  We present the first vertically resolved hydrodynamic simulations of a
laterally propagating, deflagrating flame in the thin helium ocean of a
rotating accreting neutron star. We use a new hydrodynamics solver tailored to
deal with the large discrepancy in horizontal and vertical length scales
typical of neutron star oceans, and which filters out sound waves that would
otherwise limit our timesteps. We find that the flame moves horizontally with
velocities of order $10^5$ cm s$^{-1}$, crossing the ocean in few seconds,
broadly consistent with the rise times of Type I X-ray bursts. We address the
open question of what drives flame propagation, and find that heat is
transported from burning to unburnt fuel by a combination of top-to-bottom
conduction and mixing driven by a baroclinic instability. The speed of the
flame propagation is therefore a sensitive function of the ocean conductivity
and spin: we explore this dependence for an astrophysically relevant range of
parameters and find that in general flame propagation is faster for slower
rotation and higher conductivity.
