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

Meridional circulation transports heat and AM Thermal wind

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

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

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

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)

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)

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)

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)

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: 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 Rotation axis Equator

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

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

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

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

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

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

A few results: a 5 Msun star at the TAMS Rotation axis 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 Mixed modes with mostly p-mode character Mixed modes with mostly g-mode character Rotation profiles

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

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. cg cp

Internal Gravity Waves: ray tracing High frequency Low 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 – Ωc) (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 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.

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) V = (u, v, w) X → direction of rotation (zonal) Y → meridional Z → vertical V = (u, v, w)

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

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

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

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 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!

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???