The theoretical models mentioned above have made assumptions. The validity of these assumptions needs to be examined. Besides, all these models involve parameters to be determined. These problems could be better understood through numerical simulations. Hurlburt et al. (1994) performed a two-dimensional numerical simulation of compressible flow on downward overshooting. They investigated the dependence of subadiabatic extent δ on stability parameter S. They revealed a scaling law of δ ∝ S−1 in the penetrative layer and δ ∝ S−1/4 in the overshooting layer, respectively. This result is in good agreement with Zahn’s analytic model (Zahn 1991). Freytag et al. (1996) performed two-dimensional numerical simulations with a realistic description of radiation and ionization on A-type stars and DA white dwarf stars. They described the overshooting as a diffusive process, and derived an exponential decay parameter for the diffusion. Early attempts of low-resolution three-dimensional numerical simulations on overshooting were made by Singh et al. (1994; upward overshooting), Singh et al. (1995, 1998), and Saikia et al. (2000) (downward overshooting). The scaling laws derived from the numerical simulations of downward overshooting (Singh et al. 1995, 1998; Saikia et al. 2000) agree well with Zahn’s analytical model. Only a qualitative result was given in the numerical simulations of upward overshooting (Singh et al. 1994). High-resolution numerical simulations of downward overshooting across a wide range of parameters were presented by Brummell et al. (2002). They confirmed the −1/4 scaling law of the thermal adjustment overshooting layer, while the −1 scaling law of the nearly adiabatic penetrative layer was absent in the simulations. Based on a semianalytic model, Rempel (2004) argued that the absence of the nearly adiabatic penetrative layer is caused by the large energy flux specified in the numerical simulations. Numerical experiments on Boussinesq flow were performed by Korre et al. (2019). They reported steeper scaling laws of δ ∝ S−1/3 or δ ∝ S−1/2, depending on the steepness of the background radiative temperature gradient. Simulations with realistic physical variables on stellar core convection were performed by Browning et al. (2004) and Brun et al. (2005). The effects of rotation and magnetic field are considered. They found that the penetrative convection yields a prolate shape of a nearly adiabatic region. Kitiashvili et al. (2016) performed 3D radiative hydrodynamic simulations of the outer layer of a moderate-mass star (1.47 solar mass). Their result discovered a nearly adiabatic layer and a deeper subadiabatic layer. The recent work of Brun et al. (2017) simulated the differential rotation and overshooting in solar-like stars. Their result indicated that slow rotators favor a wider overshooting region near the poles and fast rotators at mid-to-low latitude. Hotta (2017) performed numerical simulations on the solar overshooting with very low energy fluxes F. He found that the overshooting distance obeys a scaling law of δ ∝ F0.31. Käpylä (2019) conducted numerical experiments on downward overshooting by considering the effect of the smoothness of the heat conduction profiles. He discovered that the power-law index of the overshooting distance on the energy flux is smaller in the smooth heat conduction profile than in the step profile. Efforts on prediction of 321D turbulent theory were made by Arnett et al. (2015) and Arnett & Moravveji (2017). They separated the overshooting region into three layers: a fully mixed layer, a partially mixed wave layer, and an extra diffusive mixing layer. With the scale analysis of turbulent plumes and eddies, Viallet et al. (2015) discussed the three possible regimes of turbulent overshooting: a diffusion-dominated regime (only mix composition), a penetrative regime (transition within the boundary layer), and an entrainment regime (mix both entropy and composition). The selection criterion of different regimes during a stellar evolution calculation is not well defined yet.
Citation: Singh et al. 1995
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Citation: Singh et al. (1995
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