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%\chapter{Two-dimensional system of vortex filaments}

In this chapter, we discuss interaction and motion of vortices which are close to each other in the two-dimensional ideal fluid.
For simplicity, each vortex is idealized to {\it a vortex filament}, whose vorticity is concentrated in a line (a point on the two-dimensional plane).
The vortex filaments can be regarded as the singular points in a non-rotational flow of the non-divergent ideal fluid.
The velocity field induced by one vortex filament is derived analytically.
The velocity field induced by a group of vortex filaments is the sum of the velocity fields induced by each vortex filament.
The motion of the vortex filaments is governed by the velocity field.
We can analytically solve the motion of the vortex filaments in the system of two vortex filaments.
In the system of more-than-two vortex filaments, the motion of the vortex filaments is generally solved through numerical integration of the governing equations.
More detailed discussion of the following theoretical issues is in Tatsumi (1982; subsection 9-3).

%======================================================================
\section{Theoretical Background}
%------------------------------------------------------------------------------
\subsection{Motion of the system of vortex filaments}

The flow induced by one vortex filament is one of the two-dimensional potential flows which is defined as a non-rotational flow of the non-divergent ideal fluid (Tatsumi, 1982; subsection 8-2-4).
We can define a complex velocity potential, $W$, of a non-rotational flow induced by a vortex filament located at the origin as 
\begin{equation}
   W = -i \kappa \log z = \kappa \theta - i \kappa \log r,	\label{eq:vp1}
\end{equation}
where $z=r e^{i \theta}$, $(r, \theta )$ is the polar coordinates, and
$\kappa$ is a real constant.
The complex velocity potential can be represented as $W = \Phi + i \Psi$ where
$\Phi$ is the velocity potential and $\Psi$ the stream function.
The stream line of this flow is, therefore, a group of concentric
circles whose centers are at the origin. The radial component of the
flow, $u_r$, is zero. The tangential component of the flow
,$u_{\theta}$, is
\begin{equation}
   u_{\theta} = \frac{1}{r} \frac{\partial\Phi}{\partial\theta} = \frac{\kappa}{r}.
\end{equation}
The circulation $\Gamma$ along any closed curve which encloses the
origin is 
\begin{equation}
   \Gamma = 2 \pi \kappa.					\label{eq:vp2}
\end{equation}

We can represent the above velocity field also in the Cartesian coordinates
$(x,y)$. The complex velocity, $w\equiv u-iv$, where $(u,v)$ represent the
velocity field, is given by
\begin{equation}
   w  = \frac{dW}{dz},						\label{eq:vp3}
\end{equation}
where $z=x+iy$.

Consider the velocity field induced by $N$ vortex filaments in the
two-dimensional Cartesian coordinates. The complex velocity potential
of the flow induced by the $N$ vortex filaments is given by
\begin{equation}
   W = \frac{1}{2 \pi i} \sum_{n=1}^N \Gamma_n \log (z-z_n),	\label{eq:vp4}
\end{equation}
where $\Gamma_n$ is the circulation which $n$-th vortex filament has, and
$(x_n,y_n)$ is the circulation of the $n$-th vortex filaments. Substitution
(\ref{eq:vp4}) for (\ref{eq:vp3}) gives us the complex velocity at any
point $z$.

The velocity of $m$-th vortex filament, which is induced by the other
vortex filaments, is given by
\begin{equation}
   \frac{dz_m^*}{dt}
       = w_m
       = \left[ \frac{d}{dz} 
                \left( W - \frac{\Gamma_m}{2 \pi i} \log (z-z_m) \right)
                \right]_{z=z_m} 
       = \frac{1}{2\pi i} \sum_{n=1, n\neq m}^N \frac{\Gamma_n}{z_m-z_n},
							\label{eq:vp_moms}
\end{equation}
where $< >^*$ denotes the complex conjugate of $< >$. The $N$
complex simultaneous equations (\ref{eq:vp_moms}) can describe
completely the motion of the system of the $N$ vortex filaments.

%------------------------------------------------------------------------------
\subsection{The Hamilton function of the system of the vortex filaments}

Using a real function, $H$, which depends only on the relative distance between
vortex filaments which constitute the system, $r_{mn}$,
\begin{equation}
   \left . \begin{array}{l}
   H = - \frac{1}{4\pi} \sum_{m=1}^N \sum_{n=1, m \ne n}^N
         \Gamma_m \Gamma_n \log r_{mn}, 		\\
   r_{mn} = | z_m - z_n | = \sqrt{(x_m-x_n)^2+(y_m-y_n)^2},
	   \end{array} \right\}				\label{eq:vp_hamil}
%   \Ddsty H = - \frac{1}{4\pi} \sum_{m=1}^N \sum_{n=1, m \ne n}^N
%         \Gamma_m \Gamma_n \log r_{mn}, 		\\
%   \Ddsty r_{mn} = | z_m - z_n | = \sqrt{(x_m-x_n)^2+(y_m-y_n)^2},
%	   \end{array} \right\}				\label{eq:vp_hamil}
\end{equation}
we can represent the equations of motion of the system
(\ref{eq:vp_moms}) as 
\begin{equation}
   \Gamma_m \frac{dz^*_m}{dt} = 2i \frac{\partial H}{\partial z_m}.	\label{eq:vp5}
\end{equation}
Decomposition of (\ref{eq:vp5}) into real and imaginary parts yields
\begin{equation}
   \Gamma_m \frac{dx_m}{dt} = \frac{\partial H}{\partial y_m}, \qquad
   \Gamma_m \frac{dy_m}{dt} =-\frac{\partial H}{\partial x_m}.	\label{eq:vp_xymom}
\end{equation}
For the derivation of (\ref{eq:vp_xymom}), we use a relation 
$\frac{\partial}{\partial z} = \frac{1}{2} \left(
\frac{\partial}{\partial x} - i \frac{\partial}{\partial y} \right)$.
Equation (\ref{eq:vp_xymom}) is {\it a Hamilton's canonical equation}
with canonical variables $x_m$ and $y_m$, and $H$ is {\it a Hamiltonian}.

General features of the system of the vortex filaments are summarized
as follows.
\vspace{-1em}
\begin{itemize}
   \item Function $H$ only depends on the relative distance $r_{mn}$, and
         is invariant under translation and rotation of the system.
   \item In case of $\sum \Gamma_m \neq 0$, the center of the gravity of the
         system, $z_0 \equiv \sum_{m=1}^N \Gamma_m z_m / \sum_{m=1}^N \Gamma_m$,
         is stationary.
   \item In case of $\sum \Gamma_m =0$, if the vortex filaments are
         divided into any two groups, the relative distance of the centers of
         the gravity of them is constant.
   \item The moment of inertia of the system with respect to the origin,
         $I \equiv \sum_{m=1}^N \Gamma_m (x_m^2 + y_m^2)$, is constant.
   \item The angular momentum of the system with respect to the origin,
         $A \equiv \sum_{m=1}^N \Gamma_m (x_m dy_m/dt - y_m dx_m/dt)$,
         is constant.
   \item Function $H$ is independent of time.
\end{itemize}

%======================================================================
\section{Exercises}
%----------------------------------------------------------------------
\subsection{The flow induced by the system of vortex filaments}

The program "C6-1\&2" visualizes the flow induced by two
vortex filaments and their motion.
Velocity at any point $(x,y)$ is derived from (\ref{eq:vp3}) and (\ref{eq:vp4}).
\begin{description}
   \item[Example 1] The flow induced by two vortex filaments.
\end{description}
\begin{center}
\begin{tabular}{cccccc}
%\multicolumn{6}{l}{'same sign, same intensity'} \\
5.0    & -5.0 & 0.0 &  5.0   & 5.0 & 0.0 \\
$\gamma$1 &  x1  & y1  & $\gamma$2 & x2  & y2
\end{tabular}
\end{center}
Parameters $\gamma$1 and ({\it x1},{\it y1}) represents
the circulation and the position of the vortex filament 1, respectively.
Parameters $\gamma$2 and ({\it x2},{\it y2}) represents the same
except for the vortex filament 2. {\it START 1} of the program {\it uzuito\_ui}
first shows the flow induced by the vortex filament 1, a few
seconds later the flow induced by the vortex filament 2. After next
few seconds, it shows the flow which is induced by these two vortex
filaments. Change strengths and signs of the circulation and the
relative distance of these vortex filaments to examine how flow
is induced.

\begin{figure}[t]
  \begin{center}
    \includegraphics[width=10cm,keepaspectratio,clip]{fig/C05_01.epsf}
    \caption{Result of Example 1.}
    \label{fig:C05_01}
  \end{center}
\end{figure}

\subsection{Motion of the system of two vortex filaments}

In case of the system of two vortex filaments, the Hamiltonian
(\ref{eq:vp_hamil}) of the system is expressed as 
\begin{equation}
   H = - \frac{1}{2\pi} \Gamma_1 \Gamma_2 \log r_{12} = constant,    \label{eq:vp7}
\end{equation}
and thus the distance between the vortex filaments, $r_{12}$, is turned out to be
constant value of $d$. If $\Gamma_1 + \Gamma_2\neq 0$, the center of the
gravity of the system is stationary. In this case, two vortex
filaments rotate with constant speeds along circles which center is
the center of the gravity of the system. The center of the gravity
$(x_0,y_0)$ is
\begin{equation}
   (x_0,y_0) =
   \left( \frac{\Gamma_1 x_1+\Gamma_2 x_2}{\Gamma_1+\Gamma_2},
          \frac{\Gamma_1 y_1+\Gamma_2 y_2}{\Gamma_1+\Gamma_2} \right).
								 \label{eq:vp8}
\end{equation}
The angular momentum of the system is constant. The two vortices
rotate around their center of the gravity with a constant angular
velocity. Such motion of these vortices is a kind of {\it the Fujiwara
effect} since a vortex with positive circulation is a good
model of typhoon.
\begin{description}
\item[the Fujiwara effect] : When two typhoons are close to each
other, their motion is complicated because of their interaction. This
is called {\it the Fujiwara effect}. Their ideal relative motion is
that they rotate anti-clockwise (clockwise in the Southern Hemisphere)
around a certain point between the typhoons. Sometimes one typhoon
decelerates to become almost stationary while the other accelerates. 
In reality, the Fujiwara effect is recognized when the distance of two typhoons
are less than 800 km. This effect was advocated by Sakuhei Fujiwara who had
served as a director of the Japan Meteorological Agency. (Encyclopedia
of Meteorology and Atmospheric Sciences, 1998)
\end{description}

On the other hand, when $\Gamma_1+\Gamma_2=0$,
i.e. $\Gamma_1=-\Gamma_2=\Gamma$, the center of the gravity can not be
defined. Two vortex filaments move with the same velocity,
\begin{equation}
   \frac{dz_1}{dt} = \frac{dz_2}{dt} = \frac{1}{2\pi i} \frac{\Gamma}{z_2^*-z_1^*}.
\end{equation}
Such {\it vortex pair} moves with speed of $\Gamma /(2\pi d)$ and
the direction perpendicular to the straight line which connects these
vortex filaments.

Substituting (\ref{eq:vp7}) to (\ref{eq:vp_xymom}) yields
\begin{equation}
   \left \{ \begin{array}{rrr}
         \frac{dx_1}{dt} = 
\! \! \! &   \frac{1}{\Gamma_1} \frac{\partial H}{\partial y_1} = 
\! \! \! &     -\frac{\Gamma_2}{2 \pi} \frac{y_1-y_2}{d^2},      \\[0.3cm]
         \frac{dy_1}{dt} = 
\! \! \! &  -\frac{1}{\Gamma_1} \frac{\partial H}{\partial x_1} = 
\! \! \! &      \frac{\Gamma_2}{2 \pi} \frac{x_1-x_2}{d^2},      \\[0.3cm]
         \frac{dx_2}{dt} =  
\! \! \! &   \frac{1}{\Gamma_2} \frac{\partial H}{\partial y_2} = 
\! \! \! &      \frac{\Gamma_1}{2 \pi} \frac{y_1-y_2}{d^2},      \\[0.3cm]
         \frac{dy_2}{dt} = 
\! \! \! &  -\frac{1}{\Gamma_2} \frac{\partial H}{\partial x_2} = 
\! \! \! &     -\frac{\Gamma_1}{2 \pi} \frac{x_1-x_2}{d^2}.
   \end{array} \right.
\end{equation}
The motion of the vortex filaments are solved with integration of
the above simultaneous ordinary differential equations. 

\begin{description}
   \item[Example 2] The system of two vortex filaments
\end{description}
\begin{center}
\begin{tabular}{cccccc}
%\multicolumn{6}{l}{'same sign, same intensity'} \\
5.0    & -5.0 & 0.0 &  5.0   & 5.0 & 0.0 \\
$\gamma$1 &  x1  &  y1 & $\gamma$2 & x2  & y2
\end{tabular}
\end{center}
These parameters are the same as that of Example 1.
Two vortices which have the same value of the circulation rotate
along a circle which center is the intermediate point between the
vortices. The direction of the motion of the vortices is, of course,
the same as the direction of the vortices. 

If the intensity of the circulation of the two vortices are different
each other and the signs of them are the same, the center of the
gravity of them is the point which divide $r_{12}$ internally to
$\Gamma_2 : \Gamma_1$. These two vortices rotate along circles which
centers are at the center of the gravity. This is so-called the Fujiwara
effect. Change $\Gamma_2$ to examine how the position of the center of the
gravity changes.

In case of the vortex pair, the center of the gravity can not be
defined. The vortex filaments move with the same velocity each
other toward the direction perpendicular to the straight line which
connects the vortex filaments. Confirm the relationship between the
direction of the rotation of the vortices and the direction of their motion.

If two vortex filaments have different intensity and different
signs of the circulation, the center of the gravity of them is the
point which divide $r_{12}$ externally to $\Gamma_2 : \Gamma_1$. The
vortex filaments rotate along circles which centers
are at the center of the gravity. Change $\Gamma_2$ to examine how the
position of the center of the gravity changes.

%----------------------------------------------------------------------
\subsection{The system of three vortex filaments}

We can solve the motion of the system of three vortex filaments
analytically only if they have the same value of the circulation,
$\Gamma$ and locate at vertices of a equilateral triangle (see the
next subsection). Like the system of two vortex filaments, three
vortex filaments move along various loci which depend on the sign and
intensity of their circulation and the position of them.
\begin{description}
   \item[Example 3] The system of three vortex filaments.
\end{description}
\begin{center}
\begin{tabular}{ccccccccc}
%\multicolumn{9}{l}{'different sign and intensity'} \\
0.65   & -4.33 & -2.5 &   0.3  & 4.33 & -2.5 & -0.45  & 0.0 & 5.0 \\
$\gamma$1 &   x1  &  y1  & $\gamma$2 &  x2  &  y2  & $\gamma$3 & x3  & y3
\end{tabular}
\end{center}

The program "C6-3" visualizes the loci of three vortex
filaments. In its graphics, the red, blue, and green points indicate location of the
vortex filaments which have the circulation of $\gamma$1, $\gamma$2, $\gamma$3,
respectively. The system of three vortex filaments whose signs of
the circulation are different each other can move along chaotic loci.
This chaotic motion is not observed in the system of
two vortex filaments. 

\begin{figure}[t]
  \begin{center}
    \includegraphics[width=10cm,keepaspectratio,clip]{fig/C05_02.epsf}
    \caption{Result of Example 3.}
    \label{fig:C05_02}
  \end{center}
\end{figure}

On the other hand, three vortex filaments have the same value of
the circulation and locates in equal distance (vertices of a
equilateral triangle), they rotate along the same circle apart 120
degrees each other. When three vortex filaments which have the same
value of the circulation locate on a straight line at regular
intervals, the central filament is motionless while the others rotate
along the same circle which center is at the central filament. 
Examine how motion changes in these cases when position of a vortex
filament change slightly. 

Examine how is the motion of three vortex filaments which circulation
are the same signs but different intensity. Is the motion chaotic?

Examine how is the motion of three vortex filaments which circulation
are the same intensity but different signs. How is the motion of
three vortex filaments if two vortex filaments
which circulation are the same sign are located closely to each other
while the other which circulation is the different sign is located
apart?
How is the motion of three vortex filaments if two vortex filaments which circulation are the opposite signs to each other are located closely while the other is located apart ?

Change the circulation and initial position of the vortex filaments
in various ways to examine the motion.

%======================================================================
\section{Appendix}

Examine analytically the motion of three vortex filaments which have
the same value of the circulation, $\Gamma$, and are located at the
vertices of a equilateral triangle. Derive the Hamiltonian of the
system and the
equations of motion to confirm that the vortex filaments rotate along
the same circle which center is at the center of the gravity of
them. You can also discuss the stability of the circular motion.

%======================================================================
\section*{References}

\begin{description}
\labelsep  0 mm
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\item GEFD Summer School, 1994: 
      Demonstration 5: Point vortices.
      {\it Computer Demonstration Note}.

\item Meteorological Society of Japan (edit.), 1998: Encyclopedia of
Meteorology and Atmospheric Sciences, Tokyo Shoseki, 637pp (in Japanese).

\item Tomomasa Tatsumi, 1982: Fluid Mechanics, Baifukan, 453pp (in Japanese).
\end{description}

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