  The burning regimes encountered by laminar deflagrations and ZND detonations
propagating through helium-rich compositions in the presence of buoyancy-driven
turbulence are analyzed. Particular attention is given to models of X-ray
bursts which start with a thermonuclear runaway on the surface of a neutron
star, and the thin shell helium instability of intermediate-mass stars. In the
X-ray burst case, turbulent deflagrations propagating in the lateral or radial
directions encounter a transition from the distributed regime to the flamlet
regime at a density of 10^8 g cm^{-3}. In the radial direction, the purely
laminar deflagration width is larger than the pressure scale height for
densities smaller than 10^6 g cm^{-3}. Self-sustained laminar deflagrations
travelling in the radial direction cannot exist below this density. Similarily,
the planar ZND detonation width becomes larger than the pressure scale height
at 10^7 g cm^{-3}, suggesting that a steady-state, self-sustained detonations
cannot come into existance in the radial direction. In the thin helium shell
case, turbulent deflagrations travelling in the lateral or radial directions
encounter the distributed regime at densities below 10^7 g cm^{-3}, and the
flamelet regime at larger densities. In the radial direction, the purely
laminar deflagration width is larger than the pressure scale height for
densities smaller than 10^4 g cm^{-3}, indicating that steady-state laminar
deflagrations cannot form below this density. The planar ZND detonation width
becomes larger than the pressure scale height at 5 10^4 g cm^{-3}, suggesting
that steady-state, self-sustained detonations cannot come into existance in the
radial direction.
