Convection In Shells Research Articles

Turbulent Rayleigh–Bénard convection in spherical shells

We simulate numerically Boussinesq convection in non-rotating spherical shells for a fluid with a Prandtl number of unity and for Rayleigh numbers up to $10^{9}$. In this geometry, curvature and radial variations of the gravitational acceleration yield asymmetric boundary layers. A systematic parameter study for various radius ratios (from ${\it\eta}=r_{i}/r_{o}=0.2$ to ${\it\eta}=0.95$) and gravity profiles allows us to explore the dependence of the asymmetry on these parameters. We find that the average plume spacing is comparable between the spherical inner and outer bounding surfaces. An estimate of the average plume separation allows us to accurately predict the boundary layer asymmetry for the various spherical shell configurations explored here. The mean temperature and horizontal velocity profiles are in good agreement with classical Prandtl–Blasius laminar boundary layer profiles, provided the boundary layers are analysed in a dynamical frame that fluctuates with the local and instantaneous boundary layer thicknesses. The scaling properties of the Nusselt and Reynolds numbers are investigated by separating the bulk and boundary layer contributions to the thermal and viscous dissipation rates using numerical models with ${\it\eta}=0.6$ and with gravity proportional to $1/r^{2}$. We show that our spherical models are consistent with the predictions of Grossmann & Lohse’s (J. Fluid Mech., vol. 407, 2000, pp. 27–56) theory and that $\mathit{Nu}(\mathit{Ra})$ and $\mathit{Re}(\mathit{Ra})$ scalings are in good agreement with plane layer results.

Software for numerical simulation of convection in spherical shells for hybrid CPU/GPU computing systems

This paper describes software developed by the authors for numerical 3D simulation of problems on hydrodynamics and heat transfer in spherical shells. The software was based on the OpenCL standard and intended for hybrid CPU/GPU computing systems with graphical processing units. The computations were implemented on a computing system with an Nvidia GTX Titan graphics accelerator. The presented results of the tests demonstrate the correctness of the obtained numerical solutions and the efficient use of GPUs for solving problems of computational fluid dynamics.

Hybrid C–O–Ne white dwarfs as progenitors of Type Ia supernovae: dependence on Urca process and mixing assumptions

When carbon is ignited off-centre in a CO core of a super-asymptotic giant branch star, its burning in a convective shell tends to propagate to the centre. Whether the C flame will actually be able to reach the centre depends on the efficiency of extra mixing beneath the C convective shell. Whereas thermohaline mixing is too inefficient to interfere with the C-flame propagation, convective boundary mixing can prevent the C burning from reaching the centre. As a result, a C–O–Ne white dwarf (WD) is formed, after the star has lost its envelope. Such a ‘hybrid’ WD has a small CO core surrounded by a thick ONe zone. In our 1D stellar evolution computations, the hybrid WD is allowed to accrete C-rich material, as if it were in a close binary system and accreted H-rich material from its companion with a sufficiently high rate at which the accreted H would be processed into He under stationary conditions, assuming that He could then be transformed into C. When the mass of the accreting WD approaches the Chandrasekhar limit, we find a series of convective Urca shell flashes associated with high abundances of 23Na and 25Mg. They are followed by off-centre C ignition leading to convection that occupies almost the entire star. To model the Urca processes, we use the most recent well-resolved data for their reaction and neutrino-energy loss rates. Because of the emphasized uncertainty of the convective Urca process in our hybrid WD models of Type Ia supernova (SN Ia) progenitors, we consider a number of their potentially possible alternative instances for different mixing assumptions, all of which reach a phase of explosive C ignition, either off or in the centre. Our hybrid SN Ia progenitor models have much lower C-to-O abundance ratios at the moment of the explosive C ignition than their pure CO counterparts, which may explain the observed diversity of the SNe Ia.

ERRATUM: “CHARACTERIZING THE CONVECTIVE VELOCITY FIELDS IN MASSIVE STARS” (2014, ApJ, 795, 92)

We apply the mathematical formalism of vector spherical harmonics decomposition to convective stellar velocity fields from multi-dimensional hydrodynamics simulations, and show that the resulting power spectra furnish a robust and stable statistical description of stellar convective turbulence. Analysis of the power spectra help identify key physical parameters of the convective process such as the dominant scale of the turbulent motions that influence the structure of massive evolved pre-supernova stars. We introduce the numerical method that can be used to calculate vector spherical harmonics power spectra from 2D and 3D convective shell simulation data. Using this method we study the properties of oxygen shell burning and convection for a 15 Msun star simulated by the hydrodynamics code FLASH in 2D and 3D. We discuss the importance of realistic initial conditions to achieving successful core-collapse supernova explosions in multi-dimensional simulations. We show that the calculated power spectra can be used to generate realizations of the velocity fields of pre-supernova convective shells. We find that the slope of the solenoidal mode power spectrum remains mostly constant throughout the evolution of convection in the oxygen shell in both 2D and 3D simulations. We also find that the characteristic radial scales of the convective elements are smaller in 3D than in 2D while the angular scales are larger in 3D.

HYDRODYNAMIC SIMULATIONS OF H ENTRAINMENT AT THE TOP OF He-SHELL FLASH CONVECTION

We present the first three-dimensional, fully compressible gas-dynamics simulations in 4π geometry of He-shell flash convection with proton-rich fuel entrainment at the upper boundary. This work is motivated by the insufficiently understood observed consequences of the H-ingestion flash in post-asymptotic giant branch (post-AGB) stars (Sakurai's object) and metal-poor AGB stars. Our investigation is focused on the entrainment process at the top convection boundary and on the subsequent advection of H-rich material into deeper layers, and we therefore ignore the burning of the proton-rich fuel in this study. We find that for our deep convection zone, coherent convective motions of near global scale appear to dominate the flow. At the top boundary convective shear flows are stable against Kelvin–Helmholtz instabilities. However, such shear instabilities are induced by the boundary-layer separation in large-scale, opposing flows. This links the global nature of thick shell convection with the entrainment process. We establish the quantitative dependence of the entrainment rate on grid resolution. With our numerical technique, simulations with 10243 cells or more are required to reach a numerical fidelity appropriate for this problem. However, only the result from the 15363 simulation provides a clear indication that we approach convergence with regard to the entrainment rate. Our results demonstrate that our method, which is described in detail, can provide quantitative results related to entrainment and convective boundary mixing in deep stellar interior environments with very stiff convective boundaries. For the representative case we study in detail, we find an entrainment rate of 4.38 ± 1.48 × 10−13 M☉ s−1.

THE FINAL FATE OF STARS THAT IGNITE NEON AND OXYGEN OFF-CENTER: ELECTRON CAPTURE OR IRON CORE-COLLAPSE SUPERNOVA?

In the ONeMg cores of $8.8-9.5~{\rm M}_\odot$ stars, neon and oxygen burning is ignited off-center. Whether the neon-oxygen flame propagates to the center is critical to determine whether these stars undergo Fe core collapse or electron capture induced ONeMg core collapse. We present more details of stars that ignite neon and oxygen burning off-center. The neon flame is established in a similar manner to the carbon flame of super-AGB stars, albeit with a narrower flame width. The criteria for establishing a flame are able to be met if the strict Schwarzschild criterion for convective instability is adopted. Mixing across the interface of the convective shell disrupts the conditions for the propagation of the burning front and instead the shell burns as a series of inward-moving flashes. While this may not directly affect whether the burning will reach the center (as in super-AGB stars), the core is allowed to contract between each shell flash. Reduction of the electron fraction in the shell reduces the Chandrasekhar mass and the center reaches the threshold density for the URCA process to activate and steer the remaining evolution of the core. This highlights the importance of a more accurate treatment of mixing in the stellar interior for yet another important question in stellar astrophysics - determining the properties of stellar evolution and supernova progenitors at the boundary between electron capture supernova and iron core-collapse supernova.

Ultra-weak magnetic fields in Am stars: β UMa and θ Leo

AbstractAn extremely weak circularly-polarized signature was recently detected in the spectral lines of the Am star Sirius A. With a prominent positive lobe, the shape of the phase-averaged Stokes V line profile is atypical of stellar Zeeman signatures, casting doubts on its magnetic origin. We report here on ultra-deep spectropolarimetric observations of two more bright Am stars: β Uma and θ Leo. Stokes V line signatures are detected in both objects, with a shape and amplitude similar to the one observed on Sirius A. We demonstrate that the amplitude of the Stokes V line profiles depend on various line parameters (Landé factor, wavelength, depth) as expected from a Zeeman signature, confirming that extremely weak magnetic fields are likely present in a large fraction of Am stars. We suggest that the strong asymmetry of the polarized signatures, systematically observed so far in Am stars and never reported in strongly magnetic Ap stars, bears unique information about the structure and dynamics of the thin surface convective shell of Am stars.

THE INITIATION AND PROPAGATION OF HELIUM DETONATIONS IN WHITE DWARF ENVELOPES

Detonations in helium-rich envelopes surrounding white dwarfs have garnered attention as triggers of faint thermonuclear ".Ia" supernovae and double detonation Type Ia supernovae. However, recent studies have found that the minimum size of a hotspot that can lead to a helium detonation is comparable to, or even larger than, the white dwarf's pressure scale height, casting doubt on the successful ignition of helium detonations in these systems. In this paper, we examine the previously neglected effects of C/O pollution and a full nuclear reaction network, and we consider hotspots with spatially constant pressure in addition to constant density hotspots. We find that the inclusion of these effects significantly decreases the minimum hotspot size for helium-rich detonation ignition, making detonations far more plausible during turbulent shell convection or during double white dwarf mergers. The increase in burning rate also decreases the minimum shell mass in which a helium detonation can successfully propagate and alters the composition of the shell's burning products. The ashes of these low mass shells consist primarily of silicon, calcium, and unburned helium and metals and may explain the high-velocity spectral features observed in most Type Ia supernovae.

CHARACTERIZING THE CONVECTIVE VELOCITY FIELDS IN MASSIVE STARS

We apply the mathematical formalism of vector spherical harmonics decomposition to convective stellar velocity fields from multidimensional hydrodynamics simulations and show that the resulting power spectra furnish a robust and stable statistical description of stellar convective turbulence. Analysis of the power spectra helps identify key physical parameters of the convective process such as the dominant scale of the turbulent motions that influence the structure of massive evolved pre-supernova stars. We introduce the numerical method that can be used to calculate vector spherical harmonics power spectra from two-dimensional (2D) and three-dimensional (3D) convective shell simulation data. Using this method we study the properties of oxygen shell burning and convection for a 15 M {sub ☉} star simulated by the hydrodynamics code FLASH in 2D and 3D. We discuss the importance of realistic initial conditions to achieving successful core-collapse supernova explosions in multidimensional simulations. We show that the calculated power spectra can be used to generate realizations of the velocity fields of presupernova convective shells. We find that the slope of the solenoidal mode power spectrum remains mostly constant throughout the evolution of convection in the oxygen shell in both 2D and 3D simulations. We also find that the characteristic radial scalesmore » of the convective elements are smaller in 3D than in 2D, while the angular scales are larger in 3D.« less

Impact basin relaxation as a probe for the thermal history of Pluto

We investigate viscoelastic impact basin relaxation on Pluto for a variety of thermal evolution scenarios encompassing both convective and conductive ice shells. Basins smaller than 200 km in diameter do not relax appreciably, while relaxation fractions can be up to ∼60% for large impact basins. The main control on basin relaxation is the amount of radiogenic heat produced in the rocky core; our results are insensitive to the formation time of the basin, the ice reference viscosity adopted, and the presence/absence of a subsurface ocean. Other volatiles, such as CO2 or NH3, if present in the ice shell in sufficient quantities could increase the predicted relaxation fraction of basins. Relaxation causes extensional stresses interior to the basin; the orientation of the resulting tectonic features is controlled by the effective elastic thickness beneath the basin. Future observations of the relaxation states and tectonics of impact basins are therefore likely to provide a key constraint on Pluto’s thermal history.

Proper use of Schwarzschild Ledoux criteria in stellar evolution computations

The era of detailed asteroseismic analyses opened by space missions such as CoRoT and Kepler has highlighted the need for stellar models devoid of numerical inaccuracies, in order to be able to diagnose which physical aspects are being ignored or poorly treated in standard stellar modeling. We tackle here the important problem of fixing convective zone boundaries in the frame of the local mixing length theory. First we show that the only correct way to locate a convective zone boundary is to find, at each iteration step, through interpolations or extrapolations from points within the convective zone, the mass where the radiative luminosity is equal to the total luminosity. We then discuss two misuses of the boundary condition and the ways they affect stellar modeling and stellar evolution. The first consists in applying the neutrality condition for convective instability on the radiative side of the convective boundary. The second way of misusing the boundary condition comes from the process of fixing the convective boundary through the search for a change of sign of a possibly discontinuous function. We show that these misuses can lead to completely wrong estimates of convective core sizes with important consequences for the following evolutionary phases. We point out the advantages of using a double mesh point at each convective zone boundary. The specific problem of a convective shell is discussed and some remarks concerning overshooting are given.

Linking 1D Stellar Evolution to 3D Hydrodynamic Simulations

AbstractIn this contribution we present initial results of a study on convective boundary mixing (CBM) in massive stellar models using the GENEVA stellar evolution code (Eggenbergeret al.2008). Before undertaking costly 3D hydrodynamic simulations, it is important to study the general properties of convective boundaries, such as the: composition jump; pressure gradient; and “stiffness”. Models for a 15M⊙star were computed. We found that for convective shells above the core, the lower (in radius or mass) boundaries are “stiffer” according to the bulk Richardson number than the relative upper (Schwarzschild) boundaries. Thus, we expect reduced CBM at the lower boundaries in comparison to the upper. This has implications on flame front propagation and the onset of novae.

Combining observational techniques to constrain convection in evolved massive star models

AbstractRecent stellar evolution computations indicate that massive stars in the range ~ 20-30 M⊙ are located in the blue supergiant (BSG) region of the Hertzsprung-Russell diagram at two different stages of their life: immediately after the main sequence (MS, group 1) and during a blueward evolution after the red supergiant phase (group 2). From the observation of the pulsationnal properties of a subgroup of variable BSGs (α Cyg variables), one can deduce that these stars belongs to group 2. It is however difficult to simultaneously fit the observed surface abundances and gravity for these stars, and this allows to constrain the physical processes of chemical species transport in massive stars. We will show here that the surface abundances are extremely sensitive to the physics of convection, particularly the location of the intermediate convective shell that appears at the ignition of the hydrogen shell burning after the MS. Our results show that the use of the Ledoux criterion to determine the convective regions in the stellar models leads to a better fit of the surface abundances for α Cyg variables than the Schwarzschild one.

The compressional beta effect: A source of zonal winds in planets?

Giant planets like Jupiter and Saturn feature strong zonal wind patterns on their surfaces. Although several different mechanisms that may drive these jets have been proposed over the last decades, the origin of the zonal winds is still unclear. Here, we explore the possibility that the interplay of planetary rotation with the compression and expansion of the convecting fluid can drive multiple deep zonal jets by a compressional Rhines-type mechanism, as originally proposed by Ingersoll and Pollard (Ingersoll, A.P., Pollard, D. [1982]. Icarus 52(1), 62–80). In a certain limit, this deep mechanism is shown to be mathematically analogous to the classical Rhines mechanism possibly operating at cloud level. Jets are predicted to occur on a compressional Rhines length lR=(2Ω〈Hρ-1〉vjet-1)-1/2, where Ω is the angular velocity, 〈Hρ-1〉 is the mean inverse density scale height and vjet is the typical jet velocity. Two-dimensional numerical simulations using the anelastic approximation reveal that this mechanism robustly generates jets of the predicted width, and that it typically dominates the dynamics in systems deeper than O(lR). Potential vorticity staircases are observed to form spontaneously and are typically accompanied by unstably stratified buoyancy staircases. The mechanism only operates at large rotation rates, exceeding those typically reached in three-dimensional simulations of deep convection in spherical shells. Applied to Jupiter and Saturn, the compressional Rhines scaling reasonably fits the available observations. Interestingly, even weak vertical density variations such as those in the Earth core can give rise to a large number of jets, leading to fundamentally different flow structures than predicted by the Boussinesq models typically used in this context.

Zonal flow scaling in rapidly-rotating compressible convection

The surface winds of Jupiter and Saturn are primarily zonal. Each planet exhibits strong prograde equatorial flow flanked by multiple alternating zonal winds at higher latitudes. The depth to which these flows penetrate has long been debated and is still an unsolved problem. Previous rotating convection models that obtained multiple high latitude zonal jets comparable to those on the giant planets assumed an incompressible (Boussinesq) fluid, which is unrealistic for gas giant planets. Later models of compressible rotating convection obtained only few high latitude jets which were not amenable to scaling analysis.Here we present 3-D numerical simulations of compressible convection in rapidly-rotating spherical shells. To explore the formation and scaling of high-latitude zonal jets, we consider models with a strong radial density variation and a range of Ekman numbers, while maintaining a zonal flow Rossby number characteristic of Saturn.All of our simulations show a strong prograde equatorial jet outside the tangent cylinder. At low Ekman numbers several alternating jets form in each hemisphere inside the tangent cylinder. To analyze jet scaling of our numerical models and of Jupiter and Saturn, we extend Rhines scaling based on a topographic β-parameter, which was previously applied to an incompressible fluid in a spherical shell, to compressible fluids. The jet-widths predicted by this modified Rhines length are found to be in relatively good agreement with our numerical model results and with cloud tracking observations of Jupiter and Saturn.

AN AZIMUTHAL DYNAMO WAVE IN SPHERICAL SHELL CONVECTION

We report the finding of an azimuthal dynamo wave of a low-order (m=1) mode in direct numerical simulations (DNS) of turbulent convection in spherical shells. Such waves are predicted by mean field dynamo theory and have been obtained previously in mean-field models. Observational results both from photometry and Doppler imaging have revealed persistent drifts of spots for several rapidly rotating stars, but, although an azimuthal dynamo wave has been proposed as a possible mechanism responsible for this behavior, it has been judged as unlikely, as practical evidence for such waves from DNS has been lacking. The large-scale magnetic field in our DNS, which is due to self-consistent dynamo action, is dominated by a retrograde m=1 mode. Its pattern speed is nearly independent of latitude and does not reflect the speed of the differential rotation at any depth. The extrema of magnetic m=1 structures coincide reasonably with the maxima of m=2 structures of the temperature. These results provide direct support for the observed drifts being due to an azimuthal dynamo wave.

A linear stability analysis of thermal convection in spherical shells with variable radial gravity based on the Tau-Chebyshev method

The onset of thermal convection in a non-rotating spherical shell is investigated using linear theory. The Tau-Chebyshev spectral method is used to integrate the linearized equations. We investigate the onset of thermal convection by considering two cases of the radial gravitational field (i) a local acceleration, acting radially inward, that is proportional to the distance from the center r, and (ii) a radial gravitational central force that is proportional to r−n. The former case has been widely analyzed in the literature, because it constitutes a simplified model that is usually used, in astrophysics and geophysics, and is studied here to validate the numerical method. The latter case was analyzed since the case n=5 has been experimentally realized (by means of the dielectrophoretic effect) under microgravity condition, in the experimental container called GeoFlow, inside the International Space Station. Our study is aimed to clarify the role of (i) a radially inward central force (either proportional to r or to r−n), (ii) a base conductive temperature distribution provided by either a uniform heat source or an imposed temperature difference between outer and inner spheres, and (iii) the aspect ratio η (ratio of the radii of the inner and outer spheres), on the critical Rayleigh number. In all cases the surface of the spheres has been assumed to be rigid. The results obtained with the linear theory based on the Tau-Chebyshev spectral method are compared with those of the integration of the full non-linear equations solved by using the spectral element method. By using the Tau-Chebyshev method, we were able to explore new cases that have not been previously reported in the literature.

Three-dimensional quasi-geostrophic convection in the rotating cylindrical annulus with steeply sloping endwalls

AbstractThe rotating cylindrical annulus geometry was first developed by Busse (J. Fluid Mech., vol. 44, 1970, pp. 441–460) as a simplified analogue for studying convection in rapidly rotating spherical geometries. Although it has provided a more tractable two-dimensional model than the sphere, it is formally limited to asymptotically small slopes and thus weak velocities in the direction parallel to the rotation axis. We present an asymptotically reduced three-dimensional equation set to model quasi-geostrophic convection in the annulus geometry where order-one slopes are permissible; this model provides a closer analogue to quasi-geostrophic convection in spheres and spherical shells where steeply sloping boundaries are present. A linear stability analysis of the reduced equations shows that a new class of three-dimensional, convectively driven Rossby waves is present in this system. The gravest modes exhibit strong axial variations as the slope of the boundaries becomes large. In addition, higher-order eigenmodes showing increasingly complex axial dependence are found that possess critical Rayleigh numbers close to that of the gravest mode.

Anelastic dynamo models with variable electrical conductivity: An application to gas giants

The observed surface dynamics of Jupiter and Saturn are dominated by a banded system of fierce zonal winds. The depth of these winds remains unclear but they are thought to be confined to the very outer envelopes where hydrogen remains molecular and the electrical conductivity is small. The dynamo maintaining the dipole-dominated magnetic fields of both gas giants, on the other hand, likely operates in the deeper interior where hydrogen assumes a metallic state.Here, we present numerical simulations that attempt to model both the zonal winds and the interior dynamo action in an integrated approach. Using the anelastic version of the MHD code MagIC, we explore the effects of density stratification and radial electrical conductivity variations. The electrical conductivity is mostly assumed to remain constant in the thicker inner metallic region and it decays exponentially towards the outer boundary throughout the molecular envelope.Our results show that the combination of a stronger density stratification and a weaker conducting outer layer is essential for reconciling dipole dominated dynamo action and a fierce equatorial zonal jet. Previous simulations with homogeneous electrical conductivity show that both are mutually exclusive, with solutions either having strong zonal winds and multipolar magnetic fields or weak zonal winds and dipole-dominated magnetic fields. All jets tend to be geostrophic and therefore reach right through the convective shell in our simulations.The particular setup explored here allows a strong equatorial jet to remain confined to the weaker conducting outer region where it does not interfere with the deeper seated dynamo action. The flanking mid to high latitude jets, on the other hand, have to remain faint to yield a strongly dipolar magnetic field. The fiercer jets on Jupiter and Saturn only seem compatible with the observed dipolar fields when they remain confined to a weaker conducting outer layer.

On a transitional and turbulent natural convection in spherical shells

Laminar and turbulent natural convection inside concentric spherical shells with isothermal cold and hot boundaries is numerically investigated up to Rayleigh number values Ra⩽1012 and Pr=0.71. The study utilizes direct numerical simulation (DNS), large eddy simulation (LES) and Reynolds averaged Navier–Stokes (RANS) approaches for investigation of the laminar, transitional and fully developed turbulent flow regimes, respectively. Three-dimensional flow patterns for slightly supercritical oscillatory flow regime inside the shell, with internal/external diameter ratio equal to Di/Do=0.714 are presented and may be potentially useful for verification of the future linear stability analysis results. Particular attention has been given to the complex, fully three-dimensional unsteady flows occurring in narrow shell geometries characterized by 0.85⩽Di/Do⩽0.95. For this geometry considerable deviations in predicted heat flux rate through the shell boundaries are observed when compared with existing heat transfer correlations for the entire range of Ra numbers. The deviations tend to increase for transitional and fully turbulent flows. A new correlation for the heat transfer rate is suggested for laminar and transitional flow regimes.