Weakly non-Boussinesq convection in a gaseous spherical shell (original) (raw)
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Boussinesq convection in a gaseous spherical shell
Eas Publications Series, 2019
In this paper, we investigate the dynamics of convection in a spherical shell under the Boussinesq approximation but considering the compressibility which arises from a non zero adiabatic temperature gradient, a relevant quantity for gaseous objects such as stellar or planetary interiors. We find that depth-dependent superiadiabaticity, combined with the use of mixed boundary conditions (fixed flux/fixed temperature), gives rise to unexpected dynamics that were not previously reported.
Journal Of Geophysical Research: Solid Earth, 2016
We 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-meansquare (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. The goal of this study is to evaluate and expand the theoretical scaling relationships of Moore [2008]. The theory was developed for a 2-D planar system, which is designed to emulate physical tank experiments. We extend it to address a 3-D spherical shell system, designed to emulate planetary interiors. We then use a large suite of numerical experiments to test the theoretical scaling predictions. We focus on isoviscous systems in order to test the scaling theory as straight forwardly, and comparably, as possible. We will extend the theory to include temperature-and depth-dependent viscosities, as well as surface yielding in future work.
Off-centre gravity induces large-scale flow patterns in spherical Rayleigh–Bénard convection
Journal of Fluid Mechanics
We perform direct numerical simulations to study the effect of the gravity centre offset in spherical Rayleigh–Bénard convection. When the gravity centre is shifted towards the south, we find that the shift of the gravity centre has a pronounced influence on the flow structures. At low Rayleigh number RaRaRa , a steady-state large-scale meridional circulation induced by the baroclinic imbalance, created by the misalignment of the gravity potentials and isotherms, is formed. At high RaRaRa , an energetic jet is created on the northern side of the inner sphere that is directed towards the outer sphere. The large-scale circulation induces a strong co-latitudinal dependence in the local heat flux. Nevertheless, the global heat flux is not affected by the changes in the large-scale flow organization induced by the gravity centre offset.
2012
In this study we investigate the flow of a Boussinesq fluid contained in a rotating, differentially heated spherical shell. Previous work, on the spherical shell of Boussinesq fluid, differentially heated the shell by prescribing temperature on the inner boundary of the shell, setting the temperature deviation from the reference temperature to vary proportionally with − cos 2θ, from the equator to the pole. We change the model to include an energy balance equation at the earth’s surface, which incorporates latitudinal solar radiation distribution and ice-albedo feedback mechanism with moving ice boundary. For the fluid velocity, on the inner boundary, two conditions are considered: stress-free and no-slip. However, the model under consideration contains only simple representations of a small number of climate variables and thus is not a climate model per se but rather a tool to aid in understanding how changes in these variables may affect our planet’s climate. The solution of the m...
Non-Boussinesq simulations of Rayleigh–Bénard convection in a perfect gas
Physics of Fluids, 2004
We present direct numerical simulations of Boussinesq and non-Boussinesq Rayleigh-Bénard convection in a rigid box containing a perfect gas. For small stratifications, which includes Boussinesq fluids, the first instability after steady rolls was an oscillatory instability ͑a Hopf bifurcation͒. The resulting convection was characterized by two hot and two cold blobs circulating each convective roll. The same sign thermal perturbations ͑blobs͒ are at diametrically opposite points on the circular rolls, i.e., they are symmetric about the roll center. The time for a hot ͑or cold͒ blob to circulate a roll was between two and three roll turnover times. When the stratification was of sufficient strength, there was a dramatic change in the nature of the bifurcation. The sign of the thermal perturbations became antisymmetric with respect to the roll center, i.e., a hot blob was diametrically opposite a cold blob. In this case, a hot or cold blob circulated around each roll in about one turnover time. In a stratified layer, the Rayleigh number varies with height. We found that at the Hopf bifurcation, the Rayleigh number at the base was closest to the Boussinesq value. The change in instability appeared to be related to an increase in the speed ͑or Mach number͒ of the circulating rolls. It did not seem to be affected by the transport property variation with temperature. If the along roll aspect ratio was less than 2 or the walls perpendicular to the roll axis periodic, then only the symmetric instability could be found. We describe how our results might be reproduced in a laboratory experiment of convection in cryogenic helium gas.
The onset of thermo-compositional convection in rotating spherical shells
Geophysical & Astrophysical Fluid Dynamics, 2019
Double-diffusive convection driven by both thermal and compositional buoyancy in a rotating spherical shell can exhibit a rather large number of behaviours often distinct from that of the single diffusive system. In order to understand how the differences in thermal and compositional molecular diffusivities determine the dynamics of thermo-compositional convection we investigate numerically the linear onset of convective instability in a double-diffusive setup. We construct an alternative equivalent formulation of the non-dimensional equations where the linearised double-diffusive problem is described by an effective Rayleigh number, Ra, measuring the amplitude of the combined buoyancy driving, and a second parameter, α, measuring the mixing of the thermal and compositional contributions. This formulation is useful in that it allows for the analysis of several limiting cases and reveals dynamical similarities in the parameters space which are not obvious otherwise. We analyse the structure of the critical curves in this Ra−α space, explaining asymptotic behaviours in α, transitions between inertial and diffusive regimes, and transitions between large scale (fast drift) and small scale (slow drift) convection. We perform this analysis for a variety of diffusivities, rotation rates and shell aspect ratios showing where and when new modes of convection take place.
Non-Boussinesq Low-Prandtl-number Convection with a Temperature-dependent Thermal Diffusivity
The Astrophysical Journal, 2021
In an attempt to understand the role of the strong radial dependence of thermal diffusivity on the properties of convection in sun-like stars, we mimic that effect in non-Oberbeck-Boussinesq (NOB) convection in a horizontally-extended rectangular domain (aspect ratio 16), by allowing the thermal diffusivity κ to increase with the temperature (as in the case of stars). Direct numerical simulations (i.e., numerical solutions of the governing equations by resolving up to the smallest scales without requiring any modeling) show that, in comparison with Oberbeck-Boussinesq (OB) simulations (two of which we perform for comparison purposes), the symmetry of the temperature field about the midhorizontal plane is broken, whereas the velocity and heat flux profiles remain essentially symmetric. Our choice of κ(T), which resembles the variation in stars, results in the temperature field that loses its fine structures towards the hotter part of the computational domain, but the characteristic large scale of the turbulent thermal 'superstructures', which are structures whose size is typically larger than the depth of the convection domain, continue to be largely independent of the depth.
EPL (Europhysics Letters), 2009
We demonstrate the specific non-Boussinesq roles played by various fluid properties in thermal convection by allowing each of them to possess, one at a time, a temperature dependence that could be either positive or negative. The negative temperature dependence of the coefficient of thermal expansion hinders effective thermal convection and reduces the Nusselt number, whereas the negative dependence of fluid density enhances the Nusselt number. Viscosity merely smears plume generation and has a marginal effect on heat transport, whether it increases or decreases with temperature. At the moderate Rayleigh number examined here, the specific heat capacity shows no appreciable effect. On the other hand, the conductivity of the fluid near the hot surface controls the heat transport from the hot plate to the fluid, which suggests that a less conducting fluid near the bottom surface will reduce the Nusselt number and the bulk temperature.