Convection In Planetary Interiors Research Articles

Assessing the Accuracy of 2‐D Planetary Evolution Models Against the 3‐D Sphere

AbstractRegardless of the steady increase of computing power during the last decades, numerical models in a 3D spherical shell are only used in specific setups to investigate the thermochemical convection in planetary interiors, while 2D geometries are typically favored in most exploratory studies involving a broad range of parameters. The 2D cylindrical and the more recent 2D spherical annulus geometries are predominantly used in this context, but the extent to how well they reproduce the 3D spherical shell results in comparison to each other and in which setup has not yet been extensively investigated. Here we performed a thorough and systematic study in order to assess which 2D geometry reproduces best the 3D spherical shell. In a first set of models, we investigated the effects of the geometry on thermal convection in steady‐state setups while varying a broad range of parameters. Additional thermal evolution models of three terrestrial bodies, namely Mercury, the Moon, and Mars, which have different interior structures, were used to compare the 2D and 3D geometries. Our investigations show that the 2D spherical annulus geometry provides results closer to models in a 3D spherical shell compared to the 2D cylindrical geometry. Our study indicates where acceptable differences can be expected when using a 2D instead of a 3D geometry and where to be cautious when interpreting the results.

Onset of convection in internally heated, temperature-dependent, power-law viscosity fluids at large viscosity contrasts

We use two-dimensional numerical simulations to investigate the finite-amplitude onset of convection in internally heated, infinite Prandtl number fluids with strongly temperature-dependent, power-law viscosity. We focus on the stagnant-lid regime which is relevant to planetary interiors. We find that convection can occur in both the usual, widespread convection planform, where the convection cells form in the entire layer, as well as the localized convection planform characterized by a single upwelling surrounded by a nearly stationary fluid. The critical Rayleigh number for the onset of convection by finite-amplitude perturbations is nearly independent of the convection planform in a broad range of the spacings between upwellings, from of the order of the depth of the layer to up to infinity. A simple heuristic analysis suggests scaling relationships which fit the numerical results in the stagnant-lid regime for an arbitrary power-law exponent, n>1. The results of this study provide new fluid dynamical constrains on the threshold for convection in planetary interiors.

Thermo‐electrohydrodynamic convection in a differentially heated shell with electric central force field

AbstractThermo‐electrohydrodynamic convection is investigated in a shell with central force field to investigate convective pattern formation for potential heat transfer application and small scaled laboratory experiments e.g. the modelling of convection in planetary interiors or atmospheres. To induce TEHD convection the outer and inner shells are heated at different temperatures whereas an alternating electric potential is applied between both shells. The observation of the convective pattern formation suggest that dielectric fluids have a significant response to thermal and electrical forcing. The observed patterns ranged from quasi‐stationary convective modes in azimuthal direction to shape and mode amplitude vacillations with mode merging and separation that are classified into three regimes. The heat transfer was evaluated by the Nusselt number and showed an increase in heat transfer with electric forcing. However, the results indicate a much smaller heat transfer when compared to natural convection.

Scale of plume clustering in large-Prandtl-number convection.

Clustering of plumes in turbulent Rayleigh-Bénard convection has been numerically observed in low-Prandtl-number fluids. In this framework, turbulent plumes undergo a phase-separation process leading to large-scale clusters and circulations, sometimes called plume superstructures and reminiscent of solar granulation and supergranulation. On the other hand, the possible presence of large-scale plume aggregates has not been explored in the case of large values of the Prandtl number, Pr, relevant to geological settings such as convection in planetary interiors. Here we address this problem and numerically explore the behavior of plume ensembles in turbulent convection at very high Prandtl number values, including the case Pr→∞. The results indicate the presence of plume clustering, albeit at smaller scale, also for large Pr number fluids, suggesting interesting consequences for mantle convection processes.

Stability range of localized subcritical Rayleigh–Bénard convection in temperature-dependent viscosity fluids: Constraints from two-dimensional simulations

We use a finite element code CITCOM to determine the stability range of a unique planform of Rayleigh–Bénard convection in temperature-dependent viscosity fluids—a spatially localized upwelling. Such a planform can only exist at Rayleigh numbers below the critical Rayleigh number for the linear onset of convection. Its initiation requires a finite-amplitude perturbation. We perform numerical simulations in long two-dimensional rectangular boxes with free-slip boundary conditions, which are relevant to convection in planetary interiors. The main result is a stability map of spatially localized single upwellings. Localized upwellings can exist at viscosity contrasts higher than about 150. The range of Rayleigh numbers at which localized upwellings are stable increases with the viscosity contrast. At the largest viscosity contrast studied, 1010, localized upwellings are stable in the range between about 0.5 and 0.9 of the critical Rayleigh number. The horizontal scale of localized upwellings is on the order of the layer thickness and decreases with the viscosity contrast. Spatially localized upwellings can occur on icy satellites and small planets where the conditions are likely to be subcritical due to the small size of these bodies. Localized upwellings might have industrial applications as well, such as patterning of polymers.

Scaling relationships and physics for mixed heating convection in planetary interiors: Isoviscous spherical shells

AbstractWe use a suite of 3‐D numerical experiments to test and expand 2‐D planar isoviscous scaling relationships of Moore (2008) for mixed heating convection in spherical geometry mantles over a range of Rayleigh numbers (Ra). The internal temperature scaling of Moore (2008), when modified to account for spherical geometry, matches our experimental results to a high degree of fit. The heat flux through the boundary layers scale as a linear combination of internal (Q) and basal heating, and the modified theory predictions match our experimental results. Our results indicate that boundary layer thickness and surface heat flux are not controlled by a local boundary layer stability condition (in agreement with the results of Moore (2008)) and are instead strongly influenced by boundary layer interactions. Subadiabatic mantle temperature gradients, in spherical 3‐D, are well described by a vertical velocity scaling based on discrete drips as opposed to a scaling based on coherent sinking sheets, which was found to describe 2‐D planar results. Root‐mean‐square (RMS) velocities are asymptotic for both low Q and high Q, with a region of rapid adjustment between asymptotes for moderate Q. RMS velocities are highest in the low Q asymptote and decrease as internal heating is applied. The scaling laws derived by Moore (2008), and extended here, are robust and highlight the importance of differing boundary layer processes acting over variable Q and moderate Ra.

Rheological decoupling at the Moho and implication to Venusian tectonics

Plate tectonics is largely responsible for material and heat circulation in Earth, but for unknown reasons it does not exist on Venus. The strength of planetary materials is a key control on plate tectonics because physical properties, such as temperature, pressure, stress, and chemical composition, result in strong rheological layering and convection in planetary interiors. Our deformation experiments show that crustal plagioclase is much weaker than mantle olivine at conditions corresponding to the Moho in Venus. Consequently, this strength contrast may produce a mechanical decoupling between the Venusian crust and interior mantle convection. One-dimensional numerical modeling using our experimental data confirms that this large strength contrast at the Moho impedes the surface motion of the Venusian crust and, as such, is an important factor in explaining the absence of plate tectonics on Venus.

Rapidly rotating plane layer convection with zonal flow

The onset of convection in a rapidly rotating layer in which a thermal wind is present is studied. Diffusive effects are included. The main motivation is from convection in planetary interiors, where thermal winds are expected due to temperature variations on the core–mantle boundary. The system admits both convective instability and baroclinic instability. We find a smooth transition between the two types of modes, and investigate where the transition region between the two types of instability occurs in parameter space. The thermal wind helps to destabilise the convective modes. Baroclinic instability can occur when the applied vertical temperature gradient is stable, and the critical Rayleigh number is then negative. Long wavelength modes are the first to become unstable. Asymptotic analysis is possible for the transition region and also for long wavelength instabilities, and the results agree well with our numerical solutions. We also investigate how the instabilities in this system relate to the classical baroclinic instability in the Eady problem. We conclude by noting that baroclinic instabilities in the Earth's core arising from heterogeneity in the lower mantle could possibly drive a dynamo even if the Earth's core were stably stratified and so not convecting.

Using magnetic fluids to simulate convection in a central force field in the laboratory

Abstract. Large-scale convection in planetary or stellar interiors plays a significant role but it is difficult to reproduce the central force field of those systems in experimental studies. A technique to approximate a central force field through the magnetic field from magnets acting on a magnetic liquid is presented. The thermomagnetic convection in a spherical shell filled with a magnetic liquid is analyzed in the context of a terrestrial laboratory using a 2D Finite Element model. Two configurations of magnetic fields were investigated, one resulting in a radially decreasing force field, and the other in a radially increasing force field. The results suggest that, while the actual force field does not reproduce the central gravity in planetary interiors accurately, it captures the essential qualitative character of the flow unlike other terrestrial experiments, which were either dominated by gravity or by a cylindrical radial force field. It is therefore suggested that such an experiment would provide a useful tool to investigate thermal convection in planetary interiors.

Numerical approaches to solid-state convection in planetary bodies

Numerical methods suitable for the calculation of finite-amplitude thermal convection in planetary interiors are reviewed. Three methods of approach are described and compared; namely, finite-difference, finite-element and spectral procedures. Application of each approach to three-dimensional solid-state convection with complex rheology is examined. Finite-difference methods have enjoyed some success to date. However, future developments are likely to favor finite-element methods when rheology and boundaries are complex, and spectral methods when accuracy is paramount and rheology and boundaries are simple.

Evidence for convection in planetary interiors from first-order topography

Center of mass-center of figure offsets are known for the Earth, Moon, Mars and Venus. Such an offset requires a density distribution asymmetric about the center of mass. Observational evidence indicates that the terrestrial, lunar and Martian offsets result from crusts of variable thickness rather than lateral density inhomogeneities and that the thickness variations are more likely caused by internal convection than impact.

Session II: Internal Constitution and Thermal Histories of the Terrestrial Planets

Hall: On behalf of the Organizing Committee of Joint Discussion A it gives me great pleasure to introduce our next speaker, Dr B. Levin. He is Chief of the Department on the Evolution of the Earth at the Schmidt Institute of Physics of the Earth at Moscow. He will present an invited paper on the Internal Constitution and Thermal Histories of Terrestrial Planets.Levin: The problems of the internal constitution of the terrestrial planets and that of their thermal histories are closely connected. We are interested not only in giving the distribution of density in planetary interiors and their stratification into physically distinct layers, but also in giving some insight into the chemical gross composition of planets. At the same time the thermal histories of the terrestrial planets strongly depend on the content of radioactive elements and thus on the chemical composition. The differentiation of planetary interiors into chemically distinct layers depends on their temperatures which determine their viscosity, but, on the other hand, the redistribution of radioactive elements in the course of differentiation seriously change the further course of the thermal history. The controversial problem of thermal convection in planetary interiors depends both on their constitution and on their thermal history.

The dependence of convection in planetary interiors upon the magnitude of the rayleigh number

The conservation equations of mass, energy and momentum when applied to the problem of convection produced by isothermal compression as well as by thermal expansion, in self-gravitating, homogeneous, non rotating spheres with uniform viscosity and under steady flow conditions, lead to a relationship between two characteristic numbers. In the present paper these characteristic numbers are evaluated, for both rigid and free surfaces, for spherical harmonic disturbances of orders 1→6.

Permissible modes of nonthermal convection in planetary interiors

Boundary conditions are imposed upon the solutions of the conservation equations for nonthermal convective motion in self-gravitating, homogeneous, nonrotating spheres of compressible fluid, with uniform viscosity and under steady flow conditions. It is shown that the only permissible modes of convective motion are those for which n = 1 and 2.

Convection in planetary interiors

The aim of the present paper is to formulate explicitly the linearized equations safe-guarding the conservation of mass, energy, and momentum of viscous flow inside a heterogeneous, compressible, rotating fluid configuration, in which the coefficient of viscosity is an arbitrary function of radial distance, and the rotation is about a fixed z axis with angular velocity ω. These equations are then applied to the problem of convection in a homogeneous, self-gravitating, nonrotating fluid sphere of uniform viscosity in which departures from the state of equilibrium give rise to internal motion. The changes of density and radial velocity accompanying steady flow inside the sphere, caused by isothermal compression as well as by thermal expansion, are investigatedunder various boundary conditions and lead to the appearance of certain characteristic numbers.

The combined effect of thermal and nonthermal convection in planetary interiors

In the present paper convection, produced jointly by thermal and nonthermal mechanisms, is investigated in self-grabitating, homogeneous, nonrotating spheres of compressible fluid, with uniform viscosity and under steady flow conditions. It is found that applying boundary conditions leads to a certain relationship between two charteristic numbers.

Convection in planetary interiors

The aim of the present paper will be to formulate explicitly the linearized equations safeguarding the conservation of mass, energy, and momentum of viscous flow inside a heterogeneous, compressible fluid sphere, in which the coefficient of viscosity is an arbitrary function of central distance. An application of such equations to the problem of convection in lunar or planetary interiors is considered; and it is pointed out that the changes in density accompanying convective flow are caused mainly by isothermal compression of the respective rocks rather than by their thermal expansion.