% 表題  2 次元非弾性流体モデル deepconv-mars 解説: 離散モデル編
%       第 2 節  大気モデルの離散化
%
% 履歴  2001/11/01  小高正嗣
%       2007/04/25  小高正嗣: 英語版作成
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\section{\DselJ{大気モデル}{Atmospheric model}}

\subsection{\DselJ{運動方程式}{Equation of motion}} \Dseclab{離散化運動方程式}

\DselJ{
第 1 部に示した式(1)$\sim$(3)は以下のように変形した後に離散化する.
}
{
Before making finite difference equations, 
   equation (1)$\sim$(3) show in Part I are transformed as follows.
}
\begin{eqnarray*}
 \DP{u}{t} &=& -\DP{\hat{P}}{x} + \alpha , \\
 \DP{v}{t} &=& \beta , \\
 \DP{w}{t} &=& -\DP{\hat{P}}{z} 
\end{eqnarray*}
\DselJ{
ただし,
}
{
where,
}
\begin{eqnarray*}
 \hat{P} &\equiv& c_{p}\Theta _{0}\pi - \gamma , \\
 \alpha  &\equiv& - u\DP{u}{x} - w\DP{u}{z} - fv + D(u) - \DP{\beta }{x}, \\
 \beta   &\equiv& - u\DP{v}{x} - w\DP{v}{z} + fu + D(v), \\
 \gamma   &\equiv& \int _{0}^{z}
                  \left(- u\DP{w}{x} - w\DP{w}{z} + g\frac{\theta}{\Theta _{0}}
  + D(w)\right) \Dd z
\end{eqnarray*}
\DselJ{
である. このように変形するのはモデル上下端において $\pi $ の境界条件を時
間変化させないようにするためである.
}
{
In this formulation, the time dependence of upper and lower boundary
   conditions are disappeared.
}

\DselJ{
これらを以下のように離散化する. 移流項 $\mbox{D[UVW]ADV}$ は保存型と移流
型の混合型で計算する. 摩擦項 $[\mbox{D[UVW]VIS}]_{i+\frac{1}{2},j}^{N}$,
$[\mbox{D[UVW]NLV}]_{i+\frac{1}{2},j}^{N}$ の時間積分は前進差分, その他の
項の積分は leap frog スキームと前進差分の組み合わせを用いて行う. 圧力項 
$\hat{P}$ の導出について\Dsecref{離散化圧力診断式}を参照.
}
{
The advection terms $\mbox{D[UVW]ADV}$ are evaluated by the 
   combination scheme of flux and advection forms.
The time integration is performed by the forward scheme for the
   friction term $[\mbox{D[UVW]VIS}]_{i+\frac{1}{2},j}^{N}$,
   $[\mbox{D[UVW]NLV}]_{i+\frac{1}{2},j}^{N}$, and combination of
   the leap-frog and forward scheme for the other terms.
The calculation method of pressure term $\hat{P}$ are shown in 
  \Dsecref{離散化圧力診断式}.
}
\begin{eqnarray}
  u_{i+\frac{1}{2},j}^{n+1} 
   &=& u_{i+\frac{1}{2},j}^{N} + dt \left\{
         \frac{\hat{P}_{i+1,j}-\hat{P}_{i,j}}{\Delta x}
       + \alpha _{i+\frac{1}{2},j} \right\}, \Deqlab{eq: 運動量保存1} \\
  v_{i+\frac{1}{2},j}^{n+1} 
  &=& v_{i+\frac{1}{2},j}^{N} 
     + dt \beta _{i+\frac{1}{2},j}^{n},
     \Deqlab{eq: 運動量保存2}\\
  w_{i,j+\frac{1}{2}}^{n+1} 
  &=& w_{i,j+\frac{1}{2}}^{N} + dt
      \frac{\hat{P}_{i,j+1}-\hat{P}_{i,j}}{\Delta z_{j+\frac{1}{2}}}.
  \Deqlab{eq: 運動量保存3}
\end{eqnarray}

\begin{equation}
 N = \left\{
 \begin{array}{lcl}
   n-1 & \mbox{for} & \mbox{leap frog}, \\
   n   & \mbox{for} & \mbox{forward}, 
 \end{array}
 \right. \quad
 dt = \left\{
 \begin{array}{lcl}
   2\Delta t & \mbox{for} & \mbox{leap frog}, \\
   \Delta t   & \mbox{for} & \mbox{forward}. 
 \end{array}
 \right.
\end{equation}

\begin{eqnarray}
 \alpha _{i+\frac{1}{2},j}  
  &=& 
   - ( \gamma _{i+1,j} - \gamma _{i,j}) 
   + [\mbox{DUADV}]_{i+\frac{1}{2},j}^{n} 
   + [\mbox{DUPRS}]_{i+\frac{1}{2},j}^{n} 
   + [\mbox{DUCOLI}]_{i+\frac{1}{2},j}^{n} \nonumber \\
  &&
   + [\mbox{DUVIS}]_{i+\frac{1}{2},j}^{N} 
   + [\mbox{DUNLV}]_{i+\frac{1}{2},j}^{N} , \\
 \beta _{i+\frac{1}{2},j}^{n} 
  &=& 
     [\mbox{DVADV}]_{i+\frac{1}{2},j}^{n} 
   + [\mbox{DVCOLI}]_{i+\frac{1}{2},j}^{n} 
   + [\mbox{DVVIS}]_{i+\frac{1}{2},j}^{N} 
   + [\mbox{DVNLV}]_{i+\frac{1}{2},j}^{N}, \\
 \gamma _{i,j} 
  &=& \sum _{j'=0}^{j}
   \left(
     [\mbox{DWADV}]_{i,j'-\frac{1}{2}}^{n}  
   + [\mbox{DWVIS}]_{i,j'-\frac{1}{2}}^{N}  
   + [\mbox{DWNLV}]_{i,j'-\frac{1}{2}}^{N}  
   \right.
   \nonumber \\
  && 
   \left.
   + [\mbox{BUOY}]_{i,j'-\frac{1}{2}}^{n}  
   \right)\Delta z_{j'-\frac{1}{2}} 
\end{eqnarray}
      
\begin{eqnarray}
  \mbox{DUADV}_{i+\frac{1}{2},j}^{n} 
  &=& - \frac{1}{\Delta x}\left\{ 
	\left(u_{i+1,j}^{n}u_{i+1,j}^{n} - u_{i-1,j}^{n}u_{i-1,j}^{n}\right) 
        \right. \nonumber \\
  & & + 
      \left.
      \left( 
        \rho _{0,j+\frac{1}{2}}
        u_{i+\frac{1}{2},j+\frac{1}{2}}^{n}w_{i+\frac{1}{2},j+\frac{1}{2}}^{n} 
      - \rho _{0,j-\frac{1}{2}}
        u_{i+\frac{1}{2},j-\frac{1}{2}}^{n}w_{i+\frac{1}{2},j-\frac{1}{2}}^{n}
      \right)
      \Delta x\right/(\rho _{0,j}\Delta z_{j}) \nonumber \\
  && \left. 
     - u_{i+\frac{1}{2},j}^{n}
       \left(\Ddiv \rho _{0}\Dvect{v}/\rho _{0}\right)
        _{i+\frac{1}{2},j}^{n}
      \right\}, \Deqlab{DUADV}\\
  \mbox{DVADV}_{i+\frac{1}{2},j}^{n} 
  &=& - \left\{ 
	\left(v_{i+1,j}^{n}u_{i+1,j}^{n} - v_{i-1,j}^{n}u_{i-1,j}^{n}\right) 
        \right. \nonumber \\
  & & + 
      \left.
      \left( 
        \rho _{0,j+\frac{1}{2}}
        v_{i+\frac{1}{2},j+\frac{1}{2}}^{n}w_{i+\frac{1}{2},j+\frac{1}{2}}^{n} 
      - \rho _{0,j-\frac{1}{2}}
        v_{i+\frac{1}{2},j-\frac{1}{2}}^{n}w_{i+\frac{1}{2},j-\frac{1}{2}}^{n}
      \right)
      \Delta x\right/(\rho _{0,j}\Delta z_{j}) \nonumber \\
  && \left. 
     \left. 
     - v_{i+\frac{1}{2},j}^{n}
       \left(\Ddiv \rho _{0}\Dvect{v}/\rho _{0}\right)_{i+\frac{1}{2},j}^{n}
     \right\} 
     \right/ \Delta x, \Deqlab{eq: DVADV}\\
  \mbox{DWADV}_{i,j+\frac{1}{2}}^{n}   
  &=& - \frac{1}{\Delta x}\left\{ 
	\left(w_{i+\frac{1}{2},j+\frac{1}{2}}^{n}
              u_{i+\frac{1}{2},j+\frac{1}{2}}^{n} 
            - w_{i-\frac{1}{2},j+\frac{1}{2}}^{n}
              u_{i-\frac{1}{2},j+\frac{1}{2}}^{n}
	\right) 
        \right. \nonumber \\
  & & + \left.
        \left( \rho _{0,j+1}w_{i,j+1}^{n}w_{i,j+1}^{n} 
            -  \rho _{0,j}w_{i,j}^{n}w_{i,j}^{n}
        \right)
        \Delta x \right/
        (\rho _{0,j+\frac{1}{2}}\Delta z_{j+\frac{1}{2}}) \nonumber \\
  & & \left. 
      - w_{i,j+\frac{1}{2}}^{n}
        \left(\Ddiv \rho _{0}\Dvect{v}/\rho _{0,j}\right)
        _{i,j+\frac{1}{2}}^{n}
      \right\}, \Deqlab{DWADV} \\
  & & \nonumber \\
\mbox{DUVIS}_{i+\frac{1}{2},j}^{n} 
  &=& \frac{1}{(\Delta x)^{2}}\left\{ 
      \left[  K_{i+1,j}^{n}
	      \left(u_{i+\frac{3}{2},j}^{n}-u_{i+\frac{1}{2},j}^{n}\right) 
             -K_{i,j}^{n}
	      \left(u_{i+\frac{1}{2},j}^{n}-u_{i-\frac{1}{2},j}^{n}\right)
      \right]
      \right. \nonumber \\
  & & + \frac{(\Delta x)^{2}}{\rho _{0,j}\Delta z_{j}}
      \left[ \left.
      \rho _{0,j+\frac{1}{2}}
      K_{i+\frac{1}{2},j+\frac{1}{2}}^{n}
      \left(u_{i+\frac{1}{2},j+1}^{n}-u_{i+\frac{1}{2},j}^{n}\right) 
             \right/\Delta z_{j+\frac{1}{2}} -
      \right. \nonumber \\
  & &  
      \left. 
      \left. 
      \left. \rho _{0,j-\frac{1}{2}}
        K_{i+\frac{1}{2},j-\frac{1}{2}}^{n}
 	\left(u_{i+\frac{1}{2},j}^{n}-u_{i+\frac{1}{2},j-1}^{n}\right) 
             \right/\Delta z_{j-\frac{1}{2}} 
      \right]\right\}, \Deqlab{eq: DUVIS}\\
  \mbox{DVVIS}_{i+\frac{1}{2},j}^{n} 
  &=& \frac{1}{(\Delta x)^{2}}
      \left\{ 
      \left[  K_{i+1,j}^{n}
	      \left(v_{i+\frac{3}{2},j}^{n}-v_{i+\frac{1}{2},j}^{n}\right) 
             -K_{i,j}^{n}
	      \left(v_{i+\frac{1}{2},j}^{n}-v_{i-\frac{1}{2},j}^{n}\right)
      \right]
      \right. \nonumber \\
  & & + \frac{(\Delta x)^{2}}{\rho _{0,j}\Delta z_{j}}
      \left[ \left.
      \rho _{0,j+\frac{1}{2}}K_{i+\frac{1}{2},j+\frac{1}{2}}^{n}  
      \left(v_{i+\frac{1}{2},j+1}^{n}-v_{i+\frac{1}{2},j}^{n}\right) 
             \right/\Delta z_{j+\frac{1}{2}} -
      \right. \nonumber \\
  & &  
      \left. 
      \left. 
      \left. \rho _{0,j-\frac{1}{2}}K_{i+\frac{1}{2},j-\frac{1}{2}}^{n}
 	\left(v_{i+\frac{1}{2},j}^{n}-v_{i+\frac{1}{2},j-1}^{n}\right) 
             \right/\Delta z_{j-\frac{1}{2}} 
      \right]   \right\}, \Deqlab{eq: DVVIS}\\
  \mbox{DWVIS}_{i,j+\frac{1}{2}}^{n} 
  &=& \frac{1}{(\Delta x)^{2}}\left\{ 
      \left[  K_{i+\frac{1}{2},j+\frac{1}{2}}^{n}
	      \left(w_{i+1,j+\frac{1}{2}}^{n}-w_{i,j+\frac{1}{2}}^{n}\right) 
             -K_{i-\frac{1}{2},j+\frac{1}{2}}^{n}
	      \left(w_{i,j+\frac{1}{2}}^{n}-w_{i-1,j+\frac{1}{2}}^{n}\right)
      \right]
      \right. \nonumber \\
  & & + \frac{(\Delta x)^{2}}{\rho _{0,j}\Delta z_{j}}
      \left[ \left.
      \rho _{0,j+1}K_{i,j+1}^{n}
      \left(w_{i,j+1}^{n}-w_{i,j}^{n}\right) 
             \right/\Delta z_{j+1} -
      \right. \nonumber \\
  & &  
      \left. 
      \left. 
      \left. \rho _{0,j}K_{i,j}^{n}
 	\left(w_{i,j+\frac{1}{2}}^{n}-w_{i,j-\frac{1}{2}}^{n}\right) 
             \right/\Delta z_{j-1} 
      \right] \right\}. \Deqlab{eq: DWVIS}
\end{eqnarray}

\begin{eqnarray}
  \mbox{DUNLV}_{i+\frac{1}{2},j}^{n} 
  &=& \left\{ 
      \left[ 
           \left(u_{i+\frac{3}{2},j}^{n}-u_{i+\frac{1}{2},j}^{n} \right)^{3} 
          -\left(u_{i+\frac{1}{2},j}^{n}-u_{i-\frac{1}{2},j}^{n} \right)^{3} 
      \right] 
      + 0.1 
      \left[ \left(u_{i+\frac{1}{2},j+1}^{n}-u_{i+\frac{1}{2},j}^{n} 
      \right)^{3} \right. \right.
      \nonumber \\
  &&  \left. 
      \left. 
      \left. - \left(u_{i+\frac{1}{2},j}^{n}-u_{i+\frac{1}{2},j-1}^{n}
      \right)^{3} 
      \right]
      \right\} 
      \right/
      (16.0 \cdot 10^{3} \cdot \rho _{0,j}\Delta z_{j}/\Delta x ), 
      \Deqlab{eq: DUNLV}\\       
  \mbox{DVNLV}_{i+\frac{1}{2},j}^{n} 
  &=& \left\{ 
      \left[ 
           \left(v_{i+\frac{3}{2},j}^{n}-v_{i+\frac{1}{2},j}^{n}\right)^{3} 
          -\left(v_{i+\frac{1}{2},j}^{n}-v_{i-\frac{1}{2},j}^{n}\right)^{3} 
      \right] 
      + 0.1 
      \left[ \left(v_{i+\frac{1}{2},j+1}^{n}-v_{i+\frac{1}{2},j}^{n} 
      \right)^{3} \right.
      \right.
      \nonumber \\
  &&  \left. 
      \left. 
      \left. - \left(v_{i+\frac{1}{2},j}^{n}-v_{i+\frac{1}{2},j-1}^{n}
      \right)^{3} 
      \right]
      \right\} 
      \right/
      (16.0 \cdot 10^{3} \cdot \rho _{0,j}\Delta z_{j}/\Delta x ),
      \Deqlab{eq: DVNLV}\\
  \mbox{DWNLV}_{i,j+\frac{1}{2}}^{n} 
  &=& \left\{ 
      \left[ 
           \left(w_{i+1,j+\frac{1}{2}}^{n}-w_{i,j+\frac{1}{2}}^{n}\right)^{3} 
          -\left(w_{i,j+\frac{1}{2}}^{n}-w_{i-1,j+\frac{1}{2}}^{n}\right)^{3} 
      \right] 
      \right.
      \nonumber \\
  &&  \left. 
      \left. 
      + 0.1 
      \left[ \left(w_{i,j++\frac{3}{2}}^{n}-w_{i,j+\frac{1}{2}}^{n}\right)^{3}
    - \left(w_{i,j+\frac{1}{2}}^{n}-w_{i,j-\frac{1}{2}}^{n}\right)^{3} 
      \right]
      \right\} 
      \right/ \nonumber \\
  &&  (16.0 \cdot 10^{3} \cdot \rho _{0,j+\frac{1}{2}}
       \Delta z_{j+\frac{1}{2}}/\Delta x ), \Deqlab{eq: DWNLV}  \\
 \mbox{BUOY}_{i,j+\frac{1}{2}}^{n}  
 &=& \frac{g}{\Theta _{0,j+\frac{1}{2}}}
     \theta _{i,j+\frac{1}{2}}^{n}, \Deqlab{eq: BUOY}\\
  \mbox{DUCOLI}_{i+\frac{1}{2},j}^{n} 
  &=& - f v_{i+\frac{1}{2},j}^{n},\Deqlab{eq: DUCORI} \\
  \mbox{DVCOLI}_{i+\frac{1}{2},j}^{n} 
  &=& + f u_{i+\frac{1}{2},j}^{n}. \Deqlab{eq: DVCORI}
\end{eqnarray}


\begin{eqnarray*}
 && u_{i+\frac{1}{2},j+\frac{1}{2}}^{n} =
 0.5\left(u_{i+\frac{1}{2},j+1}^{n} + u_{i+\frac{1}{2},j}^{n}\right), 
 \quad 
 u_{i,j}^{n} =
 0.5\left(u_{i+\frac{1}{2},j}^{n} + u_{i-\frac{1}{2},j}^{n}\right), \\
 && v_{i+\frac{1}{2},j+\frac{1}{2}}^{n} =
 0.5\left(v_{i+\frac{1}{2},j+1}^{n} + v_{i+\frac{1}{2},j-1}^{k}\right), 
 \quad
 v_{i,j}^{n} =
 0.5\left(v_{i+\frac{1}{2},j}^{n} + v_{i-\frac{1}{2},j}^{n}\right), \\
 &&
 w_{i+\frac{1}{2},j+\frac{1}{2}}^{n} =
 0.5\left(w_{i+1,j+\frac{1}{2}}^{n} + w_{i,j+\frac{1}{2}}^{n}\right), 
 \quad
 w_{i,j}^{n} =
 0.5\left(w_{i,j+\frac{1}{2}}^{n} + w_{i,j-\frac{1}{2}}^{n}\right), \\
 && 
 \theta _{i,j+\frac{1}{2}} =
 \left( -\frac{1}{16}\theta _{i,j+2}^{n}
        +\frac{9}{16}\theta _{i,j+1}^{n}
        +\frac{9}{16}\theta _{i,j-1}^{n}
        -\frac{1}{16}\theta _{i,j-2}^{n} \right) , \\
 && 
 K_{i+\frac{1}{2},j+\frac{1}{2}}^{n} =
 0.25\left(K_{i+1,j+1}^{n} + K_{i+1,j}^{n} + 
           K_{i,j+1}^{n}   + K_{i,j}^{n}\right), \\
 && \\
 && 
 \Theta _{0,j+\frac{1}{2}} =
 0.5\left(\Theta _{0,j+1}+\Theta _{0,j+1}\right), \quad
 \rho  _{0,j+\frac{1}{2}} =
 0.5\left(\rho _{0,j+1}+\rho _{0,j+1}\right), \\
 && \\
 && 
 (\Ddiv \rho _{0}\Dvect{v}/\rho _{0})_{i+\frac{1}{2},j}^{n}
 = \frac{u_{i+1,j}^{n} - u_{i,j}^{n}}{\Delta x}
 + \frac{\rho _{0,j+\frac{1}{2}}w_{i+\frac{1}{2},j+\frac{1}{2}}^{n} -
         \rho _{0,j-\frac{1}{2}}w_{i+\frac{1}{2},j-\frac{1}{2}}^{n}}
   {\rho _{0,j}\Delta z_{j}}, \\
 &&
 (\Ddiv \rho _{0}\Dvect{v}/\rho _{0})_{i,j+\frac{1}{2}}^{n}
 = \frac{u_{i+\frac{1}{2},j+\frac{1}{2}}^{n} - 
         u_{i-\frac{1}{2},j+\frac{1}{2}}^{n}}{\Delta x}
 + \frac{\rho _{0,j+1}w_{i,j+1}^{n} - \rho _{0,j}w_{i,j}^{n}}
   {\rho _{0,j+\frac{1}{2}}\Delta z_{j+\frac{1}{2}}}.
\end{eqnarray*}

\subsection{\DselJ{熱力学の式}{Thermodynamics equation}}

\DselJ{
移流項 $[\mbox{DTADV}]_{i,j}^{n}, [\mbox{DTAD0}]_{i,j}^{n}$ は 4 次中央
差分で空間離散化する. 摩擦項 $[\mbox{DTDIF}]_{i,j}^{N}$,
$[\mbox{DTDI0}]_{i,j}^{N}$ と放射加熱項 $Q_{rad,i,j}^{N}$ 
と散逸加熱項 $Q_{dis,i,j}^{N}$ の時間積分は常に前進差分を用いて行う. 
放射加熱項の計算方法は\Dsecref{離散化放射モデル}を参照.
}
{
The advection terms 
   $[\mbox{DTADV}]_{i,j}^{n}, [\mbox{DTAD0}]_{i,j}^{n}$ 
   of equation (5) in Part I are evaluated by 
   forth order centered scheme.
In time integration, the forward scheme is adapted for the
   friction terms 
   $[\mbox{DTDIF}]_{i,j}^{N}$, $[\mbox{DTDI0}]_{i,j}^{N}$, the radiative
   heating term $Q_{rad,i,j}^{N}$ and the dissipative heating term 
   $Q_{dis,i,j}^{N}$. 
The calculation method of radiative heating term is shown in 
   \Dsecref{離散化放射モデル}.
}
\begin{eqnarray}
 \theta _{i,j}^{n+1} &=& 
	\theta _{i,j}^{N} +
        dt \left\{
	\frac{\Theta _{0,j}}{T_{0,j}}(Q_{rad,i,j}^{N} + Q_{dis,i,j}^{N})
	\right.
        \nonumber \\
       && + \left.
	[\mbox{DTADV}]_{i,j}^{n} +
        [\mbox{DTAD0}]_{i,j}^{n} +
        [\mbox{DTDIF}]_{i,j}^{N} +
        [\mbox{DTDI0}]_{i,j}^{N} \right\}
\end{eqnarray}

\begin{equation}
  Q_{dis,i,j}^{N} = \frac{C_{\epsilon}}{lc_{p}}
  (\varepsilon _{i,j}^{N})^{\frac{3}{2}},
\end{equation}

\begin{eqnarray}
  &&
  \mbox{DTADV}_{i,j}^{n} 
  = - \left\{
  \frac{1}{\rho _{0,j}\Delta x}\left[
      - \frac{1}{24}F\theta _{x(i+\frac{3}{2},j)}^{n}
      + \frac{9}{8} F\theta _{x(i+\frac{1}{2},j)}^{n}
      - \frac{9}{8} F\theta _{x(i-\frac{1}{2},j)}^{n}  
      + \frac{1}{24}F\theta _{x(i-\frac{3}{2},j)}^{n} \right] 
  \right. \nonumber \\
  && +  \left.
  \frac{1}{\rho _{0,j}\Delta z_{j}}\left[
      - \frac{1}{24}F\theta _{z(i,j+\frac{3}{2})}^{n}
      + \frac{9}{8} F\theta _{z(i,j+\frac{1}{2})}^{n}
      - \frac{9}{8} F\theta _{z(i,j-\frac{1}{2})}^{n}  
      + \frac{1}{24}F\theta _{z(i,j-\frac{3}{2})}^{n} \right] \right\}, 
  \Deqlab{スカラー移流の離散表現} \\
  &&
  F\theta _{x(i+\frac{1}{2},j)}^{n} 
  = 
  \rho _{0,j}u_{i+\frac{1}{2},j}^{n}
  \left( - \frac{1}{16}\theta _{i+2,j}^{n} 
         + \frac{9}{16}\theta _{i+1,j}^{n} 
         + \frac{9}{16}\theta _{i,j}^{n} 
         - \frac{1}{16}\theta _{i-1,j}^{n}  \right), \nonumber \\
  &&
  F\theta _{z(i,j+\frac{1}{2})}^{n} 
  = 
  \rho _{0,j+\frac{1}{2}}w_{i,j+\frac{1}{2}}^{n}
  \left( - \frac{1}{16}\theta _{i,j+2}^{n} 
         + \frac{9}{16}\theta _{i,j+1}^{n} 
         + \frac{9}{16}\theta _{i,j}^{n} 
         - \frac{1}{16}\theta _{i,j-1}^{n}  \right). \nonumber 
\end{eqnarray}
  
\begin{eqnarray}
  &&
  \mbox{DTAD0}_{i,j}^{n} 
  = - \left\{
  \frac{1}{\rho _{0,j}\Delta x}\Theta _{0,j}\left[
      - \frac{1}{24}F_{x(i+\frac{3}{2},j)}^{n}
      + \frac{9}{8} F_{x(i+\frac{1}{2},j)}^{n}
      - \frac{9}{8} F_{x(i-\frac{1}{2},j)}^{n}  
      + \frac{1}{24}F_{x(i-\frac{3}{2},j)}^{n} \right] 
  \right. \nonumber \\
  &&
  + \left.
  \frac{1}{\rho _{0,j}\Delta z_{j}}\left[
      - \frac{1}{24}F\Theta _{0,z(j+\frac{3}{2})}^{n}
      + \frac{9}{8} F\Theta _{0,z(j+\frac{1}{2})}^{n}
      - \frac{9}{8} F\Theta _{0,z(j-\frac{1}{2})}^{n}  
      + \frac{1}{24}F\Theta _{0,z(j-\frac{3}{2})}^{n} \right] \right\}, \\
  &&  
  F_{x(i+\frac{1}{2},j)}^{n} 
  = 
  \rho _{0,j}u_{i+\frac{1}{2},j}^{n},\nonumber \\
  &&
  F\Theta _{0,z(j+\frac{1}{2})}^{n} 
  = 
  \rho _{0,j+\frac{1}{2}}w_{i,j+\frac{1}{2}}^{n}
  \left( - \frac{1}{16}\Theta _{0,j+2} 
         + \frac{9}{16}\Theta _{0,j+1}
         + \frac{9}{16}\Theta _{0,j}
         - \frac{1}{16}\Theta _{0,j-1} \right). \nonumber 
\end{eqnarray}

\begin{eqnarray}
 \mbox{DTDIF}_{i,j}^{n}
 &=& \frac{1}{(\Delta x)^{2}}
 \left[
     \tilde{K}_{i+\frac{1}{2},j}^{n}(\theta _{i+1,j}^{n}-\theta _{i,j}^{n})
    -\tilde{K}_{i-\frac{1}{2},j}^{n}(\theta _{i,j}^{n}  -\theta _{i-1,j}^{n})
 \right] +\nonumber \\
 & &
 \frac{1}{\rho _{0,j}\Delta z_{j}}
 \left[ \rho _{0,j+\frac{1}{2}}
        \tilde{K}_{i,j+\frac{1}{2}}^{n}
        \frac{
        (\theta _{i,j+1}^{n}-\theta _{i,j}^{n})}{\Delta z_{j+\frac{1}{2}}}
      - \rho _{0,j-\frac{1}{2}}
        \tilde{K}_{i,j-\frac{1}{2}}^{n}
        \frac{
        (\theta _{i,j}^{n}  -\theta _{i,j-1}^{n})}{\Delta z_{j-\frac{1}{2}}}
 \right], \Deqlab{スカラー拡散の離散表現}
\end{eqnarray}

\begin{equation}
 \mbox{DTDI0}_{i,j}^{n}
 =
 \frac{1}{\rho _{0,j}\Delta z_{j}}
 \left[
     \rho _{0,j+\frac{1}{2}}\tilde{K}_{i,j+\frac{1}{2}}^{n}
     \frac{
     (\Theta _{0,j+1}^{n}-\Theta _{0,j}^{n})}{\Delta z_{j+\frac{1}{2}}}
   - \rho _{0,j-\frac{1}{2}}\tilde{K}_{i,j-\frac{1}{2}}^{n}
     \frac{
     (\Theta _{0,j}^{n}  -\Theta _{0,j-1}^{n})}{\Delta z_{j-\frac{1}{2}}}
 \right],
\end{equation}

\[
 \tilde{K}_{i+\frac{1}{2},j}^{n} 
 = 0.5(\tilde{K}_{i+1,j}^{n} +\tilde{K}_{i,j}^{n} ), \quad
 \tilde{K}_{i,j+\frac{1}{2}}^{n} 
 = 0.5(\tilde{K}_{i,j+1}^{n} +\tilde{K}_{i,j}^{n} ).
\]

\subsection{\DselJ{圧力診断式}{Diagonostic equation of pressure function}} \Dseclab{離散化圧力診断式}

\DselJ{
圧力の診断式\Deqref{圧力診断式}を\Dsecref{離散化運動方程式}で行った変形
にあわせて以下のように変形する.}
{
The diagnostic equation of nodimensional pressure function 
   is solved by using the
   dimension reduction method.
Before making the finite difference equation,
   equation (9) show in Part I is transformed as follows.
}
\[
  \left(\DP[2]{}{x} +\frac{1}{\rho _{0}}\DP{}{z}\rho _{0}\DP{}{z}\right)
  P = \DP{}{t}\left(\frac{1}{\rho _{0}}\Ddiv \rho _{0}\Dvect{v}\right) 
  + \DP{}{x}\alpha ,
\]
\DselJ{
これを dimension-reduction 法を用いて解く. 上の式を適当に離散化すると,
}
{
The finite difference form of the pressure equation can be written in 
   matrix form as follows.
}       
\begin{equation}
   \Dvect{D_{x}P} + \Dvect{D_{z}P} = \Dvect{S}
   \Deqlab{圧力方程式の行列表現}
\end{equation}
\DselJ{
と書くことができる. ただし $\Dvect{P}, \Dvect{D_{x}}, \Dvect{D_{z}},
\Dvect{S}$ はそれぞれ, 
}
{
where $\Dvect{P}, \Dvect{D_{x}}, \Dvect{D_{z}},
   \Dvect{S}$ are matrixes whose elements are finite
   difference form of following terms.
}
\[
  \hat{P}, \quad
  \DP[2]{}{x}, \quad
  \frac{1}{\rho _{0}}\DP{}{z}\left(\rho _0\DP{}{z}\right), \quad
  \DP{}{t}\left(\frac{1}{\rho _{0}}\Ddiv \Dvect{v}\right) 
  + \DP{}{x}\alpha ,      
\]
\DselJ{
を離散化した行列である. 行列 $\Dvect{D_{z}}$ の固有値を $\lambda_{i}$, 
固有ベクトルを $V_{i}(z)$ とし, 固有列ベクトルを並べた行列を $\Dvect{V}$, 
対角成分が $\lambda _{i}$ である行列を $\Dvect{\Lambda }$ とすると,
$\Dvect{D_{z}V} = \Dvect{V\Lambda }$ となる. $\Dvect{P} = \Dvect{V}\cdot
\Dvect{H}$ と展開すると, 
}
{
\Deqref{圧力方程式の行列表現}
   can be rewritten by using the eigenvalue
   matrix $\Dvect{\Lambda }$ and the eigenvector matrix 
   $\Dvect{V}$ of $\Dvect{D_{z}}$.
}
\[
  \Dvect{V}\Dvect{D_{x}H} + \Dvect{V\Lambda H} = \Dvect{S}.
\]
\DselJ{  
したがって,
}
{
where $\Dvect{D_{z}V} = \Dvect{V\Lambda }$ and 
   $\Dvect{P} = \Dvect{V}\cdot\Dvect{H}$.
The final form of matrix equation is as follows.
}
\begin{equation}
  \left(\Dvect{D_{x}} + \Dvect{\Lambda }\right) \Dvect{H}
  = \Dvect{V}^{-1}\Dvect{S},
\end{equation}
\DselJ{    
となる. 
}{
}

\DselJ{
固有値を $\lambda_{i}$ と固有ベクトルを $V_{i}(z)$ を求めるために必要な
$z$ 方向の係数行列 $\Dvect{D}_{z}$ は
}
{
In calculating elements of matrix $\Dvect{D}_{z}$, the vertical
   derivative in
}       
\[
  \frac{1}{\rho _{0}}\DP{}{z}\rho _{0}\DP{}{z} \pi 
\]
\DselJ{    
を差分化して与える. 最初の微分は 2 次中央差分, 次の微分は 4 次中央差分で
解く. これは連続の式は 4 次中央差分, 圧力の微分は 2 次中央差分であること
による. よって係数行列は 5重対角行列になる. 係数行列の成分を $A_{ij}$ と
すると以下のようになる.}
{
are evaluated by the second and forth order centered schemes because
   the space differencing in the continuity equation is evaluated
   by the forth order centered scheme while that in the  pressure gradient
   term is evaluated by the second order centered scheme.
Therefore $\Dvect{D}_{z}$ is represented as a band matrix whose
   elements $A_{i,j}$ are given as follows.
}
\begin{eqnarray}
          A_{i,i+2} &=& -\frac{1}{\rho _{0}\Delta z_{i}}\left(
            \frac{1}{24\Delta z_{i+\frac{3}{2}}}\frac{(\rho _
            {0,i+2}+\rho _{0,i+1})}{2}\right), \\
          A_{i,i+1} &=& \frac{1}{\rho _{0}\Delta z_{i}}\left(
            \frac{1}{24\Delta z_{i+\frac{3}{2}}}\frac{(\rho _
            {0,i+2}+\rho _{0,i+1})}{2} + 
            \frac{9}{8\Delta m_{i+\frac{1}{2}}}\frac{(\rho _
            {0,i+1}+\rho _{0,i})}{2}\right), \\
          A_{i,i} &=& - \frac{1}{\rho _{0}\Delta z_{i}}\left(
            \frac{9}{8\Delta zm_{i+\frac{1}{2}}}\frac{(\rho _
            {0,i+1}+\rho _{0,i})}{2} 
            + \frac{9}{8\Delta z_{i-\frac{1}{2}}}\frac{(\rho _
            {0,i}+\rho _{0,i-1})}{2}\right), \\
          A_{i+1,i} &=& \frac{1}{\rho _{0}\Delta z_{i}}\left(
            \frac{1}{24\Delta z_{i-\frac{3}{2}}}\frac{(\rho _
            {0,i-1}+\rho _{0,i-2})}{2} + 
            \frac{9}{8\Delta z_{i-\frac{1}{2}}}\frac{(\rho _
            {0,i}+\rho _{0,i-1})}{2}\right), \\
          A_{i+2,i} &=& -\frac{1}{\rho _{0}\Delta z_{i}}\left(
            \frac{1}{24\Delta z_{i-\frac{3}{2}}}\frac{(\rho _
          {0,i-1}+\rho _{0,i-2})}{2} \right).
\end{eqnarray}
\DselJ{        
境界条件は上下壁で $\DP{}{z}=0$ となるようにする.
}
{
The boundary conditions are $\DP{}{z}=0$ at the lower and upper boundary.
}

\DselJ{
$x$ 方向についても適当な固有関数に展開して各モードに対する展開係数
を求める. ここでは三角関数で展開する. 
}
{
The horizontal differencing is evaluated by using the Fourier expansion.
}
\begin{eqnarray}
  \Dvect{H} &=& \sum_{k=1}^{NX/2-1} 
     \left[\Dvect{H}\right]_{k_{x}},\\
      \Dvect{V}^{-1}\Dvect{S}
      &=& \sum_{k=1}^{NX/2-1}
          \left[\Dvect{V}^{-1}\Dvect{S}\right]_{k_{x}}, \\
      \left(\Dvect{D_{x}} + \Dvect{\Lambda }\right)
      &=& 
      \sum_{k_{x}=1}^{NX/2-1}
      \left[\Dvect{D_{x}} + \Dvect{\Lambda }\right]_{k_{x}}
\end{eqnarray}
\begin{equation}
      \left[\Dvect{H}\right]_{k_{x}} = 
      \left[\Dvect{D_{x}} + \Dvect{\Lambda }\right]_{k_{x}}^{-1}
      \left[\Dvect{V}^{-1}\Dvect{S}\right]_{k_{x}}
\end{equation}

\subsection{\DselJ{基本場}{Basic state equations}}

\DselJ{
温度分布 $T_{j}$ を与え, 静水圧平衡の式と状態方程式を用いて $P_{0,j}$ 
と $\rho _{0,j}$ を計算する. }
{
The basic state pressure ($P_{0,j}$) and density ($\rho_{0,j}$)
   are calculated by the hydrostatic equation and the equation of state
   given the basic state
   temperature $T_{0,j}$.
}
\begin{eqnarray}
  && \ln P_{0,j} = \ln P_{00}
		 - \sum _{j=1}^{j}\frac{g}{RT_{0,j}}\Delta z_{j}, \\
  && \rho _{0,j} = \frac{P_{0,j}}{RT_{0,j}}. 
\end{eqnarray}
\DselJ{
$P_{0,j}, \rho _{0,j}$ を求めた後 $\Pi _{0,j}, \Theta _{0,j}$ を計算する. 
}
{
$\Pi_{0,j}$, $\Theta _{0,j}$ are calculated by using 
   $P_{0,j}$, $\rho _{0,j}$ as follows.
}
\begin{eqnarray}
  && \Pi _{0,j}    = \left(\frac{P_{0,j}}{P_00}\right)^{\kappa }, \\
  && \Theta _{0,j} = \frac{T_{0,j}}{\Pi _{0,j}}.
\end{eqnarray}

