
As shown in Figure~\ref{fig:lambdac}, $\lambda_c$ drops quickly with radius,
mainly due to the increasing gravity as one moves away from the center of the
star, and after this due to the falling flame speed.
We take $r_{\rm bub} <\lambda_c$ in all cases, so that our assumption of a
spherical bubble at rest is physically justifiable \citep{fisher+07}.

\begin{figure*}
\plotone{tworow_reduce.eps}
\caption{\label{fig:bubble}
Stages of bubble growth at different resolutions.  The initial radius of the
bubble was 16 km centered at 40 km from the center of the star, contours are
shown for the progress variable $\phi=0.1$ (green), 0.5 (red), and 0.9
(blue).
}
\end{figure*}

Initially bubbles can be thought of as moving from lower left to upper right
(growing and rising) in Figure~\ref{fig:lambdac} (see \citealt{zingale+07}).
We distinguish three
phases of the bubble evolution and rise in terms of $\lambda_c$.  Each of
these phases can be seen in Figure~\ref{fig:bubble}, where we show the
evolution of the flame surface with time for two different resolutions: our
typical resolution, 4 km, and our highest resolution, 0.5 km.  Initially,
while small and near the center with $r_{\rm bub} < \lambda_c$, the flame
bubble will grow according to the laminar flame speed, roughly keeping a
spherical shape.  This structure is evident at $t=0.2$ seconds, where the
bubble has already grown to a radius of about 30 km, twice its original size.

Eventually as the bubble rises and grows, it will reach the point when
$r_{\rm bub}\sim \lambda_c$.  For our case this occurs at a little over 100
km radius.  As seen at $t=0.4$ sec, the bubble forms a R-T roll at
approximately its full dimensions, the first scale that is unstable.
This feature shows some differences with resolution, but
generally the same kind of feature (a roll) has appeared in both simulations.
Crossing $\lambda_c$, the bubble is now a R-T unstable flamelet.  The
subsequent evolution is resolved in our simulation until $\lambda_c$ falls
below the grid scale.  Thus we will term the second stage of evolution as the
resolved R-T stage.  The R-T structure becomes unresolved at approximately
350 and 500 km radius for 4 and 0.5 km resolution respectively.  Thus by
t=0.6 seconds, the 4 km simulation has already entered the unresolved regime,
while the 0.5 km simulation has nearly entered it.  This is evidenced by the
separation between consecutive contours in the progress variable caused by
strong advection of material within the flame front.

The rest of the bubble rise and burn is dominated by unresolved R-T burning.
The bubble continues to grow, both due to burning and expansion of the
material under decreasing pressure.  Its top generally reaches the surface at
approximately 1.0 seconds (``breakout''), after which it mostly exits the interior of the
star and expands strongly to create the flow around the stellar surface.
A notable difference between our evolution and that seen by
\citep{roepke-astroph+06}
is the presence of the ``stem'' below the rising bubble.  This stem is absent
in non-reactive rising bubbles \citep{Vlad07}, but it is
mysterious that it is completely absent in the level-set reactive flow
simulations.  This might be related to the fact that the initial bubbles
used by \citet{roepke-astroph+06} are too large and far off-center to capture
either the laminar or resolved R-T evolution stages, but more investigation
is necessary.  Though both simulations in Figure~\ref{fig:bubble} are
unresolved at $t=0.8$ seconds, comparing the two provides a good demonstration
of the enhancement in flame surface facilitated by smaller scale structure.

As mentioned in section \ref{sec:model}, any flame tracking method that
depends on the advection of a scalar field faces difficulties in the highly
turbulent flow produced in the unresolved R-T stage.  Measures must be taken
to compensate for the finite resolution of the simulation, otherwise the
flame ceases to propagate correctly because the turbulence destroys the
necessary structure for self-propagation of the scalar field.
This problem is particularly relevant to the simulation of WD deflagrations,
as has been extensively discussed by previous authors
\citep{khokhlov+95,gamezo+03,schmidt+06}.  R-T can create turbulence down to
the scale $\lambda_c$, below which the flame is able to polish out the mixing
perturbations.  It is thought \citep{khokhlov+95} that the details of this
small-scale behavior need not be followed explicitly.  If the overall
dynamics of the flame surface are determined by the large scale
perturbations, simulations of moderate resolution (which may however be only
just possible today in 3-d) can determine the evolution of the burning front.
This is called self-regulation, and the open question is: What resolution is
``enough''?  This is currently under debate, and we will argue here that
we have enough resolution for some purposes and can make statements about
others based on trends that we see with resolution.

\citet{schmidt+06} attempted to compensate for the shortcoming of the limited
resolution by artificially enhancing the propagation speed of the flame front
based on a measure of local turbulence.  Such complex measures should not be
necessary if self-regulation holds, thus we have taken what we consider to be
a conservative approach, making the smallest possible adjustment to the front
propagation speed and evaluating the inaccuracy directly via resolution
study.  We enforce a minimum flame speed (which therefore acts effectively as
an enhancement) that is intended to ensure $\tau_{\rm R-T} \gtrsim
\tau_{\rm flame}$ where, for flame width $\delta$, $\tau_{\rm R-T} \approx
\sqrt{\delta/Ag}$ is the characteristic R-T rise time and $\tau_{\rm
flame}\approx \delta/s$ is the flame self-crossing time.  Since our
flame width is proportional to the resolution, this leads to a limit $s \ge
\alpha\sqrt{Agm_f\Delta x}$, where $\Delta x$ is the resolution, $\alpha$ is a
geometrical factor that is different in two and three dimensions, and $m_f$ is a
calibrated constant.

We have determined $m_f$, the constant that determines when the flame model
``breaks'', empirically by running two-dimensional flame-in-a-box simulations like those
of \citet{khokhlov+95} and \citet{zhang+06}.  We evaluated the range of $s$
over which self-regulation holds, in which the burning rate, $\dot m_{\rm
burned} = \rho L^{(d-1)} s_{\rm eff}$ is determined by the box size, $L$,
such that $s_{\rm eff} = \alpha\sqrt{AgL}$, independent of $s$. In two 
dimensions $\alpha=0.28$ and in three dimensions, as found by 
previous authors, $\alpha=0.5$.  We
found that the self-regulated regime is bounded above by $s\simeq \sqrt{Ag
L}$, corresponding to the requirement
$\lambda_c\lesssim L$, and below by $s \simeq \alpha\sqrt{Ag0.04\Delta x}$.  In
our WD simulations we have used the value for $\alpha$ from three-dimensional
simulations, although we are
working in two-dimensional cylindrical geometry, and $m_f=0.04$. These values 
have been confirmed by
preliminary three-dimensional flame-in-a-box calculations, which will be
discussed in a
separate work.  Using this type of floor on $s$ requires that we explicitly
turn off the flame at low density.  This is done smoothly between
$\rho = 10^7\ \mathrm{g\ cm^{-3} }$ and $5\times 10^6\ \mathrm{g\ cm^{-3} }$, so that $s=0$ for densities below this.

\subsection{Resolution Study}
\label{sec:refine}

Some properties of the off-center deflagration model that we are trying to
deduce from our simulations show dependence on the simulation resolution,
while others do not.  We would like to make statements as much as
possible based on features that are not influenced by resolution, and where we
cannot avoid it, account for the dependence in other conclusions that we
draw.  Problems with resolution-dependence is not entirely unexpected, since,
as just discussed, a significant amount of our simulation is \emph{a priori}
known to be unresolved.  In summary,