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%\chapter{Rossby wave}
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\section{Theoretical Background}
\subsection{Basic equations and\\ the dispersion relation of Rossby wave}

We assume that the coriolis parameter f is written in the form (\ref{C_PARA}).
\begin{equation}
  f = 2\Omega sin \phi \approx 2\Omega sin\phi _0 + 2\Omega\frac{y}{a}cos\phi _0 = f_0 + \beta y
  \label{C_PARA}
\end{equation}
where 
\begin{equation}
  \phi = \phi _0 + \frac{y}{a}
\end{equation}
The constant $\Omega$ is the angular velocity of the planet and $\phi$ is latitude.
The constant $\phi _o$ is the latitude of the reference state.
The constant a is the planetary radius 
and the variable y is the latitudal displacement from the reference state.\\

On this {\it $\beta$ plane}, the quasi-geostrophic potential vorticity equation is given as (\ref{QGPV}).
\begin{equation}
  \frac{\partial}{\partial t}\nabla _H \psi 
  + \frac{\partial(\psi ,\Delta _H \psi)}{\partial (x,y)}
  + \beta \frac{\partial \psi}{\partial x}
  = 0
  \label{QGPV}
\end{equation}
(see Rhines(2002).)\\
%(see Pedrosky(1986) or Cushman-Roisin(1994))\\

Eaquation (\ref{QGPV}) is a non-linear partial derivative equation, in which a stream function
$\psi (x,y,t)$ is a dependent variable of x, y, and t.\\

We separate variables into basic state and perturbation. A variable $\overline{\alpha}$ is a basic 
state of $\alpha$. A variable $\alpha ^\prime$ is a perturbation part of $\alpha$.
We assume a static solution as a basic state.
\begin{eqnarray}
  &&\overline{u} = \overline{v} = 0\\
  &&\overline{\psi} = constant
\end{eqnarray}
The stream function $\psi$ is written as (\ref{STREAM_FN}).
\begin{equation}
  \psi(x,y,t) = \overline{\psi} + \psi ^\prime(x,y,t) \label{STREAM_FN}
\end{equation}
The basic equation linearized in terms of perturbation $\psi ^\prime(x,y,t)$ is (\ref{LIN_EQ}).
\begin{equation}
  \frac{\partial}{\partial t}\nabla _H \psi ^\prime 
  + \beta \frac{\partial \psi ^\prime}{\partial x}
  = 0
  \label{LIN_EQ}
\end{equation}
We assume periodicity in time and space. Give a sine wave solution as (\ref{SIN_SOL}).
\begin{equation}
  \psi ^\prime(x,y,t) = Re\left[\Psi_0e^{i(kx+ly-\omega t)}\right] \label{SIN_SOL}
\end{equation}
Substituting (\ref{SIN_SOL}) into (\ref{LIN_EQ}), we obtain the dispersion relation of two-dimensional
{\it Rossby wave} on a $\beta$ plane.
\begin{equation}
  \omega = - \frac{\beta k}{k^2 + l^2} \label{DISP}
\end{equation}
%
\begin{figure}[t]
  \begin{center}
    \includegraphics[width=12cm,keepaspectratio,clip]{fig/wr4.eps}
    \caption{Dispersion relation of two-dimensional non-diverce Rossby wave on a $\beta$ plane.
      (a) Non-dimensionalized $\omega$ by $\Omega$ 
      as a function of non-dimensionalized wavenumber $(k,l)a$ by planetaly radius ``a''.
      Bold dashed arrow is phase velocity in real space. 
      Bold solid arrow is group velocity in real space.
      (b) Meridional wavenumber $la$ as a function of $ka$ and $\omega/\Omega$.
      Shade represents area of $l^2<0$.
      We will show a structure of Rossby wave marked as $\star$ in fig. \ref{EPS:WR5}.}
    \label{EPS:WR4}
  \end{center}
\end{figure}
%
Because $\beta>0$, the sign of $\omega$ and that of $k$ is always different.
Figure~\ref{EPS:WR4} shows the dispersion relation. 
In fig. \ref{EPS:WR4}, wavenumber and frequency is non-dimensionalized by
$a^{-1}$ and $\Omega$, respectively. Figure~\ref{EPS:WR4}(a) shows non-dimensionalized freqency $\omega/\Omega$
in terms of non-dimensionalized wavenumber $(ka,la)$, where $\omega > 0$.
Equation (\ref{DISP}) can be translated into an equation below. 
\begin{equation}
  \left( k + \frac{\beta}{2\omega} \right) ^2 + l^2 = \left(\frac{\beta}{2\omega}\right)^2
\end{equation}
This equation represents circle-shape when we take a line where $\omega$ is a constant.
The center of the circle is located $(-\beta /2\omega,0)$, and the radius of it is $\beta /2\omega$.
As $\omega$ increases, the radius of the circle decreases and approaches the origin.
Figure~\ref{EPS:WR4}(b) shows non-dimensionalized meridional wavenumber $la$ in terms of $ka$ and $\omega/\Omega$.
In shaded area, while $\omega > -\beta /k$, $l$ becomes pure imaginary number.
In this case, there is no solution of meridionaly propagationg sine wave. 
The wave solution in this case is an external wave with an exponential structure in space.\\

From dispersion relation (\ref{DISP}), the phase velocity $(c_{px},c_{py})$ 
and the group velocity $(c_{gx},c_{gy})$ of Rossby wave are calculated 
as shown in (\ref{CPX}) - (\ref{CGY}).
\begin{eqnarray}
  &&c_{px} = \frac{\omega}{k} = -\frac{\beta}{k^2+l^2} \label{CPX}\\
  &&c_{py} = \frac{\omega}{l} = -\frac{k}{l} \frac{\beta}{k^2+l^2}\\
  &&c_{gx} = \frac{\partial \omega}{\partial k} = \frac{\beta (k^2-l^2)}{(k^2+l^2)^2}\\
  &&c_{gy} = \frac{\partial \omega}{\partial l} = \frac{2\beta kl}{(k^2+l^2)^2} \label{CGY}
\end{eqnarray}
Because $c_{px} < 0$, the phase of this wave always goes west. The sign of the group velocity changes
as the sign of $(k^2 - l^2)$ changes. If $(k^2 - l^2) > 0$, $c_{gx} > 0$. 
If $(k^2 - l^2) < 0$, $c_{gx} < 0$. This means that when the direction of the wave front is near 
south-north, the wave energy goes east. When the direction of it is near east-west,
the wave energy goes west.\\

\begin{figure}[t]
  \begin{center}
    \includegraphics[width=8cm,keepaspectratio,clip]{fig/wr5.eps}
    \caption{(a) Phase relation of Rossby wave 
      (In case of $\omega >0, k<0, m>0$; At the point of $\star$ in fig. \ref{EPS:WR4}).
      (b) Spacial structure of Rossby wave at t in (a). 
      Dashed arrow shows phase velocity and Bold arrow shows group velocity. 
      Solid arrows represent velocity field of fluid. Area of $\zeta ^\prime < 0$ is shaded.}
    \label{EPS:WR5}
  \end{center}
\end{figure}
When the solution is written as (\ref{SIN_SOL}), the phase relationship among velocity, vorticity, and 
stream function is written as (\ref{U_PSI}) - (\ref{ZETA_PSI}).
\begin{eqnarray}
  && u^\prime = -\frac{\partial \psi ^\prime}{\partial y} = -il\psi ^\prime \label{U_PSI}\\
  && v^\prime = \frac{\partial \psi ^\prime}{\partial x} = ik\psi ^\prime\\
  && \zeta ^\prime = \Delta \psi ^\prime = -(k^2 + l^2)\psi ^\prime \label{ZETA_PSI}
\end{eqnarray}
Since the direction of motion is perpendicular to  that of wave propagation, 
i.e., ${\bf u} \cdot {\bf k} = 0$ or ${\bf u} \cdot {\bf c} = 0$,
this kind of Rossby wave is a transverce wave. \\

Figure~\ref{EPS:WR5}(a) shows the phase relation of Rossby wave marked $\star$ in Figure~\ref{EPS:WR4}.
Figure~\ref{EPS:WR5}(b) shows spacial structure of the wave. Because $k<0$ and $l>0$, 
iso-phase line of $kx + ly = constant$ slopes upwards going from left to right.
The phase of $u ^\prime$ is the same as that of $v ^\prime$, 
which is faster than that of $\psi ^\prime$ by $\pi /2$.
The phase of vorticity is reverse of that of $\psi ^\prime$.\\

Figure~\ref{EPS:RHINES} shows the wave pattern of Rossby wave 
generated by an oscillating vorticity source. 
Iso-phase line is horseshoe-shaped.

\begin{figure}[t]
  \begin{center}
    \includegraphics[width=10cm,keepaspectratio,clip]{fig/rossbygf3.ps}
    \caption{Rossby wave generated by an oscillating vorticity source (Rhines,2002).
    Seen from southeast of the forcing region.}
    \label{EPS:RHINES}
  \end{center}
\end{figure}

%======================================================================
\section{Exercises}
Perform the program "C9" with two types of initial conditions.

\begin{figure}[t]
  \begin{center}
    \includegraphics[width=10cm,keepaspectratio,clip]{fig/CA4_01.epsf}
    \caption{Result of Example 1.}
    \label{fig:CA4_01}
  \end{center}
\end{figure}

%======================================================================
\begin{description}
\item[Example 1] Wave packet
\end{description}
\begin{center}
  \begin{tabular}[]{cc}
    1.0     & 2.0 \\
    $\beta$ & {\it k}
  \end{tabular}
\end{center}

To run this experiment, click ``start1'' button.

At an initial time, a wave packet is located at the center of the region. 
During the calculation, the wave packet goes west. 
The wave packet is reflected when it reaches boundaries.

We can change $\beta$ and the wave number in x direction, {\it k}. 
As $\beta$ or {\it k} changes, both phase verosity and group verosity change.

%======================================================================
\begin{description}
\item[Example 2] Gaussian
\end{description}
\begin{center}
  \begin{tabular}[]{cc}
    1.0     & 2.0 \\
    $\beta$ & {\it k}
  \end{tabular}
\end{center}

At an initial time, a gaussian-shaped height anomaly is located at the center of the field. 
As time goes, the anomaly is reformed into a horseshoe shape.
In this experiment the calculation field is larger than the field we can see on the screen in order to prevent reflection of the wave at boundaries.

We can change $\beta$ and the width of the gaussian-shaped anomaly {\it k}.

To run this experiment, click ``start2'' button.

%======================================================================
\section*{Reference}
\begin{description}
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\item Rhines, 2002 : {\it Rossby waves}, Encyclopedia of Atmospheric Sciences, Academic Press
(available at http://www.ocean.washington.edu/research/gfd/Rossby-final-prfdt-.pdf).
\end{description}

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