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The turbulent burning of nuclei is a common phenomenon in the evolution of stars. Here we examine a challenging case: the merging of the neon and oxygen burning shells in a 23 M$_{\odot}$ star. A previously unknown quasi-steady state is established by the interplay between mixing, turbulent transport, and nuclear burning. The resulting stellar structure has two burning shells within a single convection zone. We find that the new neon burning layer covers an extended region of the convection zone, with the burning peak occurring substantially below where the Damk\"ohler number first becomes equal to unity. These characteristics differ from those predicted by 1D stellar evolution models of similar ingestion events. We develop the mean-field turbulence equations that govern compositional evolution, and use them to interpret our data set. An important byproduct is a means to quantify sub-grid-scale effects intrinsic to the numerical hydrodynamic scheme. For implicit large eddy simulations, the analysis method is particularly powerful because it can reveal where and how simulated flows are modified by resolution, and provide straightforward physical interpretations of the effects of dissipation or induced transport. Focusing on the mean-field composition variance equations for our analysis, we recover a Kolmogorov rate of turbulent dissipation without it being imposed, in agreement with previous results which used the turbulent kinetic energy equation., Comment: 17 pages, 8 figures, Submitted to MNRAS

Renzini wrote an influential critique of “overshooting” in mixing-length theory (MLT), as used in stellar evolution codes, and concluded that three-dimensional fluid dynamical simulations were needed. Such simulations are now well tested. Implicit large eddy simulations connect large-scale stellar flow to a turbulent cascade at the grid scale, and allow the simulation of turbulent boundary layers, with essentially no assumptions regarding flow except the number of computational cells. Buoyant driving balances turbulent dissipation for weak stratification, as in MLT, but with the dissipation length replacing the mixing length. The turbulent kinetic energy in our computational domain shows steady pulses after 30 turnovers, with no discernible diminution; these are caused by the necessary lag in turbulent dissipation behind acceleration. Interactions between coherent turbulent structures give multi-modal behavior, which drives intermittency and fluctuations. These cause mixing, which may justify use of the instability criterion of Schwarzschild rather than the Ledoux. Chaotic shear flow of turning material at convective boundaries causes instabilities that generate waves and sculpt the composition gradients and boundary layer structures. The flow is not anelastic; wave generation is necessary at boundaries. A self-consistent approach to boundary layers can remove the need for ad hoc procedures of “convective overshooting” and “semi-convection.” In Paper II, we quantify the adequacy of our numerical resolution in a novel way, determine the length scale of dissipation—the “mixing length”—without astronomical calibration, quantify agreement with the four-fifths law of Kolmogorov for weak stratification, and deal with strong stratification.

We examine the physical basis for algorithms to replace mixing-length theory (MLT) in stellar evolutionary computations. Our 321D procedure is based on numerical solutions of the Navier-Stokes equations. These implicit large eddy simulations (ILES) are three-dimensional (3D), time-dependent, and turbulent, including the Kolmogorov cascade. We use the Reynolds-averaged Navier-Stokes (RANS) formulation to make concise the 3D simulation data, and use the 3D simulations to give closure for the RANS equations. We further analyze this data set with a simple analytical model, which is non-local and time-dependent, and which contains both MLT and the Lorenz convective roll as particular subsets of solutions. A characteristic length (the damping length) again emerges in the simulations; it is determined by an observed balance between (1) the large-scale driving, and (2) small-scale damping. The nature of mixing and convective boundaries is analyzed, including dynamic, thermal and compositional effects, and compared to a simple model. We find that (1) braking regions (boundary layers in which mixing occurs) automatically appear {\it beyond} the edges of convection as defined by the Schwarzschild criterion, (2) dynamic (non-local) terms imply a non-zero turbulent kinetic energy flux (unlike MLT), (3) the effects of composition gradients on flow can be comparable to thermal effects, and (4) convective boundaries in neutrino-cooled stages differ in nature from those in photon-cooled stages (different P\'eclet numbers). The algorithms are based upon ILES solutions to the Navier-Stokes equations, so that, unlike MLT, they do not require any calibration to astronomical systems in order to predict stellar properties. Implications for solar abundances, helioseismology, asteroseismology, nucleosynthesis yields, supernova progenitors and core collapse are indicated., Comment: 22 pages, 4 figures, 2 tables; significantly re-written, critique of Pasetto, et al. model added, accepted for publication by ApJ

We present 3D implicit large eddy simulations (ILES) of the turbulent convection in the envelope of a 5 Msun red giant star and in the oxygen-burning shell of a 23 Msun supernova progenitor. The numerical models are analyzed in the framework of 1D Reynolds-Averaged Navier-Stokes (RANS) equations. The effects of pressure fluctuations are more important in the red giant model, owing to larger stratification of the convective zone. We show how this impacts different terms in the mean-field equations. We clarify the driving sources of kinetic energy, and show that the rate of turbulent dissipation is comparable to the convective luminosity. Although our flows have low Mach number and are nearly adiabatic, our analysis is general and can be applied to photospheric convection as well. The robustness of our analysis of turbulent convection is supported by the insensitivity of the mean-field balances to linear mesh resolution. We find robust results for the turbulent convection zone and the stable layers in the oxygen-burning shell model, and robust results everywhere in the red giant model, but the mean fields are not well converged in the narrow boundary regions (which contain steep gradients) in the oxygen-burning shell model. This last result illustrates the importance of unresolved physics at the convective boundary, which governs the mixing there., 26 pages, 20 figures, Accepted for publication in ApJ

Shell convection driven by nuclear burning in a stellar core is a common hydrodynamic event in the evolution of many types of stars. We encounter and simulate this convection (i) in the helium core of a low-mass red giant during core helium flash leading to a dredge-down of protons across an entropy barrier, (ii) in a carbon-oxygen core of an intermediate-mass star during core carbon flash, and (iii) in the oxygen and carbon burning shell above the silicon-sulfur rich core of a massive star prior to supernova explosion. Our results, which were obtained with the hydrodynamics code HERAKLES, suggest that both entropy gradients and entropy barriers are less important for stellar structure than commonly assumed. Our simulations further reveal a new dynamic mixing process operating below the base of shell convection zones., 8 pages, 3 figures .. submitted to a proceedings of conference about "Red Giants as Probes of the Structure and Evolution of the Milky Way" which has taken place between 15-17 November 2010 in Rome

Context. The injection of hydrogen into the convection shell powered by helium burning during the core helium flash is commonly encountered during the evolution of metal-free and extremely metal-poor low-mass stars. Multidimensional hydrodynamic simulations indicate that the hydrogen injection may also occur in more metal-rich stars due to turbulent entrainment that accelerates the growth of the shell convection zone and increases its size. However, one-dimensional stellar models cast doubts that helium-flash hydrogen mixing does occur as it requires the crossing of an entropy barrier at the helium-hydrogen interface. Aims. With specifically designed multidimensional hydrodynamic simulations, we aim to prove that an entropy barrier is no obstacle to the growth of the helium-burning shell convection zone in the helium core of a metal-rich Population I star, i.e.  convection can penetrate into the hydrogen-rich layers for these stars, too. We study whether this is also possible in one-dimensional stellar evolutionary calculations. Methods. We artificially shift the hydrogen-rich layer closer to the outer edge of the helium-burning shell convection zone in a Population I star with a mass of 1.25 MâŠ(tm), and simulate the subsequent evolution in two and three dimensions, respectively. We also perform stellar evolutionary calculations of the core helium flash in metal-rich stars implementing turbulent entrainment by means of a simple prescription. These simulations were performed with the Eulerian hydrodynamical code HERAKLES and the stellar evolution code STAREVOL, respectively. Results. Our hydrodynamical simulations show that the helium-burning shell convection zone in the helium core moves across the entropy barrier and reaches the hydrogen-rich layers. This leads to a mixing of protons into the hotter layers of the core and to a rapid increase in the nuclear energy production at the upper edge of the helium-burning convection shell-the hydrogen injection flash. As a result, a second convection zone appears in the hydrogen-rich layers. In contrast to one-dimensional models, the entropy barrier separating the two convective shells from each other is largely permeable to chemical transport when allowing for multidimensional flow and consequently hydrogen is continuously mixed deep into the helium core. We find it difficult to replicate this behavior using one-dimensional stellar evolutionary calculations. © 2011 ESO., SCOPUS: ar.j, info:eu-repo/semantics/published

We continue our study of turbulent convection during the core helium flash close to its peak by comparing the results of 2D and 3D hydrodynamic simulations performed with our multidimensional Eulerian hydrodynamics code HERAKLES on the ALTIX 4700 computer of the LRZ. In our previous study based on well resolved 2D and coarsely resolved 3D models covering only the onset of convection we found that (i) the temporal evolution and properties of the convection are similar to those predicted by quasi-hydrostatic stellar evolutionary calculations, and that (ii) the core helium flash does not lead to any explosive behavior. These results are confirmed by our new well resolved 3D models covering longer evolutionary periods. They also show that the flow patterns developing in the 3D models are structurally different from those of the corresponding 2D ones, and that the typical convective velocities in these models are smaller than those in their 2D counterparts. The 3D models also tend to agree better with the predictions of mixing length theory. Our hydrodynamic simulations further show the presence of turbulent entrainment that results in a growth of the convection zone on a dynamic time scale.

We describe and discuss hydrodynamic simulations of the core helium flash using an initial model of a 1.25 M_sol star with a metallicity of 0.02 near at its peak. Past research concerned with the dynamics of the core helium flash is inconclusive. Its results range from a confirmation of the standard picture, where the star remains in hydrostatic equilibrium during the flash (Deupree 1996), to a disruption or a significant mass loss of the star (Edwards 1969; Cole & Deupree 1980). However, the most recent multidimensional hydrodynamic study (Dearborn 2006) suggests a quiescent behavior of the core helium flash and seems to rule out an explosive scenario. Here we present partial results of a new comprehensive study of the core helium flash, which seem to confirm this qualitative behavior and give a better insight into operation of the convection zone powered by helium burning during the flash. The hydrodynamic evolution is followed on a computational grid in spherical coordinates using our new version of the multi-dimensional hydrodynamic code HERAKLES, which is based on a direct Eulerian implementation of the piecewise parabolic method., 6 pages, 5 figures. IAUS 252 Conference Proceeding (Sanya, China): "The art of modeling stars in the 21st century"

We study turbulent convection during the core helium flash close to its peak by comparing the results of two and three-dimensional hydrodynamic simulations. We use a multidimensional Eulerian hydrodynamics code based on state-of-the-art numerical techniques to simulate the evolution of the helium core of a $1.25 M_{\odot}$ Pop I star. Our three-dimensional hydrodynamic simulations of the evolution of a star during the peak of the core helium flash do not show any explosive behavior. The convective flow patterns developing in the three-dimensional models are structurally different from those of the corresponding two-dimensional models, and the typical convective velocities are smaller than those found in their two-dimensional counterparts. Three-dimensional models also tend to agree better with the predictions of mixing length theory. Our hydrodynamic simulations show the presence of turbulent entrainment that results in a growth of the convection zone on a dynamic time scale. Contrary to mixing length theory, the outer part of the convection zone is characterized by a sub-adiabatic temperature gradient., 19 pages, 18 figures

We study the turbulent convection during the core helium flash at its peak with our code HERAKLES. Our 3D hydrodynamic simulations of the core helium flash performed on the LRZ’s ALTIX 4700 allowed us to investigate differences between the 3D models and our earlier models in 2D and 3D using the same initial conditions and angular resolution. We mainly studied the onset of the convection where part of the released thermonuclear energy via helium triple α burning is transported away by material flow from the burning regions inhibiting thereby a thermonuclear explosion.

It is regretted that the originally published paper (Mocák, Müller, Weiss & Kifonidis, 2008) was not the authors' final amended version. We apologise for this oversight and reproduce the entire corrected paper here on-line only, with revised figure captions.

During most stages of stellar evolution the nuclear burning of lighter to heavier elements results in a radial composition profile which is stabilizing against buoyant acceleration, with light material residing above heavier material. However, under some circumstances, such as off-center ignition, the composition profile resulting from nuclear burning can be destabilizing and characterized by an outwardly increasing mean molecular weight. The potential for instabilities under these circumstances and the consequences that they may have on stellar structural evolution remain largely unexplored. In this paper we study the development and evolution of instabilities associated with unstable composition gradients in regions that are initially stable according to linear Schwarzschild and Ledoux criteria. In particular, we study the development of turbulent flow under a variety of stellar evolution conditions with multi-dimensional hydrodynamic simulation; the phases studied include the core helium flash in a 1.25 M ⊙ star, the core carbon flash in a 9.3M ⊙ star, and oxygen shell burning in a 23M ⊙ star. The results of our simulations reveal a mixing process associated with regions having outwardly increasing mean molecular weight that reside below convection zones. The mixing is not due to overshooting from the convection zone, nor is it due directly to thermohaline mixing which operates on a timescale several orders of magnitude larger than the simulated flows. Instead, the mixing appears to be due to the presence of a wave field induced in the stable layers residing beneath the convection zone which enhances the mixing rate by many orders of magnitude and allows a thermohaline type mixing process to operate on a dynamical, rather than thermal, timescale. The mixing manifests itself in the form of overdense and cold blob-like structures originating from density fluctuations at the lower boundary of convective shell and "shooting" down into the core. They are enriched with nuclearly processed material, hence leaving behind traces of higher mean molecular weight. In these regions, we find that initially smooth composition gradients steepen into stair-step-like profiles in which homogeneous, mixed regions are separated by composition jumps. These step-like profiles are then seen to evolve by a process of interface migration driven by turbulent entrainment. We discuss our results in terms of related laboratory phenomena and associated theoretical developments. We also discuss the degree to which the simulated mixing rates depend on the numerical resolution, and what future steps can be taken to capture the mixing rates accurately. © 2011. The American Astronomical Society. All rights reserved., SCOPUS: ar.j, info:eu-repo/semantics/published

Degenerate ignition of helium in low-mass stars at the end of the red giant branch phase leads to dynamic convection in their helium cores. One-dimensional (1D) stellar modeling of this intrinsically multi-dimensional dynamic event is likely to be inadequate. Previous hydrodynamic simulations imply that the single convection zone in the helium core of metal-rich Pop I stars grows during the flash on a dynamic timescale. This may lead to hydrogen injection into the core, and a double convection zone structure as known from one-dimensional core helium flash simulations of low-mass Pop III stars. We perform hydrodynamic simulations of the core helium flash in two and three dimensions to better constrain the nature of these events. To this end we study the hydrodynamics of convection within the helium cores of a 1.25 \Msun metal-rich Pop I star (Z=0.02), and a 0.85 \Msun metal-free Pop III star (Z=0) near the peak of the flash. These models possess single and double convection zones, respectively. We use 1D stellar models of the core helium flash computed with state-of-the-art stellar evolution codes as initial models for our multidimensional hydrodynamic study, and simulate the evolution of these models with the Riemann solver based hydrodynamics code Herakles which integrates the Euler equations coupled with source terms corresponding to gravity and nuclear burning. The hydrodynamic simulation of the Pop I model involving a single convection zone covers 27 hours of stellar evolution, while the first hydrodynamic simulations of a double convection zone, in the Pop III model, span 1.8 hours of stellar life. We find differences between the predictions of mixing length theory and our hydrodynamic simulations. The simulation of the single convection zone in the Pop I model shows a strong growth of the size of the convection zone due to turbulent entrainment. Hence we predict that for the Pop I model a hydrogen injection phase (i.e. hydrogen injection into the helium core) will commence after about 23 days, which should eventually lead to a double convection zone structure known from 1D stellar modeling of low-mass Pop III stars. Our two and three-dimensional hydrodynamic simulations of the double (Pop III) convection zone model show that the velocity field in the convection zones is different from that predicted by stellar evolutionary calculations. The simulations suggest that the double convection zone decays quickly, the flow eventually being dominated by internal gravity waves.