F. Herwig 1,2 , B. Freytag 3,2 , R. M. Hueckstaedt 2 , F. Timmes 2
1. Keele Astrophysics Group, School of Physical and Geographical Sciences, Keele University, UK; fherwig@astro.keele.ac.uk
2. Los Alamos National Laboratory, Los Alamos, NM, USA; rmhx@lanl.gov, timmes@lanl.gov
3. Centre de Recherche Astronomique de Lyon- Ecole Normale Sup é rieure, Lyon, France;
Bernd.Freytag@ens-lyon.fr
The Asymptotic Giant Branch (AGB) phase is the most productive evolutionary phase in terms of nucleosynthesis for low and intermediate mass stars. Nucleosynthesis in these stars is aided by the mixing and heating triggered by recurrent He-shell flashes (Fig.1).
Extreme nuclear energies (corresponding to 10 8 L

) are generated during these thermonuclear flashes with multiple implications for nucleosynthesis and evolution. 1-d stellar evolution codes (Fig.1) have to adopt simplifying assumptions on convection induced mixing, especially at convective boundaries. Here, we report on 2D (Fig.2) and
3D (Fig.3, bottom panel) hydrodynamic models of convective mixing in the AGB He-shell
(Herwig etal. 2006, Freytag etal 2007). As opposed to the shallow surface convection in Atype stars studied by Freytag et al. (1996), coherently moving convective cells do not cross the convective boundary significantly (Fig.2). In other words, penetration is minimal for this convection zone. We find that convective motions induce a rich spectrum of internal gravity waves in the neighboring stable layers (Fig.3, top). Interactions of these
(mainly horizontal) oscillations with the convective boundary, as well as motions with convective characteristics within the stable layers, do cause a finite amount of mixing across the convective boundary. Our preliminary analysis of this mixing (Fig.4 and 5) is consistent with semi-analytical results obtained from observations and 1D stellar evolution simulations (Werner & Herwig, 2006).
convection p-modes g-modes
Fig. 3. Top: Frequency-Wave number (k-

) power diagrams for different vertical locations in a 2D
RAGE simulation. Gravity modes are strongest in the stable layers. Convective motions are evident in the stable layers (marked with red line) as well. Bottom: Horizontal slices through different heights of a 3D simulation showing vertical velocity (light for upward; dark for downward).
Fig. 2. Top: Fully-developed convection in a 2-D RAGE simulation shown as color-coded pressure fluctuations around the horizontal mean with pseudo-streamlines superimposed. Middle: The same result shown as entropy fluctuations. A brighter color indicates higher entropy (as well as higher temperature). Bottom: Different vertical slices of a 3-d RAGE simulation showing entropy fluctuations. The structure is similar to that seen in 2-d, albeit at a lower spatial resolution.
Fig.1: Evolution of convective boundaries and burning shells at the coreenvelope interface where all of the nucleosynthesis in
AGB stars takes place. Convection regions are green, the location of the H- and
He-burning shells are shown with red and purple solid lines. The data was obtained with a 1D stellar evolution code. We use the stratification from a 1D He-shell flash model (t=0yr) for the vertical stratification of the 2D and
3D hydrodynamic simulations (Fig. 2, 3).
Fig. 5. Diffusion coefficient derived with a tracer particle method from the velocity fields of a 2D simulation.
Green circles and red crosses show diffusion coefficients derived using spread evolutions of entropy and vertical coordinate, respectively. The evolution of the vertical coordinate is more appropriate in the convection zone.
Fig. 4. Mixing efficiency f versus the number of vertical grid points for two slopes in the lower overshoot region (left and center panels) and one slope in the upper overshoot region (left panel). The f-value quantifies the exponential decay of
D (see Fig. 5) across the convective boundaries. Models were run using two different Eulerian adaptive-mesh refinement codes: RAGE (LANL and SAIC) and FLASH (U. of Chicago), and at two heating rates (g-heat: realistic rate as in stellar model, d-heat: 30 x g-heat). Compared to RAGE, FLASH has a higher-order method and shows more small-scale structure; yet the f-values are comparable for the two codes with the exception of one outlier. With respect to spatial resolution, simulations with the enhanced heating rate converge more quickly than those with a nominal heating rate.
References: Freytag B., Ludwig H.-G., & Steffen M., 1996, A&A 313, 497
Freytag B., Herwig F., Hueckstaedt R.M., etal., 2007, in prep.
Herwig F., Freytag B., Hueckstaedt R.M., & Timmes F.X., 2006, ApJ 642, 1057
Herwig, F. & Werner, K., 2006, PASP 118, 138