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% Chapter 2   Wave Motion in the Atmosphere  Part I
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% 2002/10/09  ���� ����   ����
% 2002/10/13  ���� ����   ����
% 2002/11/21  ���� ����   ���ߥʡ��ˤ�뽤��
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\section{������α���}

\begin{itembox}[l]{����}
 ������μ�갷���ˤĤ��Ƶ�������.
 �ºݤ˴�¬�����, ���ѵ��Ϥˤ����Ƥϥ����ӡ��Ȥ˶ᤤ�������Ȥ�
 ��¬����뤬, ���������ȤϤ��ޤ긫���ʤ�.
 ����ͤ����ݤˤϤ��μºݤ˹礦�褦�˷���ɬ�פ�����.
 ������Ǥ�����ǰ��ä����������Ȥȥ����ӡ��Ȥ��θ���ƽ���ͤ�Ƴ��.
\end{itembox}

2-7, 2-8��Ǥ�, 11�Ϥǵ����������ͤμ�갷���˴ؤ����طʤ���������.
����Ǥ�2�ĤΥ����פ���ư�ˤĤ��ƾҲ𤷤�.
1���ܤ�®�����������Ȥ���300m sec$^{-1}$��®�٤����($H = 10$km�ξ��). 
2���ܤ��٤������ӡ��Ȥ���10m sec$^{-1}$
\footnote{
 10 m sec$^{-1}$ = 864 km day$^{-1}$ $\sim$ 1000 km day$^{-1}$�Ǥ���.
 ����Ϥ��褽�㵤����ⵤ���ΰ�ư����®�٤Ǥ���.
}
��®�٤����.
�����Ƥ��ε��ݥ����ƥ�
\footnote{
 �����Ǥ�����٤Ǥβ����㵤���ΰ�ư�Τ褦�ʵ��ݥ����ƥ��ͤ��Ƥ���.
 ����¾�ˤⵤ�ݥ����ƥ�ˤϡֹ߱��פ��ε���פȤ��ä��͡���
 ��������Τ�Τ�����.
}
�Ǥϥ����ӡ��Ȥ��餤�ΰ���®�٤����.
���Τ��Ȥ�, ���ѵ��Ϥα�ư�Ǥϴ��������ȤϾ����ʿ������������ʤ����Ȥ򼨤�
�Ƥ���.
������(�Ĥޤ��$\delta = 1$��)�������Ϥ�ͽ��˻Ȥ����ˤ�,
���������Ȥ������ʿ�������Ĥ褦�ˤ����ν���ͤ�Ȥ뤳�Ȥ����פǤ���.
���������Ǥ�դν�������Ѥ�������β��­����碌�뤳�Ȥ�
Ĵ�٤뤳�Ȥ��Ǥ���.
����Ǥ�(3-44)��괷�������Ȥ�2�ĤΥ⡼�ɤ�, (3-50)���
�����ӡ��Ȥ�1�ĤΥ⡼�ɤ�����줿.
���줾��Υ⡼�ɤ��Ф��ư���®�٤�����,
������3�Ĥΰ���®��$c_j$, $j = 1,2,3$���б�����, ®����ʬ$u_j$, $v_j$
�ȼ�ͳɽ�̤ι⤵$h_j$�˴ؤ���ۤʤ��Υ��åȤ�¸�ߤ���.
������, ��(3-41)��Ʊ���������Ǥ��뤿��, ��Ԥ�ʪ����
\footnote{
 $u_j$, $v_j$, $h_j$�Τ���.
}
�ΰ�ĤϤɤΥ��åȤǤ�Ǥ�դȤʤäƤ��ޤ�����, �򤯤���ˤ�
�ɤ줫��Ĥ����ꤷ�ʤ���Фʤ�ʤ�
\footnote{
 (3-42)��$u_0, v_0, h_0$�˴ؤ��Ʋ򤤤ƤߤƤ�, ��դ˲�������ʤ�.
 ������, $u_0, v_0, h_0$��3�ѿ��θߤ��δط����������뤿��,
 ����3�ĤΤ����ɤ줫����ꤹ���¾��2�ѿ��Ϸ��뤳�Ȥ��Ǥ���.
}.
�����Ǥ�$\mu$����ꤷ�����
$u_j$��$u_j = U - c_j$�����ꤷ��,
¾�ο���$v_j, h_j$�˴ؤ�����������(3-42)($\delta = 1$)�ξ�2�Ĥμ����.
���η�̤ϰʲ��Τ褦�ˤʤ�
\footnote{
 (3-42)��$u_0, v_0, h_0$�ˤ��줾��$u_j, v_j, h_j$����������.
 �ޤ�, $u_j = U - c_j$��(3-42)��2���ܤμ�
 $$
  f u_0 + i \mu (U - c) v_0 = 0
 $$
 ����������$v_j$���ͤ�����. ������1���ܤμ�
 $$
 (U - c) i \mu u_0 - f v_0 + gi \mu h_0 = 0
 $$
 ���������뤳�Ȥ�$h_j$��������.
}.
%
\begin{eqnarray}
 u_j &=& U - c_j,
  \nonumber \\
 v_j &=& if/ \mu,
  \\
 h_j &=& g^{-1} \left[ (f^2 / \mu^2 ) - (U - c_j)^2 \right].
  \nonumber
\end{eqnarray}
%
�����$\mu$�˴ؤ������Ū�ʲ�ϰʲ��Τ褦�ˤʤ�.
%
\begin{eqnarray}
 u &=& \sum_{j = 1}^{3} a_j u_j e^{i \mu (x - c_j t)},
  \nonumber \\
 v &=& \sum_{j = 1}^{3} a_j v_j e^{i \mu (x - c_j t)},
   \\
 h &=& \sum_{j = 1}^{3} a_j h_j e^{i \mu (x - c_j t)}.
  \nonumber
\end{eqnarray}
%
������$a_j$��Ǥ�դ�����Ǥ���.

$t = 0$�ˤ�����$u, v, h$�ν���ͤ�Ĵ�´ؿ�����ư��
$u(x,0) = u_0 e^{i \mu x}$, $v(x,0) = v_0 e^{i \mu x}$,
$h(x,0) = h_0 e^{i \mu x}$���ȹͤ���.
$u_0, v_0, h_0$��ʣ������Ǥ���.
�����ν������(3-61)�����������, �ʲ��Τ褦�ˤʤ�.
%
\begin{eqnarray}
 u_0 &=& \sum_{j = 1}^{3} a_j u_j,
  \nonumber \\
 v_0 &=& \sum_{j = 1}^{3} a_j v_j,
  \\
 h_0 &=& \sum_{j = 1}^{3} a_j h_j.
  \nonumber
\end{eqnarray}
%
����®�٤϶��Ū�˰ʲ��Τ褦�ˤʤ�
\footnote{
 1���ܤμ���(3-50)��(3-49)���������뤳�Ȥ�������.
 2,3���ܤμ���(3-44)�˴��ܾ��ʿ����®��ä�����ΤǤ���.
}.
%
\begin{eqnarray}
 c_1 &=& U - \frac{U}{1 + (gH \mu^2 / f^2)},
  \nonumber \\
 c_2 &=& U + \sqrt{gH + (f^2 / \mu^2)},
  \\
 c_3 &=& U - \sqrt{gH + (f^2 / \mu^2)}.
  \nonumber
\end{eqnarray}
%
(3-63)��(3-60)��������, �������ͤ�(3-62)����������$a_j$�˴ؤ��Ʋ򤯤�,
$a_j$��������.
%
\begin{eqnarray}
 a_1 &=& \frac{i \mu f^{-1} g H v_0 - g h_0}
  {U^2 f^4 \mu^{-4} \alpha^{-2} - \alpha},
  \nonumber \\
 a_{2,3} &=& \mp u_0 / 2 \alpha^{1/2}
  \mp \frac{if \mu^{-1} (1 \pm U \alpha^{-1/2}) v_0 + g h_0}
  {2 (U f^2 \mu^{-2} \alpha^{-1/2} \pm \alpha)}.
\end{eqnarray}
%
�ʤ�, $\alpha \equiv g H + f^2 \mu^{-2}$
\footnote{
 �Ĥޤ괷�������Ȥΰ���®�٤�2��Ǥ���.
 (3-44)�򻲾Ȥ���.
}
�Ǥ���.

������֤��Ϲ�ήŪ�Ǥ���ȹͤ����,
%
\begin{eqnarray}
 u = - \frac{g}{f} \DP{h}{y} = 0, \quad
  v = \frac{g}{f} \DP{h}{x}
  \nonumber
\end{eqnarray}
%
�Ǥ���, ������
%
\begin{eqnarray}
 u_0 = 0 , \quad v_0 = i \mu f^{-1} g h_0
\end{eqnarray}
%
���б�����
\footnote{
 ������֤�ͤ��Ƥ���Τ�����, $u(x,0) = u_0 e^{i \mu x}$,
 $v(x,0) = v_0 e^{i \mu x}$, $h(x,0) = h_0 e^{i \mu x}$���Ϲ�ήʿ�դ�
 ������������Ȥ����μ���������.
}.
�����������(3-64)�����������, �����ϰʲ��Τ褦�ˤʤ�.
%
\begin{eqnarray}
 a_1 &=& \frac{\mu^2 f^{-2} g h_0}{1 - U^2 f^4 \mu^{-4} \alpha^{-3}},
  \nonumber \\
 a_{2,3} &=& \frac{U \alpha^{-3/2} g h_0}
  {2 (1 \pm U f^2 \mu^{-2} \alpha^{-3/2})}.
\end{eqnarray}
%
���ѵ��Ϥˤ��������Ǥϰʲ��Τ褦��������������Ω��
\footnote{
 ���ѵ��ϤǤ�ʿ��Ū��ʪ���̤��餳���������ˤĤ��ƹͤ��Ƥߤ�.
 $f \sim 10^{-4}, \mu \sim 1 / 10^3$ [km$^{-1}$]�Ȥ����,
 $f \mu^{-1} \sim 10^{-4} \times 10^{6} = 100$ [m/s].
 $g \sim 10$ [m/s$^2$], $H \sim 10^4$ [m]�Ȥ����,
 $\alpha^{1/2} = \sqrt{gH + f^2 \mu^{-2}} \sim \sqrt{10^5 + 10^4}
 \sim 10^2 \sqrt{10} \sim 300$ [m/s].
 ������$U \sim 10$ [m/s]�Ǥ���.
 �����ͤ����, (3-67)�����ѵ��Ϥˤ������������Ǥ��뤳�Ȥ�ʬ����. 
}.
%
\begin{eqnarray}
 \alpha^{1/2} > f \mu^{-1} \gg U.
\end{eqnarray}
%
$v$�ν���ͤؤΥ����ӡ��Ȥδ�Ϳ[���ʤ��, (3-61)��2���ܤμ����Ф���
$i f \mu^{-1} a_1$�Τ��ȤǤ���]��$v_0$�ΰ���Ȱ��פ�,
�����礭����$v_0$����٤Ƥ鷺�����礭�������Ǥ���
\footnote{
 $if \mu^{-1} a_1$��(3-66)���,
 %
 \begin{eqnarray}
  if \mu^{-1} a_1 &=&
   if \mu^{-1} \frac{\mu^2 f^{-2} g h_0}{1 - U^2 f^4 \mu^{-4} \alpha^{-3}}
   \nonumber \\
  &=&
  i \mu f^{-1} g h_0
   \frac{1}
   {1 - \left( \dfrac{U}{\sqrt{\alpha}} \right)^2 \left(\dfrac{f \mu^{-1}}{\sqrt{\alpha}} \right)^4}
   \nonumber \\
  &\sim& 
  i \mu f^{-1} g h_0
   \left\{
    1 + \left( \dfrac{U}{\sqrt{\alpha}} \right)^2 \left(\dfrac{f \mu^{-1}}{\sqrt{\alpha}} \right)^4
   \right\}
   \nonumber
 \end{eqnarray}
 %
 �Ȥʤ�. �ʤ�, �׻��������(3-67)���Ѥ��ƶ����Ԥä�.
 ���η�̤��, $i$���Ĥ��Ƥ��뤿��$v_0 = i \mu^{-1} g h_0$��
 ���꤬���פ��뤳�Ȥ�ʬ����.
 �ޤ�, $v_0$�Ȱۤʤ�Τ�$(U/\sqrt{\alpha})^2 (f\mu^{-1}/\sqrt{\alpha})^4$
 ��ʬ�Ǥ���, �����(3-67)��������Ǥ��뤳�Ȥ��狼�äƤ���.
 ������, �����ӡ��Ȥδ�Ϳ��$v_0$�����˻��Ƥ��뤳�Ȥ��狼��.
}.
����Ͻ���β�ž��ʬ
\footnote{
 �����Ǹ����ֲ�ž��ʬ�פȤϥ����ӡ��Ȥ���ʬ�Τ��ȤǤ���.
 2-6��Ǵ��������Ȥ�ȯ�������ۤ�, �����ӡ��Ȥϲ�ž�����ۤ��뤳�Ȥ�
 ���˽Ҥ٤�줿.
 �Ĥޤꤳ���Ǥ�, $j=1$�ϥ����ӡ��Ȥ���ʬ�ʤΤǡֲ�ž��ʬ�פǤ���,
 $j=2,3$�ϴ��������Ȥ���ʬ�ʤΤǡ�ȯ����ʬ�פǤ���.

 ���, �虜�虜�Ϲ�ήʿ�դ�������Ѥ����Ȥ�, ���ѵ��Ϥˤ������ʪ���̤��礭
 ����ͤ���(����Ū�ˤ�(3-67)���Ѥ���)��, $v$�Ǥϥ����ӡ���(��ž��ʬ)��
 ����Ū�Ǥ���, �Ϥᤫ�餽�β�ž��ʬ���ۤ��Ϲ���ʿ�դʾ��֤��äƤ���
 �Ȥ������ȤǤ���.
}
���ۤ��Ϲ�ήŪ�Ǥ��뤳�Ȥ򼨺����Ƥ���.
���������ȤϽ�����(3-65)�ˤ⸽��Ƥ���Τ���,
���������Ȥο����ȥ����ӡ��Ȥο��������
%
\begin{eqnarray}
 \frac{a_{2,3}}{a_1} \simeq \frac{U}{(gH)^{1/2}}
  \frac{f^2}{\mu^2 g H}
\end{eqnarray}
%
�Ǥ���,
(3-67)��ꤳ�ο��ͤ����˾�����.

$u_0$��Ŭ�ڤ��ͤ�Ȥ��, ���������Ȥϴ����˾õ�
���뤳�Ȥ��Ǥ���.
(3-64)�ˤ�����$a_{2,3}$�ϥ����Ǥ���Ȥ�, $v_0$�˴ؤ����Ϲ�ή�δط�
���Ѥ����, $u_0$�μ���������
\footnote{
 (3-64)�ˤ�$a_{2,3}=0$�Ȥ���, ����ˤ�����(3-65)��$v_0 = i \mu f^{-1} gh_0$
 ���������ƶ�����뤳�ȤǤ���$u_0$�˴ؤ��뼰��������.
}.
%
\begin{eqnarray}
 u_0 = \frac{U g h_0}{gH + f^2 \mu^{-2}}.
\end{eqnarray}
%
�����Ǥ�ʬ��ˤ�����$\alpha$����٤ƾ����������Ȥ��Ƥ���.
������֤�$u$�ο���(3-69)��(3-61)�ˤ�����ؿ���ʬ���,
(3-58)��Ϳ����줿�褦�ʽ��Ϲ�ήȯ�������Τ�������
\footnote{
 (3-69)��(3-61)�����������,
 $$
  u = u_0 e^{i \mu (x - ct)}
   = \frac{U g h_0}{gH + f^2 \mu^{-2}} e^{i \mu (x - ct)}
 $$
 �Ȥʤ�. (���Τˤϻؿ�����$e^{i \mu (x - c_j t)}$�ʤΤǤ��μ���
 �������ʤ���, ��ñ�Τ��ᤳ�����Ҥ���.)
 �����$x$����ʬ����$h = h_0 e^{i \mu (x - ct)}$��Ȥ����Ȥ�
 (3-58)��������.
}.



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