Multidimensional low-Mach number time-implicit hydrodynamic simulations of convective helium shell burning in a massive star
Authors: Horst L. , Hirschi R. , Edelmann P. V. F. , Andrássy R. , Röpke F. K. 
Published: July 2021

Context. A realistic parametrization of convection and convective boundary mixing in conventional stellar evolution codes is still the subject of ongoing research. To improve the current situation, multidimensional hydrodynamic simulations are used to study convection in stellar interiors. Such simulations are numerically challenging, especially for flows at low Mach numbers which are typical for convection during early evolutionary stages.
Aims. We explore the benefits of using a low-Mach hydrodynamic flux solver and demonstrate its usability for simulations in the astrophysical context. Simulations of convection for a realistic stellar profile are analyzed regarding the properties of convective boundary mixing.
Methods. The time-implicit Seven-League Hydro (SLH) code was used to perform multidimensional simulations of convective helium shell burning based on a 25 M⊙ star model. The results obtained with the low-Mach AUSM+-up solver were compared to results when using its non low-Mach variant AUSMB+-up. We applied well-balancing of the gravitational source term to maintain the initial hydrostatic background stratification. The computational grids have resolutions ranging from 180 × 902 to 810 × 5402 cells and the nuclear energy release was boosted by factors of 3 × 103, 1 × 104, and 3 × 104 to study the dependence of the results on these parameters.
Results. The boosted energy input results in convection at Mach numbers in the range of 10−3–10−2. Standard mixing-length theory predicts convective velocities of about 1.6 × 10−4 if no boosting is applied. The simulations with AUSM+-up show a Kolmogorov-like inertial range in the kinetic energy spectrum that extends further toward smaller scales compared with its non low-Mach variant. The kinetic energy dissipation of the AUSM+-up solver already converges at a lower resolution compared to AUSMB+-up. The extracted entrainment rates at the boundaries of the convection zone are well represented by the bulk Richardson entrainment law and the corresponding fitting parameters are in agreement with published results for carbon shell burning. However, our study needs to be validated by simulations at higher resolution. Further, we find that a general increase in the entropy in the convection zone may significantly contribute to the measured entrainment of the top boundary.
Conclusion. This study demonstrates the successful application of the AUSM+-up solver to a realistic astrophysical setup. Compressible simulations of convection in early phases at nominal stellar luminosity will benefit from its low-Mach capabilities. Similar to other studies, our extrapolated entrainment rate for the helium-burning shell would lead to an unrealistic growth of the convection zone if it is applied over the lifetime of the zone. Studies at nominal stellar luminosities and different phases of the same convection zone are needed to detect a possible evolution of the entrainment rate and the impact of radiation on convective boundary mixing.