scholarly journals 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

2021 ◽  
Vol 73 (1) ◽  
Author(s):  
Masanori Kameyama
Keyword(s):  
Thermal Conductivity ◽  
Thermal Convection ◽  
Mantle Convection ◽  
Thermal Stratification ◽  
Variable Viscosity ◽  
Critical Rayleigh Number ◽  
Spherical Geometry ◽  
Compressible Fluids ◽  
Strong Temperature Dependence ◽  
Depth Dependence

AbstractIn 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.

scholarly journals Linear Analysis On The Onset Of Thermal Convection of Highly Compressible Fluids With Variable Physical Properties In Spherical Geometry: Implications For The Mantle Convection of Super-Earths

2021 ◽  
Author(s):  
Masanori Kameyama
Keyword(s):  
Thermal Conductivity ◽  
Physical Properties ◽  
Thermal Convection ◽  
Mantle Convection ◽  
Thermal Stratification ◽  
Critical Rayleigh Number ◽  
Linear Analysis ◽  
Compressible Fluids ◽  
Strong Temperature Dependence ◽  
Depth Dependence

Abstract In this paper we carried out a series of linear analysis 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. The variations in thermodynamic properties (thermal expansivity and reference density) with depth are taken to be relevant for the super-Earths with 10 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 to 109, 15 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 ows 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 suffciently large, the thermal stratification can be greatly stabilised 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.

Stability of Thermal Convection in a Vertical Porous Layer

1987 ◽  
Vol 109 (4) ◽  
pp. 889-893 ◽  
Author(s):  
L. P. Kwok ◽  
C. F. Chen
Keyword(s):  
Rayleigh Number ◽  
Linear Stability ◽  
Porous Layer ◽  
Thermal Convection ◽  
Variable Viscosity ◽  
Critical Rayleigh Number ◽  
Linear Stability Theory ◽  
Lateral Heating ◽  
The Stability ◽  
Allowable Temperature

Experiments were carried out to study the stability of thermal convection generated in a vertical porous layer by lateral heating in a tall, narrow tank. The porous medium, consisting of glass beads, was saturated with distilled water. It was found that the flow became unstable at a critical ΔT of 29.2°C (critical Rayleigh number of 66.2). Linear stability analysis was applied to study the effects of the Brinkman term and of variable viscosity separately using a quadratic relationship between the density and temperature. It was found that with the Brinkman term, no instability could occur within the allowable temperature difference across the tank. With the effect of variable viscosity included, linear stability theory predicts a critical ΔT of 43.4°C (Rayleigh number of 98.3).

scholarly journals On the sensitivity of 3-D thermal convection codes to numerical discretization: a model intercomparison

2014 ◽  
Vol 7 (5) ◽  
pp. 2065-2076 ◽  
Author(s):  
P.-A Arrial ◽  
N. Flyer ◽  
G. B. Wright ◽  
L. H. Kellogg
Keyword(s):  
Rayleigh Number ◽  
Numerical Simulations ◽  
Spherical Harmonics ◽  
Thermal Convection ◽  
Mantle Convection ◽  
Numerical Study ◽  
Critical Rayleigh Number ◽  
Initial Condition ◽  
Numerical Discretization ◽  
The Impact

Abstract. Fully 3-D numerical simulations of thermal convection in a spherical shell have become a standard for studying the dynamics of pattern formation and its stability under perturbations to various parameter values. The question arises as to how the discretization of the governing equations affects the outcome and thus any physical interpretation. This work demonstrates the impact of numerical discretization on the observed patterns, the value at which symmetry is broken, and how stability and stationary behavior is dependent upon it. Motivated by numerical simulations of convection in the Earth's mantle, we consider isoviscous Rayleigh–Bénard convection at infinite Prandtl number, where the aspect ratio between the inner and outer shell is 0.55. We show that the subtleties involved in developing mantle convection models are considerably more delicate than has been previously appreciated, due to the rich dynamical behavior of the system. Two codes with different numerical discretization schemes – an established, community-developed, and benchmarked finite-element code (CitcomS) and a novel spectral method that combines Chebyshev polynomials with radial basis functions (RBFs) – are compared. A full numerical study is investigated for the following three cases. The first case is based on the cubic (or octahedral) initial condition (spherical harmonics of degree ℓ = 4). How this pattern varies to perturbations in the initial condition and Rayleigh number is studied. The second case investigates the stability of the dodecahedral (or icosahedral) initial condition (spherical harmonics of degree ℓ = 6). Although both methods first converge to the same pattern, this structure is ultimately unstable and systematically degenerates to cubic or tetrahedral symmetries, depending on the code used. Lastly, a new steady-state pattern is presented as a combination of third- and fourth-order spherical harmonics leading to a five-cell or hexahedral pattern and stable up to 70 times the critical Rayleigh number. This pattern can provide the basis for a new accuracy benchmark for 3-D spherical mantle convection codes.

scholarly journals On the sensitivity of 3-D thermal convection codes to numerical discretization: a model intercomparison

2014 ◽  
Vol 7 (2) ◽  
pp. 2033-2064
Author(s):  
P.-A. Arrial ◽  
N. Flyer ◽  
G. B. Wright ◽  
L. H. Kellogg
Keyword(s):  
Rayleigh Number ◽  
Numerical Simulations ◽  
Spherical Harmonics ◽  
Thermal Convection ◽  
Mantle Convection ◽  
Numerical Study ◽  
Critical Rayleigh Number ◽  
Initial Condition ◽  
Numerical Discretization ◽  
The Impact

Abstract. Fully 3-D numerical simulations of thermal convection in a spherical shell have become a standard for studying the dynamics of pattern formation and its stability under perturbations to various parameter values. The question arises as to how does the discretization of the governing equations affect the outcome and thus any physical interpretation. This work demonstrates the impact of numerical discretization on the observed patterns, the value at which symmetry is broken, and how stability and stationary behavior is dependent upon it. Motivated by numerical simulations of convection in the Earth's mantle, we consider isoviscous Rayleigh-Bénard convection at infinite Prandtl number, where the aspect ratio between the inner and outer shell is 0.55. We show that the subtleties involved in development mantle convection models are considerably more delicate than has been previously appreciated, due to the rich dynamical behavior of the system. Two codes with different numerical discretization schemes: an established, community-developed, and benchmarked finite element code (CitcomS) and a novel spectral method that combines Chebyshev polynomials with radial basis functions (RBF) are compared. A full numerical study is investigated for the following three cases. The first case is based on the cubic (or octahedral) initial condition (spherical harmonics of degree ℓ =4). How variations in the behavior of the cubic pattern to perturbations in the initial condition and Rayleigh number between the two numerical discrezations is studied. The second case investigates the stability of the dodecahedral (or icosahedral) initial condition (spherical harmonics of degree ℓ = 6). Although both methods converge first to the same pattern, this structure is ultimately unstable and systematically degenerates to cubic or tetrahedral symmetries, depending on the code used. Lastly, a new steady state pattern is presented as a combination of order 3 and 4 spherical harmonics leading to a five cell or a hexahedral pattern and stable up to 70 times the critical Rayleigh number. This pattern can provide the basis for a new accuracy benchmark for 3-D spherical mantle convection codes.

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