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

Dynamical Shear Instabilities in Massive Stars

Philipp Edelmann

Heidelberg Institute for Theoretical Studies, Germany

Rotation in Stellar Evolution


  • 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

Surface Deformation

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

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



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


  • 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

angular velocity in GENEC

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)

Angular Velocity

Evolution of Richardson Number

Evolution of Angular Velocity


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

Source Terms

3D Shear Instability

F. Miczek

3D Shear Instability

F. Miczek

3D Shear Instability

F. Miczek

3D Shear Instability

F. Miczek

3D Shear Instability

F. Miczek

3D Shear Instability

F. Miczek

Model in 3D Shellular Rotation

L. Horst


  • 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


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