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

## Dynamical Shear Instabilities in Massive Stars

Philipp Edelmann

Heidelberg Institute for Theoretical Studies, Germany

# Rotation in Stellar Evolution

### Rotation

• in general, stars show differential rotation
• changes the internal structure of the star
• gives rise to many additional instabilities, mixing processes
• in principle: at least 2D to describe the structure

### Stellar Structure

e.g. Maeder (2009)

• $\D{r}{M} = \frac{1}{4 \pi r^2 \rho}$
• $\D{P}{M} = - \frac{G M}{4 \pi r^4}$
• $\D{L}{M} = \epsilon_\text{nuc} - \epsilon_\nu + \epsilon_\text{grav}$
• $\D{\ln T}{M} = - \frac{G M}{4 \pi r^4} \min \left[\nabad, \nabla_\text{rad} \right]$
• $\D{r_P}{M_P} = \frac{1}{4 \pi r_P^2 \bar{\rho}}$
• $\D{P}{M_P} = - \frac{G M_P}{4 \pi r_P^4} \color{red}{f_P}$
• $\D{L_P}{M_P} = \epsilon_\text{nuc} - \epsilon_\nu + \epsilon_\text{grav}$
• $\D{\ln T}{M_P} = - \frac{G M_P}{4 \pi r_P^4} \color{red}{f_P} \min \left[\nabad, \nabla_\text{rad} \color{red}{\frac{f_T}{f_P}} \right]$

### (Some) Open Questions about Shear

• How well does the diffusive approximation for shear reproduce hydrodynamic result?
• Is shellular rotation still satisfied after mixing?

# Dynamical Shear

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

### Theory

unstable if gain in kinetic energy greater than potential energy needed

Richardson number $Ri=\frac{N^2}{(\partial v/\partial z)^2}$
unstable if $Ri < Ri_c = \frac{1}{4}$

### Treatment in 1D Code

e.g. from Hirschi+ (2004)

• compute $Ri$ using finite differences on $\Omega$ profile
• zones with $Ri < Ri_c$ are unstable
• apply $D = \frac{1}{3} r \Delta \Omega \Delta r$ in unstable zones

### Caveats

• computation of $Ri$
• single unstable zones
• treatment as diffusive process

### Hydro Simulations

• follow development of shear instability in hydro
• compare outcome with 1D code
• 2D simulations first

### A Good Initial Model

• should become shear unstable in stellar evolution code
• should not show other instabilities at the same time
• should be on similar time scale in stellar evolution and hydro code
• $20\,M_\odot$ ZAMS star, 40% crit. rotation
• post core O burning phase
• C/Ne shell interface
• convectively stable
• Ri unstable

### Seven-League Hydro Code

• solves the compressible Euler equations in 1-, 2-, 3-D
• explicit and implicit time integration
• low Mach number scheme (Miczek+, 2015)
• works for low and high Mach numbers on the same grid
• hybrid (MPI, OpenMP) parallelization (up to 100 000 cores)
• several solvers for the linear system:
BiCGSTAB, GMRES, Multigrid, (direct)
• arbitrary curvilinear meshes
using a rectangular computational mesh
• gravity solver (monopole, Multigrid)
• radiation in the diffusion limit
• general equation of state
• general nuclear reaction network

### Simulations with SLH

• 2D equatorial plane
• compute isobaric shells from 1D $\Omega$ profile
• reconstruct hydrostatic profile
• use $\nabla - \nabad$ for reconstruction instead of $T$ or $s$
(different EoS)
• more than 6 hours of simulated time
• hydrodynamics only (no source/sink terms)

### Comparison

1D Stellar Evolution – GENEC

2D Hydro – SLH

### What next?

• try to calibrate $D$ to match hydro more closely
• include same physics as in SE code ($\nu$, reactions)
• 3D simulations

F. Miczek

F. Miczek

F. Miczek

F. Miczek

F. Miczek

F. Miczek

L. Horst

### Conclusions

• treatment of shear in stellar evolution still involves many uncertainties
• favourable conditions in certain models let us directly compare hydro and stellar evolution codes
• behaviour of dynamical shear instability agrees qualitatively

### Outlook

• calibration of $D$ in stellar evolution code
• 3D simulations
• influence of reactions in shear unstable zone