Prandtl number dependence of stellar convection: Flow statistics and convective energy transport

<jats:p><jats:italic>Context.</jats:italic> The ratio of kinematic viscosity to thermal diffusivity, the Prandtl number, is much smaller than unity in stellar convection zones.</jats:p> <jats:p><jats:italic>Aims.</jats:italic> The main goal of this work is to study the statistics of convective flows and energy transport as functions of the Prandtl number.</jats:p> <jats:p><jats:italic>Methods.</jats:italic> Three-dimensional numerical simulations of compressible non-rotating hydrodynamic convection in Cartesian geometry are used. The convection zone (CZ) is embedded between two stably stratified layers. The dominant contribution to the diffusion of entropy fluctuations comes in most cases from a subgrid-scale diffusivity whereas the mean radiative energy flux is mediated by a diffusive flux employing Kramers opacity law. Here, we study the statistics and transport properties of up- and downflows separately.</jats:p> <jats:p><jats:italic>Results.</jats:italic> The volume-averaged rms velocity increases with decreasing Prandtl number. At the same time, the filling factor of downflows decreases and leads to, on average, stronger downflows at lower Prandtl numbers. This results in a strong dependence of convective overshooting on the Prandtl number. Velocity power spectra do not show marked changes as a function of Prandtl number except near the base of the convective layer where the dominance of vertical flows is more pronounced. At the highest Reynolds numbers, the velocity power spectra are more compatible with the Bolgiano-Obukhov <jats:italic>k</jats:italic><jats:sup>−11/5</jats:sup> than the Kolmogorov-Obukhov <jats:italic>k</jats:italic><jats:sup>−5/3</jats:sup> scaling. The horizontally averaged convected energy flux (<jats:italic>F̅</jats:italic><jats:sub>conv</jats:sub>), which is the sum of the enthalpy (<jats:italic>F̅</jats:italic><jats:sub>enth</jats:sub>) and kinetic energy fluxes (<jats:italic>F̅</jats:italic><jats:sub>kin</jats:sub>), is independent of the Prandtl number within the CZ. However, the absolute values of <jats:italic>F̅</jats:italic><jats:sub>enth</jats:sub> and <jats:italic>F̅</jats:italic><jats:sub>kin</jats:sub> increase monotonically with decreasing Prandtl number. Furthermore, <jats:italic>F̅</jats:italic><jats:sub>enth</jats:sub> and <jats:italic>F̅</jats:italic><jats:sub>kin</jats:sub> have opposite signs for downflows and their sum <jats:italic>F̅</jats:italic><jats:sup>↓</jats:sup><jats:sub>conv</jats:sub> diminishes with Prandtl number. Thus, the upflows (downflows) are the dominant contribution to the convected flux at low (high) Prandtl numbers. These results are similar to those from Rayleigh-Benárd convection in the low Prandtl number regime where convection is vigorously turbulent but inefficient at transporting energy.</jats:p> <jats:p><jats:italic>Conclusions.</jats:italic> The current results indicate a strong dependence of convective overshooting and energy flux on the Prandtl number. Numerical simulations of astrophysical convection often use a Prandtl number of unity because it is numerically convenient. The current results suggest that this can lead to misleading results and that the astrophysically relevant low Prandtl number regime is qualitatively different from the parameter regimes explored in typical contemporary simulations.</jats:p>