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% 2002/11/05  ���� ����   ����
% 2002/11/13  ���� ����   ����
% 2002/12/27  ���� ����   ���߸�ν���
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\section{����ٲ���}

\begin{itembox}[l]{����}
 \Dparskip \vspace{-6mm}
 ���ѵ���, ����٤ǤΥ���������Ϥ�Ԥ�.

 ���β��Ϥˤ�{\bfseries ���̶��}({\itshape beta plane approximation})
 ���Ѥ���Τ������Ǥ���.
 ���̶���Ȥϥ��ꥪ���Ϥΰ��������Ѳ�Ψ�����ȶ�����Ƥ��ޤ�
 (�Ĥޤ��$df/dy \simeq \beta_0$(���)�Ȥ���)��ˡ�Ǥ���.
 ����ˤ�äƱ�����������ȯ���������ǤΥ��ꥪ���Ϥΰ�������ñ�ˤʤ�.
 ���Φ��̶�����Ѥ��Ʊ���������, ȯ��������, Ǯ�ϳؤμ����ѷ����Ƥ�����
 �Ȥǽ��Ϲ�ή��������Ƴ��.
 
 ����ˤ��α���, ȯ��, Ǯ�ϳؤμ�����
 {\bfseries ���Ϲ�ή�ݥƥ󥷥�뱲��������}
 ({\itshape quasi-geostrophic potential vorticity equation})��
 {\bfseries ���Ϲ�ή��ľ��ư������}
 ({\itshape quasi-geostrophic vertical motion equation})
 ��Ƴ��.
 ����2����򤯤���ˤ��絤�ξ�������, ���������ζ����������
 ɬ�פ�����.
 
 �Ǹ�ˤ�����᤿������(����������, Ǯ�ϳ�������,
 ���Ϲ�ή�ݥƥ󥷥�뱲��������, ���Ϲ�ή��ľ��ư������)��
 ���Ϻ�ɸ�Ϥ�ɽ������.
\end{itembox}

���ѵ��ϤǤ�����٤ξ����ͤ���.
���ξ��, $L \sim 10^6$m �Ȥʤ�Τǳ��ѿ��ϰʲ��Τ褦�ˤʤ�.
%
\begin{eqnarray}
 R_O \sim 0.1, \quad
  L/a \sim 0.1, \quad
  \varepsilon \sim 1, \quad
  \beta a / f \sim 1.
  \nonumber
\end{eqnarray}
%
�ʤ�, $a \sim 6.4 \times 10^6$m , $f \sim 10^{-4}$ sec$^{-1}$�Ǥ���.
$\varepsilon \sim 1$�Ǥ�����, ����������(4-43)��Ǯ�ϳ���1ˡ§(4-46)
����ʲ��δط���������
\footnote{
 ���δط�����, (4-46)�ˤ����ƽ��פʻ�����ʬ�ι�ȴ��ܾ�β��ٰ�ή�ι�
 $\dot{Z} \Gamma (Z) = \dot{Z} \ \partial / \partial Z
 (\partial \overline{\phi} / \partial Z + \kappa \overline{\phi})$
 ��Х�󥹤����褦�Ȥ��������������ΤǤ���.
}.
%
\begin{eqnarray}
 R_1 \lesssim R_O.
  \nonumber
\end{eqnarray}
%
���ä�, $R_1 = R_O$�Ȥ���Τ�Ŭ�ڤǤ���. (4-43), (4-44), (4-46)�ˤ�����
$R_O$��������⾮����������������Ĺ����Ȥ���, �ʲ����������Ϥ�
������.
%
\begin{eqnarray}
 \DP{\zeta}{t} + \Dvect{V}_{\Psi} \cdot \nabla (\zeta + f)
  + fD &=& 0,
  \\
 \Dlapla \phi' - f \zeta &=& 0,
  \\
 \DP{}{t} \DP{\phi'}{Z} + \Dvect{V}_{\Psi} \cdot \nabla \DP{\phi'}{Z}
  + \Gamma (Z) \dot{Z} &=& 0.
\end{eqnarray}
%
���٤μ�(4-52)�ˤ����Ƥϰʲ��ιब̵�뤵���.
̵�뤵���Τ�, ���б��٤�ȯ����ʬ�ˤ���ή
\footnote{(4-43)����3, 5��.}, ���٤α�ľ��ή\footnote{(4-43)����6��.},
$\zeta$��ޤ�ȯ���ι�\footnote{(4-43)����8��.},
������Ω���夬���\footnote{(4-43)����9, 10��.}�Ǥ���.
Ǯ�ϳ���1ˡ§(4-54)�ˤ����Ʋ��٤�ȯ����ʬ�ˤ���ή
\footnote{
 �����(4-32)���$RT = \partial \phi / \partial Z$�ʤΤ�(4-46)����3��
 ������ˤ�����.
}
�Ȱ����٤���ư
\footnote{
 (4-46)����5�ब����ˤ�����.
 �ʤ�������$\Gamma(Z) = \partial / \partial Z
 (\partial \overline{\phi} / \partial Z + \kappa \overline{\phi})$
 �Ǥ��뤳�Ȥ���,
 $\partial / \partial Z
 (\partial \phi' / \partial Z + \kappa \phi')$�������٤���ư�Ǥ���
 ���Ȥ��狼�������.
}
��̵�뤵���.

���̴����ؤ��θ���������{\bfseries ���̶��}
({\itshape beta plane approximation})��Ƴ������������Ǥ���.
�ǥ���Ⱥ�ɸ�ϰʲ��Τ褦����������.
%
\begin{eqnarray}
 \nabla = \Dvect{i} \DP{}{x} + \Dvect{j} \DP{}{y},
  \quad
  \Dvect{V} = u \Dvect{i} + v \Dvect{j}.
  \nonumber
\end{eqnarray}
%
$x$�����������礭���ʤ�, $y$�����������礭���ʤ�.
(4-52), (4-53)�ˤ�����$f$�ζ����Ѳ��ϰ�ή��ǤΤ߽��פǤ���.
��ή��Ǥ�$f$�ζ����Ѳ���
%
\begin{eqnarray}
 \Dvect{V}_{\Psi} \cdot \nabla f
  = v_{\Psi} \DD{f}{y}
  = v_{\Psi} \beta
\end{eqnarray}
%
�Ȥʤ�. �ʤ�$\beta = df/dy$
\footnote{2-6�Ỳ��.}�Ǥ���Τ�, 
%
\begin{eqnarray}
 \beta = \DD{}{y} \left( 2 \Omega \sin \varphi \right)
  = 2 \Omega \cos \varphi \DD{\varphi}{y}
  = \frac{2 \Omega \cos \varphi}{a}
\end{eqnarray}
%
�Ǥ���.
���ѵ��Ϥ�����٤Ǥ�$f$���Ѳ��Ͻ��פǤϤʤ�����$f$��ñ�ʤ뷸���Ȥʤ�.
����򼨤�����, $f$��$y=y_0$�Τޤ��ǥƥ��顼Ÿ������.
%
\begin{eqnarray}
 f = f_0 + \beta_0 (y - y_0) + \cdots
  = f_0
  \left[
   1 + \frac{\beta_0}{f_0} (y-y_0) + \cdots 
  \right].
\end{eqnarray}
%
$f_0 = f(y_0), \beta_0 = \beta(y_0) = d f_0/dt$�Ǥ���.
���ϤǤϾ���Υ��������$L$�Ȱ��äƤ���Τ�, $y$���Ѳ���
$|y-y_0| \lesssim L$�����¤���Τ�Ŭ�ڤǤ���.
����٤�$\beta$�Υ��������$\beta \sim f/a$�Ǥ��뤿��,
(4-57)����2��ϰʲ��Τ褦������������.
%
\begin{eqnarray}
 \left| \frac{\beta_0}{f_0} (y-y_0) \right| \lesssim L/a.
\end{eqnarray}
%
$L/a \sim 0.1$�Ǥ��뤿��, (4-58)���$|(\beta_0/f_0) (y-y_0)| \lesssim0.1$
�Ǥ���, ���Τ���(4-57)����2��ʹߤϾ������Ȥ���̵�뤵���.
���η��, ���ѵ��Ϥ�����٤Ǥ�$f=f_0$�ȹͤ��뤳�Ȥ��Ǥ���.
������,  $df / dy $�˴ؤ��Ƥ�$f$������Ȱ��鷺, 
$df/dy = \beta_0 =$����, �Ȱ���.
���(4-55)��$\Dvect{V}_{\Psi} \cdot \nabla f = v_{\Psi} \beta_0$�Ȥʤ�.
���̶����, �������Ϥ�ޤ����Ѻ�ɸ�Ϥ��ѷ�����, �ޥåץե�������
\footnote{
 �ޥåץե��������˴ؤ��Ƥ�1-8�Ỳ��.
 �ޥåץե��������ϴ�ñ�˸�����, ��ƿޤν̼ܤǤ���.
 ��ñ�����󤲤��, �˼ͱƿ�ˡ�ˤ�����ޥåץե���������
 $m(\varphi) = 2/(1 + \sin \varphi)$�Ǥ���. ¾����ƿޤˤ����
 �ޥåץե���������������Τ��Ѥ�뤬, ����$\varphi$�λ��Ѵؿ��˰�¸����
 ���Ȥ˴ؤ��Ƥ϶��̤��Ƥ���.
}
�ζ����Ѳ���̵�뤵��ƻϤ�������������.
�ޥåץե������������̰��٤λ��Ѵؿ��Ǥ���. $L/a$�����������ˤϤ���
���Ѵؿ���������֤���������
\footnote{
 ���ꥪ����$f = 2 \Omega \sin \varphi$����٤λ��Ѵؿ��Ǥ���.
 ���������ȶ���������Ȥ�Ʊ�ͤ�, �ޥåץե���������������֤�������
 ���Ȥ������.
}
�Τǥޥåץե��������ζ����Ѳ���̵��Ǥ���.
Phillips(1963)
���������Ϥ��륫�ȥ��ɸ�Ϥǵ��Ҥ��뤳�Ȥ�,
Ÿ�����줿���Ƥ�ʪ���̤�����ӡ����ǵ��Ҥ��뤳�Ȥ�$Z$-��������
�����Ū�ʥ���������Ϥ򤪤��ʤä�.
��β��Ϥˤ�ä�, �ǥ���Ⱥ�ɸ�Ϥλ��ѤϤ����ǹͤ��Ƥ����ѿ����Ŭ�ڤǤ�
�뤳�Ȥ������줿.

����������(4-52)�ϰʲ��Τ褦�˵��ҤǤ���.
%
\begin{eqnarray}
 \DP{\zeta}{t} + \Dvect{V}_{\Psi} \cdot \nabla (\zeta + \beta_0 y)
  - f_0 e^Z \DP{}{Z} (e^{-Z} \dot{Z}) = 0.
\end{eqnarray}
%
��2���ܤǤ�$\Dvect{V}_{\Psi} \cdot \nabla f = \Dvect{V}_{\Psi} \cdot
\nabla (\beta_0 y)$�Ȥ����ѷ��򤪤��ʤä���, �����(4-55)���
�ط���
$\Dvect{V}_{\Psi} \cdot \nabla (\beta_0 y) =
\beta_0 \Dvect{V}_{\Psi} \cdot \nabla y = \beta_0 v_{\Psi} =
\Dvect{V}_{\Psi} \cdot \nabla f$
������Ω�Ĥ���Ǥ���. Ϣ³�μ�(4-45)�򤹤����ѷ��������Ѥ��뤳�Ȥ�
��3���ܤ�$D$�Ϻ�����뤳�Ȥ������.
ȯ���μ�(4-53)��(4-14)���ʲ��Τ褦�ˤʤ�.
%
\begin{eqnarray}
 \Dlapla \phi' = f_0 \zeta = f_0 \Dlapla \Psi.
\end{eqnarray}
%
���μ���Ŭ�ڤʲ��
%
\begin{eqnarray}
 \Psi = {f_0}^{-1} \phi'
\end{eqnarray}
%
�Ǥ���. (4-13), (4-14)������β�ž��ʬ�ȱ��٤Ϥ��줾��
%
\begin{eqnarray}
 \Dvect{V}_{\Psi} = {f_0}^{-1} \Dvect{k} \times \nabla \phi',
  \quad
  \zeta = {f_0}^{-1} \Dlapla \phi'
\end{eqnarray}
%
�Ȥʤ�. (4-54), (4-59), (4-62)�򥻥åȤ�
{\bfseries ���Ϲ�ή������}({\itshape quasi-geostrophic equation})
�ȸƤ�.

(4-54)��(4-59)��$\dot{Z}$�������뤳�ȤǤ��뼰��������.
�ޤ�, (4-54)��$e^{-Z}/\Gamma$�򤫤�, $Z$������ʬ�����, $e^{-Z} \dot{Z}$��
�ؤ���ʲ��μ���������.
%
\begin{eqnarray}
 \DP{}{Z} (e^{-Z} \dot{Z})
  = - \DP{}{t} \DP{}{Z} \left( \frac{e^{-Z}}{\Gamma} \DP{\phi'}{Z} \right)
  - \Dvect{V}_{\Psi} \cdot \nabla \DP{}{Z}
  \left( \frac{e^{-Z}}{\Gamma}\DP{\phi'}{Z} \right).
\end{eqnarray}
%
�ʤ�, (4-62)���$\partial \Dvect{V}_{\Psi} / \partial Z $��ޤ���
���Ǥ���
\footnote{
 (4-54)�ΰ�ή�ι��$e^{-Z}/\Gamma$�򤫤�, $Z$������ʬ���Ƥߤ��,
 %
 \begin{eqnarray}
  \DP{}{Z}
   \left(
    \frac{e^{-Z}}{\Gamma} \Dvect{V}_{\Psi} \cdot \nabla \DP{\phi'}{Z}
   \right)
   &=& \DP{}{Z}
   \left\{
    \Dvect{V}_{\Psi} \cdot \nabla
    \left( \frac{e^{-Z}}{\Gamma} \DP{\phi'}{Z} \right)
   \right\}
   \nonumber \\
  &=& \DP{\Dvect{V}_{\Psi}}{Z}
   \cdot \nabla \left( \frac{e^{-Z}}{\Gamma} \DP{\phi'}{Z} \right)
   + \Dvect{V}_{\Psi} \cdot \nabla
   \DP{}{Z} \left( \frac{e^{-Z}}{\Gamma} \DP{\phi'}{Z} \right)
   \nonumber
 \end{eqnarray}
 %
 �Ǥ���. (4-62)���
 $\Dvect{V}_{\Psi} = {f_0}^{-1} (- \partial \phi' / \partial y \ \Dvect{i}
 + \partial \phi' / \partial x \ \Dvect{j})$�Ǥ��뤳�Ȥ���,
 $\partial \Dvect{V}_{\Psi} / \partial Z$��ޤ��Ͼ��Ǥ��뤳�Ȥ�
 �狼��.
}.
(4-63)��(4-59)��������, ����Ū�ʹ��ޤȤ��Ȱʲ��μ��ˤʤ�.
%
\begin{eqnarray}
 \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.
\end{eqnarray}
%
�����{\bfseries ���Ϲ�ή�ݥƥ󥷥�뱲��������}
({\itshape quasi-geostrophic potential vorticity equation})
�ȸƤФ�뼰��, �����������Ϥˤ�������Ϲ�ή��������(4-28)��
���椵���
\footnote{
 (4-28)��(4-64)�Ȥ�����˴ؤ��Ƥ�{\bf 4-4-A}����. 
}.
(4-62)��Ȥ��Ȥ�����������$\phi'$�ι�����ǵ��Ҥ��뤳�Ȥ�
�Ǥ���. ���줬�ʲ��μ��Ǥ���.
%
\begin{eqnarray}
 \left[ \Dlapla + {f_0}^2 e^Z \DP{}{Z}
  \left(
   \frac{e^{-Z}}{\Gamma} \DP{}{Z}
  \right)
 \right] \DP{\phi'}{t}
 \hspace{60mm}
 \nonumber \\
 = - \Dvect{k} \times \nabla \phi' \cdot \nabla
 \left[
  {f_0}^{-1} \Dlapla \phi' + \beta_0 y + e^Z \DP{}{Z}
  \left(
   \frac{f_0 e^{-Z}}{\Gamma} \DP{\phi'}{Z}
  \right)
 \right].
\end{eqnarray}
%
$\phi'$�ν�����֤ˤ�����ʬ�ۤ�����ʬ���äƤ���Ȥ����, ���դϷ׻��Ǥ�
��. ���դα黻�Ҥ��ʱ߷�
�ʤΤ�������������郎Ϳ�����Ƥ����
(4-65)��$\partial \phi' / \partial t$�˴ؤ��Ʋ򤯤��Ȥ������.

���Ϲ�ή�������Ϥ�������֤�$Z$�����Ǥ���.
�ʲ��ν�ľ���줿����������(4-59)��Ǯ�ϳ���1ˡ§�μ�(4-54)�򸫤��ߤ���.
%
\begin{eqnarray}
 \DP{\zeta}{t} = \frac{1}{f_0} \Dlapla \DP{\phi'}{t}
  = - \Dvect{V}_{\Psi} \cdot \nabla (\zeta + \beta_0 y )
  + f_0 e^Z \DP{}{Z} \left( e^{-Z} \dot{Z} \right),
  \\
 R \DP{T'}{t} = \DP{}{Z} \DP{\phi'}{t}
  = - \Dvect{V}_{\Psi} \cdot \nabla \DP{\phi'}{Z} - \Gamma (Z) \dot{Z}.
\end{eqnarray}
%
����2�����Ȥ߹�碌�뤳�Ȥ�, 
���ٰ�ή $- \Dvect{V}_{\Psi} \cdot \nabla (\zeta + \beta_0 y)$ ��
���ٰ�ή $- \Dvect{V}_{\Psi} \cdot \nabla (\partial \phi'/ \partial Z)$
 ��Ϳ�����$\dot{Z}$����դ˷���Ǥ���褦�ʼ�[(4-68)]�����뤳�Ȥ������. 
������Ϲ�ήŪ, �ſ尵Ū�ʾ���$\partial \phi' / \partial t$�ˤ�ä�
$\partial \zeta / \partial t$, �����$\partial T' / \partial t$��Ϳ����
��뤳�ȤȻ��Ƥ���.
$\dot{Z}$�˴ؤ��������������(4-66)��$\partial / \partial Z$����
(4-67)��${f_0}^{-1} \Dlapla$��������Ȥ�������.
�ʲ������ο����������Ǥ���.
%
\begin{eqnarray}
 \Gamma (Z) \Dlapla \dot{Z} + {f_0}^2 \DP{}{Z}
  \left(
   e^Z \DP{}{Z} (e^{-Z} \dot{Z})
  \right)
  \hspace{60mm}
  \nonumber \\
  = f_0 \DP{}{Z}
  \left[
   \Dvect{V}_{\Psi} \cdot \nabla (\zeta + \beta_0 y)
  \right]
  - \Dlapla
  \left[
   \Dvect{V}_{\Psi} \cdot \nabla (\partial \phi' / \partial Z)
  \right]. 
\end{eqnarray}
%
�����{\bfseries ���Ϲ�ή��ľ��ư������}
({\itshape quasi-geostrophic vertical motion equation})�ȸƤФ��.
���������ν��Ϲ�ήȯ��������(4-29)�Ȼ������Ǥ���.
���դ�(4-62)��Ȥ����Ȥ�����$\phi'$�ι�ǽ񤯤��Ȥ������.
������֤�$\phi'$��ʬ�ۤ�Ϳ�����Ƥ�����, ���դα黻�Ҥ�
�ʱ߱黻�ҤʤΤǤ�����������$\dot{Z}$�˴ؤ��Ʋ򤯤��Ȥ������.
(4-68)����$\dot{Z}$����������, (4-45)����ȯ�������뤳�Ȥ������.
���Τ褦�˹ͤ����, (4-62)��Ȥ����Ȥ�ȯ���ϱ��٤δؿ��Ǥ��뤳�Ȥ�
�狼��
\footnote{
 (4-68)�α��դϽ�ľ���ȱ���$\zeta$�ǽ񤯤��Ȥ������.
 ($ \nabla^2 [ \Dvect{V}_{\Psi} \cdot \nabla (\partial \phi' / \partial Z)]
  = {f_0}^{-1} [\Dvect{V}_{\Psi} \cdot \nabla (\partial / \partial Z) \zeta]$ ).
 ���Τ��Ȥ���$\dot{Z}$�ϱ��٤δؿ��ȸ��뤳�Ȥ������.
 (4-45)����ȯ����$\dot{Z}$�δؿ��ʤΤ�, ȯ���ϱ��٤δؿ���
 ���뤳�Ȥ������.
}.
���Τ褦��, �����ӡ��������������ϲ�ž�ˤ�붯���Ϥ�����Ư�����Ȥ�
�狼��.

��(4-68)�Ϥ��Ф������ѵ��Ϥθ���ˤ����ƿ����Ѥ�ƻ��Ȥ����Ѥ�����.
����⤷���Ͻ��Ϲ�ή�Ǥ�ͽ��򤹤���, $\phi'$�줬����ʬ���äƤ���ʤ��
(4-68)��򤯤��ȤǱ�ľ�����α�ư���Τ뤳�Ȥ������.
���α�ľ��ư�ξ�Ϲ߿��̤���ѥ�����ȴط�����.

(4-65)��(4-68)��򤳤��Ȥ���ʤ��, �絤�ξ����Ȳ����ζ����������ɬ
�פ�����.
$\dot{Z}$��ʬ���äƤ�����, (4-65)��򤯤Τ�ɬ�פ�
$(\partial / \partial Z)( \partial \phi' / \partial t)$��(4-54)����������.
�絤��ü($Z \rightarrow \infty$)�Ǥζ�������$\dot{Z}$��ͭ�¤Ǥ��뤳��
�Ǥ���\footnote{(4-34)�򻲾Ȥ���.}.
���������ˤ�����$\dot{Z}$����ꤹ�뤿��ˤ�$\dot{Z}$��$w$��ط��դ���
ɬ�פ�����.
$gw = d \phi / dt$��Ÿ������Ȥ��줾��ι�Υ�������ϰʲ��Τ褦�ˤʤ�.
%
\begin{eqnarray}
 gw \ = \ \DP{\phi'}{t} \ + \ \Dvect{V}_{\Psi} \cdot \nabla \phi'
  \ + \ \Dvect{V}_{\chi} \cdot \nabla \phi' \ + \ \dot{Z} \DP{\phi'}{Z}
  \ + \ \dot{Z} R \overline{T}.
  \hspace{8mm}
  \\
 fV^2 \hspace{12mm}
 fV^2 \hspace{13mm}
 R_1 fV^2 \hspace{10mm}
 R_1 fV^2 \hspace{6mm}
 \frac{R_1}{R_O} M fV^2
 \nonumber
\end{eqnarray}
%
�����Ǥ�$\partial \overline{\phi} / \partial Z = R \overline{T}$�δط���
���Ѥ���. �ޤ������Ǥ�$M$�ϰʲ��Τ褦����������.
%
\begin{eqnarray}
 M \equiv R \overline{T} / f^2 L^2.
\end{eqnarray}
%
����ϲ�ž�ե롼�ɿ��εտ��Ǥ���.
�����ǹͤ��Ƥ��륹������Ǥ�$R_1 = R_O$�Ǥ���, �ޤ�,
$R \overline{T} \sim 10^5$ m$^2$ sec$^{-2}$�Ǥ������
%
\begin{eqnarray}
 M \sim 10
\end{eqnarray}
%
�Ǥ���. ���ʤߤ�3-2��ǲ�ž�ե롼�ɿ�$F = f^2 L^2 / \overline{\phi}$��
�Ҳ𤷤���, $\overline{\phi} = R \overline{T}$�Ǥ�����
\footnote{
 ����������(4-32)���
 $\partial \overline{\phi} / \partial Z = R \overline{T}$�Ǥ���.

 �ʤ�, ���β������ή���Ǥϳ���������.
}, 
$M = F^{-1}$�Ȥʤ�.
�����ξ�ﲼ�Ǥ�(4-69)�α��դκǸ�ιब����Ū�Ȥʤ�,
���ϰʲ��Τ褦�ʴ�ñ�ʷ��ˤʤ�.
%
\begin{eqnarray}
 gw = R \overline{T} \dot{Z}. 
\end{eqnarray}
%
���μ������ľ®�٤�ŵ��Ū���ͤ�$w \sim g^{-1} f V^2 M \sim 1$ cm sec$^{-1}$
����ʬ����.
(4-72)��������$Z = Z_s$��Ŭ�Ѥ����,
%
\begin{eqnarray}
 \dot{Z_s} = (g/R \overline{T}) w_s = (g/R \overline{T})
  \Dvect{V} \cdot \nabla h_s
\end{eqnarray}
%
�Ȥʤ�. $h_s (x,y)$�ϲ��������ι⤵�Ǥ���
\footnote{
 ���������ϼ�ͳɽ�̤ǤϤʤ��ΤǶɽ�ˤ���������Ѳ��Ϥ��ʤ�.
 ��ä�, $d h_s / dt = \Dvect{V} \cdot \nabla h_s$�Ȥʤ�.
}.
�����ޥ��ؤξ�ü�ˤ������ľ��ư��ä����뤬,
Phillips(1963)
�ˤ�äƤ���Ͼ����᤮�Ƥ��β��Ϥˤ����ƤϽ��פǤϤʤ�
���Ȥ������줿.
�⤷��ɽ�̤˻�(�Ĥޤ�ϵ���)��̵������(4-73)��$\dot{Z_s} = 0$�Ȥʤ�.

(4-73)�����$Z$, �㤨��$Z=0$�Ȥ��Ƥߤ뤳�Ȥ�ͭ�ѤǤ���.
�ʤ�, $Z=0$��$p=p_0$���б�����.
$Z$������Ȥ��Ƥߤ�����, �����(4-73)�Ȥ� (�������ѿ���) ������,
(4-73)��$Z$�Τޤ��ǥƥ��顼Ÿ�����뤳�Ȥ��Τ뤳�Ȥ������.
���θ������礭����ɾ��������̤�$Z_s$�Ȥʤ�.
��餫�ʻ�����$H/10$�ޤ��Ϥ��褽1km����⤯�ʤ��¤��,
���ζ����1/10�ʲ��Υ��������ι��̵�뤹�뤳�Ȥ�������.

���Ϲ�ή���������Ϻ�ɸ�Ϥ�ɽ�����뤳�Ȥϸ塹�����Ǥ���.
(4-33)��(4-34)��Ȥ��ȱ���������(4-59)��Ǯ�ϳ���1ˡ§(4-54)�ϰʲ��Τ褦
�ˤʤ�.
%
\begin{eqnarray}
 \left( \DP{}{t} + \Dvect{V}_{\Psi} \cdot \nabla \right)
  (\zeta + \beta_0 y) - f_0 \DP{\omega}{p} = 0,
  \\
 \frac{\partial^2 \phi'}{\partial t \partial p} +
  \Dvect{V}_{\Psi} \cdot \nabla (\partial \phi' / \partial p)
  + \sigma (p) \ \omega = 0.
\end{eqnarray}
%
$\sigma = \Gamma / p^2 = (1/p) (d/dp) [p (d \overline{\phi} / dp)
- \overline{\phi}]$�Ǥ���.
�ݥƥ󥷥�뱲��������(4-64)�ϰʲ��Τ褦�˵��ҤǤ���.
%
\begin{eqnarray}
 \left(
  \DP{}{t} + {f_0}^{-1} \Dvect{k} \times \nabla \phi' \cdot \nabla
 \right)
 \left[
  {f_0}^{-1} \Dlapla \phi' + \beta_0 y + \DP{}{p}
  \left( \frac{f_0}{\sigma (p)} \DP{\phi'}{p} \right)
 \right]
 = 0.
\end{eqnarray}
%
��ľ������ư������(4-68)�ϰʲ��Τ褦�ʷ��ˤʤ�.
%
\begin{eqnarray}
 \sigma \Dlapla \omega + {f_0}^{2} \DP[2]{\omega}{p}
  = f_0 \DP{}{p} \left( \Dvect{V}_{\Psi} \cdot \nabla (\zeta + \beta y) \right)
  - \Dlapla \left( \Dvect{V}_{\Psi} \cdot \nabla \DP{\phi'}{p} \right).
\end{eqnarray}
%
������������{\bfseries ���Ϲ�ή��������}
({\itshape quasi-geostrophic omega equation})�ȸƤФ��.


%\begin{center}
% {\bf 
% --------------------------------------
% ��ʸ�����ޤ�
% --------------------------------------
% } 
%\end{center}


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%%%%      4-4-A      %%%%
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\subsection*{4-4-A\qquad �簵�ȷ����Ǥν��Ϲ�ή�ݥƥ󥷥�뱲��������}
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�簵(������������)�Ǥν��Ϲ�ή�ݥƥ󥷥�뱲��������(4-28)��
�����Ǥν��Ϲ�ή�ݥƥ󥷥�뱲��������(4-64)����Ӥ��Ƥߤ�.
���줾��, �簵, �������ν��Ϲ�ή�ݥƥ󥷥�뱲���������ϰʲ���
�̤�Ǥ���.

�簵, ���Ϲ�ή�ݥƥ󥷥�뱲��������
%
\begin{eqnarray}
 \Deqlab{hw-03-04-�簵���Ϲ�ή�ݥƥ󥷥�뱲��������}
  \left( \DP{}{t} + \Dvect{V}_{\Psi} \cdot \nabla \right)
  \left( \zeta + f - f \frac{\overline{\phi} + \phi'}{\overline{\phi}} \right)
  =0.
\end{eqnarray}

����, ���Ϲ�ή�ݥƥ󥷥�뱲��������
%
\begin{eqnarray}
 \Deqlab{hw-03-04-�������Ϲ�ή�ݥƥ󥷥�뱲��������}
  \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. 
\end{eqnarray}

\Deqref{hw-03-04-�������Ϲ�ή�ݥƥ󥷥�뱲��������}�ˤ�����
$\beta_0 y$�������$f$�ʤΤ�, �����Ǥϴ�ñ�Τ���$\beta_0 y \simeq f$��
�ͤ���.
����Ȥ��줾��μ��ϰʲ��Τ褦�ˤʤ�.
%
\begin{eqnarray}
 \Deqlab{hw-03-04-�簵���Ϲ�ή�ݥƥ󥷥�뱲��������2}
  \left( \DP{}{t} + \Dvect{V}_{\Psi} \cdot \nabla \right)
  \left(
   \zeta + f
   - \hspace{10mm} 
   f \frac{\overline{\phi} + \phi'}{\overline{\phi}}
   \hspace{10mm} 
  \right)
  &=& 0
  \\
 \Deqlab{hw-03-04-�������Ϲ�ή�ݥƥ󥷥�뱲��������2}
  \left( \DP{}{t} + \Dvect{V}_{\Psi} \cdot \nabla \right)
  \left[ \zeta + f + e^Z \DP{}{Z}
   \left(
    \frac{f_0 e^{-Z}}{\Gamma} \DP{\phi'}{Z}
   \right)
  \right]
  &=& 0. 
\end{eqnarray}
%
���Τ褦�˸����, ���դκǸ�ιब�簵�ȷ����ˤ��㤤����ʬ����.
��ñ�����򤹤뤿��,
\Deqref{hw-03-04-�������Ϲ�ή�ݥƥ󥷥�뱲��������2}
�κ��պǸ�ι��(4-32)
\footnote{
 $$
  RT = \partial \phi / \partial Z
  \eqno \textrm{(4-32)}
 $$
}
���Ѥ��Ʋ��٤�ɽ������.
%
\begin{eqnarray}
 e^Z \DP{}{Z}
  \left(
   \frac{f_0 e^{-Z}}{\Gamma} \DP{\phi'}{Z}
  \right)
  &=&
 e^Z \DP{}{Z}
  \left(
   \frac{f_0 e^{-Z}}{R \Gamma} T'
  \right)
  \nonumber \\
 &=& \frac{f_0}{R \Gamma} \DP{T'}{Z}
  - \frac{f_0}{R \Gamma} T'
  \nonumber \\
 \Deqlab{hw-03-04-�դ�T��}
  &\simeq& \frac{f_0}{R}
  \dfrac{\displaystyle \DP{T'}{Z}}
  {\dfrac{H^2 g}{\overline{T}} \left( \dfrac{g}{c_p} + \dfrac{1}{H}
				\displaystyle \DP{\overline{T}}{Z} \right)}
\end{eqnarray}
%
�����, $T' \ll \partial T' / \partial Z$�Ȥ��������,
(4-47)
\footnote{
 $$
  \Gamma (Z) \simeq \frac{H^2 g}{\overline{T}}
  \left( \frac{g}{c_p} + \frac{1}{H} \DP{\overline{T}}{Z} \right)
  \eqno \textrm{(4-47)}
 $$
}
���Ѥ���.
\Deqref{hw-03-04-�դ�T��}���Ѥ���,
\Deqref{hw-03-04-�簵���Ϲ�ή�ݥƥ󥷥�뱲��������2}��
\Deqref{hw-03-04-�������Ϲ�ή�ݥƥ󥷥�뱲��������2}��
��ľ����,
%
\begin{eqnarray}
 \Deqlab{hw-03-04-�簵���Ϲ�ή�ݥƥ󥷥�뱲��������3}
  \left( \DP{}{t} + \Dvect{V}_{\Psi} \cdot \nabla \right)
  \left(
   \eta
   - \hspace{13mm} 
   f \frac{\overline{\phi} + \phi'}{\overline{\phi}}
   \hspace{13mm} 
  \right)
  &=& 0
  \\
 \Deqlab{hw-03-04-�������Ϲ�ή�ݥƥ󥷥�뱲��������3}
  \left( \DP{}{t} + \Dvect{V}_{\Psi} \cdot \nabla \right)
  \left[ \eta + 
   \frac{f_0}{R}
   \dfrac{\displaystyle \DP{T'}{Z}}
   {\dfrac{H^2 g}{\overline{T}} \left( \dfrac{g}{c_p} + \dfrac{1}{H}
				 \displaystyle \DP{\overline{T}}{Z} \right)}
  \right]
  &=& 0
\end{eqnarray}
%
�Ȥʤ�. �ʤ�$\eta$�����б��٤Ǥ���.
������޼�����.
%
\begin{figure}[htbp]
 \begin{center}
  \Depsf[][]{./eps/barotropic_QG.eps}
 \end{center}
 \begin{quote}
  \caption{
  {\footnotesize �簵�Ǥα��٤ȹ⤵$h (= \phi / g)$�δط�}
  }
  \Dfiglab{hwfig-03-04-�簵�Ǥα��٤ȹ⤵�δط�}
 \end{quote}
\end{figure}
%
\Dfigref{hwfig-03-04-�簵�Ǥα��٤ȹ⤵�δط�}���,
�ؤ������ʤ�Ȥ����Ǥϱ��������ʤ�, �����ʤ�Ȥ����Ǥϱ����夯�ʤ뤳�Ȥ�
�狼��.
�����Τ��Ȥϰ����˥ݥƥ󥷥�뱲����¸�μ�(3-39b)
\footnote{
 $$
  \DD{}{t} \left( \frac{\zeta + f}{h} \right) = 0
  \eqno \textrm{(3-39b)}
 $$
}
�ǹͤ������Ȥ�Ʊ���Ǥ���.
%
\begin{figure}[htbp]
 \begin{center}
  \Depsf[][]{./eps/baroclinic_QG.eps}
 \end{center}
 \begin{quote}
  \caption{
  {\footnotesize �����Ǥα��٤Ȳ���
  $T (= \partial \phi / \partial Z = g \partial z / \partial Z)$
  �δط�}
  }
  \Dfiglab{hwfig-03-04-�����Ǥα��٤Ȳ��٤δط�}
 \end{quote}
\end{figure}
%
\Dfigref{hwfig-03-04-�����Ǥα��٤Ȳ��٤δط�}�Ǥ�
$T_2$, $T_3$�֤��ؤȹͤ����, �����ؤ������ʤ뤳�ȤǱ��������ʤ뤳�Ȥ�
�狼��. 
�����ι�¤��, �簵�ǹͤ�����Ĥ��ؤ�Ť͹�碌��¿�ع�¤�ȹͤ��뤳�Ȥ�
�Ǥ���.





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%\subsection*{Appendix \qquad }

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