%  
%  $BMzNr(B  2004/08/14  $B?y;39L0lO/(B: $B:n@.3+;O(B
%        2004/09/21  $B>.9b@5;L(B, $BKL<iB@0l(B
%
\chapter{$B1?F0J}Dx<0$H05NOJ}Dx<0$N;~4VJ}8~$NN%;62=(B}

$B6u4VN%;62=$5$l$?1?F0J}Dx<0(B\Deqref{$B6u4VN%;62=$5$l$?(B x $BJ}8~1?F0J}Dx<0(B},
\Deqref{$B6u4VN%;62=$5$l$?(B x $BJ}8~1?F0J}Dx<0(B}$B$H05NOJ}Dx<0(B
\Deqref{$B6u4VN%;62=$5$l$?05NOJ}Dx<0(B}$B$r;~4VJ}8~$KN%;62=$9$k(B. 
$B2;GH$K4XO"$9$k9`$OC;$$%?%$%`%9%F%C%W(B $\Delta \tau$ $B$GN%;62=$7(B, $B$=$NB>(B
$B$N9`$OD9$$%?%$%`%9%F%C%W(B $\Delta t$ $B$GN%;62=$9$k(B. $B2;GH$K4XO"$9$k9`$NN%(B
$B;62=$K$O(B HE-VI $BK!$r:NMQ$7(B, $u$ $B$N<0$OA0?J:9J,(B, $w, \pi$ $B$N<0$O8eB`:9J,(B
($B%/%i%s%/!&%K%3%k%=%sK!(B)$B$GN%;62=$9$k(B. $B$=$NB>$N9`$NN%;62=$K$O%j!<%W%U%m%C(B
$B%0K!$rMQ$$$k(B. $BN%;62=$7$?<0$N7W;;$O$^$:(B $u$ $B$N<0$+$i9T$&(B. $BF@$i$l$?(B 
$\tau +\Delta \tau$ $B$N(B $u$ $B$rMQ$$$F(B $\pi$ $B$r7W;;$7(B, $u, \pi$ $B$rMQ$$$F(B 
$w$ $B$r7W;;$9$k(B.


$B2r$/J}Dx<0$r0J2<$N$h$&$K=q$-D>$9(B. 
%
\begin{eqnarray}
&& \DP{u_{i(u),k}}{t} = - \left[\bar{c_{p}} \bar{\theta}_{v} 
                       \DP{(\pi - \alpha Div )}{x}\right]_{i(u),k} +
  F_{u,i(u),k},
  \Deqlab{uwpi:u}\\
&& \DP{w_{i,k(w)}}{t} = - \left[\bar{c_{p}} \bar{\theta}_{v} 
                       \DP{(\pi - \alpha Div )}{z}\right]_{i,k(w)} +
  F_{w,i,k(w)},
  \Deqlab{uwpi:w}\\
&& \DP{\pi_{i,k}}{t}  
  + \left[\frac{\bar{c}^{2}}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
  \DP{(\bar{\rho} \bar{\theta}_{v} w)}{z}\right]_{i,k}
  = 
  -  \left[\frac{\bar{c}^{2}}{\bar{c_{p}} \bar{\theta}_{v}} \DP{u}{x}\right]_{i,k}.
  \Deqlab{uwpi:pi}
\end{eqnarray}
%
$B$?$@$7(B $u, w$ $B$N<0$K$O2;GH8:?j9`(B(reference)$B$r2C$($F$"$k(B. 
$B$3$3$G(B $F_{u}, F_{w}$ $B$O2;GH$K4XO"$7$J$$9`(B, 
%
 \begin{eqnarray}
  F_{u,i(u),k} &=&
  - u_{i(u),k}\left[\DP{u}{x}\right]_{i(u),k}
  - w_{i(u),k}\left[\DP{u}{z}\right]_{i(u),k}
  + \left[D_{u}\right]_{i(u),k} 
  \\
  F_{w,i,k(w)} &=&
  - 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 \\
  &&
  + g\frac{\theta_{i,k(w)}}{\overline{\theta}_{i,k(w)}}
  + \left[D_{w}\right]_{i,k(w)}.
 \end{eqnarray}
%
$B$G$"$j(B, $B;~9o(B $t$ $B$NCM$GI>2A$9$k(B.

\section{$u$ $B$N<0$N;~4VJ}8~$NN%;62=(B}

\Deqref{uwpi:u}$B$r;~4VJ}8~$KN%;62=$9$k$H0J2<$N$h$&$K$J$k(B. 
%
\begin{eqnarray}
 u^{\tau + \Delta \tau}_{i(u),k}
  =  u^{\tau}_{i(u), k}
  -   \left[
  \bar{c_{p}} \bar{\theta}_{v} \Delta \tau
  \left\{
   \DP{\pi^{\tau}}{x} 
   - \DP{(\alpha Div)^{\tau}}{x} 
  \right\}
 \right]_{i(u),k}
  + 
     F_{u,i(u),k}^{t} \Delta \tau
  \Deqlab{uwpi:u_sabun}
\end{eqnarray}
%

\section{$w, \pi$ $B$N<0$N;~4VJ}8~$NN%;62=(B}

HE-VI $BK!$rMQ$$$k$N$G(B, $w$ $B$H(B $\pi$ $B$N<0$rO"N)$7$F2r$/(B.  $w$ $B$N<0$K$*$$(B
$B$F2;GH8:?j9`$OA0?J:9J,(B, $B05NO9`$O8eB`:9J,$GN%;62=$9$k(B.  $\pi$ $B$N<0$K$*(B
$B$$$F?eJ?HyJ,9`$O(B\Deqref{uwpi:u_sabun}$B$G5a$a$?(B $u^{\tau +\Delta \tau}$ 
$B$rMQ$$$FN%;62=$7(B, $B1tD>HyJ,9`$O8eB`:9J,$GN%;62=$9$k(B.
%
\begin{eqnarray}
 w^{\tau + \Delta \tau} 
  =  w^{\tau}
  - \bar{c_{p}} \bar{\theta}_{v} \Delta \tau
  \left\{
   \beta \DP{\pi^{\tau + \Delta \tau}}{z} 
   + (1 - \beta) \DP{\pi^{\tau}}{z} 
   - \DP{(\alpha Div)^{\tau}}{z} 
  \right\}
 + F_{w}^{t} \Delta \tau.
 \Deqlab{uwpi:w_sabun}
\end{eqnarray}
%
\begin{eqnarray}
% \frac{\pi^{\tau + \Delta \tau} - \pi^{\tau}}{\Delta \tau} 
%  + \frac{\bar{c}^{2}}{\bar{c_{p}} \bar{\rho} {\bar{\theta}_{v}}^{2}}
%  \left\{
%   \beta \DP{(\bar{\rho} \bar{\theta}_{v} w^{\tau + \Delta \tau})}{z}
%   + (1 - \beta) \DP{(\bar{\rho} \bar{\theta}_{v} w^{\tau})}{z}
%  \right\}
%  = 
%  -  \frac{\bar{c}^{2}}{\bar{c_{p}} \bar{\theta}_{v}} \DP{u^{\tau +
%  \Delta \tau}}{x}, 
%  \nonumber \\
%
 \pi^{\tau + \Delta \tau} 
 + \beta \frac{\bar{c}^{2}\Delta \tau}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
   \DP{(\bar{\rho} \bar{\theta}_{v} w^{\tau + \Delta \tau})}{z}
  = 
  \pi^{\tau}
  - (1 - \beta) \frac{\bar{c}^{2}\Delta \tau}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
    \DP{(\bar{\rho} \bar{\theta}_{v} w^{\tau})}{z}
  -  \frac{\bar{c}^{2} \Delta \tau}{\bar{c_{p}} \bar{\theta}_{v}} \DP{u^{\tau + \Delta \tau}}{x}.
 \nonumber \\ 
 \Deqlab{uwpi:pi_sabun}
\end{eqnarray}
%
$B$3$3$G$O4JC1$N$?$a3J;RE@0LCV$rI=$9E:;z$O>JN,$7$?(B. 
%
\Deqref{uwpi:pi_sabun} $B<0$K(B \Deqref{uwpi:w_sabun} $B$rBeF~$7$F(B $w^{\tau +
\Delta \tau}$ $B$r>C5n$9$k(B. 
%
\begin{eqnarray}
 \pi^{\tau + \Delta \tau} 
  &-& 
  \beta^{2} 
  \frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho} 
  \bar{\theta}_{v}^{2}}
  \DP{}{z}
  \left\{
   \left(\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}\right)
   \left(
     \DP{\pi^{\tau + \Delta \tau}}{z} 
   \right)
\right\}
  \nonumber \\
 &=& 
  \pi^{\tau}
  -(1 - \beta) 
 \frac{\bar{c}^{2}\Delta \tau}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
   \DP{(\bar{\rho} \bar{\theta}_{v} w^{\tau})}{z}
  -  \frac{\bar{c}^{2} \Delta \tau}{\bar{c_{p}} \bar{\theta}_{v}} 
  \DP{u^{\tau + \Delta \tau}}{x}
  \nonumber \\
 &&- \beta  \frac{\bar{c}^{2}\Delta \tau}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
   \left[
    \DP{}{z}
    \bar{\rho} \bar{\theta}_{v}
   \left\{
   w^{\tau}
  - \bar{c_{p}} \bar{\theta}_{v} \Delta \tau
  \left\{
    (1 - \beta) \DP{\pi^{\tau}}{z} 
   - \DP{(\alpha Div)^{\tau}}{z} 
  \right\}
 + F_{w}^{t} \Delta \tau   
\right\}
\right].
  \nonumber \\
\Deqlab{uwpi:sabun}
\end{eqnarray}
%
%$B$3$NJ}Dx<0$r2r$$$F(B $\pi^{\tau + \Delta \tau}$ $B$r5a$a$k$o$1$@$,(B, $B:8JU$N6u(B
%$B4VHyJ,$rM[$K=q$-2<$9$3$H$G(B, $B4JC1$J9TNs<0$rF3$/$3$H$,$G$-$k(B. 
\Deqref{uwpi:sabun} $B<01&JU$r6u4VJ}8~$KN%;62=$7(B, 
$B3J;RE@0LCV$rI=$9E:;z$rIU$1$FI=$9$H0J2<$N$h$&$K$J$k(B
($B7W;;$N>\:Y$O(B \Dsecref{appendix-a} $B;2>H(B). 
%
\begin{eqnarray}
&&  \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 \\   
 && \hspace{10mm}+ \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 \\   
 && \hspace{10mm}+ \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}
   \nonumber \\
 &&= 
  \pi^{\tau}_{i,k}
  - (1 - \beta)  
  \left(
   \frac{\bar{c}^{2}\Delta \tau}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
  \right)_{k}
  \left\{
    \DP{(\bar{\rho} \bar{\theta}_{v} w^{\tau})}{z}
  \right\}_{i,k}
   -\left(
     \frac{\bar{c}^{2} \Delta \tau}{\bar{c_{p}} \bar{\theta}_{v}} 
    \right)_{k}
  \left(
  \DP{u^{\tau + \Delta \tau}}{x} 
  \right)_{i,k}
  \nonumber \\
 &&\hspace{2mm} - \beta  
  \left(
   \frac{\bar{c}^{2}\Delta \tau}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
  \right)_{k}
   \left[
    \DP{}{z}
    \bar{\rho} \bar{\theta}_{v}
   \left\{
   w^{\tau}
  - \bar{c_{p}} \bar{\theta}_{v} \Delta \tau
  \left\{
    (1 - \beta) \DP{\pi^{\tau}}{z} 
   - \DP{(\alpha Div)^{\tau}}{z} 
  \right\}
 + F_{w}^{t} \Delta \tau   
\right\}
\right]_{i,k}.
\Deqlab{uwpi:sabun_ik}
\nonumber \\
\end{eqnarray}
%



\subsection{$B6-3&>r7o(B}

$B>e2<6-3&$r8GDjJI$H$9$k>l9g(B, $B6-3&>r7o$O>eIt2<It6-3&$G(B, 
%
\begin{eqnarray}
&& w(i,0(w)) = 0, \\
&& w(i,km(w)) = 0 
\end{eqnarray}
%
$B$G$"$k(B.

\subsubsection{$B2<It6-3&(B}

$B2<It6-3&(B($k(w) = 0(w)$)$B$K$D$$$F9M$($k(B. $B$3$N;~(B \Deqref{uwpi:w_sabun} $B<0$O(B, 
%
\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}
%
$B$H$J$k(B. $B$7$?$,$C$F(B \Deqref{uwpi:sabun} $B<0$O0J2<$N$h$&$K$J$k(B. 
%
\begin{eqnarray}
&&  \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}
+ 
  \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 \\
 &=& 
  \pi^{\tau}_{i,1}
 -(1 - \beta)  
  \left(
   \frac{\bar{c}^{2}\Delta \tau}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
  \right)_{1}
  \left\{
  \DP{(\bar{\rho} \bar{\theta}_{v} w^{\tau})}{z}
  \right\}_{i,1}
  -  
  \left(
   \frac{\bar{c}^{2} \Delta \tau}{\bar{c_{p}} \bar{\theta}_{v}}
  \right)_{1}
  \left(
  \DP{u^{\tau + \Delta \tau}}{x}
  \right)_{i,1}
  \nonumber \\
 &&- \beta  
  \left(
   \frac{\bar{c}^{2}\Delta \tau}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
  \right)_{1}
   \left[
    \DP{}{z}
    \bar{\rho} \bar{\theta}_{v}
   \left\{
   w^{\tau}
  - \bar{c_{p}} \bar{\theta}_{v} \Delta \tau
  \left\{
    (1 - \beta) \DP{\pi^{\tau}}{z} 
   - \DP{(\alpha Div)^{\tau}}{z} 
  \right\}
 + F_{w}^{t} \Delta \tau   
\right\}
\right]_{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)_{i,0(w)}
   E_{i,0(w)}.
\Deqlab{uwpi:sabun_kabu}
\end{eqnarray}
%

\subsubsection{$B>eIt6-3&(B}

$B>eIt6-3&(B($k(w) = km(w)$)$B$K$D$$$F9M$($k(B. $B$3$N;~(B \Deqref{uwpi:w_sabun} $B<0$O(B, 
%
\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}
%
$B$H$J$k(B. $B$7$?$,$C$F(B \Deqref{uwpi:sabun} $B<0$O0J2<$N$h$&$K$J$k(B. 
%
\begin{eqnarray}
&&  \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 \\
 &=& 
  \pi^{\tau}_{i,km}
-(1 - \beta)  
  \left(
   \frac{\bar{c}^{2}\Delta \tau}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
  \right)_{km}
  \left\{
    \DP{(\bar{\rho} \bar{\theta}_{v} w^{\tau})}{z}
  \right\}_{i,km}
-  
  \left(
   \frac{\bar{c}^{2} \Delta \tau}{\bar{c_{p}} \bar{\theta}_{v}} 
  \right)_{km}
  \left(
    \DP{u^{\tau + \Delta \tau}}{x}
  \right)_{i,km}
  \nonumber \\
 &&- \beta  
  \left(
   \frac{\bar{c}^{2}\Delta \tau}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
  \right)_{km}
   \left[
    \DP{}{z}
    \bar{\rho} \bar{\theta}_{v}
   \left\{
   w^{\tau}
  - \bar{c_{p}} \bar{\theta}_{v} \Delta \tau
  \left\{
    (1 - \beta) \DP{\pi^{\tau}}{z} 
   - \DP{(\alpha Div)^{\tau}}{z} 
  \right\}
 + F_{w}^{t} \Delta \tau   
\right\}
\right]_{i,km}
\nonumber \\
 &&+ \frac{\beta}{\Delta z}
  \left(
  \frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}  
  \right)_{km}
   \left(
   \bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
   \right)_{km(w)}
   A_{i,km(w)}.
\Deqlab{uwpi:sabun_joubu}
\end{eqnarray}
%

\subsection{$B05NOJ}Dx<0$N2r$-J}(B}

\Deqref{uwpi:sabun_ik}, \Deqref{uwpi:sabun_kabu},
\Deqref{uwpi:sabun_joubu} $B<0$rO"N)$9$k$H(B, $B0J2<$N$h$&$J9TNs<0$N7A<0$G=q$/(B
$B$3$H$,$G$-$k(B. 
%
 \begin{eqnarray}
  \left(\begin{array}{cccc}
          A_{1} & B_{2}  &        &   0      \\ 
          C_{1} & \ddots & \ddots &          \\
	 & \ddots & \ddots & B_{km}   \\
            0   &        & C_{km-1} & A_{km} \\
        \end{array}
  \right)
  \left(\begin{array}{cccc}
          \pi_{1,1} & \pi_{2,1}  & \cdots & \pi_{im,1} \\
          \pi_{1,2} & \ddots    & \ddots & \vdots        \\
          \vdots   &           & \ddots & \vdots        \\
          \pi_{1, km} & \cdots & \cdots & \pi_{im, km} \\
        \end{array}
  \right)^{\tau + \Delta \tau}
  \nonumber \\
  = 
  \left(\begin{array}{cccc}
   D_{1,1} & D_{2,1}  & \cdots & D_{im,1} \\
   D_{1,2} & \ddots    & \ddots & \vdots        \\
        \vdots   &           & \ddots & \vdots        \\
   D_{1,km} & \cdots & \cdots & D_{im,km} \\
        \end{array}
  \right)^{\tau}
  .
 \end{eqnarray}
%
$B$3$NO"N)J}Dx<0$r2r$/$3$H$G(B $\pi_{i,k}$ $B$r5a$a$k(B. $B$3$NO"N)J}Dx<0$N78?t$O0J2<$N(B
$B$h$&$K=q$1$k(B. 
%
\begin{eqnarray}
 A_{k} 
  &=& 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\} 
   \nonumber \\   
 && (k = 2, 3, \cdots km-1),
  \nonumber \\
 A_{1} 
  &=& 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)},
   \nonumber \\   
 A_{km} 
  &=& 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)},
   \nonumber \\   
 B_{k} &=&
  - \beta^{2}
  \left(
   \frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
  \right)_{k-1}
  \Dinv{\Delta z^{2}} 
  \left(
   \bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
  \right)_{k-1(w)},
  \nonumber \\
 C_{k} &=&
  - \beta^{2}
  \left(
   \frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
  \right)_{k+1}
  \Dinv{\Delta z^{2}} 
   \left(
    \bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2} 
       \right)_{k(w)},
   \nonumber \\
 D_{i,k}
 &=& \pi^{\tau}_{i,k}
   -(1 - \beta)  
  \left(
   \frac{\bar{c}^{2}\Delta \tau}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
  \right)_{k}
  \left\{
    \DP{(\bar{\rho} \bar{\theta}_{v} w^{\tau})}{z}
  \right\}_{i,k}
   -\left(
     \frac{\bar{c}^{2} \Delta \tau}{\bar{c_{p}} \bar{\theta}_{v}} 
    \right)_{k}
  \left(
  \DP{u^{\tau + \Delta \tau}}{x} 
  \right)_{i,k}
  \nonumber \\
 &&- \beta  
  \left(
   \frac{\bar{c}^{2}\Delta \tau}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
  \right)_{k}
   \left[
    \DP{}{z}
    \bar{\rho} \bar{\theta}_{v}
   \left\{
   w^{\tau}
  - \bar{c_{p}} \bar{\theta}_{v} \Delta \tau
  \left\{
    (1 - \beta) \DP{\pi^{\tau}}{z} 
   - \DP{(\alpha Div)^{\tau}}{z} 
  \right\}
 + F_{w}^{t} \Delta \tau   
\right\}
\right]_{i,k}
\nonumber \\
 && (k = 2, 3,  \cdots km-1),
\nonumber \\
 D_{i,1} &=& \pi^{\tau}_{i,1}
 -(1 - \beta)  
  \left(
   \frac{\bar{c}^{2}\Delta \tau}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
  \right)_{1}
  \left\{
    \DP{(\bar{\rho} \bar{\theta}_{v} w^{\tau})}{z}
  \right\}_{i,1}
-  
  \left(
   \frac{\bar{c}^{2} \Delta \tau}{\bar{c_{p}} \bar{\theta}_{v}}
  \right)_{1}
  \left(
  \DP{u^{\tau + \Delta \tau}}{x}
  \right)_{i,1}
  \nonumber \\
 &&- \beta  
  \left(
   \frac{\bar{c}^{2}\Delta \tau}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
  \right)_{1}
   \left[
    \DP{}{z}
    \bar{\rho} \bar{\theta}_{v}
   \left\{
   w^{\tau}
  - \bar{c_{p}} \bar{\theta}_{v} \Delta \tau
  \left\{
    (1 - \beta) \DP{\pi^{\tau}}{z} 
   - \DP{(\alpha Div)^{\tau}}{z} 
  \right\}
 + F_{w}^{t} \Delta \tau   
\right\}
\right]_{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(
   \bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
   \right)_{i,0(w)}
   E_{i,0(w)},
\nonumber \\
 D_{i,km} &=& 
  \pi^{\tau}_{i,km}
  -(1 - \beta)  
  \left(
   \frac{\bar{c}^{2}\Delta \tau}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
  \right)_{km}
  \left\{
   \DP{(\bar{\rho} \bar{\theta}_{v} w^{\tau})}{z}
  \right\}_{i,km}
-  
  \left(
   \frac{\bar{c}^{2} \Delta \tau}{\bar{c_{p}} \bar{\theta}_{v}} 
  \right)_{km}
  \left(
  \DP{u^{\tau + \Delta \tau}}{x}
  \right)_{i,km}
  \nonumber \\
 &&- \beta  
  \left(
   \frac{\bar{c}^{2}\Delta \tau}{\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}}
  \right)_{km}
   \left[
    \DP{}{z}
    \bar{\rho} \bar{\theta}_{v}
   \left\{
   w^{\tau}
  - \bar{c_{p}} \bar{\theta}_{v} \Delta \tau
  \left\{
    (1 - \beta) \DP{\pi^{\tau}}{z} 
   - \DP{(\alpha Div)^{\tau}}{z} 
  \right\}
 + F_{w}^{t} \Delta \tau   
\right\}
\right]_{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 
\end{eqnarray}
%
$B$?$@$7(B, 
%
\begin{eqnarray}
 E_{i,k(w)} = 
  \left( \DP{(\alpha Div)^{\tau}}{z} \right)_{i,k(w)}
  - (1 - \beta) \left( \DP{\pi^{\tau}}{z} \right)_{i,k(w)}
  + \left(\Dinv{\bar{c_{p}} \bar{\theta}_{v}} F_{w}^{t}\right)_{i,k(w)}
  \nonumber 
\end{eqnarray}
%
$B$G$"$k(B. 