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% 表題  2 次元非静力学モデル -- 離散モデル  付録A
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% 履歴  2004/08/14  杉山耕一朗: 作成開始
%       2005/04/13  小高正嗣
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\chapter{圧力方程式 \Deqref{uwpi:sabun} の左辺の空間微分の書き下し}
\Dchaplab{appendix-a}

\Deqref{uwpi:sabun} 左辺の変形を行う. 
%
\begin{eqnarray}
\Deqref{uwpi:sabun}\mbox{左辺} &=&
 \pi^{\tau + \Delta \tau}_{i,k}
 \nonumber \\
 &&- \beta^{2} 
  \left(
   \frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
  \right)_{k}
  \Dinv{\Delta z}
  \left\{
   \left(
   \bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
   \right)_{k(w)}
   \left(
     \DP{\pi^{\tau + \Delta \tau}}{z} 
   \right)_{i,k(w)}
\right\}
 \nonumber \\
 &&+ \beta^{2} 
  \left(
   \frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
  \right)_{k}
  \Dinv{\Delta z}
  \left\{
   \left(
   \bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
   \right)_{k-1(w)}
   \left(
     \DP{\pi^{\tau + \Delta \tau}}{z} 
   \right)_{i,k-1(w)}
\right\}
  \nonumber \\
 &=&
 \pi^{\tau + \Delta \tau}_{i,k}
 \nonumber \\
 &&- \beta^{2} 
  \left(
   \frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
  \right)_{k}
  \Dinv{\Delta z}
  \left\{
   \left(
   \bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
   \right)_{k(w)}
   \left(
    \frac{\pi^{\tau + \Delta \tau}_{i,k+1} 
    - \pi^{\tau + \Delta \tau}_{i,k}}{\Delta z} 
   \right)
\right\}
 \nonumber \\
 &&+ \beta^{2} 
  \left(
   \frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
  \right)_{k}
  \Dinv{\Delta z}
  \left\{
   \left(
   \bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
   \right)_{k-1(w)}
   \left(
    \frac{\pi^{\tau + \Delta \tau}_{i,k} 
    - \pi^{\tau + \Delta \tau}_{i,k-1}}{\Delta z} 
   \right)
\right\}
  \nonumber \\
 &=& 
  \left\{
   - \beta^{2}
  \left(
  \frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
  \right)_{k}
  \Dinv{\Delta z^{2}} 
  \left(
    \bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
  \right)_{k(w)}
  \right\}
   \pi^{\tau + \Delta \tau}_{i,k+1}
\nonumber \\   
 &&+ \left[
  1 + \beta^{2}
  \left(
   \frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
  \right)_{k}
  \Dinv{\Delta z^{2}} 
   \left\{
   \left(
   \bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2} 
   \right)_{k(w)}
   + 
   \left(
   \bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2} 
   \right)_{k-1(w)}
   \right\} 
    \right]
 \pi^{\tau + \Delta \tau}_{i,k}
   \nonumber \\   
 &&+ \left\{
   - \beta^{2}
  \left(
   \frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
  \right)_{k}
  \Dinv{\Delta z^{2}} 
   \left(
    \bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2} 
       \right)_{k-1(w)}
   \right\}
   \pi^{\tau + \Delta \tau}_{i,k-1}.
   \Deqlab{uwpi:LH_sabun}
\end{eqnarray}
%

\section{下部境界}

下部境界($k(w) = 0(w)$)について考える. この時 \Deqref{uwpi:w_sabun} 式は, 
%
\begin{eqnarray}
 \beta \left(
	\DP{\pi^{\tau + \Delta \tau}}{z}  
       \right)_{i,0(w)}
 &=&
\left[
\left( \DP{(\alpha Div)^{\tau}}{z} \right)
  - (1 - \beta) \left( \DP{\pi^{\tau}}{z} \right)
  + \left(\Dinv{\bar{c_{p}} \bar{\theta}_{v}} F_{w}^{t}\right)
\right]_{i,0(w)} 
\nonumber \\
 &\equiv& E_{i,0(w)}
  \Deqlab{uwpi:w_sabun_kabu}
\end{eqnarray}
%
となるので, \Deqref{uwpi:sabun} 式の左辺は, $k = 1$ の場合には, 
%
%
\begin{eqnarray}
\Deqref{uwpi:sabun}\mbox{左辺}
 &=&
 \pi^{\tau + \Delta \tau}_{i,1}
 \nonumber \\
 &&- \beta^{2} 
  \left(
   \frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho}
   \bar{\theta}_{v}^{2}}
  \right)_{1}
  \Dinv{\Delta z}
  \left\{
   \left(
    \bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
   \right)_{1(w)}
   \left(
    \DP{\pi^{\tau + \Delta \tau}}{z} 
   \right)_{i,1(w)}
\right\}
 \nonumber \\
 &&+ \beta^{2} 
  \left(
   \frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
  \right)_{1}
  \Dinv{\Delta z}
  \left\{
   \left(
   \bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
   \right)_{0(w)}
   \left(
     \DP{\pi^{\tau + \Delta \tau}}{z} 
   \right)_{i,0(w)}
\right\}
  \nonumber \\
 &=&
 \pi^{\tau + \Delta \tau}_{i,1}
 \nonumber \\
 &&- \beta^{2} 
  \left(
   \frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho}
   \bar{\theta}_{v}^{2}}
  \right)_{1}
  \Dinv{\Delta z}
  \left\{
   \left(
    \bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
   \right)_{1(w)}
   \left(
    \frac{\pi^{\tau + \Delta \tau}_{i,2} 
    - \pi^{\tau + \Delta \tau}_{i,1}}{\Delta z} 
   \right)
\right\}
 \nonumber \\
 &&+ \beta^{2} 
  \left(
   \frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
  \right)_{1}
  \Dinv{\Delta z}
  \left\{
   \left(
   \bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
   \right)_{0(w)}
   \left(
     \DP{\pi^{\tau + \Delta \tau}}{z} 
   \right)_{i,0(w)}
\right\}
  \nonumber \\
 &=&
  \left\{
 - \beta^{2} 
  \left(
  \frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho}
  \bar{\theta}_{v}^{2}}
  \right)_{1}
  \Dinv{\Delta z^{2}}
   \left(
   \bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
   \right)_{1(w)}
  \right\}
  \pi^{\tau + \Delta \tau}_{i,2}
 \nonumber \\
 &&+ 
  \left\{
  1 + \beta^{2} 
  \left(
  \frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho}
  \bar{\theta}_{v}^{2}}
  \right)_{1}
  \Dinv{\Delta z^{2}}
   \left(
   \bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
   \right)_{1(w)}
  \right\} \pi^{\tau + \Delta \tau}_{i,1}
 \nonumber \\
 &&+ \beta 
  \left(
   \frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho}
   \bar{\theta}_{v}^{2}}
  \right)_{1}
  \Dinv{\Delta z}
%\left[
   \left(
   \bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
   \right)_{0(w)}
   E_{i,0(w)}
  \nonumber 
\end{eqnarray}
%


\section{上部境界}

上部境界($k(w) = km(w)$)について考える. \Deqref{uwpi:sabun} 式の左辺は, 
%
\begin{eqnarray}
\beta \left(
	\DP{\pi^{\tau + \Delta \tau}}{z}  
       \right)_{i,km(w)}
 &=&
 \left[
 \left( \DP{(\alpha Div)^{\tau}}{z} \right)
  - (1 - \beta) \left( \DP{\pi^{\tau}}{z} \right)
  + \left(\Dinv{\bar{c_{p}} \bar{\theta}_{v}} F_{w}^{t}\right)
\right]_{i,km(w)}
\nonumber \\
 &\equiv& E_{i,km(w)}
  \Deqlab{uwpi:w_sabun_joubu}
\end{eqnarray}
%
となるので, \Deqref{uwpi:sabun} 式の左辺は, $k(w) = km(w)$ の場合には, 
%
%
\begin{eqnarray}
\Deqref{uwpi:sabun}\mbox{左辺}
 &=&
 \pi^{\tau + \Delta \tau}_{i,km}
 \nonumber \\
 &&- \beta^{2} 
  \left(
   \frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
  \right)_{km}
  \Dinv{\Delta z}
  \left\{
   \left(
   \bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
   \right)_{km(w)}
   \left(
     \DP{\pi^{\tau + \Delta \tau}}{z} 
   \right)_{i,km(w)}
\right\}
 \nonumber \\
 &&+ \beta^{2} 
  \left(
   \frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
  \right)_{km}
  \Dinv{\Delta z}
  \left\{
   \left(
   \bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
   \right)_{km-1(w)}
   \left(
     \DP{\pi^{\tau + \Delta \tau}}{z} 
   \right)_{i,km-1(w)}
\right\}
  \nonumber \\
 &=&
 \pi^{\tau + \Delta \tau}_{i,km}
 \nonumber \\
 &&- \beta 
  \left(
   \frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
  \right)_{km}
  \Dinv{\Delta z}
   \left(
   \bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
   \right)_{km(w)}
   E_{i,km(w)}
 \nonumber \\
 &&+ \beta^{2} 
  \left(
   \frac{\bar{c}^{2}{\Delta \tau}^{2}}
        {\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
  \right)_{km}
  \Dinv{\Delta z}
  \left\{
   \left(
   \bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
   \right)_{km-1(w)}
   \left(
    \frac{\pi^{\tau + \Delta \tau}_{i,km}
          - \pi^{\tau + \Delta \tau}_{i,km-1}}{\Delta z}
   \right)
\right\}
  \nonumber \\
 &=&
  \left\{
   1 + 
  \beta^{2} 
  \left(
   \frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
  \right)_{km}
  \Dinv{\Delta z^{2}}
   \left(
   \bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
   \right)_{km-1(w)}
  \right\}
    \pi^{\tau + \Delta \tau}_{i,km}
  \nonumber \\
 &&+ \left\{
  - \beta^{2} 
  \left(
   \frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho}
   \bar{\theta}_{v}^{2}} 
  \right)_{km}
  \Dinv{\Delta z^{2}}
   \left(
   \bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
   \right)_{km-1(w)}
  \right\}
     \pi^{\tau + \Delta \tau}_{i,km-1}
  \nonumber \\
 &&- \beta 
  \left(
   \frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
  \right)_{km}
  \Dinv{\Delta z}
   \left(
   \bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
   \right)_{km(w)}
   E_{i,km(w)}
 \nonumber 
\end{eqnarray}
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