diff --git "a/1tE0T4oBgHgl3EQfuQHv/content/tmp_files/load_file.txt" "b/1tE0T4oBgHgl3EQfuQHv/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/1tE0T4oBgHgl3EQfuQHv/content/tmp_files/load_file.txt" @@ -0,0 +1,1235 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf,len=1234 +page_content='MNRAS 000, 1–12 (2022) Preprint 9 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='0 A study of convective core overshooting as a function of stellar mass based on two-dimensional hydrodynamical simulations I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Baraffe,1,2 ★ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Clarke,1 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Morison,1 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Vlaykov,1 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Constantino,1 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Goffrey,3 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Guillet,1 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Le Saux1,2 and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Pratt4 1University of Exeter, Physics and Astronomy, EX4 4QL Exeter, UK 2École Normale Supérieure, Lyon, CRAL (UMR CNRS 5574), Université de Lyon, France 3Centre for Fusion, Space and Astrophysics, Department of Physics, University of Warwick, Coventry, CV4 7AL, UK 4Lawrence Livermore National Laboratory, 7000 East Ave, Livermore, CA 94550, USA Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Received YYY ABSTRACT We perform two-dimensional numerical simulations of core convection for zero-age-main-sequence stars covering a mass range from 3 𝑀⊙ to 20 𝑀⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The simulations are performed with the fully compressible time-implicit code MUSIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' We study the efficiency of overshooting, which describes the ballistic process of convective flows crossing a convective boundary, as a function of stellar mass and luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' We also study the impact of artificially increasing the stellar luminosity for 3 𝑀⊙ models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The simulations cover hundreds to thousands of convective turnover timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Applying the framework of extreme plume events previously developed for convective envelopes, we derive overshooting lengths as a function of stellar masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' We find that the overshooting distance (𝑑ov) scales with the stellar luminosity (𝐿) and the convective core radius (𝑟conv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' We derive a scaling law 𝑑ov ∝ 𝐿1/3𝑟1/2 conv which is implemented in a 1D stellar evolution code and the resulting stellar models are compared to observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The scaling predicts values for the overshooting distance that significantly increase with stellar mass, in qualitative agreement with observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Quantitatively, however, the predicted values are underestimated for masses >∼ 10𝑀⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Our 2D simulations show the formation of a nearly-adiabatic layer just above the Schwarzschild boundary of the convective core, as exhibited in recent 3D simulations of convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The most luminous models show a growth in size with time of the nearly-adiabatic layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' This growth seems to slow down as the upper edge of the nearly-adiabatic layer gets closer to the maximum overshooting length and as the simulation time exceeds the typical thermal diffusive timescale in the overshooting layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Key words: Convection – Hydrodynamics – Stars: evolution 1 INTRODUCTION One of the major uncertainties in stellar evolution models is the treat- ment of mixing taking place at convective boundaries (see Stancliffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Convective motions do not abruptly stop at the classical Schwarzschild boundary, but extend beyond it and lead to the pro- cess of convective boundary mixing (CBM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The complex dynamics resulting from convective flows penetrating in stable layers drives the transport of chemical species and heat, strongly affecting the structure and the evolution of stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The same complex dynamics can also drive transport of angular momentum, impacting the rotational evolution of stars, the generation of magnetic field in their interior and their magnetic activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' CBM affects the evolution of all stars that develop a convective envelope, core or shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Its treatment is one of the oldest unsolved problems of stellar structure and evolution theory (Shaviv & Salpeter 1973).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' This extra mixing could significantly alter the size of a convective core, the lifetime of major burning phases or the surface chemistry over a wide range of stellar masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' It can impact the entire evolution of massive stars (𝑀 >∼ 8𝑀⊙), determin- ing their structure before core-collapse supernova explosion and thus ★ E-mail: i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='baraffe@ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='uk affecting nucleosynthetic yields which are crucial for galactic evolu- tion studies (Arnett & Meakin 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' There is ample observational evidence pointing towards the need for extra internal mixing to ex- plain a wide range of observations, such as eclipsing binaries (Claret & Torres 2016), color-magnitude diagrams (Rosenfield et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2017) or asteroseismology (Bossini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Rosenfield et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (2017) illustrate the uncertainty due to the treatment of core overshooting on ages and on morphological changes in stellar evolution tracks, signif- icantly impacting stellar population studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' An increasing number of observational studies also suggests an increase of convective bound- ary mixing efficiency with stellar mass, using eclipsing binaries (see Claret & Torres 2019, and references therein) or Hertzsprung-Russell diagrams of massive stars (Castro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' In a recent study, John- ston (2021) confirms that current stellar models with no or with little convective boundary mixing usually under-predict the mass of con- vective cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' While such comparisons between stellar models and observations cannot identify a mechanism responsible for mixing at the convective boundaries, Johnston (2021) concludes that a range of efficiencies for the mixing mechanism(s) should be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' In addition to CBM, additional mixing could be due to rotation (Zahn 1992) or internal gravity waves (Schatzman 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The latter are connected to CBM as they are excited at convective boundaries by turbulent con- © 2022 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='02604v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='SR] 6 Jan 2023 2 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' vective motions (Press 1981;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Goldreich & Kumar 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Lecoanet & Quataert 2013) and penetrating flows (Rieutord & Zahn 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Montalbán & Schatzman 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Pinçon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' CBM is a generic term that encompasses different processes, namely penetration, overshooting or entrainment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The first term de- scribes motions that cross a convective boundary and alter the back- ground in such a way that the location of the convective boundary, defined by the Schwarzschild or the Ledoux criterion, moves inward or outward, resulting in the extension of the convective region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Over- shooting usually describes convective penetrative motions that do not alter the background but can still result in more or less efficient mixing (Zahn 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' In the literature, the terms overshooting and penetration are often used interchangeably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' These processes have been described in stellar evolution models by an overshooting distance 𝑑ov and/or a diffusion coefficient which remains constant or exponentially decays over the overshooting length (Freytag et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' These parameters are usually calibrated to fit observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The temperature gradient in the overshooting region is either set to the radiative or to the adiabatic temperature gradient (see for example Michielsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The third term entrainment is used to characterise shear-induced turbulent motions at the interface between the convectively stable and unstable regions driven by convective penetrative motions (plumes or eddies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Interfacial instabilities contribute to mixing fluids of different com- positions and/or densities, eroding the convective boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' This one can then grow in time following an entrainment rate characterised by the bulk Richardson number (Fernando 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Strang & Fernando 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Entrainment rates based on hydrodynamical simulations per- formed in a stellar context (Meakin & Arnett 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Cristini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2019) are also implemented in stellar evolution codes to describe the extension of convective cores and shells (Staritsin 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Scott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' However, as shown by Scott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (2021), adopting entrainment rates derived from existing stellar hydrodynamical sim- ulations to main sequence stellar models produces unrealistic growth of the convective cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The parameters that control the entrainment rates need to be decreased by several orders of magnitude to repro- duce observations, questioning the reliability of the quantitative rates derived from existing numerical simulations and even the existence of an entrainment process for main sequence convective cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Describing and isolating these different processes characterising CBM and at play at convective boundaries can be difficult in numer- ical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Downward flows (or plumes) crossing a convective boundary at the bottom of an envelope are clearly observed in nu- merical simulations (see for example Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Ballistic plume crossings may eventually lead to a modification of the thermal background – the so-called penetration process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' But for such modifi- cation to be observed, simulations must be run over many thousands of convective turnover timescales, as theoretically expected and re- cently demonstrated in simulations by Anders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (2022) based on 3D simulations of convection in a Cartesian box with idealised se- tups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' In a numerical study of solar-like convective envelopes, Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (2021) show that artificially boosting the luminosity of the stellar model by a factor 104 yields a significant modification of the thermal background below the convective boundary with an ex- tension of the size of the layer characterised by the penetration of convective flows, which could lead to a growth of the convectively unstable zone down to deeper levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Whether this growth stabilises or whether the convective boundary continues moving downward indefinitely is unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' For the solar-like model with realistic stellar luminosity, a slight modification of the thermal background is also observed in the simulations of Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (2021), but they show no trend of an extension of the Schwarzschild convective boundary over the simulation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Following the approach developed in Pratt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (2017) for con- vective envelopes, the most vigorous plumes can be used to define a maximal overshooting length, which can be significantly deeper than the typical length reached by the bulk of the plumes (Pratt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Vlaykov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Whether this bal- listic process is also observed for convective cores and can drive significant mixing is an open question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Arguments based on the dy- namics of convective motions and plumes suggest that mixing below a convective zone (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='g envelope overshooting) and above (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='g core overshooting) may indeed be different (Andrássy & Spruit 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Simple arguments based on the kinetic energy of a plume with typ- ical velocity and the restoring buoyancy force suggest very small overshooting lengths for the cores of low and intermediate mass zero-age-main-sequence (ZAMS) stars (Higl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' But these estimates are based on typical velocities without considering possi- ble extreme plume events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The situation could also be different for convective cores on the ZAMS and on the main-sequence respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Indeed, the building of a molecular weight gradient at the core boundary due to hydrogen burning in the core can hamper the lifting of heavier material by ballistic processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' An entrainment process slowly eroding the convective boundary may thus dominate at some point over the ballistic process during the main sequence evolution, or both processes may coexist and contribute to mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' These ques- tions are still unsettled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Existing numerical simulations of convective cores have mostly focussed on one single stellar mass model, rather than a range of stellar masses (Meakin & Arnett 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Gilet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Rogers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Edelmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Horst et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Higl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Additionally, many of these works enhance the stellar luminosity of the model, to provide numerical stability, or to accelerate the thermal relaxation or the Mach number of the con- vective flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' This artefact may artificially favour one process over the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' At this time, it is difficult to draw any firm conclusion re- garding the main mechanisms that drive CBM in stars and how their efficiency is affected with stellar mass and with the stage of evolution on the main sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' In this work devoted to convective cores, we study the efficiency for convective plumes to penetrate into the stable region as a function of stellar mass for ZAMS models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' In the following we will refer to overshooting to describe this process, since we essentially describe the ballistic process and even if a modification of the temperature gradient is observed for the most luminous models (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 5), likely leading to penetration as defined by Zahn (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' We perform two-dimensional (2D) numerical simulations of convective cores of ZAMS stellar models covering a range of stellar masses between 3 𝑀⊙ and 20 𝑀⊙ (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Our goal is to apply the framework of ex- treme plume events developed for convective stellar envelopes (Pratt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2017, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2021) to the convective cores of inter- mediate and massive stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' We analyse whether extreme events can provide overshooting lengths required for stellar models to reproduce observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' For this purpose, we derive a relationship between over- shooting length and stellar luminosity based on present numerical simulations (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' We apply the relationship to one-dimensional stellar evolution models and test them against observations (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' This is the first step for a systematic study devoted to convective core overshooting in intermediate mass and massive stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2 NUMERICAL SIMULATIONS We use the fully compressible time-implicit code MUSIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' A full description of MUSIC and of the time-implicit integration can be found in Viallet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (2011, 2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Goffrey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' MUSIC MNRAS 000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 1–12 (2022) A study of convective core overshooting as a function of stellar mass 3 solves the inviscid Euler equations in the presence of external gravity and thermal diffusion: 𝜕𝜌 𝜕𝑡 = −∇ · (𝜌v),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (1) 𝜕𝜌v 𝜕𝑡 = −∇ · (𝜌v ⊗ v) − ∇𝑝 + 𝜌g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (2) 𝜕𝜌𝑒 𝜕𝑡 = −∇ · (𝜌𝑒v) − 𝑝∇ · v + ∇ · (𝜒∇𝑇) + 𝑄nuc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (3) where 𝜌 is the density,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 𝑒 the specific internal energy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' v the velocity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 𝑝 the gas pressure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 𝑇 the temperature,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' g the gravitational accelera- tion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' and 𝜒 the thermal conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The term 𝑄nuc represents the nuclear energy rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The symbol ⊗ is the outer product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' All hydrody- namical simulations presented in this work are performed assuming spherically symmetric gravitational acceleration g, which is updated every time interval Δ𝑡1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' All simulations presented in this work are performed with Δ𝑡 = 103 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The typical dynamical timescale of the entire stellar cores analysed in this study 𝜏dyn ∼ 1/ √︁ (𝜌mean𝐺), with 𝜌mean the mean density of the core and 𝐺 the gravitational constant, is of the order of 103 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' We have checked with a number of test simulations that a variation of Δ𝑡 between 102 and 105 seconds does not impact our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' In the stellar models considered, radiative transfer is the major heat transport that contributes to the thermal conductivity, which is given for photons by 𝜒 = 16𝜎𝑇3 3𝜅𝜌 , (4) where 𝜅 is the Rosseland mean opacity, and 𝜎 the Stefan-Boltzmann constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Realistic stellar opacities and equation of states appropriate for the description of stellar interiors are implemented in MUSIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Opacities are interpolated from the OPAL tables (Iglesias & Rogers 1996) for solar metallicity and the equation of state is based on the OPAL EOS tables of Rogers & Nayfonov (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='1 Initial stellar models To provide the initial structures for the 2D simulations, we compute stellar models in the mass range 3-20 𝑀⊙ with the one-dimensional Lyon stellar evolution code (Baraffe & El Eid 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 1998), using the same opacities and equation of state as MUSIC2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The 2D simulations require as initial input a radial profile of den- sity and internal energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The 1D stellar evolution models have an initial helium abundance in mass fraction 𝑌=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='28 and solar metal- licity 𝑍=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='02 and were computed through the pre-main sequence and main sequence phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' All initial models for the 2D simulations in this study are taken at the beginning of core hydrogen burning and have a central abundance of helium 𝑌c=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='2838, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' only ∼ 1% of their central hydrogen has been depleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' There is thus a very shallow mean molecular weight gradient at the convective boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Follow-up analysis of later stages of evolution with a steeper gradient of molecular weight at the core boundary are in progress (Morison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' in prep).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Convective stability is defined by the Schwarzschild 1 Note that Δ𝑡 is the time after which the gravitational potential is updated, not the numerical timestep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The numerical timestep used for these simulations is set by the hydrodynamical CFL number varying between 10 and 50 (see Viallet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2011, for definitions) and corresponding to values for the timestep ranging between 5 s and 40 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2 The 1D initial structures are available on the repository http://perso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='ens- lyon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='fr/isabelle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='baraffe/2Dcore_overshooting_2023 criterion ∇ < ∇ad, with ∇ = d log𝑇 d log 𝑃 the temperature gradient and ∇ad = d log𝑇 d log 𝑃 |𝑆 the adiabatic gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The 1D stellar models used to generate the initial structures for the 2D simulations do not account for overshooting at the convective core boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' In the following, we define the Schwarzschild boundary as the transition layer be- tween convective instability (∇ > ∇ad) and stability (∇ < ∇ad).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The properties of the initial 1D stellar structures are provided in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Nuclear energy generated in the convective cores is accounted for in the internal energy equation (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (3)) through the term 𝑄nuc using the radial profile of the nuclear energy rate from the 1D stellar model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Given that the simulation times are orders of magnitude smaller than the nuclear timescale for H burning in the cores, the nuclear energy is assumed to remain constant with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='2 Spherical-shell geometry and boundary conditions Two-dimensional simulations are performed in a spherical shell using spherical coordinates, namely 𝑟 the radius and 𝜃 the polar angle, and assuming azimuthal symmetry in the 𝜙-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' For all models, the inner radius 𝑟in is defined at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='02 𝑅star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The choice of the outer radius 𝑟out depends on the stellar model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Since the main motivation of this work is to analyse the extent of the overshooting layer for different stellar masses, the outer radius 𝑟out is fixed at a distance of ∼ 1 × 𝐻𝑃,CB for the lowest mass (3 𝑀⊙) to ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='5 × 𝐻𝑃,CB for the highest mass (20 𝑀⊙) away from the convective boundary 𝑟conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Extension of the radial domain to analyse the generation of internal waves at the core boundary and their propagation in the radiative envelope is work in progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The angular extent ranges from 𝜃 = 0◦ to 𝜃 = 180◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The grid has uniform spacing in the r and 𝜃 coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The choice for the resolution (𝑁𝑟, 𝑁𝜃) is set by the condition to have a good resolution of the pressure scale height at the Schwarzschild boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Effective Reynolds and Prandtl numbers are commonly used to set the resolution of numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' But given that our simulations are based on an implicit Large Eddy Simulation (ILES) approach, only a rough estimate can be provided for these numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' They will in any case remain far away from the conditions prevailing in stellar interiors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' We suggest that a more relevant resolution criterion for hydrodynamical simulations devoted to the study of overshooting using realistic stellar structures should be the number of grid cells per pressure scale height at the convective boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' This should allow a more relevant comparison between the works of different groups devoted to the study of different stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' We use ∼ 110 − 140 grid cells per pressure scale height in the radial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The details of the resolution adopted in this work are provided in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' We have also performed a few tests with higher resolution and analyse the impact in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The radial boundary conditions for the density correspond to a constant radial derivative on the density (see Pratt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The energy flux at the inner and outer radial boundaries are set to the value of the energy flux at that radius in the one-dimensional stellar evolution model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' At the boundaries in 𝜃, because of the extension of the angular domain to the poles, reflective boundary conditions for the density and energy are used (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' the values are mirrored at the boundary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' For the velocity, we impose reflective conditions at the radial and polar boundaries, corresponding to: v𝑟 = 0 and 𝜕v𝜃 𝜕𝑟 = 0 at 𝑟in and 𝑟out, 𝜕v𝑟 𝜕𝜃 = 0 and v𝜃 = 0 at 𝜃 = 0◦ and 𝜃 = 180◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' We have also performed simulations for the 3 𝑀⊙ model with artificial enhancement of the stellar luminosity and the thermal dif- fusivity by factors 10, 102, 103 and 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' This covers the range of MNRAS 000, 1–12 (2022) 4 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Properties of the initial stellar models (all models have a central helium abundance 𝑌c=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='2838) used for the 2D hydrodynamical simulations: total mass, stellar luminosity, stellar radius, mass and radius of the convective core (corresponding to the location of the Schwarzschild boundary) and the pressure scale height at the Schwarzschild boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 𝑀/𝑀⊙ 𝐿star/𝐿𝑎 ⊙ 𝑅star (cm) 𝑀conv/𝑀⊙ 𝑟conv/𝑅star 𝐻𝑃,CB (cm) 3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='7673 × 101 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='3855 × 1011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='5724 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='1486 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='3 × 1010 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='2186 × 102 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='8424 × 1011 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='212 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='1814 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='8 × 1010 10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='5726 × 103 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='7295 × 1011 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='046 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='2239 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='7 × 1010 15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='9242 × 104 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='4255 × 1011 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='2580 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='3 × 1010 20 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='2962 × 104 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='0172 × 1011 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='7947 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='2869 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='7 × 1010 𝑎 We use 𝐿⊙ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='839 × 1033 erg/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Evolution of the total kinetic energy (in erg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' y-axis with a base-10 log scale) as a function of time (in s) for the simulations described in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Top panel: results for 3 𝑀⊙ models with various luminosity enhancement factors: 3L0 (black), 3L1 (blue), 3L2 (magenta), 3L3 (cyan) and 3L4 (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Bottom panel: results for a range of stellar masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The dotted line for each model corresponds to the value of the total kinetic energy at the beginning of the steady state for convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' luminosities of the stellar masses considered in this work (3-20 𝑀⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' This choice of enhancement factor allows a comparative analysis of the impact of the luminosity for fixed core mass and increasing core mass, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Note that even larger enhancement factors (up to 107) for a 3 𝑀⊙ stellar structure can be found in previous works (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Rogers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Edelmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' For the artificially boosted simulations, the energy flux (equivalently the luminosity) at the ra- dial boundaries is multiplied by the enhancement factor, the nuclear energy rate is multiplied by the same factor and the Rosseland mean opacities 𝜅 in MUSIC are decreased by the same factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Radial profile of the time averaged rms velocity (solid lines) and rms radial velocity (dashed lines) scaled by (𝐿star/1035)1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Top panel: re- sults for 3 𝑀⊙ models with various luminosity enhancement factors: 3L0 (black), 3L1 (blue), 3L2 (magenta), 3L3 (cyan) and 3L4 (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Bottom panel: results for a range of stellar masses: 3L0 (black), 5L0 (blue), 10L0 (ma- genta),15L0 (red) and 20L0 (cyan).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The convective boundary corresponding to the Schwarzschild boundary from the 1D initial model is indicated by a vertical solid line with the colour corresponding to each stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 3 RESULTS: AVERAGE DYNAMICS The properties of all simulations are summarised in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' We de- fine 𝑡steady as the time required to reach a steady state for convection, characterised by the total kinetic energy 𝐸kin of the system reaching a plateau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Before 𝑡steady, the initial relaxation phase is characterised by the propagation of strong acoustic waves and the onset of convec- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' At 𝑡steady, the value of the kinetic energy starts to stabilise and MNRAS 000, 1–12 (2022) A study of convective core overshooting as a function of stellar mass 5 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Main properties of the 2D simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Model 𝑀/𝑀⊙ 𝐿 (erg/s) 𝑁𝑟 × 𝑁𝜃 𝑟out/𝑅star 𝜏𝑎conv (s) 𝑁 𝑏 conv 𝑡𝑐 steady (s) 𝑡𝑑 sim (s) 3L0 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='981 ×1035 336 x 168 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='9 ×106 1442 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='5 ×108 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='71 ×109 3L1 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='981 ×1036 336 x 168 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='25 8 ×105 1211 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='6 ×108 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='43 ×109 3L2 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='981 ×1037 336 x 168 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='9 ×105 501 9 ×107 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='84 ×108 3L2xhres 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='981 ×1037 684 x 342 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='8 ×105 514 9 ×107 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='84 ×108 3L3 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='981 ×1038 336 x 168 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='7 ×105 1904 6 ×107 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='81×108 3L3xhres 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='981 ×1038 684 x 342 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='7 ×105 1243 6 ×107 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='71 ×108 3L4 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='981 ×1039 336 x 168 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='25 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='9 ×104 1457 3 ×107 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='60 ×108 3L4xhres 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='981 ×1039 684 x 342 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='25 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='7 ×104 1400 3 ×107 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='52 ×108 5L0 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='003 ×1036 400 x 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='4 ×106 1260 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='45 ×108 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='01 ×109 10L0 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='139 ×1037 416 x 208 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='2 ×106 1260 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='1 × 108 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='77 ×109 15L0 15 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='387 ×1037 688 x 344 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='1 ×106 875 108 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='14×109 20L0 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='649 ×1038 864 x 430 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='1 ×106 800 9 × 107 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='99 ×108 𝑎 Convective turnover time (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 3 for its definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 𝑏 Number of convective turnover times covered by the simulation once steady state convection is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 𝑐Physical time to reach a steady state for convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 𝑑Total physical runtime of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' from this time it remains roughly constant with time (following the dotted curve which corresponds to the value of 𝐸kin at 𝑡steady for each model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The simulations are stopped at time 𝑡sim provided in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' None of these simulations are thermally relaxed, given that the total simulation times for all models are orders of magnitude smaller than the relevant thermal timescale ∼ 𝐺𝑀2/(𝑅star𝐿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' As a consequence all these simulations are expected to maintain a secular drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' We have compared the radial profile of the internal energy, averaged in the angular direction, for each 2D model at time 𝑡steady and at time 𝑡sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' We find a maximum of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='5% relative difference for the internal energy at a given radius, with the largest difference found for the most luminous models (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The above-mentioned drift is thus so slow that calculating statistical or averaged data during this very slowly changing transitional state is sensible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Figure 1 shows the evolution of the total kinetic energy as a func- tion of time for all models and the plateau characterising their steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The initial transient phase can last a relatively long time, de- pending on the model studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' For the model 3L0, we note a dif- ferent behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' After the peak due to strong acoustic waves, the kinetic energy continuously decreases until 𝑡 ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='4 × 108 s (log 𝑡 ∼ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' In this regime, convection develops in the core (within the 1D Schwarzschild boundary) in two spatially separate regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The abrupt increase of 𝐸kin observed at 𝑡 ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='4×108 s marks the merging of these two convective regions and the beginning of fully developed convection in the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The Mach number characterising the con- vective velocities in model 3L0 is small, of the order of ∼ 10−4, which is numerically challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' This low Mach number explains why several previous works artificially enhance the luminosity of the model (Rogers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Horst et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' There is no need for this artefact for the model 3L0 as MUSIC’s numerical scheme allows convection to develop and eventually reach a steady state even after a long transient phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Note that this unusual transient phase observed for the model 3L0 will likely change with a different procedure for initialising the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' All simulations start without an imposed background noise (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' initial velocities are set to zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Imposing initially a background noise for the model 3L0 may change the loca- tion where convection starts and thus the behaviour of the transient phase, which is irrelevant for the analysis performed in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' A global convective turnover time 𝜏conv is estimated based on the rms velocity vrms(𝑟, 𝑡) at radius 𝑟 and time 𝑡, which characterises a bulk convective velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' We define 𝜏conv by: 𝜏conv = �∫ 𝑟conv 𝑟in d𝑟 vrms(𝑟, 𝑡) � 𝑡, (5) where the rms velocity is given by vrms(𝑟, 𝑡) = √︃ ⟨v2(𝑟, 𝜃, 𝑡)⟩𝜃, (6) with v2 = v2𝑟 + v2 𝜃, v𝑟 and v𝜃 being the radial and angular velocities, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Time averages are denoted by ⟨⟩𝑡 and calculated between 𝑡steady and 𝑡sim, the final time reached by the simulation (see values in Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' For any quantity 𝑋 we define: � 𝑋 � 𝑡 = 1 (𝑡sim − 𝑡steady) ∫ 𝑡sim 𝑡steady 𝑋d𝑡 (7) The volume-weighted average in the angular direction ⟨⟩𝜃 is defined for any quantity X as: � 𝑋(𝑟, 𝜃, 𝑡) � 𝜃 = ∫ 𝜃 𝑋(𝑟, 𝜃, 𝑡)d𝑉(𝑟, 𝜃) ∫ 𝜃 d𝑉(𝑟, 𝜃) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (8) The simulations are stopped after a time 𝑡sim when convergence of the statistics used to determine the size of the layer penetrated by plumes is obtained, as explained in the next section (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Table 2 provides the values and numbers of the convective turnover times, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Figure 2 displays the rms velocity and rms radial velocity for the 3 𝑀⊙ models with artificially enhanced luminosities (upper panel) and for the range of stellar masses investigated (lower panel), scaled by 𝐿1/3 star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' In the convective core, our simulations re- produce the expected scaling of convective velocity with luminosity vconv ∝ 𝐿1/3 recovered by many hydrodynamical simulations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Edelmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Andrassy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Horst et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Higl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' This scaling is expected from mixing-length theory based on the argument that the turbulent dissipation rate of kinetic energy in a turbulent convective zone scales with v3 (Biermann 1932).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' But a general scaling of the total flux with v3 can also be derived for the kinetic energy and the enthalpy fluxes based on simple dimensional arguments (see Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2017) MNRAS 000, 1–12 (2022) 6 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The rms velocities in the stably stratified region are due to the penetrative flows just above the convective boundary and to the prop- agation of internal waves excited by the convective motions and the penetrating plumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The top panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2 shows that these ve- locities also increase with the luminosity, suggesting more efficient overshooting of the convective motions above the convective bound- ary and thus larger overshooting length with increasing luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (2021) reports similar behaviours for convective en- velopes of solar-like models with artificially enhanced luminosities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Quantitative estimate of the overshooting lengths for all models is performed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 4 RESULTS: EXTENT OF THE OVERSHOOTING REGION 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='1 Determination of overshooting lengths To determine an overshooting length, we adopt the same approach as in Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (2021) and initially inspired by the findings of Pratt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' This approach is based on the analysis of the depth of all convective plumes that penetrate beyond the convective boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The two criteria used to determine the depth of a penetrative plume at a given angle 𝜃 and time 𝑡 are based on the first zero above the convective boundary 𝑟conv of the vertical kinetic energy flux fk and vertical heat flux f𝛿T, defined by (see Pratt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2017): fk(𝑟, 𝜃, 𝑡) = 1 2 𝜌(𝑟, 𝜃, 𝑡)v2(𝑟, 𝜃, 𝑡)v𝑟 (𝑟, 𝜃, 𝑡), (9) f𝛿T(𝑟, 𝜃, 𝑡) = 𝜌(𝑟, 𝜃, 𝑡)𝑐𝑃(𝑟, 𝜃, 𝑡)𝛿𝑇(𝑟, 𝜃, 𝑡)v𝑟 (𝑟, 𝜃, 𝑡), (10) where 𝑐𝑃 is the specific heat at constant pressure and the temperature fluctuation 𝛿𝑇 is defined by: 𝛿𝑇(𝑟, 𝜃, 𝑡) = 𝑇(𝑟, 𝜃, 𝑡) − �� 𝑇(𝑟, 𝜃, 𝑡) � 𝜃 � 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (11) The method is the same as the one developed in Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (2021) for convective envelopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' At each time 𝑡, we calculate at each angle 𝜃 the radial positions 𝑟0(𝜃, 𝑡) of a plume corresponding to the first zero of fk and f𝛿T, respectively, above the convective boundary 𝑟conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The corresponding overshooting length 𝑙0 with respect to 𝑟conv is defined by 𝑙0(𝜃, 𝑡) = 𝑟0(𝜃, 𝑡) − 𝑟conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (12) Figure 3 illustrates the angular structure of the overshooting layer at an arbitrary time for the 10 𝑀⊙ stellar model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' We then define the maximal overshooting length 𝑙max 0 at a given time by the maximum over all angles 𝜃: 𝑙max 0 (𝑡) = max(𝑙0(𝜃, 𝑡)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (13) The time average 𝑙max = ⟨𝑙max 0 (𝑡)⟩𝑡 provides an effective width for the overshooting layer where the most vigorous plumes penetrate and which we use to characterise the extension of the mixing layer over the long term evolution of the star (Pratt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Table 3 displays 𝑙max based on the criterion for fk and f𝛿T, respectively, for all models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The distributions of overshooting lengths derived from fk and f𝛿T, respectively, slowly converges with time, as found in Pratt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (2017) and Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Several hundreds to thousand convective turnover times, depending on the stellar model, are required for the statistics to converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Eventually, both criteria provide similar values for the effective overshooting width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The values of the overshooting width based on f𝛿T converge faster with time, compared to the value based on fk, as found as well for convective envelopes in Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The values of 𝑙max(f𝛿T) provided in Table 3 have reached a steady state for all Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Overshooting lengths𝑙0 defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (12) as a function of the angle 𝜃 at time 𝑡 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='3108s for the 10 𝑀⊙ model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The upper panel corresponds to 𝑙0 defined by fk and the lower panel to 𝑙0 defined by f𝛿T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The horizontal dashed line in each panel indicates the average overshooting length at this time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' models after 𝑡sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Depending on the stellar model, 𝑙max(fk) gets close to 𝑙max(f𝛿T) (difference of <∼ 20%) for all models but models 3L0 and 20L0, for which 𝑙max(fk) continues slowly decreasing even after more than 800 ×𝜏conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' We run three simulations for the 3 𝑀⊙ models with enhanced luminosity with twice the resolution in both radial and angular directions and covering about the same simulation time as their lower resolution counterpart, in order to check the sensitivity of the values of 𝑙max to the resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The properties of these higher resolution models (labelled 2xhres) are displayed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The results for the overshooting lengths are given in Table 3 and show similar values for lmax(f𝛿T) as found with a lower resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The values for 𝑙max(fk) of the higher resolution models are larger than the corresponding value for the lower resolution model, as it takes more time for 𝑙max(fk) in the high resolution models to decrease to the level of 𝑙max(f𝛿T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' But the value of 𝑙max(fk) in the high resolution models continues decreasing with time and we expect it to eventually converge and thus get much closer to 𝑙max(f𝛿T) and to the value of 𝑙max(fk) found in the lower resolution model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='2 Relationship between overshooting length and stellar luminosity The variation of 𝑙max with the stellar luminosity is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 4 for the 3𝑀⊙ models with enhanced luminosity and for the set of stellar masses with realistic luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' As expected from the behaviour of the rms velocities (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2) overshooting lengths increase with the stellar luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' To derive an approximate scaling relationship for the overshooting length 𝑑ov that can be implemented in stellar evolution codes, we use the values of 𝑙max derived from f𝛿T, since these values have converged with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' We derive the following MNRAS 000, 1–12 (2022) A study of convective core overshooting as a function of stellar mass 7 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Effective width 𝑙max of the overshooting layer in units of the total stellar radius and of the pressure scale height at the convective boundary, for all models considered in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The quantity 𝑙max(fk) is based on the criterion using fk (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 9) and 𝑙max(f𝛿T) is based on f𝛿T (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Model 𝑙max(fk)/𝑅star 𝑙max(f𝛿T)/𝑅star 𝑙max(fk)/𝐻𝑃,CB 𝑙max(f𝛿T)/𝐻𝑃,CB 3L0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='4 ×10−3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='7 ×10−3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='8 ×10−2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='9 ×10−2 3L1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='2 ×10−3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='2 ×10−3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='5 × 10−2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='5 ×10−2 3L2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='2 ×10−3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='1 ×10−3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='6 × 10−2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='5 ×10−2 3L2xhres 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='4 ×10−3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='4 ×10−3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='9 × 10−2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='8 ×10−2 3L3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='8 ×10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='6 ×10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='9 ×10−1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='7 ×10−1 3L3xhres 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='2 ×10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='6 ×10−2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='3 ×10−1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='7 ×10−1 3L4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='5 ×10−2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='8 ×10−2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='7 ×10−1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='0 ×10−1 3L4xhres 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='0 ×10−2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='0 ×10−2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='2 ×10−1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='2×10−1 5L0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='3 ×10−3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='0 ×10−3 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='5 ×10−2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='1 ×10−2 10L0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='2 ×10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='1 ×10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='2 ×10−1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='1 ×10−1 15L0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='6 ×10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='3 ×10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='66 ×10−1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='35 ×10−1 20L0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='5 ×10−2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='0 ×10−2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='8 ×10−1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='17 ×10−1 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Overshooting length 𝑙max, in units of the pressure scale height at the convective boundary, as a function of the model luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The 3 𝑀⊙ models with various luminosity enhancement factors are indicated in red (dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The results for a range of stellar masses with realistic stellar luminosity are indicated in blue (solid line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The dotted curve shows the fit for the overshooting length 𝑑𝑜𝑣/𝐻P,CB given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' expression which fits the results for the stellar mass range studied: 𝑑ov/𝐻P,CB = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='05 × 10−3 × (𝐿/𝐿⊙)1/3 × (𝑟conv/𝐻𝑃,CB)1/2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='02 (14) We find a typical scaling with the luminosity 𝑑ov ∝ 𝐿1/3 ∝ vconv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Numerical studies of convective envelopes report overshooting lengths 𝑑ov which vary with the luminosity following 𝑑ov ∝ 𝐿𝑎 with 𝑎 varying between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='08 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='31 (Hotta 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Käpylä 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The analytical model of Zahn (1991) for pene- tration, based on first order estimate of the deceleration of a plume in an adiabatically stratified penetration zone, predicts 𝑑ov ∝ v3/2 conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Our results also show that the overshooting lengths derived for a fixed stellar mass (and thus a fixed convective core size) are systematically smaller than the one derived for larger cores but similar luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Interestingly, a dependence of 𝑑ov with the size of the core 𝑟conv is also predicted by Zahn (1991) (see their Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='5)) with the same re- lation of proportionality 𝑑ov ∝ (𝑟conv/𝐻𝑃,CB)1/2 as found in present simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' This dependence in the Zahn model is derived from the strong variations with radius of various relevant quantities such as the gravitational acceleration 𝑔, the mass 𝑚(𝑟) enclosed in a sphere of radius 𝑟, the radiative conductivity 𝜒, and thus the radiative flux, close to the convective core boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' In our simulations, we expect the radial dependence of the gravitational acceleration to have the main impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' We find that the larger the core (in terms of radius and mass), the smaller the gravitational acceleration at the core boundary 𝑔conv ∼ 𝐺𝑀conv/𝑟2conv (see values in Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Therefore, the larger the stellar mass, the larger the velocities at the convective boundary and the smaller the restoring force due to gravity, implying up-flows to penetrate over larger distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' This is a plausible explanation for the dependence of 𝑑ov on the convective core radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' We analyse below (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 6) whether the expression provided by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (14) pro- vides a reasonable agreement between stellar evolution models and observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 5 THERMAL BACKGROUND EVOLUTION The prescription used in the previous section to determine overshoot- ing lengths relies on two assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Firstly, we consider that the simulations have reached a steady state for convection (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' a global dynamical steady state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' This assumption is reasonable based on the observation that the total kinetic energy of the system reaches a plateau as a function of time (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Secondly, we assume that the relevant convective boundary from which the overshooting lengths are defined is the 1D Schwarzschild boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' This is directly useful for the purpose of implementing these overshooting lengths in 1D stellar evolution codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' However, we find that in all models a small nearly adiabatic layer just above the convective boundary forms rapidly once convection steady state is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' For the most luminous models, we observe that this small layer slowly grows in size with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Anders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (2022) also find a modification of the temperature gradient which becomes close to the adiabatic gradient in the pen- etration layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' They report that their simulations exhibit the process of convective penetration as defined by e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Zahn (1991), with con- vective penetrating motions mixing entropy and establishing a nearly adiabatic stratification above the Schwarzschild boundary (see also Brummell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Anders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (2022) suggest that the extent of convective penetration is limited and derive arguments involving the MNRAS 000, 1–12 (2022) 8 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Visualisation of the radial velocity v𝑟 [cm/s] (top panel) and the relative temperature fluctuations (𝑇 −⟨𝑇 ⟩𝜃)/⟨𝑇 ⟩𝜃 (bottom panel) in a region zoomed around the convective boundary (horizontal black line) for the model 20L0 at time 𝑡 = 7 × 108 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The x-axis represents the co-latitude (in terms of cos 𝜃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Note that to the better illustrate upwellings and downwellings in the top panel, the velocity scale is saturated, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' any velocity > vr,max = 5 × 103 cm/s (< vr,min = −5×103 cm/s) are represented with the same color as vr,max (vr,min).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' convective flux, the viscous dissipation rate and the buoyancy work, providing an estimate of the penetration width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Depending on their setup, they find that penetration zones can take thousands of con- vective turnover times to saturate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' They show properties of the flow and of the temperature fluctuations close to a convective boundary (see their Figure 1) which are similar to our results, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 5 for the model 20L0 at a given time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' As expected in con- vective regions, convective upflows transport hot material from the central regions up to the top of the convective core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Inspection of temperature fluctuations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' the difference between the local tem- perature and the horizontally averaged thermal background) indeed indicates that upflows in the convective region are characterised by positive temperature fluctuations and downflows by negative tem- perature fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' When upflows cross the convective boundary, at the top of the convective core, and penetrate the stably stratified medium, they adiabatically expand and therefore get cooler (neg- ative temperature fluctuation) and denser than the subadiabatically stratified environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' To understand the establishment of a nearly adiabatic layer in the penetration region, one needs to compare the advection timescale, which characterises the process of entropy mixing by penetrating flows (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' an advection process), and the thermal diffusion timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' If penetrating flows, as illustrated in the top panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 5, can drive efficient entropy/thermal mixing, the layer characterised by pene- trating up-flows will remain nearly adiabatic if thermal diffusion is slow enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Table 4 provides estimates of the diffusive timescale Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Profile of the time and angular averages of the quantity (∇ − ∇ad) in the layers just above the convective core for the most luminous models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The 1D profile of (∇ − ∇ad) is indicated by the black dashed line and the 1D convective core boundary by the vertical solid line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The location of 𝑙max derived from f𝛿T is indicated by the vertical dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' In both panels, the solid blue line corresponds to the time average between 𝑡steady and 𝑡sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The curves in magenta correspond to time averages over 20×𝜏conv at a given time, as indicated in each panel (time 𝑡 in s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 𝜏diff = 𝐿2/𝜅rad at the core boundary, with 𝐿 a relevant lengthscale and 𝜅rad = 𝜒/(𝜌𝑐𝑃) the thermal diffusivity (which is the radiative diffusivity for present stellar models with 𝜒 defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' (4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Esti- mate of an advection timescale 𝜏adv = 𝐿/vr,rms is based on the time averaged rms radial velocity at the core boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' For the charac- teristic lengthscales at the core boundary, we use the overshooting distance 𝑙max(f𝛿T) (see Table 3) and the pressure scale height 𝐻P (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' As illustrated in Table 4, typical advection timescales are much smaller than typical thermal diffusion timescales for all models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The growth in size with time of the nearly adiabatic layer observed in the most luminous models is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 6 for the models 3L3 and 3L4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' This growth with time may also happen in the less luminous models, but their very slow evolution and less vigorous penetrating flows may prevent clearly exhibiting this feature over present simulation times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' We also note that the angular averaged temperature gradient in the models, while getting very close to the adiabatic gradient, remains stable against the Schwarzschild criterion over the simulation times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' For the purpose of analysing the time evolution of the nearly adiabatic layer, we have extended the simulation time of the models 3L3 and 3L4 beyond the value of 𝑡sim used to determine overshooting depths (see Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' 2), until 𝑡final = 5 × 108 s (∼ 2600 × 𝜏conv for 3L3 and ∼ 5300×𝜏conv for 3L4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' The aim is to reach a simulation time for these models close to or greater than the thermal diffusion timescale in the overshooting layer 𝜏diff(𝑙max).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Given the smaller grid size and larger thermal diffusivity of these models, this is still computationally affordable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content=' Figure 6 shows clearly in models 3L3 and 3L4 that the radial extension of the nearly adiabatic layer slows down with time MNRAS 000, 1–12 (2022) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='36 4000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='32 2000 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='36 10-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='34 10-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='32 10-5 10-6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfuQHv/content/2301.02604v1.pdf'} +page_content='30 T- (T)e)/