Analytical and semianalytical approaches have been developed to describe this process and estimate the width of the overshooting layer (e.g., Schmitt et al. 1984; Zahn 1991; Rempel 2004). With the improvement of computational methods and resources, an increasing number of studies have been devoted to numerical simulations of convection and overshooting using realistic stellar conditions (geometry, luminosity, thermal diffusivity, equation of state, opacities, etc.). A commonly used tactic to increase the efficiency and improve the stability of these simulations is to artificially increase the luminosity (or nuclear energy for convective cores or burning shells) and to modify the thermal diffusivity of the reference stellar model. This tactic is common and has been used, for example, in Rogers et al. (2006, 2013), Meakin & Arnett (2007), Tian et al. (2009), Brun et al. (2011, 2017), Hotta (2017), Cristini et al. (2017), Edelmann et al. (2019), Horst et al. (2020). This approach is used to increase the Mach number of the convective flow, reducing the disparity between advective and acoustic timescales, improving the efficiency of time-explicit codes limited by the Courant–Friedrich–Levy constraint. It is also used to provide numerical stability or to accelerate the thermal relaxation. But no examination of its potentially far-reaching impact has been conducted. Rempel (2004) pointed out that enhanced energy flux in numerical simulations could lead to unrealistically vigourous convection, which could impact the properties of the overshooting layer and could explain some of the discrepancies between analytical models and numerical simulations. Numerical simulations also suggest an increase in the overshooting depth with increasing flux (Hotta 2017; Käpylä 2019). Determining scaling laws of the overshooting depth as a function of the energy input could thus allow for an extrapolation of the results to more realistic stellar conditions and help to estimate the overshooting depth in real stars, as suggested by Hotta (2017), for example, for the Sun. But Käpylä (2019) also shows that an artificial modification of the heat conductivity in the radiative and overshooting regions could impact the overshooting process. In some computational studies, both the luminosity and the thermal diffusivity are enhanced by the same factor to ensure that the thermal structure is unchanged compared to the reference stellar structure and with the expectation that the larger thermal diffusivity counterbalances the larger energy flux. This procedure has been proposed as a way to provide a good representation of the true dynamics of the system (e.g., Rogers et al. 2006, 2013; Tian et al. 2009). But this expectation has never been demonstrated. Another expectation concerns internal waves, excited by convective motions and by flows penetrating the convective boundary. Simulations with artificially modified luminosity and thermal diffusivity are also used to perform the analysis of internal waves, either for convective envelopes (e.g., Rogers et al. 2006; Brun et al. 2011; Alvan et al. 2014) or for convective cores (e.g., Rogers & McElwaine 2017; Edelmann et al. 2019; Horst et al. 2020). None of these works have examined whether the wave spectrum of a realistic star is accurately predicted by such simulations.
Citation: Hotta (2017); Hotta 2017; Hotta (2017)
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