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
Caveats
- 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
Conclusions
- 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