The shape of convection in 2D and 3D global simulations of stellar interiors

Theoretical descriptions of convective overshooting often rely on a one-dimensional parameterization of the flow called the filling factor for convection. Several definitions of the filling factor have been developed, based on: (1) the percentage of the volume, (2) the mass flux, and (3) the convective flux that moves through the boundary. We examine these definitions of the filling factor with the goal of establishing their ability to explain differences between 2D and 3D global simulations of stellar interiors that include fully compressible hydrodynamics and realistic microphysics for stars. We study pairs of identical two- and three-dimensional global simulations of stars produced with MUSIC, a fully compressible, time-implicit hydrodynamics code. We examine (1) a $3 M_\odot$ red giant star near the first dredge-up point, (2) a $1 M_\odot$ pre-main-sequence star with a large convection zone, (3) the current sun, and (4) a $20 M_\odot$ main-sequence star with a large convective core. Our calculations of the filling factor based on the volume percentage and the mass flux indicate asymmetrical convection near the surface for each star with an outer convection zone. However, near the convective boundary, convective flows achieve inward-outward symmetry; for 2D and 3D simulations, these filling factors are indistinguishable. A filling factor based on the convective flux is contaminated by boundary-layer-like flows, making theoretical interpretation difficult. We present two new alternatives to these standard definitions, which compare flows at two different radial points. The first is the penetration parameter of Anders et al. (2022). The second is a new statistic, the plume interaction parameter. We demonstrate that both of these parameters capture systematic differences between 2D and 3D simulations around the convective boundary.