% 表題   AGCM5 第2部 離散化 鉛直離散化
%
% 履歴
\Drireki{93/03/18 沼口敦・保坂征宏}
%

\section{鉛直離散化}

ここでは支配方程式を鉛直方向に離散化する. 
Arakawa and Suarez(1983) に従って, 
基礎方程式を鉛直方向に差分によって離散化する\footnotemark.
     \footnotetext{
          このスキームは次のような特徴をもつ. 
          ( 2005/4/4 石渡: 初の 4 つは不正確な表現かも )
          \begin{itemize}
              \item 全領域積分した質量を保存
              \item 全領域積分したエネルギーを保存
              \item 全領域積分の角運動量を保存
              \item 全質量積分した温位を保存
              \item 静水圧の式がlocalにきまる.
                    (下層の高度が上層の温度に依存しない)
              \item 水平方向に一定の, ある特定の温度分布について,
                    静水圧の式が正確になり, 気圧傾度力が0になる.
              \item 等温位大気はいつまでも等温位に留まる
          \end{itemize}      
          }
各方程式の離散化表現は次のようになる.

\subsection{連続の式, 鉛直速度}

\begin{equation}
  \frac{\partial \pi}{\partial t}
 = - \sum_{k=1}^{K} ( D_k + \Dvect{v}_k \cdot \nabla \pi ) 
       \Delta  \sigma_k
\end{equation}
%
\begin{equation}
  \dot{\sigma}_{k-1/2}
 = - \sigma_{k-1/2} \frac{\partial \pi}{\partial t}
   - \sum_{l=k}^{K} ( D_l + \Dvect{v}_l \cdot \nabla \pi )          
       \Delta  \sigma_l
\end{equation}
%
\begin{equation}
  \dot{\sigma}_{1/2} = \dot{\sigma}_{K+1/2} = 0
\end{equation}
   
\subsection{静水圧の式}

\begin{eqnarray}
 \Phi_{1} & = & \Phi_{s} + C_{p} ( \sigma_{1}^{-\kappa} - 1  ) T_{v,1} \\
          & = & \Phi_{s} + C_{p} \alpha_{1} T_{v,1} \nonumber
\end{eqnarray}
%
\begin{eqnarray}
 \Phi_k - \Phi_{k-1} 
  & = & C_{p}
   \left[ \left( \frac{ \sigma_{k-1/2} }{ \sigma_k } \right)^{\kappa}
          - 1 \right] T_{v,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_{v,k} + C_{p} \beta_{k-1} T_{v,k-1}
               \nonumber
\end{eqnarray}
%
ここで,
%
\begin{eqnarray}
 \label{静水圧係数}
 \alpha_k  & = & \left( \frac{ \sigma_{k-1/2} }
                               { \sigma_k } \right)^{\kappa} -1 \\
 \beta_k   & = & 1- \left( \frac{ \sigma_{k+1/2} }
                               { \sigma_k } \right)^{\kappa} .
\end{eqnarray}

\subsection{運動方程式}

\begin{equation}
  \label{渦度結局}
  \frac{\partial \zeta_k}{\partial t} 
     =   \frac{1}{a(1-\mu^{2})} \frac{\partial (\mbox{\sl VA})_k}
                                      {\partial \lambda}
          - \frac{1}{a} \frac{\partial (\mbox{\sl UA})_k}{\partial \mu}
          + {\cal D}(\zeta_k) 
\end{equation}
%
\begin{equation}
  \frac{\partial D}{\partial t} 
   =   \frac{1}{a(1-\mu^{2})}
           \frac{\partial (\mbox{\sl UA})_k}{\partial \lambda}
          + \frac{1}{a} \frac{\partial (\mbox{\sl VA})_k}{\partial \mu}
          - \nabla^{2}_{\sigma}
           ( \Phi_k + C_{p} \hat{\kappa}_k \bar{T}_k \pi 
             + (\mbox{\sl KE})_k )
          + {\cal D}(D_k) 
\end{equation}
%
ここで,
%
\begin{eqnarray}
 (\mbox{\sl UA})_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} ) ]
           \nonumber \\
   &   &     - \frac{C_{p} \hat{\kappa}_k T_{v,k}'}{a} 
                  \frac{\partial \pi}{\partial \lambda} 
             + F_{\lambda k} \cos \varphi \\
\end{eqnarray}
%
\begin{eqnarray}
 (\mbox{\sl VA})_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} ) ]
           \nonumber \\
   &   &     - \frac{C_{p} \hat{\kappa}_k T_{v,k}'}{a} 
             ( 1 - \mu^{2} ) \frac{\partial \pi}{\partial \mu} 
             + F_{\varphi k} \cos \varphi 
\end{eqnarray}
%
ここで,
%
\begin{eqnarray}
   \label{はっとかっぱ}
   \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} )         } 
         \nonumber   \\
 & = & \frac{ \sigma_{k-1/2} \alpha_k + \sigma_{k+1/2} \beta_k }
            { \Delta \sigma_k                                    } 
\end{eqnarray}

\begin{equation}
T_{v,k} = T_k - \bar{T}_k
\end{equation}
%
\begin{equation}
     (\mbox{\sl KE})_k =  \frac{U^{2}_k+V^{2}_k}{2(1-\mu^{2}) } 
\end{equation}
%

\subsection{熱力学の式}

\begin{eqnarray}
  \frac{\partial T_k}{\partial t}
    & = & - \frac{1}{a(1-\mu^{2})} 
               \frac{\partial U_k T_k'}{\partial \lambda}
          - \frac{1}{a}
               \frac{\partial V_k T_k'}{\partial \mu}
          + \hat{H}_k \nonumber \\
    &   & + \frac{Q_k}{C_{p}}
          + {\cal D}(T_k) 
          + {\cal D}'(\Dvect{v})  \nonumber \\
\end{eqnarray}
%
ここで,
\begin{eqnarray}
   \hat{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} ) ]
               \nonumber \\
    &   & + \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.   \nonumber \\
    &   &   + \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_{v,k}  \nonumber \\
%
    & = & 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} ) ]
               \nonumber \\
    &   & + \hat{\kappa}_k \Dvect{v}_k \cdot \nabla \pi T_{v,k} 
               \nonumber \\
    &   & - \alpha_k \sum_{l=k}^{K} 
                           ( D_l + \Dvect{v}_l \cdot \nabla \pi )
                            \Delta  \sigma_l 
                            \frac{T_{v,k}}{\Delta \sigma_k} 
               \nonumber \\
    &   & - \beta_k \sum_{l=k+1}^{K} 
                           ( D_l + \Dvect{v}_l \cdot \nabla \pi )
                            \Delta  \sigma_l 
                            \frac{T_{v,k}}{\Delta \sigma_k} 
\end{eqnarray}
%
\begin{eqnarray} 
  \hat{T}_{k-1/2}
 &  = & \frac{ \left[ \left( \frac{ \sigma_{k-1/2} }
                               { \sigma_k } \right)^{\kappa}
                  - 1 \right] \sigma_{k-1}^{\kappa} T_k 
          + \left[ 1- 
                   \left( \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}
\end{eqnarray}
%
\begin{eqnarray}
  \label{温度補間係数}
  a_k & = & \alpha_k 
              \left[ 1- \left( \frac{ \sigma_k }{ \sigma_{k-1} }
                        \right)^{\kappa} \right]^{-1}   \\
  b_k & = & \beta_k 
              \left[ \left( \frac{ \sigma_k }{ \sigma_{k+1} } 
                     \right)^{\kappa} - 1 \right]^{-1} .  
\end{eqnarray}

\subsection{水蒸気の式}

\begin{equation}
  \label{q結局}
  \frac{\partial q_k}{\partial t}
      =   - \frac{1}{a(1-\mu^{2})} 
               \frac{\partial U_k q_k}{\partial \lambda}
          - \frac{1}{a}
               \frac{\partial V_k q_k}{\partial \mu}
          + R_k 
          + S_{q,k}
          + {\cal D}(q_k) 
\end{equation}
%
\begin{equation}
R_k  =  q_k D_k 
       - \frac{1}{2 \Delta \sigma_k} 
             [   \dot{\sigma}_{k-1/2} ( q_{k-1} - q_k   )
               + \dot{\sigma}_{k+1/2} ( q_k   - q_{k+1} ) ]
\end{equation}
