We have modelled the time-dependent dynamics of exosolar planets within the framework of a two-dimensional Cartesian model and the extended-Boussinesq approximation. The mass of the super-Earth models considered is 8 times the Earth's mass and the thickness of the mantle is 4700 km, based on a constant density approximation and a similar core mass fraction as in the Earth. The effects of depth-dependent properties have been considered for the thermal expansion coefficient, the viscosity and thermal conductivity. The viscosity and thermal conductivity are also temperature-dependent. The thermal conductivity has contributions from phonons, photons and electrons. The last dependence comes from the band-gap nature of the material under high pressure and increases exponentially with temperature and kicks in at temperatures above 5000 K. The thermal expansivity decreases by a factor of 20 across the mantle because of the high pressures, greater than 1 TPa in the deep mantle. We have varied the temperatures at the core-mantle boundary between 6000 and 10,000 K. Accordingly the Rayleigh number based on the surface values varies between 3.5 × 1 07 and 7 × 1 07 in the different models investigated. Three phase transitions have been considered: the spinel to perovskite, the post-perovskite transition and the post-perovskite decomposition in the deep lower mantle. We have considered an Arrhenius type of temperature dependence in the viscosity and have extended the viscosity contrast due to temperature to over one million. The parameter values put us well over into the stagnant lid regime. Our numerical results show that because of the multiple phase transitions and strongly depth-dependent properties, particularly the thermal expansitivity, initially most of the planetary interior is strongly super-adiabatic in spite of a high surface Rayleigh number, because of the presence of partially layered and penetrative convective flows throughout the mantle, very much unlike convection in the Earth's mantle. But with the passage of time, after several billion years, the temperature profiles become adiabatic. The notable influence of electronic thermal conductivity is to heat up the bottom boundary layer quasi-periodically, giving rise to strong coherent upwellingss, which can punch their way to the upper mantle and break up the layered convective pattern.
Bibliographical noteFunding Information:
We thank discussions with Phil Allen, Michel Jacobs, Radek Matyska, Joost van Summeren, Diane Valencia, Renata Wentcovitch and K. Umemoto. Constructive reviews of Craig O’Neil and an anonymous reviewer were very helpful for improving the manuscript. This research has been supported by the ITR grant from National Science Foundation given to the VLAB at the University of Minnesota. Computational resources were provided through the Netherlands Research Center for Integrated Solid Earth Science (ISES 3.2.5).
- Exosolar planets
- Mantle convection
- Phase transition
- Variable conductivity