% ɽ��   AGCM5 ��1�� ������ǥ� ����������
%
% ���� 
\Drireki{93/06/16  �¸��ء��ݺ�����}
%

\section{����������}

�����Ǥϻ������������˼���. 

\subsection{Ϣ³�μ�}

\begin{equation}
  \label{����}
  \frac{\partial \pi}{\partial t} 
    + \Dvect{v}_{H} \cdot \nabla_{\sigma} \pi
     =  - \nabla_{\sigma} \cdot \Dvect{v}_{H} 
          - \frac{\partial \dot{\sigma}}{\partial \sigma}
\end{equation}

\subsection{�ſ尵�μ�}

\begin{equation}
\label{�ſ尵}
  \frac{\partial \Phi}{\partial \sigma} = - \frac{RT_v}{\sigma} 
\end{equation}

\subsection{��ư������}

\begin{equation}
  \label{����}
  \frac{\partial \zeta}{\partial t} 
     =   \frac{1}{a(1-\mu^{2})}
            \frac{\partial \mbox{\sl VA}}{\partial \lambda}
          - \frac{1}{a} \frac{\partial \mbox{\sl UA}}{\partial \mu}
          + {\cal D}(\zeta) 
\end{equation}
\begin{equation}
  \label{ȯ��}
  \frac{\partial D}{\partial t} 
     =    \frac{1}{a(1-\mu^{2})}
            \frac{\partial \mbox{\sl UA}}{\partial \lambda}
          + \frac{1}{a} \frac{\partial \mbox{\sl VA}}{\partial \mu}
          - \nabla^{2}_{\sigma}
           ( \Phi + R \bar{T} \pi + \mbox{\sl KE} ) 
          + {\cal D}(D) 
\end{equation}

\subsection{Ǯ�ϳؤμ�}

\begin{eqnarray}
\label{Ǯ��}
  \frac{\partial T}{\partial t}
    & = & - \frac{1}{a(1-\mu^{2})} 
               \frac{\partial UT^{\prime}}{\partial \lambda}
          - \frac{1}{a}
               \frac{\partial VT^{\prime}}{\partial \mu}
          + T^{\prime} D \nonumber \\
    &   & - \dot{\sigma} 
              \frac{\partial T }{\partial \sigma}
          + \kappa T \left( \frac{\partial \pi}{\partial t}
                            + \Dvect{v}_{H} \cdot \nabla_{\sigma} \pi 
                            + \frac{ \dot{\sigma} }{ \sigma } 
                     \right)
          + \frac{Q}{C_{p}}
          + {\cal D}(T) 
          + {\cal D}^{\prime}(\Dvect{v}) 
\end{eqnarray}

\subsection{������μ�}

\begin{eqnarray}
\label{�����}
  \frac{\partial q}{\partial t}
    & = & - \frac{1}{a(1-\mu^{2})} 
               \frac{\partial Uq}{\partial \lambda}
          - \frac{1}{a}
               \frac{\partial Vq}{\partial \mu}
          + q D \nonumber \\
    &   & - \dot{\sigma} \frac{\partial q }{\partial \sigma}
          + S_{q}
          + {\cal D}(q) 
\end{eqnarray}

������,
%
\begin{eqnarray}
\theta & \equiv & T  \left( p/p_{0} \right)^{- \kappa} \\
\kappa & \equiv & R/C_{p} \\
  \Phi & \equiv & gz \\
   \pi & \equiv & \ln p_{S} \\
%
 \dot{\sigma} & \equiv &  \frac{d \sigma}{d t} \\
   \mu & \equiv & \sin \varphi \\
%
     T_v & \equiv & T ( 1+\epsilon_v q ) \\
     U   & \equiv & u \cos \varphi \\
     V   & \equiv & v \cos \varphi \\
%
\label{�������}
 \zeta & \equiv & \frac{1}{a ( 1-\mu^{2} ) }
                    \frac{\partial V}{\partial \lambda} 
             -    \frac{1}{a} \frac{\partial U}{\partial \mu} \\
%
\label{ȯ�����}
     D & \equiv & \frac{1}{a ( 1-\mu^{2} ) }
                    \frac{\partial U}{\partial \lambda} 
             +    \frac{1}{a} \frac{\partial V}{\partial \mu} \\
%
\label{B��}
 \mbox{\sl UA} & \equiv &  ( \zeta + f ) V 
             - \dot{\sigma} \frac{\partial U}{\partial \sigma} 
             - \frac{RT^{\prime}}{a} 
                  \frac{\partial \pi}{\partial \lambda} 
             + {\cal F}_{\lambda} \cos \varphi \\
%
\label{A��}
 \mbox{\sl VA} & \equiv & - ( \zeta + f ) U 
             - \dot{\sigma} \frac{\partial V}{\partial \sigma} 
             - \frac{RT^{\prime}}{a} ( 1 - \mu^{2} ) 
                  \frac{\partial \pi}{\partial \mu} 
             + {\cal F}_{\varphi} \cos \varphi \\
%
\label{E��}
\mbox{\sl KE} & \equiv &  \frac{U^{2}+V^{2}}{2(1-\mu^{2}) } \\
     T & \equiv &  \bar{T}(\sigma) + T^{\prime} \\
     \Dvect{v}_{H} \cdot \nabla
       & \equiv & \frac{u}{a \cos \varphi} 
         \left( \frac{\partial }{\partial \lambda} \right)_{\sigma}
     + \frac{v}{a}
         \left( \frac{\partial }{\partial \varphi} \right)_{\sigma} 
           \nonumber \\
       & = & \frac{U}{a ( 1 -\mu^{2} )} 
         \left( \frac{\partial }{\partial \lambda} \right)_{\sigma}
     + \frac{V}{a}
         \left( \frac{\partial }{\partial \mu} \right)_{\sigma} \\
     \nabla^{2}_{\sigma}  
       & \equiv & 
               \frac{1}{a^{2}(1-\mu^{2})} 
                 \frac{\partial^{2} }{\partial \lambda^{2}} 
             + \frac{1}{a^{2}} 
                 \frac{\partial }{\partial \mu}
                  \left[ (1-\mu^{2})  
                       \frac{\partial }{\partial \mu} \right]  .
\end{eqnarray}

������, 
${\cal D}(\zeta), {\cal D}(D), {\cal D}(T), {\cal D}(q)$
�Ͽ�ʿ�Ȼ���Ǥ���, \ref{��ʿ�Ȼ���} �����������. 
${\cal F}_\lambda, {\cal F}_\varphi$
�Ͼ����ϱ�ư�����ˤ���ϤǤ���. 
$Q$ ������, �ŷ�, �����ϱ�ư�������ˤ���Ǯ�������Ѳ�, 
$S_q$�϶ŷ�, �����ϱ�ư�������ˤ��������������, 
${\cal D}' (\Dvect{v})$ ���໤Ǯ�Ǥ���. 


%   �໤Ǯ������. ʪ�������˰ܤ����.
%\begin{equation}
%  {\cal D}^{\prime} (\Dvect{v}) 
% = - \frac{1}{C_{p}}
%     \Dvect{v} \cdot  ( \frac{\partial \Dvect{v}}{\partial t} )_{diff} .
%\end{equation}
%
%$( \frac{\partial \Dvect{v}}{\partial t} )_{diff} $ ��,
%��ʿ����ӱ�ľ�γȻ��ˤ�� $u,v$ �λ����Ѳ���Ǥ���.


\subsection{�������}

��ľή�˴ؤ��붭������
%
\begin{equation}
  \dot{\sigma} = 0  \ \ \ at \ \ \sigma = 0 , \ 1 .
\end{equation}
%
�Ǥ���. ��ä�(\ref{����}) ����,
��ɽ�����λ����Ѳ�����
$\sigma$�ϤǤα�ľ®��$\dot{\sigma}$�������Ǽ�
%
\begin{equation}
   \label{��������}
   \frac{\partial \pi}{\partial t}
   = - \int_{0}^{1} \Dvect{v}_{H} \cdot \nabla_{\sigma} \pi d \sigma
     - \int_{0}^{1} D  d \sigma ,
\end{equation}
%
\begin{equation}
   \label{��ľ®��}
   \dot{\sigma} 
   = - \sigma 
     \frac{\partial \pi}{\partial t}
     - \int_{0}^{\sigma} D d \sigma
     - \int_{0}^{\sigma} 
         \Dvect{v}_{H} \cdot \nabla_{\sigma} \pi d \sigma ,
\end{equation}
%
��Ƴ�����.
\vspace{1em}

������ǮŪ�������ˤĤ��Ƥ� \ref{tihyoumen} �Ϥˤ����Ƶ��Ҥ���. 

