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\chapter{$B4pACJ}Dx<07O$NF3=P(B}

\section{$B4pK\>l$HJP:9$NJ,N%(B}

 $BJQ?t$r4pK\>l$H$=$3$+$i$NJP:9$KJ,$1$k(B. $B4pK\>l$NJQ?t$O>eIU$-%P!<(B 
 $(\bar{~})$ $B$GI=$7(B, $BJP:9$r>eIU$-%W%i%$%`(B $( ^{\prime})$ $B$GI=$9(B.  $B$"$kJQ(B
 $B?t(B $\phi$ $B$O0J2<$N$h$&$KJ,N%$5$l$k(B.
 \begin{eqnarray}
  \phi = \overline{\phi} + \phi^{\prime}.
 \end{eqnarray}
 $BJ,N%$5$l$kJQ?t$O(B $u, w$, $B290L(B $\theta$, $B05NO(B $p$ $B$^$?$O(B $\Pi$,
 $BL)EY(B $\rho$$B$G$"$k(B.

 $B4pK\>l$O?eJ?0lMM(B ($\overline{\phi} = \overline{\phi}(z)$) $B$G(B, $B>qMp$N$J(B
 $B$$@E;_>uBV(B ($\overline{u} = \overline{v} = \overline{w} = 0$) $B$G$"$k$H(B
 $B$9$k(B. $B4pK\>l$NL5<!8505NO4X?t$H290L$O(B, $B$=$l$>$l(B,
 \begin{eqnarray}
  \overline{\Pi} &=& \left(\frac{\overline{p}}{p_{0}}\right)^{R_{d}/c_{p}}
  , \\
  \overline{\theta} &=& \frac{\overline{T}}{\overline\Pi}
 \end{eqnarray}
 $B$G$"$k(B. $B4pK\>l$NL)EY(B $\overline{\rho}$ $B$O>uBVJ}Dx<0$+$i(B
 \begin{equation}
   \overline{\rho} = \frac{p_{0}}{R_{d}}
                    \frac{\overline{\Pi}^{c_{v}/R_{d}}}{\overline{\theta}}
 \end{equation}
 $B$HM?$($i$l$k(B. $B$3$3$G(B $c_{v}$ $B$ODj@QHfG.$G$"$k(B.

 $B0J2<$G$O(B $\Pi^{\prime} = \pi$ $B$H$7(B,
 $B5-=R$r4JJX$K$9$k$?$a$=$NB>$NJQ?t$NJP:9NL$KIU$/>eIU$-%W%i%$%`(B $(
 ^{\prime})$ $B$O>JN,$9$k(B.
 
%--------------------------------------------------------------------
\pagebreak

\section{$B1?F0J}Dx<0$NF3=P(B}

 $B$3$3$G$O%(%/%9%J!<4X?t$rMQ$$$?1?F0J}Dx<0$NF3=P$H(B, $B?eJ?0lMM@E;_4pK\>l$H(B
 $B$=$3$+$iJP:9$H$NJ,N%$r9T$&(B. 

\subsection{$B%(%/%9%J!<4X?t$NF3F~(B}

 $B1?F0J}Dx<0$O(B
 \begin{eqnarray}
  \DD{u_{i}}{t} = - \frac{1}{\rho} \DP{p}{x_{i}}
                  - \delta_{i3} g
		  + D_{u_{i}}
 \end{eqnarray}
 $B$G$"$k(B.
 $B05NO8{G[9`$r(B $B%(%/%9%J!<4X?t$rMQ$$$F=q$-D>$9$H(B, 
 \begin{eqnarray}
  \DD{u_{i}}{t} = - \frac{c_{p} p}{\rho R_{d} \Pi}
                    \DP{\Pi}{x_{i}}
                  - \delta_{i3} g
		  + D_{u_{i}}
 \end{eqnarray}
 $B$3$3$G(B, $B>uBVJ}Dx<0(B
 \begin{eqnarray}
  p &=& \rho R_{d} \Pi \theta
 \end{eqnarray}
 $B$rMQ$$$F$5$i$K=q$-D>$9$H(B, 
 \begin{eqnarray}
  \DD{u_{i}}{t} = - c_{p} \theta \DP{\Pi}{x_{i}}
                  - \delta_{i3} g
		  + D_{u_{i}}
 \end{eqnarray}
 $B$rF@$k(B.

\subsection{$B4pK\>l$HJP:9$NJ,N%(B}

 $B1?F0J}Dx<0$r?eJ?0lMM@E;_4pK\>l$N<0$H$=$3$+$i$NJP:9$N<0(B($B>qMp>l$N<0(B)
 $B$KJ,N%$9$k(B. $B4pK\>l$NJ}Dx<0$O@E?e05$N<0(B
 \begin{equation}
  \DP{\overline{\Pi}}{z} = - \frac{g}{c_{p} \overline{\theta}}
 \end{equation}
 $B$G$"$k(B.

 $B>qMp>l$N?eJ?J}8~$N1?F0J}Dx<0(B $(i = 1)$ $B$O(B
 \begin{eqnarray}
  \DD{u}{t} &=& - c_{p}
                  \left(\overline{\theta}
	        + \theta^{\prime}
	          \right)
		  \DP{\left(\overline{\Pi} + \pi\right)}{x}
		+ D_{u}
  \nonumber \\
             &\simeq& - c_{p}
		     \left(\overline{\theta}_{v}
		           \DP{\overline{\Pi}}{x}
			   + \overline{\theta}_{v}
		             \DP{\pi}{x}
			   + \theta_{v}^{\prime}
		             \DP{\overline{\Pi}}{x}
		     \right)
		    + D_{u}
 \end{eqnarray}
 $B$H$J$k(B. $B>qMp>l$N1tD>J}8~$N1?F0J}Dx<0(B $(i = 3)$$B$O(B
 \begin{eqnarray}
  \DD{w}{t} &=& - c_{p}
                  \left(\overline{\theta}
		        + \theta^{\prime}
		  \right)
		  \DP{\left(\overline{\Pi} + \pi\right)}{z}
                -  g 
		+ D_{w}
  \nonumber \\
             &\simeq& - c_{p}
		     \left(\overline{\theta}_{v}
		           \DP{\overline{\Pi}}{z}
			   + \overline{\theta}_{v}
		             \DP{\pi}{z}
			   + \theta_{v}^{\prime}
		             \DP{\overline{\Pi}}{z}
		     \right)
                    - g
		    + D_{u}
  \Deqlab{$B4pK\>l4^$`1?F0J}Dx<0(B}
 \end{eqnarray}
 $B$H$J$k(B. $B$=$l$>$l$N<0$G05NO8{G[9`$K$*$1$k(B 2 $B<!NL$O>.$5$$$H2>Dj$7L5;k$9$k(B.

 $B4pK\>l$O?eJ?0lMM$G$"$k$3$H$H(B, $B1tD>J}8~$N1?F0J}Dx<0$+$i@E?e05$N<0$r:9$7(B
 $B0z$/$3$H$K$h$j(B, $B0J2<$N>qMp>l$N1?F0J}Dx<0$rF@$k(B.
 \begin{eqnarray}
  \DD{u}{t} &=&  - c_{p}\overline{\theta}_{v}
		         \DP{\pi}{x}
		+ D_{u}
  , \\
  \DD{w}{t} &=&  - c_{p}
                  \overline{\theta}_{v} \DP{\pi}{z}
		+ g\frac{\theta}{\overline{\theta}}
		+ D_{w}
  .
 \end{eqnarray}

 $B$^$H$a$FI=$9$H(B, 
 \begin{eqnarray}
  \DD{u_{i}}{t} = - c_{p} \overline{\theta}_{v} \DP{\pi}{x_{i}}
      + \delta_{i3} g\frac{\theta}{\overline{\theta}}
      + D_{u_{i}}
 \end{eqnarray}
 $B$H$J$k(B.

%--------------------------------------------------------------------

\clearpage
\section{$B05NOJ}Dx<0$NF3=P(B}

\subsection{$B05NOJ}Dx<0(B}

 $B05NOJ}Dx<0$O(B, $BO"B3$N<0(B
 \begin{eqnarray}
  \DD{\rho}{t}
   + \rho \DP{u_{j}}{x_{j}}
   = 0
  \Deqlab{$BO"B3$N<0(B}
 \end{eqnarray}
 $B$H%(%/%9%J!<4X?t$GI=$7$?>uBVJ}Dx<0(B
 \begin{eqnarray}
  \Pi = \left(\frac{R_{d}}{p_{0}}
              \rho \theta_{v}\right)^{R_{d}/c_{v}}
  \Deqlab{$B%(%/%9%J!<4X?t$rMQ$$$?>uBVJ}Dx<0(B}
 \end{eqnarray}
 $B$+$iF3=P$9$k(B.

 $B>uBVJ}Dx<0(B\Deqref{$B%(%/%9%J!<4X?t$rMQ$$$?>uBVJ}Dx<0(B}$B$r(B $t$ $B$GHyJ,$9$k(B:
 \begin{eqnarray}
  \DD{\Pi}{t} &=& \left(\frac{R_{d}}{p_{0}}\right)^{R_{d}/c_{v}}
                  \left\{\frac{R_{d}}{c_{v}}
		         \theta^{R_{d}/c_{v}}
		         \rho^{R_{d}/c_{v} - 1}
		         \DD{\rho}{t}
			 + \frac{R_{d}}{c_{v}}
		           \rho^{R_{d}/c_{v}}
		           \theta^{R_{d}/c_{v} - 1}
			   \DD{\theta}{t}
                  \right\}
  \nonumber \\
	      &=& \frac{R_{d}}{c_{v}}
	          \left(\frac{R_{d}}{p_{0}}
		        \rho \theta
		  \right)^{R_{d}/c_{v}}
                  \left\{\frac{1}{\rho}
		         \DD{\rho}{t}
			 + \frac{1}{\theta}
			   \DD{\theta}{t}
                  \right\}
  \nonumber \\
              &=& \frac{R_{d}}{c_{v}} \Pi
                  \left\{\frac{1}{\rho}
		         \DD{\rho}{t}
			 + \frac{1}{\theta}
			   \DD{\theta}{t}
                  \right\}
  .
 \end{eqnarray}
 $BO"B3$N<0(B\Deqref{$BO"B3$N<0(B}$B$r1&JU$KBeF~$9$k$H(B,
 \begin{eqnarray}
  \DD{\Pi}{t}  &=& \frac{R_{d}}{c_{v}} \Pi
                  \left\{- \DP{u_{j}}{x_{j}}
			 + \frac{1}{\theta}
			   \DD{\theta}{t}
                  \right\}
 \end{eqnarray}
 $B$rF@$k(B.

\subsection{$B05NOJP:9$NJ}Dx<0(B}

 $B%(%/%9%J!<4X?t$r?eJ?0lMM4pK\>l(B $\overline{\Pi}$ $B$H$=$3$+$i$NJP:9(B $\pi$ 
 $B$KJ,N%$7(B, $\pi$ $B$NJ}Dx<0$r5a$a$k(B. 
 \begin{eqnarray}
  \DP{\pi}{t} + u_{j} \DP{\pi}{x_{j}}
              + w \DP{\overline{\Pi}}{z}
	      = - \frac{R_{d}}{c_{v}}
	            \overline{\Pi} \DP{u_{j}}{x_{j}}
		  - \frac{R_{d}}{c_{v}}
		    \pi \DP{u_{j}}{x_{j}}
		  + \frac{R_{d}}{c_{v}}
		    \frac{\Pi}{\overline{\theta} +\theta}
		    \DD{(\overline{\theta}+\theta)}{t}
 .
 \end{eqnarray}
 $B2;B.(B
 \begin{eqnarray}
  c^{2} &=& \frac{c_{p} R_{d}}{c_{v}} \Pi (\overline{\theta}+\theta)
  , \\
  \overline{c^{2}} &=& \frac{c_{p} R_{d}}{c_{v}}
                  \overline{\Pi} \overline{\theta}
 \end{eqnarray}
 $B$rMQ$$$F=q$-D>$9$H(B,
 \begin{eqnarray}
  \DP{\pi}{t} + \frac{\overline{c}^{2}}{c_{p} \overline{\theta}}
	        \DP{u_{j}}{x_{j}}
              + w \DP{\overline{\Pi}}{z}
	      &=& - u_{j} \DP{\pi}{x_{j}}
		  + \frac{R_{d}}{c_{v}}\pi
		    \DP{u_{j}}{x_{j}}
		  + \frac{c^{2}}{c_{p} (\overline{\theta} +\theta)^{2}}
		    \DD{(\overline{\theta}+\theta)}{t}
 .
 \end{eqnarray}
 $B$3$3$G>uBVJ}Dx<0(B\Deqref{$B%(%/%9%J!<4X?t$rMQ$$$?>uBVJ}Dx<0(B}$B$rMQ$$$F(B
 $B:8JUBh(B 3 $B9`$rJQ7A$9$k(B.
 \begin{eqnarray*}
  w \DP{\overline{\Pi}}{z}
      &=& w \left(\frac{R_{d}}{p_{0}}\right)^{R_{d}/c_{v}}
          \DP{\left(\overline{\rho}
	            \overline{\theta}
	      \right)^{R_{d}/c_{v}}}
	     {z}
  \nonumber \\
      &=& w \left(\frac{R_{d}}{p_{0}}\right)^{R_{d}/c_{v}}
          \frac{R_{d}}{c_{v}}
	  \left(\overline{\rho}
	            \overline{\theta}
	      \right)^{R_{d}/c_{v} - 1}
          \DP{\overline{\rho} \overline{\theta}}{z}
  \nonumber \\
      &=& w \frac{R_{d}}{c_{v}}
          \overline{\Pi}
	  \frac{1}{\overline{\rho} \overline{\theta}}
          \DP{\left(\overline{\rho} \overline{\theta}_{v}\right)}{z}
  \nonumber \\
      &=& \frac{\overline{c}^{2}}
               {c_{p} \overline{\rho} \overline{\theta}_{v}^{2}}
	  w
          \DP{\left(\overline{\rho} \overline{\theta}_{v}\right)}{z}
  \nonumber \\
      &=& \frac{\overline{c}^{2}}
               {c_{p} \overline{\rho} \overline{\theta}^{2}}
	  u_{j}
          \DP{\left(\overline{\rho} \overline{\theta}\right)}{x_{j}}
  .
 \end{eqnarray*}
 $B$^$?(B, $B:8JUBh(B 2 $B9`$O(B
 \begin{eqnarray*}
   \frac{\overline{c}^{2}}{c_{p} \overline{\theta}}
    \DP{u_{j}}{x_{j}}
      = \frac{\overline{c}^{2}}
             {c_{p} \overline{\rho} \overline{\theta}^{2}}
	 \overline{\rho} \overline{\theta}
	\DP{u_{j}}{x_{j}}
 \end{eqnarray*}
 $B$H=q$-49$($k$3$H$,$G$-$k(B.
 $B:8JUBh(B 2, 3 $B9`$r$^$H$a$k$H05NOJ}Dx<0$O0J2<$N$h$&$KI=$5$l$k(B.
 \begin{equation}
  \DP{\pi}{t}
     + \frac{\overline{c}^{2}}{c_{p}
         \overline{\rho} \overline{\theta}^{2}}
         \DP{}{x_{j}}
	 \left(\overline{\rho} \overline{\theta} u_{j}\right)
     = f_{\pi}
 \end{equation}
 \begin{equation}
  f_{\pi} = - u_{j} \DP{\pi}{x_{j}}
            + \frac{R_{d} \pi}{c_{v}} \DP{u_{j}}{x_{j}}
  	    + \frac{c^{2}}{c_{p} (\overline{\theta} +\theta)^{2}}
	   \DD{(\overline{\theta}+\theta)}{t}
  .
 \Deqlab{$B05NOJ}Dx<0$N6/@)9`(B}
 \end{equation}
% $B$3$N05NOJ}Dx<0$O(B Tapp and White (1976) $B$,Hs@ENO%a%=%b%G%k$K(B
% $BMQ$$$?$N$HF1$8J}Dx<0$G$"$k(B.

 Klemp and Wilhelmson (1978) $B$K$7$?$,$$(B, $BK\%b%G%k$G$O(B $f_{\pi} = 0$ $B$H$9(B
 $B$k(B. $B$3$l$O@~7A2=$r9T$$(B, $BHsCGG.9`(B(\Deqref{$B05NOJ}Dx<0$N6/@)9`(B}
 $B1&JUBh(B 3 $B9`(B)$B$rL5;k$9$k$3$H$r0UL#$9$k(B.


