates of internal heating of 1, 5 and 20 (Models B1, B5 and B20, respectively).

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[13] Finally, we present the effect of varying the height below which the viscosity starts to decrease. On the one hand this allows us to study the effect of planet sizes. On the other we take into account that the overall mass of super-Earths will cause higher mantle pressures and increase the transition height. In Figure 4 (top) the mobilities are plotted as a function of the height zd for Models A-10 and B1 (mobile-lid convection), B5 (sluggish-lid convection) and B20 (stagnant-lid convection) for a low decrease in viscosity (Δηd = 2). In each model we find a very small decrease in surface mobility. The reduction flattens out and almost reaches an equilibrium for large zd. In Figure 4 (bottom) mobilities are plotted as a function of the height zd for cases like Model B1 but featuring Δηd = 2 (mobile-lid convection), Δηd = 200 (sluggish-lid convection) and Δηd = 2000 (stagnant-lid convection). In this suite of models we also observe a decrease in surface mobility as the height zd increases and this behaviour becomes more pronounced for larger Δηd. Note that the corresponding masses of the planets are given along the top horizontal axis. For the example super-Earth of 7.5 M, GJ876d, and Karato's [2010] argument that the lower mantle viscosity decrease will be between 100 and 1000 we find a low surface mobility (cf. dashed lines in Figure 4). Only for small viscosity decreases or masses lower than about 2 M do we still observe mobile-lid convection.

image

Figure 4. (top) Mobility as a function of the height zd (planets' size) for model runs with Δηd = 2. Models A-10 and B1 represent mobile-lid convection, B5 sluggish-lid convection and B20 stagnant-lid convection, respectively, for a case with 7.5 M and zd = 0.42 (see dashed line and Figure 3.) (bottom) Mobility as function of the height zd (planets' size) for Model B1 with Δηd = 2 (mobile lid), Δηd = 200 (sluggish lid) and Δηd = 2000 (stagnant lid).

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4. Conclusion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Models
  5. 3. Results
  6. 4. Conclusion
  7. Acknowledgments
  8. References

[14] We performed a study of mantle convection in a numerical model incorporating plates that achieve surface mobilization for Earth-like parameters and considered a decrease in lower mantle viscosity as proposed for super-Earths [Karato, 2010]. Starting from the Earth-like viscosity structure (where viscosity decreases with temperature and increases with pressure), we systematically add a viscosity decrease for the lowermost mantle. This viscosity reduction leads to a reduction in surface mobility which we attribute to the increased overall viscosity contrast (ratio of top to bottom viscosity). The effect is similar to increasing the thermal viscosity contrast where the surface viscously decouples from the interior [e.g., Moresi and Solomatov, 1996]. As a result of the increased thermal viscosity contrast the plate resisting stresses outweigh the shear stresses at the base of the plate. Therefore our result also agrees with the observation of reduced surface mobility described by O'Neill and Lenardic [2007]. When scaling mantle stress to super-Earths, O'Neill and Lenardic [2007] find that stresses decouple as the ratio of convecting to resisting stresses decreases.

[15] We have shown that this result is independent of the model parameter set chosen. To address the lack of knowledge we have of super-Earth parameter ranges, we have analysed four reference cases. These cover a wide range of parameters (a bottom Rayleigh number of Ra ≈ 106 − 108, viscosity contrasts of ΔηT = 105 − 109, Δηp = 30 − 500 and non-dimensional heating rates from H = −10 to H = 20) and represent various flow regimes (mobile-lid, sluggish-lid and stagnant-lid convection) as well as very hot and very cold planets. Starting from a parameter set that gives sluggish-lid or stagnant-lid convection for the Earth-like viscosity structure, model runs show a continued or further reduced surface mobility and remain in the stagnant-lid mode of convection when a super-Earth-like viscosity structure is added. Model runs that feature mobile-lid convection for the Earth-like viscosity structure show a drastic reduction in surface mobility once the lower viscosity is reduced, even changing from mobile-lid to stagnant-lid convection. The fact that a lower mantle viscosity reduces the surface mobility was also shown when we increase the non-dimensional heating rate. Increased mantle temperatures decrease mantle viscosity due to the temperature-dependence of viscosity [cf. also O'Neill et al., 2007].

[16] While Karato [2010] suggests that super-Earths may feature a viscosity reduction of 2–3 orders of magnitude, our results show that an even lower reduction can produce the transition to stagnant-lid convection.

[17] We also find that varying the height where the viscosity reduction occurs does not suppress the transition to stagnant-lid convection. Based on Karato's [2010] estimate that planets larger than ∼1 M have pressures exceeding the critical value and this causes pressure-weakening, we assume Earth's mass (mantle height) to correspond to the critical height zd. Accordingly, we propose that the different heights analyzed can easily be converted to planets of different sizes (MSE = 1/(1 − zd)1/0.27). We find that for masses exceeding about MSE = 2 M the surface mobility is drastically reduced and the surface moves sluggishly or even becomes immobile.

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Models
  5. 3. Results
  6. 4. Conclusion
  7. Acknowledgments
  8. References

[18] We thank Adrian Lenardic and Carl Gable for useful comments that improved the manuscript. C.S. and U.H. are grateful for financial support from the Helmholtz Alliance ‘Planetary evolution and life’. JPL thanks the NSERC of Canada for funding (327084-2009).

[19] The Editor thanks Carl W. Gable and an anonymous reviewer for their assistance in evaluating this paper.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Models
  5. 3. Results
  6. 4. Conclusion
  7. Acknowledgments
  8. References