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 of stellar convection have a higher magnitude of velocity on average than three-dimensional simulations. Boundary conditions and the extent of the spherical shell can affect the magnitude and variability of convective velocities. The difference between 2D and 3D velocities is dependent on these background points; in our simulations this can have an effect as large as the difference resulting from the dimensionality of the simulation. Nevertheless, radial velocities near the convective boundary are comparable in our 2D and 3D simulations. The average local enstrophy of the flow is lower for two-dimensional simulations than for three-dimensional simulations, indicating a different shape and structuring of 3D stellar convection. We perform a statistical analysis of the depth of convective penetration below the convection zone, using the model proposed in our recent study (Pratt et al. 2017). Here we analyze the convective penetration in three dimensional simulations, and compare the results to identically set-up 2D simulations. In 3D the penetration depth is as large as the penetration depth calculated from 2D simulations.
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…
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…
(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 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…
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…
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…
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…
(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…
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…
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 comp…
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…
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…
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 …
We present the first 3-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-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 …
A grid of numerical simulations of double-diffusive convection is presented for astrophysical conditions. As in laboratory and geophysical cases convection takes place in a layered form. A translation between the astrophysical fluid mechanics and incompressible (Boussinesq) approximation is given, valid for thin layers. Its validity is checked by comparison of the results of fully compressible and Boussinesq simulations of semiconvection. A fitting formula is given for the superadiabatic gradient as a function of this parameter. The superadiabaticity depends on the thickness $d$ of the double …
We present the first global 3D simulations of thermal convection in the oblate envelopes of rapidly-rotating solar-type stars. This has been achieved by exploiting the capabilities of the new Compressible High-ORder Unstructured Spectral difference (CHORUS) code. We consider rotation rates up to 85\% of the critical (breakup) rotation rate, which yields an equatorial radius that is up to 17\% larger than the polar radius. This substantial oblateness enhances the disparity between polar and equatorial modes of convection. We find that the convection redistributes the heat flux emitted from the …
Context: In Cepheids close to the red edge of the classical instability strip, a coupling occurs between the acoustic oscillations and the convective motions close to the surface.The best topical models that account for this coupling rely on 1-D time-dependent convection (TDC) formulations. However, their intrinsic weakness comes from the large number of unconstrained free parameters entering in the description of turbulent convection. Aims: We compare two widely used TDC models with the first two-dimensional nonlinear direct numerical simulations (DNS) of the convection-pulsation coupling in …
Spherical solar dynamo simulations are performed. Self-consistent, fully compressible magnetohydrodynamic system with a stably stratified layer below the convective envelope is numerically solved with a newly developed simulation code based on the Yin-Yang grid. The effects of penetrative convection are studied by comparing two models with and without the stable layer. The differential rotation profile in both models is reasonably solar-like with equatorial acceleration. When considering the penetrative convection, a tachocline-like shear layer is developed and maintained beneath the convectio…
We carry out non-rotating high-resolution calculations of the solar global convection, which resolve convective scales of less than 10 Mm. To cope with the low Mach number conditions in the lower convection zone, we use the reduced speed of sound technique (RSST), which is simple to implement and requires only local communication in the parallel computation. In addition, the RSST allows us to expand the computational domain upward to about $0.99 R_{\odot}$ as it can also handle compressible flows. Using this approach, we study the solar convection zone on the global scale, including small-scal…
As a step toward a complete theoretical integration of 3D compressible hydrodynamic simulations into stellar evolution, convection at the surface and sub-surface layers of the Sun is re-examined, from a restricted point of view, in the language of mixing-length theory (MLT) . Requiring that MLT use a hydrodynamically realistic dissipation length gives a new constraint on solar models. While the stellar structure which results is similar to that obtained by YREC and Garching models, the theoretical picture differs. A new quantitative connection is made between macro-turbulence, micro-turbulence…
A hypothesis for sunspot formation is the buoyant emergence of magnetic flux tubes created by the strong radial shear at the tachocline. In this scenario, the magnetic field has to exceed a threshold value before it becomes buoyant and emerges through the whole convection zone. We follow the evolution of a random seed magnetic field with the aim of study under what conditions it is possible to excite the dynamo instability and whether the dynamo generated magnetic field becomes buoyantly unstable and emerges to the surface as expected in the flux-tube context. We perform numerical simulations …
Stellar convection is a non-local process responsible for the transport of heat and chemical species. It can lead to enhanced mixing through convective overshooting and excitation of internal gravity waves (IGWs) at convective boundaries. The relationship between these processes is still not well understood and requires global hydrodynamic simulations to capture the important large-scale dynamics. The steep stratification in stellar interiors suggests that the radial extent of such simulations can affect the convection dynamics, the IGWs in the stably stratified radiative zone, and the depth o…
Realistic numerical simulations, i.e., those that make minimal use of ad hoc modeling, are essential for understanding the complex turbulent dynamics of the interiors and atmospheres of the Sun and other stars and the basic mechanisms of their magnetic activity and variability. The goal of this paper is to present a detailed description and test results of a compressible radiative MHD code, `StellarBox', specifically developed for simulating the convection zones, surface, and atmospheres of the Sun and moderate-mass stars. The code solves the three-dimensional, fully coupled compressible MHD e…
We demonstrate that a magneto-convection simulation incorporating essential physical processes governing solar surface convection exhibits turbulent small-scale dynamo action. By presenting a derivation of the energy balance equation and transfer functions for compressible magnetohydrodynamics (MHD), we quantify the source of magnetic energy on a scale-by-scale basis. We rule out the two alternative mechanisms for the generation of small-scale magnetic field in the simulations: the tangling of magnetic field lines associated with the turbulent cascade and Alfvenization of small-scale velocity …
[Abridged] Bipolar magnetic regions are formed when loops of magnetic flux emerge at the solar photosphere. Our aim is to investigate the flux emergence process in a simulation of granular convection. In particular we aim to determine the circumstances under which magnetic buoyancy enhances the flux emergence rate (which is otherwise driven solely by the convective upflows). We use three-dimensional numerical simulations, solving the equations of compressible magnetohydrodynamics in a horizontally-periodic Cartesian domain. A horizontal magnetic flux tube is inserted into fully developed hydro…
We present results from 2D radiation-hydrodynamical simulations of fully compressible convection for the surface layers of A-type stars with the ANTARES code. Spectroscopic indicators for photospheric convective velocity fields show a maximum of velocities near Teff ~8000 K. In that range the largest values are measured for the subgroup of Am stars. Thus far, no prognostic model, neither theoretical nor numerical, is able to exactly reproduce the line profiles of sharp line A and Am stars in that temperature range. In general, the helium abundance of A stars is not known from observations. Hen…
We compare simple analytical closure models of homogeneous turbulent Boussinesq convection for stellar applications with three-dimensional simulations. We use simple analytical closure models to compute the fluxes of angular momentum and heat as a function of rotation rate measured by the Taylor number. We also investigate cases with varying angles between the angular velocity and gravity vectors, corresponding to locating the computational domain at different latitudes ranging from the pole to the equator of the star. We perform three-dimensional numerical simulations in the same parameter re…
We study fully compressible convection in the context of plane-parallel, polytropically stratified atmospheres. We perform a suite of 2D and 3D simulations in which we vary the initial superadiabaticity ($ε$) and the Rayleigh number (Ra) while fixing the initial density stratification, aspect ratio, and Prandtl number. The evolved heat transport, quantified by the Nusselt number (Nu), follows scaling relationships similar to those found in the well-studied, incompressible Rayleigh-Bénard problem. This scaling holds up in both 2D and 3D and is not appreciably affected by the magnitude of $ε$.
Characterizing stellar convection in multiple dimensions is a topic at the forefront of stellar astrophysics. Numerical simulations are an essential tool for this task. We present an extension of the existing numerical tool-kit A-MaZe that enables such simulations of stratified flows in a gravitational field. The finite-volume based, cell-centered, and time-explicit hydrodynamics solver of A-MaZe was extended such that the scheme is now well-balanced in both momentum and energy. The algorithm maintains an initially static balance between gravity and pressure to machine precision. Quasi-station…
Despite the lack of a shear-rich tachocline region low-mass fully convective stars are capable of generating strong magnetic fields, indicating that a dynamo mechanism fundamentally different from the solar dynamo is at work in these objects. We present a self-consistent three dimensional model of magnetic field generation in low-mass fully convective stars. The model utilizes the anelastic magnetohydrodynamic equations to simulate compressible convection in a rotating sphere. A distributed dynamo working in the model spontaneously produces a dipole-dominated surface magnetic field of the obse…
A grid of numerical simulations of double-diffusive convection is presented for the astrophysical case where viscosity (Prandtl number Pr) and solute diffusivity (Lewis number Le) are much smaller than the thermal diffusivity. As in laboratory and geophysical cases convection takes place in a layered form. The proper translation between subsonic flows in a stellar interior and an incompressible (Boussinesq) fluid is given, and the validity of the Boussinesq approximation for the semiconvection problem is checked by comparison with fully compressible simulations. The predictions of a simplified…
We present results from compressible Cartesian convection simulations with and without imposed shear. In the former case the dynamo is expected to be of $α^2\varOmega$ type which is generally expected to be relevant for the Sun, whereas the latter case refers to $α^2$ dynamos which are more likely to occur in more rapidly rotating stars whose differential rotation is small. We perform a parameter study where the shear flow and the rotational influence are varied to probe the relative importance of both types of dynamos. Oscillatory solutions are preferred both in the kinematic and saturated re…
We explore the effects of rotation on convective carbon, oxygen, and silicon shell burning during the late stages of evolution in a 20Msun star. Using the Modules for Experiments in Stellar Astrophysics (MESA) we construct 1D stellar models both with no rotation and with an initial rigid rotation of 50% of critical. At different points during the evolution, we map the 1D models into 2D and follow the multidimensional evolution using the FLASH compressible hydrodynamics code for many convective turnover times until a quasi-steady state is reached. We characterize the strength and scale of conve…
We examine a penetration layer formed between a central radiative zone and a large convection zone in the deep interior of a young low-mass star. Using the Multidimensional Stellar Implicit Code (MUSIC) to simulate two-dimensional compressible stellar convection in a spherical geometry over long times, we produce statistics that characterize the extent and impact of convective penetration in this layer. We apply extreme value theory to the maximal extent of convective penetration at any time. We compare statistical results from simulations which treat non-local convection, throughout a large p…
We explore the effects of an outer stably stratified coronal envelope on rotating turbulent convection, differential rotation, and large-scale dynamo action in spherical wedge models of the Sun. We solve the compressible magnetohydrodynamic equations in a two-layer model with unstable stratification below the surface, representing the convection zone, and a stably stratified coronal envelope above. The interface represents a free surface. We compare our model to models that have no coronal envelope. The presence of a coronal envelope is found to modify the Reynolds stress and the $Λ$ effect re…
