% ɽ��   dcpam5-primitive  �ϳز���  ��ľΥ����
%
% ����
%\Drireki{2010/09/01 �ݹ�����}
%\Drireki{2009/03/03 ��������}
%\Drireki{2008/06/14 ��������}
%\Drireki{1993/03/18 �¸��ء��ݺ�����}
%

\section{Υ��ɽ��: ��ľΥ����}

�����Ǥϻ������������ľ������Υ��������. 
Arakawa and Suarez(1983) �˽��ä�,
\Deqref{Ϣ³�μ�}��\Deqref{Ǯ�ϳؤμ�}
���ľ�����˺�ʬ�ˤ�ä�Υ��������. 
����������Υ����ɽ���ϼ��Τ褦�ˤʤ�.


\subsection{Ϣ³�μ�, ��ľ®��}
\vspace*{-5ex}
\begin{align}
  \DP{\pi}{t}
 &= - \sum_{k=1}^{K} ( D_k + \Dvect{v}_k \cdot \nabla \pi ) 
       \Delta  \sigma_k, \\
  \dot{\sigma}_{k-1/2}
 &= - \sigma_{k-1/2} \DP{\pi}{t}
   - \sum_{l=k}^{K} ( D_l + \Dvect{v}_l \cdot \nabla \pi )          
       \Delta  \sigma_l \qquad (k = 2, \cdots, K),  \\
 \dot{\sigma}_{1/2} &= \dot{\sigma}_{K+1/2} = 0. 
\end{align}
%
������,
%
\begin{align}
 \Dvect{v}_k \cdot \nabla \pi =
  \frac{U_k}{a (1-\mu^2)} \DP{\pi}{\lambda}
  + \frac{V_k}{a (1-\mu^2)} (1-\mu^2) \DP{\pi}{\mu}. 
\end{align}
%

\subsection{�ſ尵�μ�}
\vspace*{-5ex}
\begin{align}
 \begin{split}
  \Phi_{1} & = \Phi_{s} + C_{p} ( \sigma_{1}^{-\kappa} - 1  ) T_{1} \\
           & = \Phi_{s} + C_{p} \alpha_{1} T_{1}.
 \end{split}
\end{align}
%
\begin{align}
 \begin{split}
  \Phi_k - \Phi_{k-1} 
    & = C_{p}
      \left[ \left( \frac{ \sigma_{k-1/2} }{ \sigma_k } \right)^{\kappa}
             - 1
      \right] T_{k} 
      + C_{p}
      \left[ 1- 
         \left( \frac{ \sigma_{k-1/2} }{ \sigma_{k-1} } \right)^{\kappa}
      \right] T_{v,k-1} \\
  & = C_{p} \alpha_k T_{k} + C_{p} \beta_{k-1} T_{k-1}.
 \end{split}
\end{align}
%
������,
%
\begin{align}
 \Deqlab{�ſ尵������}
 \alpha_k  & = \left( \frac{ \sigma_{k-1/2} }
                               { \sigma_k } \right)^{\kappa} -1 , \\
 \Deqlab{�ſ尵������}
 \beta_k   & = 1- \left( \frac{ \sigma_{k+1/2} }
                               { \sigma_k } \right)^{\kappa} , \\
 \Deqlab{��ɽ�����ݥƥ󥷥��}
 \Phi_{s}  & = gz_{s}
\end{align}
%
�Ǥ���, $z_{s}$����ɽ�̹��٤Ǥ���. 


\subsection{��ư������}
\vspace*{-5ex}
\begin{align}
  \Deqlab{�����������α�ľ��ʬ}
  \DP{\zeta_k}{t}
    & =   \Dinv{a}
          \left(
              \Dinv{1 - \mu^2} \DP{{V_A}_{,k}}{\lambda}
            - \DP{{U_A}_{,k}}{\mu}
          \right)
        + {\cal D}(\zeta_k), 
   \\
  \Deqlab{ȯ���������α�ľ��ʬ}
  \DP{D_k}{t} 
    & =   \Dinv{a}
          \left(
              \Dinv{1 - \mu^2} \DP{{U_A}_{,k}}{\lambda}
            + \DP{{V_A}_{,k}}{\mu}
          \right)
        - \nabla^{2}_{\sigma}
         ( \Phi_k + C_{p} \hat{\kappa}_k \overline{T}_k \pi 
           + (\mbox{\sl KE})_k )
        + {\cal D}(D_k). 
\end{align}
%
������,
%
\begin{align}
 \begin{split}
  {U_A}_{,1}
    & =   ( \zeta_1 + f ) V_1 
        - \frac{1}{2 \Delta \sigma_1} 
                 \dot{\sigma}_{3/2} ( U_1     - U_2 ) 
        - \frac{C_{p} \hat{\kappa}_1 T_{1}'}{a}
            \DP{\pi}{\lambda} 
        + {\cal F}_{\lambda, 1}  \cos \varphi, \\
  {U_A}_{,k}
    & =   ( \zeta_k + f ) V_k 
        - \frac{1}{2 \Delta \sigma_k} 
             [   \dot{\sigma}_{k-1/2} ( U_{k-1} - U_k   )
               + \dot{\sigma}_{k+1/2} ( U_k     - U_{k+1} ) ] \\
    & \quad
        - \frac{C_{p} \hat{\kappa}_k T_{k}'}{a} 
            \DP{\pi}{\lambda} 
        + {\cal F}_{\lambda, k} \cos \varphi, \qquad (k=2, \cdots, K-1) \\
  {U_A}_{,K}
    & =   ( \zeta_K + f ) V_K 
        - \frac{1}{2 \Delta \sigma_K} 
                 \dot{\sigma}_{K-1/2} ( U_{K-1} - U_K   )
        - \frac{C_{p} \hat{\kappa}_K T_{K}'}{a} 
            \DP{\pi}{\lambda} 
        + {\cal F}_{\lambda, K} \cos \varphi, 
 \end{split}
\end{align}
%
\begin{align}
 \begin{split}
  {V_A}_{,1}
    & = - ( \zeta_1 + f ) U_1 
        - \frac{1}{2 \Delta \sigma_1} 
                 \dot{\sigma}_{3/2} ( V_1 - V_2 ) 
        - \frac{C_{p} \hat{\kappa}_1 T_{1}'}{a} 
             (1-\mu^2) \DP{\pi}{\mu} 
        + {\cal F}_{\varphi, 1} \cos \varphi, \\
  {V_A}_{,k}
    & = - ( \zeta_k + f ) U_k 
        - \frac{1}{2 \Delta \sigma_k} 
             [   \dot{\sigma}_{k-1/2} ( V_{k-1} - V_k   )
               + \dot{\sigma}_{k+1/2} ( V_k   - V_{k+1} ) ] \\
    & \quad
        - \frac{C_{p} \hat{\kappa}_k T_{k}'}{a} 
             (1-\mu^2) \DP{\pi}{\mu} 
        + {\cal F}_{\varphi, k} \cos \varphi,  \qquad (k=2, \cdots, K-1) \\
  {V_A}_{,K}
    & = - ( \zeta_K + f ) U_K 
        - \frac{1}{2 \Delta \sigma_K} 
                 \dot{\sigma}_{K-1/2} ( V_{K-1} - V_K   )  \\
    & \quad
        - \frac{C_{p} \hat{\kappa}_K T_{K}'}{a} 
             (1-\mu^2) \DP{\pi}{\mu} 
        + {\cal F}_{\varphi, K} \cos \varphi, 
 \end{split}
\end{align}
%
\begin{align}
 \begin{split}
  \Deqlab{��ľ��ʬ�Φʤ����}
   \hat{\kappa}_k 
     & = \frac{  \sigma_{k-1/2}(   \sigma^{\kappa}_{k-1/2} 
                                 - \sigma^{\kappa}_k      ) 
               + \sigma_{k+1/2}(   \sigma^{\kappa}_k 
                                 - \sigma^{\kappa}_{k+1/2}  ) }
              { \sigma^{\kappa}_k ( \sigma_{k-1/2} - \sigma_{k+1/2} ) } \\
     & = \frac{ \sigma_{k-1/2} \alpha_k + \sigma_{k+1/2} \beta_k }
              { \Delta \sigma_k                                  },  
 \end{split} \\
 %
 T_{k}' & = T_{k} - \overline{T}_k,  \\
 %
 (\mbox{\sl KE})_k &= \frac{U^{2}_k + V^{2}_k}{2 (1-\mu^2)}.
\end{align}

\subsection{Ǯ�ϳؤμ�}
\vspace*{-5ex}
\begin{align}
 \begin{split}
  \DP{T_k}{t}
    & = - \Dinv{a \cos \varphi}
            \left(    \Dinv{1 - \mu^2} \DP{U_k T^{\prime}_k}{\lambda}
                    + \DP{V_k T^{\prime}_k}{\mu}
            \right)
        + H_k  \\
    & \quad
        + \frac{Q_k}{C_{p}}
        + {\cal D}(T_k) 
        + {\cal D}'(\Dvect{v}).
 \end{split}
\end{align}
%
������,
\begin{align}
 \begin{split}
   H_k 
    & \equiv
           T_k' D_k
         - \frac{1}{\Delta \sigma_k} 
             [   \dot{\sigma}_{k-1/2} ( \hat{T}_{k-1/2} - T_k   )
               + \dot{\sigma}_{k+1/2} ( T_k   - \hat{T}_{k+1/2} ) ]
               \\
    & \quad
         + \left\{ \alpha_k
                    \left[   \sigma_{k-1/2} \Dvect{v}_k \cdot \nabla \pi
                           - \sum_{l=k}^{K} 
                               ( D_l + \Dvect{v}_l \cdot \nabla \pi )
                                 \Delta  \sigma_l
                    \right]
           \right.
               \\
    & \quad
         + \quad
           \left. \beta_k
                   \left[   \sigma_{k+1/2} \Dvect{v}_k \cdot \nabla \pi
                          - \sum_{l=k+1}^{K} 
                              ( D_l + \Dvect{v}_l \cdot \nabla \pi )
                                \Delta  \sigma_l
                   \right]
           \right\} 
             \frac{1}{\Delta \sigma_k} T_{k} \\
%
    & =    T_k' D_k 
         - \frac{1}{\Delta \sigma_k} 
             [   \dot{\sigma}_{k-1/2} ( \hat{T}_{k-1/2} - T_k   )
               + \dot{\sigma}_{k+1/2} ( T_k   - \hat{T}_{k+1/2} ) ]
               \\
    & \quad
         + \hat{\kappa}_k \Dvect{v}_k \cdot \nabla \pi T_{v,k} 
               \\
    & \quad
         - \alpha_k \sum_{l=k}^{K} 
                      ( D_l + \Dvect{v}_l \cdot \nabla \pi )
                        \Delta  \sigma_l 
                          \frac{T_{k}}{\Delta \sigma_k} 
               \\
    & \quad
         - \beta_k \sum_{l=k+1}^{K} 
                     ( D_l + \Dvect{v}_l \cdot \nabla \pi )
                       \Delta  \sigma_l 
                         \frac{T_{k}}{\Delta \sigma_k}
      \qquad (k=1, \cdots, K-1), \\
%
   H_K
    & \equiv
           T_K' D_K 
         - \frac{1}{\Delta \sigma_K} 
             [   \dot{\sigma}_{K-1/2} ( \hat{T}_{K-1/2} - T_K   )
               + \dot{\sigma}_{K+1/2} ( T_K   - \hat{T}_{K+1/2} ) ]
               \\
    & \quad
         + \hat{\kappa}_K \Dvect{v}_K \cdot \nabla \pi T_{v,K} 
               \\
    & \quad
         - \alpha_K 
                      ( D_K + \Dvect{v}_K \cdot \nabla \pi )
                        \Delta  \sigma_K 
                          \frac{T_{K}}{\Delta \sigma_K} 
 \end{split}
\end{align}
%
�Ǥ���, 
%
\begin{align} 
 \begin{split}
  \hat{T}_{k-1/2}
    & = \frac{
          \left[
            \left( \displaystyle
                \frac{ \sigma_{k-1/2} }{ \sigma_k } \right)^{\kappa}
              - 1
            \right] \sigma_{k-1}^{\kappa} T_k
        + \left[
              1
            - \left( \displaystyle
                  \frac{ \sigma_{k-1/2} }{ \sigma_{k-1} }
              \right)^{\kappa}
          \right] \sigma_k^{\kappa} T_{k-1}                   }
        { \sigma_{k-1}^{\kappa} - \sigma_k^{\kappa}           }
        \\
    & = a_k T_k + b_{k-1} T_{k-1}
        \qquad  \quad (k = 2, \cdots, K), \\
  \hat{T}_{1/2} & = 0, \\
  \hat{T}_{K + 1/2} & = 0, 
 \end{split}
\end{align}
%
\begin{align}
  \Deqlab{������ַ���a}
  a_k & = \alpha_k 
            \left[ 1- \left( \frac{ \sigma_k }{ \sigma_{k-1} }
                        \right)^{\kappa} \right]^{-1},    \\
  \Deqlab{������ַ���b}
  b_k & = \beta_k 
            \left[ \left( \frac{ \sigma_k }{ \sigma_{k+1} } 
                     \right)^{\kappa} - 1 \right]^{-1} .  
\end{align}
