Mean-field dynamo theory suggests that turbulent convection in a rotating layer of electrically-conducting fluid produces a significant alpha-effect, which is one of the key ingredients in any mean-field dynamo model. Provided that this alpha-effect operates more efficiently than (turbulent) magnetic diffusion, such a system should be capable of sustaining a large-scale dynamo. However, in the Boussinesq model that was considered by Cattaneo&Hughes (2006) the dynamo produced small-scale, intermittent magnetic fields with no significant large-scale component. In this paper, we consider the compressible analogue of the rotating convective layer that was considered by Cattaneo&Hughes (2006). Varying the horizontal scale of the computational domain, we investigate the dependence of the dynamo upon the rotation rate. Our simulations indicate that these turbulent compressible flows can drive a small-scale dynamo but, even when the layer is rotating very rapidly (with a mid-layer Taylor number of Ta=10^8), we find no evidence for the generation of a significant large-scale component of the magnetic field on a dynamical timescale. Like Cattaneo&Hughes(2006), we measure a negligible (time-averaged) alpha-effect when a uniform horizontal magnetic field is imposed across the computational domain. Although the total horizontal magnetic flux is a conserved quantity in these simulations, the (depth-dependent) horizontally-averaged magnetic field always exhibits strong fluctuations. If these fluctuations are artificially suppressed within the code, we measure a significant mean electromotive force that is comparable to that found in related calculations in which the alpha-effect is measured using the test-field method, even though we observe no large-scale dynamo action.
Numerical simulations of turbulent Rayleigh-Bénard convection in an ideal gas, using either the anelastic approximation or the fully compressible equations, are compared. Theoretically, the anelastic approximation is expected to hold in weakly superadiabatic systems with $ε= ΔT / T_r \ll 1$, where $ΔT$ denotes the superadiabatic temperature drop over the convective layer and $T_r$ the bottom temperature. Using direct numerical simulations, a systematic comparison of anelastic and fully compressible convection is carried out. With decreasing superadiabaticity $ε$, the fully compressible results…
Numerical simulations of magneto-convection have greatly expanded our understanding of stellar interiors and stellar magnetism. Recently, fully compressible hydrodynamical simulations of full-star models have demonstrated the feasibility of studying the excitation and propagation of pressure and internal gravity waves in stellar interiors, which would allow for a direct comparison with asteroseismological measurements. However, the impact of magnetic fields on such waves has not been taken into account yet in three-dimensional simulations. We conduct a proof of concept for the realization of t…
(abridged) Context: Convective motions overshooting to regions that are formally convectively stable cause extended mixing. Aims: To determine the scaling of overshooting depth ($d_{\rm os}$) at the base of the convection zone as a function of imposed energy flux ($\mathscr{F}_{\rm n}$) and to estimate the extent of overshooting at the base of the solar convection zone. Methods: Three-dimensional Cartesian simulations of compressible non-rotating convection with unstable and stable layers are used. The simulations use either a fixed heat conduction profile or a temperature and density dependen…
In this paper, we calibrate the coefficients for the one-dimensional Reynolds stress model with the data generated from the three-dimensional numerical simulations of upward overshooting in turbulent compressible convection. It has been found that the calibrated convective and isotropic coefficients are almost the same as those calibrated in the pure convection zone. However, the calibrated diffusive coefficients differ significantly from those calibrated in the pure convection zone. We suspect that the diffusive effect induced by the boundary is stronger than by the adjacent stable zone. We h…
A simplified model for stellar and planetary convection is derived for the magnitude of the rms velocity, degree of superadiabaticity, and characteristic length scale with Rossby number as well as with thermal and viscous diffusivities. Integrating the convection model into a linearization of the dynamics in the transition region between convectively unstable and stably-stratified region yields a Rossby number, diffusivity, and pressure scale height dependent convective penetration depth into the stable region. This may have important consequences for mixing along the evolution of rotating sta…
Context: We study the impact of two-dimensional spherical shells on compressible convection. Realistic profiles for density and temperature from a one-dimensional stellar evolution code are used to produce a model of a large stellar convection zone representative of a young low-mass star. Methods: We perform hydrodynamic implicit large-eddy simulations of compressible convection using the MUltidimensional Stellar Implicit Code (MUSIC). Because MUSIC has been designed to use realistic stellar models produced from one-dimensional stellar evolution calculations, MUSIC simulations are capable of s…
The mechanism of large-scale dynamos in rigidly rotating stratified convection is explored by direct numerical simulations (DNS) in Cartesian geometry. A mean-field dynamo model is also constructed using turbulent velocity profiles consistently extracted from the corresponding DNS results. By quantitative comparison between the DNS and our mean-field model, it is demonstrated that the oscillatory $α^2$ dynamo wave, excited and sustained in the convection zone, is responsible for large-scale magnetic activities such as cyclic polarity reversal and spatiotemporal migration. The results provide s…
We present a novel and powerful Compressible High-ORder Unstructured Spectral-difference (CHORUS) code for simulating thermal convection and related fluid dynamics in the interiors of stars and planets. The computational geometries are treated as rotating spherical shells filled with stratified gas. The hydrodynamic equations are discretized by a robust and efficient high-order Spectral Difference Method (SDM) on unstructured meshes. The computational stencil of the spectral difference method is compact and advantageous for parallel processing. CHORUS demonstrates excellent parallel performanc…
Context. Recent, nonlinear simulations of wave generation and propagation in full-star models have been carried out in the anelastic approximation using spectral methods. Although it makes long time steps possible, this approach excludes the physics of sound waves completely and rather high artificial viscosity and thermal diffusivity are needed for numerical stability. Direct comparison with observations is thus limited. Aims. We explore the capabilities of our compressible multidimensional hydrodynamics code SLH to simulate stellar oscillations. Methods. We compare some fundamental propertie…
Within the umbrae of sunspots, convection is generally inhibited by the presence of strong vertical magnetic fields. However, convection is not completely suppressed in these regions: bright features, known as umbral dots, are probably associated with weak, isolated convective plumes. Motivated by observations of umbral dots, we carry out numerical simulations of three-dimensional, compressible magnetoconvection. By following solution branches into the subcritical parameter regime (a region of parameter space in which the static solution is linearly stable to convective perturbations), we find…
Solar dynamo models require some mechanism for magnetic field concentration near the base of the convection zone in order to generate super-kilogauss toroidal fields with sufficiently large (~10^{24} Mx) magnetic flux. We consider the downward diamagnetic pumping near the base of the convection zone as a possible concentration mechanism and derive the pumping velocities with allowance for the effect of rotation. Transport velocities for poloidal and toroidal fields differ in rotating fluid. The toroidal field is transported downward along the radius only but the pumping velocity for the poloid…
Extending our recent studies of two-dimensional stellar convection to 3D, we compare three-dimensional hydrodynamic simulations to identically set-up two-dimensional simulations, for a realistic pre-main sequence star. We compare statistical quantities related to convective flows including: average velocity, vorticity, local enstrophy, and penetration depth beneath a convection zone. These statistics are produced during stationary, steady-state compressible convection in the star's convection zone. Our simulations with the MUSIC code confirm the common result that two-dimensional simulations o…
Simulating deep solar convection and its coupled mean-field motions is a formidable challenge where few observational results constrain models that suffer from the non-physical influence of the grid resolution. We present hydrodynamic global Implicit Large-Eddy simulations (ILES) of deep solar convection performed with the EULAG-MHD code, and explore the effects of grid resolution on the properties of rotating and non-rotating convection. The results, based on low-order moments and turbulent spectra reveal that convergence could be achieved in non-rotating simulations provided sufficient resol…
Structures seen in idealized numerical experiments on compressible magnetoconvection in an imposed strong vertical magnetic field show important differences from those detected in observations or realistic numerical simulations of sunspot umbrae. To elucidate the origin of these discrepancies, we present a series of idealized 3D compressible magnetoconvection experiments that differ from previous such experiments in several details, bringing them closer to realistic solar conditions. An initially vertical magnetic field B 0 is imposed on a time snapshot of fully developed solar-like turbulent …
Twists in the flow: revisiting convective mixing in rotating stellar models. I. Effect on the stellar structure
Dartmouth College（达特茅斯学院）；Université Paris-Saclay, Université Paris Cité, CEA, CNRS, AIM（巴黎萨克雷大学，巴黎西岱大学，法国原子能和替代能源委员会，法国国家科学研究中心，天体物理学与分子间介质实验室）；KU Leuven（荷语鲁汶大学）；Radboud University Nijmegen（奈梅亨拉德堡德大学）；Max Planck Institute for Astronomy（马克斯·普朗克天文学研究所）
(abridged) Context: The ratio of kinematic viscosity to thermal diffusivity, the Prandtl number, is much smaller than unity in stellar convection zones. Aims: To study the statistics of convective flows and energy transport as functions of the Prandtl number. Methods: Three-dimensional numerical simulations convection in Cartesian geometry are used. The convection zone (CZ) is embedded between two stably stratified layers. Statistics and transport properties of up- and downflows are studied separately. Results: The rms velocity increases with decreasing Prandtl number. At the same time the fil…
We study the generation of large-scale vortices in rotating turbulent convection by means of Cartesian direct numerical simulations. We find that for sufficiently rapid rotation, cyclonic structures on a scale large in comparison to that of the convective eddies, emerge, provided that the fluid Reynolds number exceeds a critical value. For slower rotation, cool cyclonic vortices are preferred, whereas for rapid rotation, warm anti-cyclonic vortices are favored. In some runs in the intermediate regime both types of cyclones co-exist for thousands of convective turnover times. The temperature co…
We present a statistical analysis of turbulent convection in stars within our Reynolds-Averaged Navier Stokes (RANS) framework in spherical geometry which we derived from first principles. The primary results reported in this document include: (1) an extensive set of mean-field equations for compressible, multi-species hydrodynamics, and (2) corresponding mean-field data computed from various simulation models. Some supplementary scale analysis data is also presented. The simulation data which is presented includes: (1) shell convection during oxygen burning in a 23 solar mass supernova progen…
We study dynamo action in a convective layer of electrically-conducting, compressible fluid, rotating about the vertical axis. At the upper and lower bounding surfaces, perfectly-conducting boundary conditions are adopted for the magnetic field. Two different levels of thermal stratification are considered. If the magnetic diffusivity is sufficiently small, the convection acts as a small-scale dynamo. Using a definition for the magnetic Reynolds number $R_M$ that is based upon the horizontal integral scale and the horizontally-averaged velocity at the mid-layer of the domain, we find that rota…
The expansion of the rotating fluid will change the vorticity and rotational speed of the expanding region. In turbulent thermal convection, this microscopic effect is preserved. Tracking the fluid micelles shows that the average vorticity varies with density, producing vorticity transport and angular momentum transport from the low-density area to the high-density area, forming a macroscopic vorticity difference and rotational speed difference. Taking the axial thermal convection model of the solar polar region, it can generate axial differential rotation, and the centrifugal force difference…
Convection is an important physical process in astrophysics well-studied using numerical simulations under the Boussinesq and/or anelastic approximations. However these approaches reach their limits when compressible effects are important in the high Mach flow regime, e.g. in stellar atmospheres or in the presence of accretion shocks. In order to tackle these issues, we propose a new high performance and portable code, called "ARK" with a numerical solver well-suited for the stratified compressible Navier-Stokes equations. We take a finite volume approach with machine precision conservation of…
We consider the effect of stratification on systematic, large-scale flows generated in anelastic convection. We present results from three-dimensional numerical simulations of convection in a rotating plane layer in which the angle between the axis of rotation and gravity is allowed to vary. This model is representative of different latitudes of a spherical body. We consider two distinct parameter regimes: (i) weakly rotating and (ii) rapidly rotating. In each case, we examine the effect of stratification on the flow structure and heat transport properties focussing on the difference between B…
We study thermal convection in a rotating fluid in order to better understand the properties of convection zones in rotating stars and planets. We first derive mixing-length theory for rapidly-rotating convection, arriving at the results of Stevenson (1979) via simple physical arguments. The theory predicts the properties of convection as a function of the imposed heat flux and rotation rate, independent of microscopic diffusivities. In particular, it predicts the mean temperature gradient; the rms velocity and temperature fluctuations; and the size of the eddies that dominate heat transport. …
Buoyancy-driven convection is likely the dominant driver of turbulent motions in the universe, and thus, is widely studied by physicists, engineers, geophysicists and astrophysicists. Maybe unsurprisingly, these different communities discuss the gross convective behaviors in different ways, often without significant cross-talk existing between them. Here, we seek to draw connections between these communities. We do so by carrying out a set of basic scale estimations for how heat and fluid momentum transport should behave in non-rotating, slowly rotating and rapidly rotating buoyancy-driven con…
Convective dynamo simulations are performed in local Cartesian geometry. We report the first successful simulation of a large-scale oscillatory dynamo in rigidly rotating convection without stably stratified layers. A key requirement for exciting the large-scale dynamo is a sufficiently long integration time comparable to the ohmic diffusion time. By comparing two models with and without stably stratified layers, their effect on the large-scale dynamo is also studied. The spatiotemporal evolution of the large-scale magnetic field is similar in both models. However, it is intriguing that the ma…
Rayleigh-Bénard convection in rotating spherical shells can be considered as a simplified analogue of many astrophysical and geophysical fluid flows. Here, we use three-dimensional direct numerical simulations to study this physical process. We construct a dataset of more than 200 numerical models that cover a broad parameter range with Ekman numbers spanning $3\times 10^{-7} \leq E \leq 10^{-1}$, Rayleigh numbers within the range $10^3 < Ra < 2\times 10^{10}$ and a Prandtl number unity. We investigate the scaling behaviours of both local (length scales, boundary layers) and global (Nusselt an…
Motivated by the significant interaction of convection, rotation and magnetic field in many astrophysical objects, we investigate the interplay between large-scale flows driven by rotating convection and an imposed magnetic field. We utilise a simple model in two dimensions comprised of a plane layer that is rotating about an axis inclined to gravity. It is known that this setup can result in strong mean flows; we numerically examine the effect of an imposed horizontal magnetic field on such flows. We show that increasing the field strength in general suppresses the time-dependent mean flows, …
We find an instability resulting in generation of large-scale vorticity in a fast rotating small-scale turbulence or turbulent convection with inhomogeneous fluid density along the rotational axis in anelastic approximation. The large-scale instability causes excitation of two modes: (i) the mode with dominant vertical vorticity and with the mean velocity being independent of the vertical coordinate; (ii) the mode with dominant horizontal vorticity and with the mean momentum being independent of the vertical coordinate. The mode with the dominant vertical vorticity can be excited in a fast rot…
We examine the role of small-scale granulation in helping to drive supergranulation and even larger scales of convection. The granulation is modeled as localized cooling events introduced at the upper boundary of a 3-D simulation of compressible convection in a rotating spherical shell segment. With a sufficient number of stochastic cooling events compared to uniform cooling, we find that supergranular scales are realized, along with a differential rotation that becomes increasingly solar-like.
A monomodal model for stellar and planetary convection is derived for the magnitude of the rms velocity, degree of superadiabaticity, and characteristic length scale as a function of rotation rate as well as with thermal and viscous diffusivities. The convection model is used as a boundary condition for a linearization of the equations of motion in the transition region between convectively unstable and stably-stratified regions, yielding the depth to which convection penetrates into the stable region and establishing a relationship between that depth and the local convective Rossby number, di…
In this paper, we investigate the upward overshooting by three-dimensional numerical simulations. We find that the above convectively stable zone can be partitioned into three layers: the thermal adjustment layer (mixing both entropy and material), the turbulent dissipation layer (mixing material but not entropy), and the thermal dissipation layer (mixing neither entropy nor material). The turbulent dissipation layer is separated from the thermal adjustment layer and the thermal dissipation layer by the first and second zero points of the vertical velocity correlation. The simulation results a…
Gravito-inertial waves can be excited at the interface of convective and radiative regions and by the Reynolds stresses in the bulk of the convection zone. The magnitude of their energy flux will therefore vary with the properties of the convection. To assess how convection changes with rotation, a simplified local monomodal model for rotating convection is presented that provides the magnitude of the rms velocity, degree of superadiabaticity, and characteristic length scale as a function of the convective Rossby number as well as with thermal and viscous diffusivities. In the context of this …
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 analyzing spherical shell simulations in terms of their regionalized convective heat transfer properties. This is done by measuring local Nusselt numbers in two specific, …
We present an extensive study of Boussinesq thermal convection including a temperature-dependent internal heating source, based on numerical three-dimensional simulations. The temperature dependence mimics triple-$α$ nuclear reactions and the fluid geometry is a rotating spherical shell. These are key ingredients for the study of convective accreting neutron star oceans. A dimensionless parameter ${\rm Ra}_n$, measuring the relevance of nuclear heating, is defined. We explore how flow characteristics change with increasing ${\rm Ra}_n$ and give an astrophysical motivation. The onset of convect…
We develop a theory of three-dimensional slow Rossby waves in rotating spherical density stratified convection. The Rossby waves with frequencies which are much smaller than the rotating frequency, are excited by a non-axisymmetric instability from the equilibrium based on the developed convection. These waves interact with the inertial waves and the density stratified convection. The density stratification is taken into account using the anelastic approximation for very low-Mach-number flows. We study long-term planetary Rossby waves with periods which are larger than two years. We suggest th…
Gravito-inertial waves are excited at the interface of convective and radiative regions and by the Reynolds stresses in the bulk of the convection zones of rotating stars and planets. Such waves have notable asteroseismic signatures in the frequency spectra of rotating stars, particularly among rapidly rotating early-type stars, which provides a means of probing their internal structure and dynamics. They can also transport angular momentum, chemical species, and energy from the excitation region to where they dissipate in radiative regions. To estimate the excitation and convective parameter …
One prominent feature in the atmospheres of Jupiter and Saturn is the appearance of large-scale vortices. However, the sustaining mechanism of these large-scale vortices remains unclear. One possible mechanism is that these large-scale vortices are driven by rotating convection. Here we present numerical simulation results on rapidly rotating Rayleigh-Bénard convection at a small Prandtl number $Pr=0.1$ (close to the turbulent Prandtl numbers of Jupiter and Saturn). We have identified four flow regimes in our simulation: multiple small vortices, coexisted large-scale cyclone and anticyclone, l…
