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


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


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


Ω(P,θ) μ(P,θ)


Ω(P,θ) ρ(P,θ) μ(P,θ)

Thermal wind


Ω(P,θ) ρ(P,θ) μ(P,θ)

S(P,θ)

Thermal wind

EoS


Ω(P,θ)

Ω(P,θ)

ρ(P,θ) μ(P,θ)

S(P,θ)

Thermal wind

EoS


Ω(P,θ)

Ω(P,θ)

ρ(P,θ) μ(P,θ)

S(P,θ)

S(P,θ)

Thermal wind

U, Dv

EoS


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


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



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


Horizontal variations

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


star in the MS

Rota

tion a

xis

Equator


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

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???

João Pedro Marques The Stellar Evolution Code CESTAM Numerical and physical challenges ESTER...

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