João Pedro Marques The Stellar Evolution Code CESTAM Numerical and physical challenges ESTER...
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Transcript of João Pedro Marques The Stellar Evolution Code CESTAM Numerical and physical challenges ESTER...
João Pedro Marques
The Stellar Evolution Code CESTAM
Numerical and physical challenges
ESTER Workshop – 10/06/2014 Toulouse
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
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
The Stellar Evolution Code CESAM
Several EoS, opacities, nuclear reaction rates
Transport of Angular Momentum in Stellar
Radiative Zones(Zahn 1992)
Angular momentum transported by
Meridional circulation
Turbulent viscosity
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.
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)
Meridional circulation transports heat and AM
Meridional circulation transports heat and AM
Ω(P,θ) μ(P,θ)
Meridional circulation transports heat and AM
Ω(P,θ) ρ(P,θ) μ(P,θ)
Thermal wind
Meridional circulation transports heat and AM
Ω(P,θ) ρ(P,θ) μ(P,θ)
S(P,θ)
Thermal wind
EoS
Meridional circulation transports heat and AM
Ω(P,θ)
Ω(P,θ)
ρ(P,θ) μ(P,θ)
S(P,θ)
Thermal wind
EoS
Meridional circulation transports heat and AM
Ω(P,θ)
Ω(P,θ)
ρ(P,θ) μ(P,θ)
S(P,θ)
S(P,θ)
Thermal wind
U, Dv
EoS
Meridional circulation transports heat and AM
Ω(P,θ)
Ω(P,θ)
ρ(P,θ) μ(P,θ)
S(P,θ)
S(P,θ)
μ(P,θ)
Thermal wind
U, Dv U, Dh
EoS
U, div F, div Fh, ε
Meridional circulation transports heat and AM
Ω(P,θ)
Ω(P,θ)
ρ(P,θ) μ(P,θ)
S(P,θ)
S(P,θ)
μ(P,θ)
Thermal wind
U, Dv U, Dh
EoS
U, div F, div Fh, ε
Meridional circulation transports heat and AM
Ω(P,θ)
Ω(P,θ)
ρ(P,θ) μ(P,θ)
S(P,θ)
ρ(P,θ)
S(P,θ)
μ(P,θ)
Thermal wind
U, Dv U, Dh
EoS
U, div F, div Fh, ε
Meridional circulation transports heat and AM
Ω(P,θ)
Ω(P,θ)
ρ(P,θ) μ(P,θ)
S(P,θ)
ρ(P,θ)
S(P,θ)
μ(P,θ)
Thermal wind
Thermal wind
U, Dv U, Dh
EoS
U, div F, div Fh, ε
Horizontal variations
It's complicated...
It's complicated...
It's complicated...
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:
No shear. 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.
A few results: a 5 Msun
star at the ZAMS
Rota
tion a
xis
Equator
A few results: a 5 Msun
star at the ZAMS
Rota
tion a
xis
Equator
A few results: a 5 Msun
star in the MS
Rota
tion a
xis
Equator
A few results: a 5 Msun
star in the MS
Rota
tion a
xis
Equator
A few results: a 5 Msun
star at the TAMS
Rota
tion a
xis
Equator
A few results: a 5 Msun
star at the TAMS
Rota
tion a
xis
Equator
A few results: a 5 Msun
star at the TAMS
Rota
tion a
xis
Equator
A few results: a 5 Msun
star at the TAMS
Rota
tion a
xis
Equator
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.
Theoretical rotation profiles of subgiant star KIC 7341231
Best model
Theoretical average rotation rate from splittings
Rotation profiles
Mixed modes withmostly p-mode character
Mixed modes withmostly g-mode character
Theoretical splittings do not agree with observations
Observed splittings
Theoretical splittings
Factor of ~102
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)
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.
cp
cg
Internal Gravity Waves: ray tracing
Low frequency
High frequency
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 – Ω
c) (Doppler shift!)
Zahn at al. 1997
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!
Plumb-McEwan Experiment
Plumb-McEwan Experiment
A new method is needed
σ(r) = ω – m[Ω(r) – Ωc] : local methods no longer
possible! A new ''test module'' to experiment.
A finite volume method
Equation to solve:
Three fluxes: Meridional circulation. Viscosity. IGW flux.
A finite volume method
(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.
k-1 k k+1
Cell k Face k+1/2Face k-1/2
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.
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!
The role of wave heat fluxes
Simplification:
Local cartesian grid.
Boussinesq.
Quasi-geostrophic.
Adiabatic.
Cartesian grid
Coordinates: (x, y, z)
X → direction of rotation (zonal)
Y → meridional Z → vertical
V = (u, v, w)
Waves can accelerate the mean flow
Momentum equation:
Coriolis
Wave momentum flux divergence
Forcing
But heat fluxes also contribute
Momentum equation:
Heat equation:
b: buoyancy
Diabatic heating
Vertical advection
But heat fluxes also contribute
Momentum equation:
Heat equation:
b: buoyancy
Diabatic heating
Vertical advection
b(y) and u(z) connected by the thermal wind equation!
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:
Eliassen-Palm flux
The TEM equations are:
F is the Eliassen-Palm flux:
Eliassen-Palm flux
The equations become:
F is the Eliassen-Palm flux:
Heat and momentum fluxesdo not act separately:
only in the combination givenby the EP flux!
TEM in stars
We derived a formulation for stellar interiors.
We are testing in the '' test module''...
Work in progress...
Other developments
Magnetic braking by stellar winds: several models– Kawaler (1988)– Reiners and
Mohanty (2012)– Gallet and Bouvier
(2013)
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???