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\begin{center}{\bf{\Large  Project Description}}
\end{center}

% (d) Project Description - NSF Standard FastLane Form.
%
% i) Proposed Research. Narrative, not to exceed twenty pages, consisting of
% the following items:
%
%    * An explanation of the scientific context and timeliness of the
%      proposed project.
%    * A description of the proposed research.
%    * A justification for why a group effort is necessary to carry out the
%      proposed project.
%    * A timeline for the planned work and a justification for the duration.
%    * Plans for disseminating the results.
%    * Results from prior NSF support, if applicable and related to the
%      proposal.

\section{Proposed Research}

\subsection{Context and Timeliness of This Project}
% it is my view that this subsection and the project
% summary are very important - tfd

% - recent progress in the analysis of reacting flows gives
%    not only the confidence that progress can be made, but
%    also gives useful analytical techniques
%  - the availability of a robust, well-documented and supported
%    code for doing simulations of the phenomena of interest here
%    is a boon to this project and other similar projects.
% - questions are of importance in the understanding of
%   the processes associated with mixing in stars, and
%   understanding this mixing is central to explaining the
%   observed abundances of elements in the universe.
% - while our motivations arise from astrophysical considerations,
%   it is clear that reactive flows in which stratification
%   is important occur in many contexts, in particular in the
%   earth's atmosphere and oceans. We are not proposing to
%   study those problems here, but it is highly likely
%   that progress made in this project will have implications
%   in these other problems

The research proposed here is an outgrowth of recent collaborations
between three previously distinct lines of research at the University
-- in Mathematics, Astrophysics and Physics, and Computational Science
-- and our realization that further enhancements of this collaboration
can have a major impact in each of the disciplinary areas. First,
there has been recent, substantial advance in the state of the art in
the rigorous mathematical analysis of reactive flows; much of this
work involves participants in this proposal, and is outlined below in
the description of the proposed research. Second, astrophysicists
interested in nuclear ``flames" (analogous to premixed flames in
chemical combustion) have realized that the available models
previously derived from the chemical domain are not adequate for
the purpose of understanding the explosive evolution of novae and
supernovae; and have therefore begun to re-investigate nuclear flame
propagation from first principles (but in the astrophysical context).
Third, a group of astrophysicists, physicists, and computer
scientists at Chicago has been involved in development of a new
parallel, adaptive mesh computational code for reactive, compressible
hydrodynamics, funded by the Department of Energy in the form of a
center of excellence (the ``Flash Center"). The collaboration in
question has involved astrophysicists defining specific physical
circumstances in which flame propagation is problematical;
mathematicians considering what rigorous results could be obtained
for model problems that approximate more or less closely the actual
physical circumstances of interest to the astronomers; and
computational scientists who have carried out simulations that
attempt to bridge that which can be studied rigorously to that which
is of interest in the actual physical systems. The specifics of what
we are doing, and what we would like to do, will be detailed below;
here we simply note that the presently funded research is largely
focused on the astrophysics/physics and the code construction; and
that the present proposal is aimed at support of the mathematical
component of our collaboration.

The questions involved in reactive flow are in general quite
difficult; and it is therefore extremely important to narrow the
specifics of the questions that need to be answered. In the recent
past, we have focused our mathematical attention on (pre-mixed) flame
propagation in fluids with prescribed flows; and our considerable
progress in this area (described below) not only gives us confidence
that there are further successes to be had it in this area, but that
it also provides the mathematical community with well-formed
questions to investigate and additional techniques to aid in that
investigation. However, the astrophysicists have been essential in
this process because they have led the way in defining the questions
that are ultimately important to solve; and the computational
scientists have been essential in exploring the solution space, and
thereby illuminating what kinds of propagation problems (viz., what
kinds of prescribed flows) might be amenable to rigorous analysis.

The specific issue we want to address is the question of feedback, or
active combustion. In past combustion work, it has been generally
assumed that -- aside from interface dynamics controlled by pressure
and inertial forces at the flame front -- there is no feedback from
the flame back onto the flow. In the astrophysical case, there is
however a mechanism for such feedback: in the presence of gravity
(e.g., stratification), buoyancy forces resulting from density
changes at the flame front can produce feedback in the form of, for
example, Rayleigh-Taylor instability of the fuel-ash interface. We
believe that carrying our mathematical work forward to the context in
which stratification plays an important role will lead to not only
interesting mathematics, but also to insights that help us deal with
mathematical models that include parts of the physics needed to
address questions that are very natural in astrophysics. Such
questions arise not only in the context of nuclear flames, but also
for other problems in which exothermic ``reactions"  -- and thus
density changes -- occur at fronts, such as for example recombination
or ionization fronts in which the ionization state of a fluid changes.

The timeliness of the proposed program is clear: we have the
astrophysical and computational science aspects of this problem in
hand; and major progress awaits new rigorous studies of flame
propagation in this new context of stratified flows. The need for a
center of focused collaboration -- in which the whole is
substantially greater than the simple sum of the components -- is
made evident by our past work, in which the mathematical analysis was
motivated by the specifics of the astrophysics, and required the
computations in order to guide and extend the rigorous analysis.
Finally, we are fully confident in our ability to carry out the
collaboration as described here because we already have a track record
of four years of collaborative research in this general research area.
Finally, we note that while our primary motivations are from
astrophysics (and these provide the guidelines for our proposed
work), it is the case that reactive flows -- including such flows in
which stratification is important -- are ubiquitous; important
examples of stratified reactive flows are found, for example, in our
atmosphere and oceans. Thus, although we expect to concentrate our
efforts on the questions that originally generated this
collaboration, it is very probable that progress made on
understanding reactive flows here will lead to progress on related
problems.

\subsection{ Description of the Proposed Research}


\subsubsection{Speed-up and Quenching by Passive Flows}

In this section we describe results concerning reaction diffusion
equations carried by a passive flow. The subject has been  studied
recently and numerous  questions remain open and are worth
investigating. Besides being intrinsically interesting, these
problems provide both context and  validation material for our
proposed work on active models.

We consider situations in which mixtures of reactants interact in a
region that may have a rather complicated spatial structure. This
reaction region moves towards the fresh reactants. When the
reactants are carried by an ambient fluid then the process rate may
be  enhanced. The physical reason  for the observed speed-up is that
fluid advection tends to increase the area available for reaction.
Different enhancement laws have been suggested ([21], [82], [53],
[68]-[71]).

Much of the rigorous mathematical research on the subject is based
on studies of a single passive advection-reaction-diffusion equation:
\begin{equation}\label{mainmod}
T_t + u \cdot \nabla T - \kappa \Delta T = \frac{v_0^2}{4\kappa} f(T).
\end{equation}
Here $u$ is a prescribed advection velocity, $\kappa$ is the
temperature diffusivity, $v_0$ is the laminar front velocity, and
$f(T)$ is the reaction term that has steady states $f(0), f(1) = 0$.
The temperature is normalized to satisfy $0 \leq T \leq 1,$ and
remains within these bounds because of the maximum principle. The
most common reaction types considered in this framework are KPP-type
($f(T)>0$ on $(0,1),$  combustion or ignition type ($f(T)=0,$ $T_0
\geq T \geq 0,$ $f(T)>0,$ $1>T> T_0$), and bistable type (in the
simplest case, $f(T)<0$ for $T_0>T>0,$ $f(T)>0$ for $1>T>T_0$). The
equation (\ref{mainmod}) is usually  considered in a strip (or, more
generally, in a cylinder in higher  dimensions) with Neumann or
periodic conditions on the boundary,  or in all space. A very
brief and necessarily sketchy summary of the large body of knowledge
regarding the issues of front propagation and homogenization is
presented below.

The equation (\ref{mainmod}) originated ([55, 36]) in biology and is
used as a model in many areas of science (e.g., [2, 3, 43, 12, 83,
21, 10, 57, 78]). Traveling wave solutions $T(x,t) = W(x-ct)$ were
found already in the pioneering work [55] for $u=0$, $f(T) =
T(1-T)$. Substantial work relates to existence, uniqueness and
generalizations of traveling wave solutions [49, 77, 19, 35, 48, 50,
63, 75]. Higher dimensions and $u\neq 0$ situations were studied by
Berestycki and Nirenberg [13, 14].  Berestycki, Larrouturou and P-L
Lions [11] studied the existence of generalized traveling waves of the
form
\begin{equation}
   \label{tws}
   T(x,y,t)=W(x-ct,y)
\end{equation}
for shear flows $u=(u(y),0)$.  Their stability was investigated in
[12, 62, 73, 72], while in further works [9, 56, 65, 79, 80, 81]
existence and stability of traveling waves was studied for a wider
class of periodic flows.  In this latter case, the analog of the
traveling waves are the pulsating fronts
%\begin{equation}
  % \label{eq:per-front}
  \[  T(x,y,t)=U(x-ct,x,y), \]
%\end{equation}
which are periodic in the second variable, satisfy the usual
normalization  conditions and $U_t \geq 0.$  Results of Xin [79, 80]
also relate the speed of traveling waves and pulsating fronts with
the  large time propagation speed of more general initial data. These
and other results were recently reviewed in [81]. Numerical studies
of the propagation of fronts were performed for shear flows in [46]
%with ${\hbox{Le}}\ne 1$,
and for cellular flows in [47] and, more recently, in [1] and in our
group (paper [76] by Cattaneo, Malagoli, Oberman, Vladimirova
submitted to PRL).  Homogenization regimes when  $\kappa \to 0$, and
the front width tends to zero were studied extensively by Freidlin
for the KPP-type nonlinearity and velocities that vary on the
integral scale or on the diffusive scale ([38, 39, 40]).
Homogenization models involving combustion in  random velocity fields
have been studied by Majda and Souganidis in [60] and [74]. Recently,
Freidlin [41] considered the limits of small diffusion or large flow
intensity for a certain family of flows, and related the problem of
estimating the solution to the study of a certain diffusion process
on a graph. Majda and Souganidis derived an effective
Hamilton-Jacobi equation in the limit $\kappa \to 0$ for the case of
advection velocity varying on a small $\kappa-$dependent scale that
is larger or comparable to that of the front width [59]. Analytical
and  numerical studies further exploring these results have been
carried out in [32, 33, 64]. There are significant added challenges
for accurate  numerical calculation of reactive fronts in the
presence of large  scale advecting velocities. A. Oberman, a very
recent graduate of  our Ph.D.\ program has devised and successfully
implemented an effective theoretical and numerical tracking method
that  uses a curvature-dependent correction to the classical
Hamilton-Jacobi description of the propagating front, consistent
with the results of [59].  (A.\ Oberman, Thesis, The University of
Chicago 2001.)  Two major questions arise in studies of front
propagation under the  influence of fluid advection. The first one
is: Which characteristics of the  ambient fluid flow are responsible
for propagation rate enhancement? The second question is: what kinds
of flows are capable of bringing to extinction large reacting
regions?

In order to make precise the first question, we defined in [27] an
unambiguous quantity $V$  representing the reaction volume transport
rate. We provided explicit estimates of $V$ in terms of the
magnitude of the advecting velocity and  the geometry of streamlines.
In situations when the transport is done by traveling waves, $V$
coincides with the traveling wave speed and the estimates provide
thus automatically bounds  for the speed of the traveling waves.  The
main result of [27]  is the identification of a class of flows that
are particularly effective in speeding up the reaction volume
transport rate. The main feature of these ``percolating flows''  is
the presence of tubes of  streamlines connecting distant  regions of
fresh and used material.  For such flows we obtained  an optimal
linear enhancement bound $ V\ge KU $ where $U$ represents  the
magnitude of the advecting velocity and $K$ is a proportionality
factor that depends on the geometry of streamlines but not the speed
of the flow. Other flows and in particular cellular flows, which have
closed streamlines, on the other hand, produce a weaker enhancement.

The instantaneous reaction volume transport rate is defined by the
formula
$
V(t)=\int_D \frac{\partial T}{\partial t}(x,y,t)~dxdy ,
$
where the integral extends over the spatial domain $D$, taken here for
simplicity of exposition to be a two-dimensional strip of unit width
and infinite length
$
0\le y\le 1, \,\,-\infty <x< \infty.
$
The temperature $T$ is assumed to obey Neumann boundary conditions
at the finite boundaries and to obey
$
T(-\infty, y)=1,
\,\,\,\,T(\infty, y)=0.
$
The simplest non-trivial model is a passive reactive scalar with a
KPP nonlinearity
$
T_t + u \cdot \nabla T -\kappa \Delta T =
\frac{v_{0}^{2}}{4\kappa}T(1-T).
$
with prescribed velocity $u$ that satisfies
$
\int_0^1 u(x,y,t)\,dy = 0, \quad \nabla\cdot u = 0.
$
The constant $v_0$ represents the speed of a stable one-dimensional
laminar ($u=0$) traveling wave. For arbitrary
initial data obeying
$
0 \leq T_0(x,y) \leq 1.
$
one has the general lower bound
$ V(t) \geq Cv_0 \left( 1- e^{-\frac{v_{0}^{2}t}{2\kappa}} \right). $
%\vspace{1cm}
Although information about the velocity is not present in the general
result, it nevertheless shows that this model does not  permit
quenching. Also, the general lower bound applies to the  homogenized
version of the equations as well. For a very general class of
velocities $u(x,y,t)$ it can be shown that  the reaction volume
transport rate can not exceed a linear bound in the  amplitude of the
advecting velocity.  For a large  class of flows  we proved  lower
bounds on the reaction volume transport  rate that are  linear in the
magnitude of advection.  We denote by
$\langle V \rangle_\tau = \frac{1}{\tau} \int\limits_{0}^{\tau} V(t)
\,dt$
the time average of the instantaneous bulk burning rate. The result of
[27] is that the presence of coherent tubes of streamlines connecting
fresh and  consumed reactant regions enhances the reaction volume
transport rate
$$
\langle V \rangle_{\tau}  \geq K U
$$
as long as the velocity spatial scales are not too small compared to
the reaction length scale $\frac{\kappa}{v_0}$, and the time scale of
change of the advecting velocity is not too small  compared to
$\tau_0 = {\rm max}[\frac{\kappa}{v_0^2},
\frac{\tilde{H}}{v_0}]$ where $\tilde{H}$ is associated to the width
of the coherent tubes of streamlines. For instance, a result
concerning mean zero shear flow of the form $ u(x,y)=(u(y),0),
~~\int_0^1 u(y)dy=0 $ can be stated as follows.  Consider an
arbitrary partition of the interval $[0,1]$ into subintervals
$I_j=[c_j-h_j,c_j+h_j]$ on which $u(y)$ does not change sign. Denote
by $D_{-},$ $D_{+}$ the unions of intervals $I_j$ where
$u(y)>0$ and $u(y)<0$ respectively. Then there exist  constants
$C_{\pm} >0$, independent of the partition, $u(y),$ and the initial
data $T_0(x,y)$, so that the average burning rate $\langle V
\rangle_\tau$ satisfies the following estimate:
$$
   \langle V\rangle_\tau
  \ge C_+
c_+
\sum_{I_j \subset D_+}
\left(1+\frac{l^2}{h_j^2}\right)^{-1}
\int\limits_{c_j-\frac{h_j}{2}}^{c_j+\frac{h_j}{2}}|u(y)|dy
$$
$$
+C_-c_-
\sum_{I_j \subset D_-}
\left(1+\frac{l^2}{h_j^2}\right)^{-1}
\int\limits_{c_j-\frac{h_j}{2}}^{c_j+\frac{h_j}{2}}|u(y)|dy
$$
for any $\tau \geq \tau_0=\hbox{max}\left[ \frac{\kappa}{v_0^2}, \frac{H}{v_0}
  \right].$ ($H=1$).
Here $l=\kappa/v_0.$
The constants $c_\pm$ are explicit:
$$
c_\pm = \left( \sum\limits_{I_j \subset D_\mp} \frac{h_j^3}{h_j^2 +l^2}
\right) \left( \sum\limits_{I_j} \frac{h_j^3}{h_j^2 +l^2}
\right)^{-1}.
$$

The result of ([27]) applies to a large class of flows that are not
necessarily spatially periodic, nor shears, and can  have completely
arbitrary features outside the tubes of streamlines. The reaction
volume transport rate is still linear in the magnitude of the
advecting velocity, no matter what kind of behavior (closed
streamlines, regions of stagnating fluid, etc.) the flow has outside
the tubes. The proportionality coefficient depends on the geometry
of the flow in a rather complex manner. These results have been
checked numerically in calculations performed at Chicago and
agreement with the theoretical prediction is very good. The results
do depend on a number of conditions. One of them is the fact that we
are dealing with a single equation (Lewis number equal to one,
special initial conditions). The questions of propagation enhancement
in systems are more difficult to treat for both technical reasons
(no maximum principle) and physical ones (complex dynamics). We do
have some preliminary indications when such results can be
meaningfully generalized and would like to pursue this line of
research for  systems. Even in the case of Lewis number equal to one,
for flows that are not of percolating type, there are still open
problems concerning the minimal rate of enhancement. In this case, if
one keeps the  geometry of the flow fixed while increasing the speed,
on expects a  sublinear rate of enhancement. At present there is a
gap between the  lower bound one can rigorously prove (which scales
as the  1/5 power of the amplitude of velocity) and the conjectured
bound (scaling with exponent 1/4). Progress in this question will
result in a better understanding of the advection of diffusive
internal interfaces.

The second fundamental question, related to  quenching, can be
phrased in the context of simple reaction-advection-diffusion
equation  with ignition-type nonlinearity in the case the  initial
data is compactly  supported. In this case, even without advecting
velocities,  two typical behaviors occur. If the support of the
initial data is large enough, then fronts form and propagate. On the
other hand, however,  if the support of the initial data
is small enough, then molecular diffusion is sufficient to cool the
reactant below the ignition temperature, and the reaction terminates.
In the framework of  the model (\ref{mainmod}), the quenching
phenomenon is present for ignition or bistable nonlinearities, for
it is feasible only  if there is a sufficiently sharp fall-off of the
reaction rate.  This situation of $u=0$ with ignition-type
nonlinearity has been considered by Kanel [51]. He showed that,
in the absence of fluid motion, there exist two length scales
$L_0<L_1$ such that the flame becomes extinct for the initial
data coinciding with the characteristic function $\chi_{[-L,L]}$
for $L<L_0$, and propagates for $L>L_1$.  More precisely, he has shown
that there exist $L_0$ and $L_1$ such that
\begin{eqnarray}  \label{eq:2.2}
&&  T(t,x,y)\to 0~\hbox{as $t\to\infty$ uniformly in $D$ if $L<L_0$}\\
&&T(t,x,y)\to 1~\hbox{as $t\to\infty$ for all $(x,y)\in D$ if 
$L>L_1$}. \nonumber
\end{eqnarray}
thus, in the absence of advection, the flame extinction is achieved by
diffusion alone, given that the support of initial data is small
compared to the scale of the laminar flame width $l = \kappa/v_0.$

In the presence of fluid advection the question is: what kinds of
velocity  profiles are capable of quenching any given flame, provided
the velocity's amplitude is adequately large? Even for shear flows,
the answer turns out to be surprisingly subtle. In [29]  we studied
quenching in a shear flow $Uu(y)e_x$ (with fixed profile $u(y)$ and
amplitude $U$). We considered compactly supported initial data
$$
T(x,y,0)=0 \,\,\,if \,\,\, |x|>L.
$$
We used the following terminology: The profile $u(y)$ is ``quenching'' if
for any $L$ and any  initial data $T_0(x,y)$ supported inside the interval
   $[-L,L]\times[0,H],$ there exists $U_0$ such that solution becomes extinct:
\[
T(t,x,y)\to 0~\hbox{as $t\to\infty$ uniformly in $D$}
\]
for all $U\ge U_0$.  The profile $u(y)$ is termed ``strongly
quenching'' if the critical amplitude of advection $U_0$ satisfies
$U_0 \leq C L$ for some constant $C(u,\kappa,v_0,H)$ (which has the
dimension of inverse time). In [29] we showed that the ``strongly
quenching'' property is implied by the hypoellipticity of an
associated linear degenerate diffusion advection equation. Using a
well-known result of H\"ormander ([44-45]), we showed that the
hypoellipticity of the operator follows if there is no $y$ where  all
the derivatives of $u$ vanish. We termed this the H condition. So,
profiles that satisfy the H condition everywhere are strongly
quenching.   But H is not necessary: For every ignition temperature
$T_0>0$ there exists a constant $b>0$ such that, if a profile $u(y)$
satisfies  the H condition outside a small interval $y\in[a-h,a+h]$
with  $h\le b\kappa/v_0$,  then it is  strongly quenching. Actually,
the ``strong quenching'' property is generic:  in [29] we proved that
the set of all strongly quenching profiles in $C[0,H]$ contains a
dense $G_\delta$ set.  On the other hand, if the  velocity profile is
identically constant in a sizable region, then the  ensuing flow is
incapable of quenching large enough  flames, no matter how  much
larger is the amplitude of this velocity.  The constancy region  must
be wider across than a couple of laminar  propagating front-widths.

The quenching question is most puzzling when one steps away from the
idealized situation we studied so far. On one hand, the shear flow
system has no chaotic trajectories, so quenching certainly does not
require underlying dynamics with positive Lyapunov coefficients. On
the other hand, the mixing properties of the Lagrangian paths play a
fundamental role in the acceleration of flames, and they may be
expected to play a role in quenching as well. But the question
remains: Do there exist natural, sufficient conditions in terms of
the mixing properties of the underlying ODE that guarantee quenching
for more general systems? Is there a generic quenching property for
systems? Is the mechanism observed in [29] operating in systems as
well?

The study of the issue of reaction volume propagation versus
quenching, in the context of flows that have internal diffusive
interfaces is an exceedingly interesting and widely open area of
research in the passive advection case.

\subsubsection{Active Flows}

The study of models of active reaction-diffusion-advection systems is
the main new direction we wish to focus on. The prototype model
couples a variable density Navier-Stokes system to a reaction
diffusion system for several species. Gravitational forces couple the
reactants to the momentum balance equation and advection couples the
latter back to the reaction diffusion system. The general systems
have not been studied extensively theoretically because they are
very complex. We will use PDE methods in order to distinguish
between models that lead to well-posedness and regularity, and those
that need cut-offs. The variable-density models are difficult to
study quantitatively because the  fact that the density gradients
introduce multi-scale forces in the  momentum equation. A simplified
level of model is obtained in the  Boussinesq approximation but,
nevertheless, the models in this class are still quite complicated.

Rayleigh-Benard convection, where a fluid layer heated from below
produces an instability leading to convective fluid motions, has
played a central role in both the experimental and theoretical
development of the modern sciences of nonlinear dynamics and physical
pattern formation. Driven far beyond the instability, thermal
convection becomes turbulent. Heat transport by convective turbulence
is an important component of a wide variety of problems in applied
physics ranging from stellar structure in astrophysics, to mantle
convection and plate tectonics in geophysics, to transport in
physical oceanography and atmospheric science. The Boussinesq
equations, describing the physical system in a nearly uniform density
approximation consist of the heat  advection-diffusion equation for
the local temperature coupled to the  incompressible Navier-Stokes
equations via a buoyancy force proportional to  the local
temperature. In the reactive case the local temperature equation is
replaced by a nonlinear reaction diffusion equation or system of
equations.

One of the fundamental quantities of interest in these systems is the
total  heat transport across the layer and its dependence on the
nondimensional  parameters such as the Rayleigh number and the
Prandtl number. Our group has studied aspects of this problem in the
non-reactive  case (see for instance [24], [25], [28], [30], [31] and
references therein).

In the reactive case one may start by studying the case in which one
does not heat from below at the boundary of the system, but rather
the reaction itself produces the instability. A simplified model was
proposed by A. Kerstein (unpublished) and is being investigated by
our student B. Winn. In this system the gravitational force is
unidirectional but not monotonic. The model has a nondimensional
parameter that measures the  strength of the chemical reaction. In
preliminary results Ms. Winn proved  that, when this parameter is
zero, then the system possesses a  nonlinearly globally stable state,
but  when the reaction is  turned on, this state becomes unstable
while the burned state becomes  nonlinearly stable. In analogy with
the non-reacting case, one would like  to study the enhancement of
heat transport, and its dependence on  nondimensional parameters.

An ongoing  numerical study (N. Vladimirova, unpublished material) of
a reactive Boussinesq system in a strip finds a traveling front
solution. This solution is likely to be Rayleigh unstable, and the
methods of estimating heat transport based on a background flow
should be applicable for the estimate of the Nusselt number in this
situation.

The full Boussinesq model is still rather complicated for analytical
study. In the non-reactive case analytic progress has been achieved
in the study of  a convection through a porous layer ([30]), infinite
Prandtl number convection ([25]) and rotating convection ([28]). The
model of convection through a porous layer replaces the momentum
equation by Darcy's law where the velocity balances the local
temperature, $u -\nabla p = RT\hat{z}$. Here $\hat{z}$ is the
direction of gravity, $T$ is the local temperature and the
coefficient $R$ is the Rayleigh number appropriate for a porous layer,
$R=\frac{g\alpha\Delta TKh}{\nu\kappa}$, with $K$ Darcy's
permeability coefficient  (the porous medium microscale is of the
order of $K^{1/2}$), $h$ the width  of the layer, $\Delta T$ the
temperature difference across the layer and $\nu$ the kinematic
viscosity. The equations are valid in the limit of  the infinite
Prandtl-Darcy number limit $\frac{\nu h^2}{\kappa K}\to \infty$. This
incompressible velocity is advecting the local temperature
advection-diffusion equation. In the reactive case, this equation is
replaced by (\ref{mainmod}). In the infinite Prandtl number model the
velocity is obtained by a quasi-steady Stokes flow $\nabla^2 u
-\nabla p =  RT\hat{z}$  with Rayleigh number $R=\frac{g\alpha\Delta
TL^2h}{\nu\kappa}$, $L$ being the  lateral size of the system. The
equations are valid in the limit of infinite  Prandtl number
$\frac{\nu}{\kappa} \to\infty$. In the reactive case this
incompressible velocity is coupled to a (\ref{mainmod}) equation. In
the  rotating convection model the velocity is computed from the
local temperature by a similar quasi-steady law, but has an
additional Ekman number dependent factor due to the Coriolis force.
Rotation enhances stratification and has a nontrivial effect on heat
transport. The three models above are some of the simplest
incompressible, physically  relevant stratified active
reaction-diffusion-advection equations. The questions formulated in
the section concerning the passive advection cases are widely open for these
active models, but they are amenable to computational and analytical study.

\subsection{Why Is a Group Effort Required?}

The senior personnel involved in this project come from mathematics,
astrophysics, and computer science; in addition they have considerable
experience in computational physics. Expertise in each of these
areas is essential in carrying out our program. The basic questions
that we would like to answer are motivated from astrophysics;
astrophysics defines what questions are physically interesting, and
motivates the physical circumstances in which these questions are to
be answered. The modalities of knowing in which we want to
express our results involve numerical simulation and traditional
definition/theorem/proof mathematical analysis. The group of senior
personnel thus seems to represent well the called-for skill set,
which is rarely (if ever) found in any single investigator.

We have proposed having three postdocs, each of whom is only half-time
on this project. To zero-th order the postdocs will be from
mathematics, computer science, and (astro)physics. Again the mix of
skills that these individuals represent seems an appropriate match to
the proposed work.

\subsection{Timeline of The Project}

In year one we expect to focus on Boussinesq problems, these
provide feed back from the reaction to the flow, but with
as tame a mechanism as possible. In year two we will work
with anelastic fluids in which the densities are almost constant,
pre- and post-burn. In year three we will be working primarily
on examples in which the fluids are highly compressible. In
particular in year three we expect to consider ionization waves
passing through very thin interstellar gas.



\subsection{Plans For Dissemination of Results}

In addition to the standard mechanisms of publication in the public
literature and presentation at conferences, we expect to produce a
collection of documented examples of flame propagation that will
become part of the distribution of the University of Chicago FLASH
Project Code; we already know this to be valuable since the number of
examples that can be used to validate simulation studies of
combustion is small. Further, since we operate in the mode of ``open
software", our results can be expected to be used by other
simulation-oriented research groups; indeed we encourage such use by
external scientitsts.


\subsection{Results from Prior NSF Funding}
% Has there been nsf funded stuff that is related to this.
% my guess is that the answer is ``yes'' and that the
% results are already discussed above. That seems fine,
% but it seems that here we should point out the grants
% and the PI's involved. tfd/20sep01-11am

There are no NSF-funded results by the  personnel on
this proposal that are directly related
to the work that we are planning to do. Most of our
funding has been from the DOE.


\section{Modes of Collaboration}
% ii) Modes of Collaboration and Training. The following components, not to
% exceed an additional five pages total, are optional and can be included if
% appropriate:
%
%    * A description of new modes of collaboration.
%    * A description of new modes of training graduate students, postdoctoral
%      researchers, or undergraduates.
%    * A description of planned workshops and a list of tentative
%      participants.

Toward the end of the first year of this project a small (30-50
attendees) workshop, lasting one week, will be held here at the
University of Chicago. We have substantial experience with such
workshops, which are aimed at both the principal scientists working
in the specific field (here, premixed combustion science) as well as
at students and postdocs. (Our past narrowly-defined workshops at
Chicago occasionally lead to summary monographs in the subject in
question, which allow for yet greater dissemination of our results.)

% Are there other things that we want to say here?
% Do we try to bring in the collaborations with the DOE labs?
% We don't even have to include this section.

\section{Management Plan}
% iii) Management Plan. Provide a management plan, describing how the group
% effort will be coordinated and how decisions will be made regarding the
% conduct of the project. This section may not exceed one page.

A project of the type proposed here does not require much formal
management. We expect that the Co-PI's will meet quarterly to
discuss appointment and financial issues. However, the co-ordination
of technical work is an almost continuous process. There will be
weekly meetings of all the project participants to review progress
and discuss plans; many of those involved in this proposal have
been involved in collaborative projects in which this means of
technical co-ordination has been used. We find it works well.
The ``weekly meetings'' don't take place every week because
of holidays, conferences, etc., but we find that they actually
happen about 75\%  of the time.

The group of Co-PI's has considerable experience in management
and administration, and that should help in seeing to it that the
project runs smoothly and effectively.  Rosner and Dupont have served
as chairs of departments,  Rosner is currently the director of the
DOE-funded ASCI FLASH project, and Rosner has, as his CV shows,
directed other projects as well.

\end{document}