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%\chapter{Inertial Oscillation and Inertial Instability}

Motion of a free particle on a rotating frame of reference without the effect of the centrifugal force, which results from the rotation of the frame, is periodic and termed the inertial oscillation.
We can discuss stability of sheared flow on the same rotating frame of reference with the parcel method.
The stability results from a relationship between the shear of the flow and the Coriolis force.
 
%======================================================================
\section{Theoretical Background}
\subsection{Inertial oscillation}
The equation of motion of a free particle on a plane rotating with a constant angular velocity $\Omega$ is given by
\begin{eqnarray}
   \frac{du}{dt} - fv &=& 0, 	\label{i-p-i.eq:du} \\
   \frac{dv}{dt} + fu &=& 0, 	\label{i-p-i.eq:dv}
\end{eqnarray}
where $f\equiv 2\Omega$ is the Coriolis parameter, while the centrifugal force is neglected.
Eliminating $v$ from (\ref{i-p-i.eq:du}) and (\ref{i-p-i.eq:dv}), we obtain
\begin{equation}
   \frac{d^{2}u}{dt^{2}} + f^2 u= 0.		\label{i-p-i.eq:ddu}
\end{equation}
This ordinary differential equation (\ref{i-p-i.eq:ddu}) can be integrated 
with the initial condition of $u=u_0$ and $v=v_0$ at $t=0$,
\begin{equation}
   u(t) = u_0\cos ft+v_0\sin ft.		\label{i-p-i.eq:u}
\end{equation}
By similar procedure,
\begin{equation}
   v(t) = u_0\sin ft+v_0\cos ft.		\label{i-p-i.eq:v}
\end{equation}
Integration of  (\ref{i-p-i.eq:u}) and (\ref{i-p-i.eq:v}) with the initial
condition of $x=x_0$ and $y=y_0$ at $t=0$ yields
\begin{eqnarray}
   x(t) &=&  \frac{v_0}{f}(1-\cos ft) + \frac{u_0}{f}\sin ft + x_0,
	\label{i-p-i.eq:usolve}\\
   y(t) &=& -\frac{u_0}{f}(1-\cos ft) + \frac{v_0}{f}\sin ft +y_0.
	\label{i-p-i.eq:vsolve}
\end{eqnarray}
We can easily find from solutions (\ref{i-p-i.eq:usolve}) and 
(\ref{i-p-i.eq:vsolve}) that the motion of the free
particle is periodic with its period (inertial period) of $2\pi /f$,
which is a half of the period of the frame's rotation, $2\pi /\Omega$, and with its
direction reverse to the direction of the frame's rotation.

%======================================================================
\subsection{Inertial instability}
We discuss the stability of two-dimensional flow on the rotating frame of reference.
More detailed discussion is given in Yoden (1998; subsection 10.3).
If the pressure gradient has only $y$-component, the basic state field of the flow is a zonally directed geostrophic wind $\overline{u}_g(y)$ which is independent of $x$,
\begin{equation}
   \overline{u}_g(y)=-\frac{1}{\rho f}\frac{\partial p}{\partial y}.
\end{equation}
Here we study a motion of a parcel on this field.
Since we assume that the parcel displacement does not disturb the pressure field, the background geostrophic wind is stable.
The equations of motion of the parcel are given by
\begin{eqnarray}
   \frac{Du}{Dt} &=& fv,	       		\label{i-p-i.eq:du2}\\
   \frac{Dv}{Dt} &=& -fu-\frac{1}{\rho}\frac{\partial p}{\partial y}
    \;\;=\;\;-f\left\{u-\overline{u}_g(y)\right\},\label{i-p-i.eq:dv2}
\end{eqnarray}
where $u=Dx/Dt$ and $v=Dy/Dt$. If the parcel is displaced along $y$
direction by a distance $\delta y$, parcel's new $u$ is
derived through integration of (\ref{i-p-i.eq:du2}),
\begin{equation}
   u(y_0+\delta y)=\overline{u}_g(y_0)+f\delta y.	\label{i-p-i.eq:u0}
\end{equation}
On the other hand, the geostrophic wind at $y=y_0+\delta y$ can be
approximated as 
\begin{equation}
   \overline{u}_g(y_0+\delta y)=\overline{u}_g(y_0)
            +\frac{d\overline{u}_g}{dy}\delta y.	\label{i-p-i.eq:ug0}
\end{equation}
Using (\ref{i-p-i.eq:u0}) and (\ref{i-p-i.eq:ug0}), we have the $y$-component of the equation of motion of the parcel (\ref{i-p-i.eq:dv2}) at $y=y_0+\delta y$ as
\begin{equation}
   \frac{Dv}{Dt}=-f\left(f-\frac{d\overline{u}_g}{dy}\right)\delta y.
						\label{i-p-i.eq:dv3}
\end{equation}
Introducing an absolute momentum $M\equiv fy-\overline{u}_g$ to (\ref{i-p-i.eq:dv3}), we obtain
\begin{equation}
   \frac{D^2}{Dt^2}(\delta y)=-f\frac{dM}{dy}\delta y.
						\label{i-p-i.eq:d2y}
\end{equation}
Equation (\ref{i-p-i.eq:d2y} enables us to discuss the stability of the geostrophic wind $\overline{u}_g$.
If $f\frac{dM}{dy}$ is positive, $\delta y$ only oscillates with the period of $2\pi (f\frac{dM}{dy})^{-1/2}$ and thus the geostrophic wind $\overline{u}_g$ is stable.
On the other hand, if $f\frac{dM}{dy}$ is negative, $\delta y$ leaves from the
original value exponentially with time, and thus the geostrophic wind
$\overline{u}_g$ is unstable. In summary, the condition for the inertial
instability is
\begin{equation}
   \left\{
      \begin{array}{cl}
         fdM/dy > 0 & stable,\\
         fdM/dy = 0 & neutral,\\
         fdM/dy < 0 & unstable.\\
      \end{array}
   \right.
\end{equation}
\\
In the case that the background geostrophic wind is expressed as 
\begin{equation}
   \overline{u}_g(y)=py+q,			\label{i-p-i.eq:ug}
\end{equation}
we can solve (\ref{i-p-i.eq:du2}) and (\ref{i-p-i.eq:dv2}) with
the initial condition of $(x,y,u,v)=(x_0,y_0,u_0,v_0)$ at $t=0$ as follows.
\begin{description}
\item[{\sf i})] when $fdM/dy > 0$($f>p$, stable),
\begin{eqnarray}
   x(t) &=& \frac{1}{f-p}\left\{(1-\cos\omega t)v_0
       -f(t-\frac{1}{\omega}\sin\omega t)(u_0-u_g)\right\}+u_0t+x_0, \\
   y(t) &=& -\frac{1}{f-p}(1-\cos\omega t)(u_0-u_g)
       +\frac{v_0}{\omega}\sin\omega t+y_0,
\end{eqnarray}
where
\begin{equation}
   \omega =\sqrt{f(f-p)},
\end{equation}
\item[{\sf ii})] when $fdM/dy = 0$ ($f=p$, neutral),
\begin{eqnarray}
   x(t) &=& -\frac{f^2}{6}(u_0-u_g)t^3+\frac{1}{2}v_0ft^2+u_0t+x_0, \\
   y(t) &=& -\frac{f}{2}(u_0-u_g)t^2+v_0t+y_0,
\end{eqnarray}
\item[{\sf iii})] when $fdM/dy < 0$ ($f<p$, unstable),
\begin{eqnarray}
   x(t) &=& \frac{1}{2(p-f)}\left\{v_0-\frac{f}{r}(u_0-u_g)\right\}e^{rt}
            +\frac{1}{2(p-f)}\left\{v_0+\frac{f}{r}(u_0-u_g)\right\}e^{-rt}
                                                       \nonumber \\
         && +\frac{f}{p-f}(u_0-u_g)t+u_0t+x_0-\frac{1}{p-f}v_0, \\
   y(t) &=& \frac{1}{2r}\left\{v_0-\frac{f}{r}(u_0-u_g)\right\}e^{rt}
            +\frac{1}{2r}\left\{v_0+\frac{f}{r}(u_0-u_g)\right\}e^{-rt}
                                                       \nonumber \\
         && +\frac{1}{p-f}(u_0-u_g)+y_0,
\end{eqnarray}
where
\begin{equation}
   r=\sqrt{f(p-f)}.
\end{equation}
\end{description}
In the above solutions, $u_g$ implies $\overline{u}_g(y_0)$.

%======================================================================
\section{Exercises}
\subsection{Inertial oscillation}
In the rotating frame of reference, the free particle rotates along the inertial circle with the inertial period.
The program "C7-1" enables us to visualize a behavior of a free particle viewed not only on rotating frame of reference but also on inertial frame of reference.
\begin{description}
\item[Example 1]
\end{description}
\begin{center}
\begin{tabular}{ccccc}
	30.0 & 0.0 & 400.0 & 25.0 & 0.0 \\
Latitude & X0  &  Y0   &  U0  & V0
\end{tabular}
\end{center}
It will be confirmed that the free particle on the rotating frame of reference (top panel) rotates along the inertial circle with a half of the period of frame's rotation.
Select the different latitude to examine how the inertial period changes with latitude.

\begin{figure}[t]
  \begin{center}
    \includegraphics[width=10cm,keepaspectratio,clip]{fig/C10_01.epsf}
    \caption{Result of Example 1.}
    \label{fig:C10_01}
  \end{center}
\end{figure}

\subsection{Inertial instability}

Execute the program "C7-2" to examine the stability of the geostrophic
flow which has the meridional shear, {\it p}.
\begin{description}
\item[Example 2]
\end{description}
\begin{center}
\begin{tabular}{cccccccc}
5.0  & 0.0 & 0.0 & 12.676 & 0.0 & 18.984 & 0.0 & 50.0 \\
Lat. & p1  & q1  &   p2   & q2  &  p3    &  q3 & dy
\end{tabular}
\end{center}
In this example, the flow is stable (top panel), neutral (middle panel), and
unstable (bottom panel). Change {\it Lat.} or {\it p} to examine which
condition makes the flow stable (unstable).\\
Setting all three {\it p}s to 0, you can observe the inertial
oscillation in three cases of uniform background zonal flow.
\begin{description}
\item[Example 3]
\end{description}
\begin{center}
\begin{tabular}{cccccccc}
30.0  & 0.0 & 0.0 & 0.0 & 5.0 & 0.0 & 10.0 & 100.0 \\
Lat.  & p1  & q1  & p2  & q2  & p3  &  q3  & dy
\end{tabular}
\end{center}
Note that the periods of three cases in the example 3 are the same each other.\\

\begin{figure}[t]
  \begin{center}
    \includegraphics[width=10cm,keepaspectratio,clip]{fig/C10_02.epsf}
    \caption{Result of Example 2.}
    \label{fig:C10_02}
  \end{center}
\end{figure}

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

To understand the inertial oscillation in a physical sense, refer to Sakai (1995), Cushman-Roisin (1994) and Durran (1993).
Note that consideration to motion of a ball in a bowl is helpful to our understanding of the inertial oscillation.

\section*{References}

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\item Cushman-Roisin, B., 1994:{\it Introduction to Geophysical Fluid
      Dynamics}, Prentice-Hall, 16-32.
\item Dale R. Durran, 1993:{\it Is the Coriolis Force Really Responsible 
      for the Inertial Oscillation?}, America Meteorological
      Society, 2179-2184.
\item Sakai, S., 1995:{\it Movies of fundamental experiments of 
      geofluid dynamics} (in Japanese).
\item Yoden S., 1998:{\it Meteorology I}, 146pp (in Japanese).

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