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\markboth{\hfil Calculations of the hurricane eye motion
\hfil EJDE--2002/??}
{EJDE--2002/??\hfil Vladimir Danilov, Georgii Omel'yanov, \& Daniil Rozenknop
\hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc Electronic Journal of Differential Equations},
Vol. {\bf 2002}(2002), No. 16, pp. 1--17. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu (login: ftp)}
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Calculations of the hurricane eye motion based
on singularity propagation theory
%
\thanks{ {\em Mathematics Subject Classifications:} 35D99, 86A10.
\hfil\break\indent
{\em Key words:} asymptotic behavior, forecast, hurricane.
\hfil\break\indent
\copyright 2002 Southwest Texas State University. \hfil\break\indent
Submitted November 7, 2001. Published February 18, 2002.} }
\date{}
%
\author{Vladimir Danilov, Georgii Omel'yanov, \& Daniil Rozenknop}
\maketitle
\begin{abstract}
We discuss the possibility of using calculating singularities
to forecast the dynamics of hurricanes.
Our basic model is the shallow-water system.
By treating the hurricane eye as a vortex type singularity and
truncating the corresponding sequence of Hugoniot type
conditions, we carry out many numerical experiments.
The comparison of our results with the tracks of three actual
hurricanes shows that our approach is rather fruitful.
\end{abstract}
\newcommand{\Hess}{\mathop{\rm Hess}\nolimits}
\section{Introduction}
In this paper we discuss a possibility of using an approach
related to calculating singularities
for numerical modeling the dynamics of hurricanes.
It is well known that for a detailed mathematical description of
large-scale and meso-scale processes in the atmosphere one needs
to use very complicated systems of nonlinear partial
differential equations based on equations of three-dimensional
gas dynamics (e.g., see [1--5]).
So far it is impossible to solve such systems numerically in
real time, therefore, one must use different
simplifying assumptions.
The first simplification is to neglect viscosity and
heat conduction effects. As a result, the order of equations
decreases and the problem of posing boundary conditions
disappears.
Further possible simplifications
(neglect of vertical displacements, heat exchange effects, etc.)
lead to comparatively simple models, the simplest of which is
the so-called shallow-water system
\begin{eqnarray}%1
\frac{\partial U}{\partial t}+\langle U,\nabla\rangle U+\nabla z
&=&f\Pi U,\\
\frac{\partial z}{\partial t}+\langle \nabla, zU\rangle&=&0.\nonumber
\end{eqnarray}
Here $U=(u,v)$ is the vector of horizontal velocity,
$\Pi$ is the matrix of rotation through $\pi/2$
($\Pi_{11}=\Pi_{22}=0$, $\Pi_{12}=-\Pi_{21}=1$),
$f$ is the Coriolis parameter,
and~$z$ is the geopotential.
At the same time, if we neglect dissipative effects, then the
solution loses its smoothness.
The fact that the solution is singular,
allows us to calculate its dynamics by using rather powerful
tools developed within the framework of the theory of
generalized functions.
Roughly speaking,
these tools allow us to avoid finding the solution
in the entire range of variables
and thus only to determine the dynamics of the singularity support.
Besides of a natural simplification due a decrease in the
dimension of the problem,
this approach allows us to avoid a very difficult problem of
choosing the initial data in the entire range of spatial
variables.
Namely, existing monitoring facilities do not
allow one to determine the initial distributions of velocity,
density, and temperature with sufficient accuracy for
large-scale formations.
Simultaneously,
the hurricane trajectory (the singularity support) can be fixed
well by satellite imaging.
V.~P.~Maslov [6] proposed the hypothesis that a solution with a
weak singularity whose singular support is of codimension~2
corresponds to the center of a hurricane.
Such solution admits the representation
\begin{equation}%2
U=U^0(x,t)+\sqrt{S(x,t)}U^1(x,t),
\quad z=z^0(x,t)+\sqrt{S(x,t)}z^1(x,t),
\end{equation}
where $U^i$, $z^i$ and $S$ are smooth functions and
\begin{equation}
S\geq0,\quad \nabla S\Big|_{S=0}=0,\quad \Hess S\Big|_{S=0}>0.
\end{equation}
To find the trajectory $\Gamma=\{(x,t),S(x,t)=0\}$ of the
singularity support, i.e., of a moving point,
we substitute a solution of the form (2) into the equation
(e.g., into (1)), which leads to the relation
\begin{equation}
D^0+S^{-1/2}D^1=0,
\end{equation}
where $D^i$ are smooth functions.
In turn, (4) implies the relations
\begin{equation}
\frac{\partial^{|\alpha|}}{\partial x^\alpha}D^i\bigg|_{\Gamma}=0,\quad
|\alpha|=0,1, \dots, \quad i=0,1,
\end{equation}
which lead to necessary conditions for the existence of a
solution of the form~(2).
Maslov called these conditions {\em Hugoniot type conditions\/}
(by analogy with the Hugoniot condition for shock waves).
The above scheme was realized by Maslov and Zhikharev [6, 7]
for the
shallow-water system without Coriolis effects and by group of
authors [8] for the system~(1) with the Coriolis
force.
Hugoniot type conditions form an infinite
non-triangular system.
The first 14 equations of them have the form [8]
\begin{eqnarray}
\dot z^0_0&=&-2q z^0_0,\nonumber\\
\dot V_1&=&fV_2- z^0_{10},\nonumber\\
\dot V_2&=&-fV_1- z^0_{01},\nonumber\\
\dot z^0_{10}&=&-3qz^0_{10}+pz^0_{01}-z^0_0(v^0_{11}+2u^0_{20}),\nonumber\\
\dot z^0_{01}&=&-3qz^0_{01}-pz^0_{10}-z^0_0(u^0_{11}+2v^0_{02}),\nonumber\\
\dot q&=&-q^2+p^2-fp-2r,\nonumber\\
\dot p&=&-2pq+fq,\\
\dot r&=&-4qr-z^0_{10}(3u^0_{20}+v^0_{11})-z^0_{01}v^0_{20}
-\{z^0_0(3u^0_{30}+v^0_{21})\},\nonumber\\
\dot u^0_{20}&=&-3qu^0_{20}+p(u^0_{11}-v^0_{20})+fv^0_{20}-\{3z^0_{30}\},\nonumber\\
\dot u^0_{11}&=&-3qu^0_{11}+p(2u^0_{02}-2u^0_{20}-v^0_{11})+fv^0_{11}
-\{2z^0_{21}\},\nonumber\\
\dot u^0_{02}&=&-3qu^0_{02}-p(u^0_{11}+v^0_{02})+fv^0_{02}-\{z^0_{12}\},\nonumber\\
\dot v^0_{20}&=&-3qv^0_{20}+p(v^0_{11}+u^0_{20})-fu^0_{20}-\{z^0_{21}\},\nonumber\\
\dot v^0_{11}&=&-3qv^0_{11}+p(2v^0_{02}-2v^0_{20}+u^0_{11})-fu^0_{11}
-\{2z^0_{12}\},\nonumber\\
\dot v^0_{02}&=&-3qv^0_{02}-p(v^0_{11}-u^0_{02})-fu^0_{02}
-\{3z^0_{03}\}.\nonumber
\end{eqnarray}
Here $V=(V_1,V_2)$ is the velocity of the singularity support
$x=a(t)$,
$q=u^0_{10}=v^0_{01}$, $p=u^0_{01}=-v^0_{10}$,
$r=z^0_{20}=z^0_{02}$,
and $u^0_\alpha, v^0_\alpha, z^0_\alpha$ are coefficients of the
expansion of the solution~(2) in a neighborhood of the singularity
support, i.e., $U^i=(u^i, v^i)$, $i=0,1$,
$$
u^i=\sum^\infty_{k=0}\sum_{|\alpha|=k}u^i_\alpha(t)\big(x-a(t)\big)^\alpha,
\quad
v^i=\sum^\infty_{k=0}\sum_{|\alpha|=k}v^i_\alpha(t)\big(x-a(t)\big)^\alpha,
$$
$$
z^i=\sum^\infty_{k=0}\sum_{|\alpha|=k}z^i_\alpha(t)\big(x-a(t)\big)^\alpha.
$$
The truncation of the sequence at the 14th term means that we neglect
the terms in the braces in~(6).
A comparison of numerical results with the actual track
of the hurricane FORREST (21/09--31/09/1983, the Pacific Ocean)
shows that there is a qualitative coincidence between these
trajectories~[8].
It was also shown that, besides (2), Eqs. (1) do not have any other singular
solutions with pointwise support of the singularity [9] and that
the truncated system~(6)
can be reduced to the Hill equation [10].
These results stimulated us to try to forecast the dynamics of
hurricanes. It should be noted that we consider not simply a
nonsmooth solution
from a Banach space but a solution of some special structure
and try to calculate the trajectory of motion its singularity.
It is impossible to ``catch'' this solution by traditional
methods for studying PDE.
The authors developed a special method for estimating the error
of the asymptotic with
respect to smoothness solution. By using this method,
the simplest version of the shallow water
equations, i.e., the Hopf equation, has been studied [11].
Even in this special case, the estimation of the remainder
turned out to be a very nontrivial problem.
For the shallow water equations, this estimation must be more
difficult.
On the other hand, the obtained asymptotic solution
could be compared with the results of the direct numerical
computation for the shallow water equations.
However, here we have a very complicated problem of setting
the boundary conditions corresponding to the hurricane problem,
which are necessary for numerical computations.
Therefore, we decided to omit all stages that are traditional
in the mathematical study
and to test the constructed asymptotic solution
by using the forecasted hurricane motion.
We present the results of
numerical calculations for three actually existing hurricanes
(there we used the information about hurricane tracks delivered
by the National Hurricane Center, USA).
All results obtained can be judged as follows:
the present approach is reasonable and competent and allows us
to obtain a sufficiently good short-term forecast
(not less than 24 hours).
However,
system~(6) cannot be used for a long-term forecast.
The results obtained allow us to assume that, most
likely, this fact is related to defects of model~(1) used here.
Thus there is the problem of choosing an initial model that is
more adequate than model~(1), for which we need to
construct a sequence of Hugoniot type equations
and to carry out the corresponding numerical
experiments.
This research was partially supported by the Russian Foundation for
Basic Research, grants No.~99-01-01074 and No.~01-01-06057.
\section{Numerical calculations of Hugoniot type equations.
Long-term forecast}
By truncating the system (6),
we reduce the problem of calculating the dynamics
of the hurricane center to the problem of solving a system
consisting of 14 ordinary differential equations.
There is the principal problem of choosing the Cauchy data.
Our main idea is to choose the Cauchy
data so that
the trajectory calculated by the truncated system~(6) be
maximally close to a given track of an actual hurricane
during some time interval $[0,t_{K+1}]$.
Obviously,
by studying the corresponding solution of~(6) for $t>t_{K+1}$,
we can forecast the further motion of the hurricane center.
Let us describe the algorithm that realizes this idea.
By $\xi(t)=(\lambda(t),\varphi(t))$, $t\in[0,t_I]$,
we denote the trajectory of an actual hurricane.
Let $\xi_i=\xi(t_i)$ be the coordinates of the hurricane center
at time $t_i=(i-1)\Delta t$, where
$i=1,\dots,I$ is the measurement number and
$\Delta t$ is the time interval between successive measurements
(usually, $\Delta t=6$\,hrs.).
By $\mu^0=(z^0_0\big|_{t=0},\dots,v^0_{02}\big|_{t=0})$ we denote
the set of initial data for system~(6)
and by $\mu(t_i,\mu^0)$ the solution of the Cauchy problem
for~(6) with the initial data~$\mu^0$ calculated at time~$t_i$.
Here and in the following,
system~(6) means that the system is truncated.
We choose a number $K*t_{46}$ (see Fig.~12).
Curiously enough,
all these forecasts show that the hurricane makes a turn
at a longitude of $27^\circ$--$33^\circ$,
although the beach was not taken into account in model~(1).
So, we see that our {\it a priori\/} assumption that model~(1)
and system~(6) are sufficiently adequate for forecasting the
hurricane GEORGES is true.
\begin{figure}[t]
\centering
\includegraphics[width=0.7\textwidth]{fig19.eps}
\caption{\small $\xi$ is the track of NICOLE starting
at 09:00 a.m. 25/11 1998,
$\zeta$ is the artificial trajectory.
Initial data: 8.12 $10^{-5}$, ${}-2.64$, 1.33 $10^{-2}$,
0.10, 4.75,
2.22 $10^{-2}$, 0.165, 4.22 $10^{-9}$, ${}-9.8 10^{-16}$,
${}-9.8 10^{-16}$,
${}-9.8 10^{-16}$, ${}-9.8 10^{-16}$, ${}-9.8 10^{-16}$,
${}-9.8 10^{-16}$, K=3, M=1.
Distances for approximation: 0, 0.35, 0.31, 0.28.
Distances for forecast: 0.32, 1.08, 1.82, 2.77, 4.26, 6.02, 8.19, 10.8, 13.9.
}
\end{figure}
2. The behavior of the hurricane NICOLE was affected by powerful
external factors.
This hurricane, first observed on 24/11/1998
about 700 miles to the West of the Canary
Isles, initially moved to the South--West and then turned to the
West.
About 96\,hrs. later, the hurricane sharply changed the
direction of its motion and moved to the Northeast.
Next, about 30\,hrs. after the turn,
the trajectory of the hurricane started to
bend in the direction opposite to the action of
the Coriolis force.
Because of such behavior of NICOLE,
one can hardly expect that model~(1) can describe
the actual trajectory.
Nevertheless,
the numerical experiments performed
show that our algorithms provide a sufficiently good forecast
for 24\,hrs.
Figure~13 shows the artificial trajectory calculated
for $K=3$ and $M=1$, which corresponds to the forecast
for $t>t_4=18$\,hrs.
We see that the artificial and actual trajectories agree
sufficiently well until the turning point of the actual
hurricane.
\begin{figure}[t]
\centering
\includegraphics[width=0.7\textwidth]{fig20.eps}
\caption{\small Initial data: 8.12 $10^{-5}$, ${}-4.21$, 9.50 $10^{-3}$,
0.195, 2.77,
3.54 $10^{-2}$, 0.321, 4.22 $10^{-9}$, ${}-9.8 10^{-16}$,
${}-9.8 10^{-16}$,
${}-9.8 10^{-16}$, ${}-9.8 10^{-16}$, ${}-9.8 10^{-16}$,
${}-9.8 10^{-16}$, K=3, M=9.
Distances for approximation: 0, 0.03, 0.09, 0.22.
Distances for forecast: 0.41, 0.45, 0.75, 0.23, 1.44, 3.26, 5.86, 10.0, 16.3.
}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.7\textwidth]{fig21.eps}
\caption{\small Initial data: 8.12 $10^{-5}$, ${}-2.15$, ${}-1.56$,
4.47, ${}-8.10$,
0.233, 0.299, 4.84 $10^{-11}$, ${}-8.22 10^{-16}$,
${}-10^{-16}$,
${}-10^{-16}$, ${}-10^{-16}$, ${}-10^{-16}$,
${}-10^{-16}$, K=3, M=15.
Distances for approximation: 0, 0.56, 0.79, 0.84.
Distances for forecast: 2.37, 3.91, 6.30, 7.84, 10.9, 13.7.
}
\end{figure}
The calculations of the next sloping part of the trajectory
($K=3$, $M=9$, see Fig.~14)
yield extremely good qualitative forecast for 24\,hrs.
($\Delta_{16}\approx 23$\,km).
Nevertheless,
the artificial hurricane still does not forecast the turn of the
trajectory, and therefore,
the error of the forecast for the second 24\,hrs.
increases to $\Delta_{19}\approx590$\,km, while
$\Delta_{20}\approx 1000$\,km.
One can hardly say that the forecast is successful
in the region containing the turning point of
NICOLE (see Fig.~15).
The initial error is $\Delta_{18}\approx84$\,km for $t_{18}$,
then it increases approximately by 200\,km every 6\,hrs.,
so that we obtain $\Delta_{22}\approx780$\,km after 24\,hrs.
This failure can easily be explained, since the forecast shown
in Fig.~15 is based on the hurricane trajectory till the turn
at~$t_{17}$.
\begin{figure}[t]
\centering
\includegraphics[width=0.7\textwidth]{fig22.eps}
\caption{\small Initial data: 8.12 $10^{-5}$, ${}-6.63 10^{-3}$,
${}-0.891$,
0.126, ${}-12.6$,
0.363, 0.255, 4.84 $10^{-11}$, ${}-8.22 10^{-16}$,
${}-10^{-16}$,
${}-10^{-16}$, ${}-10^{-16}$, ${}-10^{-16}$,
${}-10^{-16}$, K=3, M=18.
Distances for approximation: 0, 0.09, 0.13, 0.83.
Distances for forecast: 1.24, 2.19, 3.43, 4.57, 6.32, 7.28.
}
\end{figure}
After the hurricane changes the direction of its motion,
the forecast quality improves to some extent.
Here a decisive factor is the fact that we forecast the motion
of NICOLE on the basis of its trajectory after the turning point.
The plot in Fig.~16 ($K=3$, $M=18$) shows that the trajectories
are qualitatively close to one another until the latitude
$\approx 37^\circ$ is achieved.
Next, under the action of the Coriolis force,
the artificial hurricane continues to move to the East,
while
the actual hurricane moves to the Northeast.
An attempt to improve the situation by using a longer prehistory
made the forecast quality even worse.
Similar pictures are observed for the forecasts starting
from~$t_{24}$ and from $t_{28}$:
the actual and artificial hurricanes still behave
in qualitatively different ways.
\begin{figure}[t]
\centering
\includegraphics[width=0.7\textwidth]{fig28.eps}
\caption{\small Probability that center of NICOLE will pass within
75 miles during the 72 hours starting at 10:00 a.m. 30/11 1998
(the National Hurricane Center, USA)
Contour levels shown are 10\%, 20\% 50\%, and 100\%.
$\xi$ is the track of NICOLE, curve 1 is the solution of (6)
for K=3 and M=18, curve 2 is the solution of (6) for K=3 and M=25.}
\end{figure}
\section{Conclusions}
The analysis of numerical results
shows that, on all parts of the hurricane track
corresponding to more or less stable external factors,
the artificial trajectories
calculated by the truncated system~(6)
(with the initial data corresponding to this particular part)
qualitatively and quantitatively coincide to a satisfactory
extent with the actual trajectories.
This coincidence is rather close in the low latitudes
and becomes worse as the hurricane moves to the North.
Both facts correspond to an {\it a priori\/} analysis of whether
model~(1) is adequate.
Therefore, we can draw the conclusion that the truncated system~(6)
possesses sufficiently good approximating properties.
This conclusion holds for a time interval of several days.
However,
to use the truncated system~(6) on larger time intervals
is rather problematic.
The plots in Figs.~4,~5 clearly demonstrate that there are
restrictions on the long term applicability of this system.
The trajectory ``breakdown''
(after the calculation time $\approx 150$\,hrs.)
is closely related to an increase in the error arising due to
the truncation of the infinite system of equations.
Similar ``breakdowns'' can be seen in other figures,
however,
they occur at a considerable distance from the trajectory of the
actual hurricane.
Thus the error due to the truncation manifests itself after
sufficiently large time and is unessential for short-term and
medium-term forecast.
Next, numerical results obtained do not allows us to hope that a
good forecast can be obtained by using~(6) in the case of a
sharp change in the trajectory.
The hurricane NICOLE gives the most illustrative example of
this fact.
There is no doubt that this fact is closely related to the
defects of the initial model~(1).
In order to estimate the results of numerical experiments,
let us compare the artificial trajectory of NICOLE
with the probability forecast made
by the National Hurricane Center at 10:00 a.m. 30/11/1998.
This forecast was made almost at the same time as our forecast
shown in Fig.~16, since the choice $K=3$, $M=18$ implies
the absolute time $t_{21}=\mbox{09:00}$ a.m. 30/11/1998.
The comparison of the probability forecast
with the actual NICOLE trajectory for $t>t_{21}$ (see Fig.~17)
shows
that during 36\,hrs. the actual hurricane approaches
the 20\% region and then, after 3\,days, enters the 10\% region.
This means that the predicted probability $p=0.8$
(that the eye of NICOLE stays in the 20\% region during 72\,hrs.)
is too excessive.
Only the probability $p=0.9$
(that NICOLE stays in a considerably larger 10\% region)
is adequate.
The forecast obtained by using system~(6)
(for $K=3$, $M=18$, curve~1 in Fig.~17) predicts that
the trajectory leaves the 20\% region during 36\,hrs.
In this sense
our forecast is even better than the professional forecast.
However, as shown above,
the artificial trajectory qualitatively differs
from the actual trajectory.
We can partially improve this forecast
by using the dynamical correction
(curve~2 in Fig.~17).
Thus,
although the basic model~(1) is rough,
the quality of our forecast is comparable (even if somewhat
worse) with the quality of the professional forecast.
All numerical experiments resulted in
a sufficiently good forecast for 24--48\,hrs.
Apparently,
the forecast of extremely high quality in Fig.~4
(more than for 5\,days)
is accidental.
Nevertheless,
several successful medium-term forecasts,
including the 3-day forecasts (Figs.~3,~11)
make us optimistic.
Finally, the results obtained show that
the following fundamental hypotheses hold:
\begin{itemize}
\item
Methods of the theory of generalized functions
can be used for calculating the dynamics of a hurricane;
\item
A hurricane can be treated as a weak pointwise singularity;
\item
The singularity dynamics can be calculated by using a truncated
sequence of Hugoniot type conditions.
\end{itemize}
It is highly probable that we can improve
the quality of short-term and medium-term forecasts
by choosing a basic model that is more adequate than
model~(1).
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\noindent\textsc{Vladimir Danilov} (e-mail: pm@miem.edu.ru)\\
\textsc{Georgii Omel'yanov}\\
\textsc{Daniil Rozenknop}\\[3pt]
Moscow State Institute of Electronics and Mathematics, \\
B. Trekhsvyatitel'skii per., 3/12, \\
Moscow 109028, Russia
\end{document}
*