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% Chapter 2   Wave Motion in the Atmosphere  Part I
%             2-6 INERTIAL GRAVITY WAVES AND ROSSBY WAVES
%
% 2002/09/30  ���� ����   ����
% 2002/10/02  ���� ����   �����������
% 2002/10/03  ���� ����   ����
% 2002/10/31  ���� ����   ���ߤˤ�뽤��
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\section{���������Ȥȥ����ӡ���}

\begin{itembox}[l]{����}
 �ϵ�μ�ž���θ����ȥ��ꥪ���Ϥ�Ư��, {\bfseries ����������}
 ({\itshape inertial gravity wave})��{\bfseries �����ӡ���}
 ({\itshape Rossby wave})�Ȥ��ä��Ȥ�ȯ������.
 ������ǤϤ������ȤˤĤ��Ƶ�������.
 {\bfseries ������������}({\itshape shallow-water equations})
 ��������������ư���Ϳ���뤳�Ȥˤ��,
 �������Ȥ������򼨤��ط���������.
 �ޤ�, ���������Ȥȥ����ӡ��ȤΤ��줾��ˤĤ��Ʊ��٤�ȯ���Τɤ��餬
 ����Ū����Ĵ�٤Ƥߤ�.
\end{itembox}

������Ǥ�, �����Ƴ�Ф����ſ尵����������Ĥ��ؤ˴ؤ����������Ϥ�
�ϵ�β�ž�θ��̤�ä��ƹͤ���.
���ꥪ���Ϥˤ�äư�������������Ĥߤ�����ȤǤ�������Ȥ˱ƶ���ڤܤ�.
�����, ���ꥪ��ѥ�᡼��������Ū�Ѳ�
\footnote{
 ���ꥪ��ѥ�᡼��$f = 2 \Omega \sin \varphi$�Ϥ�����������ʬ����褦
 �˰���$\varphi$�ˤ�ä��Ѳ�����.
}
���뤳�Ȥˤ�äƥ����ӡ��Ȥ�
������.
�����ϵ�β�ž��ͤ������α�ư�������ϰʲ��Τ褦�ʷ��ǽ񤯤��Ȥ��Ǥ���
\footnote{
 ������ʪ���̤Ͼ���ǤϤʤ��Τ�����.
}
\footnote{
 ���μ���(2-16), (2-17)�������뤳�Ȥ��Ǥ���.
 (2-16), (2-17)���鳰�Ϥȥ�ȥ�å����ʤ�, ����������ط���
 $g \partial h / \partial x = (1/\rho) \partial p / \partial x$, 
 $g \partial h / \partial y = (1/\rho) \partial p / \partial y$
 �����Ѥ��ư��Ϲ���ľ����(3-35)��ξ���ˤʤ�.
}.
%
\setcounter{equation}{0}
\def\theequation{3-35\alph{equation}}
%
\begin{eqnarray}
 \DP{u}{t} + u \DP{u}{x} + v \DP{u}{y} - fv + g \DP{h}{x} &=& 0,
  \\
 \DP{v}{t} + u \DP{v}{x} + v \DP{v}{y} + fu + g \DP{h}{y} &=& 0.
\end{eqnarray}
%
\setcounter{equation}{35}
\def\theequation{\arabic{chapter}-\arabic{equation}}
%

�ſ尵��̩�ٰ�����ꤷ�Ƥ��뤿��, ��ʿ��������������
\footnote{
 �Ĥޤ�, $\partial p /\partial x = 0, \partial p / \partial y =0$
 �Ȥ�������.
}
�Ϲ⤵�˰�¸���ʤ�.
�ޤ�, ®�پ�Ϲ⤵�˰�¸���ʤ��ȻϤ�˲��ꤷ�Ƥ���,
(3-35)�ˤ������ľ��ή��Ͻ�������Ƥ���.
�󰵽̤��ꤹ�뤳�Ȥ�, Ϣ³�μ��ϰʲ��Τ褦�������������.
%
\begin{eqnarray}
 \DP{u}{x} + \DP{v}{y} + \DP{w}{z} = 0.
\end{eqnarray}
%
(3-36)��$z$������$z=0$����$z=h$�ޤ���ʬ�����,
%
\begin{eqnarray}
 \left(
  \DP{u}{x} + \DP{v}{y}
 \right) h
 + w_h - w_0 = 0
\end{eqnarray}
%
�Ȥʤ�.
��ư��Ū��������ͤ����, ����������ư���ʤ�����$z=0$�Ǥ�$w$��
�ä��ʤ���Фʤ�ʤ�. (�Ĥޤ� $w_0 = 0$�Ȥʤ�).
����, ���������ˤ�����$w = dz/dt$�ϼ�ͳɽ�̤��岼�����Ѳ�Ψ��ɽ��.
���Τ���, $w_h = dh/dt$�Ȥʤ�, (3-37)�ϰʲ��Τ褦�ˤʤ�.
%
\begin{eqnarray}
 h \left( \DP{u}{x} + \DP{v}{y} \right)
  = - \DD{h}{t}
  = - \left( \DP{h}{t} + u \DP{h}{x} + v \DP{h}{y} \right).
\end{eqnarray}
%
���μ���2-5�������줿(3-34)��Ʊ��������, �����ǤϤ����餫
�㤦��ˡ������줿.
������(3-35a), (3-35b), (3-38)�ˤ�ä�
3�Ĥ�̤�ο�$u, v, h$�˴ؤ���3�Ĥ��������Ϥ����������.
�����������Ϥ�{\bfseries ������������}
({\itshape shallow-water equations})�ȸƤФ��.
�����3�Ĥμ����Ȥ߹�碌������פ���������������.
(3-35b)��$x$����ʬ��������(3-35a)��$y$����ʬ�������κ���Ȥ��,
���Υ�ǥ�Ǥα�����������������. �ʲ������μ��Ǥ���
\footnote{
 ���μ���Ƴ�Ф˴ؤ��Ƥ� {\bfseries 3-6-A}����. 
}.
%
\setcounter{equation}{0}
\def\theequation{3-39\alph{equation}}
%
\begin{eqnarray}
 \DP{\zeta}{t} + u\DP{\zeta}{x} + v\DP{\zeta}{y} + \beta v
  = -(\zeta + f) \left( \DP{u}{x} + \DP{v}{y} \right)
  = (\zeta + f) \frac{1}{h} \DD{h}{t}.
\end{eqnarray}
%
$\beta = \partial f / \partial y$�Ǥ���. ���μ��ϰʲ��Τ褦�ˤ�񤱤�.
%
\begin{eqnarray}
 \DD{}{t} \left( \frac{\zeta + f}{h} \right) = 0.
\end{eqnarray}
%
\setcounter{equation}{39}
\def\theequation{\arabic{chapter}-\arabic{equation}}
%
���μ��ˤ��{\bfseries �ݥƥ󥷥�뱲��}({\itshape potential vorticity})
\footnote{
 �ݥƥ󥷥�뱲�٤˴ؤ��Ƥ�Pedlosky(1987)�򻲾ȤΤ���.
}
�ȸƤФ����$(\zeta + f)/h$����¸����뤳�Ȥ��狼��.
�����Ertel
�ˤ�ä��󾧤��줿����Ū�ʥݥƥ󥷥�뱲������
\footnote{
 �ȳ��ˤ�ä��͡��ʡ֥ݥƥ󥷥�뱲�٤μ��פ�¸�ߤ��뤿��, 
 �����ȶ��̤��뤿��{\bfseries Ertel}�Υݥƥ󥷥�뱲�٤μ��ȸ���.
}
���餹������̤ʥ�������
����.
������(3-39b)�����гѱ�ư�̤���¸�μ����б�����.
�����, ��ž���Ƥ������ή�ΤǤι⤵���Ѳ��ˤ��ƶ���ͤ��Ƥߤ�Ȥ狼��.
$h$��������ˤĤ����������ѤϾ������ʤ�ΤǱ���Ϥ���᤯��ž����.
���Τ��Ȥϱ���γ�®�٤�$(\zeta + f)/2$�Ǥ��뤳�Ȥ���(3-39b)�Ȱ��פ���.

������(3-35)��(3-38)��$U$(����)��$H$������������.
�������̤ϰʲ��Τ褦���Ͼ���Ū�ʴط��ˤ���
\footnote{
 ���δط�����(3-35b)�ˤ�����$u=U, v=0, h=H$�Ȥ��뤳�Ȥ����뤳�Ȥ��Ǥ���.
 �ط���$fU = g (\partial H / \partial y)$�κ��դϥ��ꥪ���Ϥι�Ǥ���,
 ���դϵ��������Ϥι�Ǥ���.
 (���ι�ϸ���$(1/\rho) (\partial p /\partial y)$���ѷ�������ΤǤ���).
}.
%
\begin{eqnarray}
 U = - \frac{g}{f} \DP{H}{y}.
\end{eqnarray}
%
$H$�Ϥ����ޤǤǹͤ��Ƥ���̩�ٰ����ή�Το����Ǥ���.
����$u, v, h$
\footnote{
 ���������$u, v, h$�Ͼ�����̤ˤʤ�Τ�����.
}
��$f$��$y$�˰�¸���ʤ��Ȥ����, ��������(3-35a), (3-35b), (3-38)��
�ʲ��Τ褦�ˤʤ�.
%
\begin{eqnarray}
 \delta \left( \DP{u}{t} + U \DP{u}{x} \right) - fv + g \DP{h}{x}
  \hspace{3mm} &=& 0, 
  \nonumber \\
 \DP{v}{t} + U \DP{v}{x} \hspace{3mm} + fu \hspace{16mm} &=& 0,
  \\
 \DP{h}{t} + U \DP{h}{x} + H \DP{u}{x} + v \DP{H}{y} &=& 0.
  \nonumber
\end{eqnarray}
%
$\delta$�ΤĤ��Ƥ�����ȯ���������ˤ�����$dD/dt$
($D = \partial u / \partial x + \partial v / \partial y$)�Ȥʤ�
��Ǥ���.
����Ū��ȯ���������ϱ�ư��������$u, v$��ʬ��$x, y$����ʬ���뤳�Ȥ�
������.
����ξ��, $\partial v /\partial y = 0$�ʤΤ�$u$�ι������ȯ���ˤʤ�.
����$H$������Ȱ���
\footnote{
 �����Ǥ�¿���ü�ʲ���򤪤��ʤ�.
 $H$������Ȱ����Τ���, $\partial H / \partial y$������Ȱ����ΤǤ���.
 $H$������Ȱ����ʤ��$\partial H/ \partial y=0$�Ȥʤꤽ���ʤ�Τ���,
 �����ϰ���ʤ�.
},
�����$u_0 e^{i \mu (x - ct)}$,
$v_0 e^{i \mu (x - ct)}$, $h_0 e^{i \mu (x - ct)}$�Ȥ���Ĵ�´ؿ���
���ꤹ���, (3-41)�ϰʲ����������Ϥ��ѷ�����.
%
\begin{eqnarray}
 \delta (U - c) i \mu u_0 - f v_0 + gi \mu h_0 &=& 0, 
  \nonumber \\
 f u_0 + i \mu (U - c) v_0 &=& 0 ,
  \\
 i \mu H u_0 + \partial H /\partial y \ v_0 + i \mu (U - c) h_0 &=& 0.
  \nonumber
\end{eqnarray}
%
(3-42)��$u_0, v_0, h_0$�˴ؤ��Ƽ����Ǥʤ������Ĥ���ˤϰʲ���
��郎��������ͤФʤ�ʤ�.
%
\begin{eqnarray}
 \left|
  \begin{array}{ccc}
   \delta (U - c) i \mu & -f & g \mu i \\
   f & (U - c) i \mu & 0 \\
   i \mu H & \partial H /\partial y & (U - c) i \mu \\
  \end{array}
 \right| = 0.
 \nonumber
\end{eqnarray}
%
���ι��󼰤�Ÿ�������, �ʲ��ο�ư���˴ؤ���3������������������.
%
\begin{eqnarray}
 \delta(U - c)^3 - (gH + f^2 / \mu^2) (U - c)
  - \frac{fg}{\mu^2} \DP{H}{y} = 0.
\end{eqnarray}
%
������������2�Ĥ��ᤤ�����Ȥβ��1�Ĥ��٤����ݳ�Ū������.
�ᤤ���������Ф�����ˤ�$U=0$, $\delta = 1$�Ȥ���.
����Ȱʲ��Τ褦�ʰ���®�٤β�������.
%
\begin{eqnarray}
 c = \pm \sqrt{gH + f^2 / \mu^2}.
\end{eqnarray}
%
�⤦��Ĥβ��$c=0$�Ǥ���. ������{\bfseries ����������}
({\itshape inertial gravity wave})�ȸƤФ���ȤǤ���.
$f=0$�ˤ����(3-44)�������Ȥ˴ؤ������(3-31)�ˤʤ�.

�⤷�⺬�����1���ܤιब2���ܤι�Ȥ���Ӥ���Ȥ���Τʤ��,
�ʲ��Τ褦��{\bfseries ������ư}({\itshape inertial oscillation})�Τߤ�
���Ф����Ȥ��Ǥ���
\footnote{
 ���ޤ������, $f = 2 \Omega \sin \varphi$ [1/s]�Ǥ���.
}.
%
\begin{eqnarray}
 \nu = \mu c = \pm f.
\end{eqnarray}
%
��������$(1/\mu)$����ʬ���礭����, �⤷����$gH$����ʬ�˾���������
���ξ��֤�����Ω��.
���δ��������Ȥβ�ϰ��ϸ��ۤ�̵�뤹�뤳�Ȥ��������α�ư����������
ľ�����뤳�Ȥ��Ǥ���.
%
\begin{eqnarray}
 \DD{u}{t} = - f v, \quad
  \DD{v}{t} = f u.
\end{eqnarray}
%
�����μ����Ȥ߹�碌�뤳�Ȥǰʲ��μ���������.
%
\begin{eqnarray}
 \DD[2]{u}{t} + f^2 u = 0. 
\end{eqnarray}
%
���μ��ϰʲ��Τ褦�ʷ��β�����.
%
\begin{eqnarray}
 u = u_0 e^{\pm i f t}.
\end{eqnarray}
%
���β򤫤�$\nu = \pm f$�Ǥ��뤳�Ȥ��狼��.
���Τ��Ȥ��鴷����ư�μ�����
$2 \pi / \nu = 2 \pi / f = 12 / \sin \varphi$ [1/hr]
�Ǥ��뤳�Ȥ��狼��
\footnote{
 �ϵ��24���֤ǰ��ž����Τ�$\Omega = 2 \pi /24$ [1/hr]�Ǥ���.
 ���ꥪ��ѥ�᡼����$f = 2 \Omega \sin \varphi$�ʤΤ�,
 $2 \pi /f = \pi / \Omega \sin \varphi = 12 / \sin \varphi$�Ǥ���.
}.

%�����ޤǤβ��ϤǤϺ��Ū�˰��Ϥϰ���Ǥ���Ȳ��ꤷ�Ƶ������Ƥ���.
%�������ºݤˤϼ��̤����Ͼ�Ȥ��Զѹդˤ�ä������������Ȥˤ�ä�
%���Ͼ�ˤ��Ѳ����⤿�餵���.
%��������ˤ⤫����餺, �����ޤǤ˽Ҥ٤������αƶ��ˤ�ä�������
%���ݤ���¿���ºݤ˴�¬����Ƥ���.
%�㤨��, ���ϰ���ε���ε�ƻ�ϴľ��ˤʤ�. ����ϴ����αƶ��򼨺����Ƥ���.
%������ư�ˤ�ä����������⤦��Ĥθ��ݤȤ���
%{\bfseries ��ֶ����إ����å�}({\itshape nocturnal boundary-layer jet})
%������.
%�����Ǥ���֤α�Ǵ�����Ѳ��ˤ����֤ˤϹ���500m�ն���໤�Ϥˤ��
%���꤬������.
%�������Ū�����٤λ����Ѳ��˵��������ΤǤ���.
%������ư�θ�ŵŪ����Ȥ��ƤϥХ�ȳ���Gustafson��Kullenberg
%�ˤ�äƴ�¬���줿���ݤ�����. ����ϥ�������θ����������ѥ륹������
%�������ݤǤ��ä�.
%�Ƕ��¬��Ǥϰ���Ū���̸����ɥ�ե�
%���鴷����ư�νŤ͹�碌��Ϥä���ɽ���Ǥ���褦�ˤʤä�.

����, (3-43)���٤����ݳ�Ū�ʲ��$\delta = 0$�Ȥ��뤳�Ȥ����뤳�Ȥ��Ǥ���.
%
\begin{eqnarray}
 c = U + \frac{(f/H) \partial H / \partial y}{\mu^2 + (f^2/gH)}.
  \nonumber
\end{eqnarray}
%
���μ��ϴ��ܾ��֤Υݥƥ󥷥�뱲��$\overline{q} = f/H$�ǽ񤯤��Ȥ��Ǥ���.
�ݥƥ󥷥�뱲�٤�$y$��ʬ��,
%
\begin{eqnarray}
 \DP{\overline{q}}{y} = - \frac{f}{H^2} \DP{H}{y} = \frac{f^2 U}{g H^2}
\end{eqnarray}
%
�Ȥʤ�. ��ͳɽ�̤ι⤵$H(y)$�ϴ��ܾ�Ǥ��Ϲ�ήʿ��(3-40)������������
$y$�����˸��ۤ����. ���Τ�����ܾ��֤ǤΥݥƥ󥷥�뱲�٤θ��ۤ�
�����ˤϤʤ�ʤ�
\footnote{
 �����Ǥ�$f$��$y$�������Ѳ����ʤ����Ȥˤ��Ƥ���Τ�����.
}.
 (3-49)�ˤ�äư���®�٤ϼ��Τ褦�˽񤯤��Ȥ��Ǥ���.
%
\begin{eqnarray}
 c = U - \frac{H \ \partial \overline{q} /\partial y}{\mu^2 + (f^2 / gH)}.
\end{eqnarray}
%
���β��{\bfseries �����ӡ���}({\itshape Rossby wave})�η��Ǥ���.
���ΰ���®�٤ϴ�¬��������ѵ��ϤǤξ���Ȥ褯���פ���.
���μ���$f$�����٤ˤ�ä��Ѳ�������˰��̲������.
���ξ��ϥݥƥ󥷥�뱲�٤θ��ۤϰʲ��Τ褦�ˤʤ�.
%
\begin{eqnarray}
 \DP{\overline{q}}{y} = \left( \beta - \frac{f}{H} \DP{H}{y} \right) /H
  = [\beta + (f^2 U / g H)]/H.
\end{eqnarray}
%
$\beta = df/dy$�Ǥ���.
����ϥ����ӡ��Ȥ����ܾ��ή�������Ū������
$\Dvect{k} \times \nabla \overline{q}$�����Ť���ˡ§�ΰ�Ĥ���Ǥ���
\footnote{
 �����ӡ��Ȥ����������˴ؤ��Ƥ�{\bfseries 3-6-C}�򻲾ȤΤ���.
}.
$f$��$y$�����Ѳ�(�Ĥޤ�$\beta$)�˰�¸�������®�٤�(3-50)��(3-51)����
���뤳�Ȥ��Ǥ���.
�����(3-41)�ˤ�����$f=f(y)$��Ȥ�, $v$�α�ư������������������
�����������ˤ��뤳�Ȥǳ�ǧ�Ǥ���.

�����ӡ��Ȥ�(3-41)�ˤ�����$\delta = 0$�Ȥ��뤳�Ȥ�(3-43)��
���Ф����Ȥ��Ǥ���. ���ξ��, $v$��ʬ���Ϲ�ήŪ�Ǥ���.
�Ĥޤ�,
%
\begin{eqnarray}
 f v = g \DP{h}{x}
\end{eqnarray}
%
�Ǥ���. ������, $u$��ʬ��ȯ�������㤷,
�Ϲ�ήŪ�ǤϤʤ�.

����줬$x$��$y$���Ѳ�������, ���Ϥ򤪤��ʤ��ˤ�
��(3-35a)��(3-35b)���٤�ȯ���μ����ѷ�����ɬ�פ�����.
���ξ��, ȯ���������ˤ�����ȯ���λ�����ʬ
($dD/dt$)��̵�뤵����,
{\bfseries ����}�Ȥ��ӽ�����뤳�Ȥ�ʬ����
\footnote{
 (3-41)�ǹͤ��Ƥߤ�. $u$��ʬ�μ���$x$����ʬ����ȯ���μ�����.
 �ʤ�, �����$y$�����ˤ��Ѳ����ʤ�����, ȯ��$D$��$D= \partial u /
 \partial x $�Ǥ���.
 %
 \begin{eqnarray}
  \DP{D}{t} + U \DP{D}{x} - f \DP{v}{x} + g \DP[2][]{h}{x} = 0.
   \nonumber
 \end{eqnarray}
 %
 ������$D$�λ�����ʬ($= dD/dt$)��̵�뤹���, �ط���
 $fv = g \partial h /\partial x$������Ω��. �����(3-52)��Ʊ�����Ǥ���.
 �Ĥޤ�, $D$�λ�����ʬ��̵�뤹�뤳�Ȥ�(3-41)�ˤ�$\delta$�ι�
 ���ӽ����뤳�Ȥ�Ʊ����̤ˤʤ�.
}.
����ˤ�ä����β�ž�����Ǥȹ⤵�Ȥδط���ʬ����.
����Ū�ˤ��Ϲ�ή�ξ��֤�ɽ����(3-52)�����δط�����������.

�����ޤǤǹͤ��Ƥ���2������Ȥα��٤�ȯ������Ӥ��������.
������ӤϺ���Τ褦��ʪ���̤�$y$�������Ѳ����ʤ����ˤ�
�ä˴�ñ��, �ʲ��Τ褦�ˤʤ�.
%
\begin{eqnarray}
 \left| \frac{D}{\zeta} \right| =
  \left| \frac{\partial u /\partial x}{\partial v/ \partial x} \right|.
\end{eqnarray}
%
����������($U = 0$)�ξ�����Ӥ򤪤��ʤ������, (3-41)��
$v$��ʬ�μ��ˤ�����$v = e^{i \mu (x - ct)}$�Ȥ����,
�ʲ��δط�����������
\footnote{
 $U = 0$�Ǥ��뤿��, (3-41)��$v$��ʬ�μ���
 $\partial v / \partial t + fu = 0$�Ȥʤ�.
 $v = e^{i \mu (x - ct)}$�Ȥ����,
 $\partial v / \partial t = - i \mu c e^{i \mu (x - ct)} = - i \mu c v$
 �Ȥʤ�, �ط���(3-45)��������.
}.
%
\begin{eqnarray}
 v = - \frac{i f u}{\mu c}.
\end{eqnarray}
%
(3-54)��(3-44)�����(3-53)�ϰʲ��Τ褦�ˤʤ�.
%
\begin{eqnarray}
 \left| \frac{D}{\zeta} \right|_{I - G}
  = (1 + \mu^2 g h f^{-2})^{1/2}.
\end{eqnarray}
%
�����̤Ͼ��1�����礭��. �Ĥޤ괷�������Ȥˤ����Ƥ�ȯ���ϱ��٤��Ф���
����Ū�Ǥ���Ȥ������Ȥ�ʬ����.

�Ϲ����⤷���ϥ����ӡ���($\delta = 0$)�ξ��, ���٤�ȯ����$h$�ι��
���Ҥ���������Ǥ���. (3-52)���鱲�٤����뤳�Ȥ��Ǥ���
\footnote{
 $h$��$e^{i \mu (x - ct)}$�Ȥ�����(3-52)��$x$����ʬ����Ȥ��μ���
 ������.
 ���ʤߤ�$\partial u / \partial y = 0$�ʤΤǱ��٤�
 $\zeta = \partial v / \partial x$�Ǥ���.
}.
%
\begin{eqnarray}
 \DP{v}{x} = - \frac{\mu^2 g}{f} h.
\end{eqnarray}
%
�ʲ���$u$��$h$�˴ؤ������������
\footnote{
 ������ʬ�ι��ޤޤʤ����Τ���.
}
��$\delta = 0$�Ȥ���(3-41)����Ƴ�ФǤ���
\footnote{
 ���μ���Ƴ�Ф˴ؤ��Ƥ�{\bfseries 3-6-B}����.
}.
%
\begin{eqnarray}
 \DP[2]{u}{x} - \frac{f^2}{gH} u = \frac{U}{H} \DP[2]{h}{x}.
\end{eqnarray}
%
���μ�������ȯ�����ǤˤĤ��Ƥ�{\bfseries ���Ϲ�ή������}
({\itshape quasi-geostrophic equation})�Ǥ���.
�ؿ�Ū���ȷ���Ϳ��
\footnote{
 ����Ū�ˤ�$u = u_0 e^{i \mu (x - ct)}$, $h = h_0 e^{i \mu (x - ct)}$
 �Τ褦���ȷ����ꤹ��.
},
�����$x$����ʬ�����
\footnote{
 �ؿ�Ū���ȷ���Ϳ�������Ȥ�$\partial^2 u / \partial x^2 = - \mu^2 u$,
 $\partial^2 h / \partial x^2 = - \mu^2 h$�Ȥʤ�.
 ����ˤ�ä�$u = Ugh /(gH + f^2/\mu^2)$�Ȥ����ط���������Ω��.
 ���ˤ��μ���$x$����ʬ����.
},
�ʲ��μ���������.
%
\begin{eqnarray}
 \DP{u}{x} = \frac{i \mu U g h}{g H + (f^2 / \mu^2)}.
\end{eqnarray}
%
(3-56)��(3-58)��(3-53)��������������
%
\begin{eqnarray}
 \left| \frac{D}{\zeta} \right|_{R} =
  \frac{(U \mu /f)}{(\mu^2 g H /f^2) + 1}
\end{eqnarray}
%
�Ȥʤ�. �����̤Ͽ��ͷ׻��ˤ����Ƥ�{\bfseries �����ӡ���}
({\itshape Rossby number})�ȸƤФ�, ���ѵ��Ϥα�ư�ǤϤ�������
1/10 ���餤���ͤ����.
ʬ���1��������礭������, �����ӡ��ȤǤ�ȯ���ϱ��٤���٤ƾ�����.
�������4�ϤΥ���������Ϥ��꼰���ˤ��������Ω��.



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


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%%%%%%%%             �ʲ���Appendix             %%%%%%%%
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\def\thefigure{A-\arabic{figure}}
\def\thetable{A-\arabic{table}}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%      3-6-A      %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{3-6-A\qquad �����������ȥݥƥ󥷥�뱲����¸�μ���Ƴ��}
\setcounter{equation}{0}
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\def\theequation{3-6-A-\arabic{equation}}
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����������(3-39a)�ȥݥƥ󥷥�뱲����¸�μ�(3-39b)�μ���Ƴ�Ф���.

\subsubsection*{����������(3-39a)}

(3-35b)��$x$����ʬ��������(3-35a)��$y$����ʬ�������Ϥ��줾��
%
\begin{eqnarray}
 \DP{}{t} \DP[][]{v}{x} + u \DP[2][]{v}{x} + \DP{u}{x} \DP{v}{x}
  + v \DP{}{y} \DP[][]{v}{x} + \DP{v}{x} \DP{v}{y} + f \DP{u}{x}
  + g \DP{}{x} \DP[][]{h}{y} = 0,
  \nonumber \\
 \Deqlab{hw-02-06-y��ư������x��ʬ}
  \\
 \DP{}{t} \DP[][]{u}{y} + u \DP{}{x} \DP[][]{u}{y} + \DP{u}{y} \DP{u}{x}
  + v \DP[2][]{u}{y} + \DP{v}{y} \DP{u}{y} - f \DP{v}{y} - v \DP{f}{y}
  + g \DP{}{y} \DP[][]{h}{x} = 0
  \nonumber \\
 \Deqlab{hw-02-06-x��ư������y��ʬ}
\end{eqnarray}
%
�Ȥʤ�.
\Deqref{hw-02-06-y��ư������x��ʬ}����
\Deqref{hw-02-06-x��ư������y��ʬ}�������,
%
\begin{eqnarray}
 \DP{}{t} \left( \DP{v}{x} - \DP{x}{y} \right)
  + u \DP{}{x} \left( \DP{v}{x} - \DP{x}{y} \right)
  + \DP{u}{x} \left( \DP{v}{x} - \DP{x}{y} \right)
  \hspace{30mm}
  \nonumber \\
 \Deqlab{hw-02-06-����������Ƴ��1}
  + v \DP{}{y} \left( \DP{v}{x} - \DP{x}{y} \right)
  + \DP{v}{y} \left( \DP{v}{x} - \DP{x}{y} \right)
  + f \left( \DP{u}{x} + \DP{v}{y} \right)
  - v \DP{f}{y} = 0
\end{eqnarray}
%
�Ȥʤ�.
���٤�$\zeta = \Dvect{k} \cdot (\nabla \times \Dvect{V}) =
\partial v / \partial x - \partial u / \partial y$�ʤΤ�,
\Deqref{hw-02-06-����������Ƴ��1}��
%
\begin{eqnarray}
 \DP{\zeta}{t} + u \DP{\zeta}{x} + v \DP{\zeta}{y}
  + \zeta \DP{u}{x} + \zeta \DP{v}{y}
  + f \left( \DP{u}{x} + \DP{v}{y} \right) - \beta v = 0
  \nonumber \\
 \Deqlab{hw-02-06-����������Ƴ��2}
  \DP{\zeta}{t} + u\DP{\zeta}{x} + v\DP{\zeta}{y} + \beta v
  = -(\zeta + f) \left( \DP{u}{x} + \DP{v}{y} \right)
\end{eqnarray}
%
�Ȥʤ�. $\beta = \partial f /\partial y$�Ǥ���.
�Ǹ��(3-38)��
\Deqref{hw-02-06-����������Ƴ��2}�α��դ˻Ȥ���,
�ʲ��α��٤μ��Ȥʤ�.
%
\begin{eqnarray}
 \Deqlab{hw-02-06-����������}
 \DP{\zeta}{t} + u\DP{\zeta}{x} + v\DP{\zeta}{y} + \beta v
  = (\zeta + f) \frac{1}{h} \DD{h}{t}. 
\end{eqnarray}


\subsubsection*{�ݥƥ󥷥�뱲����¸�μ�(3-39b)} 

\Deqref{hw-02-06-����������}�μ����ѷ����ƥݥƥ󥷥�뱲����¸��
����Ƴ�Ф���.
$f$��$y$�Τߤ˰�¸��,
$\partial f / \partial t = \partial f / \partial x = 0$�Ǥ��뤳�Ȥ�
���դ��뤳��.
%
\begin{eqnarray}
 \DP{\zeta}{t} + u\DP{\zeta}{x} + v\DP{\zeta}{y} + v \DP{f}{y}
  + (\zeta + f) h \DD{}{t} \left( \frac{1}{h} \right)
  &=& 0
  \nonumber \\
 \DP{}{t} (\zeta + f) + u \DP{}{x} (\zeta + f) + v \DP{}{y} (\zeta + f)
  + (\zeta + f) h \DD{}{t} \left( \frac{1}{h} \right)
  &=& 0
  \nonumber \\
 \frac{1}{h} \DD{}{t} (\zeta + f)
  + (\zeta + f) \DD{}{t} \left( \frac{1}{h} \right)
  &=& 0
  \nonumber \\
 \Deqlab{hw-02-06-�ݥƥ󥷥�뱲����¸�μ�}
  \DD{}{t} \left( \frac{\zeta + f}{h} \right) &=& 0.
\end{eqnarray}
%



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%      3-6-B      %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{3-6-B\qquad ���Ϲ�ή��������Ƴ��}
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\def\theequation{3-6-B-\arabic{equation}}
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������������������������������(3-41)������Ϲ�ή������(3-57)��Ƴ��.

$\delta = 0$�ʤΤ�, (3-41)��1���ܤμ���$fv = g \partial h / \partial x$
�Ȥʤ�. ���μ��ȴ��ܾ���Ϲ�ήʿ�դμ�(3-40)
\footnote{
 $U = - (g/f) \partial H / \partial y$
}
��(3-41)��2, 3���ܤμ���������, 3���ܤμ��˴ؤ��ƤϹ���$x$����ʬ�����,
�ʲ��Τ褦�ˤʤ�.
%
\begin{eqnarray}
 \Deqlab{hw-02-06-���Ϲ�ή������Ƴ��1}
  \DP{}{t} \DP[][]{h}{x} + U \DP[2]{h}{x} + \frac{f^2}{g} u
  \hspace{20mm}&=& 0,
  \\
 \Deqlab{hw-02-06-���Ϲ�ή������Ƴ��2}
  \DP{}{t} \DP[][]{h}{x} + U \DP[2]{h}{x} + H \DP[2]{u}{x}
  - U \DP[2]{h}{x} &=& 0.
\end{eqnarray}
%
\Deqref{hw-02-06-���Ϲ�ή������Ƴ��1}����
\Deqref{hw-02-06-���Ϲ�ή������Ƴ��2}��������Ȥ�
���Ϲ�ή��������������.
%
\begin{eqnarray}
 \Deqlab{hw-02-06-���Ϲ�ή������}
  \DP[2]{u}{x} - \frac{f^2}{gH} u = \frac{U}{H} \DP[2]{h}{x}.
\end{eqnarray}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%      3-6-C      %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{3-6-C\qquad �����ӡ��Ȥ����¤�������}
\setcounter{equation}{0}
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\def\theequation{3-6-C-\arabic{equation}}
\def\thefigure{3-6-C-\arabic{figure}}

��ʸ��ˤϡ֥����ӡ��Ȥϴ��ܾ��ή�������Ū������
$\Dvect{k} \times \nabla \overline{q}$�����Ť����
�Ȥ���ɽ��������Τ���, ����ˤĤ��Ʋ��⤹��.

$\overline{q} = f/H$�ʤΤ�$\overline{q}$��$y$�����ˤΤ߰�¸����.
��ä�,
%
\begin{eqnarray}
 \nabla \overline{q} =
  \left(
   \begin{array}{c}
    0 \\ \DP{\overline{q}}{y} \\ 0
   \end{array}
  \right)
  \nonumber
\end{eqnarray}
%
�Ǥ���. �����,
%
\begin{eqnarray}
 \Dvect{k} \times \nabla \overline{q} =
  \left(
   \begin{array}{ccc}
    \Dvect{i} & \Dvect{j} & \Dvect{k} \\
    0 & 0 & 1\\
    0 & \partial \overline{q} / \partial y & 0 \\
   \end{array}
  \right)
  = - \DP{\overline{q}}{y} \Dvect{i}
  \nonumber
\end{eqnarray}
%
�Ȥʤ�.
�Ĥޤ�ºݤˤϥ����ӡ��Ȥ��������������¤��뤳�Ȥ�ʬ����.
(3-49)����$U$�ε����������¤��뤳�Ȥ��狼��.







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%%%%      Appendix        %%%%
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%\subsection*{Appendix \qquad }

%\subsubsection*{}


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