%
% Chapter 1  The Governing Equations
%            1-8 MAP PREJECTIONS
%
% 2002/06/07 ���� ����   ����
% 2002/06/13 ���� ����   �ǽ�����
% 2002/07/11 ���� ����   ����
%
%
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\markboth{2-8 �Ͽ����ˡ}{2-8 �Ͽ����ˡ}
%\markright{1-8 �Ͽ����ˡ}
%\Ddate{2002/07/11}
%\Dfile{hw-01-08.tex}

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\section{�Ͽ����ˡ}

���̤Ǥ����ϵ�ɽ�̤�����, �ޤ��ϰ�����ʿ���Ͽޤ�ɽ�����Ȥ�
���ݸ��ݤβ���, ͽ��, ����ˤ�����ͭ�ѤǤ���.
�Ͽޤκ����ˤ����äƤ�ʿ���Ͽޤ˽񤭽Ф�����Τ�
�Ǥ���������̤˶ᤤ��ΤǤʤ���Фʤ�ʤ���,
ʿ�̤ˤ���ʾ�, �������ٵ��̤���ħ�ϼ����Ƥ��ޤ�.
�򺹤�������γ��٤�ݻ�����Ͻ��פʤ��Ȥ�,
�㤨�б߾��ΰ����ȻҸ�����ľ�Ѥ˸򺹤���Τ⤽���Ǥ���.
�������ä����פ�����������Ͽ����ˡ��
{\bf �������ˡ}({\it conformal\/})�ȸƤФ��.
���̤ˤ������Υ��ʿ���Ͽޤ���Ƥ���Ƥ�ݻ�����Ƥ���
�褦�����ˡ��{\bf �������ˡ}({\it isometric\/}) �ȸƤФ��.
�����Ͽޤϵ��ݳؼԤδ֤ǻȤ��Ƥ���ʬ�ˤ�
���ޤ���������Ƥ��ʤ�.
���ΰ�����, ��Υ���ĤߤϤ��ʤ��ɤ���٥�ޤǲ������Ƥ���.

����Ū�˻Ȥ��Ƥ����Ͽޤˤ�3���ढ��.
1�Ĥ�{\bf �˼ͱƿ�ˡ}({\it polar stereographic\/})�ȸƤФ�,
Ⱦ����Ͽޤ��������Τ˸����Ƥ���.
2�Ĥ��{\bf ��륫�ȥ뼰������ˡ}
({\it Mercator cylindrical projection\/})�Ȥ���,
�ä���ƻ�Ӥ��Ͽޤκ����������ʿ�ˡ�Ǥ���.
3�Ĥ��{\bf ���٥�ȱ߿���ˡ}({\it Lambert conical projection\/})
�Ǥ���.
���������ƤϿޤ�1-2�˼�����Ƥ���.
%
\begin{figure}[htb]
 \begin{center}
  \Depsf[][]{./eps/hwfig-01-02.eps}
 \end{center}
 \begin{quote}
  \caption{
  {\footnotesize (a) ��륫�ȥ������ˡ�ϰ��٤�20$^{\circ}$�Ǥ�������,
  (b) ���٥�ȱ߿���ˡ�ϰ���30$^{\circ}$��60$^{\circ}$��������,
  (c) �˼ͱƿ�ˡ��90$^{\circ}$��������.
  (Haltiner, G. J., and Williams, R. T., 1979:
      Numerical Prediction and Dynamic Meteorology.
      {\itshape John Wiley \& Sons,} 477pp)}
  }
  \Dfiglab{hwfig-01-02}
 \end{quote}
\end{figure}
%
�˼ͱƿ�ˡ�ȥ�륫�ȥ��ˡ�ϥ��٥�ȱ߿���ˡ���ü�ʤ�Τ�
��ĤǤ���.
���̤�ʿ�̤���Ƥ����Ͽޤ���Ѥ���ˤ����äƤ�,
ʿ���Ѥ˿����ʺ�ɸ�Ϥ��Ѱդ�,
����˱�ư�������⤳�κ�ɸ���Ѥ��ѷ����뤳�Ȥ�˾�ޤ��.

�ʹߤǤ��ɤ����Ѥ�������ˡ�ˤ�����
�����������Ϥ򰷤�.
�ʤ�, �Ͽ޳ؤξܤ�������� ��¼ ���� �����Ͽ����ˡ�פʤɤ򻲹ͤˤ��줿��
(������, �����ܤ�����).


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% 2-8-1 Polar Stereographic Projection %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{�˼ͱƿ�ˡ}

���ο�ˡ�Ͽ��͵���ͽ��ˤ����ƹ����Ȥ��Ƥ���.
���̤���ʿ�̤ؤ���Ƥϰʲ����ѷ��ˤ�ä�
�Ԥ���.
%
\begin{eqnarray}
 r = 2a \tan (\phi/2) \qquad and \qquad \theta = \lambda.
\end{eqnarray}
%
����ܤθ����Ͽ�2-2���餹����������.
����$a$���ϵ�Ⱦ�¤�ʿ���ͤ�ɽ��,
$r$���Ͽ޾�ΰ����αߤ�Ⱦ�¤Ǥ���.
$\phi$��;����({\it colatitude\/})�ȸƤФ�,
����ΰ���$\varphi$�Ȥ�
$\phi = \pi/2 - \varphi$�Ȥ����ط��ˤ���.
��(2-21)�ϰʲ��Τ褦�˽񤯤��Ȥ��Ǥ���
\footnote{
���Ѵؿ��δط���
\begin{eqnarray*}
 \tan ^2 \alpha = \frac{1- \cos 2 \alpha}{1+ \cos 2 \alpha}
\end{eqnarray*}
���,
\begin{eqnarray*}
 \tan (\phi /2) = \sqrt {\frac{1- \cos \phi}{1+ \cos \phi}}
  = \sqrt{\frac{\sin ^2 \phi}{(1+ \cos \phi)^2}}
  = \frac{\sin \phi}{1+ \cos \phi}
  = \frac{\cos \varphi}{1 + \sin \varphi}.
\end{eqnarray*}
�ʤ�, $\phi = \pi/2 -\varphi$���,
$\sin \varphi = \cos \phi$, $\cos \varphi = \sin \phi$�Ǥ���.
}.
%
\begin{eqnarray}
 r = a \ m(\varphi) \cos \varphi \qquad
     \theta = \lambda \qquad
     m(\varphi) = 2/(1+\sin \varphi).
\end{eqnarray}
%
�����Ǥ�$m(\varphi)$��{\bf �ޥåץե�������}({\it map factor\/})
�ȸƤФ���̤���ƿޤν̼ܤǤ���
\footnote{
$m(\varphi)$�ΰ�̣�ˤĤ��Ƥ�
{\bf 2-8-A}����.
}.

���ζ˼ͱƿޤ�$x=r \cos \theta, \ y = r \sin \theta$�Ȥ���
�ǥ���Ⱥ�ɸ��Ƴ������.
$x, \ y$��$\varphi, \ \lambda$���Ѥ��ưʲ��Τ褦��
���Ҥ��뤳�Ȥ��Ǥ���.
%
\begin{eqnarray}
 x=\frac{2a \cos \varphi \cos \lambda}{1+ \sin \varphi},
 \qquad
 y=\frac{2a \cos \varphi \sin \lambda}{1+ \sin \varphi},
 \qquad
 z=z.
\end{eqnarray}
%
�����μ�����ʬ���뤳�ȤǶ˼ͱƿ޾�Ǥε�Υ($dx, dy$)��
���̾�Ǥε�Υ($dx_s, dy_s$)�Ȥδط�����Ƴ�����Ȥ��Ǥ���.
�ط����ϰʲ��Τ褦�ʹ�������ˤʤ�.
\footnote{
Ƴ�Ф� {\bf 2-8-B}����
}
%
\begin{eqnarray}
 \left(
  \begin{array}{c}
   dx \\
   dy
  \end{array}
 \right)
  =
 \frac{2}{1+\sin \varphi}
 \left(
  \begin{array}{cc}
   - \sin \lambda & -\cos \lambda \\
     \cos \lambda & -\sin \lambda \\
  \end{array}
 \right)
 \left(
  \begin{array}{c}
   dx_s \\
   dy_s \\
  \end{array}
 \right).
\end{eqnarray}
%
����$2/(1+ \sin \varphi)$�ϵ��̤���˼ͱƿޤ���Ƥ��줿�ݤ�
����Ψ$m(\varphi)$��ɽ��.
�ޤ�, ����ϵ��̺�ɸ�Ϥβ�ž��ɽ��.
���Τ���, �˼ͱƿ�ˡ�ǤΥ����������$h_x, \ h_y$��
%
\begin{eqnarray}
 h_x = h_y = 1/m(\varphi) = (1+ \sin \varphi)/2
\end{eqnarray}
%
�Ȥʤ�.
��(2-24)�������餫�ʤ��Ȥ���,
�˼ͱƿ޺�ɸ�ϤǤ���®($U, \ V$)��
�ϵ��Ǥ���������̸�����®��($u_s, \ v_s$)�ȴط����Ƥ���.
���δط����ϼ�(2-24)��Ʊ�ͤ˰ʲ��Τ褦��ɽ�����Ȥ��Ǥ���.
%
\begin{eqnarray}
 \left(
  \begin{array}{c}
   U \\
   V \\
  \end{array}
 \right)
 =
 \left(
  \begin{array}{cc}
   - \sin \lambda & - \cos \lambda \\
     \cos \lambda & - \sin \lambda \\
  \end{array}
 \right)
 \left(
  \begin{array}{c}
   u_s \\
   v_s \\
  \end{array}
 \right),
\end{eqnarray}
%
\begin{eqnarray}
 U = h_x \dot{x} \quad and \quad V = h_y \dot{y}.
\end{eqnarray}

�˼ͱƿ޺�ɸ�ϤǤα�ư������������ˤ�
���Ҥδط�����(2-14)������������ɤ�.
���η�̤ϰʲ��Τ褦�ˤʤ�
\footnote{
Ƴ�Ф���ˡ�� {\bf 2-8-C}����.
}.
%
\begin{eqnarray}
 & &
 \DD{U}{t}-V
  \left(
   f - \frac{xV-yU}{2 a^2}
  \right)
  - \frac{W}{a}
  \left[
   (1+ \sin \varphi) \Omega y -U
  \right]
  = - m \alpha \DP{p}{x},  \\
 & &
 \DD{V}{t}+U
  \left(
   f - \frac{xV-yU}{2 a^2}
  \right)
  + \frac{W}{a}
  \left[
   (1+ \sin \varphi) \Omega x +V
  \right]
  = - m \alpha \DP{p}{y},  \\
 & &
 \DD{W}{t} - \frac{1}{a}
  \left[
   U^2+V^2+ \Omega(1+ \sin \varphi)(xV-yU)
  \right]
 = - \alpha \DP{p}{z} - g.
\end{eqnarray}
%
�����Ǥ� $f=2\Omega \sin \varphi$�Ǥ���,
$F$�Ͻ������Ƥ���.
������,
%
\begin{eqnarray}
 \DD{U}{t} = \DP{U}{t} + m
             \left(
	      U \DP{U}{x} + V \DP{U}{y}
	     \right)
	     + W \DP{U}{z}
\end{eqnarray}
%
�Ȥʤ�
\footnote{
����ʸ�κǸ�˷Ǻܤ��Ƥ�����ظ���(A-21)����
}.

�ſ尵���������Ǥ�����ϼ�(2-11)��ɽ�����褦��
®��$U, V$�λ����Ѳ���®��$W$�˰�¸���Ƥ��ʤ�����,
��(2-28), (2-29)�α��դ�$W$��ޤ��Ͻ�������ʤ���Фʤ�ʤ�.
�ޤ�, ��ư���ͥ륮����¸§�ǹͤ����(2-28)��(2-30)�κ��դϰʲ��Τ褦��
���Ȥʤ�.
%
\begin{eqnarray*}
 \DD{}{t} \left( U^2 + V^2 + W^2 \right) = \cdots .
\end{eqnarray*}
%
���ͥ륮����¸§��ͤ������, (2-28), (2-29)��$W$��ޤ�ब������������
(2-30)�� \\
$- (1/a) \left[U^2+V^2+\Omega(1+\sin\varphi)(xV-yU)\right]$
�ι����������ɬ�פ�����
\footnote{
������ؤ�ζ���Ū�ʼ���{\bfseries 2-7�� ��ɸ��}��
{\bf 2-7-H ���������פΰ�̣}�򻲾ȤΤ���
}.
���Τ褦�˼�(2-28)��(2-30)�Τ����Ĥ��ι����Ȥ���,
(2-16)��(2-18)�Ȼ����褦�ʰʲ��η��ˤʤ�.
%
\begin{eqnarray}
 \DD{U}{t}-V
  \left(
   f - \frac{xV-yU}{2a^2}
  \right)
 &=& -m \alpha \DP{p}{x}, \\
 \DD{V}{t}+U
  \left(
   f - \frac{xV-yU}{2a^2}
  \right)
 &=& -m \alpha \DP{p}{y}, \\
 0 \ = \  - \frac{1}{\rho} \DP{p}{z} -g.
\end{eqnarray}
%
�ʤ�, Ϣ³�μ��ˤϾ����ѹ����ä��, �ʲ��Τ褦�ˤʤ�
\footnote{
��Ͽ�ο��ظ���(A-18)����.
}.
%
\begin{eqnarray}
 \DP{\rho}{t}+m^2
  \left[
   \DP{(\rho U/m)}{x} + \DP{(\rho V/m)}{y}
  \right]
 + \DP{(\rho W)}{z} = 0.
\end{eqnarray}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% 2-8-2 Mercator Projection %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{��륫�ȥ��ˡ}

��륫�ȥ��ˡ�ϰʲ��μ�����������.
%
\begin{eqnarray*}
 x = (a \cos \varphi_0) \lambda,  \qquad
 y = a \cos \varphi_0
     \ln \left[(1+ \sin \varphi)/ \cos \varphi \right].
\end{eqnarray*}
%
������, $\varphi_0$����ƿޤ�''���Τ�''���٤Ǥ���
\footnote{
�����Ǥ�����{\bfseries ���Τ�}���١פȤ�,
���̤��Ф���ƿޤΥ������뤬���������٤Τ��Ȥ򼨤�.

��ª�����, ���ΰ��٤��⾮�������٤Ǥ���ƿޤϽ̾�����,
�礭�����٤Ǥϳ��礵���.
}.
�ޥåץե�������$m(\varphi)$��
$m(\varphi) = \cos \varphi_0 / \cos \varphi = 1 / h_x = 1/h_y$�Ȥʤ�.
������, ��ʿ�̤Ǥα�ư�������ϰʲ��Τ褦�ˤʤ�
\footnote{
���μ���Ƴ�Ф˴ؤ��Ƥ�{\bf 2-8-D}����.
}.
%
\begin{eqnarray}
 \DP{U}{t}
  + m
   \left(
    U \DP{U}{x} + V \DP{U}{y}
   \right)
  + w \DP{U}{z}
  -
    \left(
     f + \frac{U \tan \varphi}{a}
    \right) V
 &=&
  - \frac{m}{\rho} \DP{p}{x},
 \\
 \DP{V}{t}
  + m
   \left(
    U \DP{V}{x} + V \DP{V}{y}
   \right)
  + w \DP{V}{z}
  +
    \left(
     f + \frac{U \tan \varphi}{a}
    \right) U
 &=&
  - \frac{m}{\rho} \DP{p}{x}.
\end{eqnarray}
%
$U=h_x \ dx/dt, \ V=h_y \ dy/dt$�Ǥ���.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% 2-8-3 Lambert Conformal Projection %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{���٥��������ƿ�ˡ}

���٥��������ƿޤϰʲ��μ�����������.
%
\begin{eqnarray}
 r = r_0
     \left[
      \tan
      \left(
       \frac{\pi}{4} - \frac{\varphi}{2}
      \right)
     \right]^K,
 \qquad
 \theta = K (\lambda - \lambda_0).
\end{eqnarray}
%
2�Ĥ����$r_0$��$K$��
����$\varphi_1$�����$\varphi_2$�ˤ�
��ƿޤ�''����''�ˤʤ�褦�����ꤵ���.
���������Ԥ���, (2-38)�ϰʲ��Τ褦�˽񤯤��Ȥ��Ǥ���.
%
\begin{eqnarray}
 r=(a/K) \ m(\varphi) \cos \varphi,
  \qquad
 \theta = K (\lambda - \lambda_0).
\end{eqnarray}
%
�����Ǥ�
%
\begin{eqnarray}
 m(\varphi) &=&
  \left(
   \frac{\cos \varphi}{\cos \varphi_1}
  \right)
  (K-1)
  \left(
   \frac{1+\sin \varphi_1}{1+\sin \varphi}
  \right)
  K
  \nonumber, \\
 & &
 \hspace{10mm}
  K = \ln
      \left(
       \frac{\cos \varphi_1}{\cos \varphi_2}
      \right)
      \bigg/
      \ln
      \left\{
       \frac{
        \tan
         \left[
	  (\pi / 4) - (\varphi_1 / 2)
	 \right]
       }
       {
        \tan
	 \left[
	  (\pi / 4) - (\varphi_2 / 2)
	 \right]
       }
      \right\}.
\end{eqnarray}
%
��ư��������(2-32)��(2-34)�η��Ȱ��פ���.





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% 2-8-4 Additional Remarks %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{�ɵ�}

ͽ��Τ�����������Ϥ�������Ū�������
���ݳؤθ���Τ������10ǯ������Ԥ��Ƥ���.
����������Ū�ʵ���ͽ��Ϥ��Τ褦�ʸ¤�줿���̤ʴĶ�����
�����ϰϤ��鹭�ϰϤˤ錄�äƹԤ��Ƥ���.
������, �Ƕ�ϥ���ԥ塼���Υѥ�����Ϥˤʤ�,
���̤δĶ����Ǥ�
100km ���� 200km��å���Ǥ�����Ū�ʵ���ͽ���͡��ʷ���
�Ԥ���褦�ˤʤä�.
�����ǥ�ΰ�Ĥ�������
��ʿ�̤Ǥζ������򵤤ˤ��ʤ����ɤ����Ǥ���.
������Ф�, �¤�줿�ΰ���ǤϿ�ʿ�̤ζ��������θ����
ɬ�פ�����.
���줫��������ǥ��
�������������᥽��������θ��ݤ��������뤿���
�����ΰ�ǤΥ�ǥ���ɤ��Ȥ�����ۼ����Ƥ���������.




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

%%%%%%%%           ���ֹ�Υ������饤��         %%%%%%%% 
\setcounter{equation}{0}
\def\theequation{A-\arabic{equation}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%\begin{tabular}{p{\linewidth}}
% \hline \\
%\end{tabular}

\subsection*{2-8-A \qquad ��ƿޤν̼�$m(\varphi)$�ˤĤ���}
\setcounter{equation}{0}
\setcounter{figure}{0}
\def\theequation{2-8-A-\arabic{equation}}
\def\thefigure{2-8-A-\arabic{figure}}



$m(\varphi) = 2/(1+ \sin \varphi)$����ƿ�(���ξ��϶˼ͱƿ�)
�Ǥμºݤε��̤��Ф������Ψ��ɽ��.
$\varphi$���ѿ��˻����ͤʤ���, ���٤˰�¸���뤳�Ȥ��狼��.
$\varphi$�򲣼��ˤȤ�, $m(\varphi)$��ļ��ˤȤä�
$0 \leq \varphi \leq \pi /2$���ϰϤǴ�ñ�ʥ���դ�񤯤�
�ʲ���
\Dfigref{hwfig-01-08-�ޥåץե�������}�Τ褦�ˤʤ�.
�Ĥޤ�, �˼ͱƿޤ��濴($\varphi = \pi/2$)�Ǥ�
�̼ܤ�1�ʤΤǶ˼ͱƿޤǤε�Υ�ȵ��̾�Ǥε�Υ�ȤϤ��������פ���.
�դ˶˼ͱƿޤ�ü($\varphi = 0$)�ǤϽ̼ܤ�2�Ȥʤ�,
�˼ͱƿ޾�Ǥε�Υ�ϵ��̾�Ǥμºݤε�Υ��2�ܤȤʤäƤ���.
�����, �˼ͱƿ޾�Ǥ��ϵ�α߼���$2\pi \times 2a = 4\pi a$
�ȤʤäƤ���, �ºݤα߼�$2\pi a$��2�ܤȤʤäƤ��뤳�Ȥ����
����Ǥ���.
%
\begin{figure}[htb]
 \begin{center}
  \Depsf[][]{./eps/image_scale.eps}
 \end{center}
 \begin{quote}
  \caption{
  {\footnotesize �ޥåץե�������$m(\varphi)$}
  }
  \Dfiglab{hwfig-01-08-�ޥåץե�������}
 \end{quote}
\end{figure}
%



\subsection*{2-8-B \qquad ���̤Ǥε�Υ�ȶ˼ͱƿ޾�Ǥε�Υ�δط���}
\setcounter{equation}{0}
\setcounter{figure}{0}
\def\theequation{2-8-B-\arabic{equation}}
\def\thefigure{2-8-B-\arabic{figure}}

��(2-23)����ʬ���Ƽ�(2-24)��Ƴ�Ф���.
%
\begin{eqnarray}
 \Deqlab{x������ʬ}
 dx
  &=& \DP{x}{\lambda} d \lambda + \DP{x}{\varphi} d \varphi,
       \\
 \Deqlab{y������ʬ}
 dy
  &=& \DP{y}{\lambda} d \lambda + \DP{y}{\varphi} d \varphi.
\end{eqnarray}
%
������, ���$\varphi$��ʬ����ʬ��ԤäƤ���.
%
\begin{eqnarray}
 \DP{}{\varphi} \left(\frac{\cos \varphi}{1 + \sin \varphi}\right)
  &=&
  \DP{}{\varphi}
   \left\{
    \cos \varphi
    \left(
     1 + \sin \varphi
    \right)^{-1}
   \right\}
  \nonumber \\
  &=&
  - \sin \varphi
    \left(
     1 + \sin \varphi
    \right)^{-1}
  - \cos ^2 \varphi
   \left(
    1 + \sin 
   \right) ^{-2}
  \nonumber \\
  &=&
  - \frac{\sin \varphi + \sin ^2 \varphi + \cos ^2 \varphi}
         {\left(1+ \sin \varphi \right)^2}
  \nonumber \\
  &=&
  - \frac{1 + \sin \varphi}
         {\left(1+ \sin \varphi \right)^2}
  \nonumber \\
  \Deqlab{�̼ܤ���ʬ}
  &=&
  - \frac{1}{1 + \sin \varphi}.
\end{eqnarray}
%
\Deqref{x������ʬ}���.
$\varphi$����ʬ�ˤ�\Deqref{�̼ܤ���ʬ}���Ѥ���.
%
\begin{eqnarray}
 dx
  &=&
  \DP{}{\lambda}
   \left(
    \frac{2 a \cos \varphi \cos \lambda}{1 + \sin \varphi}
   \right)
  d \lambda
  + \DP{}{\varphi}
   \left(
    \frac{2 a \cos \varphi \cos \lambda}{1 + \sin \varphi}
   \right)
  d \varphi
  \nonumber \\ 
  &=&
   - \frac{2 a \cos \varphi \sin \lambda}{1 + \sin \varphi}
  d \lambda
   - \frac{2 a \cos \lambda}{1 + \sin \varphi}
  d \varphi
  \nonumber \\ 
 \Deqlab{x������ʬ�β�}
  &=&
   \frac{2}{1 + \sin \varphi}
    \left(
     - a \cos \varphi \sin \lambda
     d \lambda
     - a \cos \lambda
     d \varphi
    \right).
\end{eqnarray}
%
\Deqref{y������ʬ}��Ʊ�ͤ˷׻�����.
%
\begin{eqnarray}
 dy
  &=&
  \DP{}{\lambda}
   \left(
    \frac{2 a \cos \varphi \sin \lambda}{1 + \sin \varphi}
   \right)
  d \lambda
  + \DP{}{\varphi}
   \left(
    \frac{2 a \cos \varphi \sin \lambda}{1 + \sin \varphi}
   \right)
  d \varphi
  \nonumber \\ 
 \Deqlab{y������ʬ�β�}
  &=&
   \frac{2}{1 + \sin \varphi}
    \left(
     a \cos \varphi \cos \lambda
     d \lambda
     - a \sin \lambda
     d \varphi
    \right).
\end{eqnarray}
%
������, ���̾�Ǥε�Υ($dx_s, \ dy_s$)��
�ʲ��Τ褦��ɽ����.
(��(2-13)����).
%
\begin{eqnarray*}
 dx_s &=& a \cos \varphi \ d \lambda, \\
 dy_s &=& a \ d \varphi.
\end{eqnarray*}
%
�����δط�����\Deqref{x������ʬ�β�}, \Deqref{y������ʬ�β�}
���������ƹ���������ѷ������, ��(2-24)��
Ƴ�ФǤ���.



\subsection*{2-8-C \qquad �˼ͱƿ޺�ɸ�ϤǤα�ư��������Ƴ��}
\setcounter{equation}{0}
\setcounter{figure}{0}
\def\theequation{2-8-C-\arabic{equation}}
\def\thefigure{2-8-C-\arabic{figure}}

�˼ͱƿ޺�ɸ�ϤǤα�ư������(2-28)��(2-30)��,
���̺�ɸ�Ǥα�ư������(2-14)
\footnote{
%
\begin{eqnarray*}
 \DD{u_s}{t} &=& - \frac{1}{\rho}\DP{p}{x_s}
               + (2\Omega + \frac{u_s}{r \cos \varphi})(v_s \sin \varphi - w_s \cos \varphi)
	       + F_{\lambda} \nonumber, \\
 \DD{v_s}{t} &=& - \frac{1}{\rho}\DP{p}{y_s}
               - (2\Omega + \frac{u_s}{r \cos \varphi})u_s \sin \varphi - \frac{v_s w_s}{r}
	       + F_{\varphi}, \\
 \DD{w_s}{t} &=& - \frac{1}{\rho}\DP{p}{z_s} - g
               + (2\Omega + \frac{u_s}{r \cos \varphi})u_s \cos \varphi + \frac{v_s^2}{r}
	       + F_{z} \nonumber .
\end{eqnarray*}
%
}
��,
���̤Ǥ�®��($u_s, v_s$)�ȶ˼ͱƿ޾�Ǥ�®��($U, V$)�Ȥδط���
(2-26)
\footnote{
%
\begin{eqnarray*}
 \left(
  \begin{array}{c}
   U \\
   V \\
  \end{array}
 \right)
 =
 \left(
  \begin{array}{cc}
   - \sin \lambda & - \cos \lambda \\
     \cos \lambda & - \sin \lambda \\
  \end{array}
 \right)
 \left(
  \begin{array}{c}
   u_s \\
   v_s \\
  \end{array}
 \right).
\end{eqnarray*}
%
}
���������뤳�Ȥ�Ƴ�ФǤ���.
��(2-14)���������뤿��˼�(2-26)��ʲ��Τ褦���ѷ����Ƥ���.
%
\begin{eqnarray}
 \Deqlab{���̾�®�٤ȶ˼ͱƿ�®��}
 u_s = - U \sin \lambda + V \cos \lambda
 , \quad
 v_s = - U \cos \lambda - V \sin \lambda
 , \quad
 w_s = W.
\end{eqnarray}
%
�����ܤδط����ϼ�(2-23)���Ƴ�����.


{\bf 2-8-C-a \quad (2-28)��Ƴ��}

��(2-14)��$\Dvect{i}$��ʬ��$\Dvect{j}$��ʬ�μ���
\Deqref{���̾�®�٤ȶ˼ͱƿ�®��}����������
­����碌�뤳�Ȥ�, ��(2-28)��Ƴ�Ф���.
�ʤ�, ��ʸ�ˤ⤢��Ȥ���
$r=a$�Ȥ�, ����$\Dvect{F}$�Ͻ�������.

�ޤ�, $\Dvect{i}$�����������,
%
\begin{eqnarray}
 \lefteqn{\DD{}{t} \left( - U \sin \lambda + V \cos \lambda  \right)}
  \hspace{10mm}
 \nonumber \\
 &=&
  - \frac{1}{\rho}\DP{p}{x_s}
  + \left(
     2 \Omega + \frac{- U \sin \lambda + V \cos \lambda}{a \cos \varphi}
    \right)
    \left(
     - U \cos \lambda - V \sin \lambda
    \right) \sin \varphi
 \nonumber \\
 & & \hspace{10mm}
  - \left(
     2 \Omega + \frac{- U \sin \lambda + V \cos \lambda}{a \cos \varphi}
    \right)
    W \cos \varphi,
 \nonumber \\
 \nonumber \\
 \lefteqn{
  - \sin \lambda \DD{U}{t} - U \cos \lambda \DD{\lambda}{t}
  + \cos \lambda \DD{V}{t} - V \sin \lambda \DD{\lambda}{t}
 }
  \hspace{10mm}
 \nonumber \\
 &=&
  - \frac{1}{\rho} \DP{p}{x_s}
  + \left(
     2 \Omega + \frac{- U \sin \lambda + V \cos \lambda}{a \cos \varphi}
    \right)
    \left(
     - U \cos \lambda - V \sin \lambda
    \right) \sin \varphi
 \nonumber \\
 \Deqlab{i��ʬ�μ�}
 & & \hspace{10mm}
  - 2 \Omega W \cos \varphi
  - \left(
     \frac{- U \sin \lambda + V \cos \lambda}{a}
    \right)
    W.
\end{eqnarray}
%
����, $\Dvect{j}$�����������,
%
\begin{eqnarray}
 \lefteqn{\DD{}{t} \left( - U \cos \lambda - V \sin \lambda  \right)}
  \hspace{10mm}
 \nonumber \\
 &=&
  - \frac{1}{\rho}\DP{p}{y_s}
  - \left(
     2 \Omega + \frac{- U \sin \lambda + V \cos \lambda}{a \cos \varphi}
    \right)
    \left(
     - U \sin \lambda + V \cos \lambda
    \right) \sin \varphi
 \nonumber \\
 & & \hspace{10mm}
  - \left(
     \frac{- U \cos \lambda - V \sin \lambda}{a}
    \right)
    W, 
 \nonumber \\
 \nonumber \\
 \lefteqn{
  - \cos \lambda \DD{U}{t} + U \sin \lambda \DD{\lambda}{t}
  - \sin \lambda \DD{V}{t} - V \cos \lambda \DD{\lambda}{t}
 }
  \hspace{10mm}
 \nonumber \\
 &=&
  - \frac{1}{\rho} \DP{p}{y_s}
  + \left(
     2 \Omega + \frac{- U \sin \lambda + V \cos \lambda}{a \cos \varphi}
    \right)
    \left(
     U \sin \lambda - V \cos \lambda
    \right) \sin \varphi
 \nonumber \\
 \Deqlab{j��ʬ�μ�}
 & & \hspace{10mm}
  - \left(
     \frac{- U \cos \lambda - V \sin \lambda}{a}
    \right)
    W. 
\end{eqnarray}
%
\Deqref{i��ʬ�μ�} $\times \sin \lambda \ +$
\Deqref{j��ʬ�μ�} $\times \cos \lambda$
�Ȥ����,
%
\begin{eqnarray}
 - \DD{U}{t} - V \DD{\lambda}{t}
 &=& -
     \frac{1}{\rho}
     \left(
      \sin \lambda\DP{}{x_s}
    + \cos \lambda\DP{}{y_s}
     \right) p
 \nonumber \\
 & & {}
   + \left(
     2 \Omega + \frac{- U \sin \lambda + V \cos \lambda}{a \cos \varphi}
     \right)
     (-V) \sin \varphi 
 \nonumber \\
 \Deqlab{U�μ�1}
 & & {}
   - 2 \Omega W \sin \lambda \cos \varphi
   - \frac{- U W}{a}
\end{eqnarray}
%
�Ȥʤ�.
��(2-13)���
%
\begin{eqnarray*}
 \DD{\lambda}{t}
  = \frac{u_s}{a \cos \varphi}
  = \frac{- U \sin \lambda + V \cos \lambda}{a \cos \varphi}
\end{eqnarray*}
%
�Ȥʤ�Τ�, �����\Deqref{U�μ�1}�����������,
%
\begin{eqnarray}
 - \DD{U}{t}
 &=& -
     \frac{1}{\rho}
     \left(
      \sin \lambda\DP{}{x_s}
    + \cos \lambda\DP{}{y_s}
     \right) p
 \nonumber \\
 & & {}
   - 2 \Omega V \sin \varphi
   + V
     \left(
     \frac{- U \sin \lambda + V \cos \lambda}{a \cos \varphi}
     \right)
     \left( 1 - \sin \varphi \right)
 \nonumber \\
 \Deqlab{U�μ�2}
 & & {}
   + 2 \Omega W \sin \lambda \cos \varphi
   - \frac{- U W}{a}.
\end{eqnarray}
%
�����˻��ˤʤ�Τ�, \Deqref{U�μ�2}��ƹ��ʬ���Ʒ׻�����.
�ޤ�, \Deqref{U�μ�2}�α�����1��ˤĤ��ƹͤ���.

��(2-24)���, �ʲ��δط���������Ω��
($m = 1/(1 + \sin \varphi)$�Ǥ���).
%
\begin{eqnarray*}
 dx &=& -m \ \sin \lambda \ dx_s -m \cos \lambda \ dy_s , \\
 dy &=& \hspace{3.5mm} m \ \cos \lambda \ dx_s -m \sin \lambda \ dy_s , \\
 dz &=& dz_s.
\end{eqnarray*}
%
�ѷ������,
%
\begin{eqnarray*}
 dx_s &=& \frac{1}{m}
           \left(
	    - \sin \lambda \ dx + \cos \lambda \ dy
	   \right) , \\
 dy_s &=& \frac{1}{m}
           \left(
	    - \cos \lambda \ dx - \sin \lambda \ dy
	   \right) , \\
 dz_s &=& dz
\end{eqnarray*}
%
�����μ����, �ʲ��δط���������Ω��.
%
\begin{eqnarray}
 \DP{x_s}{x} = - \frac{\sin \lambda}{m}
 , \qquad
 \DP{x_s}{y} =   \frac{\cos \lambda}{m}
 , \qquad
 \DP{x_s}{z} =   0 \nonumber , \\
 \Deqlab{��ž��ɸ������ʬɽ��}
 \DP{y_s}{x} = - \frac{\cos \lambda}{m}
 , \qquad
 \DP{y_s}{y} = - \frac{\sin \lambda}{m}
 , \qquad
 \DP{y_s}{z} =   0 , \\
 \DP{z_s}{x} =   0
 , \qquad  \hspace{10mm}
 \DP{z_s}{y} =   0
 , \qquad  \hspace{10mm}
 \DP{z_s}{z} =   0. \nonumber
\end{eqnarray}
%
��ä�,
%
\begin{eqnarray}
 \DP{p}{x}
  &=&
  \DP{p}{x_s}\DP{x_s}{x} + \DP{p}{y_s}\DP{y_s}{x} + \DP{p}{z_s}\DP{z_s}{x}
  \nonumber , \\
  &=&
   - \frac{\sin \lambda}{m}\DP{p}{x_s}
   - \frac{\cos \lambda}{m}\DP{p}{y_s}
  \nonumber , \\
 \Deqlab{��1���ܤη�}
 m \DP{p}{x}
  &=&
   - \sin \lambda \DP{p}{x_s}
   - \cos \lambda \DP{p}{y_s}.
\end{eqnarray}

����, \Deqref{U�μ�2}�α�����2, 3��ˤĤ��ƹͤ���.
(2-23)���ѷ������ʲ��μ������Ѥ���.
%
\begin{eqnarray*}
 \sin \lambda = y \frac{1 + \sin \varphi}{2 a \cos \varphi}, \quad
 \cos \lambda = x \frac{1 + \sin \varphi}{2 a \cos \varphi}.
\end{eqnarray*}
%
�����μ����դ���2, 3������������,
%
\begin{eqnarray}
 \lefteqn{
   - 2 \Omega V \sin \varphi
   + V
     \left(
     \frac{- U \sin \lambda + V \cos \lambda}{a \cos \varphi}
     \right)
     \left( 1 - \sin \varphi \right)
 } \hspace{20mm} \nonumber \\
 &=&
   - 2 \Omega V \sin \varphi
  + V
   \left(
    \frac{x V - y U}{2 a^2 \cos ^2 \varphi}
    \left( 1 + \sin \varphi \right) 
   \right)
   \left( 1 - \sin \varphi \right)
 \nonumber \\
 \Deqlab{��2,3���ܤη�}
 &=&
   - V
     \left(
      f
    + \frac{xV - yU}{2a^2}
     \right).
\end{eqnarray}
%
�����ˤ⤢�ä��Ȥ���, $f=2\Omega \sin \varphi$�Ǥ���.

�Ǹ��, \Deqref{U�μ�2}�α�����4, 5��ˤĤ��ƹͤ���.
%
\begin{eqnarray}
   - 2 \Omega W \sin \lambda \cos \varphi
   - \frac{- U W}{a}
 &=&
   - \frac{W}{a} (2 a \Omega \sin \lambda \cos \varphi - U)
 \nonumber \\  
 \Deqlab{��4,5���ܤη�}
 &=&
   - \frac{W}{a} \left[(1 + \sin \varphi)\Omega y - U) \right].
\end{eqnarray}
%
\Deqref{��1���ܤη�}, \Deqref{��2,3���ܤη�}, \Deqref{��4,5���ܤη�}
��\Deqref{U�μ�2}���������뤳�Ȥǰʲ��μ�(2-28)��Ƴ�ФǤ���.
%
\begin{eqnarray*}
 \DD{U}{t}-V
  \left(
   f - \frac{xV-yU}{2 a^2}
  \right)
  - \frac{W}{a}
  \left[
   (1+ \sin \varphi) \Omega y -U
  \right]
  = - m \alpha \DP{p}{x}  .
\end{eqnarray*}


{\bf 2-8-C-b \quad (2-29)��Ƴ��}

��(2-29)��Ƴ�Фϼ�(2-28)��Ƴ�Ф���ˡ�ȤۤȤ��Ʊ���褦�ˤ����ʤ�.
�����
\Deqref{i��ʬ�μ�} $\times - \cos \lambda \ +$
\Deqref{j��ʬ�μ�} $\times \sin \lambda$
�Ȥ���.
�����,
%
\begin{eqnarray}
 - \DD{V}{t} + U \DD{\lambda}{t}
 &=& 
     \frac{1}{\rho}
     \left(
      \cos \lambda\DP{}{x_s}
    - \sin \lambda\DP{}{y_s}
     \right) p
 \nonumber \\
 & & {}
   + \left(
     2 \Omega + \frac{- U \sin \lambda + V \cos \lambda}{a \cos \varphi}
     \right)
     U \sin \varphi 
 \nonumber \\
 \Deqlab{V�μ�1}
 & & {}
   + 2 \Omega W \cos \lambda \cos \varphi
   - \frac{- V W}{a}
\end{eqnarray}
%
�Ȥʤ�.
���μ���(2-28)��Ƴ�Ф�Ʊ�ͤ��ѷ���Ԥ���,
�ʲ��μ�(2-29)���ѷ��Ǥ���.
%
\begin{eqnarray*}
 \DD{V}{t}+U
  \left(
   f - \frac{xV-yU}{2 a^2}
  \right)
  + \frac{W}{a}
  \left[
   (1+ \sin \varphi) \Omega x +V
  \right]
  = - m \alpha \DP{p}{y} .
\end{eqnarray*}


{\bf 2-8-C-c \quad (2-30)��Ƴ��}

(2-30)��(2-14)��$\Dvect{k}$��ʬ��\Deqref{���̾�®�٤ȶ˼ͱƿ�®��}
����������Ƴ�Ф��뤳�Ȥ��Ǥ���.
%
\begin{eqnarray*}
 \DD{W}{t}
  &=& - \frac{1}{\rho}\DP{p}{z} -g
      + 2 \Omega \cos \varphi
        \left( - U \sin \lambda + V \cos \lambda \right) \\
  & & + \frac{\left( - U \sin \lambda + V \cos \lambda \right)^2}{a}
      + \frac{\left( - U \cos \lambda - V \sin \lambda \right)^2}{a}
  \\
  &=& - \frac{1}{\rho}\DP{p}{z} -g
      + \frac{\Omega}{a} \left( x V - y U \right)
        \left( 1 + \sin \varphi \right)
      + \frac{U^2 + V^2}{a},
\end{eqnarray*}
\begin{eqnarray*}
 \DD{W}{t} - \frac{1}{a}
  \left[
   U^2+V^2+ \Omega(1+ \sin \varphi)(xV-yU)
  \right]
 = - \alpha \DP{p}{z} - g .
\end{eqnarray*}




\subsection*{2-8-D \qquad ��륫�ȥ��ˡ��ɸ�ϤǤα�ư������}
\setcounter{equation}{0}
\setcounter{figure}{0}
\def\theequation{2-8-D-\arabic{equation}}
\def\thefigure{2-8-D-\arabic{figure}}

��륫�ȥ��ˡ�Ǥ�˼ͱƿ�ˡ�ξ���Ʊ����ˡ�Ǳ�ư�����������.
�ޤ�, ($x_s, y_s$)��($x, y$)�Ȥδط���
(�ޤ���($u_s, v_s$)��($U, V$)�Ȥδط���)
��(2-14)���������ƥ�륫�ȥ�޺�ɸ�Ϥα�ư�����������.
���θ�, ��ʿ�̶�������Ѥ��� $w$��ޤ���������.

�ޤ�, ��ʸ�ΰʲ��δط�������
($x_s, y_s$)��($x, y$), �����
($u_s, v_s$)��($U, V$)�Ȥδط��������.
%
\begin{eqnarray*}
 x = (a \cos \varphi_0) \lambda  \qquad
 y = a \cos \varphi_0
     \ln \left[(1+ \sin \varphi)/ \cos \varphi \right].
\end{eqnarray*}
%
�嵭�μ�����
($x_s, y_s$)��($x, y$)�δط�����ͤ���.
%
\begin{eqnarray}
 dx
  &=& \DP{x}{\lambda} d \lambda + \DP{x}{\varphi} d \varphi
  \nonumber \\
  &=& (a \cos \varphi_0)d \lambda
  \nonumber \\
  &=& \frac{\cos \varphi_0}{\cos \varphi} dx_s ,
  \\
  \nonumber \\
 dy 
  &=& \DP{y}{\lambda} d \lambda + \DP{y}{\varphi} d \varphi
  \nonumber \\
  &=& a \cos \varphi_0
      \DP{}{\varphi}
       \left\{
	\ln
	 \left(
	  \frac{1 + \sin \varphi}{\cos \varphi}
	 \right)
       \right\} d \varphi
  \nonumber \\
  &=& a \cos \varphi_0
      \frac{\cos \varphi}{1 + \sin \varphi}
      \frac{1}{1 - \sin \varphi} d \varphi
  \nonumber \\
  &=& a 
      \frac{\cos \varphi_0}{\cos \varphi}
      d \varphi
  \nonumber \\
  &=& \frac{\cos \varphi_0}{\cos \varphi}
      dy_s.
\end{eqnarray}
%
���η׻��������(2-13)�����Ѥ��Ƥ���.
����餫��
($u_s, v_s$)��($U, V$)�Ȥδط��������.
%
\begin{eqnarray}
 \Deqlab{����®�٤ȥ�륫�ȥ�®��u}
 u_s = \DD{x_s}{t}
     = \frac{\cos \varphi}{\cos \varphi_0} \DD{x}{t}
     = h_x \DD{x}{t}
     = U ,\\
 \Deqlab{����®�٤ȥ�륫�ȥ�®��v}
 v_s = \DD{y_s}{t}
     = \frac{\cos \varphi}{\cos \varphi_0} \DD{y}{t}
     = h_y \DD{y}{t}
     = V.
\end{eqnarray}
%
�Ǹ�˰��Ϲ�δط��������.
%
\begin{eqnarray}
 \Deqlab{���̰��Ϥȥ�륫�ȥ밵��u}
 \DP{p}{x_s}
  = \DP{p}{x}\DP{x}{x_s}
  = m \DP{p}{x}, \\
 \Deqlab{���̰��Ϥȥ�륫�ȥ밵��v}
 \DP{p}{y_s}
  = \DP{p}{y}\DP{y}{y_s}
  = m \DP{p}{y}.
\end{eqnarray}
%
\Deqref{����®�٤ȥ�륫�ȥ�®��u},
\Deqref{����®�٤ȥ�륫�ȥ�®��v},
\Deqref{���̰��Ϥȥ�륫�ȥ밵��u},
\Deqref{���̰��Ϥȥ�륫�ȥ밵��v}
����� (2-31)
��(2-14)����������.
�ʤ�, ��ʿ�̶���ˤ�ä�$w$��ޤ��Ͻ������Ƥ���.
%
\begin{eqnarray*}
 \DP{U}{t}
  + m
   \left(
    U \DP{U}{x} + V \DP{U}{y}
   \right)
  + w \DP{U}{z}
  -
    \left(
     f + \frac{U \tan \varphi}{a}
    \right) V
 &=&
  - \frac{m}{\rho} \DP{p}{x},
 \\
 \DP{V}{t}
  + m
   \left(
    U \DP{V}{x} + V \DP{V}{y}
   \right)
  + w \DP{V}{z}
  +
    \left(
     f + \frac{U \tan \varphi}{a}
    \right) U
 &=&
  - \frac{m}{\rho} \DP{p}{x}.
\end{eqnarray*}
%









%\subsection*{Appendix \qquad}

%\subsubsection*{}

%%%%%%%%          ���ֹ�Υ������Х饤��        %%%%%%%% 
\setcounter{equation}{40}
\def\theequation{\arabic{chapter}-\arabic{equation}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%              text end                  %%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
