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%             4-7 BAROCLINIC INSTABILITY
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\section{�����԰���}

\begin{itembox}[l]{����}
 \Dparskip \vspace{-6mm}
 ����ޤǤϽ簵�絤�Ǥ��԰���ˤĤ��ư��ä���, ������Ǥ�
 �����絤�Ǥ��԰���ˤĤ��ư���.

 �簵�԰���Ǥ�, $\beta$���̤��θ�������������������ä��Ϥޤä���,
 ������Ǥ�, 3-4�������줿�ݥƥ󥷥�뱲�������������ä�Ϥ��.
 ������Ǽ�갷�����Ȥ����ʬ����褦��, �ݥƥ󥷥�뱲����������
 �����絤�򵭽Ҥ���.

 ������ǤϤ��Υݥƥ󥷥�뱲���������� 1)������, ��
 2)�Ρ��ޥ�⡼�ɤβ��Ƴ��, �򤪤��ʤ�.

 ����Ū���԰�����˴ؤ��Ƥϼ�갷��ʤ�.
 ���������ˤ������԰���˴ؤ��Ƥϼ�����Ǽ�갷��.
\end{itembox}

��ž�絤�ˤ������ľ��������®�������������뷹���԰����
Charney(1947)��, 
���θ�Eady(1949)
�ˤ�äƻϤ�Ƹ��椵�줿.
���Ϲ�ή�������Ϥ����԰�����������Τ˻Ȥ��Ƥ���.
�ݥƥ󥷥�뱲��������(4-64)
\footnote{
 $$
  \left( \DP{}{t} + \Dvect{V}_{\Psi} \cdot \nabla \right)
  \left[ \zeta + \beta_0 y + e^Z \DP{}{Z}
   \left(
    \frac{f_0 e^{-Z}}{\Gamma} \DP{\phi'}{Z}
   \right)
  \right] = 0.
  \eqno \textrm{(4-64)}
 $$ 
}
�ϰʲ��Τ褦�˵��Ҥ����.
%
\begin{eqnarray}
 \left(
  \DP{}{t} + \Dvect{k} \times \nabla \psi^{*} \cdot \nabla
 \right) q = 0.
\end{eqnarray}
%
�ʤ�, {\bfseries �ݥƥ󥷥�뱲��}({\itshape potential vorticity})
$q$��
%
\begin{eqnarray}
 q = \Dlapla \psi^{*}
  + e^Z \DP{}{Z}
  \left(
   \frac{{f_0}^2 e^{-Z}}{\Gamma} \DP{\psi^{*}}{Z}
  \right) + \beta y
\end{eqnarray}
%
�Ǥ���.
�ޤ�, $\psi^{*} = {f_0}^{-1} \phi'$���Ϲ�ήή���ؿ�[(4-61)
\footnote{
 $$
  \psi = {f_0}^{-1} \phi'.
  \eqno \textrm{(4-61)}
 $$  
}����]
�Ǥ���, $\Gamma$����Ū������(4-47)
\footnote{
 $$
  \Gamma (Z) = \DP{}{Z}
   \left(
    \DP{\overline{\phi}}{Z} + \kappa \overline{\phi}
   \right)
   =
   \left(
    \DP{\overline{T}}{Z} + \kappa \overline{T}
   \right)
   =
   \frac{H^2 g}{T}
   \left(
    \frac{g}{c_p} + \frac{1}{H}\DP{\overline{T}}{Z}
   \right).
   \eqno \textrm{(4-47)}
 $$  
}
�Ǥ���.
��������ɬ�פȤ����Ǯ�ϳ���1ˡ§(4-54)
\footnote{
 $$
  \DP{}{t} \DP{\phi'}{Z} + \Dvect{V}_{\Psi} \cdot \nabla \DP{\phi'}{Z}
   + \Gamma (Z) \dot{Z} = 0.
   \eqno \textrm{(4-54)}
 $$
}
�ϰʲ��Τ褦�˵��Ҥ���
\footnote{����Ū�ˤ���Ǯ�ϳ���1ˡ§�򶭳����Ȥ��ư����Τ�,
����Ǥ����ʤ�.}.
%
\begin{eqnarray}
 \left(
  \DP{}{t} + \Dvect{k} \times \nabla \psi^{*} \cdot \nabla
 \right)
 \DP{\psi^{*}}{Z} + {f_0}^{-1} \Gamma \dot{Z} = 0.
\end{eqnarray}
%
$Z$��ɸ�Ϥ������԰���θ���������Ǥ���.
�ʤ��ʤ�, $Z$��ɸ�ϤϹ��ٺ�ɸ�Ϥ�̩�ܤʴط��ˤ���
\footnote{
 $Z$��ɸ��$p$(����)��ɸ��̩�ܤʴط��ˤ��뤳�Ȥ�(4-33)
 $$
  \DP{}{p} = - \frac{1}{p} \DP{}{Z}
 $$
 ����狼��. ���Ϻ�ɸ�Ϥȹ��ٺ�ɸ�ϤȤ�̩�ܤʴط��ˤ��뤳�Ȥ�
 ���� 1-9��ǽҤ٤��Ƥ��뤳�ȤǤ���.
},
����Τ˲��ٸ�Ψ$- \partial T / \partial z$�������
���ˤ�$\Gamma$������ˤʤ뤫��Ǥ���.
���Ϻ�ɸ�Ϥˤ�������Ū�����٤�����Ǥ��뤳�Ȥ�, ���ʤ��󸽼�Ū�Ǥ���.

�ݥƥ󥷥�뱲����������, ή���ؿ���ʲ��Τ褦�ˤ��뤳�Ȥ�����������.
%
\begin{eqnarray}
 \psi^{*} (x,y,Z,t) = \overline{\psi} (y,Z) + \psi (x,y,Z,t).
\end{eqnarray}
%
$|\psi |/ | \overline{\psi} |$�Ͼ������Ȳ��ꤹ��.
���δ��ܾ��$U = - \partial \overline{\psi} / \partial y$��Ϳ������.
(5-33)���Ѥ����, (5-30)�ϰʲ��Τ褦�ʴ�ñ�ʷ��ˤʤ�
\footnote{�ݥƥ󥷥�뱲�٤��������˴ؤ��Ƥ�{\bf 5-7-A}����.}.
%
\begin{eqnarray}
 \left( \DP{}{t} + U \DP{}{x} \right) q'
  + \DP{\psi}{x} \DP{\overline{q}}{y} = 0.
\end{eqnarray}
%
�ʤ�,
%
\begin{eqnarray}
 q' &=& \Dlapla \psi + e^{Z} \DP{}{Z}
  \left( \frac{{f_0}^2 e^{-Z}}{\Gamma} \DP{\psi}{Z} \right)
  \\
 \DP{\overline{q}}{y} &=& \beta - \DP[2]{U}{y}
  - e^{Z} \DP{}{Z} \left( \frac{{f_0}^2 e^{-Z}}{\Gamma} \DP{U}{Z} \right)
\end{eqnarray}
%
�Ǥ���.
(5-36)�Ͻ簵�θ���(��2��)�ȷ����θ���(��3��)��ξ����ޤ�.
����������Ǯ�ϳ���1ˡ§��,
%
\begin{eqnarray}
 \left( \DP{}{t} + U \DP{}{x} \right) \DP{\psi}{Z}
  - \DP{\psi}{x} \DP{U}{Z} + {f_0}^{-1} \Gamma \dot{Z} = 0
\end{eqnarray}
%
�Ǥ���. $\dot{Z}$�ϱ�ľ������ư�ξ���Ǥ���.
���μ��ϱ�ľ�ζ�������ɬ�פȤ���.

������$\psi = \Psi (y,Z) e^{i \mu (x - ct)}$��(5-34)��Ƴ�������,
�ʲ��μ���������.
%
\begin{eqnarray}
 (U - c)
  \left[
   \DP[2]{\Psi}{y}
   + e^{Z} \DP{}{Z} \left( \frac{{f_0}^2 e^{-Z}}{\Gamma} \DP{\Psi}{Z} \right)
   - \mu^2 \Psi
  \right]
  + \DP{\overline{q}}{y} \Psi = 0.
\end{eqnarray}
%
Ǯ�ϳ���1ˡ§����ϰʲ��μ���������.
%
\begin{eqnarray}
 (U - c) \DP{\Psi}{Z} - \DP{U}{Z} \Psi + {f_0}^{-1} \Gamma W = 0.
\end{eqnarray}
%
�ʤ�, $\dot{Z} = W (y,z) e^{i \mu (x - ct)}$�Ǥ���.



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\subsection*{5-7-A\qquad �ݥƥ󥷥�뱲����������������}
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�ݥƥ󥷥�뱲�٤��������򤪤��ʤ�.
����Ū�ˤ�, �ݥƥ󥷥�뱲��������
%
\begin{eqnarray}
 \Deqlab{hw-04-07-�ݥƥ󥷥�뱲��������1}
  & &
  \left(
   \DP{}{t} + \Dvect{k} \times \nabla \psi^{*} \cdot \nabla
  \right) q = 0
 \\
 & &
  q = \Dlapla \psi^{*}
  + e^Z \DP{}{Z}
  \left(
   \frac{{f_0}^2 e^{-Z}}{\Gamma} \DP{\psi^{*}}{Z}
  \right) + \beta y
  \nonumber
\end{eqnarray}
%
��
%
\begin{eqnarray}
 \Deqlab{hw-04-07-�ݥƥ󥷥�뱲��������������1}
  \psi^{*} (x,y,Z,t) = \overline{\psi} (y,Z) + \psi (x,y,Z,t),
  \\
 |\psi |/ | \overline{\psi} | \ll 1, \qquad
  U = - \partial \overline{\psi} / \partial y
  \hspace{8mm}
  \nonumber
\end{eqnarray}
%
�ˤ�ä���������,
%
\begin{eqnarray}
 \Deqlab{hw-04-07-�����ݥƥ󥷥�뱲��������1}
 & &
  \left( \DP{}{t} + U \DP{}{x} \right) q'
  + \DP{\psi}{x} \DP{\overline{q}}{y} = 0
  \\
 & &
  q' (x,y,Z,t)= \Dlapla \psi + e^{Z} \DP{}{Z}
  \left( \frac{{f_0}^2 e^{-Z}}{\Gamma} \DP{\psi}{Z} \right)
  \nonumber \\ 
 & &
  \DP{\overline{q} (y,Z)}{y} = \beta - \DP[2]{U}{y}
  - e^{Z} \DP{}{Z} \left( \frac{{f_0}^2 e^{-Z}}{\Gamma} \DP{U}{Z} \right)
  \nonumber
\end{eqnarray}
%
��Ƴ��.

\Deqref{hw-04-07-�ݥƥ󥷥�뱲��������1}����ʬ�黻�Ҥ�,
\Deqref{hw-04-07-�ݥƥ󥷥�뱲��������������1}���,
%
\begin{eqnarray}
 \DP{}{t} + \Dvect{k} \times \nabla \psi^{*} \cdot \nabla
  &=&
  \DP{}{t} +
  \left(
   - \DP{\psi^{*}}{y} \Dvect{i} + \DP{\psi^{*}}{x} \Dvect{j}
  \right) \cdot \nabla
  \nonumber \\
 &=& 
  \DP{}{t} +
  \left(
   - \DP{\psi^{*}}{y} \DP{}{x} + \DP{\psi^{*}}{x} \DP{}{y}
  \right) \cdot \nabla
  \nonumber \\
 &=& 
  \DP{}{t} +
  \left(
   - \DP{\overline{\psi}}{y} \DP{}{x} - \DP{\psi}{y} \DP{}{x}
   + \DP{\psi}{x} \DP{}{y}
  \right) 
  \nonumber \\
 \Deqlab{hw-04-07-�ݥƥ󥷥�뱲������������ʬ�黻��}
  &=& 
  \left( \DP{}{t} + U \DP{}{x} \right)
   + \left( - \DP{\psi}{y} \DP{}{x} + \DP{\psi}{x} \DP{}{y} \right)
\end{eqnarray}
%
�Ȥʤ�.
�����2��Ͼ����ޤळ�Ȥ��狼��.
�ޤ�, \Deqref{hw-04-07-�ݥƥ󥷥�뱲��������1}��$q$��,
\Deqref{hw-04-07-�ݥƥ󥷥�뱲��������������1}���,
%
\begin{eqnarray}
 q &=&
  \Dlapla \left( \overline{\psi} + \psi \right)
  + e^Z \DP{}{Z}
  \left(
   \frac{{f_0}^2 e^{-Z}}{\Gamma} \DP{}{Z} \left( \overline{\psi} + \psi \right)
  \right) + \beta y
  \nonumber \\
 &=&
  \Dlapla \psi
  + e^Z \DP{}{Z}
  \left(
   \frac{{f_0}^2 e^{-Z}}{\Gamma} \DP{\psi}{Z} 
  \right)
  \nonumber \\
 & &  \hspace{10mm} + 
  \Dlapla \overline{\psi}
  + e^Z \DP{}{Z}
  \left(
   \frac{{f_0}^2 e^{-Z}}{\Gamma} \DP{\overline{\psi}}{Z} 
  \right) + \beta y
  \nonumber \\
 \Deqlab{hw-04-07-�ݥƥ󥷥�뱲�ټ��Τ�������}
  &=& q'(x,y,Z,t) +
  \left\{
   \DP[2]{\overline{\psi}}{y}
   + e^Z \DP{}{Z}
   \left(
    \frac{{f_0}^2 e^{-Z}}{\Gamma} \DP{\overline{\psi}}{Z} 
   \right) + \beta y
  \right\}
\end{eqnarray}
%
�Ȥʤ�.
1���ܤι�Ͼ���Ǥ���, �Ĥ�ι�ϴ��ܾ�ι�Ǥ���.
\Deqref{hw-04-07-�ݥƥ󥷥�뱲������������ʬ�黻��}��
\Deqref{hw-04-07-�ݥƥ󥷥�뱲�ټ��Τ�������}���,
���ܾ��$x$�˰�¸���ʤ����Ȥ����դ����,
\Deqref{hw-04-07-�����ݥƥ󥷥�뱲��������1}��������.

;�̤���, ���������줿Ǯ�ϳ���1ˡ§(5-37)��Ʊ�ͤ���ˡ����������.


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