The Hydrodynamic Environment for the s Process in the He-Shell Flash of AGB Stars

Abstract

The He-shell flash convection in AGB stars is the site for the high-temperature component of the s-process in low- and intermediate mass giants, driven by the Ne22 neutron source. [...] The upper convection boundary plays a critical role during the H-ingestion episode that may lead to neutron-bursts in the most metal-poor AGB stars. We address these problems through global 3-dimensional hydrodynamic simulations including the entire spherical He-shell flash convection zone (as oposed to the 3D box-in-a-star simulations). An important aspect of our current effort is to establish the feasibility of our appoach. We explain why we favour the explicit treatment over the anelastic approximation for this problem. The simulations presented in this paper use a Cartesian grid of 512^3 cells and have been run on four 8-core workstations for four days to simulate ~5000s, which corresponds to almost ten convective turn-over times. The convection layer extends radially at the simulated point in the flash evolution over 7 H_p pressure scale-heights and exceeds the size of the underlying core. Convection is dominated by large convective cells that fill more than an entire octant. [...]

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The Hydrodynamic Environment for the s Process in
the He-Shell Flash of AGB Stars
Paul R. Woodward∗,a, David H. Portera, Falk Herwigb,c, Marco Pignataric, Jagan
Jayaraja, and Pei-Hung Lina
aUniversity of Minnesota, Laboratory for Computational Science & Engineering 499 Walter
Library, 117 Pleasant St. S. E., Minneapolis, Minnesota 55455, USA
bUniversity of Victoria, Dept. of Physics and Astronomy, Victoria, BC, V8W 3P6 Canada
cKeele University, Astrophysics Group, North Staffordshire, ST5 5BG, UK
E-mail: paul@lcse.umn.edu, fherwig@uvic.ca
The He-shell flash convection in AGB stars is the site for the high-temperature component of
the s-process in low- and intermediate mass giants, driven by the22Ne neutron source. During
this phase several s-process branchings are activated, including some with time scales similar
to the convective turn-over time scale. In addition, uncertainties regarding convective mixing
at both the bottom and the top of the convective shell are preventing accurate predictions of s-
processyields. TheupperconvectionboundaryplaysacriticalroleduringtheH-ingestionepisode
that may lead to neutron-bursts in the most metal-poor AGB stars. We address these problems
through global 3-dimensional hydrodynamic simulations including the entire spherical He-shell
flash convection zone (as oposed to the 3D box-in-a-star simulations). An important aspect of
our current effort is to establish the feasibility of our appoach. We explain why we favour the
explicit treatment over the anelastic approximation for this problem. The simulations presented
in this paper use a Cartesian grid of 5123cells and have been run on four 8-core workstations for
four days to simulate ∼ 5000s, which corresponds to almost ten convective turn-over times. The
convection layer extends radially at the simulated point in the flash evolution over 7Hppressure
scale-heights and exceeds the size of the underlying core. Convection is dominated by large
convective cells that fill more than an entire octant. In order to better understand the conditions
of the s-process branchings in this environment we have extracted particle tracers, and we discuss
the thermodynamic trajectories along those paths.
10th Symposium on Nuclei in the Cosmos
July 27 - August 1 2008
Mackinac Island, Michigan, USA
∗Speaker.
c ? Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence.
http://pos.sissa.it/
arXiv:0901.1414v1 [astro-ph.SR] 11 Jan 2009

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Hydrodynamics of He-shell flash convectionPaul R. Woodward
1. Introduction
The main component of the s process nucleosynthesis originates in AGB stars, and is in-
timately linked to the mixing induced by convection. In the He-shell flash convection zone of
stars with core-masses > 0.6M?the22Ne(α,n)25Mg reaction releases a burst of neutrons (Nn∼
1011cm−3). This activates s-process branchings sensitive to the peak temperature reached during
the flash, for example at95Zr [6] or128I [9], which in turn depen on the hydrodynamic properties
of this convection, including its boundaries [4]. Observables derived from such branchings make
He-shell flash convection a unique laboratory for investigating hydrodynamic flows in the deep
stellar interior where the elements are made.
However, our premier motivation are those He-shell flashes that ingest H from the overlying,
unprocessed and stable layers. These have been found in stellar evolution calculations of AGB
stars of extremely low metal content (e.g. [1]), and in other situations [3]. The convective-reactive
nature of the H-ingestion flash as well as its dependence on the exact nature of convective bound-
ary mixing, invalidated the one-dimensional stellar evolution approach, as it for example falsly
assumes that the abundance of fuel is constant on spheres. This assumption may lead to predictions
like a split of convection zones which are most likely artifacts of the inappropriate assumption of
spherical symmetry [11]. Here we focus on the entrainment process in full 3D geomertry that leads
to the convective-reactive case, thereby improving on previous non-reactive simulations of He-shell
flash convection, which where in plane-parallel geometry and only in two dimensions [5]. First we
explain why the explicit hydrodynamic treatment is justified for this problem.
2. Hydrodynamic simulations, computational method and implications for the s
process
The PPM gas dynamics scheme [10, 2, 12, 13] has been adapted over a decade ago to simulate
on 3-D Cartesian meshes AGB star envelope convection flows in their spherical geometry [8, 15].
In that work the roughly sonic to supersonic motions near the surface of the star made the explicit
formulation of this numerical scheme highly appropriate. In the present work, we adopt a similar
computational approach to the simulation of the helium shell flash convection zone in AGB stars.
The principal difficulty introduced through our numerical approach is our code’s explicit for-
mulation, which performs each grid cell update using only local data. We follow sound waves in
the flow, and these make the flow tend toward hydrostatic equilibrium in the radial dimension with-
out our needing to enforce this constraint explicitly. This feature of our code is a disadvantage only
because of its computational cost. Mach numbers in the helium shell flash convection flow tend to
range up to about 1/301, which means that our explicit treatment of sound wave propagation forces
us to take many more time steps than would be needed in an anelastic treatment.
However, this numerical approach is affordable because of two aspects. The first is directional
operator splitting, that an anelastic simulation on the same uniform Cartesian grid could not em-
ploy, because of its much larger time step. This already reduces the the work difference between
the two schemes down by a factor of two to three. The reason for this is that directional splitting in
1This is for convective gusts actually observed in our simnulations, and not for the much smaller mean convective
velocity that mixing-length would indicate.
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Hydrodynamics of He-shell flash convectionPaul R. Woodward
Figure 1: Snapshots of a hemisphere of the AGB star’s interior taken from PPM simulations on uniform
Cartesian grids of 5123cells, corresponding to time −0.07yr on the left and 0.2yr on the right (cf. Fig. 2
in [5]). The outer radius of each image is the upper boundary of the convection zone, located on the left at
15,200km and on the right at 30,000km. The bottom of the convection zone is at 9,150km and 10,000km
in the left and right panel, respectively. As before [5, 11] the convection layer is sandwiched by a stable
layer inside and outside to capture the behavior of stable oscillatory motions there that are driven by the
convection, and artificially damped beyond a certain radius in our simulations. Although we do not expect
the H-ingestion to occur in the high-metallicity AGB star that we used as a template for these simulations, we
nevertheless chose in both cases the entrained material from above the convection zone to visualise the flows.
Concentrations of entrained fluid from above that are near but not quite equal to unity are blue, while red and
yellow correspond to concentrations of around 1% and 0.1 to 0.01%. Both flows display the dominant role
of very large convection cells, which tends to invalidate the assumptions of the mixing length theory and to
support the need for detailed treatment of these flows in 3-D in order to determine the s-process branching
nucleosynthesis. On the left, visualisation focuses on a vertical slice, while on the right the hemisphere
facing out of the page has been cut away so that we look from the inside out at the far hemisphere of the star.
These images are taken from movie animations that can be viewed from the ’MOVIES’ link on the main
LCSE web site at http://www.lcse.umn.edu.
an explicit method accurately captures the corner transport for both sound waves and fluid advec-
tion with no additional computational labor over a 1-D scheme. The second issue is the implicit
Poisson solve, or equivalent, required by the anelastic scheme. On today’s large machines, global
synchronization – of which many are required per Poisson solve – is the most expensive funda-
mental operation that can be performed. The second most expensive thing on a modern machine
is communication not overlapped with computation, including – and not sufficiently appreciated in
the computing community – a CPU’s communication with its locally attached main memory. The
standard iteration of a Poisson solver offers little computation to perform while the necessary data
is being provided to the CPU by the hardware. As a result a Poisson solver will run at a fraction of
the Gflop/s/CPU of an explicit gas dynamics scheme like PPM. These factors together practically
eliminate the cost advantage of the anelastic scheme at Mach numbers seen in helium shell flash
convection flows. In addition, comparisons have shown that perturbations and instabilities at the
boundaries are less prounounced and probably underestimated in anelastic simulations compared
to the explicit treatment [7]. Since the convective boundaries are so important in our simulations
we conclude that the explicit treatment is the superior method for our problem.
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Hydrodynamics of He-shell flash convectionPaul R. Woodward
Figure 2: Left: three arbitrary particle traces over 4176s of simulated time in the flow; middle and right: the
radial position and the temperature of the same particle traces.
We have implemented our PPM codes on both 4-core Intel CPUs and 8-core IBM Cell pro-
cessors (cf. [14]). The simulation shown in the right panel of Fig.1 ran over 80,000 time steps
to simulate 5147 seconds of problem time, running for about 4 days on 32 CPU cores in 4 work-
stations at a sustained performance of 3.4 Gflop/s/core. An early test of essentially this same
two-fluid PPM code ran at 2.64 Gflop/s/core on 5720 CPU cores (IBM Cell processor cores) in the
Roadrunner machine at Los Alamos, delivering 15 Tflop/s, which should improve as we perfect
our implementation for the Roadrunner platform. We expect this code to scale linearly to the full
Roadrunner machine which has over 100,000 CPU cores. Even if we used only 4% of that machine
we could do this same simulation on a 10243grid and with the correct heat injection rate in just 21
hours, which shows that these computations are eminently affordable.
In the new global simulations of He-shell flash convection, in particular at the later time (right
panel in Fig.1) when the geometric extent of the unstable layer is close to the maximum, the
large-scale behaviour of the convection is revealed. Material from above the convection zone is
entrained in a large-scale and irregular way. Large cells, that occupy a full quadrant of the thick
shell, dominate the flow, indicating that the mixing patterns relevant for T-sensitive branchings may
be more complicated.
We can interpolate representative particle paths and the fluid states along those paths. Using
the fluid states along these particle paths we can solve a complete nuclear reaction network to ob-
tain the behavior along this path. In the simulation, turbulent mixing of tracked constituents, in
this case the original gas of the convection zone and that entrained from the stably stratified region
above it, is carefully accounted for using a moment-conserving advection scheme that we call PPB
[12, 11]. The PPB scheme has roughly double or triple the linear resolving power of the PPM
advection scheme, which accounts for the level of fine detail that is evident in the results shown
in Fig.1. However, the nuclear reaction network will track a great number of additional concen-
trations. By following many different particle paths simultaneously, we can use this information
to estimate gradients of these concentrations in a manner similar to the interpolation methods of
smoothed particle hydrodynamics algorithms. Putting this together with the PPM simulation’s de-
tailed information on the turbulent kinetic energy along the particle path, an accurate estimate of
s-process yields in this flow can be computed.
Acknowledgments
This work has been supported by the U. S. Department of Energy through a contract with the
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Hydrodynamics of He-shell flash convectionPaul R. Woodward
Los Alamos National Laboratory and a grant from the MICS program of the Office of Science,
DE-FG02-03ER25569. It has also been supported by NSF Computer Research Infrastructure grant
CNS-0708822, a donation of equipment from IBM, and support from the Minnesota Supercomput-
ing Institute. FH was supported in part by Marie Curie grant MIRG-CT-2006-046520, and JINA.
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