  We develop a one-dimensional theoretical model for thermals burning in Type
Ia supernovae based on the entrainment assumption of Morton, Taylor and Turner.
Extensions of the standard model are required to account for the burning and
for the expansion of the thermal due to changes in the background
stratification found in the full star. The model is compared with
high-resolution three-dimensional numerical simulations, both in a uniform
environment, and in a full-star setting. The simulations in a uniform
environment present compelling agreement with the predicted power-laws and
provide model constants for the full-star model, which then provides excellent
agreement with the full-star simulation. The importance of the different
components in the model are compared, and are all shown to be relevant. An
examination of the effect of initial conditions was then conducted using the
one-dimensional model, which would have been infeasible in three dimensions.
More mass was burned when the ignition kernel was larger and closer to the
center of the star. The turbulent flame speed was found to be important during
the early-time evolution of the thermal, but played a diminished role at later
times when the evolution is dominated by the large-scale hydrodynamics
responsible for entrainment. However, a higher flame speed effectively gave a
larger initial ignition kernel and so resulted in more mass burned. This
suggests that future studies should focus on the early-time behavior of these
thermals (in particular, the transition to turbulence), and that the choice of
turbulent flame speed does not play a significant role in the dynamics once the
thermal has become established.
