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\section{����Υ������������������}

\subsection{�ſ尵�μ�}
 \begin{equation}
  \left[\DP{\overline{\pi}}{z}\right]_{i,j,k} = 
    - \frac{g}{{c_{p}}_{d} [\overline{\theta_{v}}]_{i,j,k}}
 \end{equation}
 ���ܾ��̩�� $\overline{\rho}_{i,j,k}$ �ϰʲ��Τ褦�˷׻�����.
 \begin{equation}
   \overline{\rho}_{i,j,k} = \frac{p_{0}}{R_{d}}
                    \frac{[\overline{\pi}^{c_{v}/R_{d}}]_{i,j,k}}
		    {[\overline{\theta_{v}}]_{i,j,k}}
 \end{equation}

\subsection{��ư������}
 \begin{eqnarray}
  \DP{u_{i(u),j,k}}{t} &=&
  - (\bar{u} + u)_{i(u),j,k}\left[\DP{u}{x}\right]_{i(u),j,k}
  - (\bar{v} + v)_{i(u),j,k}\left[\DP{u}{y}\right]_{i(u),j,k}
  - w_{i(u),j,k}\left[\DP{u}{z}\right]_{i(u),j,k}\nonumber \\
 && 
  - w_{i(u),j,k}\left[\DP{\bar{u}}{z}\right]_{i(u),j,k}\nonumber \\
 && 
  - {c_{p}}_{d} [\overline{\theta_{v}}]_{i(u),j,k}
  \left[\DP{\pi}{x}\right]_{i(u),j,k}
  + \left[{\rm Turb}.{\bar{u}}\right]_{i(u),j,k} 
  + \left[{\rm Turb}.{u}\right]_{i(u),j,k} 
  \Deqlab{����Υ�������줿 x ������ư������}
  \\
  \DP{v_{i,j(v),k}}{t} &=&
  - (\bar{u} + u)_{i,j(v),k}\left[\DP{v}{x}\right]_{i,j(v),k}
  - (\bar{v} + v)_{i,j(v),k}\left[\DP{v}{y}\right]_{i,j(v),k}
  - w_{i,j(v),k}\left[\DP{v}{z}\right]_{i,j(v),k}\nonumber \\
 && 
  - w_{i,j(v),k}\left[\DP{\bar{v}}{z}\right]_{i,j(v),k}\nonumber \\
 && 
  - {c_{p}}_{d} [\overline{\theta_{v}}]_{i(u),j,k}
  \left[\DP{\pi}{y}\right]_{i,j(v),k}
  + \left[{\rm Turb}.{\bar{v}}\right]_{i,j(v),k} 
  + \left[{\rm Turb}.{v}\right]_{i,j(v),k} 
  \Deqlab{����Υ�������줿 y ������ư������}
  \\
  \DP{w_{i,j,k(w)}}{t} &=&
  - (\bar{u} + u)_{i,j,k(w)}\left[\DP{w}{x}\right]_{i,j,k(w)}
  - (\bar{v} + v)_{i,j,k(w)}\left[\DP{w}{y}\right]_{i,j,k(w)}
  - w_{i,j,k(w)}\left[\DP{w}{z}\right]_{i,j,k(w)} \nonumber \\
 &&
  - {c_{p}}_{d} [\overline{\theta_{v}}]_{i,j,k(w)}
      \left[\DP{\pi}{z}\right]_{i,j,k(w)}
  + \left[{\rm Turb}.{w}\right]_{i,j,k(w)} 
\nonumber \\
&&  
 + g \frac{\theta_{i,j,k(w)}}{\overline{\theta}_{i,j,k(w)}}
\nonumber \\
&&  
 + g \frac{\sum [q_{v}]_{i,j,k(w)}/M_{v}}{1/M_{d} 
      + \sum  [\bar{q_{v}}]_{i,j,k(w)}/M_{v}} 
\nonumber \\
&&  
    - g \frac{\sum [q_{v}]_{i,j,k(w)} + \sum [q_{c}]_{i,j,k(w)} + \sum [q_{r}]_{i,j,k(w)}}
    {1 + \sum [\bar{q_{v}}]_{i,j,k(w)}}
  \Deqlab{����Υ�������줿 z ������ư������}
 \end{eqnarray}

\subsection{����������}
 \begin{eqnarray}
    \DP{\pi_{i,j,k}}{t}
     + \frac{\overline{c}_{i,j,k}^{2}}{{c_{p}}_{d}
         [\overline{\theta_{v}}]_{i,j,k}}
         \left[
           \DP{u}{x} 
         + \DP{v}{y} 
         \right]_{i,j,k}
     + \frac{\overline{c}_{i,j,k}^{2}}{{c_{p}}_{d}
         \overline{\rho}_{i,j,k} [\overline{\theta_{v}}^{2}]_{i,j,k}}
         \left[
         \DP{\overline{\rho} \overline{\theta_{v}} w}{z}
         \right]_{i,j,k}
     = \left[F_{\pi}\right]_{i,j,k},
  \Deqlab{����Υ�������줿����������}
 \end{eqnarray}
%
\begin{eqnarray}
 \left[F_{\pi}\right]_{i,j,k} 
&=& 
- \left(
       (\bar{u} + u) \DP{\pi}{x} + (\bar{v} + v) \DP{\pi}{y} + w \DP{\pi}{z} 
  \right) 
  + \frac{R_{d} \pi}{{c_{v}}_{d}} \left(\DP{u}{x} + \DP{v}{y} +
				   \DP{w}{z}\right)
\nonumber \\
&&
+ \frac{\overline{c}_{i,j,k}^{2}}{{c_{p}}_{d}
        [\overline{\theta_{v}}]_{i,j,k}}
   \left\{
    \frac{1}{[\overline{T}]_{i,j,k}}([Q_{cond}]_{i,j,k} +
       [Q_{rad}]_{i,j,k} + [Q_{dis}]_{i,j,k})\right. \nonumber \\
&& \left.- \left(
     \frac{\sum [\dot{q}_{v}]_{i,j,k} + \sum [\dot{q}_{c}]_{i,j,k} + \sum [\dot{r}_{v}]_{i,j,k}}{1 + \sum [\overline{q_{v}}]_{i,j,k}}
    - 
    \frac{\sum [\dot{q}_{v}]_{i,j,k}/M_{v}}{1/M_{d} + \sum [\overline{q_{v}}]_{i,j,k}/M_{v}}
    \right)
   \right\}
\end{eqnarray}
���ܾ�β�® $\overline{c}$ �ϰʲ��Τ褦�˷׻�����. 
 \begin{equation}
  \overline{c}_{i,j,k}^{2} = \frac{{c_{p}}_{d} R_{d}}{c_{v}}
                  \overline{\pi}_{i,j,k} [\overline{\theta_{v}}]_{i,j,k}.
 \end{equation}


\subsection{Ǯ�ϳؤμ�}
 \begin{eqnarray}
  \DP{\theta_{i,j,k}}{t} &=& - (\bar{u} + u)_{i,j,k}\left[\DP{\theta}{x}\right]_{i,j,k}
                         - (\bar{v} + v)_{i,j,k}\left[\DP{\theta}{z}\right]_{i,j,k}
                         -
			 w_{i,j,k}\left[\DP{\theta}{z}\right]_{i,j,k}
                         - w_{i,j,k}\left[\DP{\overline{\theta}}{z}\right]_{i,j,k}
 \nonumber \\
 &&
                         + \left[{\rm Turb}.{\theta}\right]_{i,k}
                         + \left[{\rm Turb}.{\overline{\theta}}\right]_{i,k}
 \nonumber \\
 &&                 + \Dinv{\overline{\pi}_{i,j,k}}
	\left([Q_{cnd}]_{i,j,k} + [Q_{rad}]_{i,j,k} + [Q_{dis}]_{i,j,k}\right)
  \Deqlab{����Υ�������줿Ǯ�ϳؤμ�}
 \end{eqnarray}


\subsection{�Ž���ʬ�κ��������¸��}
%
\begin{eqnarray}
\DP{[q_{v}]_{i,j,k}}{t} 
&=& 
  - (\bar{u} + u)_{i,j,k} \left[\DP{q_{v}}{x} \right]_{i,j,k}
  - (\bar{v} + v)_{i,j,k} \left[\DP{q_{v}}{y} \right]_{i,j,k}
  - w_{i,j,k} \left[\DP{q_{v}}{z} \right]_{i,j,k}
  - w_{i,j,k} \left[\DP{\bar{q_{v}}}{x} \right]_{i,j,k}
\nonumber  \\
&& 
  + [{\rm Src}.q_{v}]_{i,j,k} 
  + [{\rm Turb}.{\overline{q_{v}}}]_{i,j,k} + [{\rm Turb}.{q_{v}}]_{i,j,k}, 
  \Deqlab{quasi-equations:u-w-pi-theta:qv} \\
%
\DP{[q_{c}]_{i,j,k}}{t} 
&=&
  - (\bar{u} + u)_{i,j,k} \left[ \DP{q_{c}}{x} \right]_{i,j,k}
  - (\bar{v} + v)_{i,j,k} \left[ \DP{q_{c}}{y} \right]_{i,j,k}
  - w_{i,j,k} \left[ \DP{q_{c}}{z} \right]_{i,j,k}
\nonumber \\
&&
  + [{\rm Src}.q_{c}]_{i,j,k} + [{\rm Turb}.{q_{c}}]_{i,j,k}, 
  \Deqlab{quasi-equations:u-w-pi-theta:qc} \\
%
\DP{[q_{r}]_{i,j,k}}{t} 
&=& 
  - (\bar{u} + u)_{i,j,k} \left[ \DP{q_{r}}{x} \right]_{i,j,k}
  - (\bar{v} + v)_{i,j,k} \left[ \DP{q_{r}}{y} \right]_{i,j,k}
  - w_{i,j,k} \left[ \DP{q_{r}}{z} \right]_{i,j,k}
\nonumber \\
 &&
  + [{\rm Src}.q_{r}]_{i,j,k} 
  + [{\rm Fall}.q_{r}]_{i,j,k} 
  + [{\rm Turb}.{q_{r}}]_{i,j,k}
  \Deqlab{quasi-equations:u-w-pi-theta:qr}
\end{eqnarray}
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