Plate tectonics on super-Earths: Equally or more likely than on Earth
Highlights
► We studied the likelihood of plate tectonics, as function of planet size. ► Results of numerical calculations are combined with analytical scaling laws. ► The size of planets is only weakly related to the likelihood of plate tectonics.
Introduction
Table 1
Fig. 5
Super-Earths are terrestrial planets outside our solar system with masses up to about ten times the mass of Earth. Any planet with more mass than this is unlikely to be terrestrial (Ida and Lin, 2004). Observations of super-Earths are indirect and the best constrained properties of such planets are basic properties like their masses and distance to their star. In recent years many super-Earths have been found, and many more are expected to be found in the near future, in particular by the NASA Kepler mission (kepler.nasa.gov), which is searching the skies for planets that are the same size as Earth. So far, over 700 targets with viable exoplanet candidates with sizes as small as that of Earth to larger than that of Jupiter have been found. Arguably the most fundamental question one can ask about super-Earths is: "Do we expect super-Earths to have active plate tectonics, like the Earth, or do we expect them to be in the stagnant lid mode, like present day Mars and Venus?" Since both data about super-Earths is scarce, and the complex processes of plate tectonics and subduction initiation are not very well understood it is unlikely that, at the moment, a robust prediction can be made about the convective regime of one particular exoplanet. We can however build a general theory about terrestrial planets of all sizes. Previous efforts in this direction were made by Valencia et al., 2007, O'Neill and Lenardic, 2007, Tackley and van Heck, 2008, Valencia and O'Connell, 2009, Korenaga, 2010a, Korenaga, 2010b. Valencia et al. (2007) used a parameterized convection model, based on scaling laws, to conclude that the likelihood of plate tectonics increases with planet size. Their model predicts a decrease in plate thickness and an increase of shear stresses with increasing planet size, leading them to conclude that plate tectonics is inevitable on super-Earths. O'Neill and Lenardic (2007) reached opposite conclusions, stating that stagnant lid convection is the more likely convective regime on super-Earths. The approach they used is similar to ours, using a numerical model to simulate convection on Earth and scale that to bigger planets. O'Neill and Lenardic (2007) found that the dominant effect was the more rapid increase in pressure with depth on more massive planets. This effect leads to an increase in fault strength hence a stronger lithosphere and thus a lower likelihood of plate tectonics. Korenaga (2010a) shows that, (based on empirical analytical scaling relationships for temperature dependent convection with a visco-plastic rheology, Korenaga, 2010b) plate tectonics becomes more likely with increasing planet size.
As oceanic plates act as the upper thermal boundary layer of mantle convection, and continents are formed from the mantle, it is desirable to treat mantle and plates as a single, integrated system rather than two separate entities. Although in recent years progress has been made in both the comprehensiveness and clarity of numerical models, (see Bercovici, 2003 for a review) basic questions about why the Earth is currently the only terrestrial planet with active plate tectonics, which parameters control the formation of plates, and which processes are responsible for the creation of subduction zones and spreading centers, remain without definitive answers, although several hypotheses exist.
Convection with temperature dependent viscosity displays three different regimes, none of which is plate-like. Solomatov, 1995, Moresi and Solomatov, 1995 conducted numerical experiments of convection with large viscosity contrasts. They found distinct different regimes in which the cold upper boundary layer did or did not participate in convection, separating a stagnant lid regime from a mobile and a sluggish lid regime. Models got much closer to including plate-like behavior when Moresi and Solomatov (1998) introduced a yield stress. When the stress reaches a certain critical stress, the lithospere is weakened by yielding, allowing spreading centers and subduction-like features to form. This approach was used in 3D Cartesian geometry by Trompert and Hansen, 1998, Tackley, 2000a. In general, these studies found three distinctive convective regimes: mobile lid, where the lithosphere is constantly yielding, allowing zones of subduction and spreading centers to be present at all times, a stagnant lid, where one continuous plate covers the whole domain, and episodic lid, where the regime keeps changing from mobile lid to stagnant lid, back to mobile lid over time. Later some studies included more Earth-like features such as history dependent weakening (Tackley, 2000b, Ogawa, 2003) and a low viscosity asthenosphere (Tackley, 2000b, Richards et al., 2001). Stein et al. (2004) used a similar but more extensive approach to study, among others, the influence of temperature, stress and pressure dependence of the viscosity on plate-like behavior. Solomatov (2004) discussed how subduction can be initiated by small scale convection. Muhlhaus and Regenauer-Lieb (2005) discussed the importance of elasticity and non-Newtonian rheology. More recently, Loddoch et al. (2006) argued that a fourth regime exists between the stagnant and episodic lid, where different scalings apply. van Heck and Tackley, 2008, Foley and Becker, 2009 applied these techniques to study the behavior of self consistent plate tectonics in fully 3D-spherical geometry, and Yoshida (2008) used a similar model to study the wavelength of convection, and later continental breakup (Yoshida, 2010). This type of model has also been applied to other terrestrial planets; for example, Fowler and O'Brien (2003) used a model similar to the ones mentioned above to investigate the frequency of resurfacing events on Venus.
In the present study we derive analytical scalings, accompanied by results of numerical calculations, for a basic upscaling of the Earth to obtain first order predictions about the likelihood of plate tectonics on super-Earths. In upscaling planet size, care needs to be taken to scale all parameters that change with planet size consistently throughout the equations, particularly if using nondimensional units, as we do here. Some scalings are not immediately obvious, because they appear in the scaling factors used to convert non-dimensional parameter values to their dimensional equivalents. The numerical model we use solves for the equations of thermal convection where plate tectonics and large scale mantle convection are treated as a single self consistent system, similar to van Heck and Tackley (2008).
Section snippets
Model description
The physical model we choose here to use to study the likelihood of plate tectonics on terrestrial planets is a general one to model fluid flow under the Boussinesq approximation, i.e. all material properties are assumed to be constant in both space and time except viscosity, which depends on temperature and stress through plastic yielding. This is a gross simplification of material properties on Earth, which also depend on pressure, but is in the spirit of the only other numerical studies on
Analytical scalings
Here, S is the ratio of the planet's radius to Earth's radius. Four non-dimensional parameters need to be scaled with planet size (S): Rayleigh number (Ra), internal heating rate (H), yield stress (σy) and yield stress gradient (dσy/dz).
The Rayleigh number can be expressed as:where ρ, g, α, △ T, D, κ and η0 are density, gravitational acceleration, temperature scale, depth of the mantle, thermal diffusivity and reference viscosity respectively.
In our model, the only two parameters
Numerical model and rheology
To perform the numerical experiments we use the latest version of the code StagYY (Tackley, 2008), solving the equations for momentum, energy and conservation of mass in the Bousinesq approximation. For the effect of temperature on viscosity we used an Arrhenius law, choosing activation energy such that there are nine orders of magnitude variation in viscosity from non-dimensional temperatures in the range 0 to 1. The reference viscosity was chosen to be the viscosity at a non-dimensional
Discussion
While the obtained scaling relationships seem reliable and robust, care should be taken in interpreting absolute values of (dimensional) critical yield stresses and friction coefficients, because these values depend heavily on our choice of parameter values listed in Table 2, and in any case it is a well-established problem that the values required for mobile lid behavior are lower than what is expected from laboratory rock deformation experiments.
Our results agree in essence with the scaling
Conclusions
Here we have solved the fundamental physics of convection on super-Earths for simplified, end-member cases of basally-heated and internally-heated convection with strongly temperature-dependent viscosity and using either a constant yield stress or a constant yield stress gradient. Analytical scalings and numerical calculations agree. For basally heated convection, plate tectonics is more likely on super-Earths then it is on an Earth-sized planet, becoming increasingly likely with increasing
Acknowledgments
This research was supported by SNF grant numbers 200021–112137 and 200020–126773.
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