  In the distributed burning regime, turbulence disrupts the internal structure
of the flame, and so the idea of laminar burning propagated by conduction is no
longer valid. The nature of the burning depends on the turbulent Damkohler
number (Da), which steadily declines from much greater than one to less that
one as the density decreases to a few 10^6 g/cc. Scaling arguments predict that
the turbulent flame speed s, normalized by the turbulent intensity u, follows
s/u=Da^1/2 for Da<1. The flame in this regime is a single turbulently-broadened
structure that moves at a steady speed, and has a width larger than the
integral scale of the turbulence. The scaling is predicted to break down at
Da=1, and the flame burns as a turbulently-broadened effective unity Lewis
number flame. We refer to this kind of flame as a lambda-flame. The burning
becomes a collection of lambda-flames spread over a region approximately the
size of the integral scale. While the total burning rate continues to have a
well-defined average, s_{T} ~ u, the burning is unsteady. We present a
theoretical framework, supported by both 1D and 3D numerical simulations, for
the burning in these two regimes. Our results indicate that the average value
of s can actually be roughly twice u for Da>1, and that localized excursions to
as much as five times u can occur. The lambda-flame speed and width can be
predicted based on the turbulence in the star and the turbulent nuclear burning
time scale of the fuel. We propose a practical method for measuring these based
on the scaling relations and small-scale computationally-inexpensive
simulations. This suggests that a simple turbulent flame model can be easily
constructed suitable for large-scale distributed supernovae flames.
