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Abstract

The extent of mixed regions around convective zones is one of the biggest uncertainties in stellar evolution. One-dimensional overshooting descriptions introduce a free parameter (f_ov) that is, in general, not well constrained from observations. Especially in small central convective regions, the value is highly uncertain due to its tight connection to the pressure scale height. Long-term multi-dimensional hydrodynamic simulations can be used to study the size of the overshooting region as well as the involved mixing processes. Here we show how one can calibrate an overshooting parameter by performing two-dimensional Maestro simulations of zero-age-main-sequence stars ranging from 1.3 to 3.5 M_⊙. The simulations cover the convective cores of the stars and a large fraction of the surrounding radiative envelope. We follow the convective flow for at least 20 convective turnover times, while the longest simulation covers 430 turnover time scales. This allows us to study how the mixing as well as the convective boundary itself evolve with time, and how the resulting entrainment can be interpreted in terms of overshooting parameters. We find that increasing the overshooting parameter f_ov beyond a certain value in the initial model of our simulations changes the mixing behaviour completely. This result can be used to put limits on the overshooting parameter. We find 0.010 <  f_ov <  0.017 to be in good agreement with our simulations of a 3.5 M_⊙ mass star. We also identify a diffusive mixing component due to internal gravity waves that is active throughout the convectively stable layer, but it is most likely overestimated in our simulations. Furthermore, applying our calibration method to simulations of less massive stars suggests a need for a mass-dependent overshooting description where the mixing in terms of the pressure scale height is reduced for small convective cores.