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João Pedro Marques

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

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João Pedro Marques

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  1. João Pedro Marques The Stellar Evolution Code CESTAM Numerical and physical challenges 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 Ω(P,θ) μ(P,θ)

  10. Meridional circulation transports heat and AM Ω(P,θ) ρ(P,θ) μ(P,θ) Thermal wind

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

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

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

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

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

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

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

  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 Theoretical splittings 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 (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 Face k-1/2 Cell k Face k+1/2

  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) • 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

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