• B.1 $B%5%V%0%j%C%I%9%1!<%k$N1?F0%(%M%k%.!

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B.1 $B%5%V%0%j%C%I%9%1!<%k$N1?F0%(%M%k%.!

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)

$B$H$9$k(B. $BC"$7(B $l$ $B$O:.9g5wN%$G$"$j(B $l = \left(\Delta x \Delta z \right)^{1/2}$ $B$H$9$k(B. $B$ $B$H(B $S$ $B$O(B $B$=$l$>$lIbNO$HN.$l$NJQ7AB.EY$K$h$kMpN.%(%M%k%.!<@[email protected]`(B, $D_{E}$ $B$OMpN.%((B $B%M%k%.!<3H;69`(B, $BBh(B 4 $B9`$OMpN.%(%M%k%.!<$N>C;69`$G$"$j(B,

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

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

$$D_{E} = \DP{}{x_{j}} \left(K_{m} \DP{E}{x_{j}} \right)$$ (B.2)

$B$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.3)

$$\overline{u_{j}^{\prime} \theta } = K_{h}\DP{\theta}{x_{j}}$$ (B.4)

$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.5)

$$K_{h} = 3 K_{m}.$$ (B.6)

(B.1) $B<0$N3F9`$r=q$-2<$9(B. $BIbNO$K$h$kMpN.%(%M%k%.!<@[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.7)

$B$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.8)

$B$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.9)

$B$G$"$k(B. $B0J>e$N(B (B.7), (B.8), (B.9) $B<0$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.10)

$B$5$i$K(B (B.10) $B<0$r(B (B.5) $B<0$rMQ$$$F(B $K_{m}$ $B$K4X$9$k<0(B $B$KJQ7A$9$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.10) $B<0$rJQ7A$9$k$H(B,

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

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

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

$$- \frac{C_{\varepsilon}}{C_{m}^{3} {l}^{4}} K_{m}^{3}$$

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

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

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

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

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

$B$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.11)

$B$H$J$k(B.