B. $BMpN.%Q%i%a%?%j%

B.1 $BMpN.%Q%i%a%?%j%

Klemp and Wilhelmson (1978) $B$*$h$S(B CReSS $B$GMQ$$$i$l$F$$$k(B 1.5 $B

$$\DD{E}{t} = B + S + D_{E} - \left(\frac{C_{\varepsilon}}{l}\right) E^{\frac{3}{2}}$$ (B.1)

$B$HM?$($i$l$k(B. $l$ $B$O:.9g5wN%$G(B, $l = \left(\Delta x \Delta z \right)^{1/2}$ $B$H$9$k(B. $B$ $B$H(B $S$ $B$O$=$l$>$lIbNO$HN.$l$NJQ7AB.EY$K$h(B $B$kMpN.%(%M%k%.!<@[email protected]`(B, $D_{E}$ $B$OMpN.%(%M%k%.!<3H;69`(B, $BBh(B 4 $B9`$OMpN.(B $B%(%M%k%.!<$N>C;69`$G$"$j(B,

$$B = \frac{g_{j}}{\overline{\theta}} \overline{u^{\prime}_{j} \theta^{\prime}} ,$$ (B.2)

$$S = - \overline{(u_{i}^{\prime} u_{j}^{\prime})} \DP{u_{i}}{x_{j}} ,$$ (B.3)

$$D_{E} = \DP{}{x_{j}} \left(K_{m} \DP{E}{x_{j}} \right)$$ (B.4)

$B$G$"$k(B. 1.5 $B

$$\overline{(u_{i}^{\prime} u_{j}^{\prime})} = - K_{m} \left(\DP{u_{i}}{x_{j}} + \DP{u_{j}}{x_{i}}\right) + \frac{2}{3} \delta_{ij} E,$$ (B.5)

$$\overline{u_{j}^{\prime} \theta } = - K_{h}\DP{\theta}{x_{j}}.$$ (B.6)

$B$3$3$G(B $K_{m}$ $B$O1?F0NL$KBP$9$k12G4@-78?t$G$"$j(B, $E$ $B$O%5%V%0%j%C%I%9(B $B%1!<%k$NMpN.1?F0%(%M%k%.!<(B, $K_{h}$ $B$O123H;678?t$G$"$k(B. $K_{m}$, $K_{h}$ $B$O(B $E$ $B$rMQ$$$F0J2<$N$h$&$KM?$($i$l$k(B.

$$K_{m} = C_{m} E^{\frac{1}{2}} l,$$ (B.7)

$$K_{h} = 3 K_{m}.$$ (B.8)

$B%Q%i%a!<%?(B $C_{\varepsilon}, C_{m}$ $B$O$H$b$K(B 0.2 $B$G$"$k(B. a

B.1.1 $BMpN.1?F0%(%M%k%.!

Klemp and Wilhelmson (1978) $B$G$O(B(B.1)$B$K$D$$$F(B, $B!V(BDeardroff (1975), Mellor and Yamada (1974), Schemm and Lipps (1976) $B$GMQ$$$i$l(B $B$F$$$k<0$HN`;w$N$b$N$G$"$k!W$H$@$15-=R$5$l(B, $B$=$NF3=P$N>\:Y$K$D$$$F$O2r(B $B@b$5$l$F$$$J$$(B. $B$=$l$f$(Bg5$Bg=[4D%b%G%k$G$h$/MQ$$$i$l$F$$$k(B Mellor and Yamada (1974, 1982) $B$N%Q%i%a%?%j%B.1), (B.5), () $B$NF3=P$r9T$&(B.

$B9M$($F$$$k%5%V%0%j%C%I%9%1!<%kFb$K$*$$$F(B, $BL)EY$O0lDj(B, $BF0G4@-78?t$d3H;6(B $B78?t$J$I$NJ*M}Dj?t$O0lDj$H$9$k(B. $B=PH/E@$H$J$kJ}Dx<0$O(B, Mellor and Yamada (1973) $B$N<0(B (7) $B$*$h$S(B (8)

$$\DP{\overline{u^{\prime}_{i}u^{\prime}_{j}}}{t} + \DP{}{x_{k}}\left[ u_{k}\overline{u^{\prime}_{i}u^{\prime}_{j}} +... ...^{\prime}_{j}} - \nu \DP{}{x_{k}}\overline{u^{\prime}_{i}u^{\prime}_{j}}\right]$$

$$+ \DP{}{x_{j}}\overline{pu^{\prime}_{i}} + \DP{}{x_{i}}\overline{pu... ...me}u^{\prime}_{i}} + \varepsilon _{ikl}\overline{u_{l}^{\prime}u^{\prime}_{j}})$$

$$= -\overline{u^{\prime}_{k}u^{\prime}_{i}}\DP{u_{j}}{x_{k}} -\overl... ...(g_{j}\overline{u^{\prime}_{i}\theta } + g_{i}\overline{u^{\prime}_{j}\theta })$$

$$+ \overline{p\left(\DP{u^{\prime}_{i}}{x_{j}} + \DP{u^{\prime}_{j}}... ...right)} - 2\nu \overline{\DP{u^{\prime}_{i}}{x_{k}}\DP{u^{\prime}_{j}}{x_{k}}},$$ (B.9)

$$\DP{\overline{u^{\prime}_{j}\theta^{\prime}}}{t} + \DP{}{x_{k}}\left[ u_{k}\overline{\theta^{\prime} u^{\prime}_{j}}... ...a^{\prime} } + \varepsilon _{jkl}f_{k}\overline{u^{\prime}_{l}\theta^{\prime} }$$

$$= -\overline{u^{\prime}_{j}u^{\prime}_{k}}\DP{\theta }{x_{k}} -\ove... ...lpha + \nu ) \overline{\DP{u^{\prime}_{j}}{x_{k}}\DP{\theta^{\prime} }{x_{k}}}.$$ (B.10)

$B$*$h$S(B, (B.9)$B$K$*$$$F(B $i=j$ $B$H$7$?<0(B

$$\DP{q^{2}}{t} + u_{k}\DP{q^{2}}{x_{k}} + \DP{}{x_{k}}\left(\overline{u^{\prime}_{k}u^{\prime}_{j}u^{\prime}_{j}} - \nu \DP{q^{2}}{x_{k}}\right) = - \overline{u^{\prime}_{j}u^{\prime}_{k}}\DP{u_{j}}{x_{k}} - \DP{}{x_{k}}\overline{pu^{\prime}_{j}}$$

$$+ g_{j}\beta\overline{u^{\prime}_{j}\theta^{\prime}} - 2\nu \overline{\left(\DP{u^{\prime}_{j}}{x_{k}}\right)}$$ (B.11)

$B$3$3$G(B

$$\begin{eqnarray*} q &\equiv& \sqrt{\overline{(u^{\prime}_{i})^{2}}} \\ & = & \sqrt{2E} \end{eqnarray*}$$

$B$G(B, $\nu, \alpha, \beta$ $B$O$=$l$>$lF0G4@-78?t(B, $B3H;678?t$*$h$SG.KD(B $BD%N((B, $g_{j}$ $B$O=ENO2CB.EY%Y%/%H%k$NBh(B $j$ mail protected],$G$"$k(B.

(B.9)$B$*$h$S(B(B.10)$B$K8=$l$k05NO$K4X$9$kAj4X9`(B $B$*$h$S(B 3 $BDj$r$*$/(B.

1. $\overline{p\left(\DP{u^{\prime}_{i}}{x_{j}} + \DP{u^{\prime}_{i}}{x_{j}}\right)}$ ($B05NO$K$h$k1?F0%(%M%k%.!<$N:FJ,G[(B)

   
   

   $$= - \frac{q}{3l_{1}}(\overline{u^{\prime}_{i}u^{\prime}_{j}... ...+ Cq^{2}\left(\DP{u_{i}}{x_{j}} + \DP{u_{j}}{x_{i}}\right),$$
   
   
   $B$H$*$/(B. $B$3$3$G(B $l_{1}$ $B$OMpN.$NFCD'E*$J%9%1!<%k(B, $C$ $B$OL5
2. $\overline{p\DP{\theta^{\prime} }{x_{k}}}$ ($B05NO$K$h$kG.%(%M%k%.!<:FJ,G[(B)

   1. $B$NF3=P$HF1MM$N9M;!$K$h$C$F(B,
   

   $$= -\frac{q}{3l_{2}}\overline{u^{\prime}_{i}\theta^{\prime} }$$
   
   
   $B$H$*$/(B. $B$3$3$G$NMp$l$N%9%1!<%k$O(B $l_{2}$ $B$H$9$k(B.

3. $2\nu \overline{\DP{u^{\prime}_{i}}{x_{k}}\DP{u^{\prime}_{j}}{x_{k}}}$ ($BG4@-$K$h$k;60o(B)

   $BG4@-$K4XM?$9$k$h$&$J>.%9%1!<%k$N8=>]$OEyJ}E*$H$_$F(B $q$ $B$N$_(B $B$GI=8=$9$k(B.
   

   $$= \frac{2}{3}\frac{q^{3}}{\Lambda _{1}}\delta _{ij}.$$
   
   
   $B$3$3$G(B $\Lambda _{1}$ $B$OG4@-$N5Z$VFCD'E*%9%1!<%k$G$"$k(B.

4. $(\alpha + \nu ) \overline{\DP{u^{\prime}_{j}}{x_{k}}\DP{\theta^{\prime} }{x_{k}}}$

   
   

   $$= 0$$
   
   
   $B$H$*$/(B.

5. $\overline{u^{\prime}_{k}u^{\prime}_{i}u^{\prime}_{j}}, \overline{u^{\prime}_{... ...{\prime}_{j}\theta^{\prime} }, \overline{u^{\prime}_{k}(\theta^{\prime})^{2}}$

   $BB.EYJQF0$K$h$k(B $\overline{u^{\prime}_{k}u^{\prime}_{i}u^{\prime}_{j}}, \overline{u^{\prime}_{... ...{\prime}_{j}\theta^{\prime} }, \overline{u^{\prime}_{k}(\theta^{\prime})^{2}}$ $B$H9M$(

   $$\begin{eqnarray*} \overline{u^{\prime}_{k}u^{\prime}_{i}u^{\prime}_{j}} &=& -... ... -q\lambda _{3} \DP{\overline{(\theta^{\prime})^{2}}}{x_{k}}. \end{eqnarray*}$$
   
   
   $B$3$3$G(B $\lambda _{i}(i=1,2,3)$ $B$O$=$l$>$l$NFCD'E*%9%1!<%k$G$"$k(B.

6. $\overline{pu^{\prime}_{i}}, \overline{p\theta^{\prime} }$ ($B05NOJQF0$K$h$k3H;6(B)

   
   

   $$\overline{pu^{\prime}_{i}} = \overline{p\theta^{\prime} } = 0$$
   
   
   $B$H$9$k(B. $B$3$N6a;w$O(B Deardroff (1975), Schemm and Lipps (1976) $B$G$b9T$o$l$F$$$k(B.

7. $f_{k}(\varepsilon _{jkl}\overline{u_{l}^{\prime}u^{\prime}_{i}} + \varepsilon ... ...me}_{j}}), \; f_{k}\varepsilon _{jkl}\overline{u_{l}^{\prime}\theta^{\prime} }$ ($B%3%j%*%j9`(B)

   
   

   $$f_{k}\varepsilon _{jkl}\overline{u_{l}^{\prime}u^{\prime}_{... ...varepsilon _{ikl}\overline{u_{l}^{\prime}u^{\prime}_{j}} = 0,$$
   
   
   
   
   $$f_{k}\varepsilon _{jkl}\overline{u_{l}^{\prime}\theta^{\prime}} = 0$$
   
   
   $B$H$9$k(B. $B$3$N6a;w$O(B Deardroff (1975), Schemm and Lipps (1976) $B$G$b9T$o$l$F$$$k(B.

8. $\alpha \overline{u^{\prime}_{j}\DP{\theta^{\prime} }{x_{k}}}, \; \nu \overline{\theta^{\prime} \DP{u^{\prime}_{j}}{x_{k}}}$

   
   

   $$= 0$$ (B.12)

   
   
   $B$H$9$k(B.

$B0J>e$N6a;w$r(B(B.9), (B.10), (B.11) $B$KBP$7$F9T$&$H(B, $B0J2<$N<0$rF@$k(B.

$$\DD{\overline{u^{\prime}_{i}u^{\prime}_{j}}}{t} - \DP{}{x_{k}}\left[ q\lambda _{1}\left( \DP{\overline{u^{\prime}_{... ...}}{x_{i}}\right) -\nu\DP{}{x_{k}}\overline{u^{\prime}_{i}u^{\prime}_{j}}\right]$$

$$= - \overline{u^{\prime}_{k}u^{\prime}_{i}}\DP{u_{j}}{x_{k}} - \ove... ...{\prime}_{i}\theta^{\prime} } + g_{i}\overline{u^{\prime}_{j}\theta^{\prime} })$$

$$- \frac{q}{3l_{1}}(\overline{u^{\prime}_{i}u^{\prime}_{j}} - \fra... ...+ \DP{u_{j}}{x_{i}}\right) - \frac{2}{3}\frac{q^{3}}{\Lambda _{1}}\delta _{ij},$$ (B.13)

$$\DD{\overline{u^{\prime}_{j}\theta^{\prime} }}{t} - \DP{}{x_{k}}\left[q\lambda _{2}\left( \DP{\overline{u^{\prime}_{k... ...}}{x_{j}} + \DP{\overline{u^{\prime}_{j}\theta^{\prime} }}{x_{k}}\right)\right]$$

$$= - \overline{u^{\prime}_{j}u^{\prime}_{k}}\DP{\theta }{x_{k}} - \o... ...eta^{\prime})^{2}} - \frac{q}{3l_{2}}\overline{u^{\prime}_{j}\theta^{\prime} },$$ (B.14)

$$\DD{q^{2}}{t} + \DP{}{x_{k}}\left[ q\lambda _{1}\left( 2\DP{q^{2}}{x_{k}} + \DP{\... ...ine{u^{\prime}_{j}u^{\prime}_{k}}}{x_{j}}\right) - \nu \DP{q^{2}}{x_{k}}\right]$$

$$= - 2\overline{u^{\prime}_{j}u^{\prime}_{k}}\DP{u_{j}}{x_{k}} + 2g_{j}\beta\overline{u^{\prime}_{j}\theta^{\prime}} - 2\frac{q^{3}}{\Lambda _{1}}$$ (B.15)

$B$3$3$G(B

$$\DD{}{t} \equiv \DP{}{t} + u_{k}\DP{}{x_{k}}$$

$B$G$"$k(B. $B$3$l$i$O(B Mellor and Yamada (1974) $B$N(B Level 4 $B%b%G%k$N<0$KBP1~(B $B$9$k<0$G$"$k(B.

$B<0(B(B.13), (B.14), (B.15)$B$KBP$7(B, $B$5$i$K0J2<$N6a;w$r2C$($k(B.

• $B<0(B(B.13)$B$O(B, $B1&JU$NBh(B 4 $B9`$HBh(B 5 $B9`$@$19MN8$9$k(B. $B$5$i$K(B(B.16)$B$G$O(B $C=1/3$ $B$H$9$k(B.

• $B<0(B(B.14)$B$O(B, $B1&JU$NBh(B 1 $B9`$HBh(B 4 $B9`$@$19MN8$9$k(B. $B$5$i$K(B $\overline{u^{\prime}_{j}u^{\prime}_{k}}\sim q^{2}\delta_{jk}/3$ $B$H$9$k(B.

• $B<0(B(B.15)$B$O(B, $B:8JU$N(B 3 $B

$B$3$l$i$N6a;w$r9T$&$H(B, $B<0(B(B.13), (B.14), (B.15)$B$O(B

$$\overline{u^{\prime}_{i}u^{\prime}_{j}} = \frac{\delta _{ij}}{3}q^{2} - ql_{1}\left(\DP{u_{i}}{x_{j}} + \DP{u_{j}}{x_{i}}\right)$$ (B.16)

$$\overline{u^{\prime}_{j}\theta^{\prime}} = - ql_{2}\DP{\theta }{x_{j}}$$ (B.17)

$$\DD{q^{2}}{t} = - 2\overline{u^{\prime}_{j}u^{\prime}_{k}}\DP{u_{j}}{x_{k}} + 2g_... ... + \DP{}{x_{k}}\left[\nu \DP{q^{2}}{x_{k}}\right] - 2\frac{q^{3}}{\Lambda _{1}}$$ (B.18)

$B$H$J$k(B. (B.16)$B$O(B Mellor and Yamada (1974) $B$N(B Level 1 $B%b%G%k$N(B $\overline{u^{\prime}_{i}u^{\prime}_{j}}$ $B$N<0$G$"$k(B. (B.17)$B$O(B Mellor and Yamada (1974) $B$N(B Level 1 $B%b%G%k(B $B$N(B $\overline{u^{\prime}_{j}\theta^{\prime}}$ $B$N<0$G(B $\overline{(\theta^{\prime})^{2}}$ $B$N9`$rL5;k$7$?$b$N$KBP1~$9$k(B. (B.18) $B$O(B Mellor and Yamada (1974) $B$N(B Level 3 $B%b%G(B $B%k$N(B $q^{2}$ $B$N<0$K$*$$$F(B, 3 $B $ql_{1}=K_{m}, ql_{2}=K_{h}$ $B$H$7(B, $q$ $B$r(B $E$ $B$GI=$7F0G4@-78?t$rMpN.3H;678?t$GCV$-49$($k$H(B

$$\overline{u^{\prime}_{i}u^{\prime}_{j}} = \frac{2}{3}\delta _{ij}E - K_{m}\left(\DP{u_{i}}{x_{j}} + \DP{u_{j}}{x_{i}}\right)$$ (B.19)

$$\overline{u^{\prime}_{j}\theta^{\prime}} = - K_{h}\DP{\theta }{x_{k}}$$ (B.20)

$$\DD{E}{t} = - \overline{u^{\prime}_{j}u^{\prime}_{k}}\DP{u_{j}}{x_{k}} + g_{j... ..._{k}}\left[K_{m} \DP{q^{2}}{x_{k}}\right] - \frac{2^{3/2}}{\Lambda _{1}}E^{3/2}$$ (B.21)

$B$H$J$k(B. $BM}A[5$BN$N>l9g(B $\beta = 1/\theta$ $B$G$"$k$3$H$KCm0U$9$k$H(B, $B<0(B (B.21)$B$O;60o9`$N78?t$r=|$-(B(B.1)$B$K0lCW(B $B$9$k(B.

$B0J>e$h$j(B, Klemp and Wilhelmson (1978) $B$NMpN.%Q%i%a%?%j%$NAj4XNL$O?GCGE*$K5a$a$k%b%G%k$H$7$F(B Mellor and Yamada (1974) $B$N(B Level 2.5 $B%b%G%k$,$"$k(B. $B$7$+$7(B Level 2.5 $B%b%G%k$O(B Level 3 $B%b%G%k$H(B Level 2 $B%b%G%k$H$NAH9g$;$G$"$k$3$H$KCm0U$,I,(B $BMW$G$"$k(B.

B.1.2 2 $Bl9g$NI=8=(B

2 $Bl9g$N(B(B.1)$B<0$N3F9`$r=q$-2<$9(B. $BIbNO$K$h$kMpN.%(%M%k(B $B%.!<@[email protected]`$O(B,

$$B = \frac{g_{j}}{\overline{\theta}} \overline{u^{\prime}_{j} \theta^{\prime}}$$

$$= - \frac{g}{\overline{\theta}} \overline{w^{\prime} \theta^{\prime}}$$

$$= - \frac{g}{\overline{\theta}} \left( K_{h} \DP{\theta}{z} \right)$$ (B.22)

$B$G$"$k(B. $B $B$O(B,

$$S = - \overline{(u_{i}^{\prime} u_{j}^{\prime})} \DP{u_{i}}{x_{j}}$$

$$= - \left\{ - K_{m} \left(\DP{u_{i}}{x_{j}} + \DP{u_{j}}{x_{i}}\right) + \frac{2}{3} \delta_{ij} E \right\} \DP{u_{i}}{x_{j}}$$

$$= \left\{ K_{m} \left(\DP{u_{i}}{x_{j}} + \DP{u_{j}}{x_{i}}\right) - \frac{2}{3} \delta_{ij} E \right\} \DP{u_{i}}{x_{j}}$$

$$= \left\{ K_{m} \left(\DP{u}{x_{j}} + \DP{u_{j}}{x}\right) - \frac{2}{3} \delta_{1j} E \right\} \DP{u}{x_{j}}$$

$$+ \left\{ K_{m} \left(\DP{w}{x_{j}} + \DP{u_{j}}{z}\right) - \frac{2}{3} \delta_{3j} E \right\} \DP{w}{x_{j}}$$

$$= \left\{ 2 K_{m} \left(\DP{u}{x} \right) - \frac{2}{3} E \right\} \DP{u}{x} + K_{m} \left( \DP{w}{x} + \DP{u}{z} \right) \DP{u}{z}$$

$$+ K_{m} \left(\DP{w}{x} + \DP{u}{z}\right) \DP{w}{x} + \left\{ 2 K_{m} \left(\DP{w}{z} \right) - \frac{2}{3} E \right\} \DP{w}{z}$$

$$= 2 K_{m} \left\{ \left( \DP{u}{x} \right)^{2} + \left( \DP{w}{z} \right)^{2} \right\} + K_{m} \left(\DP{u}{z} + \DP{w}{x}\right)^{2}$$

$$- \frac{2}{3} E \left( \DP{u}{x} + \DP{w}{z} \right)$$ (B.23)

$B$G$"$k(B. $BMpN.%(%M%k%.!<3H;69`(B $D_{E}$ $B$O(B,

$$D_{E} = \DP{}{x_{j}} \left(K_{m} \DP{E}{x_{j}} \right),$$

$$= \DP{}{x} \left(K_{m} \DP{E}{x} \right) + \DP{}{x} \left(K_{m} \DP{E}{x} \right)$$ (B.24)

$B$G$"$k(B. $B0J>e$N(B (B.22), (B.23), (B.24) $B<0$r(B (B.1) $B<0(B $B$KBeF~$9$k$3$H$G0J2<$N<0$rF@$k(B.

$$\DD{E}{t} = - \frac{g}{\overline{\theta}} \left( K_{h} \DP{\theta}{z} \right)$$

$$+ 2 K_{m} \left\{ \left( \DP{u}{x} \right)^{2} + \left( \DP{w}{z}... ...{z} + \DP{w}{x}\right)^{2} - \frac{2}{3} E \left( \DP{u}{x} + \DP{w}{z} \right)$$

$$+ \DP{}{x} \left(K_{m} \DP{E}{x} \right) + \DP{}{x} \left(K_{m} \DP{E}{x} \right) - \left(\frac{C_{\varepsilon}}{l}\right) E^{\frac{3}{2}}.$$ (B.25)

B.1.3 $BMpN.3H;678?t$rMQ$$$?I=8=(B

(B.25) $B<0$r(B (B.7) $B<0$rMQ$$$F(B $K_{m}$ $B$K4X$9$k<0$KJQ7A(B $B$9$k(B. $B1&JU$NMpN.%(%M%k%.!<3H;69`$r=q$-2<$9$H(B,

$$\DP{}{x} \left(K_{m} \DP{E}{x}\right) + \DP{}{z} \left(K_{m} \DP{E}{z}\right)$$

$$= \frac{1}{C_{m}^{2} l^{2}} \Biggl\{\DP{}{x} \left(K_{m} \DP{K_{m}^{2}}{x}\right) + \DP{}{z} \left(K_{m} \DP{K_{m}^{2}}{z}\right) \Biggr\}$$

$$= \frac{1}{C_{m}^{2} l^{2}} \Biggl\{K_{m} \DP[2]{K_{m}^{2}}{x} + \D... ...{2}}{x} + K_{m} \DP[2]{K_{m}^{2}}{z} + \DP{K_{m}}{z} \DP{K_{m}^{2}}{z} \Biggr\}$$

$$= \frac{K_{m}}{C_{m}^{2} l^{2}} \left(\DP[2]{K_{m}^{2}}{x} + \DP[2]... ...Biggl\{\left(\DP{K_{m}}{x}\right)^{2} + \left(\DP{K_{m}}{z}\right)^{2} \Biggr\}$$

$B$H$J$k$N$G(B, (B.25) $B<0$rJQ7A$9$k$H(B,

$$\frac{2 K_{m}}{C_{m}^{2} l^{2}} \DD{K_{m}}{t} = - \frac{g}{\overline{\theta}} \left( K_{h} \DP{\theta}{z} \right)... ...m} \left\{ \left( \DP{u}{x} \right)^{2} + \left( \DP{w}{z} \right)^{2} \right\}$$

$$+ K_{m} \left(\DP{u}{z} + \DP{w}{z}\right)^{2} - \frac{2}{3} \frac{K_{m}^{2}}{C_{m}^{2} l^{2}} \left( \DP{u}{x} + \DP{w}{z} \right)$$

$$+ \frac{K_{m}}{C_{m}^{2} l^{2}} \left(\DP[2]{K_{m}^{2}}{x} + \DP[... ...Biggl\{\left(\DP{K_{m}}{x}\right)^{2} + \left(\DP{K_{m}}{z}\right)^{2} \Biggr\}$$

$$- \frac{C_{\varepsilon}}{C_{m}^{3} {l}^{4}} K_{m}^{3}.$$ (B.26)

$B78?t$r@0M}$9$k$H(B,

$$\DP{K_{m}}{t} = - \left( u \DP{K_{m}}{x} + w \DP{K_{m}}{z} \right) - \frac{g C_{m... ...} l^{2}}{ 2 \overline{\theta}} \frac{K_{h}}{K_{m}} \left(\DP{\theta}{z} \right)$$

$$+ \left( C_{m}^{2} l^{2} \right) \left\{ \left( \DP{u}{x} \right)^{2} + \left( \DP{w}{z} \right)^{2} \right\}$$

$$+ \frac{ C_{m}^{2} l^{2} }{2} \left( \DP{u}{z} + \DP{w}{z}\right)^{2} - \frac{K_{m}}{3} \left( \DP{u}{x} + \DP{w}{z} \right)$$

$$+ \Dinv{2} \left(\DP[2]{K_{m}^{2}}{x} + \DP[2]{K_{m}^{2}}{z} \right) + \left(\DP{K_{m}}{x}\right)^{2} + \left(\DP{K_{m}}{z}\right)^{2}$$

$$- \frac{C_{\varepsilon}}{2 C_{m} l^{2}} K_{m}^{2}$$

$B$H$J$j(B, $B$3$3$G(B $C_{m} = C_{\varepsilon} = 0.2$ $B$H(B $K_{h} = 3 K_{m}$ $B$H$$$&(B $B4X78$rMQ$$$k$H(B,

$$\DP{K_{m}}{t} = - \left( u \DP{K_{m}}{x} + w \DP{K_{m}}{z} \right) - \frac{3 g C_{m}^{2} l^{2}}{ 2 \overline{\theta}} \left(\DP{\theta}{z} \right)$$

$$+ \left( C_{m}^{2} l^{2} \right) \left\{ \left( \DP{u}{x} \right)^{2} + \left( \DP{w}{z} \right)^{2} \right\}$$

$$+ \frac{ C_{m}^{2} l^{2} }{2} \left( \DP{u}{z} + \DP{w}{x}\right)^{2} - \frac{K_{m}}{3} \left( \DP{u}{x} + \DP{w}{z} \right)$$

$$+ \Dinv{2} \left(\DP[2]{K_{m}^{2}}{x} + \DP[2]{K_{m}^{2}}{z} \right) + \left(\DP{K_{m}}{x}\right)^{2} + \left(\DP{K_{m}}{z}\right)^{2}$$

$$- \Dinv{2 l^{2}} K_{m}^{2}$$ (B.27)

$B$H$J$k(B.