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

Standard Roe Scheme

Gresho vortex

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

Gresho vortex

kinetic energy

Kelvin–Helmholtz Instability

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

F. Miczek, F. K. Röpke, P. V. F. Edelmann

Alejandro Bolaños, Aron Michel, Jonas Berberich, Florian Lach

The Grid

  • Cartesian grids are badly adapted to spherical stars
  • 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

Simulation with GENEC

Simulations with SLH

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

Evolution of Angular Momentum

Evolution of Mean Atomic Mass

Evolution of Richardson Number

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.