Turbulence and nuclear reactions in 3D hydrodynamics simulations of massive stars

. Our knowledge of massive star evolution and nucleosynthesis is limited by uncertainties related to multi-dimensional processes taking place in stellar interiors. Recently, theoretical works have started to improve 1D stellar evolution codes through the implementation of results from 3D hydrodynamics models, which are used to study multi-D processes on a short time range (minutes or hours) and improve 1D prescriptions. In these proceedings, we present results coming from a new set of high-resolution hydrodynamics simulations of the neon-burning shell in a massive star, employing the PROMPI code. We focus in particular on the interplay between turbulence and nuclear reactions, discussing the impact that di ff erent boosting factors of the nuclear rates have on the results. This has important implications for supernova studies, nucleosynthesis, the physics of neutron stars and black holes.


Introduction
The lifetime of a star can be modelled employing one-dimensional (1D) stellar evolution models. These simulations are run efficiently thanks to the simplified prescriptions that describe the physics of the processes inside the star. Such prescriptions need to be carefully calibrated to observations. One example is the mixing length theory [1], used in stellar models to reproduce the convective regions of the star. To better understand and reproduce these phenomena, multi-dimensional hydrodynamics models can be employed. They can simulate the fluid motions in great detail, and combine it with physical processes like nuclear burning, rotation, and magnetic fields. However, the large computing power required to run these simulations sets a strict limit on the time scales and spatial extents that can be reproduced. It is the combined strength of 1D and multi-D models that can really push forward progress in understanding stellar evolution. One of the most interesting aspects of this field is the interplay between turbulent convection and nuclear reactions. We present in these proceedings results from a new set of hydrodynamics models run with the PROMPI code [2], simulating the convective neon-burning shell of a 15 M ⊙ star. Similar studies have been published with the same code for the oxygen-and carbon-burning shells [2,3], but not yet using a realistic nuclear burning network combined with initial conditions extracted from state-of-the-art 1D stellar evolution models, as we do for this new set [4]. We study here the effects of the nuclear burning on the convective shell and its evolution with time.

Methodology
The new set of simulations has been run with the stellar hydrodynamical code PROMPI [2]. All simulations are started from the same initial conditions, taken from a 1D stellar evolution simulation of a 15 M ⊙ star calculated with the GENEC code [5], a state-of-the-art code widely used to produce large grids of 1D models [6]. The most important assumptions are solar metallicity, Schwarzschild criterion for convective boundaries, and penetrative overshoot for the hydrogen-and helium-burning cores (see [4] for more details). The fundamental variables (e.g., density, temperature, composition) that characterize the second neon-burning shell that forms in 1D, chosen for its fast evolution and later phase, have been mapped onto a cubic box of side 0.64 × 10 8 cm, from which the 3D simulations are started with a numerical resolution of 512 3 cells. We include in our simulations an explicit nuclear network that fuels convection through energy generation, with the key isotopes for neon burning: 4 He, 16 O, 20 Ne, 24 Mg, 28 Si (the 4 He abundance is assumed to be at nuclear equilibrium, since this is a late burning stage). This simplified network includes only the main reactions for neon burning: 20 Ne(γ, α) 16 O and 20 Ne(α, γ) 24 Mg(α, γ) 28 Si. The individual simulations have been run with modifications to the nuclear reaction rates. Specifically, we included a constant "boosting factor" that multiplies the rates for the primary reactions involved in the energy generation, i.e. 20 Ne(γ, α) 16 O and 20 Ne(α, γ) 24 Mg, with respect to the nominal case. Such boosting factors were originally introduced to accelerate the simulations by increasing their convective speed, and therefore reducing the computing cost. In addition, they allow us to study the different evolution of the burning shell with time, tracking the amount of material that is entrained from the stable regions into the convective zone ("turbulent entrainment", see [7,8]). We studied the effect of the boosting factors by running three different simulations with boosting factor equal 1, 10, and 100 (see [4] for more details). The introduction of a boosting factor is common practice in stellar hydrodynamics simulations, so it is important here to consider also the case where it has not been used (boosting factor = 1), and the simulation has been run exactly as taken from the 1D model, without changing its original state.

Results
As a visual representation of our simulations, we present in Figure 1 the vertical cross-section of the nominal-luminosity simulation (boosting factor = 1) at time t = 2730 seconds, showing the velocity magnitude in colour scale. From this image, it is easy to recognize the convective region, i.e. the central zone where the fluid is dominated by plumes and eddies with high speed. The rest of the domain is a stable, radiative region characterized by low-speed gravity waves. To study the time evolution of the convective zone, we plot in Figure 2 the horizontallyaveraged mean atomic mass (in colour scale) as a function of time, for the 10-times boosted simulation. We can see the convective zone (in yellow) growing due to entrainment, easily distinguished from the upper and lower stable regions using the chemical composition. The fact that the chemical composition distinguishes so easily the convective zone from the stable  layers shows the importance of implementing nuclear abundances in hydrodynamics simulations of stellar interiors. We are interested in analysing the effects of the nuclear energy on the convective shell growth. For this reason, we parametrize turbulent entrainment with a simple law expressing the entrainment rate E (i.e., the entrainment over the convective velocity = v e /v rms ) as a function of the bulk Richardson number Ri B (a measure of the convective boundary stiffness): E = A Ri −n B (as defined by [2,9]). Then, we use the data from our hydrodynamics simulations to estimate the free parameters A, n, by measuring E from the time evolution of the bottom and top boundary locations visible in Figure 2, and computing Ri B by integrating the Brunt-Väisälä frequency across the convective boundary (see [4] for the detailed implementation). Figure 3 shows our measurements of entrainement rate versus bulk Richardson number for the three neon-shell simulations, alongside the line of best fit in log scale (in blue, solid). In addition to the present study [4], we compare results to other two PROMPI simulations for the carbon [3] and oxygen [2] shells of massive stars. In the legend of the figure, we give the estimates of parameters A and n for each study. Comparing the different tracks, we conclude that turbulent entrainment has a similar behaviour in different burning shells of late-phase massive stars. In particular, fitting the en-  Figure 3. Entrainment rate vs. bulk Richardson number and linear regressions: Ne-shell from this study (blue, solid), C-shell from [3] (green, dashed), O-shell from [2] (orange, dot-dashed). Triangles are for the lower convective boundaries, circles for the upper ones. The boosting factor is inversely proportional to Ri B , equal to 100, 10, 1 for the two triplets of blue circles and triangles. Error bars are standard deviations, computed in this work for the first time. Figure  taken from [4]. trainment law to the data from our new set of Ne-shell simulations, the estimated parameter n = 0.96 ± 0.19 is compatible with the value of 1 expected from geophysics studies [9]. Estimating A is more difficult, considering the fitting in log scale and the dispersion of the data: a reasonable estimate from our data is A ∼ 0.1 − 1.0. This is in agreement with other results from stellar hydrodynamics simulations, but disagrees with 1D stellar evolution simulations, that predict a much smaller amplitude of about ∼ 10 −4 for turbulent entrainment in the main sequence core [10,11]. This divergence motivates the need for a more detailed investigation of convection and entrainment in 1D models, especially in the late phases of massive stars.

Conclusions
We presented a new set of 3D hydrodynamics simulations of the neon-burning shell in a 15 M ⊙ star, run with the PROMPI code. We study in detail the interplay between nuclear reactions and turbulent convection, focusing on the impact of the nuclear energy generation on the entrainment of stable material into the convective zone.
The key result of this study, as described in the work of [4], is that a strong entrainment is present in the simulations also when employing the unaltered input conditions from a stateof-the-art 1D stellar model, and in particular when the nuclear energy generation rate is not modified (nominal-luminosity simulation). We see that turbulent entrainment in our simulations follows the entrainment law in agreement with previous hydrodynamics studies [2,3], showing this is an accurate and reliable way of modelling entrainment that can also be used for 1D stellar evolution models. The discrepancy between 3D hydrodynamics and 1D stellar models on entrainment, especially concerning the estimate of parameter A in the entrainment law, can be traced back to a different behaviour during the advanced phases, or to not having implemented entrainment in the 1D input models. In both cases, investigating the effects of entrainment in stellar evolution models is crucial for understanding the origin of this discrepancy, and eventually solving it. To date, few studies [10,11] have been implementing the entrainment law in a 1D stellar model, and only for the main sequence convective core. A complete understanding of the complex phenomena that takes place in the stellar interiors can only be achieved by synergy and convergence between one-and multi-dimensional stellar models, and thanks to the implementation of the nuclear reactions that fuel convection in these environments. This will have an important impact on many different fields in astrophysics, like supernova studies, nucleosynthesis and chemical enrichment, but also the physics of neutron stars and black holes.