1. The Stellar Evolution Code CESTAM Numerical and physical challenges
João Pedro Marques
ESTER Workshop – 10/06/2014 Toulouse

2. The Stellar Evolution Code CESAM
Collocation method based on piecewise polynomial approximations projected on their B-spline basis
Stable and robust calculations
Restitution of the solution not only at grid points
Automatic mesh refinement

3. The Stellar Evolution Code CESAM
Precise restoration of the atmosphere
Modular in design
Evolution of the chemical composition:
Without diffusion: implicit Runge-Kutta scheme
With diffusion: solution of the diffusion eq. using the Galerkin method

4. The Stellar Evolution Code CESAM
Several EoS, opacities, nuclear reaction rates

5. Transport of Angular Momentum in Stellar Radiative Zones (Zahn 1992)
Angular momentum transported by
Meridional circulation
Turbulent viscosity

6. Turbulence is anisotropic in RZs
Radiative zones are stably stratified:
Turbulence much stronger in the horizontal direction.
“Shellular” rotation: Ω~constant in isobars.
Lots of simplifications possible.

7. Turbulence models: a weak spot
Horizontal viscosity: various approaches.
Richard and Zahn (1999), Mathis, Palacios and Zahn (2004).
Maeder (2009).
Vertical viscosity:
Secular instability
Talon and Zahn (1997)

8. Meridional circulation transports heat and AM

9. Meridional circulation transports heat and AM

10. Meridional circulation transports heat and AM
Thermal wind

11. Meridional circulation transports heat and AM
EoS
Thermal wind
S(P,θ)

12. Meridional circulation transports heat and AM
EoS
Thermal wind
S(P,θ)
Ω(P,θ)

13. Meridional circulation transports heat and AM
EoS
Thermal wind
S(P,θ)
U, Dv
Ω(P,θ)
S(P,θ)

14. Meridional circulation transports heat and AM
EoS
Thermal wind
S(P,θ)
U, Dv
U, Dh
U, div F, div Fh, ε
Ω(P,θ)
μ(P,θ)
S(P,θ)

15. Meridional circulation transports heat and AM
EoS
Thermal wind
S(P,θ)
U, Dv
U, Dh
U, div F, div Fh, ε
Ω(P,θ)
μ(P,θ)
S(P,θ)

16. Meridional circulation transports heat and AM
EoS
Thermal wind
S(P,θ)
U, Dv
U, Dh
U, div F, div Fh, ε
Ω(P,θ)
ρ(P,θ)
μ(P,θ)
S(P,θ)

17. Meridional circulation transports heat and AM
EoS
Thermal wind
S(P,θ)
U, Dv
U, Dh
U, div F, div Fh, ε
Ω(P,θ)
ρ(P,θ)
μ(P,θ)
Thermal wind
S(P,θ)

18. Horizontal variations

19. It's complicated...

20. It's complicated...

21. It's complicated...

22. It's complicated... 4th order problem in Ω
2 boundary conditions at the top:
No shear.
Angular momentum in the external CZ changes due to advection by U + external torque.
2 boundary conditions at the bottom:
Angular momentum in the central CZ changes due to advection by U, or
U=0.
Solved by a finite-difference scheme (relaxation method), fully implicit in time.

23. A few results: a 5 Msun star at the ZAMS
Rotation axis
Equator

24. A few results: a 5 Msun star at the ZAMS
Rotation axis
Equator

25. A few results: a 5 Msun star in the MS
Rotation axis
Equator

26. A few results: a 5 Msun star in the MS
Rotation axis
Equator

27. A few results: a 5 Msun star at the TAMS
Rotation axis
Equator

28. A few results: a 5 Msun star at the TAMS
Rotation axis
Equator

29. A few results: a 5 Msun star at the TAMS
Rotation axis
Equator

30. A few results: a 5 Msun star at the TAMS
Rotation axis
Equator

31. However... It doesn't really work!
Model fails to reproduce radial differential rotation in subgiant and red-giant stars.
Model does not reproduce solid-body rotation in solar models.
Changing model parameters does not solve the problem.
Therefore, new AM transport mechanisms needed:
Internal gravity waves (IGW).
Magnetic fields.

32. Theoretical rotation profiles of subgiant star KIC 7341231
Best model

33. Theoretical average rotation rate from splittings
Mixed modes with
mostly p-mode character
Mixed modes with
mostly g-mode character
Rotation profiles

34. Theoretical splittings do not agree with observations
Factor of ~102
Observed splittings

35. Internal Gravity Waves
ω does not depend on on magnitude of k.
Only on the angle between k and the vertical.
Therefore, cg orthogonal to cp.
Holton (2009)

36. Internal Gravity Waves
ω does not depend on on magnitude of k.
Only on the angle between k and the vertical.
Therefore, cg orthogonal to cp. cg cp

37. Internal Gravity Waves: ray tracing
High frequency
Low frequency

38. IGWs can accelerate the mean flow
When they are transient and/or they are dissipated/excited.
Radiative damping: a factor exp(-τ) appears with:
It depends strongly on the intrinsic frequency σ,
σ = ω – m(Ω – Ωc) (Doppler shift!)
Zahn at al. 1997

39. IGWs can accelerate the mean flow
Prograde and retrograde waves are Doppler shifted if there is differential rotation.
They are damped at different depths.
Retrograde waves brake the mean flow when they are damped.
Prograde waves accelerate the mean flow when they are damped.
Angular momentum is transported!

40. Plumb-McEwan Experiment

41. Plumb-McEwan Experiment

42. A new method is needed
σ(r) = ω – m[Ω(r) – Ωc] : local methods no longer possible!
A new ''test module'' to experiment.

43. A finite volume method Equation to solve: Three fluxes:
Meridional circulation.
Viscosity.
IGW flux.

44. A finite volume method
k-1 k
k+1
Face k-1/2
Cell k
Face k+1/2
(Variation of AM in cell k during Δt) = [(flux through face k-1/2) – (flux through face k+1/2)] Δt.
Viscous flux evaluated at present time step.
IGW and U fluxes extrapolated from previous time steps using a 3rd order Adams-Bashforth method.

45. Adams-Bashforth method for wave and meridional circ fluxes
Stable, accurate.
Needs fluxes at 3 previous time steps.
Prototypical eq:
Solution:
ωi = time step ratios.

46. How does it come together
Once we have calculated Ω at the present time step:
Compute wave fluxes at the faces of cells;
Compute meridional circulation flux.
Next time step:
Extrapolate IGW and U fluxes using 3rd order A-B.
Solve diffusion eq. for Ω using these fluxes.
Implemented in test module and it works!

47. The role of wave heat fluxes
Simplification:
Local cartesian grid.
Boussinesq.
Quasi-geostrophic.
Adiabatic.

48. Cartesian grid Coordinates: (x, y, z) V = (u, v, w)
X → direction of rotation (zonal)
Y → meridional
Z → vertical
V = (u, v, w)

49. Waves can accelerate the mean flow
Momentum equation:
Coriolis
Forcing
Wave momentum flux divergence

50. But heat fluxes also contribute
Momentum equation:
Heat equation:
Diabatic heating
Vertical advection
b: buoyancy

51. But heat fluxes also contribute
Momentum equation:
Heat equation:
b(y) and u(z) connected by the
thermal wind equation!
Diabatic heating
Vertical advection
b: buoyancy

52. Residual circulation: the transformed eulerian mean (TEM)
The problem:
Wave fluxes and circulation nearly cancel.
It is not clear what is driving.
A solution: use residual circulation:

53. Eliassen-Palm flux
The TEM equations are:
F is the Eliassen-Palm flux:

54. Eliassen-Palm flux Heat and momentum fluxes do not act separately:
The equations become:
F is the Eliassen-Palm flux:
Heat and momentum fluxes
do not act separately:
only in the combination given
by the EP flux!

55. TEM in stars We derived a formulation for stellar interiors.
We are testing in the '' test module''...
Work in progress...

56. Other developments Magnetic braking by stellar winds: several models
Kawaler (1988)
Reiners and Mohanty (2012)
Gallet and Bouvier (2013)

57. Problems A ''shear layer oscillation'' just below the CZ
We are developing numerical methods to handle this.
Is rotation really shellular in RZ?
Lessons from geophysics seem say ''no''!
Transfer of AM between CZ and RZ.
What about magnetic fields???