  We present the results of two-dimensional calculations of turbulent nuclear
burning of hydrogen-rich material accreted onto a white dwarf of 1 solar mass.
The main aim of the present paper is to investigate the question as to whether
and how the general properties of the burning are affected by numerical
resolution effects. In particular, we want to see whether or not convective
overshooting into the surface layers of the C+O white dwarf can lead to
self-enrichment of the initially solar composition of the hydrogen-rich
envelope with carbon and oxygen from the underlying white dwarf core. Our
explicit hydrodynamic code is based on the PPM-method and computes the onset of
the thermonuclear runaway on a Cartesian grid. In contrast to previous works we
do not observe fast mixing of carbon and oxygen from the white dwarf's surface
into the envelope by violent overshooting of large eddies. The main features of
the flow fields in our simulations are the appearance of small persistent
coherent structures of very high vorticity (and velocity) compared to the
background flow. Their typical linear scales are about 10 to 20 grid zones and
thus their physical size depends on the numerical resolution, i.e, their size
decreases with increasing resolution. The two simulations (low and high
resolution) which are presented here show only moderate differences in
spatially integrated quantities such as laterally averaged temperature, energy
generation rate, and chemical composition. We have not expanded both
simulations equally long, but for the physical time under consideration the
major difference seems to be that the highly resolved simulation is a bit less
violent. In conclusion, we do find some self-enrichment, but on time-scales
much longer than in previous calculations.
