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% Chapter 4   Atmospheric Waves: Part II
%             4-10 WAVE STRUCTURE
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% 2003/01/09  ���� ����   ����
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\section{�Ȥι�¤}

\begin{itembox}[l]{����}
 \Dparskip \vspace{-6mm}
 ������Ǥ�, ����ǰ��ä�2�إ�ǥ���Ȥι�¤��ͤ���.
 ������, �����Ȥι�¤�����ͥ륮���Ѵ��˵ڤܤ����̤��������.
 �ʤ�, �Ϥ���˰�����Ȥ�, �����԰�����Ȥ򰷤�,
 �Ǹ�ˤ���������������, �絤���ΤǤΥ��ͥ륮���Ѵ��ˤĤ��Ƶ�������.

 �ޤ�, 1, 3�ؤ�ή���ؿ��ξ���ο�����, 2�ؤα�ľ��ư�ξ���ο����Ȥ�
 �ط�����ͤ���
 (�����ͤ��ʤ��ȥ��ͥ륮���Ѵ��˴ؤ������������ʤ�).

 �Ϥ����, �嵭�δط������Ѥ�, ��ΩŪ��(�����)�Ȥι�¤��ͻ�����.
 �����ʤ��饨�ͥ륮�����Ѵ��Ϥ�����ʤ���, ���Τ��Ȥ�
 ����Υ��ͥ륮���Ѵ��μ�(5-65), (5-66)���Ѥ�, ��������.

 ����, �԰�����Ȥ�ͤ���.
 ����2�إ�ǥ�ˤ������԰����Ȥοޤ�����, ������ë������,
 �徺����ή, �絤�δ��ȤȤ��ä��絤���ݤ��Ȥι�¤�Ȥδط�����������.
 ������, �԰�����Ȥˤ�륨�ͥ륮�����Ѵ�����⤹��.

 �Ǹ��, Oort(1964)�ˤ���絤�Υ��ͥ륮���������������Ѥ�,
 �絤���ΤǤΥ��ͥ륮���Ѵ��ˤĤ��Ʋ��⤷, 2�إ�ǥ�˴ؤ����äΤޤȤ��
 ����.
\end{itembox}

������Ǥ�2�إ�ǥ�ˤ������Ȥι�¤�����.
����ˤ�ꥨ�ͥ륮���μ�(5-65), (5-66)�ˤ�����ƹ���Ѥ�뤳�Ȥ������.
(5-70)��2�Ĥβ����äƤ���, ���줾��$c^{+}$, $c^{-}$�Ȥ���.
������(5-70)�ˤ�����$+$, $-$��������ä���Τ��б�����.
2-6��ˤ�����, �����������Ϥΰ��̲��3�Ĥΰ���®�٤ȸ�ͭ��ǵ��Ҥ���.
2�إ�ǥ�Ǥϰ��̲�ϰʲ��Τ褦�˵��ҤǤ���.
%
\setcounter{equation}{0}
\def\theequation{5-76\alph{equation}}
%
\begin{eqnarray}
 \psi_1 &=& Re
  \left[
   a^{+} {\Psi_1}^{+} e^{i \mu (x - c^{+} t)}
   + a^{-} {\Psi_1}^{-} e^{i \mu (x - c^{-} t)}
  \right]
  \\
 \psi_3 &=& Re
  \left[
   a^{+} {\Psi_3}^{+} e^{i \mu (x - c^{+} t)}
   + a^{-} {\Psi_3}^{-} e^{i \mu (x - c^{-} t)}   
  \right]
  \\
 {\omega'}_2 &=& \Delta p {f_0}^{-1} \ Re
  \left[
   a^{+} W^{+} e^{i \mu (x - c^{+} t)}
   + a^{-} W^{-} e^{i \mu (x - c^{-} t)}   
  \right]
\end{eqnarray}
%
\setcounter{equation}{76}
\def\theequation{\arabic{chapter}-\arabic{equation}}
\hspace{-4mm}
%
$Re$��ʣ�ǿ��μ¿���ʬ�򼨤�. �����顼�μ�
%
\begin{eqnarray}
 e^{ix} = \cos x + i \sin x
\end{eqnarray}
%
��(5-76)�����Τ������Ǥ���.
�ʤ��ʤ�, �����ʤ�ʣ�ǿ���$\Psi = |\Psi| e^{i \theta}$�Τ褦��
��ɽ�����Ȥ�����뤫��Ǥ���.
��(5-76c)�ϼ¤�ɬ�פʤ�.
�ʤ��ʤ�, ${\omega'}_2$��$\omega$�������ˤ�äƿ���Ū��$\psi_1, \psi_3$
�ȴط����Ƥ��뤫��Ǥ���
\footnote{
 $\omega$��������3-4��ǵ�᤿�ʲ��μ��Ǥ���.
 $$
  \sigma \Dlapla \omega + {f_0}^{2} \DP[2]{\omega}{p}
   = f_0 \DP{}{p} \left( \Dvect{V}_{\Psi} \cdot \nabla (\zeta + \beta y) \right)
   - \Dlapla \left( \Dvect{V}_{\Psi} \cdot \nabla \DP{\phi'}{p} \right).
   \eqno \textrm{(4-75)}
 $$
 $\zeta = \Dlapla \psi^{*} = \Dlapla (U y + \psi)$,
 $\phi' = f_0 \psi^{*} = f_0 (U y + \psi)$�ʤΤ�,
 $\omega$��$\psi$�ˤ�äƷ��뤳�Ȥ������.
}.
$c^{+}$��$c^{-}$�˴ؤ��뿶��Ʊ�Τδط���(5-69)���������Ϥ����Τ뤳�Ȥ������.
�ä�, (5-69a)��(5-69b)��ä��뤳�Ȥǰʲ��μ���������.
%
\begin{eqnarray}
 {\Psi_1}^{\pm} =
  - \frac{c^{\pm} - \left( U_3 - \beta / \mu^2 \right)}{c^{\pm} - \left( U_1 - \beta / \mu^2 \right)} {\Psi_3}^{\pm}.
\end{eqnarray}
%
��ľ��ư��(5-69b)��ʲ��η��ˤ��뤳�Ȥ��Τ뤳�Ȥ������.
%
\begin{eqnarray}
 W^{\pm} = - i \mu^3
  \left[ c^{\pm} - \left( U_3 - \beta / \mu^2 \right) \right] {\Psi_3}^{\pm}.
\end{eqnarray}
%

�⤷��$\psi_1$��$\psi_3$�ο����Ȱ���ν����郎Ϳ�����Ƥ���ΤǤ����,
�����ν������(5-76a)��(5-76b)�˻Ȥ����Ȥˤ�ä�$a^{+}$��$a^{-}$��
�Τ뤳�Ȥ������.

��5-9�ˤ������԰����ΰ���⳰�ˤ�����ΩŪ���Ȥ�, $c$��(ʣ�ǿ��ǤϤʤ�)
�¿��Ȥ��ƻ���, $c^{+} > c^{-}$��������
\footnote{$c$���¿��Ǥ��������$c^{+} > c^{-}$�Ȥʤ�.}.
���ξ��, (5-78)�α��դη����ϴ����˼¿��Ǥ��뤳�Ȥ���,
${\Psi_1}^{\pm}$��${\Psi_3}^{\pm}$��Ʊ������Ǥ��뤫,
�ޤ���180$^{\circ}$���꤬����Ƥ��뤳�Ȥ��狼��.
�ޤ�(5-79)�����, ��ľ��ư$W$��ή���ؿ�${\Psi_3}^{\pm}$�Ȥϰ��꤬
90$^{\circ}$����Ƥ��뤳�Ȥ��狼��.
�����ΰ���δط�����, �ɤ���ΩŪ���Ȥˤ����Ƥ�,
(5-65), (5-66)�α��դΥ��ͥ륮���Ѵ��ι�ϥ����ˤʤ�.
�⤷���(5-76a,b,c)��­����碌����Τ�Ȥä��ʤ��,
�Ѵ��ϥ����ˤĤ��Ƽ���Ū���Ѳ�����.

��Ĺ��û����ΩŪ���Ȥξ��, (5-70)�ˤ�����$\beta / \mu^2$�Υ���������
�������Ȥ��Ǥ���
\footnote{
 ��Ĺ��û���ʤ뤳�Ȥ��ȿ�$\mu$���礭���ʤ뤳�Ȥ�����������,
 ��Ĺ��û���Ȥ�ͤ�����ˤ�$\beta /\mu^2$���������Ȥ���̵�뤹��.
}.
���Τ褦�ˤ������(5-70)�ϰʲ��Τ褦�ˤʤ�.
%
\begin{eqnarray}
 c^{\pm} = \left( U_1 + U_3 \right) /2
  \pm \frac{1}{2}
  \left[
   \frac{\mu^2 - \lambda}{\mu^2 + \lambda} \left( U_1 - U_3 \right)^2
  \right]^{1/2}.
\end{eqnarray}
%
���ξ��, (5-78)����ʲ��Τ��Ȥ�ʬ����
\footnote{�ܺ٤�{\bfseries 5-10-A}����.}.
%
\begin{eqnarray}
 {\Psi_1}^{+} / {\Psi_3}^{+} > 1,
  \quad
  {\Psi_1}^{-} / {\Psi_3}^{-} < 1.
  \nonumber
\end{eqnarray}
%
���Τ褦��, ����®�٤����ܾ����®�˺Ǥ�ᤤ�褦�ʹ��٤ˤ�����,
�礭�ʿ�����ȯ������
\footnote{
 (5-80)�ˤ�����, $\lambda / \mu^2 \sim 0$�ȶ�������,
 $$
  c^{+} \simeq U_1
 $$
 $$
  c^{-} \simeq U_3
 $$
 �Ǥ��뤳�Ȥ��狼��.
}.
$W = |W| e^{- i \theta}$ ($\theta$��$t=0$�ˤ��������)�ȵ��Ҥ�,
$\Psi_3 > 0$�����֤�, (5-79)��(5-80)����$\theta = 90^{\circ}$�Ȥʤ�
.
�����, ������ë�����̤ǤϾ徺ή($\omega < 0$)��,
������ë�����̤Ǥϲ���ή($\omega > 0$)��ȯ�����뤳�Ȥ��̣���Ƥ���
.
�����Ȥι�¤��, 4-8��ǰ��ä�Eady��ǥ����ΩŪ���ȤȤ����Τ��Ƥ���
.
�����ǹͤ����褦�ʰ���Ū�ʱ�ľ��¤��, �����簵��ǥ�Τ�ΤȻ��Ƥ���.


��ľ������̵�����, (5-70)�ˤ�����$U_1 - U_3$�ϥ����Ȥʤ�.
�����, (5-70)�ϰʲ��Τ褦�ˤʤ�.
%
\begin{eqnarray}
 \begin{array}{l}
  c^{+} = (U_1 + U_3) / 2 - \beta / \left( \mu^2 + \lambda \right), \\
  c^{-} = (U_1 + U_3) / 2 - \beta / \mu^2.
 \end{array}
\end{eqnarray}
%
$c^{-}$�Ͻ��ϥ����ӡ���((5-10)
\footnote{
 $$
  c = U - \beta / \mu^2.
  \eqno \textrm{(5-10)}
 $$
}
����)�Ǥ���.
�ʤ��ʤ�, (5-79)��(5-69c)���$W^{-} = 0$, ${\Psi_1}^{-} = {\Psi_3}^{-}$
�Ǥ��뤫��Ǥ���.
$c^{+}$�ϴ��������ȤǤ���.
�ʤ��ʤ�, (5-78)��(5-79)���${\Psi_1}^{+} = - {\Psi_3}^{+}$,
$W^{+} \neq 0$�Ǥ��뤫��Ǥ���.
$(U_1 - U_3)^2$�����˾�������, Ĺ�Ȥ����Τ褦�ʿ����񤤤򤹤�Τ�
ͽ�ۤ���Ƥ������ȤǤ���.

��5-9�ˤ������԰����ΰ�Ǥΰ���®�٤ϰʲ��Τ褦�˵��ҤǤ���.
%
\begin{eqnarray}
 c^{\pm} = c_r \pm i \ n / \mu.
  \nonumber
\end{eqnarray}
%
�ʤ�,
%
\begin{eqnarray}
 c_r &=& (U_1 + U_3) / 2
  - \beta \left( 2 \mu^2 + \lambda \right) \big/
  \left[ 2 \mu^2 \left( \mu^2 + \lambda \right) \right],
  \nonumber \\
 \\
 n &=&
  \left[
   \mu^4 \left( \lambda^2 - \mu^4 \right) (U_1 - U_3)^2
   - \lambda^2 \beta^2
  \right]^{1/2}
  \Big/ \left[ 2 \mu \left( \mu^2 + \lambda \right) \right]
  \nonumber
\end{eqnarray}
%
�Ǥ���.
�����, ��ϰʲ��Τ褦�˽񤯤��Ȥ������.
%
\begin{eqnarray}
 \psi_{1,3} = Re
  \left[
   a^{+} \Psi_{1,3}^{+} e^{i \mu (x - c_r t)} e^{nt}
   + a^{-} \Psi_{1,3}^{-} e^{i \mu (x - c_r t)} e^{nt}
  \right].
\end{eqnarray}
%
$a^{+} = 0$�Ȥ������̤ʾ������, ��Ĺ�����($e^{nt}$��ޤ��)
�ϸ��ꤹ���($e^{-nt}$��ޤ��)���®�˻��ۤ��Ƥ���
\footnote{$n>0$�Ǥ��뤳�Ȥ�����.}.
���Τ��Ȥ���, ��Ĺ�����ι�¤��ܺ٤�Ĵ�٤뤳�Ȥ�˾�ޤ���.

$c$��ʣ�ǿ��ξ��, (5-78)��(5-79)�α��դη�����ʣ�ǿ��Ȥʤ�.
���Τ���, �԰�����Ȥΰ��깽¤��, �����˵���������ΩŪ���Ȥι�¤�Ȥ�
�ۤʤäƤ���.
(5-78)��(5-79)��(5-82)���Ѥ��뤳�Ȥǰʲ��Τ褦�˽�ľ�����
\footnote{
 (5-84)��, (5-78)��(5-82)������������, ʬ���ʬ�Ҥ�ξ����
 $c_r \pm i \ n/ \mu$��ݤ������뤳�Ȥ������.
}.
%
\begin{eqnarray}
 {\Psi_1}^{\pm}
  = -
  \frac
   {\left\{ c_r - \left[ (U_1 + U_3) / 2 - \beta / \mu^2 \right] \right\}^2
   - (U_1 - U_3)^2 / 4 + n^2 / \mu^2 \mp i \ n (U_1 - U_3) / \mu
   }
   {\left[ c_r - (U_1 - \beta / \mu^2) \right]^2 + n^2 / \mu^2}
   {\Psi_3}^{\pm},
   \nonumber
\end{eqnarray}
%
\vspace{-10mm}
%
\begin{eqnarray}
 \\
 W^{\pm} = \mu^3
  \left\{
   \pm n / \mu - i \left[ c_r - \left( U_3 - \beta / \mu^2 \right) \right]
  \right\}
  {\Psi_3}^{\pm}.
\end{eqnarray}
%
�����Ǥ��Ȥΰ��깽¤��, ��Ĺ�����ȼ�ޤ��Ȥδ֤ǵ�ž���Ƥ��뤳�Ȥ�
���դ����ߤ���
\footnote{
 (5-84)�ˤ�������Ĺ����˴ؤ��$i \ n / \mu$�ι����椬
 $\pm$�ǤϤʤ�$\mp$�ˤʤäƤ��뤳�Ȥ������Ǹ��äƤ��뤳�ȤǤ���.
}.
�������������������ޤ��ϸ��ꤹ���Ȥι�¤��׻�����Τ��Ѥ��뤳�Ȥ������.
2�ؤˤ����벹�٤ξ����, �ʲ��Τ褦���ſ尵�μ�����ʬ����
�׻����뤳�Ȥ������
\footnote{
 �ޤ�, �ſ尵�μ��Ⱦ������������Ȥ߹�碌����
 $$
  RT' = - p \ \partial \phi' / \partial p.
  \eqno \textrm{(4-32)}
 $$
 ��ͤ���.
 �����, �Ϲ�ήή���ؿ��δط���$\phi' = f_0 \psi^{*} = f_0 (U y + \psi)$
 �ˤ�������뼰
 $\partial \phi' / \partial p = \partial \psi / \partial p$
 ���Ѥ����ѷ�����Ȱʲ��μ���������.
 $$
  RT' = f_0 \ p \ \partial \psi / \partial p.
 $$
 ���μ���1�ؤ���3�ؤޤ���ʬ���뤳�Ȥ�(5-86)��������.
 �ʤ�, ${T_2}'$��$T'$��1�ء�3�ش֤�ʿ���ͤȤ��ư���.
}.
%
\begin{eqnarray}
 {T_2}' = f_0 (\psi_1 - \psi_3) /
  \left[ R \ \ln \left( p_3 / p_1 \right) \right].
\end{eqnarray}
%

��Ȥ���, ���������ι�¤��ʲ���ŵ��Ū���ͤǷ׻����Ƥߤ�.
$f_0 = 10^{-4}$ sec$^{-1}$, $\beta = 1.67 \times 10^{-11}$ m$^{-1}$ sec$^{-1}$,
$\mathcal{L} = 2 \pi / \mu = 4000$ km,
$\lambda = 5.88 \times 10^{-12}$ m$^{-2}$,
$p_s = 1000$ mb, $\Delta p = 400$ mb,
$U_1 - U_3 = 20$ m sec$^{-1}$, $a^{-} = 0$,
$a^{+} {\Psi_3}^{+} = 10$ m sec$^{-1} / \mu$.
�Ǹ�δط����ˤ��, 3�ؤ���Ĺ�����ο����Ȱ��꤬��ޤ�.
(5-83)��(5-84)��(5-77)���Ѥ��뤳�Ȥ�, �ʲ���$\psi_1$��$\psi_3$�˴ؤ���
������뤳�Ȥ������.
%
\begin{eqnarray}
 \begin{array}{l}
  \psi_1 = \left( 14.2 \textrm{ m sec}^{-1} \right) \mu^{-1}
   \cos \left[ \mu (x - c_r t) + 64^{\circ} \right] e^{nt}, \\
  \psi_3 = \left( 10 \textrm{ m sec}^{-1} \right) \mu^{-1}
   \cos \left[ \mu (x - c_r t) \right] e^{nt}.
 \end{array} 
\end{eqnarray}
%
����, ��ľ��ư��(5-76c)��(5-77), (5-85)�����Ѥ��뤳�Ȥ����뤳�Ȥ������.
%
\begin{eqnarray}
 {\omega'}_{2} = -1.36 \times 10^{-3} \textrm{ mb sec}^{-1}
  \cos \left[ \mu (x - c_r t) + 116^{\circ} \right] e^{nt}.
\end{eqnarray}
%
������(5-86)���������뤳�Ȥ�, �ʲ��β��٤�������.
%
\begin{eqnarray}
 {T'}_2 = (4.35C) \cos \left[ \mu (x - c_r t) + 108^{\circ} \right]
  e^{nt}.
\end{eqnarray}
%
���ʤߤ�, ή���ؿ��ο�����$v$��ʬ�ο������ȿ��ǳ�ä���Τ�
������.

(5-87), (5-88), (5-89)���������������̿ޤ��5-10�˼���.
%
\begin{figure}[htb]
 \begin{center}
  \Depsf[][]{./eps/hwfig-04-10.eps}
 \end{center}
 \begin{quote}
  \caption{
  {\footnotesize (5-87), (5-88), (5-89)�����������԰����Ȥ����̿�.
  (Haltiner, G. J., and Williams, R. T., 1979:
      Numerical Prediction and Dynamic Meteorology.
      {\itshape John Wiley \& Sons,} 477pp)}
  }
  \Dfiglab{hwfig-04-10}
 \end{quote}
\end{figure}
%
��Ȳ��ζ����Ϥ��줾��$\psi_1$��$\psi_3$��ɽ���Ƥ���.
�ޤ�, ������ë($T$)�ȵ���������($R$)
�ˤ�äƼ�����Ƥ���褦��, ����Ϲ��٤ȶ������˷���.
(5-87)����, ����ؤξ���ο����������礭�����Ȥ�ʬ����.
���θ�����$\beta$��¸�ߤˤ���.
���٤ξ���κ���ȺǾ��Ϥ��줾��W��C�Ǽ�����Ƥ���
\footnote{
 (5-89)����$\mu (x - c_r t) = 72^{\circ} + 180^{\circ} \times n$�ˤ�����
 �Ǿ��ͤ�, $\mu (x - c_r t) = -108^{\circ} + 180^{\circ} \times n$
 �ˤ����ƺ����ͤ��뤳�Ȥ�ʬ����.
}.
�����������ؤˤ������ľ��ư��ξ��ü��ɽ���Ƥ���
\footnote{
 (5-88)����$\mu (x - c_r t) = 64^{\circ} + 180^{\circ} \times n$�ˤ�����
 �����ͤ�, $\mu (x - c_r t) = -116^{\circ} + 180^{\circ} \times n$
 �ˤ����ƺǾ��ͤ��뤳�Ȥ�ʬ����.
 ((5-88)�α��դˤϥޥ��ʥ����Ĥ��Ƥ��뤳�Ȥ�����).
 �ʤ�, $\omega < 0$�λ��Ͼ徺ή, $\omega > 0$�λ��ϲ���ή�Ǥ��뤳�Ȥ�
 ����.
}.
�������Ͽ�ʿ��«����ʬ��, �����Ǥʤ���ʬ�Ͽ�ʿȯ������ʬ��ɽ��.
��������ʬ�Ǥϼ��̤�Ϣ³��������ȹͤ��Ƥ���.

�ºݤ˴�¬��������ѵ��Ϥξ����, 2�إ�ǥ�ˤ������԰����Ȥȶ��̤���
��ħ��¿������.
��¬���������⤵�ȶ������˷���,
���ؤε�����ë�����̤Ǿ徺��ư�򤪤��ʤ�.
�ޤ���¬�Ǥ�, ��ɽ�̤��㤤���֤���Ǻ���ι߱��������뷹���ˤ���.
��5-10�ˤ����Ƥ�, ��ľ��ư�κ������ɽ�ε�����ë�ΰ��֤�
���礦����Ǥ���.
����ϴ�¬�����߱��ο����񤤤Ȱ��פ���.

(5-65)��(5-66)�ˤ����륨�ͥ륮���Ѵ���, ��5-10�ޤ���(5-87), (5-88),
(5-89)�����ñ�˸��Ѥ�뤳�Ȥ������.
�ޤˤ�����, �Ȥ��������Ͼ徺��, �䤿�������ϲ��ߤ��Ƥ��뤳�Ȥ����餫�Ǥ���.
���Τ��Ȥ�������ͭ�����֥��ͥ륮���Ͼ���α�ư���ͥ륮�����Ѵ�����Ƥ���.
�Ȥ��������ϼ�˵����������ȵ�����ë�δ֤ˤ��뤿��, �̸�����ư��
\footnote{
 $v = \partial \psi / \partial x$�Ǥ��뤳�Ȥ����դ��뤳��.
 ���δط����ȿ�5-10����, �Ȥ����������̸�����, �䤿���������������
 ư�����Ȥ�ʬ����.
}.
�դ�, �䤿���������������ư��.
���Τ��Ȥ��������δ��ܾ��ͭ�����֥��ͥ륮��������ͭ�����֥��ͥ륮����
�Ѵ�����.
��ư���ͥ륮���ⲹ�٤���ư����Ĺ���Ƥ����԰����ȤǤ�,
�����������Ѵ���ɬ�פȤʤ�.

Oort(1964)
���絤�Υ��ͥ륮�������������Υǡ�����ޤȤ᤿.
����Ͽ�5-11�˼�����Ƥ���.
%
\begin{figure}[htb]
 \begin{center}
  \Depsf[][]{./eps/hwfig-04-11.eps}
 \end{center}
 \begin{quote}
  \caption{
  {\footnotesize Oort�Υǡ����ˤ���絤�Υ��ͥ륮���۴�.
  �ͳѤ���ο��ͤ�, ��¬���줿��Ⱦ��Υ��ͥ륮����ǯʿ���ͤǤ���,
  ñ�̤�10$^{5}$ joules m$^{-2}$�Ǥ���.
  �ޤ�, ���ͥ륮���Ѵ��������ñ�̤� watts m$^{-2}$�Ǥ���.
  (From Oort, A.H., ``On Estimates of the Atmospheric Energy Cycle.''
  {\itshape American Meteorological Society,} 1964.)
  �ʤ�, ���οޤ�
  Haltiner, G. J., and Williams, R. T., 1979:
      Numerical Prediction and Dynamic Meteorology.
      {\itshape John Wiley \& Sons,} 477pp ����ž��.}
  }
  \Dfiglab{hwfig-04-11}
 \end{quote}
\end{figure}
%
���Υ����������Ǥ�, $\overline{K}$������ʿ�����α�ư���ͥ륮���Ǥ���,
$\overline{P}$������ʿ�Ѳ��پ��ͭ�����֥��ͥ륮���Ǥ���.
����, $K'$������ʿ�Ѿ줫�����®�Τ���α�ư���ͥ륮���Ǥ���,
$P'$������ʿ�Ѿ줫��β��٤Τ����ͭ�����֥��ͥ륮���Ǥ���.
���Υǡ�������Ⱦ���ǯʿ�ѤΤ�ΤǤ���.
���ͥ륮���Ѵ�\{$\overline{P} \cdot P'$\}��\{$P' \cdot K'$\}��
��ñ�ʷ��Ϥ��줾��, (5-66)��(5-65)�α��դκǽ�ιफ��������.
����, \{$K' \cdot \overline{K}$\}��(5-17)
\footnote{
 $$
  \DD{}{t} \int_{-d}^{d} \frac{\overline{{u'}^2 + {v'}^2}}{2} dy
  = - \int_{-d}^{d} \overline{u' v'} \DD{U}{y} dy.
  \eqno \textrm{(5-17)}
 $$
}
�α��դȻ����褦�ʤ�ΤǤ���.
�Ѵ�\{$\overline{P} \cdot \overline{K}$\}��
$- \overline{\omega} \overline{T}$��ޤ�.
��������̽۴Ĥ�ʿ�Ѿ줫����뤳�Ȥ��Ǥ���.
��5-11�ˤ�����, $\overline{P}$��ή���������ʿ�Ѳ�Ǯ$\overline{Q}$��,
�ޤ���$P'$���Ѵ�����, ���θ�$K'$���Ѵ������.
�����԰����Ȥ������Υ��ͥ륮���Ѵ���������������ȤϤ����ޤǤ�
�����줿.
���ڥ��ȥ�θ���ˤ��. �Ѵ��Ϥޤ��������ѵ��Ϥ��Ȥˤ�äư������������
���Ȥ������줿��, Ʊ�����������Ϥ��Ȥ�ޤ���Ϳ���뤳�Ȥ�ʬ���ä�.
�簵�ˤ�����$K'$����$\overline{K}$�ؤ��Ѵ���4-6��ǵ������줿�褦�ˤʤ�
��ͽ�ۤ����.
���οޤǤ�\{$\overline{P} \cdot \overline{K}$\}�����˾�����.
���Τ��Ȥ�, ���ѵ��Ϥˤ����뷹���԰����Ȥ���۴ĤΥХ�󥹤ΰݻ���
�����Ƥ��뤳�Ȥ򶯤��������Ƥ���.
�������Ȥˤ�ä����߽Ф��줿��ư���ͥ륮����, �絤���໤�ˤ�꼺��줿
��ư���ͥ륮���ΥХ�󥹤�Ȥ뤳�Ȥ�����Ƥ���.
�̸�����Ǯ�ե�å����ˤ�ä�, 
���˲�Ǯ����Ƥ�������٤�������䤨�Ƥ������٤�Ǯ��͢������,
Ǯ�Х�󥹤ϰݻ�����Ƥ���.
���ѵ��Ϥξ����ޤ�, �簵Ū�ʸ�����̤���, �������δ��ܾ�α�ư���ͥ륮��
��ݻ�����Τ�����Ƥ���.

�����������ȵ�����ë�Ϥ��줾��, ��ʿȯ�����Ƥ����ΰ��
��ʿ��«���Ƥ����ΰ�ˤ���.
��(5-57)��, (����������������ˤ���)���٤κ����ͤ�,
��ʿ��«�ξ��($\partial \omega / \partial p > 0$)
�ˤΤ����ä��뤳�Ȥ򼨤��Ƥ���. (��ή������������Ǥ���).
Ʊ����ˡ��, ���٤��Ǿ��Ǥ��뵤����ë�ˤ�Ŭ�ѤǤ���.
���٤κ����ͤȺǾ��ͤκ������礷�ʤ��¤�, ���餫���Ȥ���Ĺ�����ʤ�.
Eady(1949)��Charney(1947)�Τ褦��Ϣ³��ǥ�Ǥ�,
�԰�����������Ƥι��٤ˤ�����,
���������������褤�˼�«��ȯ������.

Gall(1976)��
Simmons and Hoskins(1977)��, 
��Ⱦ���ʿ�Ѥξ��֤�ɽ���褦��, �������ץ��ե�����ΰ�������������
����Ū�˵�᤿.
���ξϤǹͤ���ñ��ʥץ��ե�����ˤ�����Ǥ��԰�����Ȥ���Ĺ��
���褽4000km���ä���, ��餬��᤿�Ǥ��԰�����Ȥ���Ĺ��2000km
�᤯�Ǥ��ä�.
������û���԰����Ȥ���ή���β��ؤǤΤ��礭�ʿ��������.
Staley and Gall(1977)
��, �������԰����Ȥ�, ��ή���β��ؤ���Ū�����٤�
��®�����������Ҵ��Ǥ��뤳�Ȥ򼨤���.
Gall, Blakeslee, et al(1979)��
�໤��ͭ�¿����θ��̤���,
Ĺ�Ȥ�����Ū�ˤʤ뤳�Ȥ򼨤���.
�������ʤ���, û�ȤϤ��ޤ�, ��ɽ�Ǥξ����Ϥʾ������Ĺ�˽��פ�
����̤����Ƥ���.


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


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%%%%      5-10-A      %%%%
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\subsection*{5-10-A\qquad ����Ʊ�Τ��羮���}

(5-80)��(5-78)���Ѥ��ƿ���${\Psi_1}^{\pm}$��${\Psi_3}^{\pm}$����Ӥ�
�����ʤ�.

�ޤ�, ${\Psi_1}^{+}$��${\Psi_3}^{-}$����Ӥ򤪤��ʤ�.
(5-78)��(5-80)�����������,
%
\begin{eqnarray}
 \frac{{\Psi_1}^{+}}{{\Psi_3}^{+}}
  &=&
  - \frac{\dfrac{1}{2} \left( U_1 + U_3 \right)
      + \dfrac{1}{2}
      \left[
       \dfrac{\mu^2 - \lambda}{\mu^2 + \lambda} \left( U_1 - U_3 \right)^2
      \right]^{1/2}
      - U_3}
      {\dfrac{1}{2} \left( U_1 + U_3 \right)
      + \dfrac{1}{2}
      \left[
       \dfrac{\mu^2 - \lambda}{\mu^2 + \lambda} \left( U_1 - U_3 \right)^2
      \right]^{1/2}
      - U_1}
      \nonumber
\end{eqnarray}
%
�Ȥʤ�. �ʤ�, ��Ĺ��û���Ȥ�ͤ��Ƥ���Τ�$\beta / \mu^2$�������Ǥ���
�Ȥ���̵�뤷��.
���μ����������Ƥ�����,
%
\begin{eqnarray}
 \frac{{\Psi_1}^{+}}{{\Psi_3}^{+}}
  &=& 
  - \frac{ \left( U_1 - U_3 \right)
      + 
       \sqrt{\dfrac{\mu^2 - \lambda}{\mu^2 + \lambda}}
       \left( U_1 - U_3 \right)
      }
      {- \left( U_1 - U_3 \right)
      + 
       \sqrt{\dfrac{\mu^2 - \lambda}{\mu^2 + \lambda}}
       \left( U_1 - U_3 \right)
      }
      \nonumber \\
 &=&
  - \frac{ \left( U_1 - U_3 \right)
      \left\{
       \dfrac{\sqrt{\mu^2 - \lambda} + \sqrt{\mu^2 + \lambda}}
         {\sqrt{\mu^2 + \lambda}}
      \right\}
      }
      {\left( U_1 - U_3 \right)
      \left\{
       \dfrac{\sqrt{\mu^2 - \lambda} - \sqrt{\mu^2 + \lambda}}
         {\sqrt{\mu^2 + \lambda}}
      \right\}
      }
      \nonumber \\
 &=&
  - \frac
      {\sqrt{1 - \dfrac{\lambda}{\mu^2}} + \sqrt{1 + \dfrac{\lambda}{\mu^2}}}
      {\sqrt{1 - \dfrac{\lambda}{\mu^2}} - \sqrt{1 + \dfrac{\lambda}{\mu^2}}}
      \nonumber
\end{eqnarray}
%
�Ȥʤ�. ���Ҥ�����, ��Ĺ��û���Ȥ�ͤ��Ƥ���Τ�
$\lambda / \mu^2 \rightarrow 0$�Ȥ���.
�����, ${\Psi_1}^{+} / {\Psi_3}^{+} \rightarrow - \infty$
�Ȥʤ뤳�Ȥ��狼��.
���Τ��Ȥ���, ${\Psi_1}^{+}$��${\Psi_3}^{+}$����٤������礭�����Ȥ�
�狼�ä�.
${\Psi_1}^{-}$��${\Psi_3}^{-}$�˴ؤ��Ƥ�Ʊ�ͤ˷׻����뤳�Ȥ�
�����羮�ط������뤳�Ȥ������.





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%\subsection*{Appendix \qquad }

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