%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% 表題  2 次元非静力学モデル -- 離散モデル  空間離散化した基礎方程式
%
%  
% 履歴  2004/09/21  小高正嗣, 北守太一
%       2005/04/13  小高正嗣
%       2005/04/13  小高正嗣
%       2005/08/25  小高正嗣
%       2006/03/12  杉山耕一朗: 湿潤大気版に修正
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{空間離散化した基礎方程式}

\subsection{静水圧の式}
%\footnote{
% ここには基本場の数値積分方法を記載する.}
 \begin{equation}
  \left[\DP{\overline{\pi}}{z}\right]_{i,k} = 
    - \frac{g}{{c_{p}}_{d} [\overline{\theta_{v}}]_{i,k}}
 \end{equation}
 基本場の密度 $\overline{\rho}_{i,k}$ は以下のように計算する.
 \begin{equation}
   \overline{\rho}_{i,k} = \frac{p_{0}}{R_{d}}
                    \frac{[\overline{\pi}^{c_{v}/R_{d}}]_{i,k}}
		    {[\overline{\theta_{v}}]_{i,k}}
 \end{equation}

\subsection{運動方程式}
 \begin{eqnarray}
  \DP{u_{i(u),k}}{t} &=&
  - u_{i(u),k}\left[\DP{u}{x}\right]_{i(u),k}
  - w_{i(u),k}\left[\DP{u}{z}\right]_{i(u),k}\nonumber \\
 && 
  - {c_{p}}_{d} [\overline{\theta_{v}}]_{i(u),k}
  \left[\DP{\pi}{x}\right]_{i(u),k}
  + \left[{\rm Turb}.{u}\right]_{i(u),k} 
  \Deqlab{空間離散化された x 方向運動方程式}
  \\
  \DP{w_{i,k(w)}}{t} &=&
  - u_{i,k(w)}\left[\DP{u}{x}\right]_{i,k(w)}
  - w_{i,k(w)}\left[\DP{u}{z}\right]_{i,k(w)} \nonumber \\
 &&
  - {c_{p}}_{d} [\overline{\theta_{v}}]_{i,k(w)}\left[\DP{\pi}{z}\right]_{i,k(w)}
  + \left[{\rm Turb}.{w}\right]_{i,k(w)} 
\nonumber \\
&&  
 + g \frac{\theta_{i,k(w)}}{\overline{\theta}_{i,k(w)}}
\nonumber \\
&&  
 + g \frac{\sum [q_{v}]_{i,k(w)}/M_{v}}{1/M_{d} 
      + \sum  [\bar{q_{v}}]_{i,k(w)}/M_{v}} 
\nonumber \\
&&  
    - g \frac{\sum [q_{v}]_{i,k(w)} + \sum [q_{c}]_{i,k(w)} + \sum [q_{r}]_{i,k(w)}}
    {1 + \sum [\bar{q_{v}}]_{i,k(w)}}
  \Deqlab{空間離散化された z 方向運動方程式}
 \end{eqnarray}

\subsection{圧力方程式}
 \begin{eqnarray}
    \DP{\pi_{i,k}}{t}
     + \frac{\overline{c}_{i,k}^{2}}{{c_{p}}_{d}
         \overline{\rho}_{i,k} [\overline{\theta_{v}}^{2}]_{i,k}}
         \left[
         \DP{\overline{\rho} \overline{\theta_{v}} u}{x} +
         \DP{\overline{\rho} \overline{\theta_{v}} u}{z}
         \right]_{i,k}
     = 0.
  \Deqlab{空間離散化された圧力方程式}
 \end{eqnarray}
 基本場の音速 $\overline{c}$ は以下のように計算する. 
 \begin{equation}
  \overline{c}_{i,k}^{2} = \frac{{c_{p}}_{d} R_{d}}{c_{v}}
                  \overline{\pi}_{i,k} [\overline{\theta_{v}}]_{i,k}.
 \end{equation}


\subsection{熱力学の式}
 \begin{eqnarray}
  \DP{\theta_{i,k}}{t} &=& - u_{i,k}\left[\DP{\theta}{x}\right]_{i,k}
                         - w_{i,k}\left[\DP{\theta}{z}\right]_{i,k}
                         - w_{i,k}\left[\DP{\overline{\theta}}{z}\right]_{i,k}
                         + \left[{\rm Turb}.{\theta}\right]_{i,k}
                         + \left[{\rm Turb}.{\overline{\theta}}\right]_{i,k}
 \nonumber \\
 &&                 + \Dinv{\overline{\pi}_{i,k}}
	\left([Q_{cnd}]_{i,k} + [Q_{rad}]_{i,k} + [Q_{dis}]_{i,k}\right)
  \Deqlab{空間離散化された熱力学の式}
 \end{eqnarray}


\subsection{凝縮成分の混合比の保存式}
%
\begin{eqnarray}
\DP{[q_{v}]_{i,k}}{t} 
&=& 
  - u_{i,k} \left[\DP{q_{v}}{x} \right]_{i,k}
  - w_{i,k} \left[\DP{q_{v}}{x} \right]_{i,k}
  - w_{i,k} \left[\DP{\bar{q_{v}}}{x} \right]_{i,k}
\nonumber  \\
&& 
  + [{\rm Src}.q_{v}]_{i,k} 
  + [{\rm Turb}.{\overline{q_{v}}}]_{i,k} + [{\rm Turb}.{q_{v}}]_{i,k}, 
  \Deqlab{quasi-equations:u-w-pi-theta:qv} \\
%
\DP{[q_{c}]_{i,k}}{t} 
&=&
  - u_{i,k} \left[ \DP{q_{c}}{x} \right]_{i,k}
  - w_{i,k} \left[ \DP{q_{c}}{x} \right]_{i,k}
  + [{\rm Src}.q_{c}]_{i,k} + [{\rm Turb}.{q_{c}}]_{i,k}, 
  \Deqlab{quasi-equations:u-w-pi-theta:qc} \\
%
\DP{[q_{r}]_{i,k}}{t} 
&=& 
  - u_{i,k} \left[ \DP{q_{r}}{x} \right]_{i,k}
  - w_{i,k} \left[ \DP{q_{r}}{x} \right]_{i,k}
  + [{\rm Src}.q_{r}]_{i,k} 
\nonumber \\
 &&
  + [{\rm Fall}.q_{r}]_{i,k} 
  + [{\rm Turb}.{q_{r}}]_{i,k}
  \Deqlab{quasi-equations:u-w-pi-theta:qr}
\end{eqnarray}
%