9 $B;YG[J}Dx<07O$NF3=P(B

$B;YG[J}Dx<07O$NF3=P(B

6 66 6 66

A. $B@_Dj(B

$BA4Bg5$$O(B, $B$H$b$KM}A[5$BN$G$"$k4%Ag6u5$$*$h$S?e>x5$$+$i@.$k:.9gBg5$$H$9(B $B$k(B. $B1@?eNL$OL5;k$9$k(B. $B$^$?(B, $B?e>x5$NL$,A4Bg5$$K@j$a$k3d9g$O>.$5$$$H2>Dj(B $B$7(B, $BA4Bg5$$NDj05HfG.$r4%AgBg5$$NCM$G6a;w$9$k(B.

$B?e>x5$NL$NJ]B8$K$D$$$F$O(B, $B6E7k$*$h$S>xH/$K$h$k@8@.>CLG$r9MN8$9$k(B. $B$7$+(B $B$7(B, $B$3$NNL$,A4Bg5$$KM?$($k8z2L$O>.$5$$$H$7(B, $BA4Bg5$$N $B=ENO2CB.EY$OCO5eCf?4$K8~$$$F$$$k$H2>Dj$9$k(B. $B$^$?(B, $B1?F0$N?eJ?%9%1!<%k$,(B $B1tD>%9%1!<%k$h$j$b$+$J$jBg$-$$1?F0$rA[Dj$7(B, $B@ENO3XJ?9U6a;w$r9T$J$&(B. $B$5(B $B$i$K(B, $B1?F0$OCO5eI=LLIU6a$K8B$i$l$k$3$H$r2>Dj$7$F6a;w$r9T$J$&(B. 6 66 6 66

B. $B4pACJ}Dx<07O$NF3=P(B

$BJ}Dx<07O$O(B 6 $BK\$NM=JsJ}Dx<0$H(B 1 $BK\$N?GCGJ}Dx<0$+$i$J$k(B. $BM=JsJ}Dx<0$O(B, $BA4x5$NL$N<0(B, $B1?F0J}Dx<0(B(3 mail protected],(B), $BG.NO3X$N<0$+$i$J$k(B. $B$3$l$i$O(B, $B$=$l$>$l(B, $BA4x5$NL$NJ]B8B'(B, $BA4uBVJ}Dx<0$rMQ$$$k(B. ⁷

$B!*!*Cm0U(B: $B$3$N(B Appendix $BCf$G$OF3=P$NET9g>e(B, $B4%Ag6u5$$N5$BNDj?t$r(B $R^d$ $BDj05HfG.$r(B $c_p^d$ $B$H$7(B, $BA4Bg5$$N5$BNDj?t$r(B $R$ $B$H$*$$$?(B. $B$7$+$7(B, $BK\J8(B $BCf$G$O(B, $B4%Ag6u5$$N5$BNDj?t$r(B $R$, $BDj05HfG.$r(B $c_p$ $B$HI=5-$7$F$$$k(B.

B..1 $B>uBVJ}Dx<0(B

$B4%Ag6u5$(B, $B?e>x5$$N>uBVJ}Dx<0$O$=$l$>$l(B

$$p^d = \rho^{d} R^d T,$$ (99)

$$p^v = \rho^{v} R^v T,$$ (100)

$B$G$"$k(B. $B$3$3$G(B $\bullet^d$, $\bullet^v$ $B$O$=$l$>$l4%Ag6u5$$*$h$S?e(B $B>x5$$K4X$9$kNL$G$"$k$3$H$r<($9(B. $B$7$?$,$C$F(B, $BA405(B $p=p^d+p^v$ $B$O(B,

$$p = (\rho^d R^d + \rho^v R^v) T$$ (101)

$$= \rho R^d ( 1 + \epsilon_v q ) T,$$ (102)

$B$H$J$k(B. $B$3$3$G(B, $q=\rho_v/\rho$ $B$OHf<>(B, $B$G$"$j(B, $\epsilon_v \equiv 1/\epsilon -1$, $\epsilon \equiv R^d/R^v(=0.622)$ $B$G$"$k(B. $B$7$?$,$C$F(B, $BA4Bg5$$N>uBVJ}Dx<0$O(B,

$$p = \rho R T.$$ (103)

$B$?$@$7(B, $R \equiv R^d ( 1+\epsilon_v q )$ $B$G$"$k(B. $B$"$k$$$O(B, $B2>29EY(B $T_v \equiv T ( 1 + \epsilon_v q )$ $B$rMQ$$$l$P(B,

$$p = \rho R^d T_v.$$ (104)

B..2 $BO"B3$N<0(B

$BA4Bg5$$Nx5$$N@8@.>CLG$rL5;k$9$l$P(B, ⁸

$$\DP{\rho}{t} + \DP{}{x_j}( \rho v_j ) = 0.$$ (105)

$B$"$k$$$O(B, $B%i%0%i%s%8%e7A<0$G5-=R$9$l$P(B,

$$\DD{\rho}{t} + \rho \Ddiv \Dvect{v} = 0.$$ (106)

B..3 $B?e>x5$$N<0(B

$B?e>x5$L)EY(B $\rho^v$ $B$KBP$9$kCLGNL$r(B $S$ $B$H$9$l$P(B,

$$\DP{\rho^v}{t} + \DP{}{x_j} ( \rho^v v_j ) = S.$$ (107)

$BHf<>(B $q=\rho^v/\rho$ $B$K4X$9$k<0$O(B, $B86M}E*$K$O<0(B() $B$H<0(B(A.9) $B$+$iF@$k$3$H$,$G$-$k(B. $B$7$+$7(B, $B:#$N>l9g(B, $B<0(B(A.7)$B$G?e>x5$$N@8@.>CLG$rL5;k$7$?$N$G(B, $B@5$7$/$OF@$i$l$J$$(B. $B$=$3$GHf<>$N@8@.>CLG$K4X$9$k9`$r2~$a$F(B $S_q$ $B$HDj(B $B5A$9$k(B.

$$\DD{q}{t} = S_q.$$ (108)

B..4 $B1?F0J}Dx<0(B

$B1?F0NLJ]B8B'$O(B, $B?e>x5$$N@8@.>CLG$K$H$b$J$&1?F0NLJQ2=$rL5;k$9$l$P

$$\DP{}{t}(\rho v_i) + \DP{}{x_j}( \rho v_i v_j ) + \DP{p}{x_i} - \DP{\sigma_{ij}}{x_j} + \rho \DP{\Phi^*}{x_i} = F'_i.$$ (109)

$B$3$3$G(B, $p$ $B$O05NO(B, $\sigma_{ij}$ $B$OG4@-1~NO%F%s%=%k(B, $\Phi^*$ $B$OCO5e(B $B$N0zNO$K$h$k%]%F%s%7%c%k(B ⁹ , $F'_i$ $B$O$=$NB>$N30NO9`$G$"$k(B. $B$"$k$$$OO"B3$N<0(B $B$rMQ$$$F%i%0%i%s%8%e7A<0$G5-=R$9$k$H(B

$$\rho \DD{v_i}{t} + \DP{p}{x_i} - \DP{\tau_{ij}}{x_j} + \rho \DP{\Phi^*}{x_i} = F'_i,$$ (110)

$B$H$J$k(B. $B$3$3$G(B, $BG4@-9`$H30NO9`$r(B $F_i$ $B$H$*$-(B, $B$5$i$K%Y%/%H%kI=<($9$k(B

$$\rho \DD{\Dvect{v}}{t} + \Dgrad p + \rho \Dgrad \Phi^* = \Dvect{F}.$$ (111)

B..5 $BG.NO3X$N<0(B

$BC10L $B$HFbIt%((B $B%M%k%.!<(B $\varepsilon$ $B$*$h$\%]%F%s%7%c%k%(%M%k%.!<(B $\Phi^*$ $B$NOB$GI=(B $B8=$5$l$k(B. $B$3$N;~4VJQ2=N($N<0$O(B, $B?e>x5$$N@8@.>CLG$K$h$k1F6A$rL5;k$9$l$P(B,

$$\DP{}{t} \left[ \rho \left( \frac{1}{2} \Dvect{v}^2 +... ...v_j + p v_j - \sigma_{ij}v_i \right] = \rho Q + F'_i v_i,$$ (112)

$B$G$"$k(B. $B$3$3$G(B, $Q$ $B$O303&$+$i$N2CG.N($G$"$k(B. $B0lJ}(B, $B1?F0%(%M%k%.!<$H%](B $B%F%s%7%c%k%(%M%k%.!<$NOB$NJ]B8<0$O(B, $B1?F0NLJ]B8<0(B () $B$K(B $v_i$ $B$r$+$1O"B3$N<0$rMQ$$$FJQ7A$9$k$3$H$GF@$i$l$k(B. $BJQ7A$N:]$K(B $B$O(B $\DP{\Phi^*}{t}=0$ $B$G$"$k$H$7$F$$$k(B. ¹⁰

$$\DP{}{t} \left( \frac{1}{2} \rho v_i^2 + \rho \Phi^* \right... ...ht) = p \DP{v_j}{x_j} - \sigma_{ij} \DP{v_i}{x_j} + F'_i v_i,$$ (113)

$B$H$J$k(B. $B<0(B (A.14) $B$+$i<0(B () $B$r0z$-5n$k$H(B, $B

$$\DP{}{t} ( \rho \varepsilon ) + \DP{}{x_j} ( \rho \varepsi... ... ) = - p \DP{v_j}{x_j} + \sigma_{ij} \DP{v_i}{x_j} + \rho Q.$$ (114)

$BO"B3$N<0$rMQ$$$F%i%0%i%s%8%e7A<0$K=q$-D>$;$P(B

$$\rho \DD{\varepsilon}{t} = \frac{p}{\rho} \left( \DD{\rho}{t} \right) + \rho Q.$$ (115)

$B$3$3$G(B, $B303&$+$i$N2CG.$N9`$HG4@-$K$h$k2CG.$N9`$r$^$H$a$F(B $Q^*$ $B$H$*$$(B $B$?(B.

$BFbIt%(%M%k%.!<$r29EY$rMQ$$$FI=8=$9$k$H(B $\varepsilon = c_v T$ $B$G$"(B $B$k(B. $B$5$i$K>uBVJ}Dx<0(B (A.5) $B$rMQ$$$F<0(B (A.17) $B$rJQ7A$9$k(B. $c_p = c_v + R$ $B$G$"$k$3$H$KCm0U$9$l$P(B

$$\DD{c_p T}{t} = \frac{1}{\rho} \DD{p}{t} + Q^*,$$ (116)

$B$H$J$k(B. $B$3$3$G(B, $c_p$ $B$r4%Ag6u5$$NDj05HfG.(B $c_p^d$ ($BDj?t(B) $B$G6a;w$9$k$H(B ¹¹$B

$$\DD{T}{t} = \frac{1}{c_p^d \rho} \DD{p}{t} + \frac{Q^*}{c_p^d}.$$ (117)

11 1111 11 1111

C. $B2sE>7O$X$NJQ49(B

C..1 $B2sE>7O$X$NJQ498x<0(B

$BJ}Dx<07O$r(B, $B0lDj$N<+E>3QB.EY(B $\Dvect{\Omega}$ $B$G2sE>$9$k2sE>7O$KJQ49$9(B $B$k(B.

C..2 $B%9%+%i!<$NJQ498x<0(B

$B47@-7O$K$*$1$k;~4VHyJ,$rE:;z(B a $B$G(B, $B2sE>7O$rE:;z(B r $B$GI=8=$9$k(B. $B$3$N$H$-(B, $BG$0U$N%9%+%i!<(B $\psi$ $B$KBP$7$F(B,

$$\left( \DD{\psi}{t} \right)_{\rm a} = \left( \DD{\psi}{t} \right)_{\rm r},$$ (118)

$B$,@.$j$?$D(B. ¹²

C..3 $B%Y%/%H%k$NJQ498x<0(B

$BG$0U$N%Y%/%H%k(B $\Dvect{A}$ $B$KBP$9$k47@-7O$*$h$S2sE>7O$G$NHyJ,$O

$$\left( \DD{\Dvect{A}}{t} \right)_{\rm a} = \left( \DD{\Dvect{A}}{t} \right)_{\rm r} + \Dvect{\Omega} \times \Dvect{A}.$$ (119)

($B>ZL@(B) $BG$0U$N%Y%/%H%k(B $\Dvect{A}$ $B$r(B, $B47@-7O$G$O(B

$$\Dvect{A} = \Dvect{i} A_x + \Dvect{j} A_y + \Dvect{k} A_z$$ (120)

$B$HI=$7(B, $B2sE>7O$G$O(B

$$\Dvect{A} = \Dvect{i}' A'_x + \Dvect{j}' A'_y + \Dvect{k}' A'_z$$ (121)

$B$HI=$9(B. $B;~4VHyJ,$r$H$k$H(B

$$\left( \DD{\Dvect{A}}{t} \right)_{\rm a} = \Dvect{i} \left( \DD{A_x}{t} \right)_{\rm a} + \Dvect{j} \left( \DD{A_y}{t} \right)_{\rm a} + \Dvect{k} \left( \DD{A_z}{t} \right)_{\rm a}$$

$$= \Dvect{i}' \left( \DD{A'_x}{t} \right)_{\rm a} + \Dvect{j}' \left... ...t{j}'}{t} \right)_{\rm a} A'_y + \left( \DD{\Dvect{k}'}{t} \right)_{\rm a} A'_z$$

$$= \Dvect{i}' \left( \DD{A'_x}{t} \right)_{\rm r} + \Dvect{j}' \left... ...+ \Dvect{\Omega} \times \Dvect{j}' A'_y + \Dvect{\Omega} \times \Dvect{k}' A'_z$$

$$= \left( \DD{\Dvect{A}}{t} \right)_{\rm r} + \Dvect{\Omega} \times \Dvect{A}.$$ (122)

($B>ZL@=*$j(B)

$B$3$3$G(B $\Dvect{A}=\Dvect{r}$ ( $\Dvect{r}$ $B$O0LCV%Y%/%H%k(B ) $B$H$*$1$P47(B $B@-7O$G$NB.EY(B $\Dvect{v}_a \equiv (d\Dvect{r}/dt)_{\rm a}$ ($B$3$l$^$G$N(B $\Dvect{v}$) $B$O2sE>7O$G$NB.EY(B $\Dvect{v} \equiv (d\Dvect{r}/dt)_{\rm r}$ $B$rMQ$$$F

$$\Dvect{v}_a = \Dvect{v} + \Dvect{\Omega} \times \Dvect{r}.$$ (123)

$B$5$i$K(B, $B<0(B(A.21) $B$G(B $\Dvect{A}=\Dvect{v}_{\rm a}$ $B$H$*(B $B$1$P(B, $BB.EY$N;~4VHyJ,9`$O(B

$$\DD{\Dvect{v}_a}{t} = \DD{\Dvect{v}}{t} + 2 \Dvect{\Omega}... ... + \Dvect{\Omega} \times ( \Dvect{\Omega} \times \Dvect{r} ),$$ (124)

$B$HJQ49$G$-$k(B.

C..4 $B2sE>7O$X$NJQ49(B

$BJQ49$N<0(B (A.26) $B$rMQ$$$F1?F0J}Dx<0$r2sE>7O$G5-=R$9$k(B.

$$\DD{\Dvect{v}}{t} = - \frac{1}{\rho} \Dgrad p - 2 \Dvect... ...vect{\Omega} \times \Dvect{r} ) + \Dgrad \Phi^* + \Dvect{F}.$$ (125)

$B$3$3$G(B, $B=ENO2CB.EY(B $\Dvect{g} \equiv \Dgrad \Phi^* - \Dvect{\Omega} \times ( \Dvect{\Omega} \times \Dvect{v})$ $B$rDj5A$9$l$P(B, $B1?F0J}Dx<0$O(B

$$\DD{\Dvect{v}}{t} = - \frac{1}{\rho} \Dgrad p - 2 \Dvect{\Omega} \times \Dvect{v} + \Dvect{g} + \Dvect{F},$$ (126)

$B$H$J$k(B.

$BO"B3$N<0$*$h$SG.NO3X$N<0$K$*$$$F$O(B, $B%i%0%i%s%8%eHyJ,$,:nMQ$7$F$$$kL)EY(B $B$*$h$S29EY$O:BI8JQ49$KL54X78$J%9%+%i!<$G$"$k$?$a(B, $B$=$N;~4VHyJ,$N7A$OJQ(B $B$o$i$J$$(B. $BO"B3$N<0$O(B, $BB.EY>l$NH/;6$r4^$`$,(B, $B$3$l$O:BI8JQ49$K$h$C$F$bCM(B $B$OJQ$o$i$J$$(B. $B$7$?$,$C$F(B, $B$3$l$i$N<0$O7A$rJQ$($J$$(B. ¹³ 13 1313 13 1313

D. $B5e:BI8$X$NJQ49(B

D..1 $BD>8r6J@~:BI87O$K$*$1$kHyJ,(B

$B0lHL$ND>8r6J@~:BI8(B $(\xi_1, \xi_2, \xi_3)$ $B$K$*$$$F(B, $B%9%+%i!<(B $\bullet$ $B$*$h$S%Y%/%H%k(B $\Dvect{A}=(A_1, A_2, A_3)$ $B$O $B$O3F<4J}8~$N5,LO0x;R$G$"$j(B, $B3F<4J}8~$N4pDl%Y%/%H%k(B $B$O(B $\Dvect{e}_i$ $B$H$9$k(B.

$$\Dgrad \bullet = \left( \frac{1}{h_1} \DP{\bullet}{\xi_1},... ...{\bullet}{\xi_2}, \frac{1}{h_3} \DP{\bullet}{\xi_3} \right),$$ (127)

$$\Ddiv \Dvect{A} = \frac{1}{h_1 h_2 h_3} \left[ \DP{}{\x... ...\xi_2} ( h_1 h_3 A_2) + \DP{}{\xi_3} ( h_1 h_2 A_3) \right],$$ (128)

$$\nabla^2 \bullet = \frac{1}{h_1 h_2 h_3} \left[ \DP{}{\... ...eft( \frac{h_1 h_2}{h_3} \DP{\bullet}{\xi_3} \right) \right],$$ (129)

$$\Drot \Dvect{A} = \left( \frac{1}{h_2 h_3} \left[ \DP{(h_... ...DP{(h_2 A_2)}{\xi_1} - \DP{(h_1 A_1)}{\xi_2} \right] \right),$$ (130)

$$\DD{\bullet}{t} = \DP{\bullet}{t} + \frac{v_1}{h_1} \DP{... ...2} \DP{\bullet}{\xi_2} + \frac{v_3}{h_3} \DP{\bullet}{\xi_3},$$ (131)

$$\DD{\Dvect{v}}{t} = \sum^3_{k=1} \Dvect{e}_k \left[ \DP{v... ...{v_k}{h_k} \frac{1}{h_j} \DP{h_k}{\xi_j} \right) v_j \right].$$ (132)

D..2 $B5e:BI87O$K$*$1$kHyJ,(B

$B=ENO2CB.EY(B $\Dvect{g}$ $B$,CO5eCf?4$r8~$$$F$$$k$H$_$J$7$F(B, $BJ}Dx<07O$r5e(B $B:BI8(B $(\xi_1, \xi_2, \xi_3) = (\lambda, \varphi, r)$ $B$KJQ49$9$k(B. $B2sE>(B $B7O$K8GDj$7$?D>8rD>@~:BI8(B $(x_1, x_2, x_3)$ $B$H$N4X78$O(B

$$x_1 = r \cos \varphi \cos \lambda,$$ (133)

$$x_2 = r \cos \varphi \sin \lambda,$$ (134)

$$x_3 = r \sin \varphi,$$ (135)

$B$G$"$k(B. $B$3$3$G(B, $\lambda$ $B$O0^EY(B, $\varphi$ $B$O7PEY(B, $r$ $B$O1tD>:BI8$G$"(B $B$k(B. $B$^$?(B, $B4pDl%Y%/%H%k$r(B $(\Dvect{e}_{\lambda}, \Dvect{e}_{\varphi}, \Dvect{e}_{r})$, $BB.EY%Y%/%H%k$r(B $(u, v, w)$ $B$GI=$9(B.

$B3FJ}8~$N5,LO0x;R(B( scale factor )$B$O(B

$$h_\lambda = r \cos \varphi, \ \ h_\varphi = r, \ \ h_r = 1.$$ (136)

$B$7$?$,$C$F(B, $B%9%+%i!<(B $\bullet$ $B$*$h$S%Y%/%H%k(B $\Dvect{A}=(A_{\lambda}, A_{\varphi}, A_r)$ $B$K4X$9$kHyJ,I=8=$O

$$\Dgrad \bullet = \Dvect{e}_{\lambda} \frac{1}{r \cos \varp... ...c{1}{r} \DP{\bullet}{\varphi} + \Dvect{e}_r \DP{\bullet}{r},$$ (137)

$$\Ddiv \Dvect{A} = \frac{1}{r^2 \cos \varphi} \left[ r \... ...hi A_{\varphi}) + \cos \varphi \DP{}{r} ( r^2 A_r ) \right],$$ (138)

$$\nabla^2 \bullet = \frac{1}{r^2 \cos \varphi} \left[ \D... ...}{r} \left( r^2 \cos \varphi \DP{\bullet}{r} \right) \right],$$ (139)

$$\Drot \Dvect{A} = \Dvect{e}_{\lambda} \frac{1}{r} \left[ \DP{A_r}{\varphi} - \DP{}{r}(r A_{\varphi}) \right]$$

$$+ \Dvect{e}_{\varphi} \frac{1}{r \cos \varphi} \left[ \DP{}{r} (r \cos \varphi A_{\lambda}) - \DP{A_r}{\lambda} \right]$$

$$+ \Dvect{e}_r \frac{1}{r \cos \varphi} \left[ \DP{A_{\varphi}}{\lambda} - \DP{}{\varphi} (\cos \varphi A_{\lambda}) \right],$$ (140)

$$\DD{\bullet}{t} = \DP{\bullet}{t} + \frac{u}{r \cos \varp... ...bda} + \frac{v}{r} \DP{\bullet}{\varphi} + w \DP{\bullet}{r},$$ (141)

$$\DD{\Dvect{A}}{t} = \Dvect{e}_{\lambda} \left[ \DP{A_{\lambda}}{t} + \frac{u}{r \cos ... ..._{\lambda}}{r} + \frac{u}{r} A_r - \frac{u \tan \varphi}{r} A_{\varphi} \right]$$

$$+ \Dvect{e}_{\varphi} \left[ \DP{A_{\varphi}}{t} + \frac{u}{r \co... ..._{\varphi}}{r} + \frac{v}{r} A_r + \frac{u \tan \varphi}{r} A_{\lambda} \right]$$

$$+ \Dvect{e}_r \left[ \DP{A_r}{t} + \frac{u}{r \cos \varphi} \DP{A... ...i} + w \DP{A_r}{r} - \frac{v}{r} A_{\varphi} - \frac{u}{r} A_{\lambda} \right].$$ (142)

D..3 $B5e:BI8$X$NJQ49(B

$B<+E>3QB.EY%Y%/%H%k$NI=8=$O

$$2 \Dvect{\Omega} \times \Dvect{v} = 2 \Omega ( \Dvect{e}_{\varphi} \cos \varphi + \Dvect{e}_r \sin \varphi) \times ( u \Dvect{e}_{\lambda} + v \Dvect{e}_{\varphi} + w \Dvect{e}_r)$$ (143)

$$= ( 2 \Omega \cos \varphi w - 2 \Omega \sin \varphi v) \Dvect{e}_{\... ...Omega \sin \varphi u \Dvect{e}_{\varphi} - 2 \Omega \cos \varphi u \Dvect{e}_r.$$ (144)

$B$7$?$,$C$F(B, $B1?F0J}Dx<0$O(B

$$\DD{u}{t} = - \frac{1}{\rho r \cos \varphi } \DP{p}{\lambda} + 2 \Omega v \si... ...\Omega w \cos \varphi + \frac{u v}{r} \tan \varphi - \frac{u w}{r} + F_\lambda,$$ (145)

$$\DD{v}{t} = - \frac{1}{\rho r} \DP{p}{\varphi} - 2 \Omega u \sin \varphi - \frac{u^2}{r} \tan \varphi - \frac{v w}{r} + F_\varphi,$$ (146)

$$\DD{w}{t} = - \frac{1}{\rho} \DP{p}{r} -g + 2 \Omega u \cos \varphi + \frac{u^2}{r} + \frac{v^2}{r} + F_r.$$ (147)

$BO"B3$N<0$O(B

$$\DD{\rho}{t} + \frac{1}{r \cos \varphi} \DP{}{\lambda} ( ... ...i} ( \cos \varphi v) + \frac{1}{r^2} \DP{}{r} ( r^2 w ) = 0.$$ (148)

$BG.NO3X$N<0$O(B

$$\DD{}{t} T = \frac{1}{c_p^d \rho} \DD{p}{t} + \frac{Q^*}{c_p^d}.$$ (149)

$B>uBVJ}Dx<0$O(B

$$p = \rho R T.$$ (150)

$B?e>x5$$N<0$O(B

$$\DD{q}{t} = S_q.$$ (151)

$B$3$3$G(B,

$$\DD{}{t} = \DP{}{t} + \frac{u}{r \cos \phi} \DP{}{\lambda} + \frac{v}{r} \DP{}{\phi} + w \DP{}{r},$$ (152)

$B$G$"$k(B. 13 1313 13 1313 13 1313

E. $z$-$B:BI8%W%j%_%F%#%VJ}Dx<0(B

E..1 $B@ENO3XJ?9U6a;w(B

$B1tD>J}8~$N1?F0J}Dx<0$KBP$7(B, $B@ENO3XJ?9U6a;w$r9T$J$&(B.

$$0 = - \frac{1}{\rho} \DP{p}{z} - g.$$ (153)

$B$3$N$H$-(B, $B1?F0%(%M%k%.!<$NJ]B8B'$r9MN8$7$F(B, $B?eJ?J}8~$N1?F0J}Dx<0$KBP$7(B $B$F$b6a;w$r;\$9(B. $B1?F0%(%M%k%.!<$N<0$O(B, $B1?F0J}Dx<[email protected],$K$=$l$>$l(B $u, v, w$ $B$r$+$1$k$3$H$GF@$i$l$k(B.

$$\DD{}{t} \left( \frac{1}{2} \Dvect{v}^2 \right) = u \DD{u}{t} + v \DD{v}{t} + w \DD{w}{t}$$

$$= u \biggl\{ - \frac{1}{\rho r \cos \varphi } \DP{p}{\lambda} + \un... ... \tan \varphi }_{(3)} - \underbrace{ \frac{u w}{r} }_{(4)} + F_\lambda \biggl\}$$

$$+ v \biggl\{ - \frac{1}{\rho r} \DP{p}{\varphi} - \underbrace{ 2 ... ... \tan \varphi }_{(3)} - \underbrace{ \frac{v w}{r} }_{(5)} + F_\varphi \biggl\}$$

$$+ w \biggl\{ - \frac{1}{\rho} \DP{p}{r} -g + \underbrace{ 2 \Omeg... ...race{ \frac{u^2}{r} }_{(4)} + \underbrace{ \frac{v^2}{r} }_{(5)} + F_r \biggl\}$$

$$= - \frac{1}{\rho} \Dvect{v} \Dgrad{p} - g w - \Dvect{v} \cdot \Dvect{F}.$$ (154)

$B%3%j%*%j$NNO$*$h$S%a%H%j%C%/9`$OF1$8HV9f$N$b$NF1;N$GBG$A>C$7$"$C$F(B, $B1?(B $BF0%(%M%k%.!<$N;~4VJQ2=$K4sM?$7$J$$$3$H$,$o$+$k(B. <sup>14</sup>$B$7$?$,$C$F(B, $B@ENO3XJ?9U6a;w$N:]$K1tD>@.J,$N<0$+$iMn$H$7$?9`(B(2),(4),(5) mail protected],$N<0$N9`$b

$$\DD{u}{t} = \frac{uv \tan \varphi}{r} + fv - \frac{1}{\rho r \cos \varphi} \DP{p}{\lambda} + F_{\lambda}$$ (155)

$$\DD{v}{t} = - \frac{u^2 \tan \varphi}{a} - fu - \frac{1}{\rho r } \DP{p}{\varphi} + F_{\varphi}.$$ (156)

$B$3$3$G(B, $f$ $B$O%3%j%*%j%Q%i%a!<%?(B $f \equiv 2\Omega \sin \varphi$ $B$G$"(B $B$k(B.

E..2 $BGv$$5e3L6a;w(B

$BBg5$$NAX$,CO5eH>7B$KHf$Y$FGv$$$3$H$r2>Dj$7(B, $BJ}Dx<0Cf$N(B $r$ $B$r(B, $BBeI=E*(B $B$JCO5eH>7B(B $a$ $B$G$*$-$+$($k(B. $B$^$?(B, $r$ $B$K$h$kHyJ,$O$9$Y$F3$H49bEY(B $z$ $B$K$h$kHyJ,$G$*$-$+$($k(B. $B$3$N$H$-4pACJ}Dx<0$O

$$\DD{\rho}{t} = - \rho \Ddiv \Dvect{v},$$ (157)

$$\DD{q}{t} = S_q,$$ (158)

$$\DD{u}{t} = \frac{uv \tan \varphi}{a} + fv - \frac{1}{\rho a \cos \varphi} \DP{p}{\lambda} + F_{\lambda},$$ (159)

$$\DD{v}{t} = - \frac{u^2 \tan \varphi}{a} - fu - \frac{1}{\rho a } \DP{p}{\varphi} + F_{\varphi},$$ (160)

$$0 = - \frac{1}{\rho} \DP{p}{z} - g,$$ (161)

$$\DD{T}{t} = \frac{1}{c_p^d \rho} \DD{p}{t} + \frac{Q^*}{c_p^d},$$ (162)

$$p = \rho R^d T_v.$$ (163)

$B$3$3$G(B,

$$\DD{}{t} = \DP{}{t} + \frac{u}{a \cos \varphi} \DP{}{\lambda} + \frac{v}{a} \DP{}{\varphi} + w \DP{}{z},$$ (164)

$$\Ddiv{\Dvect{v}} \equiv \frac{1}{a \cos \varphi} \DP{u}{\l... ...cos \varphi} \DP{v}{\varphi} ( v \cos \varphi ) + \DP{w}{z}.$$ (165)

derivation/derivation-sigmacoord 14 1414 14 1414

F. $B%b%G%k;YG[J}Dx<0(B

F..1 $B12EYJ}Dx<0$HH/;6J}Dx<0(B

$B12EY(B:

$$\zeta \equiv \frac{1}{a \cos \varphi} \DP{v}{\lambda} - \frac{1}{a \cos \varphi} \DP{}{\varphi} ( u \cos \varphi).$$ (166)

$BH/;6(B:

$$D \equiv \frac{1}{a \cos \varphi} \DP{u}{\lambda} + \frac{1}{a \cos \varphi} \DP{}{\varphi} ( v \cos \varphi).$$ (167)

F..1.1 $B12EYJ}Dx<0(B

$B1?F0J}Dx<0$N(B $u$ $B$N<0$K(B $\frac{1}{a \cos \varphi} \DP{}{\varphi} \cos \varphi$ $B$r(B $B:nMQ$7(B, $v$ $B$N<0$K(B $\frac{1}{a \cos \varphi} \DP{}{\lambda}$ $B$r(B $B:nMQ$7:9$r$H$C$FJQ7A$9$l$P

$$\DP{\zeta}{t} = - \frac{1}{a \cos \varphi} \DP{}{\varphi} ( \zeta v \cos \varphi ) - \frac{1}{a \cos \varphi} \DP{}{\lambda} ( \zeta u )$$

$$- \frac{1}{a \cos \varphi} \DP{}{\lambda} \left[ \dot{\sigma} \DP{v}{\sigma} + \frac{R^d T_v}{a p_s} \DP{p_s}{\varphi} - F_{\varphi} + f u \right]$$

$$- \frac{1}{a \cos \varphi} \DP{}{\varphi} \left[ - \cos \varphi \... ...a p_s} \DP{p_s}{\lambda} + F_{\lambda} \cos \varphi + f v \cos \varphi \right].$$ (168)

F..1.2 $BH/;6J}Dx<0(B

$B1?F0J}Dx<0$N(B $u$ $B$N<0$K(B $\frac{1}{a \cos \varphi} \DP{}{\lambda}$ $B$r:n(B $BMQ$7(B, $v$ $B$N<0$K(B $\frac{1}{a \cos \varphi} \DP{}{\varphi} \cos \varphi$ $B$r:nMQ$7OB$r$H$C$FJQ7A$9$k$H

$$\DP{D}{t} = \frac{1}{a \cos \varphi} \DP{}{\lambda} ( \zeta v ) - \frac{1}{a \cos \varphi} \DP{}{\varphi} ( \zeta u \cos \varphi)$$

$$- \frac{1}{a \cos \varphi} \DP{}{\lambda} \left[ \dot{\sigma} \DP... ...frac{R^d T_v}{a \cos \varphi p_s} \DP{p_s}{\lambda} - F_{\lambda} - f v \right]$$

$$- \frac{1}{a \cos \varphi} \DP{}{\varphi} \left[ \cos \varphi \do... ..._s}{\varphi} \cos \varphi - F_{\varphi} \cos \varphi + f u \cos \varphi \right]$$

$$- \nabla^2_{\sigma} ( \Phi + KE).$$ (169)

$B$3$3$G(B,

$$\nabla^2_{\sigma} = \frac{1}{a^2 \cos^2 \varphi} \DP[2]{}{\lambda} + \frac{1}{a^2 \cos \varphi} \DP{}{\varphi} ( \cos \varphi \DP{}{\varphi} ),$$ (170)

$$KE = \frac{u^2 + v^2}{2}.$$ (171)

F..2 $BJQ?tJQ49(B

$B;YG[J}Dx<07O$K$*$1$kJQ?t$r(B, $B%b%G%kFbIt$GMQ$$$F$$$kJQ?t$KJQ49$9$k(B. $B$^$:(B, $\mu \equiv \sin \varphi$ $B$rF3F~$9$k(B. $B$^$?B.EY>l(B $u, v$ $B$O(B $U \equiv u \cos \phi$, $V \equiv \cos \phi$ $B$KJQ49$9$k(B. ¹⁵ $B$3$N$H$-(B, $B?eJ?Iw$N12EY(B $\zeta$ $B$HH/;6(B $D$ $B$O $B$*$h$S(B $D$ $B$HDj5A$9$k(B.

$$\zeta = \frac{1}{a \cos \varphi} \DP{v}{\lambda} - \frac{1}{a \cos \varphi} \DP{}{\varphi} ( u \cos \varphi)$$

$$= \frac{1}{a \cos^2 \varphi} \DP{v \cos \phi}{\lambda} - \frac{1}{a \cos \varphi} \DP{}{\varphi} ( u \cos \varphi)$$

$$= \frac{1}{a ( 1- \mu^2 )} \DP{V}{\lambda} - \frac{1}{a} \DP{U}{\mu},$$ (172)

$$D = \frac{1}{a \cos \varphi} \DP{u}{\lambda} + \frac{1}{a \cos \varphi} \DP{}{\varphi} ( v \cos \varphi)$$

$$= \frac{1}{a \cos^2 \varphi} \DP{u \cos \phi}{\lambda} + \frac{1}{a \cos \varphi} \DP{}{\varphi} ( v \cos \varphi)$$

$$= \frac{1}{a ( 1-\mu^2)} \DP{U}{\lambda} + \frac{1}{a} \DP{V}{\mu}.$$ (173)

$B?eJ?Iw$K$h$k0\N.$O

$$\frac{u}{a \cos\phi}\DP{\bullet}{\lambda} + \frac{v}{a} \DP{\bullet}{\phi} = \frac{1}{a \cos^2 \phi} \left\{ \DP{}{\lambda} (u \cos \phi \bull... ...DP{}{\phi} (v \cos \phi \bullet) - \bullet \DP{}{\phi} ( v \cos \phi ) \right\}$$

$$= \frac{1}{a (1-\mu^2)} \DP{}{\lambda} (U\bullet) -\frac{\bullet}{a... ...\lambda} +\frac{1}{a \mu} \DP{}{\phi} (V\bullet) -\frac{\bullet}{a} \DP{V}{\mu}$$

$$= \frac{1}{a (1-\mu^2)} \DP{}{\lambda} (U\bullet) +\frac{1}{a \mu} \DP{}{\phi} (V\bullet) +\bullet D.$$ (174)

$B?eJ?Iw$K$h$k0\N.$N$b$&$R$H$D$N5-=R$rO"B3$N<0$NJQ49$N$?$a$K<($9(B.

$$\frac{u}{a \cos\phi}\DP{\bullet}{\lambda} + \frac{v}{a} \DP{\bullet}{\phi} = + \frac{u \cos \phi }{a \cos^2 \phi}\DP{\bullet}{\lambda} + \frac{v \cos \phi }{a \cos \phi } \DP{\bullet}{\phi}$$

$$= + \frac{U}{a (1-\mu^2)} \DP{\bullet}{\lambda} + \frac{V}{a} \DP{\bullet}{\mu}$$

$$\equiv \Dvect{v}_H \cdot \Dgrad_{\sigma} \bullet.$$ (175)

$B$3$l$i$rMQ$$$F(B, $BJ}Dx<07O$r $BO"B3$N<0(B ¹⁶

$$\DP{\pi}{t} + \Dvect{v}_H \cdot \Dgrad_{\sigma} \pi = -\Dgrad_{\sigma} \cdot \Dvect{v}_H - \DP{\dot{\sigma}}{\sigma}.$$ (176)

$B12EYJ}Dx<0(B

$$\DP{\zeta}{t} = -\frac{1}{a}\DP{}{\mu} ( \zeta V ) -\frac{1}{a (1-\mu^2)} \DP{}{\lambda} ( \zeta U )$$

$$- \frac{1}{a (1-\mu^2)} \DP{}{\lambda} \left[ \dot{\sigma} \DP{V}... ...ac{R^d T_v}{a} (1-\mu^2) \DP{\pi}{\mu} - F_{\varphi} \cos \varphi + f U \right]$$

$$- \frac{1}{a} \DP{}{\mu} \left[ - \dot{\sigma} \DP{U}{\sigma} - \frac{R^d T_v}{a} \DP{\pi}{\lambda} + F_{\lambda} \cos \varphi + fV \right].$$ (177)

$BH/;6J}Dx<0(B

$$\DP{D}{t} = \frac{1}{a (1-\mu^2)} \DP{}{\lambda} ( \zeta V ) - \frac{1}{a} \DP{}{\mu} ( \zeta U )$$

$$- \frac{1}{a (1-\mu^2)} \DP{}{\lambda} \left[ \dot{\sigma} \DP{U}... ... + \frac{R^d T_v}{a} \DP{\pi}{\lambda} - F_{\lambda} \cos \varphi - f V \right]$$

$$- \frac{1}{a} \DP{}{\mu} \left[ \dot{\sigma} \DP{V}{\sigma} + \frac{R^d T_v}{a} ( 1-\mu^2 ) \DP{\pi}{\mu} - F_{\varphi} \cos \varphi + f U \right]$$

$$- \nabla^2_{\sigma} ( \Phi + KE).$$ (178)

$BG.NO3X$N<0(B

$$\DP{T}{t} = - \frac{1}{a(1-\mu^{2})} \DP{UT}{\lambda} - \frac{1}{a} \DP{VT}{\mu} + T D$$

$$- \dot{\sigma} \DP{T}{\sigma} + \kappa T \left( \DP{\pi}{t} + \Dv... ...bla_{\sigma} \pi + \frac{ \dot{\sigma} }{ \sigma } \right) + \frac{Q^*}{C_{p}}.$$ (179)

$B?e>x5$$N<0(B

$$\DP{q}{t} = - \frac{1}{a(1-\mu^{2})} \DP{Uq}{\lambda} - \frac{1}{a} \DP{Vq}{\mu} + q D$$

$$- \dot{\sigma} \frac{\partial q }{\partial \sigma} + S_{q}.$$ (180)

$B2>29EY(B $T_v$ $B$r $B$N$_$K0MB8$9$k>l(B $\bar{T}_v(\sigma)$ $B$H(B, $B$=$3$+$i$N$:[email protected],(B $T'_v$ $B$K$o$1$F5-=R$9$k(B.

$B12EYJ}Dx<0$G(B $T_v$ $B$r4^$`9`$O

$$- \frac{1}{a (1-\mu^2)} \DP{}{\lambda} \left[ + \frac{R^d T_v}{a}... ...] - \frac{1}{a} \DP{}{\mu} \left[ - \frac{R^d T_v}{a} \DP{\pi}{\lambda} \right]$$

$$= - \frac{1}{a (1-\mu^2)} \DP{}{\lambda} \left[ + \frac{R^d \bar{T}... ...^2)} \DP{}{\lambda} \left[ + \frac{R^d T'_v}{a} (1-\mu^2) \DP{\pi}{\mu} \right]$$

$$- \frac{1}{a} \DP{}{\mu} \left[ - \frac{R^d \bar{T}_v}{a} \DP{\pi... ... - \frac{1}{a} \DP{}{\mu} \left[ - \frac{R^d T'_v}{a} \DP{\pi}{\lambda} \right]$$

$$= - \frac{1}{a} \frac{R^d \bar{T}_v}{a} \frac{\partial^2 \pi}{\part... ...^2)} \DP{}{\lambda} \left[ + \frac{R^d T'_v}{a} (1-\mu^2) \DP{\pi}{\mu} \right]$$

$$+ \frac{1}{a} \frac{R^d \bar{T}_v}{a} \frac{\partial^2 \pi}{\part... ... - \frac{1}{a} \DP{}{\mu} \left[ - \frac{R^d T'_v}{a} \DP{\pi}{\lambda} \right]$$

$$= - \frac{1}{a (1-\mu^2)} \DP{}{\lambda} \left[ + \frac{R^d T'_v}{a... ...- \frac{1}{a} \DP{}{\mu} \left[ - \frac{R^d T'_v}{a} \DP{\pi}{\lambda} \right].$$ (181)

$BH/;6J}Dx<0$G(B $T_v$ $B$r4^$`9`$O

$$- \frac{1}{a (1-\mu^2)} \DP{}{\lambda} \left[ \frac{R^d T_v}{a} \... ...rac{1}{a} \DP{}{\mu} \left[ \frac{R^d T_v}{a} ( 1-\mu^2 ) \DP{\pi}{\mu} \right]$$

$$= - \frac{1}{a (1-\mu^2)} \DP{}{\lambda} \left[ \frac{R^d \bar{T}_v... ...a (1-\mu^2)} \DP{}{\lambda} \left[ \frac{R^d T'_v}{a} \DP{\pi}{\lambda} \right]$$

$$- \frac{1}{a} \DP{}{\mu} \left[ \frac{R^d \bar{T}_v}{a} ( 1-\mu^2... ...ac{1}{a} \DP{}{\mu} \left[ \frac{R^d T'_v}{a} ( 1-\mu^2 ) \DP{\pi}{\mu} \right]$$

$$= - \frac{1}{a^2 (1-\mu^2)} \DP[2]{}{\lambda} ( R^d \bar{T}_v \pi )... ...a (1-\mu^2)} \DP{}{\lambda} \left[ \frac{R^d T'_v}{a} \DP{\pi}{\lambda} \right]$$

$$- \frac{1}{a^2} \DP{}{\mu} \left[ (1-\mu^2) \DP{}{\mu} ( R^d \bar... ...ac{1}{a} \DP{}{\mu} \left[ \frac{R^d T'_v}{a} ( 1-\mu^2 ) \DP{\pi}{\mu} \right]$$

$$= - \Dgrad^2_{\sigma} ( R^d \bar{T}_v \pi ) - \frac{1}{a (1-\mu^2)}... ...c{1}{a} \DP{}{\mu} \left[ \frac{R^d T'_v}{a} ( 1-\mu^2 ) \DP{\pi}{\mu} \right].$$ (182)

$BG.NO3X$N<0$N1&JUBh(B1-3$B9`$O

$$- \frac{1}{a(1-\mu^{2})} \DP{UT}{\lambda} - \frac{1}{a} \DP{VT}{\mu} + T D$$

$$= - \frac{1}{a(1-\mu^{2})} \DP{U \bar{T}}{\lambda} - \frac{1}{a(1-\... ... \frac{1}{a} \DP{V \bar{T}}{\mu} - \frac{1}{a} \DP{VT'}{\mu} + \bar{T} D + T' D$$

$$= - \frac{\bar{T}}{a(1-\mu^{2})} \DP{U}{\lambda} - \frac{1}{a(1-\mu... ...1}{a(1-\mu^{2})} \DP{U}{\lambda} + \frac{\bar{1}}{a} \DP{V}{\mu} \right] + T' D$$

$$= - \frac{1}{a(1-\mu^{2})} \DP{UT'}{\lambda} - \frac{1}{a} \DP{VT'}{\mu} + T' D.$$ (183)

$B0J>e$rMQ$$$FDx<07O$r5-=R$9$l$P $BO"B3$N<0(B

$$\DP{\pi}{t} + \Dvect{v}_H \cdot \Dgrad_{\sigma} \pi = -\Dgrad_{\sigma} \cdot \Dvect{v}_H - \DP{\dot{\sigma}}{\sigma}.$$ (184)

$B@E?e05$N<0(B

$$\DP{\Phi}{\sigma} = - \frac{R^d T_v}{\sigma}.$$ (185)

$B1?F0J}Dx<0(B¹⁷

$$\DP{\zeta}{t} = -\frac{1}{a (1-\mu^2)} \DP{VA}{\lambda} -\frac{1}{a} \DP{UA}{\mu},$$ (186)

$$\DP{D}{t} = \frac{1}{a (1-\mu^2)} \DP{UA}{\lambda} - \fra... ...A}{\mu} - \Dgrad^2_{\sigma} ( \Phi + R \bar{T}_v \pi + KE ).$$ (187)

$B$3$3$G(B,

$$UA \equiv ( \zeta + f ) V - \dot{\sigma} \DP{U}{\sigma} - \frac{RT'}{a} \DP{\pi}{\lambda} + F_{\lambda} \cos \varphi$$ (188)

$$VA \equiv - ( \zeta + f ) U - \dot{\sigma} \DP{V}{\sigma} - \frac{RT'}{a} ( 1 - \mu^{2} ) \DP{\pi}{\mu} + F_{\varphi} \cos \varphi.$$ (189)

$BG.NO3X$N<0(B

$$\DP{T}{t} = - \frac{1}{a(1-\mu^{2})} \DP{UT'}{\lambda} - \frac{1}{a} \DP{VT'}{\mu} + T' D - \dot{\sigma} \DP{T}{\sigma}$$

$$\qquad + \kappa T \left( \DP{\pi}{t} + \Dvect{v}_{H} \cdot \nabla_{\sigma} \pi + \frac{ \dot{\sigma} }{ \sigma } \right) + \frac{Q^*}{C_p^d}.$$ (190)

$B?e>x5$$N<0(B

$$\DP{q}{t} = - \frac{1}{a(1-\mu^{2})} \DP{Uq}{\lambda} -... ... - \dot{\sigma} \frac{\partial q }{\partial \sigma} + S_{q}.$$ (191)

$B<0(B(A.17) $B$GF3F~$7$?(B $Q^*$ $B$+$iG4@-$K$h$k4sM?(B $c_p \mathcal{D}(\Dvect{v})$ $B$r:F$SJ,N%$7(B, $Q^*=Q+c_p \mathcal{D}(\Dvect{v})$ $B$H$9$k(B. $B0lHL$KG4(B $B@-$O1?F0J}Dx<0$K$*$$$FE,Ev$J%Q%i%a%?%j%x5$$N<0$KBP$7$F$=$l$>$l?eJ?3H;69`(B $\mathcal{D}(\zeta)$, $\mathcal{D}(D)$, $\mathcal{D}(T)$, $\mathcal{D}(q)$ $B$r$D$1$k(B. $B$3$N9`$NIU2C$O, $c_p$ $B$r$=$l$>$l(B $R$, $c_p$ $B$N$h$&$K$"$i$?$a$FCV$-$J$*$;$P(B, $B;Y(B $BG[J}Dx<07O(B (3.1) -- (3.6) $B$rF@$k(B.

... $BM}A[5$BN$N>uBVJ}Dx<0$rMQ$$$k(B.⁷

$B4%Ag6u5$$H?e>x5$$O(B, $BF1$8B.EY$H29EY$r$b$D$3$H$r0EL[$N$&$A$K2>Dj(B $B$7$F$$$k(B. $B$7$?$,$C$F(B, $B?e>x5$$K4X$9$k1?F0NLJ]B8B'$*$h$SA4%(%M%k%.!uBVJ}Dx<0$r9MN8$9$kI,MW$,$J$$(B.

... $B?e>x5$$N@8@.>CLG$rL5;k$9$l$P(B,⁸

$Bx5$<0$G$O@8@.>CLG$r4^$a$F$$$k(B. $B$7$?$,$C$F(B, $BA4Bg5$$Nx5$$N@8@.>CLG$,5/$-$F$bA4

... $B$N0zNO$K$h$k%]%F%s%7%c%k(B⁹

$B$3$l$O1s?4NO$r9MN8$7$J$$CO5e$N

... $B$G$"$k$H$7$F$$$k(B.¹⁰

$BF3=P$N2aDx$r<($9(B. $B:8JUBh(B1$B9`$HBh(B2$B9`$O

$$v_i \DP{}{t} ( \rho v_i ) + v_i \DP{}{x_j} ( \rho v_j v_i ) = \DP{}{t} ( \rho v_i^2 ) + \DP{}{x_j} ( \rho v_j v_i^2 ) - \rho \D... ...frac{1}{2} v_i^2 \right) - \rho v_j \DP{}{x_j} \left( \frac{1}{2} v_i^2 \right)$$

$$= \DP{}{t} ( \rho v_i^2 ) + \DP{}{x_j} ( \rho v_j v_i^2 ) - \DP{}{t... ...1}{2} \rho v_i^2 \right) - \DP{}{x_j} \left( \frac{1}{2} v_i^2 \rho v_j \right)$$

$$+ \frac{1}{2} v_i^2 \DP{\rho}{t} + \frac{1}{2} v_i^2 \DP{}{x_j} ( \rho v_j )$$

$$= \DP{}{t} \left( \frac{1}{2} \rho v_i^2 \right) + \DP{}{x_j} ( \fr... ...2 ) + \frac{1}{2} v_i^2 \left\{ \DP{\rho}{t} + \DP{}{x_j} ( \rho v_j ) \right\}$$

$$= \DP{}{t} \left( \frac{1}{2} \rho v_i^2 \right) + \DP{}{x_j} ( \frac{1}{2} \rho v_j v_i^2 ).$$

$B$^$?(B, $B:8JUBh(B4$B9`$O

$$v_i \rho \DP{\Phi^*}{x_i} = \Phi^* \left\{ \DP{\rho}{t} + \DP{}{x_i}(\rho v_i) \right\} + \rho \DP{\Phi^*}{t} + v_i \rho \DP{\Phi^*}{x_i}$$

$$= \DP{}{t} ( \rho \Phi^* ) + \DP{}{x_i} ( \rho \Phi^* v_i ).$$

... $B$G6a;w$9$k$H(B¹¹

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