% ɽ��   dcpam4  ��ɸ�ϡ��Ѵ������˴ؤ������ --  GCM �Ǹ�������Ĵ��ȡ���ζ�����ʬ
%
% ����
%\Drireki{1991/12/04 �ݺ�����}
%\Drireki{2008/06/15 ��������}


\section{��ʬ����, GCM���ѿ�����ʬ�ط���}

�����Ǥ�, 
�����顼��, �٥��ȥ����ʬ��׻�����. 
����ˤ����򸵤�, 
ȯ��$D$, ���� $\zeta$, ®�٥ݥƥ󥷥��$\chi$, ή���ؿ�$\psi$ ��
$(u,v)$ �Ȥδط����դ���. 
%$(U,V)$ �Ȥδط����դ���. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{�����顼�̤���ʬ}
   
   �����顼�� $f(\lambda,\varphi)$ �� $x$ ��ʬ��
    ${\displaystyle \frac{1}{r \cos \varphi} \DP{f}{\lambda} }$
   ��Ϳ������. 
   
   $f$ �� $y$ ��ʬ��
    ${\displaystyle \frac{1}{r} \DP{f}{\varphi}
             \left( = \frac{\cos \varphi}{r} \DP{f}{\mu} \right) }$
   ��Ϳ������. 
   
   $f$ ��2������ץ饷�����
   \begin{align}
    \begin{split}
            \nabla^2_H f  
        &\equiv 
           \frac{1}{r^2}
           \left[ \frac{1}{\cos \varphi} 
                  \DP{}{\varphi} \left( \cos \varphi \DP{}{\varphi} \right)
                  + \frac{1}{\cos^2 \varphi} \DP[2]{}{\lambda} 
           \right] f  \\
        &= 
         \frac{1}{r^2}
          \left[
            \DP{}{\mu} 
               \left\{ (1-\mu^2) \DP{}{\mu} \right\}
          + \frac{1}{1-\mu^2} \DP[2]{}{\lambda}
          \right] f 
    \end{split}   
   \end{align}
   ��Ϳ������. 
   
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{�٥��ȥ��̤���ʬ}
   
   2�����٥��ȥ�� $\Dvect{v}=(v_1,v_2)$ �ο�ʿȯ���� 
   %
   \begin{align}
    \begin{split}
      \mbox{div}_H \Dvect{v}
         &\equiv     \frac{1}{r \cos \varphi} \DP{v_1}{\lambda} 
         +  \frac{1}{r \cos \varphi} \DP{}{\varphi} (v_2 \cos \varphi)     \\
     &=     \frac{1}{r \sqrt{1-\mu^2} } \DP{v_1}{\lambda} 
         +  \frac{1}{r} \DP{}{\mu}(  \sqrt{1-\mu^2} v_2 )    
    \end{split}   
   \end{align}
   %
   ��Ϳ������. 

   $\Dvect{v}$ �β�ž�� $r$ ��ʬ��,
   %
   \begin{align}
    \begin{split}
      (\mbox{rot} \Dvect{v})_r
      &\equiv    \frac{1}{r \cos \varphi} \DP{v_2}{\lambda} 
         -  \frac{1}{r \cos \varphi} \DP{}{\varphi}(v_1 \cos \varphi)    \\
     &=     \frac{1}{r \sqrt{1-\mu^2} } \DP{v_2}{\lambda} 
         -  \frac{1}{r} \DP{}{\mu}(\sqrt{1-\mu^2}v_1)      
    \end{split}   
   \end{align}
   %
   ��Ϳ������. 
   
�ʾ������줿��ʬ�����򸵤�, 
�ʲ��˼ºݤ�GCM�ǻ��Ѥ�����������ʬ�θ������¤٤Ƥ���. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{ȯ��}

   ��ʿʬ�ۤ���®�پ�ο�ʿȯ�� $D$ �� $u,\ v$ ���Ѥ���ɽ��
%
%   ��ʿʬ�ۤ���®�پ�
%   \begin{align}
%     (u,v) \equiv 
%        \left( \frac{U}{\cos \varphi}, \frac{V}{\cos \varphi} \right) 
%   \end{align}
%   �ο�ʿȯ�� $D$ ��, $U,V$ ���Ѥ���ɽ��.  
%
   \begin{align}
    \begin{split}
     D   &=     \frac{1}{r \cos \varphi} \DP{u}{\lambda} 
         +  \frac{1}{r \cos \varphi} \DP{}{\varphi} (v \cos \varphi)
%     \\
%     &=     \frac{1}{r \cos^2 \varphi} \DP{U}{\lambda} 
%         +  \frac{1}{r \cos \varphi} \DP{V}{\varphi}                 \\
%     &=     \frac{1}{r (1-\mu^2)} \DP{U}{\lambda} 
%         +  \frac{1}{r} \DP{V}{\mu}
     .
    \end{split}   
   \end{align}
   
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{����}
   
   ��ʿʬ�ۤ���®�پ�α��� $\zeta$ �� $u,\ v$ ���Ѥ���ɽ��
%
%   ��ʿʬ�ۤ���®�پ�
%   \begin{align}
%     (u,v) = \left( \frac{U}{\cos \varphi}, \frac{V}{\cos \varphi} \right) 
%   \end{align}
%   �α��� $\zeta$ ��, $U,V$ ���Ѥ���ɽ��. 
%
   \begin{align}
    \begin{split}
      \zeta 
     &=     \frac{1}{r \cos \varphi} \DP{v}{\lambda} 
         -  \frac{1}{r \cos \varphi} \DP{}{\varphi}(u \cos \varphi)
%     \\
%     &=     \frac{1}{r \cos^2 \varphi} \DP{V}{\lambda} 
%         -  \frac{1}{r \cos \varphi} \DP{U}{\varphi}                \\
%     &=     \frac{1}{r (1-\mu^2)} \DP{V}{\lambda} 
%         -  \frac{1}{r} \DP{U}{\mu}
     .
    \end{split}   
   \end{align}
   
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{®�٥ݥƥ󥷥��, ή���ؿ��� $(U,V)$}
\subsection{®�٥ݥƥ󥷥��, ή���ؿ��� $(u,v)$}
   
   ®�٥ݥƥ󥷥�� $\chi$, ή���ؿ� $\psi$ ��
   %
   \begin{align}
     D     &\equiv  \nabla_H^2 \chi , \\
     \zeta &\equiv  \nabla_H^2 \psi
   \end{align}
   %
   ����������. 
   $(u,v)$ �� $\chi,\psi$ ��ɽ��. 
%   $(U,V)$ �� $\chi,\psi$ ��ɽ��. 
   %
   \begin{align}
     u  &=  - \frac{1}{r}             \DP{\psi}{\varphi}
            + \frac{1}{r\cos\varphi}  \DP{\chi}{\lambda} , \\
     v  &=    \frac{1}{r\cos\varphi}  \DP{\psi}{\lambda}
            + \frac{1}{r}             \DP{\chi}{\varphi}
   \end{align}
   %
%   \begin{align}
%     U  &=  - \frac{1-\mu^2}{r}   \DP{\psi}{\mu}
%            + \frac{1}{r}             \DP{\chi}{\lambda} \\
%     V  &=    \frac{1}{r}             \DP{\psi}{\lambda}
%            + \frac{1-\mu^2}{r}   \DP{\chi}{\mu}
%   \end{align}
   %
   �Ȥʤ�. 
   

