%
% Chapter 4   Atmospheric Waves: Part II
%             4-9 TWO-LEVEL MODEL
%
% 2002/12/31  ���� ����   ����
% 2003/01/06  ���� ����   ����
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\markboth{5-9 2�إ�ǥ�}{5-9 2�إ�ǥ�}
%\markright{4-9 2�إ�ǥ�}
%\Ddate{2003/01/06}
%\Dfile{hw-04-09.tex}

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\section{2�إ�ǥ�}

\begin{itembox}[l]{����}
 \Dparskip \vspace{-6mm}
 ����κǸ�ˤ����ʤä�Charney(1947)��ǥ��ʣ�����ä������,
 �����԰����������Ū�˲򤯤��ȤϤ��ʤ��ä�.
 ����Ǥ�, ���Υ�ǥ���ϴ�ñ�����줿��ľ��¤�����, ���
 Charney��ǥ��Ʊ���褦�ʷ�̤����뤳�Ȥ������
 {\bfseries 2�إ�ǥ�}�ȸƤФ���ǥ���԰������ˤĤ��Ƽ�갷��.

 2�إ�ǥ�Ȥ�, 1)�絤��4�ؤ�ʬ��, 2)��֤��ؤǤ�����Ϳ��,
 3)������ؤǤϱ�ľ��ư��Ϳ����, ��ǥ�Ǥ���.
 (��5-8�˼����Ƥ���).

 ���Ѥ�����������$p$��ľ��ɸ�Ϥα���������, �����Ǯ�ϳ���1ˡ§�Ǥ���.
 (�����μ���3-4��κǸ��Ƴ�Ф��줿).
 �����μ����������������2�إ�ǥ��Ŭ�Ѥ���.
 ����ˤ����μ�����ư���Ŭ�Ѥ��ư���®�٤����.
 ��Ϥ���ޤǤ���Ǥ����ʤä��Τ�Ʊ���褦��, ����®�٤���������Ĥ�
 �ɤ�����Ĵ�٤뤳�Ȥ��԰������ξ���Ƴ��.

 �����������԰������ξ�狼��, 
 �԰������ϼ���Ϲ����α�ľ�������Ȥ���Ĺ�ˤ�äƻ��ۤ���뤳�Ȥ��狼��.

 2�إ�ǥ���԰������ξ���Charney��ǥ���԰������ξ��ϰ츫�ޤä���
 �㤦�褦�˸����뤬, ������Ĵ�٤Ƥߤ�Ȥ���2�Ĥ��ɤ����Ƥ��뤳�Ȥ�
 �狼�äƤ���.
 �����Τ��Ȥ���2�إ�ǥ�����ѵ��Ϥˤ����뷹���԰����Ȥο����񤤤�
 ��������ɽ���Ƥ���ȸ�����.
 �ޤ�, ��¬����⤳�Τ��Ȥ�΢�դ�����Ƥ��뤳�Ȥ��狼�äƤ���.
 
 �ʤ�, ���������ǥ��ͥ륮���μ���Ƴ���Ƥ���.
 ���μ���2�إ�ǥ�ˤ�����ͭ�����֥��ͥ륮���ȱ�ư���ͥ륮����
 �Ѵ��ˤĤ������򤹤�Τ������ʤ���Ǥ���.
\end{itembox}

Charney(1947)��ǥ�ΰ������������ϲ���Ū�˲򤯤Τ������񤷤�����,
���˴�ñ��������ľ��¤����Ĥ褦�ʥ�ǥ��ͤ�������ͭ�ѤǤ���.
{\bfseries 2�إ�ǥ�}��, �絤��4�ؤ�ʬ��, 1,3�ؤǤ�����, ������ؤǤ�
��ľ��ư��Ϳ����.
���������, ����Ū�ˤ����˴�ñ�Ǥ���, �����Ƥ��η�̤�Charney��ǥ��
���פ���.
��ή���˴�ñ�˱��Ѥ��뤿��, ���β��Ϥϰ��Ϻ�ɸ�ϤǤ����ʤ�.
����������(4-74)
\footnote{
 $$
  \left( \DP{}{t} + \Dvect{V}_{\Psi} \cdot \nabla \right)
  (\zeta + \beta_0 y) - f_0 \DP{\omega}{p} = 0.
  \eqno \textrm{(4-74)}
 $$
}
��Ǯ�ϳ���1ˡ§(4-75)
\footnote{
 $$
 \frac{\partial^2 \phi}{\partial t \partial p} +
  \Dvect{V}_{\Psi} \cdot \nabla (\partial \phi / \partial p)
  + \sigma (p) \ \omega = 0.
  \eqno \textrm{(4-75)}
 $$
}
�ϰʲ��Τ褦�˵��ҤǤ���
\footnote{
 �����ѷ���4-7���(5-30), (5-32)���᤿��ˡ��Ʊ���Ǥ���.
 �ѷ���ˡ�ξܺ٤Ϥ�����򻲾ȤΤ���.
}.
%
\begin{eqnarray}
 \left(
  \DP{}{t} + \Dvect{k} \times \nabla \psi^{*} \cdot \nabla
 \right)
 \left(
  \Dlapla \psi^{*} + \beta y
 \right)
 - f_0 \DP{\omega}{p} &=& 0,
 \\
 \left(
  \DP{}{t} + \Dvect{k} \times \nabla \psi^{*} \cdot \nabla
 \right)
 \DP{\psi^{*}}{p} + \frac{\sigma}{f_0} \omega
 &=& 0.
\end{eqnarray}
%
2�إ�ǥ�Ǥ�, (5-57)��(5-58)��ͭ�º�ʬ��Ȥ�������Ǥ���.
�����, �������Ȥ߹�碌�뤳�Ȥǥݥƥ󥷥�뱲����������ͭ�º�ʬ��
ľ�ܵ��뤳�Ȥ������.
�Ϥ�����ˤ�ä�, ���������������ϼ�ưŪ�˴ޤޤ�뤳�Ȥˤʤ�.

�ǽ��(5-57)��(5-58)���������򤪤��ʤ�.
�������Τ���, �ʲ��Τ褦�����ꤹ��
\footnote{
 �����, 4-7���(5-33)
 $$
  \psi^{*} (x,y,Z,t) = \overline{\psi} (y,Z) + \psi (x,y,Z,t)
 $$
 ���������ȼ¼�Ū�ˤϤۤ�Ʊ�����Ȥ򤷤Ƥ���.
 4-7��Ǥ�$U = - \partial \overline{\psi} / \partial y$�Ȳ��ꤷ�Ƥ���,
 �����񤭴������$\overline{\psi} = - U y + c$�Ȥʤ������.
 ��ʬ�ؿ���$c = 0$�Ȥ����, (5-33)��(5-59)��Ʊ�����Ȥ���ʬ����.
}.
%
\begin{eqnarray}
 \left.
 \begin{array}{l}
  \psi^{*} = - U (p) y + \psi (x,p,t) \\
  \omega = \omega' (x,p,t)
 \end{array}
 \right\}
\end{eqnarray}
%
�����μ��Ǥ�, �������δ��ܾ�Ȥ���$U(p)$��¸�ߤ�,
����˲ä�$y$����Ω�ʾ���줬¸�ߤ��뤳�Ȥ򼨤��Ƥ���.
(5-59)��(5-57), (5-58)���������뤳�Ȥǰʲ��μ���������.
%
\begin{eqnarray}
 \left( \DP{}{t} + U \DP{}{x} \right) \DP[2]{\psi}{x}
  + \beta \DP{\psi}{x} - f_0 \DP{\omega'}{p}
  &=& 0
  \\
 \left( \DP{}{t} + U \DP{}{x} \right) \DP{\psi}{p}
  + \DD{U}{p} \DP{\psi}{x} - \frac{\sigma}{f_0} \omega'
  &=& 0 
\end{eqnarray}
%
��������������, $\psi$��$\omega'$���������ʤ����Ǥ��äƤ�
�����Ǥ���.

2�إ�ǥ�α�ľ��¤����5-8�˼�����Ƥ���.
%
\begin{figure}[htbp]
 \begin{center}
  \Depsf[][]{./eps/hwfig-04-08.eps}
 \end{center}
 \begin{quote}
  \caption{
  {\footnotesize 2�إ⡼�ɤα�ľ��¤
  (Haltiner, G. J., and Williams, R. T., 1979:
      Numerical Prediction and Dynamic Meteorology.
      {\itshape John Wiley \& Sons,} 477pp)}
  }
  \Dfiglab{hwfig-04-08}
 \end{quote}
\end{figure}
%
��������, ��ɽ��$p=p_s$����ή������$p=p_T$�ˤ�����$\omega' = 0$
�Ǥ��뤳�ȤǤ���.
��Ԥξ��������ʶ���ȸ�����.
�ʤ��ʤ�, ���ط��ˤ����Ƥ���Ū�����٤��礭��,
���Τ��Ȥ����ľ�α�ư�Ͼ������ʤ뤫��Ǥ���.

1��, 3�ؤ˱���������(5-60)���Ѥ�, �Ǹ�ι�˽���ͭ�º�ʬ
���Ѥ���Ȱʲ��μ���������.
%
\begin{eqnarray}
 \left( \DP{}{t} + U_1 \DP{}{x} \right) \DP[2]{\psi_1}{x}
  + \beta \DP{\psi_1}{x} - f_0 \frac{{\omega'}_2}{\Delta p}
  &=& 0,
  \\
 \left( \DP{}{t} + U_3 \DP{}{x} \right) \DP[2]{\psi_3}{x}
  + \beta \DP{\psi_3}{x} + f_0 \frac{{\omega'}_2}{\Delta p}
  &=& 0.
\end{eqnarray}
%
�ĤäƤ��뼰��, Ǯ�ϳ���1ˡ§��2�ؤ����Ѥ��뤳�ȤǤ��뤳�Ȥ������:
%
\begin{eqnarray}
 \left[ \DP{}{t} + \left( \frac{U_1 + U_3}{2} \right) \DP{}{x} \right]
  (\psi_1 - \psi_3)
  \hspace{60mm}
  \nonumber \\
  - \left( \frac{U_1 - U_3}{2} \right)
  \left( \DP{\psi_1}{x} + \DP{\psi_3}{x} \right)
  - \left( \frac{\Delta p \sigma_2}{f_0} \right)
  {\omega'}_2 = 0.
\end{eqnarray}
%
���μ���$- \Delta p$��ݤ�����ΤǤ���.
�����Ĥ����̤�, 1�ؤ�3�ؤ��ͤ�ʿ�Ѥ�������
\footnote{
 $(U_1 + U_3) / 2$��$\partial / \partial x \{(\psi_1 + \psi_3)/2\}$�Τ���
 �Ǥ���.
}.
(5-62), (5-63), ������(5-64)��̤�Τδؿ�$\psi_1, \psi_3, {\omega'}_2$
�˴ؤ����������Ǥ���.

�����μ��������, ���ͥ륮���μ���Ƴ��.
���ͥ륮���μ��ϲ�ΰ�̣���᤹��Τ˻Ȥ����Ȥ��Ǥ�, �����ʤ���Ǥ���.
(5-62)��$- \psi_1$��ݤ�, (5-63)��$- \psi_3$��ݤ�, ­����碌�Ƥ���
$x$�˴ؤ���ʿ�Ѥ�Ȥ��, �ʲ��μ���������.
%
\begin{eqnarray}
 \DD{}{t}
  \left[
   \frac{\overline{ \left(\partial \psi_1 / \partial x \right)^2
   + \left(\partial \psi_3 / \partial x \right)^2}}{2}
  \right]
  = - \left( f_0 / \Delta p \right)
  \overline{{\omega'}_2 (\psi_1 - \psi_3)}.
\end{eqnarray}
%
���դϾ���α�ư���ͥ륮���λ����Ѳ�Ψ�Ǥ���
\footnote{2�إ�ǥ�Ǥ�, $u' = 0, v' = \partial \psi / \partial x$�Ǥ���.}.
���֥��ͥ륮���μ���(5-64)��$\lambda (\psi_1 - \psi_3)$��ݤ�,
$x$�˴ؤ���ʿ�Ѥ�Ȥ뤳�Ȥ����뤳�Ȥ������:
%
\begin{eqnarray}
 \DD{}{t}
  \left[
   \frac{\overline{\lambda (\psi_1 - \psi_3)^2}}{4}
  \right]
  = \frac{\lambda}{4} (U_1 - U_3)
  \overline{(\psi_1 - \psi_3) \left( \DP{\psi_1}{x} + \DP{\psi_3}{x} \right)}
  \hspace{20mm}
  \nonumber \\
  + \left( f_0 / \Delta p \right)
  \overline{{\omega'}_2 (\psi_1 - \psi_3)}.
\end{eqnarray}
%
�ʤ�,
%
\begin{eqnarray}
 \lambda = 2 {f_0}^2 / (\Delta p^2 \sigma_2)
\end{eqnarray}
%
�Ǥ���.
$\psi_1 - \psi_3$�ϲ��٤ξ���α�ľ������ʿ���ͤ����㤹�뤳�Ȥ�
���դ����ߤ���
\footnote{
 $\psi^{*} = {f_0}^{-1} \phi$���$\psi = {f_0}^{-1} \phi'$�Ǥ���.
 �����, (4-32)���, $RT' = -p \ \partial \phi' / \partial p$�Ǥ���.
 ���Τ��Ȥ���, $\psi_1 - \psi_3$�ϲ��٤ξ���α�ľ������ʿ���ͤ�
 ���㤹��ȸ�����.
}.
(5-66)�κ��դϾ����ͭ���ݥƥ󥷥�륨�ͥ륮���λ����Ѳ�Ψ�Ǥ���.
1-11��ˤ�����, ͭ�����֥��ͥ륮����, �����̾�Ǥβ��٤��Ѳ���
��Ū�����٤�ʿ���ͤǳ�ä���Τ˶���Ǥ��뤳�Ȥϼ�����
\footnote{
 $\overline{\lambda (\psi_1 - \psi_3)^2}$��ͭ�����֥��ͥ륮���˴ؤ��Ƥ�
 {\bfseries 5-9-A}�򻲾�.
}.
(5-65)�κǸ�ι��, �����ͭ�����֥��ͥ륮���������α�ư���ͥ륮���ؤ�
�Ѵ��ι�Ǥ���.
�ʤ��ʤ�, (5-66)����ˤ���εդ����Τ�Τ�¸�ߤ��뤫��Ǥ���.
���ι��, �Ȥ����絤($\psi_1 - \psi_3 > 0$)��
�徺����($\omega_2 < 0$
\footnote{
 $\omega = dp/dt$�Ǥ���, $p$��ɸ�ϤǤα�ľ®�٤Ǥ���.
 ����$p$����ɽ�̤Ǥ��礭��, ����ˤ椯�ˤĤ쾮�����ʤ�Τ�,
 �絤�ξ徺��$\omega < 0$��ɽ�����. 
})
�����䤿������($\psi_1 - \psi_3 < 0$)�����ߤ���($\omega_2 > 0$)
�������ˤʤ�
\footnote{
 ���Τ��Ȥ˴ؤ���ľ��Ū�ʳ�ǰ�ޤ�{\bfseries 5-9-B}�˺ܤ���.
}.
���ξ��, �絤�νſ��������˲�����Τ�, ���֥��ͥ륮���ϱ�ư���ͥ륮����
�Ѵ������.
�Ϲ����α�ľ����$U_1 - U_3$��¸�ߤ�, ���٤δ��ܾ��$y$�ޤ�������������
���ۤ��̣����\footnote{�פ����, �������Τ��Ȥ���äƤ���.}.
(5-66)�α��դ���1��ι��($U_1 - U_3$�˴ط�����)���ܾ��
ͭ�����֥��ͥ륮����������ͭ�����֥��ͥ륮���ؤ��Ѵ���ɽ���Ƥ���.
���ι��, �̸�����ư��
($(\partial \psi_1 / \partial x + \partial \psi_3 / \partial x)/2 > 0$
\footnote{$\partial \psi / \partial x = v$�Ǥ���.})
�絤���Ȥ���($\psi_1 - \psi_3 > 0$)����
�������ư��
($(\partial \psi_1 / \partial x + \partial \psi_3 / \partial x)/2 < 0$)
�絤���䤿��($\psi_1 - \psi_3 > 0$)�������ˤʤ�.
�⤷��$y$�ζ�����¸�ߤ�����, ���Υץ�������$U_1 - U_3$�Ȥʤ�.

�Ǥ�, �����(5-62)��(5-64)���.
�ʲ�����ư����������ηϤ��Ѥ���.
%
\begin{eqnarray}
 \psi_1 &=& \Psi_1 e^{i \mu (x - ct)},
  \nonumber \\
 \psi_3 &=& \Psi_3 e^{i \mu (x - ct)},
  \\
 {\omega'}_2 &=& \left( \Delta p {f_0}^{-1} \right)
  W e^{i \mu (x - ct)}.
  \nonumber
\end{eqnarray}
%
(5-68)��(5-62), (5-63), (5-64)�����������, �ʲ��μ���������.
%
\setcounter{equation}{0}
\def\theequation{5-69\alph{equation}}
%
\begin{eqnarray}
 \mu \left[ \beta - \mu^2 (U_1 - c) \right] \Psi_1 + i W &=& 0,
  \\
 \mu \left[ \beta - \mu^2 (U_3 - c) \right] \Psi_1 - i W &=& 0,
  \\
 \mu [U_3 - c] \Psi_1 - \mu [U_1 - c] \Psi_3 + i 2 W / \lambda &=& 0.
\end{eqnarray}
%
\def\theequation{\arabic{chapter}-\arabic{equation}}
\setcounter{equation}{69}
\hspace{-4mm}
%
����$\Psi_1, \Psi_2, W$�������Ǥʤ��ʤ��, (5-69)�η����ι��󼰤�
�ä��ʤ���Фʤ�ʤ�.
���ξ��ˤ��, �ʲ���$c$�˴ؤ���2����������Ƴ�����.
%
\begin{eqnarray}
 2 \mu^2 \left(\mu^2 + \lambda \right) c^2
  + \left[
     2 \beta \left( 2 \mu^2 + \lambda \right)
     - 2 \mu^2 \left( \mu^2 + \lambda \right) \left( U_1 + U_3 \right)
    \right] c
  \hspace{20mm}
  \nonumber \\
 + 2 \beta^2
  - \beta \left( 2 \mu^2 + \lambda \right) \left( U_1 + U_3 \right)
  + 2 \mu^4 U_1 U_3
  + \mu^2 \lambda \left( {U_1}^2 + {U_3}^2 \right) = 0.
  \nonumber
\end{eqnarray}
%
2���������β�θ�����Ȥ����Ȥ�, �ʲ���$c$�˴ؤ��뼰��������
\footnote{
 2��������$y = ax^2 + 2bx + c$�β�θ����ϰʲ��Ǥ���.
 $$
  x = - \frac{- b \pm \sqrt{b^2 - ac}}{a}.
 $$
}.
%
\begin{eqnarray}
 c = (U_1 + U_3) / 2
  - \frac{\beta \left( 2 \mu^2 + \lambda \right)}{2 \mu^2 \left( \mu^2 + \lambda\right)}
  \pm \frac{\left[ \lambda^2 \beta^2 - \mu^4 \left( \lambda^2 - \mu^4 \right) (U_1 - U_3)^2 \right]^{1/2}}
  {2 \mu^2 \left( \mu^2 + \lambda \right)}.
\end{eqnarray}
%
(5-70)��ʿ�����⤬��ˤʤ��, $c$��ʣ�ǿ��Ȥʤ�, ���β���԰���ˤʤ�.
�ºݤ�, $\mu^4$��$\lambda^2$��꾮����, $(U_1 - U_3)^2$����ʬ���礭����,
�԰���ϵ�����.
���椬�¿��ʤ��, ��Ω��������.
��Ĺ($\mu^{-1}$)�ȱ�ľ����($U_1 - U_3$)
�κ�ɸ�Ϥˤ�����������Υ����������Ǥ�,
���礦�ɺ��椬�ä���������԰�����ΰ�ζ����ȤʤäƤ���.
���Ρֺ��椬�ä���׾��֤ϰʲ��Τ褦�˵��ҤǤ���.
%
\begin{eqnarray}
 (U_1 - U_3)^2 =
  \frac{\lambda^2 \beta^2}{\mu^4 \left( \lambda^2 - \mu^4 \right)}.
\end{eqnarray}
%
���ζ����Ͽ�5-9�Ǽ�����Ƥ���.
(���οޤǤ�$\lambda$, $\beta$�ϸ��ꤵ��Ƥ���).
%
\begin{figure}[htb]
 \begin{center}
  \Depsf[][]{./eps/hwfig-04-09.eps}
 \end{center}
 \begin{quote}
  \caption{
  {\footnotesize 2�إ�ǥ�ˤ�����, $U_1 - U_3$��$\mu^{-1}$�δؿ��Ȥ��Ƥ�
  ��ĹΨ$n = \mu |c_i|$�Υ����.
  $\lambda = 5.88 \times 10^{-12}$ m$^{-2}$,
  $\beta = 1.67 \times 10^{11}$ m$^{-1}$ s$^{-1}$.
  $\mu^{-1}$, $U_1 - U_3$, $n$��ñ�̤Ϥ��줾��,
  $10^6$ m, m s$^{-1}$, $10^{-5}$ s$^{-1}$.
  (Haltiner, G. J., and Williams, R. T., 1979:
      Numerical Prediction and Dynamic Meteorology.
      {\itshape John Wiley \& Sons,} 477pp)}
  }
  \Dfiglab{hwfig-04-09}
 \end{quote}
\end{figure}
%
�ʲ������������԰�����ΰ��û��¦($\mu^{-1} \rightarrow 0$)��
�����ȤʤäƤ���.
%
\begin{eqnarray}
 \mu^{-1} = \lambda^{-1/2} =
  \Delta p {\sigma_2}^{1/2} / \left( f_0 \sqrt{2} \right).
\end{eqnarray}
%
Ĺ��¦($\mu^{-1} \rightarrow \infty$)�Ǥ�,
$\mu^2 \ll \lambda$�ʤΤ�, (5-71)�ϰʲ��Τ褦�ˤʤ�.
%
\begin{eqnarray}
 |U_1 - U_3| = \beta / \mu^2.
\end{eqnarray}
%
(5-71)��$\mu$�˴ؤ��Ƥ���ʬ�������ˤʤ�褦�ˤ��뤳�Ȥ�,
(5-71)�κǾ��ͤ�, �Ǿ��ͤ�Ȥ���Ĺ$\mu^{-1}$���Τ뤳�Ȥ������.
$\mu^{-1} = 2^{1/4} \lambda^{-1/2}$�λ���(5-71)�ϰʲ��κǾ��ͤ�Ȥ�.
%
\begin{eqnarray}
 (U_1 - U_3)_{\textrm{min}} = 2 \beta / \lambda
  = \beta \frac{\Delta p^2 \sigma_2}{2 {f_0}^2}.
\end{eqnarray}
%
2�ؤν��Ϲ�ή��ǥ�ΰ������������⤳����Ʊ���ˤʤ�.
�ʤ��ʤ�, �ѿ���Ŭ�ڤ�Ʊ��뤹���, ���Ϲ�ή��ǥ��ޤ�
�󰵽̤�������2�Ĥ��ؤ���Ĥ���Ǥ���.

û�Ȥ�����(5-72)����Ū�����٤ȴط�����
\footnote{$\sigma = \Gamma / p^2$�Ǥ���.}.
�ޤ�, Ĺ�Ȥΰ�����(5-73)��$\beta$�θ��̤ȴط�����.
�ʤ��ʤ�(5-73)�ϥ����ӡ��Ȥμ���$\beta$�ι�˻��Ƥ��뤫��Ǥ���.
((5-10)
\footnote{
 $$
  c = U - \beta / \mu^2.
  \eqno \textrm{(5-10)}
 $$
}
����).
�԰����ľ�����κǾ��ͤ�$\beta$����Ū�����٤��Ѥ����㤹��.
���򤤤Τ�, �ϵ�β�ž�򥼥��Ȥ���Ȥ����԰��������ä��Ƥ��ޤ����ȤǤ���.
$\beta = 0$�Ȥ����ü�ʾ��ˤ�, �԰�������ɬ�׾��ϰʲ��Τ褦�˽񤱤�.
%
$$
 \mu^{-1} > \Delta p {\sigma_2}^{1/2} / \left( f_0 \sqrt{2} \right),
  \quad
  U_1 \neq U_3.
  \eqno \textrm{(5-75a)}
$$
%
�����, �ʲ���Eady(1949)�β�Ǥ��б�������(5-50)����Ӥ��뤳�Ȥ������.
%
$$
 \mu^{-1} > \Gamma^{1/2} / \left( f_0 \ 2.4 \right),
  \quad
  S = dU/dZ \neq 0.
  \eqno \textrm{(5-75b)}
$$
%
�����ξ������餫��Ʊ���褦�ʷ��򤷤Ƥ���.
$\beta \neq 0$�Ǥ������Ū�ʾ��ˤ�, 2�إ�ǥ�ˤ�����������Υ����������
(��5-9)��, Charney��Ϣ³��ǥ�(��5-7)�Υ����������Ȥ�ɽ��Ū�ˤ�����
�ۤʤäƸ�����.
��ԤǤ�, ����Ω�Ǥ�������ζ����ʳ���, ���ƤΥ������������԰���Ǥ���.
�������ʤ���, Ϣ³��ǥ�Υ����������ΤۤȤ�ɤ���ĹΨ�����˾�����.
�⤷���໤�����ä����ˤ�, �������Ȥ���Ⱦ�ϼ����ƾä��Ƥ��ޤ�.
������ĹΨ�Ǹ��Ƥߤ���, ���¾���԰����ΰ�Ͽ�5-7�ˤ�������ĹΨ��������
�ΰ�Ĥˤ�äưϤޤ��.
������, 2�إ�ǥ�Υ����������Ǥ����5-9�ˤ�����������ζ���(5-71)��
Ʊ���褦�ʷ��򤷤Ƥ���.
���Τ��Ȥ���, 2�إ�ǥ�����ѵ��Ϥˤ����뷹���԰����Ȥο����񤤤�
�绨�Ĥˤ�ɽ���Ƥ���ȸ�����.
�Ǥ��԰������Ĺ�Ϥ��褽4000��5000km�Ǥ���, ������Ĺ�ϴ�¬��������ѵ���
���ȤΤ�ΤȰ��פ���.


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


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%             �ʲ���Appendix             %%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%   Appendix �Ѥ� Counter, OutputFormat ������   %%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%   Set Counter (chapter, section etc. ) %%%%%%%%
%\setcounter{chapter}{2}
%\setcounter{section}{0}
%\setcounter{footnote}{0}
\setcounter{equation}{0}
%\setcounter{page}{1}
\setcounter{figure}{0}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%        Counter Output Format           %%%%%%%%
\def\theequation{A-\arabic{equation}}
%\def\thesection{\arabic{chapter}-\arabic{section}}
%\def\thesubsection{\arabic{chapter}-\arabic{section}-\arabic{subsection}}
\def\thefigure{A-\arabic{figure}}
\def\thetable{A-\arabic{table}}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%      5-9-A      %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{5-9-A\qquad ͭ�����֥��ͥ륮��}
\setcounter{equation}{0}
\setcounter{figure}{0}
\def\theequation{5-9-A-\arabic{equation}}
\def\thefigure{5-9-A-\arabic{figure}}

(5-66)�ˤ�����$\overline{\lambda (\psi_1 - \psi_3)^2}$��
1-11���ͭ�����֥��ͥ륮���μ�(2-98)���б����뤳�Ȥ򼨤�.
�ʤ�, �ʲ���(2-98)��ƷǤ���.
%
\begin{eqnarray}
 \Deqlab{hw-04-09-ͭ�����֥��ͥ륮���μ�}
  \overline{A}
  \doteq
  \frac{1}{2} \int^{p_s}_{0}
  \frac{\overline{T}}{\gamma_d - \overline{\gamma}}
  \overline{
  \left(
   \frac{T'}{\overline{T}}
  \right)^2
  }
  \ dp  \ .
  \\
 \left(
  \overline{\gamma} = - \partial \overline{T} /\partial z , \quad
  \gamma_d = g/c_p
 \right)
  \hspace{2mm}
  \nonumber 
\end{eqnarray}
%
$\overline{\lambda (\psi_1 - \psi_3)^2}$��(5-67)���Ѥ���Ȱʲ��Τ褦��
��ľ�����Ȥ��Ǥ���.
%
\begin{eqnarray}
 \Deqlab{hw-04-09-ή����ɽ������ͭ�����֥��ͥ륮��}
  \overline{2 {f_0}^2 (\psi_1 - \psi_3)^2 \frac{p^2}{\Gamma \Delta p^2}}
  = 2 {f_0}^2 \overline{\frac{p^2}{\Delta p^2}} \ 
  \overline{\frac{(\psi_1 - \psi_3)^2}{\Gamma}}.
\end{eqnarray}
%
�ʤ�, $\sigma = \Gamma / p^2$�Ǥ���. (4-4�Ỳ��).
\Deqref{hw-04-09-ͭ�����֥��ͥ륮���μ�}�ˤ�����
������Ǯ��Ψ�Ȳ��ٸ�Ψ�κ�$\gamma_d - \overline{\gamma}$��
\Deqref{hw-04-09-ή����ɽ������ͭ�����֥��ͥ륮��}��
��Ū������$\Gamma$���б����뤳�Ȥ�, ��Ū�����٤��������狼��.
((4-47)����).
�ޤ�, 
\Deqref{hw-04-09-ͭ�����֥��ͥ륮���μ�}�ˤ�����
�����̾�Ǥβ����Ѳ�������ʿ��$\overline{{T'}^2}$��
\Deqref{hw-04-09-ή����ɽ������ͭ�����֥��ͥ륮��}��
$(\psi_1 - \psi_3)^2$���б�����.
����餫��, $\overline{\lambda (\psi_1 - \psi_3)^2}$��ͭ�����֥��ͥ륮��
�ȸƤ֤��Ȥ�����뤳�Ȥ������줿.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%      5-9-B      %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{5-9-B\qquad
  �絤�ΰ�ư�ˤ����֥��ͥ륮���ȱ�ư���ͥ륮�����Ѵ�}
\setcounter{equation}{0}
\setcounter{figure}{0}
\def\theequation{5-9-B-\arabic{equation}}
\def\thefigure{5-9-B-\arabic{figure}}

�Ȥ����������徺��, �䤿�����������ߤ�(ͭ��)���֥��ͥ륮���ϸ���,
��ư���ͥ륮���������뤳�Ȥ���ʸ�ǽҤ٤�.
���Τ��Ȥ�
\Dfigref{hwfig-04-07-conversion.eps}�˿޼�����. 
%
\begin{figure}[htbp]
 \begin{center}
  \Depsf[][]{./eps/conversion.eps}
 \end{center}
 \begin{quote}
  \caption{
  {\footnotesize �Ȥ�������, �䤿�����������줾��岼�˰�ư���뤳�Ȥ�,
  ���֥��ͥ륮���ϸ�����, ��ư���ͥ륮�������ä���.}
  }
  \Dfiglab{hwfig-04-07-conversion.eps}
 \end{quote}
\end{figure}
%






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%%%%      Appendix        %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\subsection*{Appendix \qquad }

%\subsubsection*{}


%%%%%%%%              Text End                  %%%%%%%%
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