We present the first hydrodynamic , multi-dimensional simulations of He-shell flash convection .
Specifically , we investigate the properties of shell convection at a time immediately before the He-luminosity peak during the 15 ^ { \mathrm { th } } thermal pulse of a stellar evolution track with initially two solar masses and metallicity Z = 0.01 .
This choice is a representative example of a low-mass asymptotic giant branch thermal pulse .
We construct the initial vertical stratification with a set of polytropes to resemble the stellar evolution structure .
Convection is driven by a constant volume heating in a thin layer at the bottom of the unstable layer .
We calculate a grid of 2D simulations with different resolutions and heating rates .
Our set of simulations includes one low-resolution 3D run .
The computational domain includes 11.4 pressure scale heights .
He-shell flash convection is dominated by large convective cells that are centered in the lower half of the convection zone .
Convective rolls have an almost circular appearance because focusing mechanisms exist in the form of the density stratification for downdrafts and the heating of localized eddies that generate upflows .
Nevertheless , downdrafts appear to be somewhat more focused .
The He-shell flash convection generates a rich spectrum of gravity waves in both stable layers above and beneath the convective shell .
The magnitude of the convective velocities from our 1D mixing-length theory model and the rms-averaged vertical velocities from the hydrodynamic model are consistent within a factor of a few .
However , the velocity profile in the hydrodynamic simulation is more asymmetric , and decays exponentially inside the convection zone .
An analysis of the oscillation modes shows that both g-modes and convective motions cross the formal convective boundaries , which leads to mixing across the boundaries .
Our resolution study shows consistent flow structures among the higher resolution runs , and we see indications for convergence of the vertical velocity profile inside the convection zone for the highest resolution simulations .
Many of the convective properties , in particular the exponential decay of the velocities , depend only weakly on the heating rate .
However , the amplitudes of the gravity waves increase with both the heating rate and the resolution .