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{\bf Research Proposal}\\
Roger Thelwell ({\tt thelwell@math.colostate.edu})
\end{center}

Given a velocity field, it's fairly easy to produce a matching 
pressure field.  The inverse problem is more difficult. It is this 
inverse problem that I'd
like to work on.  
I'd like to to recover accurate velocity measures from a
pressure snapshot, or series of snapshots.  


{\it \bigskip Can a (unique) velocity field be
recovered that matches a known pressure gradient field? \bigskip }

Specifically, I'd like to find a means to recover hurricane wind velocities 
from remotely sensed temperature data.  Currently, accurate estimates are 
available only through aircraft and surface observations. As a sailor, I
have a great interest in this topic, as do my friends still in the
Caribbean. Theoretically, this project might provide a deeper
understanding of cyclone dynamics, particularly in the spatial region 
containing the maximal pressure gradient. This understanding could
lead to robust numerical methods of Potential Vorticity (PV) inversion.  

I worked for six months on this problem as my masters
project for my MS in math  with Drs. Wayne Schubert and Mark DeMaria, both 
associated with the Atmospheric Science Department at Colorado State
University.  I'll be graduating with a PhD in applied
math in the spring of 2004.  
I've been intrigued by this question for several years, but haven't 
had the opportunity to apply several new ideas to
develop a more complete mathematical understanding of the balance
equation.

\section{Introduction}

The full set of atmospheric equations are used to independently
predict mass and
 momentum fields. There is a diagnostic relationship between these two
quantities in some cases.  The {\it geostrophic wind} is one such example, and
is fairly accurate for synoptic scale motion away from the equator.  The
earliest numerical weather models were created for large scale 
mid-latitude flow, and so the simplifying assumptions incorporated 
into these models caused little problem. As computer power increased 
and domains began to include the tropics, more general balance 
conditions were needed.  The {\it momentum equations} (ME) and
the {\it nonlinear balance equation} (NLB) are models thought to accurately
describe cyclonic flow. 

On board several NOAA satellites, the Advanced Microwave Sounding Unit 
(AMSU) is able to produce microwave radiation data that in turn is used 
to create high resolution temperature profiles on multiple levels of the 
atmosphere.  Current temperature plots are produced with a resolution of 
as fine as $40 \ km^2$, a great improvement of earlier (MSU) sensors.  
This highly refined data set might allow accurate wind fields to be found 
in areas that are inaccessible to other sensing methods.  

\section{Mathematical formulations} 

Consider the quasi-static primitive equations on an $f$-plane.
The momentum equations are 
\begin{eqnarray}
\label{meu}
\p_t u + u \p_x u + v \p_y u - fv + \p_x \Phi = 0\\
\label{mev}
\p_t v + u \p_x v + v \p_y v + fu + \p_y \Phi = 0. 
\end{eqnarray}

Computing the curl of this system produces the divergence equation,

\begin{equation}
\label{mediv}
\frac{D}{Dt} \delta + \delta^2 -2 \frac{\p(u,v)}{\p(x,y)} - f \zeta +
 \nabla^2 \Phi = 0 
\end{equation}
where $\delta := \p_x u + \p_y v$ is divergence, and $\zeta := \p_x v -
\p_y u$ is vorticity.

Define the horizontal wind in terms of the streamfunction
$\psi$ and velocity potential $\chi$, such that
\begin{equation}
\label{psichi}
u := -\p_y \psi + \p_x \chi
\quad \mbox { and } \quad v := \p_x \psi + \p_y \chi.
\end{equation}
Recasting the divergence equation in a streamfunction form and
putting all terms involving divergence in on the right produces the
nonlinear balance equation 
\[
2(\p_{xx} \psi \p_{yy} \psi - (\p_{xy}\psi)^2) + f \nabla^2 \psi -
\nabla^2 \Phi = F.
\]
 
It can be shown there are at most two solutions to 
nonlinear balance equation boundary value problem if 
$$\nabla^2 \Phi + F > - \frac{ f^2}{2}.$$
One solution corresponds to positive absolute vorticity and the second to
negative absolute vorticity.
The {\it realizability condition} can now be stated.  It is
\begin{equation}
\label{realiz}
\nabla^2 \Phi + \frac{f^2}{2} + F > 0.
\end{equation}
Neglecting divergent terms by assuming that $F := 0$, the realizability 
condition reduces to the {\it ellipticity condition}
\begin{equation}
\label{ec}
\nabla^2 \Phi + \frac{f^2}{2} > 0.
\end{equation}

\section{Motivation}

If the {\it realizability~} and/or {\it ellipticity~} conditions are met, 
then there are several numerical schemes that can recover velocity from 
pressure gradient information. As the resolution of remote sensing devices
has improved, however, there have been an increasing number of examples
where these conditions have been violated. Notably, this occurs in
strong cyclonic flows in regions of large pressure gradient.
While data smoothing  `fixes' these regions, a real question remains - Does the
solution to the smoothed problem reasonably approximate the true
solution?

Tribbia (1982) explores the failure of nonlinear normal-mode
initialization.  The model discussed describes the flow of a rotating 
fluid in a right-circular cylinder.  The upper surface is free and 
the fluid is assumed to have uniform density.  The shallow water 
equations are used, and a low order spectral
model is used to linearize the system.  The solution to the iterative scheme
is seen to be exactly the discrete gradient wind balance.  It appears from
Tribbia's analysis that there exists a true threshold for a real balancing 
wind to exist.  The pressure gradient is only able to balance a certain 
portion of the centripetal and Coriolis forces.  Once this maximum is 
reached, the mathematical model breaks down, and this is reflected  
in non-convergence of the scheme.  He makes the following three conclusions:
\begin{enumerate}
\item
There are limits on geostrophic vorticity (and therefore rotational
mode
amplitude).  Unconstrained balancing breaks down at this point.
\item
This limit corresponds to realizable wind field (the ellipticity
condition in the NLB).
\item
Current iteration methods work in cases where a real balance wind
field exists.
\end{enumerate}

This work would suggest that the search for stable numerical methods
for the normal mode initialization (and, similarly, for the nonlinear balance
equation) should remain restricted to {\it realizable} domains.  Regions of
non-ellipticity correspond to regions of non-convergence.  It appears from these low order
spectral methods that the transition from elliptic to hyperbolic regions violate
the physical constraints of physical balance in the shallow water system, and
also produces a numerical instability in the nonlinear normal mode balance scheme.
The existence of non-elliptic ground data would suggest that these models require
additional features in order to more accurately represent the true physics of
flow in regions of non-ellipticity.

The current PV inversion schemes based on the
streamfunction formulation of the divergence equation typically
require 
balanced fields, no PV anomalies, for convergence. By the numerical  
addition of PV, the problem becomes increasing elliptic and therefore
easier to solve.   

\section{Research Proposal}

{\it Can remote sensing provide enough information to recover accurate wind
velocity estimates?}

\noindent 
I believe that this answer is {\bf YES}.  

I suspect that earlier methods based on the nonlinear balance equation will
prove to be infeasible. The momentum equations, since they are quasi-linear,
might provide a better foundation for future research. I also suspect
that `non-ellipticity' classification is a feature of the 2d model
that might fail to persist in a full 3d time dependent flow.  I suspect
that incorporating vertical flow might help explain the difficulty of
`non-elliptic' (`hyperbolic' in the time dependent form) regions. When
the flow becomes hyperbolic, that is when the pressure gradient can no
longer contain the rotational flow, traveling waves are formed.  Ignoring
the vertical scale flow means that these waves must leave the computation
region. But perhaps the $z$ component of the pressure gradient is less
than the horizontal component, and therefore would allow the flow to 
`escape' vertically. In mesocale modeling, the slow mode is often
considered to be the important physical flow, while the fast mode is treated
minimally.  Perhaps the fast mode doesn't `propagate to infinity'
but instead is captured by the system.  
 
A 3d cylindrical formulation might allow these conjectures to be analyzed.  
I'd like to work with three sets of equations:

\begin{itemize}
\item Navier-Stokes 
\item Momentum 
\item Nonlinear Balance  
\end{itemize}

I propose to work on several aspects of this problem, and their
application to PV inversion schemes.

\begin{enumerate}

\item 
Survey of current cyclonic flow models. Do these produce `non-elliptic'
regions? 

\item 
Validate the Navier-Stokes, momentum and nonlinear balance systems using available ground data.

\item 
If necessary, use scale analysis to test that the momentum equations and
nonlinear balance are feasible approximations.  Explore both 2d and 3d
formulations; cartesian, polar and cylindrical.

\item
Collect and analyze physical `non-elliptic' data (ie Paegle 83). Is this a 2d phenomenon, or 3d? Compare to simulation data.  Consider both a radially 
symmetric flow, which would allow a dimension reduction, as well 
as non symmetric flows.

\item 
Linearize both the momentum equations and the nonlinear balance
equation around a known velocity field (both  `elliptic' and
`non-elliptic', and solve the resulting linear convection-diffusion system analytically. Do these differ?  If so, how?   

\item
Explore numerical techniques for the linear and nonlinear recovery in 2 and 3
dimensions. 
Output Least Squares (OLS) techniques (coupled with regularization) should be feasible, but robustness and accuracy is a concern.  Also, minimization doesn't
provide access to the true dynamics.  An OLS solution to a smoothed
system should be a good initialization to other types of numerical methods,
ie Finite Difference evolution schemes.
For this phase, use simulation data initially, then move to AMSU data.

\item
Apply numerical methods to PV inversion schemes in regions containing
PV anomalies, including highly convective regions.

\item
A bifurcation occurs in the nonlinear balance equation.  In the
literature, this the point where `ellipticity' or `realizability' condition is violated, and amounts to a region where the prescribed pressure gradient
(the bifurcation parameter in this case) can no
longer balance the velocity field. Do the momentum equations exhibit
a bifurcation as well? Numerical continuation techniques can
be used to explore this parameter space.   

\item (possible)
Consider existence and uniqueness for 2d and 3d flow regime for
with a prescribed forcing (pressure gradient).
A survey of work on Taylor-Couette flow might provide insight.     

\end{enumerate}


\begin{thebibliography}{333}

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Courant, R. and Hilbert,D., 1962: {\emph Methods of Mathematical Physics, volume 2,} Wiley and Sons

\bibitem{k82}
Kasahara, A., 1982: Significance of non-elliptic regions in balanced flows of the tropical atmosphere. {\it Mon. Wea. Rev.} {\bf 110}, 1956-1967

\bibitem{l80}
Leith, C. E., 1980: Nonlinear normal mode initialization and the quasi-geostrophic theory. {\it J. Atmos. Sci.}, {\bf 37}, 958-968

\bibitem{m56}
Miyakoda, K. 1956: On a method of solving the balance equation. {\it J. Meteor. Soc. Japan} {\bf 34} 364-367

\bibitem{m68}
Miyakoda, K. and Moyer, R., 1968: A method of initialization for dynamical weather forecasting. {\it Tellus} {\bf XX, 1} 115-127

\bibitem{p83}
Paegle, J., J.N. Paegle, and Dood G., 1983: On the occurrence of atmospheric states that are non-elliptic for the balance equation. {\it Mon. Wea. Rev.} {\bf 111}, 1709-1723

\bibitem{s57}
Schuman, F., 1957: Numerical methods in weather prediction: the balance equation, {\it Mon. Wea. Rev} {\bf 85:10} 329-332

\bibitem{t81}
Tribbia, J., 1982: Nonlinear normal-mode balancing and the ellipticity condition. {\it Mon. Wea. Rev.} {\bf 109}, 1751-1761

\bibitem{z00}
Zhang, F., S. Koch C. Davis and M. Kaplan, 2000: A survey of
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Atmospheric Sciences.} {\bf 17}, 165-183. 
 
\end{thebibliography}

\end{document}

