Convection scaling and subduction on Earth and super-Earths
Introduction
Despite the formidable observational challenges, fifteen super-Earths have been discovered from the ground in the last four years. Super-Earths are extrasolar planets made primarily of rock and/or ices and voided of a massive hydrogen or helium envelope. There is no hard limit on planetary mass that separates the gaseous from the (mostly) solid planets, but a reasonable value may be in the vicinity of 10 M⊕ (Ida and Lin, 2004). Two important characteristics make super-Earths interesting objects to study: the ones that are rocky might be similar to Earth, and depending on their thermal state be habitable; and, the bias in detection ensures that larger Earth-like planets will be discovered before a true Earth analog. Furthermore, in the near future several dozen super-Earths will be discovered with space missions underway—CoRoT (Borde et al., 2003), Kepler (Borucki et al., 2003)—and others under construction (Automated Planet Finder, JWST, etc.). To interpret the data that will be available, it is timely to set the framework for understanding the properties of these planets. In this study we address the relevant topic of tectonics in rocky massive planets.
The thermal evolution of a planet is intrinsically related to the mode of convection, which can be in a state with plate tectonics or else with an immobile stagnant lid. There have been attempts to predict the mode of convection of rocky super-Earths with opposite conclusions drawn. Valencia et al. (2007) show that terrestrial super-Earths can exhibit plate tectonics, while O'Neill and Lenardic (2007) (hereonafter OL07) determine that super-sized Earths will most likely be in a stagnant lid regime and that only in some cases will they experience episodic plate tectonics.
Plate tectonics is a complicated process and one that we only evidence on Earth. Despite the geological data available on Earth, the details of active lid tectonics are not completely understood. Nevertheless, the general features of plate tectonics have been laid out over the past 30 years and are well recognized. Determining how the onset of plate tectonics takes places is a challenge still unresolved. But once subduction has started, it is much easier to maintain because deformation can happen along pre-existing faults. Thus, we investigate if plate tectonics can be sustained in massive versions of Earth. We approach three questions: is the increase in fault strength larger than the convective stress as to hinder subduction (as suggested by O'Neill and Lenardic (2007))? Is negative buoyancy reached at subduction zones in rocky super-Earths to ensure foundering? Is dissipation during subduction large enough to slow down plates to the point of halting plate tectonics?
The discovery of super-Earths has recently opened an opportunity for comparative planetology. It promises to widen the current geophysical theories that have been developed to explain Earth's and the other terrestrial planets' properties. This places thinking about the solid planets in our solar system within a wider, more complete planetary context. We consider this study to be a step in that direction.
Section snippets
Different approaches
The subject of plate tectonics on Earth has been approached in three different ways: with analytical theories, numerical modeling and experimental work, all complementary to one another. The approach by OL07 to study tectonics on massive Earth-like planets was to adapt the numerical model by Moresi and Solomatov (1998). This finite element code was developed to reproduce plate-like behaviour on Earth. It considered a temperature-dependent viscous mantle heated from below overlain by a
Convective parameters
We elaborate on the parameterized convection analysis to show the implicit dependence of the convective parameters on the mass of the planet (M). This is done via the Rayleigh number (Ra). In the case of a fluid layer of depth D that is internally heated, the Rayleigh number can be defined in terms of the surface heat flux q:where g is the gravity and the material properties are: the density ρ, coefficient of thermal expansion α, thermal diffusivity κ, thermal conductivity k, and
Planetary heat flux
A planet's surface heat flow Q, reflects the amount of heat being transported into the mantle from the core Qcore, generated within from radioactive heat sources Qrh and its cooling rate at a given point in time dT(t)/dt:
The contribution from radioactive sources to the heat flow iswhere Ri is the amount of heat produced per mass by each of the major radioactive elements 238U, 235U, 232Th, and 40 K, γi is the amount of each
Scaling with mass
To get the dependence of plate thickness and shear stress on mass, we use a detailed internal structure model to obtain the mantle size, average mantle density, average gravity, and viscosity under the plate for rocky super-Earths. Since the details of the model have been explained elsewhere (Valencia et al., 2006, Valencia et al., 2007c), we only briefly describe it here. It solves the differential equations of density, gravity, mass, pressure and temperature with depth. It uses a Vinet
Fault strength
We arrive at the discussion of fault strength and investigate whether or not it outweighs the increase in convective stresses as suggested by OL07. Pressure does increases the frictional strength on faults and hence, can potentially hinder deformation on large planets.
We determine how the stress and the strength on the faults varies with planetary mass. For deformation to happen, the shear stress on the fault (τ) has to overcome its strength (τrock). In the same manner as (Moresi and Solomatov,
Lithospheric density
At convergent margins on Earth, the oldest, coldest and densest plate is expected to subduct under the younger, hotter and buoyant plate. Knowing that super-Earths have faster convective velocities (Valencia et al., 2007a) and hence, younger plates in general, it is appropriate to establish the conditions in which negative buoyancy can still be achieved at subduction zones.
We calculate the mean density of a plate (ρ¯¯lit) composed of basaltic crust (ρbas = 2880 kg/m3) and lithospheric mantle (ρman =
Energy dissipation during subduction
In this section we explore the effects of energy dissipation during subduction as a scenario that might halt plate tectonics. Conrad and Hager (1999a) recognized that a major source of dissipation can be the subduction of thick strong plates. If the energy required in subducting the plate is large, the plate will slow down, thicken even more making subduction more difficult, until eventually it can no longer be sustained. We investigate the effects of this process by modeling subduction in two
Discussion
Plate tectonics is an evolutionary phenomenon and the conditions for it will change as the planet evolves and cools. At present, Earth has enough heat flux to drive vigorous convection needed for an active plate state. A planet's heat flux varies with time and while on short timescales it might increase (Sleep, 2000, Van Keken et al., 2001), over long timescales it decreases as the planet cools. This means, that at some stage, when the heat flux falls below a threshold, plate tectonics will
Summary and conclusions
It is important to establish the tectonic regime of a planet when determining its thermal state, which in turn is fundamental to the question of habitability. Here, we examine from classical boundary layer theory, how the convective parameters and fault properties scale for terrestrial super-Earths. We address three key issues of plate tectonics: deformation on faults, negative buoyancy, and energy dissipation during subduction. Even though the mantle is a complicated system (with chemical
References (43)
- et al.
A climate induced transition in the tectonic style of a terrestrial planet
EPSL
(2008) - et al.
Conditions for the onset of plate tectonics on terrestrial planets and moons
EPSL
(2007) - et al.
The density structure of subcontinental lithosphere through time
Earth Planet. Sci. Lett.
(2001) - et al.
Internal structure of massive terrestrial planets
Icarus
(2006) - et al.
A dynamical investigation of the heat and helium imbalance
Earth Planet. Sci. Lett.
(2001) - et al.
Density structure and buoyancy of the oceanic lithosphere revisited
JRL
(2007) Gross thermodynamics of heat engines in deep interior of Earth
Proc. Nat. Acad. Sci.
(1975)- et al.
The development of slabs in the upper mantle: insights from numerical and laboratory experiments
JGR
(1999) Testing hypotheses on plate-driving mechanisms with global lithosphere models including topography, thermal structure, and faults
JGR
(1998)- et al.
Exoplanet detection capability of the COROT space mission
Astron. Astrophys.
(2003)