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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)}
 \vspace{\bigskipamount} \\
 %
  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 $K<I$ and carry out the following calculations.
\begin{enumerate}
\item %{\bf1.} 
Let us  choose a point $\mu^0\in\mathbb{R}^{14}$. 
We solve  system~(6) numerically using the Runge-Kutta method 
of order 4 and form the set 
$\{\mu(t_i,\mu^0)=(z^0_i(\mu^0),
V_{1,i}(\mu^0),V_{2,i}(\mu^0),\dots )\}$, $i=1,\dots,K+1$.

\item %{\bf2.}
 We calculate the artificial trajectory of the hurricane:
$\{\zeta_i(\mu^0)=(\overline{\lambda}_i,\overline{\varphi}_i)\}$, $i=1,\dots,K+1$, solving 
numerically the equations
\begin{eqnarray}
\frac{d\overline{\lambda}}{dt}=V_1,\quad
\frac{d\overline{\varphi}}{dt}=V_2, \quad
\overline{\lambda}|_{t=0}=\lambda (0), \quad \overline{\varphi}|_{t=0}=\varphi (0).
\end{eqnarray}

\item %{\bf3.} 
We calculate the error function \begin{equation}
J(K,\mu^0)=\sum^{K+1}_{i=1}\big|\xi_i-\zeta_i(\mu^0)\big|^2.
\end{equation}

\item %{\bf4.} 
We find the value $\mu^0=\mu^0_K$ at which $J(K,\mu^0)$ attains its minimum. 
\end{enumerate}
In numerical experiments we, first of all, verify the following
hypothesis: 
if we increase the number~$K$ successively with the motion of
the hurricane,  then the trajectories $\zeta(t,\mu^0_K)$
obtained become closer and closer to the actual
trajectory~$\xi(t)$.  
In other words, the more hurricane prehistory we take into
account, the better we forecast the motion of the hurricane. 
Obviously, this hypothesis  assumes that there exist an initial
point $\mu^0_I$ such that the trajectories $\zeta(t,\mu^0_I)$
and $\xi(t)$ are close to each other on the entire time interval
$[0,t_I]$.


\begin{figure}[t]
\centering
\includegraphics[width=0.7\textwidth]{fig1.eps}
\caption{\small $\xi$ is the track of HORTENSE starting
at 12:00 a.m. 3/11 1996, $\zeta$ is the artificial trajectory. 
Initial data: 8.12 $10^{-5}$, ${}-0.373$, 3.46, 12.9, 20.3, 
1, 1.15 $10^{-3}$, 7.16 $10^{-11}$, ${}-8.22$ $10^{-16}$, $-10^{-16}$,
$-10^{-16}$, $-10^{-16}$, $-10^{-16}$, $-10^{-16}$.
Distances: 0, 0.99, 1.40, 1.61, 1.71, 2.04, 2.72, 3.11, 3.00, 3.07, 
3.85, 5.03, 6.16, 7.53, 8.18, 8.98, 9.94, 11.4, 12.8, 14.6, 16.5, 18.5, 
20.6, 22.7, 24.6, 26.3, 27.7, 28.7. }
\end{figure}

For a test example, we considered the hurricane HORTENSE
(03/09--16/09, 1996) and carried out numerous numerical
experiments.
The best result is shown in Fig.~1.
Here and in subsequent figures the initial data are listed in
the same order as the unknown functions in the left-hand side
in~(6), i.e., 
$z^0_0\big|_{t=t_1}=8.12\cdot 10^{-5}$, 
$V_1\big|_{t=t_1}=-0.373,\dots,
v^0_{02}\big|_{t=t_1}=-1\cdot10^{-16}$.
The unit of measurement of the distance on the
spherical surface is 100\,km.

One can readily see that, indeed,  the artificial curve~$\zeta$
is sufficiently close (from a qualitative geometrical viewpoint) 
to the actual trajectory~$\xi$ of the hurricane.
Nevertheless,
from the forecast viewpoint, 
the obtained ``optimal'' curve~$\zeta$ is absolutely
inconsistent, since, starting from the $14$th node,
the distances between $\xi(t_j)$ and $\zeta(t_j,\mu^0_I)$,
$j\geq 14$, are greater than $750$\,km. 
This effect is caused by the fact that the velocity of the
artificial hurricane is greater than the velocity of the actual
hurricane HORTENSE. 
Due to this fact, the distance $|\xi(t_j)-\zeta(t_j,\mu^0_I)|$
increases with~$j$.  
The dynamics of an increase in the error
$|\xi(t_j)-\zeta(t_j,\mu^0_I)|$ for HORTENSE is shown in the
comment to Fig.~1. 
Similar calculations for the hurricane GEORGES shown in Fig.~8
lead to the same result. 
Thus the above hypothesis is inconsistent and the probability of
an appropriate long-term forecast is very small. 



\section{Short-term forecast}

There is good reason to hope that a short-term forecast will be
successful, since all calculations carried out according to the
algorithm in Sec.~2 show that
if $|\xi(t_{K+1})-\zeta(t_{K+1},\mu^0_K)|$ is small,
then the distance between $\xi(t_{K+i})$ and
$\zeta(t_{K+i},\mu^0_K)$  for $i=4\div 8$
is comparable with the actual diameter of the hurricane eye
($i=4\div8$ means the forecast for $24\div48$\,hrs.).

To carry out a short-term  dynamical forecast, we must slightly
change the algorithm  in Sec.~2.
Namely,
we choose a time instant ~$t_M$ and write
$\xi_M=(\lambda(t_M),\varphi(t_M))$ as the coordinates
of the hurricane center for $t=t_M$.
We choose the number~$K$ so that $K+M\leq I$ and perform the
calculations similar to those in Sec.~2 but starting from $t=t_M$.

In contrast to the algorithm in Sec.~2,
the main advantage is that $\zeta_M=\xi_M$ for all~$t_M$.
This means that we do not extremely accumulate the error
arising in calculations of~$\zeta_{M+K}$.
Moreover, we partly take into account the latitude variation of
the Coriolis parameter by setting $f=2\sin\varphi(t_M)$ in
our formulas.

\begin{figure}[t]
\centering
\includegraphics[width=0.7\textwidth]{fig2.eps}
\caption{\small Initial data: 8.12 $10^{-5}$, ${}-5.14$, 
0.43, 6.43 $10^{-3}$, 5.57, 
0.26, 0.21, 4.46 $10^{-10}$, $-10^{-15}$, $-10^{-15}$,
$-10^{-15}$, $-10^{-15}$, $-10^{-15}$, $-10^{-15}$, K=3, 
M=1.
Distances for approximation: 0, 0.433, 0.505, 0.458.
Distances for forecast: 0.408, 0.958, 1.77, 2.27, 2.28, 2.52, 3.40, 
4.64, 5.82, 7.23.}
\end{figure}

\begin{figure}[t]
\centering
\includegraphics[width=0.7\textwidth]{fig3.eps}
\caption{\small Initial data: 8.12 $10^{-5}$, ${}-4.40$, 0.673, 
3.01 $10^{-3}$, 4.77, 
0.336, 8.22 $10^{-2}$, 5.34 $10^{-10}$, $-10^{-15}$, $-10^{-15}$,
 $-10^{-15}$,  $-10^{-15}$, $-10^{-15}$, $-10^{-15}$, K=3, 
M=9.
Distances for approximation: 0, 0.49, 0.32, 0.28.
Distances for forecast: 0.67, 1.32, 1.41, 1.43, 1.40, 1.67, 1.99, 
2.19, 2.50, 2.92, 3.74, 4.70, 5.47, 6.09, 6.65, 6.64, 7.05.}
\end{figure}

Let us discuss the results of numerical experiments (Figs.~2--7)
for the hurricane HORTENSE.
In order to study the results obtained,
we briefly consider the actual track.
We conditionally divide the trajectory of the actual hurricane
into three regions.
In the first part that lasts
from $t_1=0$ to $\approx t_{18}=4.25 \sim 102$\,hrs.,
the hurricane moves practically to the West  over the ocean.
In the second region
(from $\approx t_{18}$ to $\approx t_{37}=9\sim 216$\,hrs.),
the hurricane changes the direction of motion
from the  West to the North--Northwest.
In the vicinity of the 37th node,
the hurricane makes a sharp turn to the Northeast.
From the geographic viewpoint, for $t\approx t_{18}$,
the hurricane comes to the Lesser Antilles Isles,
moves over the Caribbean Sea towards the Haiti Isle,
and then to the Bahamas.
Another sharp turn is made approximately on the boundary between
the Bahamas and the Atlantic Ocean.
The third region of the hurricane trajectory
(from $\approx t_{37}$ to $t_{52}= 13\sim 306$\,hrs.)
is over the Atlantic Ocean.
During approximately two days
(from $\approx t_{37}$ to $\approx t_{47}$),
the hurricane moves to the Northwest.
Then it smoothly turns to the East.
Of course, the boundaries of the regions are rather rough.

To the first region, there correspond results shown in
Figs.~2,~3.
We see that the artificial trajectory coincides well with the
actual trajectory.
Moreover, except for the velocity,
the trajectories are astonishingly close to each other up to the
turning point $\xi_{37}$.
However the artificial and actual velocities are different,
and thus only a short-term forecast can be reliable.
So, for $t_M=0$ (see Fig.~2),
which corresponds to the forecast starting from $t_4\sim 18$\,hrs.,
the initial error is $|\Delta_4|\approx46$\,km
(here and in the following,
$\Delta_i=|\xi_i-\zeta_i|$ is the distance between~$\xi_i$
and~$\zeta_i$).
The error increases to $\Delta_8\approx 227$\,km during the
first 24\,hrs. and to $\Delta_{12}\approx 464$\,km
during the next 24\,hrs.
\begin{figure}[t]
\centering
\includegraphics[width=0.7\textwidth]{fig4.eps}
\caption{\small Initial data: 8.12 $10^{-5}$, ${}-3.19$, 
0.891, 3.99 $10^{-3}$, 3.46, 
0.446, 0.133, 5.00 $10^{-10}$, $-10^{-15}$, $-10^{-15}$,
 $-10^{-15}$,  $-10^{-15}$, $-10^{-15}$, $-10^{-15}$, K=3, 
M=18.
Distances for approximation: 0, 0.19, 0.09, 0.12.
Distances for forecast: 0.27, 0.45, 0.84, 1.06, 1.16, 1.21, 0.81, 
0.63, 0.53, 0.51, 0.47, 0.41, 0.59, 0.89, 1.08, 1.40, 1.42, 1.37, 1.87, 
2.90, 4.40, 6.36, 8.77.}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.7\textwidth]{fig6.eps}
\caption{\small Initial data: 8.12 $10^{-5}$, ${}-0.489$, 4.54, 9.26, 7.98, 
1.09, 2.75 $10^{-3}$, 5.98 $10^{-11}$, ${}-8.22 10^{-16}$, $-10^{-16}$,
 $-10^{-16}$,  $-10^{-16}$, $-10^{-16}$, $-10^{-16}$, K=3, 
M=33.
Distances for approximation: 0, 0.14, 0.19, 0.14.
Distances for forecast: 0.04, 0.29, 0.95, 1.44, 2.58, 3.94, 5.68, 
7.85, 10.1.}
\end{figure}
For $M=9$, $t_9\sim 48$\,hrs. (see Fig.~3),
which corresponds to the forecast starting from
$t_{12}\sim 66$\,hrs., the initial error is
$\Delta_{12}\approx28$\,km.
The error increases to $\Delta_{16}\approx 140$\,km during the
first 24\,hrs.,  to $\Delta_{20}\approx 220$\,km
during the next 24\,hrs.,
and to $\Delta_{24}\approx 470$\,km
during the next 24\,hrs..
If we take into account the fact that the maximum distance
between the trajectories does not exceed 120\,km till the
node~$\xi_{37}$,
then the forecast is apparently successful.


Numerical results shown in Figs.~4, 5 correspond to
the second part of the hurricane trajectory.
We see that the approximating properties of the artificial
trajectory become worse
as the hurricane approaches the turning point $\xi_{37}$.
So, the artificial trajectory in Fig.~4
practically coincides
with the actual trajectory
until the latitude $20^\circ$
(while the longitude varies from $-60^\circ$ to $-70^\circ$)
and then sufficiently well approximates the actual trajectory
until the latitude $30^\circ$.

In Fig.~5 the artificial and actual
trajectories agree less closely
(from $\lambda\approx-70^\circ$, $\varphi\approx20^\circ$ to
         $\lambda\approx-71.8^\circ$, $\varphi\approx24.7^\circ$)
 and then they diverge.
Nevertheless,
all artificial trajectories show that the hurricane tends
to change the direction of its motion from the North--West to
the North--East.

Numerical results shown in Figs.~6, 7 correspond to
the third part of the actual trajectory.
We see that the trajectories  coincide qualitatively well, and
this coincidence becomes more close with increasing~$M$.
Nevertheless, 
the artificial trajectory forecasts a more smooth turn to the
East than that made by the actual hurricane 
(at a latitude of $\approx 46^\circ$).

\begin{figure}[t]
\centering
\includegraphics[width=0.7\textwidth]{fig8.eps}
\caption{\small Initial data: 8.12 $10^{-5}$, 0.449, 1.09, 8.51, 40.3, 
1.21, 5.63 $10^{-3}$, 6.64 $10^{-11}$, ${}-8.22 10^{-16}$, $-10^{-16}$,
 $-10^{-16}$,  $-10^{-16}$, $-10^{-16}$, $-10^{-16}$, K=3, 
M=42.
Distances for approximation: 0, 0.44, 0.69, 0.84.
Distances for forecast: 0.26, 1.22, 2.65, 3.84, 3.88, 6.37.}
\end{figure}

\begin{figure}[t]
\centering
\includegraphics[width=0.7\textwidth]{fig9.eps}
\caption{\small Initial data: 8.12 $10^{-5}$, 1.38, 3.36, 1.29, 30.4, 
0.913, 1.16 $10^{-2}$, 7.95 $10^{-11}$, ${}-8.22 10^{-16}$, $-10^{-16}$,
 $-10^{-16}$,  $-10^{-16}$, $-10^{-16}$, $-10^{-16}$, K=3, 
M=45.
Distances for approximation: 0, 1.46, 1.22, 0.23.
Distances for forecast: 1.40, 2.28, 3.71, 3.60.}
\end{figure}

Let us sum up our numerical experiments.
The first part of the trajectory, where
the hurricane moves over the Atlantic Ocean,
agrees very well with the assumptions under which Eqs.~(1) were
derived:
the trajectory lies in the low latitudes ($\approx15^\circ$)
almost parallel to the equator, and
the external factors seem to be constant.
In numerical calculations according to the truncated sequence
of Hugoniot type conditions~(6),
the artificial and actual trajectory practically coincide
(see Figs.~1--3) and
the velocity is sufficiently close to the actual velocity of the
hurricane.
These results prove that
the representation of the solution in the form of the expansion
with respect to smoothness (2)
possesses good  approximating properties
and it is possible to use the truncated Hugoniot sequence~(6)
for calculating actual hurricane tracks.

The second part of the hurricane trajectory goes over
the Ca\-rib\-be\-an Sea. There is an increasing deviation to the North
in the actual trajectory, and the hurricane velocity decreases.
Apparently,
this is due to a change in the water temperature.
The trajectory of the artificial hurricane
calculated according to the initial data
corresponding to the first region (Figs.~2 and ~3)
is close to the actual trajectory.
However,
the velocity of the artificial hurricane does not decrease.
If the initial data are chosen according to the measurements in
the second region (Fig.~4),
then the velocity of the artificial hurricane turns out to be close
to the velocity of the actual hurricane (in the second region).
However,
the approximating properties of the numerical trajectory become
worse as the hurricane moves to the North.
Apparently,
this is due to the fact that model~(1) becomes less adequate.
It was assumed  that $\varphi$ is ``constant'' and small,
while
the coordinates of the actual hurricane vary
by $\approx 10^\circ$ in the latitude.

Over the boundary between the Bahamas and the Atlantic Ocean,
the actual hurricane passes from the second part of the
trajectory to the third part.
This passages is accompanied by a sharp change in the
direction of motion and an increase in the velocity.
Apparently,
the difference between the water temperature
in the Bahamas region and in the Atlantic Ocean
(at latitudes higher than $\approx 25^\circ$)
is a decisive factor here.
The artificial hurricane
calculated according to the initial
data measured in the second region (Fig.~5)
moves significantly lower than the actual hurricane
(as if the hurricane still remained in the second region),
and its trajectory is qualitatively close to the actual
trajectory only on a comparatively small region.
By choosing the initial data so that they
correspond to the conditions in the third region (Figs.~6, 7),
we obtain
a closer correspondence between the velocities of the
actual and artificial hurricanes. In this case we have 
a qualitative coincidence between the trajectories on a larger
region. 

\section{Other examples}

The test example of the hurricane HORTENSE is a typical example.
Namely, most of the hurricanes actually existing in the Atlantic
Ocean have similar trajectories. Now we consider two
hurricanes, which are exceptional in some sense.

\begin{figure}[t]
\centering
\includegraphics[width=0.7\textwidth]{fig10.eps}
\caption{\small $\xi$ is the track of GEORGES starting 
at 03:00 a.m. 16/09 1998, 
$\zeta$ is the artificial trajectory. 
Initial data: 8.12 $10^{-5}$, ${}-5.99$, 4.06 $10^{-2}$, 
1.33 $10^{-8}$, 1.32, 
5.04 $10^{-2}$, 7.52 $10^{-2}$, 6.98 $10^{-12}$, ${}-2.08 10^{-14}$,
 ${}-1.89 10^{-14}$,
 ${}-1.72 10^{-14}$,  ${}-1.91 10^{-14}$, ${}-1.74 10^{-14}$,
        ${}-1.74 10^{-14}$.
Distances: 0, 0.45, 1.13, 1.26, 1.64, 1.93, 2.34, 
2.82, 3.42, 4.22, 4.33, 4.99, 5.31, 
5.46, 5.83, 6.33, 6.84, 7.27, 7.90, 8.60, 9.32, 10.2, 10.7.}
\end{figure}

The trajectory of the hurricane GEORGES (15/09--28/09/1998)
almost completely lies in the tropical belt and is practically
in a straight line to the West--Northwest. 
We {\em a priori\/} believe that the forecast of its dynamics
will be successful. 

The trajectory of the hurricane NICOLE (24/11--01/12/1998) 
lies further north in the Atlantic Ocean, 
makes a sufficiently sharp turn (through $\approx130^\circ$), 
and has a bend  towards the West.
We {\em a priori\/} expect that the quality of the forecast of
its dynamics will be bad.

1. The hurricane GEORGES was first observed on 15/09/1998 in the
far eastern Atlantic. 
Moving practically in a straight line
on a general West--Northwest course,
the hurricane entered the Lesser Antilles Isles in 5\,days
and passed over practically all the Greater Antilles Isles.
Then, still moving practically in a straight line,  
the hurricane crossed the Gulf of Mexico and decayed  to the
North of  New Orleans.  
The fact that the hurricane did not make sharp turns can be
treated as ``stability'' of external factors. 

The plot in Fig.~8 shows the results 
(similar to those in Fig.~1)
obtained in calculating the artificial trajectory that is
maximally close to the entire actual trajectory of the hurricane.
We see that 
the difference between the trajectories 
exceeds 850\,km on the middle part and,
on the final stage, the artificial
hurricane moves to the South--West, whereas
the actual hurricane moves to the North--West
and the difference between the trajectories exceeds 1000\,km.
If we take into account the velocity of the hurricane,
then the difference between the actual and artificial hurricanes
is more significant for large~$t$.
This result confirms the conclusion drawn in Sec.~2 that
system~(6) does not work well for long-term forecasts.

\begin{figure}[t]
\centering
\includegraphics[width=0.7\textwidth]{fig13.eps}
\caption{\small Initial data: 8.12 $10^{-5}$, ${}-6.86$, 0.155, 
2.04 $10^{-9}$, 1.84, 
2.58 $10^{-2}$, 8.62 $10^{-2}$, 1.46 $10^{-11}$, ${}-1.29 10^{-14}$,
 ${}-1.29 10^{-14}$,
 ${}-1.29 10^{-14}$,  ${}-1.43 10^{-14}$, ${}-1.30 10^{-14}$,
        ${}-1.30 10^{-14}$, K=15, M=4.
Distances for approximation: 0, 0.24, 0.50, 1.16, 1.26, 1.53, 
1.21, 1.57, 1.38, 1.08, 1.10, 1.27, 0.81, 
0.44, 0.26.
Distances for forecast: 0.54, 0.67, 0.73, 0.92, 1.18, 1.64, 2.31, 2.86, 3.24, 
3.50, 4.09, 5.48, 7.11, 9.43, 11.7, 14.5.}
\end{figure}


\begin{figure}[t]
\centering
\includegraphics[width=0.7\textwidth]{fig14.eps}
\caption{\small Initial data: 8.12 $10^{-5}$, ${}-6.12$, 0.113, 
1.82 $10^{-9}$, 2.45, 
2.30 $10^{-2}$, 9.38 $10^{-2}$, 1.59 $10^{-11}$, ${}-1.29 10^{-14}$,
 ${}-1.29 10^{-14}$,
 ${}-1.29 10^{-14}$,  ${}-1.43 10^{-14}$, ${}-1.30 10^{-14}$,
        ${}-1.30 10^{-14}$, K=9, M=16.
Distances for approximation: 0, 0.13, 0.38, 0.46, 0.53, 0.50, 0.38, 0.48, 0.53, 0.62.
Distances for forecast: 1.07, 1.41, 1.46, 1.26, 1.17, 1.70, 2.31, 3.33, 4.22, 
5.50, 6.42, 8.93.}
\end{figure}

Let us discuss the results of the short-term forecast.
Approximately at $t_{32}$ GEORGES made landfall in the Cuba isles.
Undoubtedly, this was the reason for a slight change in his
trajectory.
The artificial hurricane based on the hurricane prehistory
cannot predict this turning point.
Moreover, it predicts a smooth turn to the South.
Correspondingly,
we can see in Figs.~9, 10 that the long-term forecasts are
qualitatively wrong.
However, all these forecasts are good for $\approx60$\,hrs.

\begin{figure}[t]
\centering
\includegraphics[width=0.7\textwidth]{fig16.eps}
\caption{\small Initial data: 8.12 $10^{-5}$, ${}-6.18$, 0.171, 
1.84 $10^{-9}$, 2.03, 
2.33 $10^{-2}$, 7.75 $10^{-2}$, 1.60 $10^{-11}$, ${}-1.29 10^{-14}$,
 ${}-1.29 10^{-14}$,
 ${}-1.29 10^{-14}$,  ${}-1.43 10^{-14}$, ${}-1.30 10^{-14}$,
        ${}-1.30 10^{-14}$, K=6, M=28.
Distances for approximation: 0, 0.12, 0.15, 0.54, 1.17, 1.73, 1.95.
Distances for forecast: 2.35, 2.26, 2.02, 1.80, 1.68, 1.34, 1.85, 1.53, 1.23, 
0.87, 0.54, 1.45, 2.08, 3.34, 4.74, 6.60, 8.26, 10.1.}
\end{figure}


\begin{figure}[t]
\centering
\includegraphics[width=0.7\textwidth]{fig18.eps}
\caption{\small Initial data: 8.12 $10^{-5}$, ${}-4.50$, 0.227, 
1.34 $10^{-9}$, 1.48, 
2.07 $10^{-2}$, 5.65 $10^{-2}$, ${}-1.29 10^{-14}$, ${}-1.29 10^{-14}$,
  ${}-1.29 10^{-14}$,  ${}-1.43 10^{-14}$, ${}-1.30 10^{-14}$,
        ${}-1.30 10^{-14}$, K=6, M=40.
Distances for approximation: 0, 0.56, 0.48, 0.29, 0.50, 0.57, 0.33.
Distances for forecast: 0.45, 0.39, 0.50, 0.24, 0.33, 0.72, 0.49, 0.66.}
\end{figure}

The last group of plots (see Figs.~11, 12)
corresponds to the forecast for $t\geq t_{34}$.
This means that by choosing $\mu^0_{K,M}$ we take into account
the turns made by GEORGES at $t=t_{32}$ and $t=t_{33}$.
We see that the forecast quality sharply increases
in contrast to Figs.~9, 10.
Moreover,
we see that the forecast quality increases with $t_{K+M}$
and
is practically ideal for $t>t_{46}$ (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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\end{thebibliography}



\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}