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\section{$BMpN.%Q%i%a%?%j%<!<%7%g%s(B}

%----------------------------------------------------------------------
%\newpage

\subsection{$B%l%$%N%k%:1~NOJ}Dx<0(B}


%----------------------------------------------------------------------
\newpage

\subsection{$B%5%V%0%j%C%I%9%1!<%k$NJ}Dx<0(B}

 $B%5%V%0%j%C%I%9%1!<%k$N1?F0%(%M%k%.!<$NJ}Dx<0$K4X$7$F(B,
 Klemp and Wilhelmson (1978) $BCf$N<0(B (2.21) $B$+$i(B (3.12) $B$N(B
 $BF3=P$r9T$C$?(B.
 (2.21) $B$+$iF3=P$7$?<0$HHf$Y$k$H(B, (3.21) $B$K$OB-$j$J$$9`$,(B
 2 $B$D$"$C$?(B.

 $B$3$3$G$O%5%V%0%j%C%I%9%1!<%k$N1?F0%(%M%k%.!<$NJ}Dx<0$r(B
 $B=q$-2<$9(B.
 $B:BI87O$OD>8rD>@~:BI87O$G$"$k(B.

 Klemp and Wilhelmson (1978) $B$N<0(B (2.21):
 \begin{eqnarray}
  \DD{E}{t} &=& g
                \overline{w
		          \left(\frac{\theta^{\prime}}
			             {\bar{\theta}}
				+ 0.61 q_{v}^{\prime}
				- q_{c}^{\prime}
			  \right)}
		- \overline{u_{i}^{\prime} u_{j}^{\prime}}
		  \DP{u_{i}}{x_{j}}
		+ \DP{}{x_{j}}
		  \left(K_{m} \DP{E}{x_{j}}\right)
		- \frac{C_{\varepsilon}}{l}
		  E^{\frac{3}{2}}
  \Deqlab{$B%5%V%0%j%C%I(BE}
  .
 \end{eqnarray}
 $C_{\varepsilon}, l$ $B$ODj?t$G$"$k(B.

 $B$3$3$G(B
 \begin{eqnarray}
  \overline{w \left(\frac{\theta^{\prime}}{\bar{\theta}}
	            + 0.61 q_{v}^{\prime}
		    - q_{c}^{\prime}
	    \right)}
        &=& - A K_{h} \DP{\theta_{e}}{z} + K_{h} \DP{q_{l}}{z}
  \Deqlab{$B%5%VIbNO(B}
  , \\
  \overline{u_{i}^{\prime} u_{j}^{\prime}}
        &=& - K_{m} \left(\DP{u_{i}}{x_{j}}
		          + \DP{u_{j}}{x_{i}}
		    \right)
	    + \frac{2}{3} \delta_{ij} E
  \Deqlab{$B%5%VB.EY(B}
  , \\
  K_{m} &=& C_{m} l E^{\frac{1}{2}}
 \Deqlab{$B%5%VG4@-$H(BE$B$N4X78(B}
  .
 \end{eqnarray}
 $B$?$@$7(B
 \begin{eqnarray}
  q_{l} &=& q_{v} + q_{c}
  , \\
  A &=& \frac{1}{\bar{\theta}}
        \left\{\frac{1 + \dfrac{1.61
	             \varepsilon L q_{v}}{R_{d} T}}
	            {1 + \dfrac{\varepsilon L^{2} q_{v}}
		              {c_{p} R_{d} T^{2}}}\right\}
  , \\
  K_{h} &=& 3 K_{m}
  .
 \end{eqnarray}

 Klemp and Wilhelmson (1978) $B$N<0(B (3.12):
 \begin{eqnarray}
  \delta_{2t} K_{m}
       &=& - \frac{1}{3} \overline{u}^{x}
             \left(4 \delta_{2x} K_{m}
	           - \delta_{4x} K_{m}\right)
	   - \frac{1}{3} \overline{u}^{y}
             \left(4 \delta_{2y} K_{m}
	           - \delta_{4y} K_{m}\right)
 \nonumber \\
       & & \overline{w \delta_{z} K_{m}}^{z}
	   + \frac{C_{m}^{2} l^{2}}{2 K_{m}}
	     \left(B + S\right)
 \nonumber \\
       & & + \frac{1}{2}
	     \left(\delta_{xx} K_{m}^{2}
	           + \delta_{yy} K_{m}^{2}
	           + \delta_{zz} K_{m}^{2}
	     \right)
	   - \frac{C_{\varepsilon} K_{m}^{2}}{2 C_{m} l^{2}}
  \Deqlab{$B%5%V%0%j%C%IN%;6(BE}
  .
 \end{eqnarray}
 $B$?$@$7(B,
 \begin{eqnarray}
  B &=& - g A K_{h} \delta_{2z} \theta_{e}
        + g K_{h} \delta_{2z} q_{l}
  , \\
  S &=& - \overline{u_{i}^{\prime} u_{j}^{\prime}}
          \DP{u_{i}}{u_{j}}
  \nonumber \\
    &=& K_{m}
        \Biggl\{2 \left[\left(\delta_{x} u\right)^{2}
		       + \left(\delta_{y} v\right)^{2}
		       + \left(\delta_{z} w\right)^{2}
		 \right]
  \nonumber \\
    & & \hspace*{4em}
               + \left(\overline{\delta_{x} v
		                 + \delta_{y} w}
		 \right)^{2}
	       + \left(\overline{\delta_{x} w
		                 + \delta_{z} u}
		 \right)^{2}
	       + \left(\overline{\delta_{y} w
		                 + \delta_{z} v}
		 \right)^{2}
        \Biggr\}
  \Deqlab{$BIbNO9`(BS}
  .
 \end{eqnarray}
 $B<0(B\Deqref{$BIbNO9`(BS}$B$N1&JU$N(B$\{ \}$ $BFbBh(B 3 $B9`$O8m?"$+$b$7$l$J$$(B.
 $\left(\overline{\delta_{x} v + \delta_{y} u}\right)^{2}$ 
 $B$G$"$k$H;W$o$l$k(B.
 $B<0(B (3.12) $B$O(B 4 $B<!@:EY$GN%;62=$5$l$?8e$N<0$G$"$k(B.
 $\delta_{2t}$ $B$J$I$N0UL#$O0J2<$N$h$&$G$"$k(B.
  $\xi$ $B$r$"$kFHN)JQ?t(B, $\phi$ $B$r$"$k=>B0JQ?t$H$7(B,
 $n \Delta \xi$ $B$O$"$kFHN)JQ?t$N4V3V$H$9$k$H(B,
 \begin{eqnarray}
  \delta_{n \xi} \phi (\xi)
     &=& \frac{1}{n \Delta \xi}
         \left[\phi
	       \left(\xi + \frac{n \Delta \xi}{2}\right)
	       - \phi
	         \left(\xi - \frac{n \Delta \xi}{2}\right)
	 \right]
 , \\
 \overline{\phi \left(\xi\right)}^{n \xi}
     &=& \frac{1}{2}
         \left[\phi
	       \left(\xi + \frac{n \Delta \xi}{2}\right)
	       - \phi
	         \left(\xi - \frac{n \Delta \xi}{2}\right)
	 \right]
  .
 \end{eqnarray}
 
 \Deqref{$B%5%VIbNO(B}$B$H(B\Deqref{$B%5%VB.EY(B}$B$rMQ$$$F(B
 \Deqref{$B%5%V%0%j%C%I(BE}$B$r=q$-D>$9$H(B,
 \begin{eqnarray}
  \DP{E}{t} &=& - \left(u \DP{E}{x} + v \DP{E}{y}
		        + w \DP{E}{z}
		  \right)
  \nonumber \\
            & & + \left(- g A K_{h} \DP{\theta_{e}}{z}
		        + g K_{h} \DP{q_{l}}{z}
		  \right)
  \nonumber \\
	    & & - \left\{- K_{m}
		          \left(\DP{u_{i}}{x_{j}}
		                + \DP{u_{j}}{x_{i}}
			  \right)
			 + \frac{2}{3}
			   \delta_{ij} E
		  \right\}
		\DP{u_{i}}{x_{j}}
  \nonumber \\
	    & & + \DP{}{x_{j}}
		  \left(K_{m} \DP{E}{x_{j}}\right)
		- \frac{C_{\varepsilon}}{l}
		  E^{\frac{3}{2}}
  .
 \end{eqnarray}

\pagebreak
 $B1&JUBh(B 3 $B9`(B, $BBh(B 4 $B9`$r=q$-2<$9$H(B, $B$=$l$>$l(B,
 \begin{eqnarray}
  &-& \Biggl\{- K_{m}
	     \Biggl(\DP{u_{i}}{x_{j}}
	           + \DP{u_{j}}{x_{i}}
	     \Biggr)
	   + \frac{2}{3} \delta_{ij} E
    \Biggr\}
		\DP{u_{i}}{x_{j}}
  \nonumber \\
   &=& K_{m}
       \Biggl\{2 \left(\DP{u}{x}\right)^{2}
	       + \left(\DP{u}{y} + \DP{v}{x}\right)\DP{u}{y}
	       + \left(\DP{u}{z} + \DP{w}{x}\right)\DP{u}{z}
  \nonumber \\
   & & \hspace*{3em}
             + \left(\DP{v}{x} + \DP{u}{y}\right)\DP{v}{x}
             + 2 \left(\DP{v}{y}\right)^{2}
	     + \left(\DP{v}{z} + \DP{w}{y}\right)\DP{v}{z}
  \nonumber \\
   & & \hspace*{3em}
	     + \left(\DP{w}{x} + \DP{u}{z}\right)\DP{w}{x}
	     + \left(\DP{w}{y} + \DP{v}{z}\right)\DP{w}{y}
             + 2 \left(\DP{w}{z}\right)^{2}
  \nonumber \\
   & & \hspace*{3em}
             - \frac{2 K_{m}^{2}}{3 C_{m}^{2} l^{2}}
	       \left(\DP{u}{x} + \DP{v}{y} + \DP{w}{z}\right)
     \Biggr\}
  \nonumber \\
    &=& 2 K_{m}
       \Biggl\{\left(\DP{u}{x}\right)^{2}
             + \left(\DP{v}{y}\right)^{2}
             + \left(\DP{w}{z}\right)^{2}
     \Biggr\}
  \nonumber \\
    & & + K_{m}
         \Biggl\{\left(\DP{u}{y}\right)^{2}
                 + \DP{v}{x} \DP{u}{y}
		 + \left(\DP{u}{z}\right)^{2}
                 + \DP{w}{x} \DP{u}{z}
  \nonumber \\
    & & \hspace*{4em}
		 + \left(\DP{v}{x}\right)^{2}
                 + \DP{u}{y} \DP{v}{x}
		 + \left(\DP{v}{z}\right)^{2}
                 + \DP{w}{y} \DP{v}{z}
  \nonumber \\
    & & \hspace*{4em}
		 + \left(\DP{w}{x}\right)^{2}
                 + \DP{u}{z} \DP{w}{x}
		 + \left(\DP{w}{y}\right)^{2}
                 + \DP{v}{z} \DP{w}{y}
         \Biggr\}
  \nonumber \\
    & & - \frac{2 K_{m}^{2}}{3 C_{m}^{2} l^{2}}
	       \left(\DP{u}{x} + \DP{v}{y} + \DP{w}{z}\right)     
  \nonumber \\
    &=& 2 K_{m}
       \Biggl\{\left(\DP{u}{x}\right)^{2}
             + \left(\DP{v}{y}\right)^{2}
             + \left(\DP{w}{z}\right)^{2}
     \Biggr\}
  \nonumber \\
    & & + K_{m}
         \Biggl\{\left(\DP{v}{x}\right)^{2}
                 + 2 \DP{v}{x} \DP{u}{y}
		 + \left(\DP{u}{y}\right)^{2}
		 + \left(\DP{w}{x}\right)^{2}
                 + 2 \DP{w}{x} \DP{u}{z}
		 + \left(\DP{u}{z}\right)^{2}
  \nonumber \\
    & & \hspace*{4em}
		 + \left(\DP{w}{y}\right)^{2}
                 + 2 \DP{w}{y} \DP{v}{z}
		 + \left(\DP{v}{z}\right)^{2}
         \Biggr\}
  \nonumber \\
    & & - \frac{2 K_{m}^{2}}{3 C_{m}^{2} l^{2}}
	       \left(\DP{u}{x} + \DP{v}{y} + \DP{w}{z}\right)     
  \nonumber \\
    &=& 2 K_{m}
       \Biggl\{\left(\DP{u}{x}\right)^{2}
             + \left(\DP{v}{y}\right)^{2}
             + \left(\DP{w}{z}\right)^{2}
     \Biggr\}
  \nonumber \\
    & & + K_{m}
         \Biggl\{\left(\DP{v}{x} + \DP{u}{y}\right)^{2}
		 + \left(\DP{w}{x} + \DP{u}{z}\right)^{2}
		 + \left(\DP{w}{y} + \DP{v}{z}\right)^{2}
         \Biggr\}
  \nonumber \\
    & & - \frac{2 K_{m}^{2}}{3 C_{m}^{2} l^{2}}
	       \left(\DP{u}{x} + \DP{v}{y} + \DP{w}{z}\right)
 \end{eqnarray}
 \begin{eqnarray}
  \DP{}{x_{j}} \left(K_{m} \DP{E}{x_{j}}\right)
   &=& \DP{}{x} \left(K_{m} \DP{E}{x}\right)
       + \DP{}{y} \left(K_{m} \DP{E}{y}\right)
       + \DP{}{z} \left(K_{m} \DP{E}{z}\right)
  \nonumber \\
   &=& \frac{1}{C_{m}^{2} l^{2}}
       \Biggl\{\DP{}{x}
               \left(K_{m} \DP{K_{m}^{2}}{x}\right)
	      + \DP{}{y}
	        \left(K_{m} \DP{K_{m}^{2}}{y}\right)
  \nonumber \\
   & & \hspace*{4em}
              + \DP{}{z}
	        \left(K_{m} \DP{K_{m}^{2}}{z}\right)
       \Biggr\}
  \nonumber \\
   &=& \frac{1}{C_{m}^{2} l^{2}}
       \Biggl\{K_{m} \DP[2]{K_{m}^{2}}{x}
               + \DP{K_{m}}{x}
                 \DP{K_{m}^{2}}{x}
  \nonumber \\
   & & \hspace*{4em}
	       + K_{m} \DP[2]{K_{m}^{2}}{y}
               + \DP{K_{m}}{y}
                 \DP{K_{m}^{2}}{y}
  \nonumber \\
   & & \hspace*{4em}
	       + K_{m} \DP[2]{K_{m}^{2}}{z}
               + \DP{K_{m}}{z}
                 \DP{K_{m}^{2}}{z}
       \Biggr\}
  \nonumber \\
   &=& \frac{K_{m}}{C_{m}^{2} l^{2}}
       \left(\DP[2]{K_{m}^{2}}{x}
	       + \DP[2]{K_{m}^{2}}{y}
	       + \DP[2]{K_{m}^{2}}{z}
       \right)
  \nonumber \\
   & & + \frac{1}{C_{m}^{2} l^{2}}
         \Biggl(\DP{K_{m}}{x} 2 K_{m} \DP{K_{m}}{x}
	        + \DP{K_{m}}{y} 2 K_{m} \DP{K_{m}}{y}
  \nonumber \\
   & & \hspace*{5em}
	        + \DP{K_{m}}{z} 2 K_{m} \DP{K_{m}}{z}
	 \Biggr)
  \nonumber \\
   &=& \frac{K_{m}}{C_{m}^{2} l^{2}}
       \left(\DP[2]{K_{m}^{2}}{x}
	       + \DP[2]{K_{m}^{2}}{y}
	       + \DP[2]{K_{m}^{2}}{z}
       \right)
  \nonumber \\
   & & + \frac{2 K_{m}}{C_{m}^{2} l^{2}}
         \Biggl\{\left(\DP{K_{m}}{x}\right)^{2}
	        + \left(\DP{K_{m}}{y}\right)^{2}
	        + \left(\DP{K_{m}}{z}\right)^{2}
	 \Biggr\}
  .
 \end{eqnarray}
 $B$7$?$,$C$F(B,
 $E = \left(\frac{K_{m}}{C_{m} l}\right)^{2}, K_{h} = 3 K_{m}$ $B$h$j(B
 \begin{eqnarray}
  \DP{K_{m}}{t}
     &=& - \left(u \DP{K_{m}}{x} + v \DP{K_{m}}{y}
		 + w \DP{K_{m}}{z}
	   \right)
            + \frac{3 g C_{m}^{2} l^{2}}{2}
	      \left(- A \DP{\theta_{e}}{z}
	            + \DP{q_{l}}{z}
		  \right)
  \nonumber \\
            & & + C_{m}^{2} l^{2}
	          \Biggl\{\left(\DP{u}{x}\right)^{2}
		          + \left(\DP{v}{y}\right)^{2}
			  + \left(\DP{w}{z}\right)^{2}
		  \Biggr\}
  \nonumber \\
            & & + \frac{C_{m}^{2} l^{2}}{2}
              	  \Biggl\{\left(\DP{v}{x} + \DP{u}{y}\right)^{2}
		          + \left(\DP{w}{x} + \DP{u}{z}\right)^{2}
			  + \left(\DP{w}{y} + \DP{v}{z}\right)^{2}
		  \Biggr\}
  \nonumber \\
            & & - \frac{K_{m}}{3}
		  \left(\DP{u}{x} + \DP{v}{y}
		        + \DP{w}{z}
		  \right)
  \nonumber \\
	    & & + \frac{1}{2}
   		 \left(\DP[2]{K_{m}^{2}}{x}
 			 + \DP[2]{K_{m}^{2}}{y}
 			 + \DP[2]{K_{m}^{2}}{z}
   		 \right)
                + \left(\DP{K_{m}}{x}\right)^{2}
	                  + \left(\DP{K_{m}}{y}\right)^{2}
			  + \left(\DP{K_{m}}{z}\right)^{2}
  \nonumber \\
	    & & - \frac{C_{\varepsilon}}{2 C_{m} l^{2}}
		  K_{m}^{2}
  \Deqlab{$B%5%V(BK}
  .
 \end{eqnarray}
 \Deqref{$B%5%V(BK}$B$H(B\Deqref{$B%5%V%0%j%C%IN%;6(BE}$B$H$rHf$Y$k$H(B,
 \Deqref{$B%5%V%0%j%C%IN%;6(BE}$BCf$K$O>e(B\Deqref{$B%5%V(BK}$B$N1&JUBh(B 4 $B9`(B
 \begin{eqnarray}
 - \frac{K_{m}}{3}
   \left(\DP{u}{x} + \DP{v}{y}+ \DP{w}{z}\right)
 \end{eqnarray}$B$HBh(B 6 $B9`(B
 \begin{eqnarray}
  \left(\DP{K_{m}}{x}\right)^{2}
      + \left(\DP{K_{m}}{y}\right)^{2}
      + \left(\DP{K_{m}}{z}\right)^{2}
 \end{eqnarray}
 $B$,$J$$(B.
 $B=`05=L7O$r9M$($F$$$k$N$G(B, $BBh(B 4 $B9`$O(B 0 $B$K$J$k(B.
 $BBh(B 6 $B9`$,$J$$$N$OITL@$G$"$k(B.
 $( )^{2}$ $B$J$N$GHy>.9`$H9M$($k$N$@$m$&$+(B.
 $B>C5n$9$kM}M3$,J,$+$i$J$$$N$G(B, $BBh(B 6 $B9`$O;D$9(B.

 $B$h$C$F(B, $BD>8rD>@~:BI87O$K$*$1$k%5%V%0%j%C%I%9%1!<%k$N(B
 $B1?F0%(%M%k%.!<$NJ}Dx<0$r(B $E$ $B$K4X$9$k<0$K=q$-49$($k$H(B,
 \begin{eqnarray}
  \DP{E}{t} &=& - \left(u \DP{E}{x}
		        + v \DP{E}{y} + w \DP{E}{z}
		  \right)
                + 3 g C_{m} l E^{1/2}
		  \left(- A \DP{\theta_{e}}{z}
		        +  \DP{q_{l}}{z}
		  \right)
  \nonumber \\
            & & + 2 C_{m} l E^{1/2}
	            \Biggl\{\left(\DP{u}{x}\right)^{2}
			   + \left(\DP{v}{y}\right)^{2}
			   + \left(\DP{w}{z}\right)^{2}
		    \Biggr\}
  \nonumber \\
	    & & + C_{m} l E^{1/2}
	          \Biggl\{\left(\DP{v}{x} + \DP{u}{y}\right)^{2}
			 + \left(\DP{w}{x} + \DP{u}{z}\right)^{2}
			 + \left(\DP{w}{y} + \DP{v}{z}\right)^{2}
	          \Biggr\}
  \nonumber \\
              & & + 2 C_{m} l E^{1/2}
	            \left[\frac{1}{2}
	             	    \Biggl(\DP[2]{E}{x}
		    		   + \DP[2]{E}{y}
		 		   + \DP[2]{E}{z}
	            	    \Biggr)
			    + \left(\DP{E^{1/2}}{x}\right)^{2}
			    + \left(\DP{E^{1/2}}{y}\right)^{2}
			    + \left(\DP{E^{1/2}}{z}\right)^{2}
		     \right]
  \nonumber \\
              & & - \frac{C_{\varepsilon}}{l} E^{3/2}
 \end{eqnarray}
 $B$H$J$k(B.
 $B$3$N<0$O(B Deardorff (1975), Mellor and Yamada (1974), Schemm and
 Lipps (1976) $B$J$I$GMQ$$$i$l$F$$$k$b$N$HF1$8J}Dx<0$G$"$k(B.

