  The Low Mach Number Approximation (LMNA) is applied to 2D hydrodynamical
modeling of Type I X-ray bursts on a rectangular patch on the surface of a
non-rotating neutron star. Because such phenomena involve decidedly subsonic
flows, the timestep increase offered by the LMNA makes routine simulations of
these deflagrations feasible in an environment where strong gravity produces
significant stratification, while allowing for potentially significant lateral
differences in temperature and density. The model is employed to simulate the
heating, peak, and initial cooling stages in the deep envelope layers of a
burst. During the deflagration, Benard-like cells naturally fill up a
vertically expanding convective layer. The Mach number is always less than 0.15
throughout the simulation, thus justifying the low Mach number approximation.
While the convective layer is superadiabatic on average, significant
fluctuations in adiabaticity occur within it on subconvective timescales. Due
to convective layer expansion, significant compositional mixing naturally
occurs, but tracer particle penetration through the convective layer boundaries
on convective timescales is temporary and spatially limited. Thus, mixing
occurs on the relatively slow burst timescale through thermal expansion of the
convective layer rather than from mass penetration of the convective layer
boundary through particle convection. At the convective layer boundaries where
mixing is less efficient, the actual temperature gradient more closely follows
the Ledoux criteria.
