<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">% 表題   AGCM5 第2部 離散化 水平離散化
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% 履歴
\Drireki{93/03/18 沼口敦・保坂征宏}
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\section{水平離散化}

ここでは支配方程式を水平離散化する. 
水平方向の離散化はスペクトル変換法を用いる(Bourke, 1988).
経度, 緯度に関する微分の項は直交関数展開によって評価し,
一方, 非線型項は格子点上で計算する.
各方程式のスペクトル表現は以下のようになる. 

\subsection{連続の式}

\begin{eqnarray}
  \DP{\tilde{\pi}_n^m}{t}
 &amp; = &amp; - \sum_{k=1}^{K} (\tilde{D}_n^m)_k \Delta  \sigma_k \nonumber \\
 &amp;   &amp; + \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}  
               Z_{ij} Y_n^{m *} ( \lambda_i, \mu_j ) w_j  ,
\end{eqnarray}
%
ここで,
\begin{equation}
Z \equiv - \sum_{k=1}^{K} \Dvect{v}_k \cdot \nabla \pi .
\end{equation}

\subsection{運動方程式}

\begin{eqnarray}
  \DP{\tilde{\zeta}_n^m}{t} 
  &amp; = &amp; \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}  
          im (\mbox{\sl VA})_{ij}
          Y_n^{m *} ( \lambda_i, \mu_j )
         \frac{w_j}{a(1-\mu_j^{2})} 
         \nonumber \\
  &amp; + &amp;   \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}  
          (\mbox{\sl UA})_{ij}
          (1-\mu_j^2) 
          \frac{\partial }{\partial \mu}  Y_n^{m *} ( \lambda_i, \mu_j )
          \frac{w_j}{a(1-\mu_j^{2})} 
         \nonumber \\ 
  &amp; + &amp;  \tilde{\cal D}_{M,n}^m \tilde{\zeta}_n^m  ,
\end{eqnarray}
%
\begin{eqnarray}
  \DP{\tilde{D}_n^m}{t} 
  &amp; = &amp; \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}  
          im (\mbox{\sl UA})_{ij}
          Y_n^{m *} ( \lambda_i, \mu_j )
         \frac{w_j}{a(1-\mu_j^{2})} 
         \nonumber \\
  &amp; - &amp;   \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}  
          (\mbox{\sl VA})_{ij}
          (1-\mu_j^2) 
          \frac{\partial }{\partial \mu}  Y_n^{m *} ( \lambda_i, \mu_j )
          \frac{w_j}{a(1-\mu_j^{2})} 
         \nonumber \\
  &amp; - &amp;  \frac{n(n+1)}{a^{2}} 
         \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}  
          (\mbox{\sl KE})_{ij}
          Y_n^{m *} ( \lambda_i, \mu_j )
          w_j
         \nonumber \\ 
  &amp; + &amp;  \frac{n(n+1)}{a^{2}} 
          ( \Phi_n^m + C_{p} \hat{\kappa}_k \bar{T}_k \pi_n^m ) 
          +  \tilde{\cal D}_{M,n}^m \tilde{D}_n^m  ,
\end{eqnarray}
%
ただし,
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\begin{equation}
\tilde{\cal D}_{M,n}^m = - K_{HD} \left[ 
                            \left( \frac{-n(n+1)}{a^{2}} \right)^{N_D/2}
                            - \left( \frac{2}{a} \right)^{N_D}
                            \right]  .
\end{equation}


\subsection{熱力学の式}

\begin{eqnarray}
  \DP{\tilde{T}_n^m}{t}
  &amp; = &amp; - \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}  
          im U_{ij} T'_{ij}
          Y_n^{m *} ( \lambda_i, \mu_j )
         \frac{w_j}{a(1-\mu_j^{2})} 
         \nonumber \\
  &amp;  &amp; + \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}  
          V_{ij} T'_{ij}
          (1-\mu_j^2) 
          \frac{\partial }{\partial \mu}  Y_n^{m *} ( \lambda_i, \mu_j )
          \frac{w_j}{a(1-\mu_j^{2})} 
         \nonumber \\
  &amp;  &amp; + \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}  
          \left( \hat{H}_{ij} + \frac{Q_{ij}}{C_{p}} \right)
          Y_n^{m *} ( \lambda_i, \mu_j )
          w_j
         \nonumber \\ 
  &amp;  &amp; + \tilde{\cal D}_{H,n}^m \tilde{T}_n^m
         \nonumber \\ 
  &amp;  &amp; + \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}  
          {\cal D}'_{ij}(\Dvect{v})
          Y_n^{m *} ( \lambda_i, \mu_j )
          w_j ,
\end{eqnarray}
%
ただし,
%
\begin{equation}
\tilde{\cal D}_{H,n}^m 
   = - K_{HD} \left( \frac{-n(n+1)}{a^{2}} \right)^{N_D/2} .
\end{equation}


\subsection{水蒸気の式}

\begin{eqnarray}
  \DP{\tilde{q}_n^m}{t}
  &amp; = &amp; - \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}  
          im U_{ij} q_{ij}
          Y_n^{m *} ( \lambda_i, \mu_j )
         \frac{w_j}{a(1-\mu_j^{2})} 
         \nonumber \\
  &amp;  &amp; + \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}  
          V_{ij} q_{ij}
          (1-\mu_j^2) 
          \frac{\partial }{\partial \mu}  Y_n^{m *} ( \lambda_i, \mu_j )
          \frac{w_j}{a(1-\mu_j^{2})} 
         \nonumber \\
  &amp;  &amp; + \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}  
          \left( \hat{R}_{ij} + S_{q,ij} \right)
          Y_n^{m *} ( \lambda_i, \mu_j )
          w_j
         \nonumber \\ 
  &amp;  &amp; + \tilde{\cal D}_{H,n}^m \tilde{q}_n^m
\end{eqnarray}


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