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

A hydrodynamical perspective on shear instabilities in massive stars

Philipp Edelmann

Heidelberg Institute for Theoretical Studies, Germany

collaborators: F. Röpke (HITS), R. Hirschi (Keele), C. Georgy (Geneva), S. Jones (HITS), L. Horst (HITS)

Why are rotating stars interesting?

  • in general, most stars rotate
  • many show differential rotation (e.g. Kepler data)
  • changes the internal structure of the star
  • gives rise to many additional instabilities, mixing processes

In principle,
…we need at least 2D to describe the hydrostatic structure
…instabilities are hydrodynamic phenomena

Shellular rotation

  • assumption: isobaric shells rotate on same $\Omega$
  • $\rho$, $T$, … vary on shell
  • Roche approximation: gravity of enclosed mass treated as of a point mass
  • stellar structure equations retain 1D form
  • possible qualitatively behaviour at different latitude
  • treatment of instabilities

Rotational instabilities and their treatment in 1D codes

  • instabilities: dynamical shear, secular shear, GSF, ABCD,…
  • usually treated as diffusion in SE codes

e.g. dynamical shear

  • Richardson number $Ri=\frac{N^2}{(\partial v/\partial r)^2}$
  • unstable if $Ri < Ri_c = \frac{1}{4}$
  • apply $D = \frac{1}{3} r \Delta \Omega \Delta r$ in unstable zones

Challenges for multidimensional hydrodynamics

  • some instabilities act on secular time scales
  • differences in microphysics (e.g. EoS)
    $\Rightarrow$ careful which quantities to take as input to retain same position of convection zones etc.
  • need to reconstruct a 3D hydrostatic model from a 1D profile
    $\Rightarrow$ might lead to inconsistencies

A testbed for dynamical shear

The stellar 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
  • Geneva stellar evolution code
  • $20\,M_\odot$ ZAMS star, 40% crit. rotation
  • post core O burning phase
  • C/Ne shell interface
  • convectively stable
  • Ri unstable

Seven-League Hydrocode (SLH)

more information on slh-code.org

  • compressible Euler equations in 1, 2, 3D
  • explicit or implicit time stepping
  • scales to more than 100,000 cores
  • low and high Mach numbers on the same grid
    (Miczek+, 2015)
  • arbitrary curvilinear meshes
  • gravity solver
  • general EoS
  • nuclear reaction network
  • neutrino losses

First setup

  • pure hydrodynamics
  • explicit time stepping
  • no source terms
  • Helmholtz EoS
  • 2D in equatorial plane
  • $512 (r) \times 1024 (\varphi)$
    polar grid
  • 5 hours of simulated time

Evolution of Richardson number

Comparison with GENEC model

Diffusion coefficient (preliminary)

effective diffusion coefficients from averaged 2D simulations

method as in Jones+ (2016)

3D models (work in progress)

  • not all latitudes necessarily stable (e.g. RT unstable)
  • naïve reconstruction might lead to Solberg–Høiland instability

credit: L. Horst


  • rotation is common in stars and has an important effect on stellar models
  • 1D codes can treat rotation using shellular approximation
  • rotational instabilities in 1D are a major uncertainty
  • certain cases accessible to hydro simulations
  • details of temporal evolution and extent of shear mixing differ between 1D and 2D
  • effective diffusion coefficient is remarkably similar
  • straightforward 3D mapping is not always consistent for all shellular rotation models

Evolution of Richardson number

Source Terms (reactions & $\nu$ losses)

Mapping from GENEC to SLH

reconstruction with $\nabla - \nabad$ from input model and additional equation for T