<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% 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
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%  
% 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
%       2009/09/30  山下達也
%       2010/03/17  山下達也
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\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}{\langle c_p \rangle [\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}}{\langle R \rangle}
                    \frac{[\overline{\Pi}^{ \langle c_{v} \rangle 
		    / \langle R \rangle}]_{i,k}}
		    {[\overline{\theta_{v}}]_{i,k}}
 \end{equation}

\subsection{運動方程式}
 \begin{eqnarray}
  \DP{u_{i(u),k}}{t} &amp;=&amp;
  - 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 \\
 &amp;&amp; 
  - \langle c_p \rangle [\overline{\theta_{v}}]_{i(u),k}
  \left[\DP{\Pi}{x}\right]_{i(u),k}
  + \left[ D_{\overline{u}} \right]_{i(u),k} 
  + \left[ D_{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} &amp;=&amp;
  - u_{i,k(w)}\left[\DP{w}{x}\right]_{i,k(w)}
  - w_{i,k(w)}\left[\DP{w}{z}\right]_{i,k(w)} 
  \nonumber \\
 &amp;&amp;
  - \langle c_p \rangle
  [\overline{\theta_{v}}]_{i,k(w)}\left[\DP{\Pi}{z}\right]_{i,k(w)} 
  + \left[ D_{w} \right]_{i,k(w)} 
\nonumber \\
&amp;&amp;  
 + g \frac{\theta_{i,k(w)}}{\overline{\theta}_{i,k(w)}} g
\nonumber \\
&amp;&amp;  
  + \frac{ [q_d]_{i,k(w)} \langle M \rangle /M_d + \sum [q_v]_{i,k(w)}
    \langle M \rangle / M_v }
     { [\overline{q_d}]_{i,k(w)} \langle M \rangle  /M_d +
    \sum [\overline{q_v}]_{i,k(w)} \langle M \rangle / M_v} g
  \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}}{\langle c_p \rangle
%         \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.
%
\DP{ \Pi_{i,k}}{t}
 &amp;=&amp;
- \frac{\overline{  \langle {c_{s}} \rangle^{2} }_{i,k} }
  { \langle c_p \rangle \bar{\rho}_{i,k}
 [\bar{\theta_v}^{2}]_{i,k} }
 [ \Ddiv \left(
        \bar{\rho} \bar{\theta_v} \Dvect{u}
       \right) ]_{i,k} \nonumber \\
&amp; &amp;
 + \frac{ \overline{\langle c_s \rangle^2}_{i,k} }{\langle c_p \rangle
 [ \bar{\theta_v} ]_{I,k} }
\Bigg[ \Dinv{\bar{\rho}_{i,k}} \sum [ M_{fall} (\rho_r) ]_{i,k}
 \nonumber \\
&amp; &amp;
  + \Dinv{ \bar{\theta}_{i,k} }
    \left\{ \Dinv{\bar{\Pi}_{i,k}} 
     ( [Q_{cond}]_{i,k} + [Q_{rad}]_{i,k} + [Q_{dis}]_{i,k} )
  + [D_{\theta}]_{i,k} \right\}
 \nonumber \\
&amp; &amp;
+ \Dinv{ [\bar{q_d}]_{i,k}  \langle M \rangle / M_d
    + \sum [\bar{q_v}]_{i,k} \langle M \rangle / M_v} \nonumber \\
&amp; &amp;
  \Bigg\{ \frac{\langle M \rangle}{M_d}
 \left( - \frac{ [\bar{q_d}]_{i,k}}{\bar{\rho}_{i,k}} 
  \sum [ M_{fall} (\rho_r) ]_{i,k} 
  + [ D_{q_d} ]_{i,k} \right)
 \nonumber \\
&amp; &amp;
 + \sum \frac{\langle M \rangle}{M_v}
 \left( - \frac{[\bar{q_v}]_{i,k}}{\bar{\rho}_{i,k}} 
  \sum [M_{fall} (\rho_r)]_{i,k} 
  + [M_{src} (q_v)]_{i,k} +  [ D_{q_v} ]_{i,k} \right)
\Bigg\}
\Bigg]. \nonumber \\
  \Deqlab{空間離散化された圧力方程式}
 \end{eqnarray}
 基本場の音速 $\overline{ \langle c_s \rangle}$ は以下のように計算する. 
 \begin{eqnarray}
  \overline{ \langle c_s \rangle}_{i,k}^{2} 
    = \frac{\langle c_p \rangle \langle R \rangle}
       { \langle c_{v} \rangle} 
         \overline{\Pi}_{i,k} [\overline{\theta_{v}}]_{i,k}.
 \end{eqnarray}


\subsection{熱力学の式}
 \begin{eqnarray}
  \DP{\theta_{i,k}}{t} &amp;=&amp; - 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}
 \nonumber \\
 &amp;&amp;                 + \Dinv{\overline{\Pi}_{i,k}}
	\left([Q_{cond}]_{i,k} + [Q_{rad}]_{i,k} +
	 [Q_{dis}]_{i,k}\right)
         + \left[ D_{\theta} \right]_{i,k}
         + \left[ D_{\overline{\theta}}\right]_{i,k}
  \Deqlab{空間離散化された熱力学の式}
 \end{eqnarray}


\subsection{比湿の式}
%
\begin{eqnarray}
\DP{ [q_d]_{i,k}}{t}
 &amp;=&amp; - u_{i,k} \left[ \DP{q_d}{x} \right]_{i,k}
     - w_{i,k} \left[ \DP{q_d}{z} \right]_{i,k}
     - w_{i,k} \left[ \DP{\overline{q_d}}{z} \right]_{i,k} \nonumber \\
 &amp; &amp; - \frac{ [\overline{q_d}]_{i,k}}{\overline{\rho}_{i,k}} 
      \sum [ M_{fall} (\rho_r) ]_{i,k}
      + [ D_{q_d} ]_{i,k}, \Deqlab{quasi-equations:u-w-pi-theta:qd} \\
%
\DP{ [q_v]_{i,k}}{t}
 &amp;=&amp; - u_{i,k} \left[ \DP{q_v}{x} \right]_{i,k}
     - w_{i,k} \left[ \DP{q_v}{z} \right]_{i,k}
     - \overline{u}_{i,k} \left[ \DP{q_v}{x} \right]_{i,k}
     - w_{i,k} \left[ \DP{\overline{q_v}}{z} \right]_{i,k} \nonumber \\
 &amp; &amp;  - \frac{ [\overline{q_v}]_{i,k}}{\overline{\rho}_{i,k}} 
     \sum [ M_{fall} (\rho_r) ]_{i,k}
     + \Dinv{\overline{\rho}_{i,k}} [ M_{src} (\rho_v) ]_{i,k}
     + [ D_{\overline{q_v}} ]_{i,k} + [D_{q_v}]_{i,k}, \nonumber \\ \\
  \Deqlab{quasi-equations:u-w-pi-theta:qv}
%
\DP{ [q_c]_{i,k}}{t}
 &amp;=&amp; - u_{i,k} \left[ \DP{q_c}{x} \right]_{i,k}
     - w_{i,k} \left[ \DP{q_c}{z} \right]_{i,k}
     - \frac{ [\overline{q_c}]_{i,k}}{\overline{\rho}_{i,k}} 
     \sum [ M_{fall} (\rho_r)]_{i,k} \nonumber \\
 &amp; &amp; + \Dinv{ \overline{\rho}_{i,k}} [M_{src} (\rho_c)]_{i,k}
     + [D_{q_c}]_{i,k},   \Deqlab{quasi-equations:u-w-pi-theta:qc} \\
%
\DP{ [q_r]_{i,k}}{t}
 &amp;=&amp; - u_{i,k} \left[ \DP{q_r}{x} \right]_{i,k}
     - w_{i,k} \left[ \DP{q_r}{z} \right]_{i,k} 
 \nonumber \\
 &amp; &amp; + \Dinv{\overline{\rho}_{i,k}} [ M_{fall} (\rho_r) ]_{i,k}
     + \Dinv{\overline{\rho}_{i,k}} [ M_{src} (\rho_r) ]_{i,k}
     + [ D_{q_r}]_{i,k}. \Deqlab{quasi-equations:u-w-pi-theta:qr}
\end{eqnarray}

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