Spherical-shell boundaries for two-dimensional compressible convection in a star

Abstract

Context. Studies of stellar convection typically use a spherical shell geometry. The radial extent of the shell and the boundary conditions applied are based on the model of the star investigated. We study the impact of different two-dimensional spherical shells on compressible convection. Realistic profiles for density and temperature from an established one-dimensional stellar evolution code are used to produce a model of a large stellar convection zone representative of a young low-mass star, such as our sun at 10 6 years of age. Aims. We analyze how the radial extent of the spherical shell changes the convective dynamics that result in the deep interior of the young sun model, far from the surface. In the near-surface layers, simple smaller-scale convection develops from the profiles of temperature and density. A central radiative zone below the convection zone provides a lower boundary on the convection zone. The inclusion of either of these physically distinct layers in the spherical shell can potentially affect the characteristics of deep convection. 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 seamlessly modeling a whole star. Simulations in two-dimensional spherical shells that have different radial extents are performed over tens or even hundreds of convective turnover times, permitting the collection of well-converged statistics. Results. To measure the impact of the spherical shell geometry and our treatment of boundaries, we evaluate basic statistics of the convective turnover time, the convective velocity, and the overshooting layer. These quantities are selected for their relevance to one-dimensional stellar evolution calculations, so that our results are focused toward studies exploiting the so-called 321D link. We find that the inclusion in the spherical shell of the boundary between the radiative and convection zones decreases the amplitude of convective velocities in the convection zone. The inclusion of near-surface layers in the spherical shell can increase the amplitude of convective velocities, although the radial structure of the velocity profile established by deep convection is unchanged. The impact from including the near-surface layers depends on the speed and structure of small-scale convection in the near-surface layers. Larger convective velocities in the convection zone result in a commensurate increase in the overshooting layer width and decrease in the convective turnover time. These results provide support for non-local aspects of convection.

A well-balanced scheme for the simulation tool-kit A-MaZe: implementation, tests, and first applications to stellar structure

“…In many astrophysically relevant cases, the convective motion is very subsonic and thus τ c >> τ s . For instance, in their study of a young sun in 2D with M ≤ 0.05, Pratt et al (2016) needed an integration time of several hundred τ c , translating into several thousand τ s , for robust statistics. This sets the time scale of interest, over which Eqs.…”

“…3.1.2 is used: (x, y) → (r, θ) in the 2D axisymmetric case and (x, y, z) → (r, θ, ϕ) in the 3D case with uniform grid spacing. In the radial direction and in units of stellar radius R star , the computational domain extends from R min = 0.21 to R max = 0.94 ('Low 3' case in Pratt et al (2016)), thereby comprising both, the convective envelope and parts of the radiative core. The domain extends from 0.2π to 0.8π in polar angle θ (and in azimuth angle ϕ in 3D).…”

“…7. Axi-symmetric young sun simulations, 2D, with A-MaZe (left panels, 256 2 mesh unless otherwise stated) and MUSIC (right panels, 'Low 3' case in Pratt et al (2016)). Shown are the total kinetic energy in the domain, scaled to a full spherical layer (0 ≤ θ ≤ π, 0 ≤ ϕ ≤ 2π, 0.21 ≤ R ≤ 0.94), as function of time (different resolutions color coded for A-MaZe, shifted in time such as to ramp up simultaneously), as well as radial profiles of volume-weighted v rms and its decomposition into radial and tangential component (middle and bottom panels, solid lines give time averages, dotted lines indicate the range) after convection has become quasi-stationary.…”

Lithium Depletion in Solar-like Stars: Effect of Overshooting Based on Realistic Multi-dimensional Simulations

“…We have recently explored hydrodynamic simulations of two-dimensional compressible convection in a premain sequence 1 M star using a new time-implicit code, the MUltidimensional Stellar Implicit Code (MU-SIC) (Viallet et al 2011(Viallet et al , 2013Geroux et al 2016;Goffrey et al 2017). The 2D models are in spherical geometry and are calculated using realistic stellar interior conditions (including a realistic stellar equation of state and opacities, see Pratt et al 2016Pratt et al , 2017. These simulations cover up to 525 convective turnovers, producing statistically robust data that characterises the extent and impact of convective penetration at the bottom of the convective envelope.…”

“…Note that preliminary 3D simulations show the same patterns as found in 2D and confirm the existence of extreme plume events. Associating mixing process with plume penetration, we apply a statistical method based on extreme value theory to derive the cumulative distribution function (CDF) of the maximal penetration depth obtained in our numerical simulations (see Pratt et al 2017). We derive a new form for a diffusion coefficient D EX characterised by this CDF and describing mixing driven by the convective plumes in the penetration layer.…”