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% Chapter 4   Atmospheric Waves: Part II
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\section{�����ӡ���}

\begin{itembox}[l]{����}
 \Dparskip \vspace{-6mm}
 2-6��ǰ��ä������ӡ��ȤˤĤ��ƺ��ٵ�������.
 ������, ����Ϧ��̶����Ȥ�, ��ͳɽ�̤�̵��ή���ؤǤΥ����ӡ��Ȥ�
 �ͤ��Ƥ���.

 �ޤ�, 3-4����������Ϲ�ή�ݥƥ󥷥�뱲������������簵������������Ƴ��.
 ���μ����������������, ή���ؿ�$\psi$��Ƴ�����ƽ簵������������
 $\psi$�ǵ��Ҥ���. ���θ�$\psi$����ư��ȹͤ�, ������ư
 (�Ĥޤ�ϥ����ӡ���)�ΰ���®�٤����.
 ���, 2-6�����������®�٤ȤۤȤ��Ʊ������®�٤����뤳�Ȥ������.
 ���ΰ���®�٤����ѵ��Ϥξ���ˤĤ��Ƥ��ɤ�ɽ����, �����Ĺ����Ĺ���Ȥ�
 �ؤ��ƤϤ��ޤ�ɽ�����Ȥ�����ʤ�.
\end{itembox}

���̶����Ԥä��簵�����������ϰʲ��Τ褦�˵��Ҥ����.
%
\begin{eqnarray}
 \DP{\zeta}{t} + \Dvect{V} \cdot \nabla (\zeta + \beta y) = 0.
\end{eqnarray}
%
�ʤ�,
%
\begin{eqnarray}
 \Ddiv \Dvect{V} = 0
\end{eqnarray}
%
�Ǥ���.
��(5-1)�Ͻ��Ϲ�ή�ݥƥ󥷥�뱲��������(4-64)�ˤƲ��٤ξ����
$R^{-1} \partial \phi' / \partial Z = 0$
\footnote{
 $R^{-1} = f_0 e^{-Z} / \Gamma$�Ǥ���.
 �ʤ���ʸ�ǤϤ��ι��ֲ��١פȸƤ�Ǥ��뤬, �����(4-32)�δط�����
 ����Ȥ狼�������.
}
�Ȥ��뤫, ��Ū������$\Gamma$�������礭���Ȥ��뤳�Ȥ�������.
���ν簵������������2�Ĥη�����ʿ�̥ץ졼�Ȥˤ�äƶ��ޤ줿���ͤ�ή�Τ�,
$F \ll 1$ (4-2�Ỳ��)
\footnote{
 $F = f^2 L^2 / \overline{\phi}$�Ǥ���. $F \ll 1$�Ǥ���Ȥ������Ȥ�,
 ���ꥪ��ѥ�᡼�������Υ�����������, ή���ؤθ����������礭���Ȥ�
 �����Ȥ�ɽ���Ƥ���.
}
�ξ��μ�ͳɽ�̤��ä�ή�ΤΥ�ǥ��Ŭ�ڤ��������Ǥ���.

�簵�԰���������Ĵ�٤뤿��, ���٤ˤ�äƤΤ��Ѳ�����褦��������
(�Ĥޤ�, $U = U(y)$)��ͤ���.
(5-1)��$\zeta = \overline{\zeta} + \zeta'$, $u = U + u'$,
$v = v'$�Ȥ�������������Ȱʲ��μ���������.
%
\begin{eqnarray}
 \DP{\zeta'}{t} + U \DP{\zeta'}{x} + v'
  \left(
   \DP{\overline{\zeta}}{y} + \beta
  \right)
  = 0.
\end{eqnarray}
%
$\overline{\zeta} = - dU/dy$
\footnote{
 ���δ��ܾ��$\overline{\Dvect{V}} = (U,0)$�Ȥ����,
 $\overline{\zeta} = \Dvect{k} \cdot \nabla \times \overline{\Dvect{V}}
 = - dU/dy$�Ȥʤ�.
}
�Ǥ���.
������ή���ؿ�$\psi$��Ƴ������.
$u' = - \partial \psi / \partial y$, 
$v' = \partial \psi / \partial x$�ȵ��Ҥ��뤳�Ȥ�
(5-2)����­�����.
�����δط����Ѥ���ή���ؿ���(5-3)�򵭽Ҥ���Ȱʲ��Τ褦�ˤʤ�.
%
\begin{eqnarray}
 \DP{}{t} \Dlapla \psi + U \DP{}{x} \Dlapla \psi +
  \left(
   \beta - \DD[2]{U}{y}
  \right)
  \DP{\psi}{x} = 0.
\end{eqnarray}
%
�����ǤϾ���η���$\psi = \Psi (y) e^{i \mu (x - ct)}$
(�֥Ρ��ޥ�⡼�ɤβ��
�ȸƤ�)�ȹͤ�, �����(5-4)�����������,
%
\begin{eqnarray}
 (U - c)
  \left(
   \DD[2]{\Psi}{y} - \mu^2 \Psi
  \right)
  -
  \left(
   \DD[2]{U}{y} - \beta
  \right)
  \Psi = 0
\end{eqnarray}
%
�Ȥʤ�.
ή��ˤ�ͭ�¤���������, $y=0$���濴�Ȥ�, ����������$y = \pm d$�Ȥ���.
�����Ǥ�ˡ��������®�٤ϥ����ˤʤ�ʤ���Фʤ�ʤ��Τ�,
%
\begin{eqnarray}
 v(\pm d) = \DP{\psi}{x} (\pm d) = - i \mu \Psi (\pm d) \ e^{i \mu (x - ct)}
  = 0
  \nonumber
\end{eqnarray}
%
�Ǥ���. ��ä�, �ʲ��δط���������Ω�Ļ��Ƕ����������������.
%
\begin{eqnarray}
 \Psi (d) = \Psi (-d) = 0.
\end{eqnarray}
%
(5-6)�򶭳����Ȥ���(5-5)��$c$�˴ؤ����ͭ�������Ȥʤ�.

����Ū�ʥ������Ǥ���$U = U(y)$�򰷤�����, $U =$���� �ξ���ͤ����
�����Ǥ���.
���ξ��ˤ�(5-5)�ϰʲ��Τ褦�˴�ñ�ˤʤ�.
%
\begin{eqnarray}
 \DD[2]{\Psi}{y} -
  \left(
   \mu^2 - \frac{\beta}{U - c} 
  \right)
  \Psi = 0.
\end{eqnarray}
%
�����������ϰʲ��Τ褦�ʷ��β�����.
%
\begin{eqnarray}
 \Psi_n = A \cos
  \left[
   \frac{(2n - 1) \pi y}{2d}
  \right],
  \quad n = 1,2, ...
\end{eqnarray}
%
�����Ǥ�
$[(2n - 1) \pi / 2d]^2 = - \mu^2 + \beta / (U - c)$�Ȥ��Ƥ���.
(5-8)�����餫�˶������(5-6)��������.
���ξ��, ����®�٤ϰʲ��Τ褦�ˤʤ�.
%
\begin{eqnarray}
 c = U - \beta /
  \left\{
   \mu^2 + [(2n - 1) \pi / 2d]^2
  \right\}.
\end{eqnarray}
%
����ϲ�������ͭ�¤����������ȯ�������ӡ��ȤǤ���,
2-6��ˤ�����줿��
\footnote{
 ����Ū�ˤ�(3-50)��(3-51)�����������ʲ��μ��Ǥ���.
 %
 \begin{eqnarray}
  c = U - \frac{\beta - (f/H)(\partial H / \partial y)}{\mu^2 + (f^2/gH)}.
   \nonumber
 \end{eqnarray}
}
���б������ΤǤ���.
(5-8)��������줿��$\Psi_n$�ϴ����ϤǤ���Τ�
\footnote{
 �Ĥޤ�, ���μ������Ƥξ��֤򵭽ҤǤ�, ¾�������������Ѥ���ɬ�פ�
 ̵���櫓�Ǥ���.
 {\bfseries Ϣ³�⡼�ɤβ�}�ǤϤ��Τ褦�ˤϤʤ�ʤ�.
},
������­����碌��Ǥ�դν�����֤򵭽Ҥ�������Ǥ�,
��̤Ȥ��Ƥ�������֤Ǥο�ʿ��¤�򵭽Ҥ��뤳�Ȥ������.
$n=1$, $d \rightarrow \infty$�ξ��, ���μ��ϰʲ��Τ褦�˴�ñ�ˤʤ�.
%
\begin{eqnarray}
 c = U - \beta / \mu^2.
\end{eqnarray}
%
���μ���(3-50)����Ӥ��Ƥߤ� ($\partial \overline{q} / \partial y$��
(3-51)�Ǹ��Ѥ����).
(3-50)��(3-51)���ȹ礻��(5-10)�Ȥϰۤʤ�. ���θ����������������Ϥ�
��ͳɽ�̤�¸�ߤˤ���.
����;ʬ�ʹ��, ���Υ������뤬(3-50)��ʬ��ˤ����������礭���ʤ�
�¤�Ͻ��פǤϤʤ�.
�����ˤ⼨������, �����ӡ��Ȥ�ʿ��ή���$U$���Ф���
$\Dvect{k} \times \nabla \overline{q}$�����������¤���\footnote{(3-50)����.}.
������ǤΥ����ӡ��ȤǤ�$\overline{q} = f/ H_0$ ($H_0$������Ǥ���)�ʤΤ�
$\nabla \overline{q}$���̸����Ǥ���, �����ӡ��Ȥ�ʿ��ή����Ф�������
���Ť���
\footnote{
 $\Dvect{k} \times \nabla \overline{q}
 = - (\partial \overline{q} / \partial y) \Dvect{i}$.
}.
��Ĺ��$\mathcal{L} = \mathcal{L}_s = 2 \pi (U/ \beta^{1/2})$�Ǥ������
��, (5-10)����$U > 0$�˴ؤ����Ż߾��֤β�������
\footnote{
 $\mathcal{L} = 2 \pi (U/ \beta^{1/2})$����ȿ���
 $\mu = (\beta / U)^{1/2}$�Ȥʤ�. �����(5-10)�����������
 $c=0$, �Ĥޤ��Ż߾��֤Ȥʤ�. 
}.
�Ż߾��֤���Ĺ��$\mathcal{L}_s$�Ȥ����, ��Ĺ$\mathcal{L}$��
$\mathcal{L} < \mathcal{L}_s$�ξ��ˤ����¤���ظ�����
\footnote{
 $\mathcal{L} < \mathcal{L}_s$�Ȥ������Ȥ��ȿ���$\mu > (\beta /U)^{1/2}$
 �Ȥʤ�. ����ȷ��Ū��$U - \beta / \mu^2 > 0$, �Ĥޤ�$c > 0$�Ȥʤ뤳��
 �����Ȥ�������¤������ʬ����.
}.
�դ�$\mathcal{L} > \mathcal{L}_s$�ξ��ˤ����¤����ظ�����.
500 mb�ˤ��������ѵ��Ϥξ����(5-10)��������줿�ΤȤܻۤ���®�٤ǰ�ư��
��. ������, ����Ĺ����Ĺ������Ȥ�(5-10)�Ǽ������褦��®��
�����������¤򤹤�櫓�ǤϤʤ�
\footnote{
 �ºݤˤ�, ��������®�٤�®�����Ť򤹤�Ĺ�Ȥ�¸�ߤ���.
 ����, ����������ˤ��Ƥ���Τ�, (5-10)��$c$���礭���ʤ�᤮��Ȥ�������
 �Ǥ���.
 
 (3-50)�Ǥ�, ��Ĺ���ɤ�ʤ�Ĺ���ʤäƤ�(�ȿ����ɤ�ʤ˾������ʤäƤ�),
 ʬ���$f^2/gH$��¸�ߤ��뤿��, ����®�٤Ϥ������٤ΤȤ�����Ƭ�Ǥ��ˤʤ�.
 ������Ф���(5-10)��ʬ�줬$\mu$�����Τ���, ��Ĺ�����˾������ʤ��
 $c$���Ȥ�Ǥ�ʤ��礭���ʤäƤ��ޤ�.
 ���Τ��Ȥ���, (5-10)��������Ĺ��Ĺ���Ȥ�ɽ���Ǥ��ʤ��ΤǤ���.
}.




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