A. $B=`05=LJ}Dx<07O$NF3=P(B

A.1 $B4pACJ}Dx<0(B

$BCO5eBg5$$K$*$1$k<>=aBPN.$NDj<02=F1MM(B, $BBg5$$N4%[email protected],$H<>[email protected],$N(B $BJ,;RNL$N:9$OL)EY$N<0$K$O9MN8$9$k$,(B, $BG.$N<0$K$O9MN8$7$J$$$h$&$J(B $B7O$r9M$($k(B. $B$3$N7O$G$OBg5$$NG.%(%M%k%.!<$O4%AgBg5$$NG.%(%M%k%.!<$G(B $B7h$^$k$3$H$K$J$k(B. $B$3$N$h$&$J7O$G$O290L(B $\theta$ $B$,J]B8NL$H$7$F;H$($k(B.

A.1.1 $B5$29(B $T$, $BL)EY(B $\rho$, $BIwB.(B $u, w$ $B$rM=JsJQ?t$H$9$k>l9g(B

$B?eJ?1tD>(B 2 $BuBV$r(B $B5$29(B $T$, $B05NO(B $p$, $BIwB.(B $u, w$, $BL)EY(B $\rho$ $B$GI=8=$9$k>l9g(B, $B4pACJ}Dx<07O$O0J2<$N$h$&$K$J$k(B.

$B1?F0J}Dx<0(B

 

$$\DD{u}{t} + \Dinv{\rho}\DP{p}{x} = Turb.u$$ (A.1)

$$\DD{w}{t} + \Dinv{\rho}\DP{p}{z} = -g + Turb.w$$ (A.2)

$BO"B3$N<0(B

 

$$\DD{\rho}{t} + \rho\left( \DP{u}{x} + \DP{w}{z}\right) = 0$$ (A.3)

$BL)EY$N<0(B($B>uBVJ}Dx<0(B)

 

$$\rho = \frac{p}{R_{d} T_{v}}$$ (A.4)

$BG.$N<0(B

 

$${c_{p}}_{d}\DD{T}{t} - \Dinv{\rho_{d}} \DD{p}{t} = Q + Turb.T$$ (A.5)

mail protected],$N:.9gHfJ]B8<0(B

 

$$\DD{q_{v}}{t} = Src.q_{v} + Turb.q_{v}$$ (A.6)

$$\DD{q_{c}}{t} = Src.q_{c} + Turb.q_{c}$$ (A.7)

$$\DD{q_{r}}{t} = Src.q_{r} + Fall.q_{r} + Turb.q_{r}$$ (A.8)

$B$3$3$G(B $R_{d}$, ${c_{p}}_{d}$, $\rho_{d}$ $B$OC10L $B$OHsCGG.2CG.(B, $q_{v}$ mail protected],$N:.9gHf(B, $q_{c}$ $B$O1@?e:.9gHf(B, $q_{r}$ $B$O1+?e:.9gHf$G$"$k(B. $q_{v}, q_{r}, q_{c}$ $B$O(B, mail protected],$N?t$@$1B8:_$9$k(B. $Turb$, $Src$, $Fall$ $B$rIU$1$?9`$O$=$l$>$l(B $B3H;69`(B, $B@8@.>CLG9`(B, $BMn2<9`$r0UL#$9$k(B.

$BL)EY$N<[email protected],$N:.9gHf$,9MN8$5$l$F$$$k(B.

$$\rho = \rho_{d} + \sum \rho_{v} + \sum \rho_{c} + \sum \rho_{r}$$

$$= \rho_{d} (1 + \sum q_{v} + \sum q_{c} + \sum q_{r} ).$$ (A.9)

$B$?$@$7(B, $q_v = \rho_v/\rho_{d}$, $q_{c}$, $q_{r}$ $B$O$=$l$>$l(B, $B6E=L@-5$BN(B, $B1@?e(B, $B1+?e$N:.9gHf$r0UL#$9$k(B. $B$3$3$G4%[email protected],$NJ,05(B $p_{d}$ $B$O(B.

$$p_{d} = p \left( 1 - \frac{\sum p_{v}}{p} \right)$$

$$= p \left( 1 - \frac{\sum p_{v}}{p_{d} + \sum p_{v} }\right)$$

$$= p \left( 1 - \frac{\sum \rho_{v} R_{v} T}{\rho_{d} R_{d}T + \sum \rho_{v} R_{v} T }\right)$$

$$= p \left( 1 - \frac{\sum q_{v}/M_{v}}{1/M_{d} + \sum q_{v}/M_{v} }\right)$$

$B$H$J$k$N$G(B,

$$\rho_{d} = \frac{p_{d}}{R_{d} T} = \frac{p}{R_{d} T} \left(\frac{1/M_{d}}{1/M_{d} + \sum q_{v}/M_{v} }\right)$$ (A.10)

$B$G$"$k(B. $BC"$7(B $M$ $B$OJ,;RNL$rI=$7(B, mail protected],$NBN@Q$OL5;k$G$-$k$b$N$H8+$J$7$?(B. (A.9), (A.10) $B<0$h$j(B,

$$\rho = \frac{p}{R_{d}T} \left( \frac{1/M_{d}}{1/M_{d} + \sum q_{v}/M_{v} }\right) (1 + \sum q_{v} + \sum q_{c} + \sum q_{r} )$$ (A.11)

$B$H$J$k(B.

$$f \equiv \left(\frac{1/M_{d}}{1/M_{d} + \sum q_{v}/M_{v} }\right) (1 + \sum q_{v} + \sum q_{x} )$$

$B$HDj5A$9$k$H(B, (A.11) $B<0$O0J2<$N$h$&$K=q$1$k(B.

$$\rho = \frac{p}{R_{d} (T/f)}$$ (A.12)

$B$^$?(B, $B290L$H%(%/%9%J!<4X?t$rMQ$$$FI=8=$9$k$H(B,

$$\rho = \frac{p}{R_{d} \pi (\theta /f )}$$

$$= \frac{p_{0} \pi^{{c_{v}}_{d}/R_{d}}}{R_{d} (\theta/f)}$$ (A.13)

$B$G$"$k(B. $BC"$7%(%/%9%J!<4X?t(B $\pi$ $B$O(B $\pi = T/\theta$ $B$N4X78$rK~$?$9(B.

$B290L$O4%AgCGG.>uBV$K$*$1$kJ]B8NL$G$"$k(B. $B4%AgCGG.>uBV$rI=$9G.NO3X$N<0$O(B

$$c_{p}dT - \alpha dp = 0$$ (A.14)

$B$G$"$k(B. $B$3$3$G(B $T$ $B$O29EY(B, $p$ $B$O05NO(B, $c_{p}$ $B$OC10L $B$OHfMF$G$"$k(B. (A.14) $B<0$N(B $\alpha$ $B$O(B, $BM}A[5$BN$N>uBVJ}Dx<0$rMQ$$$k$H(B,

$$\alpha = \frac{RT}{p}$$ (A.15)

$B$H=q$1$k(B. $B$3$3$G(B $M$ $B$OJ,;RNL(B, $R$ $B$O5$BNDj?t$G$"$k(B. (A.14) $B<0$K(B (A.15) $B<0$rBeF~$7@0M}$9$k$H(B,

$$\frac{c_{p}}{T}dT - \frac{R}{p} dp = 0$$ (A.16)

$B$H$J$k(B. $B6E=L$r@8$8$J$$>l9g$K$O5$2t$NAH@.$OJQ2=$7$J$$$N$G(B $c_{p}$ $B$H(B $R$ $B$O6&$K(B $p$ $B$K0MB8$7$J$$(B. $B0lHL$K(B $c_{p}$ $B$O(B $T$ $B$N4X?t$G$"$k$,(B, $c_{p}$ $B$rDj?t$H$_$J$9$H(B,

$$\int^{T_{0}}_{T} \Dinv{T}dT = \frac{R}{c_{p}} \int^{p_{0}}_{p} \Dinv{p} dp$$

$$\ln{(T_{0}/T)} = \frac{R}{c_{p}} \ln{(p_{0}/p)}$$

$$\theta = T \left(\frac{p_{0}}{p}\right)^\frac{R}{c_{p}}$$ (A.17)

$B$H$J$j(B, $B290L$,F@$i$l$k(B.

A.1.2 $B290L(B $\theta$, $B05NO(B $p$, $BIwB.(B $u, w$ $B$rM=JsJQ?t$H$9$k>l9g(B

$B?eJ?1tD>(B 2 $BuBV$r(B $B290L(B $\theta$, $B05NO(B $p$, $BIwB.(B $u, w$, $BL)EY(B $\rho$ $B$GI=8=$9$k>l9g(B, $B4pACJ}Dx<07O$O0J2<$N$h$&$K$J$k(B. CReSS($BDZLZ$H:g86(B, 2001)$B$G$O(B, $B$3$N4pACJ}Dx<0$rMQ$$$F$$$k(B.

$B1?F0J}Dx<0(B

 

$$\DD{u}{t} + \Dinv{\rho}\DP{p}{x} = Turb.u$$ (A.18)

$$\DD{w}{t} + \Dinv{\rho}\DP{p}{z} = -g + Turb.w$$ (A.19)

$B05NOJ}Dx<0(B

 

$$\DD{p}{t} = \rho {C_{s}}^{2} \left\{ - \Ddiv{\Dvect{u}} + \Dinv{\... ...\DP{f}{q_{c}} \DD{q_{c}}{t} + \sum \DP{f}{q_{r}} \DD{q_{r}}{t} \right) \right\}$$ (A.20)

$BL)EY$N<0(B($B>uBVJ}Dx<0(B)

 

$$\rho = \frac{p_{0}}{R_{d} \theta_{v}} \left( \frac{p}{p_{0}}\right)^{{c_{v}}_{d}/{c_{p}}_{d}}$$ (A.21)

$BG.$N<0(B

 

$$\DD{\theta}{t} = Q + Turb.\theta$$ (A.22)

mail protected],$N:.9gHf$NJ]B8<0(B

 

$$\DD{q_{v}}{t} = Src.q_{v} + Turb.q_{v}$$ (A.23)

$$\DD{q_{c}}{t} = Src.q_{c} + Turb.q_{c}$$ (A.24)

$$\DD{q_{r}}{t} = Src.q_{r} + Fall.q_{r} + Turb.q_{r}$$ (A.25)

$B$?$@$7290L(B $\theta$ $B$O(B

$$\theta \equiv T \left(\frac{p_{0}}{p}\right)^{R_{d}/{c_{p}}_{d}}$$ (A.26)

$B$G$"$j(B, $B2>290L(B $\theta_{v}$ $B$O(B,

$$\theta_{v} \equiv \frac{\theta}{f}$$ (A.27)

$B$G$"$k(B. $B2;B.(B $C_{s}$ $B$O(B

$${C_{s}}^{2} \equiv \frac{{c_{p}}_{d}}{{c_{v}}_{d}} R_{d} T_{v} = \frac{{c_{p}}_{d}}{{c_{v}}_{d}} R_{d} \frac{T}{f}$$ (A.28)

$B$G$"$k(B. ${c_{p}}_{d}$ $B$H(B ${c_{v}}_{d}$ $B$O$=$l$>$lC10L ${c_{v}}_{d} + R_{d} = {c_{p}}_{d}$ $B$H$$$&(B $B4X78$K$"$k(B.

$B05NOJ}Dx<0$OL)EY$N<0$HO"B3$N<0$rAH$_9g$o$;$k$3$H$GF@$i$l$k(B. $B$^$:L)EY$r(B $\rho= \rho(\theta,p,q_{v},q_{x})$ $B$H$7$F(B $\rho$ $B$NA4HyJ,$r5a$a$k(B.

$$d\rho = d \left[ \frac{p_{0}}{R_{d} \theta_{v}} \left( \frac{p}{p_{0}} \right)^{{c_{v}}_{d}/{c_{p}}_{d}} \right]$$

$$= d \left[ \frac{p_{0}}{R_{d} (\theta / f) } \left( \frac{p}{p_{0}} \right)^{{c_{v}}_{d}/{c_{p}}_{d}} \right]$$

$$= \frac{p_{0} }{R_{d} (\theta/f)} \frac{{c_{v}}_{d}}{{c_{p}}_{d}} \... ...t( \frac{p}{p_{0}} \right)^{{c_{v}}_{d}/{c_{p}}_{d}} \frac{d\theta}{\theta^{2}}$$

$$+ \frac{p_{0}}{R_{d} \theta} \left( \frac{p}{p_{0}} \right)^{{c_{... ...{q_{v}} dq_{v} + \sum \DP{f}{q_{c}} dq_{c} + \sum \DP{f}{q_{r}} dq_{r} \right\}$$

$$= \Dinv{{C_{s}}^{2}} dp - \frac{\rho}{\theta} d\theta + \sum \frac{... ... \frac{\rho}{f} \DP{f}{q_{c}} dq_{c} + \sum \frac{\rho}{f} \DP{f}{q_{r}} dq_{r}$$ (A.29)

$B$H$J$k(B. (A.29) $B<0$r05NO$N<0$H$7$F@0M}$9$k$H(B,

$$\DD{p}{t} = {C_{s}}^{2} \left( \DD{\rho}{t} + \frac{\rho}{\theta}... ...rho}{f} \DP{f}{q_{c}} dq_{c} - \sum \frac{\rho}{f} \DP{f}{q_{r}} dq_{r} \right)$$

$B$G$"$j(B, $BO"B3$N<0$rMQ$$$k$H(B,

$$\DD{p}{t} = \rho {C_{s}}^{2} \left( - \Ddiv \Dvect{u} + \Dinv{\th... ...\sum \Dinv{f} \DP{f}{q_{c}} dq_{c} - \sum \Dinv{f} \DP{f}{q_{r}} dq_{r} \right)$$

$B$H$J$j(B, $B05NOJ}Dx<0$,F@$i$l$k(B.

A.1.3 $B290L(B $\theta$, $BL5, $BIwB.(B $u, w$ $B$rM=JsJQ?t$H$9$k>l9g(B

$B?eJ?1tD>(B 2 $BuBV$r(B $B290L(B $\theta$, $BL5, $BIwB.(B $u, w$, $BL)EY(B $\rho$ $B$GI=8=$9$k>l9g(B, $B4pACJ}Dx<07O$O0J2<$N$h$&$K$J$k(B. $BO"B3$N<0(B (A.3) $B$H>uBVJ}Dx<0(B (A.21) $B$rMQ$$$k$3$H$GF@$i$l$k05NOJ}Dx<0$rMxMQ$9$k(B. Klemp and Willhelmson (1978)$B$G$O(B, $B$3$N4pACJ}Dx<0$rMQ$$$F$$$k(B.

$B1?F0J}Dx<0(B

 

$$\DD{u}{t} + {{c_{p}}_{d}} \theta_{v} \DP{\pi}{x} = Turb.u$$ (A.30)

$$\DD{w}{t} + {{c_{p}}_{d}} \theta_{v} \DP{\pi}{z} = -g + Turb.w$$ (A.31)

$B05NOJ}Dx<0(B

 

$$\DD{\pi}{t} = \frac{{C_{s}}^{2}}{ {c_{p}}_{d} \rho \theta_{v} } \... ...\DP{f}{q_{c}} \DD{q_{c}}{t} + \sum \DP{f}{q_{r}} \DD{q_{r}}{t} \right) \right\}$$ (A.32)

$B>uBVJ}Dx<0(B

 

$$\rho = \frac{p_{0} \pi^{{c_{v}}_{d}/R_{d}}}{R_{d} \theta_{v}}$$ (A.33)

$BG.$N<0(B

 

$$\DD{\theta}{t} = Q + Turb.\theta$$ (A.34)

$B?e>x5$$*$h$S?eJ*

 

$$\DD{q_{v}}{t} = Src.q_{v} + Turb.q_{v}$$ (A.35)

$$\DD{q_{x}}{t} = Src.q_{c} + Turb.q_{c}$$ (A.36)

$$\DD{q_{x}}{t} = Src.q_{r} + Fall.q_{x} + Turb.q_{r}$$ (A.37)

$B$?$@$7(B, $B%(%/%9%J!<4X?t(B $\pi$ $B$O(B,

$$\pi \equiv \frac{T}{\theta} = \left( \frac{p}{p_{0}} \right)^{R_{d}/{c_{p}}_{d}}$$ (A.38)

$B$G$"$j(B, $B2;B.(B $C_{s}$ $B$O(B

$${C_{s}}^{2} \equiv \frac{{c_{p}}_{d}}{{c_{v}}_{d}} R_{d} \pi \theta_{v} = \frac{{c_{p}}_{d}}{{c_{v}}_{d}} R_{d} \pi \left( \frac{\theta}{f} \right)$$ (A.39)

$B$G$"$k(B.

$B1?F0J}Dx<0$N05NO8{G[$O(B, $B290L$H%(%/%9%J!<4X?t$rMQ$$$k$3$H$GF@$i$l$k(B.

$$\Dinv{\rho} dp = \frac{R_{d} \pi (\theta/f)}{p} d \left( p_{0} \pi^{{c_{p}}_{d}/R_{d}} \right)$$

$$= \frac{R_{d} \pi (\theta/f) }{p} \left( \frac{p_{0} {c_{p}}_{d}}{R_{d}} \pi^{{c_{p}}_{d}/R - 1} \right) d\pi$$

$$= \frac{R_{d} \pi (\theta/f) }{p} \left( \frac{{c_{p}}_{d}}{R_{d}} p \pi^{-1} \right)d\pi$$

$$= {c_{p}}_{d} (\theta/f) d\pi$$

$$= {c_{p}}_{d} \theta_{v} d\pi$$ (A.40)

$B05NOJ}Dx<0$OL)EY$N<0$HO"B3$N<0$rAH$_9g$o$;$k$3$H$GF@$i$l$k(B. $B$^$:L)EY$r(B $\rho= \rho(\theta, \pi, q_{v}, q_{x})$ $B$H$7$F(B $\rho$ $B$NA4HyJ,$r7W;;$9$k(B.

$$d\rho = d \left[ \frac{p_{0} \pi^{c_{v/R_{d}}}}{R_{d} \theta_{v}} \right]$$

$$= d \left[ \frac{p_{0} \pi^{c_{v/R_{d}}}}{R_{d} (\theta/f)} \right]$$

$$= \frac{p_{0}}{R_{d} (\theta/f) } \pi^{(c_{v/R_{d}}-1)} \frac{{c_{v... ..._{d}} d\pi - \frac{p_{0} \pi^{c_{v/R_{d}}} f}{R_{d}} \frac{d\theta}{\theta^{2}}$$

$$+ \frac{p_{0} \pi^{c_{v/R_{d}}} }{R_{d} \theta} \left( \sum \DP{f}{q_{v}} dq_{v} + \sum \DP{f}{q_{c}} dq_{c} + \sum \DP{f}{q_{r}} dq_{r} \right)$$

$$= {c_{p}}_{d} (\theta/f) \left( \frac{{c_{v}}_{d}}{{c_{p}}_{d} R_{d... ...d\pi - \frac{p_{0} \pi^{c_{v/R_{d}}} }{R_{d} (\theta/f)} \frac{d\theta}{\theta}$$

$$+ \Dinv{f} \left( \frac{p_{0} \pi^{c_{v/R_{d}}} }{R_{d} (\theta /... ...}{q_{v}} dq_{v} + \sum \DP{f}{q_{c}} dq_{c} + \sum \DP{f}{q_{r}} dq_{r} \right)$$

$$= \frac{ {c_{p}}_{d} \rho (\theta/f)}{{C_{s}}^{2} } d\pi - \frac{\r... ...}{q_{v}} dq_{v} + \sum \DP{f}{q_{c}} dq_{c} + \sum \DP{f}{q_{r}} dq_{r} \right)$$

$$= \frac{ {c_{p}}_{d} \rho \theta_{v}}{{C_{s}}^{2} } d\pi - \frac{\r... ...}{q_{v}} dq_{v} + \sum \DP{f}{q_{c}} dq_{c} + \sum \DP{f}{q_{r}} dq_{r} \right)$$ (A.41)

$B$H$J$k(B. (A.41) $B<0$r05NO$N<0$H$7$F@0M}$9$k$H(B,

$$\DD{\pi}{t} = \frac{{C_{s}}^{2} }{{c_{p}}_{d} \rho \theta_{v} } \... ...dq_{v} + \sum \DP{f}{q_{c}} dq_{c} + \sum \DP{f}{q_{r}} dq_{r} \right) \right\}$$ (A.42)

$B$H$J$j(B, $BO"B3$N<0$rMQ$$$k$H(B,

$$\DD{\pi}{t} = \frac{{C_{s}}^{2} }{{c_{p}}_{d} \rho \theta_{v} } \... ...dq_{v} + \sum \DP{f}{q_{c}} dq_{c} + \sum \DP{f}{q_{r}} dq_{r} \right) \right\}$$ (A.43)

$B$H$J$j(B, $B05NOJ}Dx<0$,F@$i$l$k(B.

A.2 $B=`05=LJ}Dx<07O$NF3=P(B

$B=`05=LJ}Dx<07O$G$O(B, $BJQ?t$r4pK\>l$H>qMp>l$KJ,N%$7(B, $B@~7A2=$r9T$&(B.

A.2.1 $B4pK\>l$H>qMp>l$NJ,N%(B

$BJQ?t$r4pK\>l$H>qMp>l$KJ,N%$7(B, $B4pK\>l$O@E?e05J?9U$K$"$k$H2>Dj$9$k(B. $B$3$N;~(B, $BJQ?t$O0J2<$N$h$&$K=q$1$k(B.

$$u = u^{'}(x,z,t)$$

$$w = w^{'}(x,z,t)$$

$$\pi = \bar{\pi}(z) + \pi^{'}(x,z,t)$$

$$\theta_{v} = \bar{\theta_{v}}(z) + {\theta_{v}}^{'}(x,z,t)$$

$$\rho = \bar{\rho}(z) + \rho^{'}(x,z,t)$$

$$q_{v} = \bar{q_{v}}(z) + q_{v}^{'}(x,z,t)$$

$$q_{r} = q_{r}^{'}(x,z,t)$$

$$q_{c} = q_{c}^{'}(x,z,t)$$

$BC"$7(B, $\theta_{v} = \theta/f$ $B$H$7(B, $B4pK\>l$NIwB.(B $u, w$ $B$H1@N3:.9gHf$H1+N3:.9g$O%<%m$H8+$J$7$?(B. $B$=$7$F4pK\>l$K$O@E?e05J?9U(B,

$$\DP{\bar{\pi}}{z} = - \frac{g}{{c_{p}}_{d} \bar{\theta_{v}}} = - \frac{g}{{c_{p}}_{d} (\bar{\theta}/\bar{f})}$$ (A.44)

$B$N4X78$,@.$jN)$D$b$N$H$9$k(B.

A.2.2 $B?eJ?J}8~$N1?F0J}Dx<0$N@~7A2=(B

$B?eJ?J}8~$N1?F0J}Dx<0$r4pK\>l$H>qMp>l$KJ,N%$9$k(B.

$$\DP{u^{'}}{t} = - \left( u^{'} \DP{u^{'}}{x} + w^{'} \DP{u^{'}}{z... ..._{v}}^{'} \DP{\bar{\pi}}{x} + {\theta_{v}}^{'} \DP{\pi^{'}}{x} \right) + Turb.u$$

$B>e<0$K$*$$$F0\N.9`0J30$N(B 2 $B.9`$r>C5n$7(B, $B$5$i$K4pK\>l$O(B $x$ $BJ}8~$K(B $B$OJQ2=$7$J$$$3$H$rMxMQ$9$k$H(B, $B0J2<$N>[email protected],$N<0$,F@$i$l$k(B.

$$\DP{u^{'}}{t} = - \left( u^{'} \DP{u^{'}}{x} + w^{'} \DP{u^{'}}{z} \right) - {c_{p}}_{d} \bar{\theta_{v}} \DP{\pi^{'}}{x} + Turb.u$$

$$= - \left( u^{'} \DP{u^{'}}{x} + w^{'} \DP{u^{'}}{z} \right) -{c_{p}}_{d} \left( \frac{\bar{\theta}}{\bar{f}}\right) \DP{\pi^{'}}{x} + Turb.u$$ (A.45)

$B$3$3$G(B $\bar{f}$ $B$O(B,

$$\bar{f} = {\left( 1 - \frac{\sum \bar{q_{v}}/M_{v}}{1/M_{d} + \sum \bar{q_{v}}/M_{v}} \right) (1 + \sum \bar{q_{v}} )}$$ (A.46)

$B$G$"$k(B.

A.2.3 $B1tD>J}8~$N1?F0J}Dx<0$N@~7A2=(B

$B1tD>J}8~$N1?F0J}Dx<0$r4pK\>l$H>qMp>l$KJ,N%$9$k(B.

$$\DP{w^{'}}{t} = - \left( u^{'} \DP{w^{'}}{x} + w^{'} \DP{w^{'}}{z... ...}^{'} \DP{\bar{\pi}}{z} + {\theta_{v}}^{'} \DP{\pi^{'}}{z} \right) - g + Turb.w$$

$B>e<0$K$*$$$F0\N.9`0J30$N(B 2 $B.9`$r>C5n$9$k$H0J2<$H$J$k(B.

$$\DP{w^{'}}{t} = - \left( u^{'} \DP{w^{'}}{x} + w^{'} \DP{w^{'}}{z... ...v}} \DP{\pi^{'}}{z} + {\theta_{v}}^{'} \DP{\bar{\pi}}{z} \right) - g + Turb.w .$$

$B$5$i$K@E?e05$N<0$rMxMQ$9$k$H0J2<$H$J$k(B.

$$\DP{w^{'}}{t} = - \left( u^{'} \DP{w^{'}}{x} + w^{'} \DP{w^{'}}{z} \right) + {c_{... ...eta_{v}}^{'} \left( \frac{g}{{c_{p}}_{d} \bar{\theta_{v}}} \right) - g + Turb.w$$

$$= - \left( u^{'} \DP{w^{'}}{x} + w^{'} \DP{w^{'}}{z} \right) - {c_{... ...eta_{v}} \DP{\pi^{'}}{z} + \frac{{\theta_{v}}^{'}}{\bar{\theta_{v}}} g + Turb.w$$

$B$3$3$G(B ${\theta_{v}}^{'}$ $B$O(B,

$${\theta_{v}}^{'} = \Dinv{f} \left\{ \theta^{'} - \sum \frac{\theta}{f} \DP{f}{q_{v}}... ...DP{f}{q_{c}} q_{c}^{'} - \sum \frac{\theta}{f} \DP{f}{q_{r}} q_{r}^{'} \right\}$$

$$= \Dinv{f} \left\{ \frac{\theta^{'}}{\theta} - \sum \Dinv{f} \DP{f}... ...inv{f} \DP{f}{q_{c}} q_{c}^{'} - \sum \Dinv{f} \DP{f}{q_{r}} q_{r}^{'} \right\}$$ (A.47)

$B$G$"$j(B, (A.47) $B<0$NBh(B 2 $B9`$r7W;;$9$k$H(B,

$$\sum \Dinv{f}\DP{f}{q_{v}} = \Dinv{\frac{1/M_{d}}{1/M_{d} + \sum q_{v}/M_{v}} (1 + \sum q_{v} ... ...{1/M_{d} + \sum q_{v}/M_{v}} (1 + \sum q_{v} + \sum q_{c} + \sum q_{r})}{q_{v}}$$

$$= \Dinv{\frac{1/M_{d}}{1/M_{d} + \sum q_{v}/M_{v}} (1 + \sum q_{v} + \sum q_{c} + \sum q_{r})}$$

$$\left[ \frac{1/M_{d}}{1/M_{d} + \sum q_{v}/M_{v}} - \left\{ \frac... ...m q_{v}/M_{v})^{2}} (1 + \sum q_{v} + \sum q_{c} + \sum q_{r}) \right\} \right]$$

$$= \Dinv{1 + \sum q_{v} + \sum q_{c} + \sum q_{r}} - \frac{\sum 1/M_{v}}{1/M_{d} + \sum q_{v}/M_{v}}$$

$B$G$"$j(B, (A.47) $B<0$NBh(B 3 $B9`$r7W;;$9$k$H(B,

$$\sum \Dinv{f}\DP{f}{q_{c}} = \Dinv{\frac{1/M_{d}}{1/M_{d} + \sum q_{v}/M_{v}} (1 + \sum q_{v} ... ...{1/M_{d} + \sum q_{v}/M_{v}} (1 + \sum q_{v} + \sum q_{c} + \sum q_{r})}{q_{c}}$$

$$= \Dinv{1 + \sum q_{v} + \sum q_{c} + \sum q_{r}}$$

$B$G$"$j(B, (A.47) $B<0$NBh(B 4 $B9`$r7W;;$9$k$H(B,

$$\sum \Dinv{f}\DP{f}{q_{r}} = \Dinv{\frac{1/M_{d}}{1/M_{d} + \sum q_{v}/M_{v}} (1 + \sum q_{v} ... ...{1/M_{d} + \sum q_{v}/M_{v}} (1 + \sum q_{v} + \sum q_{c} + \sum q_{r})}{q_{r}}$$

$$= \Dinv{1 + \sum q_{v} + \sum q_{c} + \sum q_{r}}$$

$B$H$J$k$N$G(B,

$${\theta_{v}}^{'} = \Dinv{f} \left\{ \frac{\theta^{'}}{\theta} + \... ...q_{c}^{'} + \sum q_{r}^{'}} {1 + \sum q_{v} + \sum q_{c} + \sum q_{r}} \right\}$$ (A.48)

$B$G$"$k(B. $B$3$3$G>[email protected],[email protected],$KHf$Y$F==J,$K>.$5$$$N$G(B, mail protected],$KCV$-49$($k$3$H$G(B,

$${\theta_{v}}^{'} = \frac{\bar{\theta}}{\bar{f}} \left\{ \frac{\th... ...um q_{v}^{'} + \sum q_{c}^{'} + \sum q_{r}^{'}} {1 + \sum \bar{q_{v}}} \right\}$$ (A.49)

$B$H$J$k(B. $B$3$l$rMQ$$$k$H(B, $B>[email protected],$NB.EY(B $w$ $B$N<0$O0J2<$N$h$&$K=q$1$k(B.

$$\DP{w^{'}}{t} = - \left( u^{'} \DP{w^{'}}{x} + w^{'} \DP{w^{'}}{z} \right) - {c_{p}}_{d} \bar{\theta_{v}} \DP{\pi^{'}}{z}$$

$$+ \left( \frac{\theta^{'}}{\bar{\theta}} + \frac{\sum q_{v}^{'}/M... ...'} + \sum q_{c}^{'} + \sum q_{r}^{'}} {1 + \sum \bar{q_{v}}} \right) g + Turb.w$$

A.2.4 $B05NOJ}Dx<0$N@~7A2=(B

Klemp and Wilhelmson (1978) $B$G$O(B, $BHsCGG.E*$J2CG.$K$h$kG.KDD%$H(B $B6E=L$KH<$&05NOJQ2=$rL5;k$7(B,

$$\DD{\pi}{t} = - \frac{{C_{s}}^{2}}{ {c_{p}}_{d} (\theta/f)} \Ddiv \Dvect{u}$$

$B$H$7$FDj<02=$7$?(B. $BK\%b%G%k$G9M$($k7O$G$O(B, mail protected],$,==J,$K>.$5$$$N$G(B, $B$3$N6a;w$rMQ$$$k$3$H$H$7$?(B.

$B05NOJ}Dx<0$K4X$7$F(B, mail protected],$H>[email protected],$KJ,$1$k(B. $B$?$@$7(B, $B>[email protected],$OJ?6Q@.(B $BJ,$h$j$b==J,>.$5$$$H$$$&2>Dj$rMQ$$(B, $1/\theta = 1/\bar{\theta}$, $1/f = 1/\bar{f}$ $B$H$9$k(B.

$$\DP{\bar{\pi} + \pi^{'}}{t} + u^{'} \DP{\bar{\pi}+\pi^{'}}{x} + w... ...verline{{C_{s}}^{2} }}{ {c_{p}}_{d} (\bar{\theta}/\bar{f})} \Ddiv \Dvect{u^{'}}$$

$B>e<0$G$O(B ${C_{s}}^{2}$ mail protected],$H>[email protected],$KJ,N%$7$F(B 2 $B.9`$r(B $BL5;k$9$k$H(B, $\overline{{C_{s}}^{2}}$ $B$HEy$7$/$J$k$3$H$rMxMQ$7$F$$$k(B.

$${C_{s}}^{2} = \frac{{c_{p}}_{d}}{{c_{v}}_{d}} R_{d} (\bar{\pi} + \pi^{'}) \left( \frac{(\bar{\theta} + \theta^{'})}{\bar{f}} \right)$$

$$\approx \frac{{c_{p}}_{d}}{{c_{v}}_{d}} R_{d} \left( \bar{\pi} \frac{\bar... ...{\pi} \frac{\theta^{'}}{\bar{f}} + \pi^{'} \frac{\bar{\theta}}{\bar{f}} \right)$$

$$= \frac{{c_{p}}_{d}}{{c_{v}}_{d}} R_{d} \bar{\pi} \frac{\bar{\theta... ...left( 1 + \frac{\theta^{'}}{\bar{\theta}} + \frac{ \pi^{'} }{\bar{\pi}} \right)$$

$$\approx \frac{{c_{p}}_{d}}{{c_{v}}_{d}} R_{d} \bar{\pi} \frac{\bar{\theta}}{\bar{f}} \equiv \overline{{C_{s}}^{2}}$$ (A.50)

$B$?$@$7(B $\theta^{'}/\bar{\theta} \ll 1$, $\pi^{'}/\bar{\pi} \ll 1$ $B$G$"$k$3$H$rMQ$$$?(B. mail protected],$O(B $z$ $B$K$N$_0MB8$9$k$3$H$rMxMQ$7(B, $B$^$?(B 2 $B.9`$rL5;k$9$k(B.

$$\DP{\pi^{'}}{t} = - w^{'} \DP{\bar{\pi}}{z} - \frac{\overline{{C_{s}}^{2}} }{ {c_{p}}_{d} (\bar{\theta} /\bar{f})} \Ddiv \Dvect{u^{'}}$$

$B$5$i$K(B $\pi$ $B$rM}A[5$BN$N>uBVJ}Dx<0$GJQ7A$7$F$^$H$a$k$H(B, $B05NO$N>[email protected],$N;~4VH/E8J}Dx<0$,F@$i$l$k(B.

$$\DP{\pi^{'}}{t} = - w^{'} \DP{}{z} \left( \frac{\bar{\rho} R_{d}(\bar{\theta}/\bar{... ...{p}}_{d} (\bar{\theta}/\bar{f})} \left( \DP{ u^{'}}{x} + \DP{ w^{'}}{z} \right)$$

$$= - w^{'} \frac{R_{d}}{{c_{v}}_{d}} \bar{\pi} \Dinv{\left( \frac{\b... ...c_{p}}_{d} (\bar{\theta}/\bar{f})} \left( \DP{u^{'}}{x} + \DP{w^{'}}{z} \right)$$

$$= - \frac{\overline{{C_{s}}^{2}}}{{c_{p}}_{d} \bar{\rho} (\bar{\the... ...o} (\bar{\theta}/\bar{f}) \left( \DP{u^{'}}{x} + \DP{w^{'}}{z} \right) \right\}$$

$$= - \frac{\overline{{C_{s}}^{2}}}{{c_{p}}_{d} \bar{\rho} (\bar{\the... ...f})^{2}} \Ddiv \left\{ \bar{\rho} (\bar{\theta}/\bar{f}) \Dvect{u^{'}} \right\}$$

$B0J>e$h$j(B,

$$\DP{\pi^{'}}{t}= - \frac{\overline{{C_{s}}^{2}}}{{c_{p}}_{d} \bar... ...f})^{2}} \Ddiv \left\{ \bar{\rho} (\bar{\theta}/\bar{f}) \Dvect{u^{'}} \right\}$$ (A.51)

$B$G$"$k(B.

A.2.5 $BG.$N<0$N@~7A2=(B

$BG.$N<[email protected],$H>[email protected],$KJ,N%$9$k(B.

$$\DP{(\bar{\theta} + \theta^{'})}{t} = - u^{'}\DP{(\bar{\theta} + ... ...w^{'}\DP{(\bar{\theta} + \theta^{'})}{x} + Q + Turb.(\bar{\theta} + \theta^{'})$$

$B$3$3$GJ?6Q>l$NNL$O(B $z$ $B$N4X?t$G$"$k$3$H$rMQ$$$k$H(B,

$$\DP{\theta^{'}}{t} = - \left( u^{'}\DP{\theta^{'}}{x} + w^{'}\DP{... ...} \right) - w^{'}\DP{\bar{\theta}}{x} + Q + Turb.\bar{\theta} + Turb.\theta^{'}$$ (A.52)

$B$H$J$k(B.

A.2.6 $B:.9gHf$NJ]B8<0$N@~7A2=(B

mail protected],$N:.9gHf$NJ]B8<0$K$D$$$F$b(B, mail protected],$H>[email protected],$KJ,N%$9$k(B. $BG.$N<0$HF1MM$K(B, $B0J2<$N$h$&$K=q$1$k(B. $BC"$7(B, $B@[email protected]`(B, $BMn2<9`$O>[email protected],$N$_(B $BB8:_$9$k$H2>Dj$9$k(B. $B$3$N2>Dj$OJ?6Q>l$G$O6E=L$O@8$8$F$$$J$$$H9M$($k$3$H$K(B $BEy$7$$(B.

$$\DP{q_{v}^{'}}{t} = - \left( u^{'}\DP{q_{v}^{'}}{x} + w^{'}\DP{q_... ...- w^{'}\DP{\bar{q_{v}}}{x} + Src.q_{v}^{'} + Turb.\bar{q_{v}} + Turb.q_{v}^{'},$$ (A.53)

$$\DP{q_{c}^{'}}{t} = - \left( u^{'}\DP{q_{c}^{'}}{x} + w^{'}\DP{q_{c}^{'}}{x} \right) + Src.q_{c}^{'} + Turb.q_{c}^{'},$$ (A.54)

$$\DP{q_{r}^{'}}{t} = - \left( u^{'}\DP{q_{r}^{'}}{x} + w^{'}\DP{q_{r}^{'}}{x} \right) + Src.q_{r}^{'} + Fall.q_{r}^{'} + Turb.q_{r}^{'}$$ (A.55)

$BC"$71@?eNL$H1+?eNL$O>[email protected],$N$_$NNL$G$"$k(B.

A.3 $B$^$H$a(B

$B=`05=LJ}Dx<07O$O0J2<$N$h$&$K$^$H$a$i$l$k(B. $B$?$@$7(B, $B>qMp$r<($9(B $~^{'}$ $B$O(B $B=|$$$?(B.

$B1?F0J}Dx<0(B

 

$$\DP{u}{t} = - \left( u \DP{u}{x} + w \DP{u}{z} \right) - {c_{p}}_{d} \bar{\theta_{v}} \DP{\pi}{x} + Turb.u$$ (A.56)

$$\DP{w}{t} = - \left( u \DP{w}{x} + w \DP{w}{z} \right) - {c_{p}}_{d} \bar{\theta_{v}} \DP{\pi}{x} + Turb.w$$

$$+ \left( \frac{\theta}{\bar{\theta}} + \frac{\sum q_{v}/M_{v}}{1/... ... - \frac{\sum q_{v} + \sum q_{c} + \sum q_{r}} {1 + \sum \bar{q_{v}}} \right) g$$ (A.57)

$B05NOJ}Dx<0(B

 

$$\DP{\pi}{t}= - \frac{\overline{{C_{s}}^{2}}}{{c_{p}}_{d} \bar{\rh... ...bar{f})^{2}} \Ddiv \left\{ \bar{\rho} (\bar{\theta}/\bar{f}) \Dvect{u} \right\}$$ (A.58)

$BG.$N<0(B

 

$$\DP{\theta}{t} = - \left( u\DP{\theta}{x} + w\DP{\theta}{x} \right) - w\DP{\bar{\theta}}{x} + Q + Turb.\bar{\theta} + Turb.\theta$$ (A.59)

mail protected],$N:.9gHf$NJ]B8<0(B

 

$$\DP{q_{v}}{t} = - \left( u\DP{q_{v}}{x} + w\DP{q_{v}}{x} \right) - w\DP{\bar{q_{v}}}{x} + Src.q_{v} + Turb.\bar{q_{v}} + Turb.q_{v},$$ (A.60)

$$\DP{q_{c}}{t} = - \left( u\DP{q_{c}}{x} + w\DP{q_{c}}{x} \right) + Src.q_{c} + Turb.q_{c},$$ (A.61)

$$\DP{q_{r}}{t} = - \left( u\DP{q_{r}}{x} + w\DP{q_{r}}{x} \right) + Src.q_{r} + Fall.q_{r} + Turb.q_{r}$$ (A.62)