Simulation of Convection in Stellar Interiors

**Principal Investigator:**

Friedrich Röpke

**Affiliation:**

Zentrum für Astronomie der Universität Heidelberg, Institut für Theoretische Astrophysik und Heidelberger Institut für Theoretische Studien

**Local Project ID:**

chwb07

**HPC Platform used:**

JUQUEEN and JUWELS of JSC

**Date published:**

Classical one-dimensional models of stellar structure and evolution have been very successful in explaining the different states and evolutionary phases of stars observed in a qualitative way. However, the limited computational power in the second half of the 20^{th} century, when most of these models were developed, combined with the extreme multi-scale multi-physics challenges that the description of stars faced required gross assumptions: stars were modeled as spherical objects (reducing the dimensionality of the problem to essentially 1D) in hydrostatic equilibrium (so that dynamical effects were excluded from direct modeling). This allowed us to follow the evolution of stars of different masses and compositions from their formation to their end of life, when they form a white dwarf star, explode as a supernova or collapse into a black hole. Even today, covering the whole lifespan of stars is only feasible in essentially one-dimensional models.

The assumptions made in these, however, prevent a consistent treatment of important physical mechanisms. This is usually compensated by parameterized descriptions that mimic the underlying physics. These parameters allow us to fit observations well, but they reduce the predictive power of the models. Moreover, new and detailed observations that have become available in the first decades of the 21^{st} century are in tension with such models, calling for a fundamental update of theory.

The goal of project hwb07 at Jülich Supercomputing Centre is therefore to use improved modeling techniques and the vastly increased power of supercomputers to perform simulations of stellar gas flows in the deep interior of stars that threat the physical mechanisms self-consistently. This requires three-dimensional hydrodynamic simulations. Following the entire stellar live in those is unrealistic in the foreseeable future, but certain aspects and stages in the evolution where 3D dynamical effects are most critical have become accessible to simulations. The goal is to understand the physical effects and to formulate new effective descriptions that can be used in the next generation of stellar evolution models.

Such simulations are still extremely demanding because of the multitude of physical effects at play – in addition to fluid dynamics gravity, nuclear reactions, and diffusion processes have to be taken into account. The scale problem is severe both in space and time: Sufficient numerical resolution of the regions of interest can often only be reached if the simulations are restricted to a certain sub-volume of the star. The flows in stellar interiors are typically very slow compared to the speed of sound. Therefore, special numerical techniques are required to follow them over substantial periods of time. Such methods (specifically implicit time integration and specialized low-Mach-number fluid dynamics solvers) have been developed by the Heidelberg group and implemented into their “Seven-League Hydro” (SLH) code. Equipped with this tool, large-scale simulations of processes in the stellar interior are performed in the framework of project hwb07.

An example is the simulation of convective helium shell burning in a massive star. Energy is generated by nuclear fusion of helium in a shell-like layer. Radiation is not sufficient to transport this energy towards the stellar surface and therefore convection takes over. Similar to boiling water, energy is transported by a fluid flow: Hot fluid elements rise upward and deposit energy in outer layers while cool material sinks down to the region where it is heated by nuclear reactions. This leads to an overall cyclic (but in detail unordered and highly turbulent) motion of stellar gas that efficiently transports thermal energy. Obviously, this process is highly dynamical and multidimensional. Its parametrization in a 1D stellar evolution model is one of the main sources of uncertainty.

Figure 1 shows the radial velocity for a 3D simulation of convective helium shell burning. Blue color indicates outgoing fluid; red color stands for parts of the fluid that move inwards. The flow is clearly restricted to a shell with stable zones below and above it. Gray shades above the convection zone indicate the excitation of waves by convective plumes hitting the interface to the stable layer. This is also illustrated in the animation available below:

Apart from the technical challenges involved in the 3D simulations, the analysis of the complicated time-dependent turbulent flow structure poses another problem. Extracting information guiding new one-dimensional descriptions is highly non-trivial. Therefore, an analysis pipeline was set up that allows for a detailed analysis of the flow structure and the energy transport and conversion in it in the so-called “Reynolds-Averaged Navier-Stokes” (RANS) framework which decomposes the flow field into its different contributions. As one example, the RANS formalism for the kinetic energy equation helps to quantify the amount of numerical dissipation of the applied hydrodynamic solver.

Figure 2 compares the result of such an analysis for a simulation as shown in Figure 1. The analysis is done for a scheme specialized for low-Mach flows (left) and a conventional approach (right). The gray dashed line represents the amount of dissipation. The specialized solver shows less *numerical* dissipation, especially at the bottom of the convection zone. This is desirable because it allows us to resolve more turbulent (i.e. more realistic) flows at the same computational cost. The colored solid lines correspond to the different contributions to the flow, e.g. work done by buoyancy (orange) or the gradient of the acoustic flux (blue). There are no considerable amplitudes at lower and upper radii because the fluid is approximately at rest there.

Simulations such as the discussed example carried out within project hwb07 improve our understanding of how stars evolve and pave the way to the next generation of stellar models.

**Scientific Contact**

Friedrich Röpke

Professor of Theoretical Astrophysics, Zentrum für Astronomie der Universität Heidelberg

Group Leader, Physics of Stellar Objects, Heidelberg Institute for Theoretical Studies (HITS)

Heidelberger Institut für Theoretische Studien

HITS gGmbH

Schloss-Wolfsbrunnenweg 35, D-69118 Heidelberg (Germany)

http://www.h-its.org

e-mail: friedrich.roepke@h-its.org

http://www.h-its.org/pso

*JSC project ID: chwb07*

*December 2019*