Linear analysis on the onset of thermal convection of highly compressible fluids with variable viscosity and thermal conductivity in spherical geometry: implications for the mantle convection of super-Earths

Masanori Kameyama

doi:10.1186/s40623-021-01499-w

Masanori Kameyama

Open Access

https://doi.org/10.1186/s40623-021-01499-w

Copy DOI

Abstract

In this paper, we carried out a series of linear analyses on the onset of thermal convection of highly compressible fluids whose physical properties strongly vary in space in convecting vessels either of a three-dimensional spherical shell or a two-dimensional spherical annulus geometry. The variations in thermodynamic properties (thermal expansivity and reference density) with depth are taken to be relevant for the super-Earths with ten times the Earth’s mass, while the thermal conductivity and viscosity are assumed to exponentially depend on depth and temperature, respectively. Our analysis showed that, for the cases with strong temperature dependence in viscosity and strong depth dependence in thermal conductivity, the critical Rayleigh number is on the order of 108–109, implying that the mantle convection of massive super-Earths is most likely to fall in the stagnant-lid regime very close to the critical condition, if the properties of their mantle materials are quite similar to the Earth’s. Our analysis also demonstrated that the structures of incipient flows of stagnant-lid convection in the presence of strong adiabatic compression are significantly affected by the depth dependence in thermal conductivity and the geometries of convecting vessels, through the changes in the static stability of thermal stratification of the reference state. When the increase in thermal conductivity with depth is sufficiently large, the thermal stratification can be greatly stabilized at depth, further inducing regions of insignificant fluid motions above the bottom hot boundaries in addition to the stagnant lids along the top cold surfaces. We can therefore speculate that the stagnant-lid convection in the mantles of massive super-Earths is accompanied by another motionless regions at the base of the mantles if the thermal conductivity strongly increases with depth (or pressure), even though their occurrence is hindered by the effects the spherical geometries of convecting vessels.

Highlights

Super-Earths are extra-solar planets which have small masses and high mean density larger than 5000 kg/m3 (Howard et al 2010)
To see how the depth dependence in thermal conductivity k affects the critical states of stagnant-lid convection, we show in Fig. 3 the schematic illustration on two-dimensional planes of the incipient flows obtained for several values of rk and various model geometries together with rη = 108 . (For the cases in 3D spherical geometry, we show the flow structures in meridian planes assuming the incipient flows occurring with only zonal components whose spherical harmonic order is m = 0 .) Shown in color are the distributions in the perturbations in temperature T ′ whose color scales are indicated at the bottom of the figure
In this paper, we carried out a series of linear analyses on the onset of thermal convection of highly compressible fluids whose physical properties strongly vary in space in convecting vessels of either a three-dimensional (3D) spherical shell or a two-dimensional (2D) spherical annulus, by extending our earlier work in the Cartesian geometry (Kameyama 2016): the variations in thermodynamic properties with depth are taken to be relevant for the super-Earths with ten times the Earth’s mass, while the thermal conductivity and viscosity are assumed to exponentially depend on depth and temperature, respectively

Summary

Introduction

Super-Earths are extra-solar planets which have small masses (compared to that of Jupiter) and high mean density larger than 5000 kg/m3 (Howard et al 2010). 83.34 45.43 20.09 10 8.360 2.829 flows with those in the static stability of thermal stratification for the reference state, to investigate how the nature of incipient convection of compressible fluids with spatial variations in physical properties is affected by the adiabatic temperature change and, in other words, the occurrence of regions of stable stratification.

Results

Conclusion