## Modeling Dynamic Phases in Stellar Evolution using Multidimensional Hydrodynamics Simulations

Philipp Edelmann

Heidelberg Institute for Theoretical Studies, Germany

$$\newcommand{\PD}[2]{\frac{\partial#1}{\partial#2}} \renewcommand{\vec}[1]{\mathbf{#1}} \newcommand{\ord}{\mathcal{O}} \newcommand{\Mr}{M_\text{r}} \newcommand{\msol}{M_{\odot}} \newcommand{\rsol}{R_{\odot}} \newcommand{\nabad}{\nabla_\text{ad}} \newcommand{\HP}{H_P} \newcommand{\CFL}{\text{CFL}}$$

# Motivation

• spherical symmetry
• no dynamical effects
• turbulence model with free parameters
• no enforced symmetry
• full equations of fluid dynamics
• turbulence from first principles

# Low Mach Number Hydrodynamics

## What?

Mach number $M = \frac{u}{c} = \frac{\text{fluid velocity}}{\text{speed of sound}}$

## Why?

Flows in the stellar interior are usually at low Mach numbers.

speed of sound $c = \sqrt{\gamma \frac{p}{\rho}} \propto \sqrt{\frac{T}{\mu}}$

### Gresho vortex

 Mach $10^{-1}$ $10^{-2}$ $10^{-3}$ prec. Roe Roe

### Other Approaches

• modify underlying equations
• e.g. anelastic approximation, Maestro, …

works well for flows with only low Mach numbers

intermediate Mach numbers ($\sim 10^{-1}$) or mixed case needs the full Euler equations

# The Tool

## Seven-League Hydro (SLH) Code

### The Grid

• Spherical grids have singularities (center, axis)
• Map Cartesian computational grid to curvilinear grid
• Code stays simple, geometry encoded in metric terms

### Implicit Hydrodynamics

explicitimplicit
time step constraint for stability
$\Delta t_\text{explicit} \le \CFL{} \frac{\Delta x}{|u + c|} \stackrel{u \ll c}{\approx} \CFL{} \frac{\Delta x}{c}$
sound crosses one cell per step
time step constraint for accuracy
$\Delta t_\text{implicit} \le \CFL{} \frac{\Delta x}{|u|}$
fluid crosses one cell per step
• Implicit time steps are larger by a factor of $1/M$.
• At each step a non-linear system has to be solved using Newton–Raphson.
• We need iterative linear solvers to invert the huge Jacobian.
• In SLH implicit time-stepping is more efficient for $M\lesssim0.1$.

# Dynamical Shear

collaborators:
Raphael Hirschi (Keele) and Cyril Georgy (Geneva)
Friedrich Röpke (HITS), Leonhard Horst (Würzburg)

Maeder (2009)

### Dynamical Shear

image credit: Maeder (2009), originally Talon (1997)

### The Quest for a Good Initial Model

• should become shear unstable in stellar evolution code
• should not show other instabilities at the same time
• ideally similar time scale in stellar evolution and hydro code

#### a lot of work by R. Hirschi

• $20\,M_\odot$ ZAMS star, 40% crit. rotation
• core O burning phase
• Ne burning shell
• convectively stable
• Ri unstable

### Simulations with SLH

• 2D equatorial plane
• more than 6 hours of physical time
• special mapping of GENEC data to keep convective stability

### First 3D work

by Leonhard Horst

• not straightforward to map model to 3D
• strict shellular rotation cannot always be upheld, while keeping a stable model
• some modifications to $\Omega$ profile to get a stable model in the equatorial plane

$\bar{A}$

# Convective Overshooting in Pop III stars

collaborators:
Alexander Heger (Monash), Friedrich Röpke (HITS)

### Setting

• zero metallicity initial model
• core He burning produced already significant amount of $^{12}\text{C}$
• H burning shell with $X(^{12}\text{C})=10^{-9}$
• convective core grows, reaches H burning shell

$250\,\msol$ star ($Z=0$) during core He burning

### 3D box (2013)

$128^3$ grid for about 4 days

Mach number

14N

energy release

### 3D wedge (2014)

$512^3$ grid

### Next Steps

• start simulation right before overshooting reaches H shell
• include core using cubed sphere

# Conclusions

• In many aspects stars should be treated as 3D, dynamical objects.
• SE codes are still needed to cover evolutionary timescale.
• Low Mach numbers require special numerical methods.
• Fully implicit, 3D hydro is possible and scales well to large supercomputers.
• We can now look at many poorly understood phenomena from stellar evolution in greater detail using hydro simulations.