Convection In Spherical Shells Research Articles

Numerical study of granulation in anelastic thermal convection in spherical shells

The present work investigates granulation or convective flow patterns in density-stratified (or anelastic) convection in spherical shells. The density-stratified thermal convection is typically present in astrophysical systems (such as solar convection); motivated by this, we performed a series of three-dimensional anelastic convection simulations in a spherical shell geometry using an in-house developed hybrid solver. We explored the effect of Rayleigh number and density scale height on the convective flow patterns. The granulation (or cell-like structures) are more prominent at higher density scale height and Rayleigh number. The granulation is further characterized by kinetic energy and helicity spectra. Our results support the argument that the convective flow patterns (or granulation) emerge due to inverse cascade owing to the presence of density stratification. Convective patterns (or granulation) are identified based on length scales, time scales, and flow velocity. The length scale of granules is further verified using a solar granulation model. Our analysis suggests the existence of inverse cascade and supergranulation on the spherical surface due to density-stratified thermal convection in spherical shells.

Fingering convection in a spherical shell

We use $123$ three-dimensional direct numerical simulations to study fingering convection in non-rotating spherical shells. We investigate the scaling behaviour of the flow length scale, the non-dimensional heat and compositional fluxes $Nu$ and $Sh$ and the mean convective velocity over the fingering convection instability domain defined by $1 \leq R_\rho < Le$ , $R_\rho$ being the ratio of density perturbations of thermal and compositional origins and $Le$ the Lewis number. We show that the chemical boundary layers are marginally unstable and adhere to the laminar Prandtl–Blasius model, hence explaining the asymmetry between the inner and outer spherical shell boundary layers. We develop scaling laws for two asymptotic regimes close to the two edges of the instability domain, namely $R_\rho \lesssim Le$ and $R_\rho \gtrsim 1$ . For the former, we develop novel power laws of a small parameter $\epsilon$ measuring the distance to onset, which differ from theoretical laws published to date in Cartesian geometry. For the latter, we find that the Sherwood number $Sh$ gradually approaches a scaling $Sh\sim Ra_\xi ^{1/3}$ when $Ra_\xi \gg 1$ ; and that the Péclet number accordingly follows $Pe \sim Ra_\xi ^{2/3} |Ra_T|^{-1/4}$ , $Ra_T$ and $Ra_{\xi}$ being the thermal and chemical Rayleigh numbers. When the Reynolds number exceeds a few tens, we report on a secondary instability which takes the form of large-scale toroidal jets which span the entire spherical domain. Jets distort the fingers, resulting in Reynolds stress correlations, which in turn feed the jet growth until saturation. This nonlinear phenomenon can yield relaxation oscillation cycles.

Asymptotic profiles of mean zonal flows generated by thermal convection of Boussinesq fluid in a rapidly rotating thin spherical shell

The asymptotic behavior of the profiles of mean zonal flows generated by thermal convection of a Boussinesq fluid in a rapidly rotating thin spherical shell was studied by extending the integration time of the numerical experiments of a previous study under similar conditions. Calculations were performed for the whole globe, as well as for a one-eighth sector region, with various values of the hyper-diffusion parameters. For sufficiently weak hyper-diffusion, a prograde zonal jet and alternating narrow zonal jets appeared at the equator and in the middle and high latitudes, respectively, as reported in the previous study. However, further integrations showed that the middle and high latitude flows are asymptotically accelerated eastward, jet peaks merge, and the banded zonal flow structure eventually disappears. Thus, the rotating thin spherical shell convection model of a Boussinesq fluid used in the previous studies seems inadequate to explain the surface banded structures of Jovian planetary atmospheres without modification of the standard setups. To maintain the alternating jet structure in the middle and high latitudes over the long-term, some sort of scale-independent dissipation process should be specified that prevents the evolution of zonal flows toward the long-term asymptotic state.

A hybrid discrete exterior calculus and finite difference method for anelastic convection in spherical shells

The present work develops, verifies, and benchmarks a hybrid discrete exterior calculus and finite difference (DEC-FD) method for density-stratified thermal convection in spherical shells. Discrete exterior calculus (DEC) is notable for its coordinate independence and structure preservation properties. The hybrid DEC-FD method for Boussinesq convection has been developed by Mantravadi et al. (2023). Motivated by astrophysics problems, we extend this method assuming anelastic convection, which retains density stratification; this has been widely used for decades to understand thermal convection in stars and giant planets. In the present work, the governing equations are splitted into surface and radial components and discrete anelastic equations are derived by replacing spherical surface operators with DEC and radial operators with FD operators. The novel feature of this work is the discretization of anelastic equations with the DEC-FD method and the assessment of a hybrid solver for density-stratified thermal convection in spherical shells. The discretized anelastic equations are verified using the method of manufactured solution (MMS). We performed a series of three-dimensional convection simulations in a spherical shell geometry and examined the effect of density ratio on convective flow structures and energy dynamics. The present observations are in agreement with the benchmark models.

Arbitrarily high-order accurate simulations of compressible rotationally constrained convection using a transfinite mapping on cubed-sphere grids

We present two major improvements over the Compressible High-ORder Unstructured Spectral difference (CHORUS) code published in Wang et al., “A compressible high-order unstructured spectral difference code for stratified convection in rotating spherical shells,” J. Comput. Phys. 290, 90–111 (2015). The new code is named CHORUS++ in this paper. Subsequently, we perform a series of efficient simulations for rotationally constrained convection (RCC) in spherical shells. The first improvement lies in the integration of the high-order spectral difference method with a boundary-conforming transfinite mapping on cubed-sphere grids, thus ensuring exact geometric representations of spherical surfaces on arbitrary sparse grids. The second improvement is on the adoption of higher-order elements (sixth-order) in CHORUS++ vs third-order elements for the original CHORUS code. CHORUS++ enables high-fidelity RCC simulations using sixth-order elements on very coarse grids. To test the accuracy and efficiency of using elements of different orders, CHORUS++ is applied to a laminar solar benchmark, which is characterized by columnar banana-shaped convective cells. By fixing the total number of solution degrees of freedom, the computational cost per time step remains unchanged. Nevertheless, using higher-order elements in CHORUS++ resolves components of the radial energy flux much better than using third-order elements. To obtain converged predictions, using sixth-order elements is 8.7 times faster than using third-order elements. This significant speedup allows global-scale fully compressible RCC simulations to reach equilibration of the energy fluxes on a small cluster of just 40 cores. In contrast, CHORUS simulations were performed by Wang et al. on supercomputers using approximately 10 000 cores. Using sixth-order elements in CHORUS++, we further carry out global-scale solar convection simulations with decreased rotational velocities. Interconnected networks of downflow lanes emerge and surround broader and weaker regions of upflow fields. A strong inward kinetic energy flux compensated by an enhanced outward enthalpy flux appears. These observations are all consistent with those published in the literature. Furthermore, CHORUS++ can be extended to magnetohydrodynamic simulations with potential applications to the hydromagnetic dynamo processes in the interiors of stars and planets.

A hybrid discrete exterior calculus and finite difference method for Boussinesq convection in spherical shells

We present a new hybrid discrete exterior calculus (DEC) and finite difference (FD) method to simulate fully three-dimensional Boussinesq convection in spherical shells subject to internal heating and basal heating, relevant in the planetary and stellar phenomenon. We employ DEC to compute the surface spherical flows, taking advantage of its unique features including coordinate system independence to preserve the spherical geometry, while we discretize the radial direction using FD method. The grid employed for this novel method is free of problems like the coordinate singularity, grid non-convergence near the poles, and the overlap regions. We have developed a parallel in-house code using the PETSc framework to verify the hybrid DEC-FD formulation and demonstrate convergence. We have performed a series of numerical tests which include quantification of the critical Rayleigh numbers for spherical shells characterized by aspect ratios ranging from 0.2 to 0.8, simulation of robust convective patterns in addition to stationary giant spiral roll covering all the spherical surface in moderately thin shells near the weakly nonlinear regime, and the quantification of Nusselt and Reynolds numbers for basally heated spherical shells.

Direct driving of simulated planetary jets by upscale energy transfer

Context. The precise mechanism that forms jets and large-scale vortices on the giant planets is unknown. An inverse cascade has been suggested by several studies. Alternatively, energy may be directly injected by small-scale convection. Aims. Our aim is to clarify whether an inverse cascade feeds zonal jets and large-scale eddies in a system of rapidly rotating, deep, geostrophic spherical-shell convection. Methods. We analyze the nonlinear scale-to-scale transfer of kinetic energy in such simulations as a function of the azimuthal wave number, m. Results. We find that the main driving of the jets is associated with upscale transfer directly from the small convective scales to the jets. This transfer is very nonlocal in spectral space, bypassing large-scale structures. The jet formation is thus not driven by an inverse cascade. Instead, it is due to a direct driving by Reynolds stresses, statistical correlations of velocity components of the small-scale convective flows. Initial correlations are caused by the effect of uniform background rotation and shell geometry on the flows and provide a seed for the jets. While the jet growth suppresses convection, it increases the correlation of the convective flows, which further amplifies the jet growth until it is balanced by viscous dissipation. To a much smaller extent, energy is transferred upscale to large-scale vortices directly from the convective scales, mostly outside the tangent cylinder. There, large-scale vortices are not driven by an inverse cascade either. Inside the tangent cylinder, the transfer to large-scale vortices is even weaker, but more local in spectral space, leaving open the possibility of an inverse cascade as a driver of large-scale vortices. In addition, large-scale vortices receive kinetic energy from the jets via forward transfer. We therefore suggest a jet instability as an alternative formation mechanism of large-scale vortices. Finally, we find that the jet kinetic energy scales approximatively as ℓ−5, the same as for the so-called zonostrophic regime.

Rapidly rotating self-gravitating Boussinesq fluid. III. A previously unknown zonal oscillation at the onset of rotating convection

Rotating convection in spheres or spherical shells has been modeled in order to understand dynamics in planetary or stellar interiors. Existing theories have predicted various different types of symmetry for the dynamics at the onset of thermal convection. The only form of global convection not found so far is axisymmetric and equatorially symmetric. In this Letter, we for the first time, report the discovery of a solution bearing such symmetry. It is the geometric oblateness due to rotational flattening that is the key factor, which had not previously been considered.

Latitudinal regionalization of rotating spherical shell convection

Convection occurs ubiquitously on and in rotating geophysical and astrophysical bodies. Prior spherical shell studies have shown that the convection dynamics in polar regions can differ significantly from the lower latitude, equatorial dynamics. Yet most spherical shell convective scaling laws use globally-averaged quantities that erase latitudinal differences in the physics. Here we quantify those latitudinal differences by analysing spherical shell simulations in terms of their regionalized convective heat-transfer properties. This is done by measuring local Nusselt numbers in two specific, latitudinally separate, portions of the shell, the polar and the equatorial regions, $Nu_p$ and $Nu_e$ , respectively. In rotating spherical shells, convection first sets in outside the tangent cylinder such that equatorial heat transfer dominates at small and moderate supercriticalities. We show that the buoyancy forcing, parameterized by the Rayleigh number $Ra$ , must exceed the critical equatorial forcing by a factor of ${\approx }20$ to trigger polar convection within the tangent cylinder. Once triggered, $Nu_p$ increases with $Ra$ much faster than does $Nu_e$ . The equatorial and polar heat fluxes then tend to become comparable at sufficiently high $Ra$ . Comparisons between the polar convection data and Cartesian numerical simulations reveal quantitative agreement between the two geometries in terms of heat transfer and averaged bulk temperature gradient. This agreement indicates that rotating spherical shell convection dynamics is accessible both through spherical simulations and via reduced investigatory pathways, be they theoretical, numerical or experimental.

Polar and mid-latitude vortices and zonal flows on Jupiter and Saturn

Zonal flow on Jupiter and Saturn consists of equatorial super-rotation and alternating East–West jet streams at higher latitudes. Interacting with these zonal flows, numerous vortices occur with various sizes and lifetimes. The Juno mission and Cassini’s grand finale have shown that the zonal jets of Jupiter and Saturn extend deeply into their molecular envelopes. On Jupiter, the vast majority of low and mid-latitude jovian vortices are anticyclonic, whereas cyclones appear at polar latitudes. Cassini mission observations revealed a similar pattern on Saturn; its North and South polar vortices are cyclonic, whereas anticyclones occur at mid-latitudes. We use the recently developed code Rayleigh to run high-resolution simulations of rotating convection in 3D spherical shells. Four models are presented that result in dynamical flows that are comparable to those on the giant planets. We confirm previous results, finding that deep convective turbulence can explain the structure of jets. However, the latitude and the strength and depth of stable stratification can modify jet morphologies and affect the formation and dynamics of vortices. Lower latitudes favour shallow anticyclonic vortices that form due to upward and divergent flow near the outer boundary. These anticyclones are typically shielded by cyclonic filaments associated with downwelling return flow. In contrast, a single polar cyclone, or clusters of cyclones form near the poles. All of our simulations have this global pattern; a strong preference for shallow anticyclones in the first anticyclonic shear zone away from the equatorial jet (corresponding to the region of the Great Red Spot on Jupiter and Storm Alley on Saturn), cyclonic and anticyclonic vortices at higher mid-latitudes, and a deeply seated cyclone or cyclone clusters at each pole. Our results show that Juno and Cassini observations of cloud-level flow can be explained in terms of deep convective dynamics in the molecular envelopes of Jupiter and Saturn.

Evolution of Lateral Tectonophysical Stresses in the Spherical Shell Convection with an Immobile Supercontinent

Evolution of Lateral Tectonophysical Stresses in the Spherical Shell Convection with an Immobile Supercontinent

Convection and Dynamo in Newly Born Neutron Stars

Abstract To study properties of magnetohydrodynamic (MHD) convection and resultant dynamo activities in proto-neutron stars (PNSs), we construct a “PNS in a box” simulation model and solve the compressible MHD equation coupled with a nuclear equation of state (EOS) and simplified leptonic transport. As a demonstration, we apply it to two types of PNS model with different internal structures: a fully convective model and a spherical-shell convection model. By varying the spin rate of the models, the rotational dependence of convection and the dynamo that operate inside the PNS is investigated. We find that, as a consequence of turbulent transport by rotating stratified convection, large-scale structures of flow and thermodynamic fields are developed in all models. Depending on the spin rate and the depth of the convection zone, various profiles of the large-scale structures are obtained, which can be physically understood as steady-state solutions to the “mean-field” equation of motion. Additionally to those hydrodynamic structures, a large-scale magnetic component of  ( 10 15 ) G is also spontaneously organized in disordered tangled magnetic fields in all models. The higher the spin rate, the stronger the large-scale magnetic component grows. Intriguingly, as an overall trend, the fully convective models have a stronger large-scale magnetic component than that in the spherical-shell convection models. The deeper the convection zone extends, the larger the size of the convective eddies becomes. As a result, rotationally constrained convection seems to be more easily achieved in the fully convective model, resulting in a higher efficiency of the large-scale dynamo there. To gain a better understanding of the origin of the diversity of a neutron star’s magnetic field, we need to study the PNS dynamo in a wider parameter range.

On the Dynamics of Overshooting Convection in Spherical Shells: Effect of Density Stratification and Rotation.

Overshooting of turbulent motions from convective regions into adjacent stably stratified zones plays a significant role in stellar interior dynamics, as this process may lead to mixing of chemical species and contribute to the transport of angular momentum and magnetic fields. We present a series of fully nonlinear, three-dimensional (3D) anelastic simulations of overshooting convection in a spherical shell that are focused on the dependence of the overshooting dynamics on the density stratification and the rotation, both key ingredients in stars that however have not been studied systematically together via global simulations. We demonstrate that the overshoot lengthscale is not simply a monotonic function of the density stratification in the convective region, but instead it depends on the ratio of the density stratifications in the two zones. Additionally, we find that the overshoot lengthscale decreases with decreasing Rossby number Ro and scales as Ro0.23 while it also depends on latitude with higher Rossby cases leading to a weaker latitudinal variation. We examine the mean flows arising due to rotation and find that they extend beyond the base of the convection zone into the stable region. Our findings may provide a better understanding of the dynamical interaction between stellar convective and radiative regions, and motivate future studies particularly related to the solar tachocline and the implications of its overlapping with the overshoot region.

Bifurcations in rotating spherical shell convection under the influence of differential rotation.

The bifurcations of thermal convection in a rotating spherical shell heated from the inner sphere and driven by the buoyancy of a central gravity field are studied numerically. This model of spherical Rayleigh-Bénard convection describes large-scale convection in planets and in the outer zones of celestial bodies. In this work, the influence of an additionally imposed differential rotation of the inner sphere with respect to the outer one on the heat transfer and, more generally, on the whole bifurcation structure is investigated. In addition to numerical simulations, path-following techniques are applied in order to compute both stable and unstable solution branches. The dynamics and the heat transfer are essentially determined by a global bifurcation, which we have identified as a homoclinic bifurcation that consists of a collision of a stable modulated rotating with an unstable rotating wave.

Relating force balances and flow length scales in geodynamo simulations

SUMMARY In fluid dynamics, the scaling behaviour of flow length scales is commonly used to infer the governing force balance of a system. The key to a successful approach is to measure length scales that are simultaneously representative of the energy contained in the solution (energetically relevant) and also indicative of the established force balance (dynamically relevant). In the case of numerical simulations of rotating convection and magnetohydrodynamic dynamos in spherical shells, it has remained difficult to measure length scales that are both energetically and dynamically relevant, a situation that has led to conflicting interpretations, and sometimes misrepresentations of the underlying force balance. By analysing an extensive set of magnetic and non-magnetic models, we focus on two length scales that achieve both energetic and dynamical relevance. The first one is the peak of the poloidal kinetic energy spectrum, which we successfully compare to crossover points on spectral representations of the force balance. In most dynamo models, this result confirms that the dominant length scale of the system is controlled by a previously proposed quasi-geostrophic (QG-) MAC (Magneto-Archimedean-Coriolis) balance. In non-magnetic convection models, the analysis generally favours a QG-CIA (Coriolis-Inertia-Archimedean) balance. Viscosity, which is typically a minor contributor to the force balance, does not control the dominant length scale at high convective supercriticalities in the non-magnetic case, and in the dynamo case, once the generated magnetic energy largely exceeds the kinetic energy. In dynamo models, we introduce a second energetically relevant length scale associated with the loss of axial invariance in the flow. We again relate this length scale to another crossover point in scale-dependent force balance diagrams, which marks the transition between large-scale geostrophy (the equilibrium of Coriolis and pressure forces) and small-scale magnetostrophy, where the Lorentz force overtakes the Coriolis force. Scaling analysis of these two energetically and dynamically relevant length scales suggests that the Earth’s dynamo is controlled by a QG-MAC balance at a dominant scale of about $200 \, \mathrm{km}$, while magnetostrophic effects are deferred to scales smaller than $50 \, \mathrm{km}$.

Sensitivity to luminosity, centrifugal force, and boundary conditions in spherical shell convection

ABSTRACTWe test the sensitivity of hydrodynamic and magnetohydrodynamic turbulent convection simulations with respect to Mach number, thermal and magnetic boundary conditions, and the centrifugal force. We find that varying the luminosity, which also controls the Mach number, has only a minor effect on the large-scale dynamics. A similar conclusion can also be drawn from the comparison of two formulations of the lower magnetic boundary condition with either vanishing electric field or current density. The centrifugal force has an effect on the solutions, but only if its magnitude with respect to acceleration due to gravity is by two orders of magnitude greater than in the Sun. Finally, we find that the parameterisation of the photospheric physics, either by an explicit cooling term or enhanced radiative diffusion, is more important than the thermal boundary condition. In particular, runs with cooling tend to lead to more anisotropic convection and stronger deviations from the Taylor-Proudman state. In summary, the fully compressible approach taken here with the Pencil Code is found to be valid, while still allowing the disparate timescales to be taken into account.

The Influence of Curvature on Convection in a Temperature‐Dependent Viscosity Fluid: Implications for the 2‐D and 3‐D Modeling of Moons

AbstractConvection in terrestrial bodies occurs within spherical shells described by the ratio, f, of their bounding radii. Previous studies that have modeled convection with a temperature‐dependent viscosity noted the strong effect of f on transition to the stagnant‐lid regime. Here we analyze stagnant‐lid convection in 2‐D and 3‐D systems with curvatures including relatively small‐core shells (f as small as 0.2) as well as in thin shell and plane‐layer cases. Several peculiarities of convection in a strongly temperature‐dependent viscosity fluid are identified for both high and low curvature systems. We demonstrate that effective Rayleigh numbers may differ by orders of magnitude in systems with different curvatures, when all other parameters are maintained at fixed values. Furthermore, as f is decreased, the nature of stagnant‐lid convection in small‐core bodies shows a divergence in the temperature and velocity fields found for 2‐D annulus and 3‐D spherical shell systems. In addition, substantial differences in the behavior of thin shell (f = 0.9) and plane‐layer (Cartesian geometry) models occur in both 2‐D and 3‐D, indicating that the latter (emulating a toroidal topology rather than spherical) may be inappropriate approximations for modeling variable viscosity convection in thin spherical shells. Our findings are especially relevant to understanding and accurately modeling the thermal structure that may exist in bodies characterized by thin shells (e.g., f = 0.9) or relatively small cores, such as shells comprising the Galilean satellites and other moons.

Onset of low Prandtl number thermal convection in thin spherical shells

This study considers the onset of stress-free Boussinesq thermal convection in rotating spherical shells with an aspect ratio η = ri/ro = 0.9 (ri and ro being the inner and outer radius), Prandtl numbers Pr ∈ [10−4,10−1], and Taylor numbers Ta ∈ [104,1012]. We are particularly interested in the form of the convective cell pattern that develops, and in its time scales, since this may have observational consequences. For a fixed Ta 3 × 109, the unicellular polar modes become also preferred at moderate Pr ∼ 10−2 because two new transition curves between EA and AP/SP and between AP/SP and SC modes are born at a triple-point bifurcation. The dependence on Pr and Ta of the transitions is studied to estimate the types of modes, and their critical parameters, preferred at different stellar regimes.

On the effect of laterally varying boundary heat flux on rapidly rotating spherical shell convection

The onset of convection in a rotating spherical shell subject to laterally varying heat flux at the outer boundary is considered in this paper. The focus is on the geophysically relevant regime of rapid rotation (low Ekman number) where the natural length scale of convection is significantly smaller than the length scale imposed by the boundary heat flux pattern. Contrary to earlier studies at a higher Ekman number, we find a substantial reduction in the onset Rayleigh number Rac with increasing lateral variation. The decrease in Rac is shown to be closely correlated to the equatorial heat flux surplus in the steady, basic state solution. The consistency of such a correlation makes the estimation of Rac possible without solving the full stability problem. The steady baroclinic flow has a strong cyclone–anticyclone asymmetry in the kinetic helicity only for equatorially symmetric lateral variations, with possible implications for dynamo action. Equatorially antisymmetric variations, on the other hand, break the symmetry of the mean flow, in turn negating its helicity. Analysis of the perturbation solution reveals strongly localized clusters through which convection rolls drift in and out at a frequency higher than that for the reference case with homogeneous boundary heat flux. Large lateral variations produce a marked decrease in the azimuthal length scale of columns, which indicates that small-scale motions are essential to the transport of heat in rapidly rotating, localized convection. With an equatorially antisymmetric heat flux pattern, convection in individual clusters goes through an asynchronous wax-wane cycle whose frequency is much lower than the drift rate of the columns. These continual variations in convection intensity may in turn result in fluctuations in the magnetic field intensity, an effect that needs to be considered in dynamo models. Finally, there is a notable analogy between the role of a laterally varying boundary heat flux and the role of a laterally varying magnetic field in confining small-scale convection.

Hysteresis of dynamos in rotating spherical shell convection

Bifurcations of dynamos in rotating and buoyancy-driven spherical Rayleigh-Benard convection in an electrically conducting fluid are investigated numerically. Both nonmagnetic and magnetic solution branches comprised of rotating waves are traced by path-following techniques, and their bifurcations and interconnections for different Ekman numbers are determined. In particular, the question of whether the dynamo branches bifurcate super- or subcritically and whether a direct link to the primary pure convective states exists is answered.