: A. $B;HMQ>e$NCm0U$H%i%$%;%s%95,Dj(B : dcpam5 $B;YG[J}Dx<07O$NF3=P$K4X$9$k;29M;qNA(B : 2. $B:BI87O$N

• 3.1 $B$O$8$a$K(B
• 3.2 $B@_Dj(B
• 3.3 $B4pACJ}Dx<07O$NF3=P(B
  • 3.3.1 $B>uBVJ}Dx<0(B
  • 3.3.2 $BO"B3$N<0(B
  • 3.3.3 $B?e>x5$$N<0(B
  • 3.3.4 $B1?F0J}Dx<0(B
  • 3.3.5 $BG.NO3X$N<0(B
  
  
• 3.4 $B2sE>7O$X$NJQ49(B
  • 3.4.1 $B%9%+%i!<$NJQ498x<0(B
  • 3.4.2 $B%Y%/%H%k$NJQ498x<0(B
  • 3.4.3 $B2sE>7O$X$NJQ49(B
  
  
• 3.5 $B5e:BI8$X$NJQ49(B
  • 3.5.1 $BD>8r6J@~:BI87O$K$*$1$kHyJ,(B
  • 3.5.2 $B5e:BI87O$K$*$1$kHyJ,(B
  • 3.5.3 $B5e:BI8$X$NJQ49(B
  
  
• 3.6 $z$-$B:BI8%W%j%_%F%#%VJ}Dx<0(B
  • 3.6.1 $B@ENO3XJ?9U6a;w(B
  • 3.6.2 $BGv$$5e3L6a;w(B
  
  
• 3.7 $\sigma$-$B:BI8%W%j%_%F%#%VJ}Dx<0(B
  • 3.7.1 $\sigma$-$B:BI8JQ498x<0(B
  • 3.7.2 $\sigma$-$B:BI8%W%j%_%F%#%VJ}Dx<07O(B
    • 3.7.2.1 $B@ENO3XJ?9U$N<0(B
    • 3.7.2.2 $B1?F0J}Dx<0(B
    • 3.7.2.3 $BO"B3$N<0(B
    • 3.7.2.4 $BG.NO3X$N<0(B
  • 3.7.3 $B6-3&>r7o(B
    • 3.7.3.1 $BCOI=LL9bEY(B
    • 3.7.3.2 $\sigma$ $B:BI81tD>B.EY(B
    • 3.7.3.3 $B?eJ?N.$*$h$SG.NO3XJQ?t(B
  • 3.7.4 $B798~J}Dx<0(B
  
  
• 3.8 $B%b%G%k;YG[J}Dx<0(B
  • 3.8.1 $B12EYJ}Dx<0(B
  • 3.8.2 $BH/;6J}Dx<0(B
  • 3.8.3 $BG.NO3X$N<0(B
  • 3.8.4 $B29EY$N4pK\>l$H$:$l$NJ,N%(B
  • 3.8.5 $BJQ?tJQ49(B
  • 3.8.6 $B;YG[J}Dx<0$N:G=*7A(B
  
  
• 3.9 $B;29MJ88%(B

3. $BNO3X2aDx$N;YG[J}Dx<07O$NF3=P(B

3.1 $B$O$8$a$K(B

dcpam5$B$G$ONO3X7W;;$H$7$F5eLL0^EY7PEY:BI8(B, $B1tD>(B $\sigma$ $B:BI8$N(B $B%W%j%_%F%#%VJ}Dx<07O$r2r$$$F$$$k(B. $B0J2<$G$O(B, $B$^$:A[Dj$9$kBg5$$K$D$$$F(B $B$N2>Dj$r9T$C$?8e(B, $BA4x5$NL$N<0(B, $B1?F0J}Dx<0(B (3 mail protected],(B), $BG.NO3X$N<0$N(B 6 $B$D$NJ}Dx<0$+$i(B, dcpam5$B$G3.8.6$B@a(B $B$r;2>H$N$3$H(B.

3.2 $B@_Dj(B

dcpam5$B$G$OCO5eBg5$$rA[Dj$7(B, $BA4Bg5$$O$H$b$KM}A[5$BN$G$"$k4%Ag6u5$$*$h$S?e>x5$$+$i@.$k:.9gBg5$$H$9$k(B. $B1@?eNL$OL5;k$9$k(B. $B$^$?(B, $B?e>x5$NL$,A4Bg5$$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 $B=ENO2CB.EY$OOG@1Cf?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, $B1?F0$OOG@1I=LLIU6a$K8B$i$l$k$3$H$r2>Dj$7$F6a;w$r9T$J$&(B.

3.3 $B4pACJ}Dx<07O$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 ^3.1.

$B!*!*Cm0U(B: $B$3$NIUO?Cf$G$OF3=P$NET9g>e(B, $B4%Ag6u5$$N5$BNDj?t$r(B $R^d$, $BDj05HfG.$r(B $C_p^d$, $BA4Bg5$$N5$BNDj?t$r(B $R$ $B$H$*$/(B. $B$7$+$7(B, $B%b%G%k$N$B;YG[J}Dx<07O$H$=$NN%;62=(B$B!Y(B $B$N!XNO3X2aDx!Y(B $B$G$O(B, $B4%Ag6u5$$N5$BNDj?t$r(B $R$, $BDj05HfG.$r(B $C_p$ $B$HI=5-$7$F$$$k(B $B$N$GN10U$$$?$@$-$?$$(B.

3.3.1 $B>uBVJ}Dx<0(B

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

$$p^d = \rho^{d} R^d T,$$ (3.1)

$$p^v = \rho^{v} R^v T$$ (3.2)

$B$G$"$k(B. $B$3$3$G(B$p$ $B$O05NO(B, $\rho$$B$OL)EY(B, $R$$B$O5$BNDj?t(B, $T$$B$O29EY$G$"$j(B, $\bullet^d$, $\bullet^v$ $B$O$=$l$>$l4%Ag6u5$$*$h$S?e>x5$$K(B $B4X$9$kNL$G$"$k$3$H$r<($9(B. $B$7$?$,$C$F(B, $BA405(B $p=p^d+p^v$ $B$O(B,

$$\begin{align*}\begin{split}p & = (\rho^d R^d + \rho^v R^v) T \\ & = \rho R^d ( 1 + \epsilon_v q ) T \end{split}\end{align*}$$ (3.3)

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

$$p = \rho R T.$$ (3.4)

$B$?$@$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,

$$p = \rho R^d T_v$$ (3.5)

$B$HI=$5$l$k(B.

3.3.2 $BO"B3$N<0(B

$BA4Bg5$$Nx5$$N@8@.>CLG$rL5;k$9$l$P(B ^3.2,

$$\DP{\rho}{t} + \DP{}{x_j}( \rho v_j ) = 0.$$ (3.6)

$B$3$3$G(B, $v$$B$OIwB.$G$"$k(B. $B%i%0%i%s%8%e7A<0$G5-=R$9$l$P(B,

$$\DD{\rho}{t} + \rho \Ddiv \Dvect{v} = 0.$$ (3.7)

3.3.3 $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,

$$\DP{\rho^v}{t} + \DP{}{x_j} ( \rho^v v_j ) = S.$$ (3.8)

$BHf<>(B $q=\rho^v/\rho$ $B$K4X$9$k<0$O(B, $B86M}E*$K$O(B() $B$H(B(3.8) $B$+$iF@$k$3$H$,$G$-$k(B. $B$7$+$7(B, $B:#$N>l9g(B, (3.6)$B$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.

$$\DD{q}{t} = S_q.$$ (3.9)

3.3.4 $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

$$\DP{}{t}(\rho v_i) + \DP{}{x_j}( \rho v_i v_j ) + \DP{p}{x_i} - \DP{\sigma_{ij}}{x_j} + \rho \DP{\Phi^*}{x_i} = {\cal F}_i^{\prime}.$$ (3.10)

$B$3$3$G(B, $\sigma_{ij}$ $B$OG4@-1~NO%F%s%=%k(B, $\Phi^*$ $B$OOG@1(B $B$N0zNO$K$h$k%]%F%s%7%c%k(B ^3.3, ${\cal F}_i^{\prime}$ $B$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

$$\rho \DD{v_i}{t} + \DP{p}{x_i} - \DP{\sigma_{ij}}{x_j} + \rho \DP{\Phi^*}{x_i} = {\cal F}_i^{\prime }$$ (3.11)

$B$H$J$k(B. $B$3$3$G(B, $BG4@-9`$H30NO9`$r(B ${\cal F}_i$ $B$H$*$-(B, $B$5$i$K%Y%/%H%kI=<($9$k(B.

$$\rho \DD{\Dvect{v}}{t} + \Dgrad p + \rho \Dgrad \Phi^* = \Dvect{\cal F}.$$ (3.12)

3.3.5 $BG.NO3X$N<0(B

$BC10L $\Dvect{v}^2/2$ $B$HFbIt%((B $B%M%k%.!<(B $\varepsilon$ $B$*$h$S%]%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,

$$\DP{}{t} \left[ \rho \left( \frac{1}{2} \Dvect{v}^2 + \varepsilon... ... \right)v_j + p v_j - \sigma_{ij}v_i \right] = \rho Q + {\cal F}_i^{\prime} v_i$$ (3.13)

$B$G$"$k(B. $B$3$3$G(B, $Q$ $B$O30It$+$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$+$1(B, $BO"B3$N<0$rMQ$$$FJQ7A$9$k$3$H$GF@$i$l$k(B ^3.4.

$$\DP{}{t} \left( \frac{1}{2} \rho v_i^2 + \rho \Phi^* \right) + \D... ...ight) = p \DP{v_j}{x_j} - \sigma_{ij} \DP{v_i}{x_j} + {\cal F}_i^{\prime} v_i .$$ (3.14)

$B$3$3$G(B, $BJQ7A$N:]$K$O(B $\DP{\Phi^*}{t}=0$ $B$G$"$k$H$7$F$$$k(B. (3.13)$B$H(B (3.14) $B$H$N:9$r$H$k$H(B, $B

$$\DP{}{t} ( \rho \varepsilon ) + \DP{}{x_j} ( \rho \varepsilon v_j ) = - p \DP{v_j}{x_j} + \sigma_{ij} \DP{v_i}{x_j} + \rho Q .$$ (3.15)

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

$$\rho \DD{\varepsilon}{t} = \frac{p}{\rho} \left( \DD{\rho}{t} \right) + \rho Q.$$ (3.16)

$B0J9_$G$O(B, $B30It$+$i$N2CG.$N9`$HG4@-$K$h$k2CG.$N9`$r(B $B$^$H$a$F(B $Q^*$ $B$H$*$/$3$H$H$9$k(B.

$BFbIt%(%M%k%.!<$r29EY$rMQ$$$FI=8=$9$k$H(B $\varepsilon = C_v T$ $B$G$"$k(B. $C_v$$B$ODj05HfG.$G$"$k(B. $B$5$i$K>uBVJ}Dx<0(B (3.4) $B$rMQ$$$F(B(3.16) $B$rJQ7A$9$k(B. $C_p = C_v + R$ $B$G$"$k$3$H$KCm0U$9$l$P(B

$$\DD{C_p T}{t} = \frac{1}{\rho} \DD{p}{t} + Q^*,$$ (3.17)

$B$H$J$k(B. $B$3$3$G(B, $C_p$ $B$r4%Ag6u5$$NDj05HfG.(B $C_p^d$ $B$G6a;w$9$k$H(B ^3.5, $B

$$\DD{T}{t} = \frac{1}{C_p^d \rho} \DD{p}{t} + \frac{Q^*}{C_p^d}.$$ (3.18)

3.4 $B2sE>7O$X$NJQ49(B

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

3.4.1 $B%9%+%i!<$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,

$$\left( \DD{\psi}{t} \right)_{\rm a} = \left( \DD{\psi}{t} \right)_{\rm r}$$ (3.19)

$B$,@.$j$?$D(B ^3.6.

3.4.2 $B%Y%/%H%k$NJQ498x<0(B

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

$$\left( \DD{\Dvect{A}}{t} \right)_{\rm a} = \left( \DD{\Dvect{A}}{t} \right)_{\rm r} + \Dvect{\Omega} \times \Dvect{A}.$$ (3.20)

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

$$\Dvect{A} = \Dvect{i} A_x + \Dvect{j} A_y + \Dvect{k} A_z$$ (3.21)

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

$$\Dvect{A} = \Dvect{i}' A'_x + \Dvect{j}' A'_y + \Dvect{k}' A'_z$$ (3.22)

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

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

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

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

$$= \left( \DD{\Dvect{A}}{t} \right)_{\rm r} + \Dvect{\Omega} \times \Dvect{A}.$$ (3.23)

($B>ZL@=*$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

$$\Dvect{v}_a = \Dvect{v} + \Dvect{\Omega} \times \Dvect{r}.$$ (3.24)

$B$5$i$K(B, (3.20) $B$G(B $\Dvect{A}=\Dvect{v}_{\rm a}$ $B$H$*(B $B$1$P(B, $BB.EY$N;~4VHyJ,9`$O(B

$$\DD{\Dvect{v}_a}{t} = \DD{\Dvect{v}}{t} + 2 \Dvect{\Omega} \times \Dvect{v} + \Dvect{\Omega} \times ( \Dvect{\Omega} \times \Dvect{r} )$$ (3.25)

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

3.4.3 $B2sE>7O$X$NJQ49(B

$BJQ49$N(B(3.25)$B$rMQ$$$F1?F0J}Dx<0$r2sE>7O$G5-=R$9$k(B.

$$\DD{\Dvect{v}}{t} = - \frac{1}{\rho} \Dgrad p - 2 \Dvect{\Omega} ... ...a} \times ( \Dvect{\Omega} \times \Dvect{r} ) + \Dgrad \Phi^* + \Dvect{\cal F}.$$ (3.26)

$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

$$\DD{\Dvect{v}}{t} = - \frac{1}{\rho} \Dgrad p - 2 \Dvect{\Omega} \times \Dvect{v} + \Dvect{g} + \Dvect{\cal F}$$ (3.27)

$B$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.

3.5 $B5e:BI8$X$NJQ49(B

3.5.1 $BD>8r6J@~:BI87O$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.

$$\Dgrad \bullet = \left( \frac{1}{h_1} \DP{\bullet}{\xi_1}, \frac{1}{h_2} \DP{\bullet}{\xi_2}, \frac{1}{h_3} \DP{\bullet}{\xi_3} \right),$$ (3.28)

$$\Ddiv \Dvect{A} = \frac{1}{h_1 h_2 h_3} \left[ \DP{}{\xi_1} ( h_2 h_3 A_1) + \DP{}{\xi_2} ( h_1 h_3 A_2) + \DP{}{\xi_3} ( h_1 h_2 A_3) \right],$$ (3.29)

$$\nabla^2 \bullet = \frac{1}{h_1 h_2 h_3} \left[ \DP{}{\xi_1} \left( \frac{h_2 h_3}... ... + \DP{}{\xi_3} \left( \frac{h_1 h_2}{h_3} \DP{\bullet}{\xi_3} \right) \right],$$ (3.30)

$$\Drot \Dvect{A} = \left( \frac{1}{h_2 h_3} \left[ \DP{(h_3 A_3)}{\xi_2} - \DP{(h_... ...{h_1 h_2} \left[ \DP{(h_2 A_2)}{\xi_1} - \DP{(h_1 A_1)}{\xi_2} \right] \right),$$ (3.31)

$$\DD{\bullet}{t} = \DP{\bullet}{t} + \frac{v_1}{h_1} \DP{\bullet}{\xi_1} + \frac{v_2}{h_2} \DP{\bullet}{\xi_2} + \frac{v_3}{h_3} \DP{\bullet}{\xi_3},$$ (3.32)

$$\DD{\Dvect{v}}{t} = \sum^3_{k=1} \Dvect{e}_k \left[ \DP{v_k}{t} + \sum^3_{j=1} \fra... ...h_j}{\xi_k} +\frac{v_k}{h_k} \frac{1}{h_j} \DP{h_k}{\xi_j} \right) v_j \right].$$ (3.33)

3.5.2 $B5e:BI87O$K$*$1$kHyJ,(B

$B=ENO2CB.EY(B $\Dvect{g}$ $B$,OG@1Cf?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

$$x_1 = r \cos \varphi \cos \lambda,$$ (3.34)

$$x_2 = r \cos \varphi \sin \lambda,$$ (3.35)

$$x_3 = r \sin \varphi$$ (3.36)

$B$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.

$B3FJ}8~$N5,3J2=0x;R(B (scale factor) $B$O(B

$$h_\lambda = r \cos \varphi, \ \ h_\varphi = r, \ \ h_r = 1.$$ (3.37)

$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

$$\Dgrad \bullet = \Dvect{e}_{\lambda} \frac{1}{r \cos \varphi} \DP{\bullet}{\lamb... ...t{e}_{\varphi} \frac{1}{r} \DP{\bullet}{\varphi} + \Dvect{e}_r \DP{\bullet}{r},$$ (3.38)

$$\Ddiv \Dvect{A} = \frac{1}{r^2 \cos \varphi} \left[ r \DP{A_{\lambda}}{\lambda} +... ...arphi} ( \cos \varphi A_{\varphi}) + \cos \varphi \DP{}{r} ( r^2 A_r ) \right],$$ (3.39)

$$\nabla^2 \bullet = \frac{1}{r^2 \cos \varphi} \left[ \DP{}{\lambda} \left( \frac{1... ...hi} \right) + \DP{}{r} \left( r^2 \cos \varphi \DP{\bullet}{r} \right) \right],$$ (3.40)

$$\begin{align*}\begin{split}\Drot \Dvect{A} & = \quad \Dvect{e}_{\lambda} \frac{1... ...da} - \DP{}{\varphi} (\cos \varphi A_{\lambda}) \right], \end{split}\end{align*}$$ (3.41)

$$\DD{\bullet}{t} = \DP{\bullet}{t} + \frac{u}{r \cos \varphi} \DP{\bullet}{\lambda} + \frac{v}{r} \DP{\bullet}{\varphi} + w \DP{\bullet}{r},$$ (3.42)

$$\begin{align*}\begin{split}\DD{\Dvect{A}}{t} & = \quad \Dvect{e}_{\lambda} \left... ...rac{v}{r} A_{\varphi} - \frac{u}{r} A_{\lambda} \right]. \end{split}\end{align*}$$ (3.43)

3.5.3 $B5e:BI8$X$NJQ49(B

$B%3%j%*%j9`$NI=8=$O

$$\begin{align*}\begin{split}2 \Dvect{\Omega} \times \Dvect{v} & = 2 \Omega ( \Dve... ...vect{e}_{\varphi} - 2 \Omega \cos \varphi u \Dvect{e}_r. \end{split}\end{align*}$$ (3.44)

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

$$\DD{u}{t} = - \frac{1}{\rho r \cos \varphi } \DP{p}{\lambda} + 2 \Omega v \... ...w \cos \varphi + \frac{u v}{r} \tan \varphi - \frac{u w}{r} + {\cal F}_\lambda,$$ (3.45)

$$\DD{v}{t} = - \frac{1}{\rho r} \DP{p}{\varphi} - 2 \Omega u \sin \varphi - \frac{u^2}{r} \tan \varphi - \frac{v w}{r} + {\cal F}_\varphi,$$ (3.46)

$$\DD{w}{t} = - \frac{1}{\rho} \DP{p}{r} -g + 2 \Omega u \cos \varphi + \frac{u^2}{r} + \frac{v^2}{r} + {\cal F}_r.$$ (3.47)

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

$$\Dinv{\rho} \DD{\rho}{t} + \frac{1}{r \cos \varphi} \DP{}{\lambda... ...arphi} \DP{}{\varphi} ( \cos \varphi v) + \frac{1}{r^2} \DP{}{r} ( r^2 w ) = 0.$$ (3.48)

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

$$\DD{}{t} T = \frac{1}{C_p^d \rho} \DD{p}{t} + \frac{Q^*}{C_p^d}.$$ (3.49)

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

$$p = \rho R^d T_v.$$ (3.50)

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

$$\DD{q}{t} = S_q.$$ (3.51)

$B$3$3$G(B,

$$\DD{}{t} = \DP{}{t} + \frac{u}{r \cos \varphi} \DP{}{\lambda} + \frac{v}{r} \DP{}{\varphi} + w \DP{}{r}$$ (3.52)

$B$G$"$k(B.

3.6 $z$-$B:BI8%W%j%_%F%#%VJ}Dx<0(B

3.6.1 $B@ENO3XJ?9U6a;w(B

$B1tD>J}8~$N1?F0J}Dx<0$KBP$7(B, $B0J2<$N$h$&$K@ENO3XJ?9U6a;w$r9T$J$&(B.

$$0 = - \frac{1}{\rho} \DP{p}{z} - g.$$ (3.53)

$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.

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

$$= \quad u \biggl\{ - \frac{1}{\rho r \cos \varphi } \DP{p}{\lambd... ...varphi }_{(3)} - \underbrace{ \frac{u w}{r} }_{(4)} + {\cal F}_\lambda \biggl\}$$

$$\quad + v \biggl\{ - \frac{1}{\rho r} \DP{p}{\varphi} - \underbra... ...varphi }_{(3)} - \underbrace{ \frac{v w}{r} }_{(5)} + {\cal F}_\varphi \biggl\}$$

$$\quad + w \biggl\{ - \frac{1}{\rho} \DP{p}{r} -g + \underbrace{ 2... ...frac{u^2}{r} }_{(4)} + \underbrace{ \frac{v^2}{r} }_{(5)} + {\cal F}_r \biggl\}$$

$$= - \frac{1}{\rho} \Dvect{v} \Dgrad{p} - g w - \Dvect{v} \cdot \Dvect{\cal F}.$$ (3.54)

$B%3%j%*%j$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>3.7</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

$$\DD{u}{t} = \frac{uv \tan \varphi}{r} + fv - \frac{1}{\rho r \cos \varphi} \DP{p}{\lambda} + {\cal F}_{\lambda} ,$$ (3.55)

$$\DD{v}{t} = - \frac{u^2 \tan \varphi}{a} - fu - \frac{1}{\rho r } \DP{p}{\varphi} + {\cal F}_{\varphi} .$$ (3.56)

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

3.6.2 $BGv$$5e3L6a;w(B

$BBg5$$NAX$,OG@1H>7B$KHf$Y$FGv$$$3$H$r2>Dj$7(B, $BJ}Dx<0Cf$N(B $r$ $B$r(B, $BBeI=E*(B $B$JOG@1H>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

$$\DD{\rho}{t} = - \rho \Ddiv \Dvect{v},$$ (3.57)

$$\DD{q}{t} = S_q,$$ (3.58)

$$\DD{u}{t} = \frac{uv \tan \varphi}{a} + fv - \frac{1}{\rho a \cos \varphi} \DP{p}{\lambda} + {\cal F}_{\lambda},$$ (3.59)

$$\DD{v}{t} = - \frac{u^2 \tan \varphi}{a} - fu - \frac{1}{\rho a } \DP{p}{\varphi} + {\cal F}_{\varphi},$$ (3.60)

$$= - \frac{1}{\rho} \DP{p}{z} - g,$$ 0 (3.61)

$$\DD{T}{t} = \frac{1}{C_p^d \rho} \DD{p}{t} + \frac{Q^*}{C_p^d},$$ (3.62)

$$p = \rho R^d T_v.$$ (3.63)

$B$3$3$G(B,

$$\DD{}{t} = \DP{}{t} + \frac{u}{a \cos \varphi} \DP{}{\lambda} + \frac{v}{a} \DP{}{\varphi} + w \DP{}{z},$$ (3.64)

$$\Ddiv{\Dvect{v}} \equiv \frac{1}{a \cos \varphi} \DP{u}{\lambda} + \frac{1}{a \cos \varphi} \DP{v}{\varphi} ( v \cos \varphi ) + \DP{w}{z}.$$ (3.65)

3.7 $\sigma$-$B:BI8%W%j%_%F%#%VJ}Dx<0(B

$B@ENO3XJ?9U$N$b$H$G$O(B, $B5$05(B $p$ $B$O1tD>:BI8(B $z$ $B$KBP$7C1D48:>/$9$k4X?t$G(B $B$"$k(B. $B$=$3$G(B, $B1tD>:BI8$r(B $z$ $B$+$i(B, $BCOI=LL5$05(B $p_s$ $B$G5,3J2=$7$?5$05:BI8(B,

$$\sigma \equiv \frac{p}{p_s}$$ (3.66)

$B$KJQ49$9$k(B. $\sigma$ $B$H(B $z$ $B$N4X78$O(B, $B@ENO3XJ?9U$N<0(B (3.5)$B$rJQ7A$7$FF@$i$l$k(B.

$$\DP{\sigma}{z} = - \frac{g \sigma}{R^d T_v}.$$ (3.67)

3.7.1 $\sigma$-$B:BI8JQ498x<0(B

$z$- $B:BI8$+$i(B $\sigma$- $B:BI8$X$NJQ498x<0$r<($9(B.

$B1tD>HyJ,(B

$$\begin{align*}\begin{split}\DP{\bullet}{z} & = \DP{\sigma}{z} \DP{\bullet}{\sigma} \\ & = - \frac{g \sigma}{R^d T_v} \DP{\bullet}{\sigma}. \end{split}\end{align*}$$ (3.68)

$B?eJ?HyJ,(B

$$\begin{align*}\begin{split}\left( \DP{\bullet}{\lambda} \right)_z & = \left( \DP... ...bullet}{\sigma} \left( \DP{z}{\lambda} \right)_{\sigma}, \end{split}\end{align*}$$ (3.69)

$$\begin{align*}\begin{split}\left( \DP{\bullet}{\varphi} \right)_z & = \left( \DP... ...bullet}{\sigma} \left( \DP{z}{\varphi} \right)_{\sigma}. \end{split}\end{align*}$$ (3.70)

$B;~4VHyJ,(B

$$\begin{align*}\begin{split}\left( \DP{\bullet}{t} \right)_z & = \left( \DP{\bull... ... \DP{\bullet}{\sigma} \left( \DP{z}{t} \right)_{\sigma}. \end{split}\end{align*}$$ (3.71)

$B%i%0%i%s%8%eHyJ,$O$3$l$i$rMQ$$$F(B,

$$\left( \DD{\bullet}{t} \right)_z = \left( \DP{\bullet}{t} \right)_z + \frac{u}{a \cos \varphi} \le... ...{a} \left( \DP{\bullet}{\varphi} \right)_z + w \left( \DP{\bullet}{z} \right)_z$$

$$= \left( \DP{\bullet}{t} \right)_{\sigma} + \frac{u}{a \cos \varp... ...a} \right)_{\sigma} + \frac{v}{a} \left( \DP{\bullet}{\varphi} \right)_{\sigma}$$

$$\quad + \frac{g \sigma}{R^d T_v} \left\{ \left( \DP{z}{t} \right)... ...{v}{a} \left( \DP{z}{\varphi} \right)_{\sigma} -w \right\} \DP{\bullet}{\sigma}$$

$$= \left( \DD{\bullet}{t} \right)_{\sigma}.$$ (3.72)

$B$3$3$G(B, $\sigma$-$B:BI81tD>B.EY(B $\dot{\sigma}$ $B$rDj5A$9$k(B.

$$\dot{\sigma} \equiv \frac{g \sigma}{R^d T_v} \left\{ \left( \DP{z... ...ht)_{\sigma} + \frac{v}{a} \left( \DP{z}{\varphi} \right)_{\sigma} -w \right\}.$$ (3.73)

3.7.2 $\sigma$-$B:BI8%W%j%_%F%#%VJ}Dx<07O(B

3.7.2.1 $B@ENO3XJ?9U$N<0(B

(3.67)$B$r=ENO%]%F%s%7%c%k(B $\Phi=gz$ $B$rMQ$$$F=q$1$P(B,

$$\DP{\Phi}{\sigma}=-\frac{R^d T_v}{\sigma}.$$ (3.74)

3.7.2.2 $B1?F0J}Dx<0(B

$B?eJ?$N05NO8{G[$O(B, (3.69)$B$*$h$S(B(3.70) $B$r(B $p$ $B$KBP$7$FE,MQ$7(B, (3.66) $B$rMQ$$$l$P

$$\frac{1}{\rho} \left( \DP{p}{\lambda} \right)_z = \frac{1}{\rho} \left\{ \DP[][\sigma]{p}{\lambda} + \frac{g \sigma}{R^d T_v} \DP{p}{\sigma} \DP[][\sigma]{z}{\lambda} \right\}$$

$$= \frac{R^d T_v}{p_s} \DP{p_s}{\lambda} + \frac{R^d T_v}{p} \frac{g \sigma}{R^d T_v} p_s \DP[][\sigma]{z}{\lambda}$$

$$= R^d T_v \DP[][\sigma]{\pi}{\lambda} + \DP{\Phi}{\lambda},$$ (3.75)

$$\frac{1}{\rho} \left( \DP{p}{\varphi} \right)_z = R^d T_v \DP[][\sigma]{\pi}{\varphi} + \DP{\Phi}{\varphi}.$$ (3.76)

$B$3$3$G(B $\pi \equiv \ln p_s$ $B$G$"$k(B. $B$7$?$,$C$F(B, $B1?F0J}Dx<[email protected],$O(B,

$$\DD{u}{t} -f v - \frac{uv}{a} \tan \varphi = - \frac{1}{a \cos \varphi} \DP{\Phi}{\lambda} - \frac{R^d T_v}{a \cos \varphi} \DP{\pi}{\lambda} + {\cal F}_{\lambda},$$ (3.77)

$$\DD{v}{t} + fu + \frac{u^2}{a} \tan \varphi = - \frac{1}{a} \DP{\Phi}{\varphi} - \frac{R^d T_v}{a} \DP{\pi}{\varphi} + {\cal F}_{\varphi}.$$ (3.78)

3.7.2.3 $BO"B3$N<0(B

$BB.EY$NH/;6$O(B,

$$\left( \Ddiv \Dvect{v} \right)_z = \frac{1}{a \cos \varphi} \left[ \DP[][\sigma]{u}{\lambda} + \frac{g \sigma}{R^d T_v} \DP{u}{\sigma} \DP[][\sigma]{z}{\lambda} \right]$$

$$\quad + \frac{1}{a \cos \varphi} \left[ \left( \DP{}{\varphi} (v ... ...ght] - \frac{g \sigma}{R^d T_v} \DP{}{\sigma} \left( \DD{z}{t} \right)_{\sigma}$$

$$= \frac{1}{a \cos \varphi} \left[ \DP[][\sigma]{u}{\lambda} + \frac{g \sigma}{R^d T_v} \DP{u}{\sigma} \DP[][\sigma]{z}{\lambda} \right]$$

$$\quad + \frac{1}{a \cos \varphi} \left[ \left( \DP{}{\varphi} (v ... ...igma}{R^d T_v}\DP{}{\sigma} ( v \cos \varphi) \DP[][\sigma]{z}{\lambda} \right]$$

$$\quad - \frac{g \sigma}{R^d T_v} \DP{}{\sigma} \left[ \DP[][\sigm... ...} + \frac{v}{a} \DP[][\sigma]{z}{\varphi} + \dot{\sigma} \DP{z}{\sigma} \right]$$

$$= \frac{1}{a \cos \varphi} \DP[][\sigma]{u}{\lambda} + \frac{1}{a... ...t( \DP{}{\varphi} (v \cos \varphi) \right)_{\sigma} + \DP{\dot{\sigma}}{\sigma}$$

$$\quad - \frac{g \sigma}{R^d T_v} \left[ \DP{}{\sigma} \DP[][\sigm... ...P[][\sigma]{z}{\varphi} + \dot{\sigma} \DP{}{\sigma} \DP[][]{z}{\sigma} \right]$$

$$= ( \Ddiv{\Dvect{v}_H})_{\sigma} + \DP{\dot{\sigma}}{\sigma} + \DP{\sigma}{z} \left( \DD{}{t} \DP{z}{\sigma} \right)_{\sigma}.$$ (3.79)

$B$3$3$G(B,

$$\Ddiv{\Dvect{v}_H} \equiv \frac{1}{a \cos \varphi} \DP[][\sigma]{... ...ac{1}{a \cos \varphi} \left( \DP{}{\varphi} (v \cos \varphi ) \right)_{\sigma}.$$ (3.80)

$B$f$($K(B, $z$- $B:BI8O"B3$N<0$O

$$\frac{1}{\rho} \left( \DD{\rho}{t} \right)_z + \left( \Ddiv{\Dvect{v}} \right)_z = \frac{1}{\rho} \left( \DD{\rho}{t} \right)_{\sigma} + \left( \D... ...igma}}{\sigma} + \DP{\sigma}{z} \left( \DD{}{t} \DP{z}{\sigma} \right)_{\sigma}$$

$$= \frac{1}{\rho} \left( \DD{\rho}{t} \right)_{\sigma} + \left( \D... ...}}{\sigma} + \frac{\rho}{p_s} \left( \DD{}{t} \frac{p_s}{\rho} \right)_{\sigma}$$

$$= \left( \DD{\ln p_s}{t} \right)_{\sigma} + \left( \Ddiv{\Dvect{v}_H} \right)_{\sigma} + \DP{\dot{\sigma}}{\sigma}.$$ (3.81)

$B$7$?$,$C$F(B $\pi \equiv \ln p_s$ $B$rMQ$$$F5-=R$9$l$P

$$\DD{\pi}{t} + \Ddiv{\Dvect{v}_H} + \DP{\dot{\sigma}}{\sigma} = 0.$$ (3.82)

3.7.2.4 $BG.NO3X$N<0(B

(3.62)$B$N1&JUBh(B1$B9`$O

$$\frac{1}{C_p^d \rho} \DD{p}{t} = \frac{1}{C_p^d \rho} \left\{ \DP{p}{t} + \Dvect{v}_H \cdot \nabla_{\sigma} p + \dot{\sigma} \DP{p}{\sigma} \right\}$$

$$= \frac{1}{C_p^d \rho} \left\{ \sigma \DP{p_s}{t} + \sigma \Dvect{v}_H \cdot \nabla_{\sigma} p_s + \dot{\sigma} p_s \right\}$$

$$= \frac{R^d T_v}{C_p^d} \left\{ \DP{\pi}{t} + \Dvect{v}_H \cdot \nabla_{\sigma} \pi + \frac{\dot{\sigma}}{\sigma} \right\}.$$ (3.83)

$B$3$3$G(B,

$$\Dvect{v}_H \cdot \nabla_{\sigma} = \frac{u}{a \cos \varphi} \DP{}{\lambda} + \frac{v}{a} \DP{}{\varphi}.$$ (3.84)

$B$7$?$,$C$F(B, $BG.NO3X$N<0$O

$$\DD{T}{t} = \frac{R^d T_v}{C_p^d} \left\{ \DP{\pi}{t} + \Dvect{v}... ...\nabla_{\sigma} \pi + \frac{\dot{\sigma}}{\sigma} \right\} + \frac{Q^*}{C_p^d}.$$ (3.85)

3.7.3 $B6-3&>r7o(B

$B$3$3$G(B, $\sigma$ $B:BI8$K$*$1$k6-3&>r7o$K$D$$$F=R$Y$k(B.

3.7.3.1 $BCOI=LL9bEY(B

$$\Phi = \Phi_s (\lambda, \varphi) \ \ \ \ {\rm at} \ \ \sigma=1.$$ (3.86)

$B$9$J$o$A(B, $\Phi_s$ $B$OI=LLCO7A$rI=$9(B. $B$3$N6-3&>r7o$rMQ$$$F(B, $B@ENO3XJ?9U(B $B$N<0$r1tD>@QJ,$9$k$3$H$G(B, $BG$0U$N(B $\sigma$ $B$K$*$1$k9bEY(B $\Phi$ $B$r5a$a$k(B $B$3$H$,$G$-$k(B.

3.7.3.2 $\sigma$ $B:BI81tD>B.EY(B

$$\dot{\sigma} = 0 \ \ \ at \ \ \sigma = 0, \ 1.$$ (3.87)

3.7.3.3 $B?eJ?N.$*$h$SG.NO3XJQ?t(B

$B$3$3$G$O=R$Y$J$$(B.

3.7.4 $B798~J}Dx<0(B

$BO"B3$N<0$r1tD>J}8~$K(B $\sigma=0$ $B$+$i(B $\sigma=1$ $B$^$G@QJ,$7(B, $\dot{\sigma}$ $B$K4X$9$k6-3&>r7o$rMQ$$$l$P(B, $B798~J}Dx<0$H$h$P$l$k(B $\pi$ $B$N;~4VJQ2=$K4X$9$k<0$,F@$i$l$k(B.

$$\frac{\partial \pi}{\partial t} = - \int_{0}^{1} \Dvect{v}_{H} \cdot \nabla_{\sigma} \pi d \sigma - \int_{0}^{1} D d \sigma.$$ (3.88)

$B$3$N<0$rMQ$$$l$P(B, $\dot{\sigma}$ $B$N>pJs$,$J$/$F$bCOI=LL5$05$N;~4VJQ2=(B $B$r5a$a$k$3$H$,$G$-$k(B. $B$J$*(B, $B$3$3$G$O8e$N$3$H$r9M$($F(B $\Ddiv{\Dvect{v}_H}$ $B$r(B $D$ $B$HI=8=$7$F$$$k(B. $D$ $B$K$D$$$F$O $B1tD>B.EY(B $\dot{\sigma}$$B$O(B, $BO"B3$N<0$r1tD>J}8~$K(B $\sigma=0$ $B$+$i(B $\sigma=\sigma$ $B$^$G@QJ,$9$k$3$H$G?GCGE*$KF@$i$l$k(B.

$$\dot{\sigma} = - \sigma \frac{\partial \pi}{\partial t} - \int_{0... ... d \sigma - \int_{0}^{\sigma} \Dvect{v}_{H} \cdot \nabla_{\sigma} \pi d \sigma.$$ (3.89)

3.8 $B%b%G%k;YG[J}Dx<0(B

3.8.1 $B12EYJ}Dx<0(B

$B12EY$NDj5A$r:F7G$9$k(B.

$$\zeta \equiv \frac{1}{a \cos \varphi} \DP{v}{\lambda} - \frac{1}{a \cos \varphi} \DP{}{\varphi} ( u \cos \varphi).$$ (3.90)

$B1?F0J}Dx<0$N(B $u$ $B$N<0(B (3.77) $B$K(B $\frac{1}{a \cos \varphi} \DP{}{\varphi} \cos \varphi$ $B$r(B $B:nMQ$7(B, $v$ $B$N<0(B (3.78) $B$K(B $\frac{1}{a \cos \varphi} \DP{}{\lambda}$ $B$r(B $B:nMQ$7(B, $B$3$NN><0$N:9$r$H$C$FJQ7A$9$l$P

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

$$\quad - \frac{1}{a \cos \varphi} \DP{}{\lambda} \left[ \dot{\sigm... ...sigma} + \frac{R^d T_v}{a} \DP{\pi}{\varphi} - {\cal F}_{\varphi} + f u \right]$$

$$\quad - \frac{1}{a \cos \varphi} \DP{}{\varphi} \left[ - \cos \va... ...\DP{\pi}{\lambda} + {\cal F}_{\lambda} \cos \varphi + f v \cos \varphi \right].$$ (3.91)

($B>ZL@(B) (3.77), (3.78)$B$N$=$l$>$l:8JUBh(B1$B9`$r(B, (3.72), (3.73) $B$rMQ$$$FE83+$9$k$H0J2<$N$h$&$K$J$k(B.

$$\DP{u}{t} + \frac{u}{a \cos \varphi} \DP{u}{\lambda} + \frac{v}{a} \DP{u}{\varphi} + \dot{\sigma} \DP{u}{\sigma} - fv - \frac{uv}{a} \tan \varphi = - \frac{1}{a \cos \varphi} \DP{\Phi}{\lambda} - \frac{R^d T_v}{a \cos \varphi} \DP{\pi}{\lambda} + {\cal F}_{\lambda},$$ (3.92)

$$\DP{v}{t} + \frac{u}{a \cos \varphi} \DP{v}{\lambda} + \frac{v}{a} \DP{v}{\varphi} + \dot{\sigma} \DP{v}{\sigma} + fu + \frac{u^2}{a} \tan \varphi = - \frac{1}{a} \DP{\Phi}{\varphi} - \frac{R^d T_v}{a} \DP{\pi}{\varphi} + {\cal F}_{\varphi}.$$ (3.93)

(3.93) $B$K(B $\DP{}{\lambda}$ $B$r:nMQ$7$?<0$+$i(B (3.92) $B$K(B $\DP{}{\varphi} \cos \varphi$ $B$r(B $B:nMQ$7$?<0$r0z$/$3$H$G(B,

$$\quad \DP{}{\lambda} \left( \DP{v}{t} \right) + \DP{}{\lambda} \l... ...v}{\varphi} \right) + \DP{}{\lambda} \left( \dot{\sigma} \DP{v}{\sigma} \right)$$

$$\hspace{5em} + \DP{}{\lambda} \left( fu \right) + \DP{}{\lambda} ... ...a} \DP{\pi}{\varphi} \right) - \DP{}{\lambda} \left( {\cal F}_{\varphi} \right)$$

$$- \DP{}{\varphi} \left( \cos \varphi \DP{u}{t} \right) - \DP{}{\v... ...right) - \DP{}{\varphi} \left( \cos \varphi \dot{\sigma} \DP{u}{\sigma} \right)$$

$$\hspace{5em} + \DP{}{\varphi} \left( \cos \varphi fv \right) + \D... ...} \right) + \DP{}{\varphi} \left( \cos \varphi {\cal F}_{\lambda} \right) = 0 .$$ (3.94)

(3.94)$B$N(B$\Phi$$B$K4X$9$k(B $B9`(B ($B:8JUBh(B7$B9`$HBh(B16$B9`(B) $B$OBG$A>C$7$"$C$F>C$($k(B. $B$=$NB>$N9`$O0J2<$N$h$&$K@0M}$5$l$k(B.

$B;~4VHyJ,$N9`(B ($BBh(B1$B9`$HBh(B10$B9`(B):

$$\DP{}{\lambda} \left( \DP{v}{t} \right) - \DP{}{\varphi} \left( \cos \varphi \DP{u}{t} \right)$$

$$= \DP{}{t} \left\{ \DP{v}{\lambda} \right\} - \DP{}{t} \left\{ \DP{}{\varphi} \left( u \cos \varphi \right) \right\}$$

$$= \DP{}{t} \left\{ \DP{v}{\lambda} - \DP{}{\varphi} \left( u \cos \varphi \right) \right\}$$

$$= a \cos \varphi \DP{\zeta}{t}.$$ (3.95)

$BB.EY$N(B2$B3,?eJ?HyJ,$N9`$=$N(B1 ($BBh(B3, 12, 15$B9`(B):

$$\DP{}{\lambda} \left( \frac{v}{a} \DP{v}{\varphi} \right) - \DP{}... ...P{u}{\varphi} \right) + \DP{}{\varphi} \left( \frac{uv}{a} \sin \varphi \right)$$

$$= \frac{1}{a} \DP{v}{\lambda} \DP{v}{\varphi} + \frac{v}{a} \DP{{}^... ...t( \cos \varphi \DP{u}{\varphi} + u \DP{\cos \varphi}{\varphi} \right) \right\}$$

$$= \DP{}{\varphi} \left( \frac{v}{a} \DP{v}{\lambda} \right) - \DP{}... ...rphi} \left\{ \frac{v}{a} \DP{}{\varphi} \left( u \cos \varphi \right) \right\}$$

$$= \DP{}{\varphi} \left\{ \left( \frac{1}{a \cos \varphi} \DP{v}{\la... ...i} \DP{}{\varphi} \left( u \cos \varphi \right) \right) v \cos \varphi \right\}$$

$$= \DP{}{\varphi} \left( \zeta v \cos \varphi \right) .$$ (3.96)

$BB.EY$N(B2$B3,?eJ?HyJ,$N9`$=$N(B2 ($BBh(B2, 6, 11$B9`(B):

$$\DP{}{\lambda} \left( \frac{u}{a \cos \varphi} \DP{v}{\lambda} \r... ...tan \varphi \right) - \DP{}{\varphi} \left( \frac{u}{a} \DP{u}{\lambda} \right)$$

$$= \DP{}{\lambda} \left( \frac{u}{a \cos \varphi} \DP{v}{\lambda} \r... ...tan \varphi \right) - \DP{}{\lambda} \left( \frac{u}{a} \DP{u}{\varphi} \right)$$

$$= \DP{}{\lambda} \left( \frac{u}{a \cos \varphi} \DP{v}{\lambda} \r... ... \varphi} \left( u \sin \varphi - \cos \varphi \DP{u}{\varphi} \right) \right\}$$

$$= \DP{}{\lambda} \left( \frac{u}{a \cos \varphi} \DP{v}{\lambda} \r... ... - u \DP{\cos \varphi}{\varphi} - \cos \varphi \DP{u}{\varphi} \right) \right\}$$

$$= \DP{}{\lambda} \left( \frac{u}{a \cos \varphi} \DP{v}{\lambda} \r... ... \frac{u}{a \cos \varphi} \DP{}{\varphi} \left( u \cos \varphi \right) \right\}$$

$$= \DP{}{\lambda} \left( \zeta u \right) .$$ (3.97)

$B$3$3$G(B, 2 $B9TL\$NBh(B3$B9`$NJQ7A$K$O(B, (3.96)$B$N(B 2, 3 $B9TL\$GBh(B1$B9`$KBP$7$FMQ$$$?JQ7A$rMQ$$$?(B.

(3.94)$B$r(B (3.95), (3.96), (3.97) $B$rMQ$$$F@0M}$7(B, $BN>JU$K(B $\frac{1}{a \cos \varphi}$ $B$r3]$1$k$3$H$G(B, (3.91)$B$,F@$i$l$k(B.

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

3.8.2 $BH/;6J}Dx<0(B

$BH/;6$NDj5A$r:F7G$9$k(B.

$$D \equiv \frac{1}{a \cos \varphi} \DP{u}{\lambda} + \frac{1}{a \cos \varphi} \DP{}{\varphi} ( v \cos \varphi).$$ (3.98)

$B1?F0J}Dx<0$N(B $u$ $B$N<0(B (3.77) $B$K(B $\frac{1}{a \cos \varphi} \DP{}{\lambda}$ $B$r:nMQ$7(B, $v$ $B$N<0(B (3.78) $B$K(B $\frac{1}{a \cos \varphi} \DP{}{\varphi} \cos \varphi$ $B$r:nMQ$7(B, $BN><0$NOB$r$H$C$FJQ7A$9$k$H

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

$$\quad - \frac{1}{a \cos \varphi} \DP{}{\lambda} \left[ \dot{\sigm... ...c{R^d T_v}{a \cos \varphi} \DP{\pi}{\lambda} - {\cal F}_{\lambda} - f v \right]$$

$$\quad - \frac{1}{a \cos \varphi} \DP{}{\varphi} \left[ \cos \varp... ... \DP{\pi}{\varphi} - {\cal F}_{\varphi} \cos \varphi + f u \cos \varphi \right]$$

$$\quad - \nabla^2_{\sigma} ( \Phi + KE).$$ (3.99)

$B$3$3$G(B,

$$\nabla^2_{\sigma} = \frac{1}{a^2 \cos^2 \varphi} \DP[2]{}{\lambda} + \frac{1}{a^2 \cos \varphi} \DP{}{\varphi} \left( \cos \varphi \DP{}{\varphi} \right),$$ (3.100)

$$KE = \frac{u^2 + v^2}{2}.$$ (3.101)

($B>ZL@(B) (3.92) $B$K(B $\DP{}{\lambda}$ $B$r:nMQ$7$?<0$H(B (3.93) $B$K(B $\DP{}{\varphi} \cos \varphi$ $B$r:nMQ$7$?<0$H$NOB$r$H$k$3$H$G(B,

$$\quad \DP{}{\lambda} \left( \DP{u}{t} \right) + \DP{}{\lambda} \l... ...u}{\varphi} \right) + \DP{}{\lambda} \left( \dot{\sigma} \DP{u}{\sigma} \right)$$

$$\hspace{5em} - \DP{}{\lambda} \left( fv \right) - \DP{}{\lambda} ... ...i} \DP{\pi}{\lambda} \right) - \DP{}{\lambda} \left( {\cal F}_{\lambda} \right)$$

$$+ \DP{}{\varphi} \left( \cos \varphi \DP{v}{t} \right) + \DP{}{\v... ...right) + \DP{}{\varphi} \left( \cos \varphi \dot{\sigma} \DP{v}{\sigma} \right)$$

$$\hspace{5em} + \DP{}{\varphi} \left( \cos \varphi fu \right) + \D... ...ht) + \DP{}{\varphi} \left( \cos \varphi \frac{1}{a} \DP{\Phi}{\varphi} \right)$$

$$\hspace{10em} + \DP{}{\varphi} \left( \cos \varphi \frac{R^d T_v}... ...} \right) - \DP{}{\varphi} \left( \cos \varphi {\cal F}_{\varphi} \right) = 0 .$$ (3.102)

$B$3$N<0$O0J2<$N$h$&$K@0M}$5$l$k(B.

$B;~4VHyJ,$N9`(B ($BBh(B1$B9`$HBh(B10$B9`(B):

$$\DP{}{\lambda} \left( \DP{u}{t} \right) + \DP{}{\varphi} \left( \cos \varphi \DP{v}{t} \right)$$

$$= \DP{}{t} \left\{ \DP{u}{\lambda} \right\} + \DP{}{t} \left\{ \DP{}{\varphi} \left( v \cos \varphi \right) \right\}$$

$$= \DP{}{t} \left\{ \DP{u}{\lambda} + \DP{}{\varphi} \left( v \cos \varphi \right) \right\}$$

$$= a \cos \varphi \DP{D}{t}.$$ (3.103)

$\Phi$$B$K4X$9$k9`(B ($BBh(B7$B9`$HBh(B16$B9`(B):

$$\DP{}{\lambda} \left( \frac{1}{a \cos \varphi} \DP{\Phi}{\lambda}... ...ht) + \DP{}{\varphi} \left( \cos \varphi \frac{1}{a} \DP{\Phi}{\varphi} \right)$$

$$= a \cos \varphi \left\{ \frac{1}{a^2 \cos^2 \varphi} \DP[2]{}{\lam... ...varphi} \DP{}{\varphi} \left( \cos \varphi \DP{}{\varphi} \right) \right\} \Phi$$

$$= a \cos \varphi \ \nabla^2_{\sigma} \Phi .$$ (3.104)

$BB.EY$N(B2$B3,?eJ?HyJ,$N9`$=$N(B1 ($BBh(B2, 12$B9`(B):

$$\DP{}{\lambda} \left( \frac{u}{a \cos \varphi} \DP{u}{\lambda} \right) + \DP{}{\varphi} \left( \cos \varphi \frac{v}{a} \DP{v}{\varphi} \right)$$

$$= \frac{1}{2 a \cos \varphi} \DP[2]{u^2}{\lambda} + \frac{1}{2a} \DP{}{\varphi} \left( \cos \varphi \DP{v^2}{\varphi} \right)$$

$$= \left\{ \frac{1}{a \cos \varphi} \DP[2]{}{\lambda} + \frac{1}{a} ... ...right) - \DP{}{\varphi} \left( \cos \varphi \frac{u}{a} \DP{u}{\varphi} \right)$$

$$= a \cos \varphi \ \nabla^2_{\sigma} KE - \DP{}{\lambda} \left( \fr... ...ght) - \DP{}{\varphi} \left( \cos \varphi \frac{u}{a} \DP{u}{\varphi} \right) .$$ (3.105)

$BBh(B2$B9`$HBh(B3$B9`$K$D$$$F$O(B, $B$3$l0J9_$N9`$N@0M}$N:]$K:FEP>l$9$k(B.

$BB.EY$N(B2$B3,?eJ?HyJ,$N9`$=$N(B2 ($BBh(B3, 6$B9`(B, (3.105)$B$NBh(B2$B9`(B):

$$\DP{}{\lambda} \left( \frac{v}{a} \DP{u}{\varphi} \right) - \DP{}... ...right) - \DP{}{\lambda} \left( \frac{v}{a \cos \varphi} \DP{v}{\lambda} \right)$$

$$= \DP{}{\lambda} \left\{ \frac{v}{a \cos \varphi} \DP{}{\varphi} \l... ...ight\} - \DP{}{\lambda} \left( \frac{v}{a \cos \varphi} \DP{v}{\lambda} \right)$$

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

$$= - \DP{}{\lambda} (\zeta v) .$$ (3.106)

$BB.EY$N(B2$B3,?eJ?HyJ,$N9`$=$N(B3 ($BBh(B11, 15$B9`(B, (3.105)$B$NBh(B3$B9`(B):

$$\DP{}{\varphi} \left( \frac{u}{a} \DP{v}{\lambda} \right) + \DP{}... ...right) - \DP{}{\varphi} \left( \cos \varphi \frac{u}{a} \DP{u}{\varphi} \right)$$

$$= \DP{}{\varphi} \left( \frac{u}{a} \DP{v}{\lambda} \right) + \DP{}... ...rac{u}{a} \left( u \sin \varphi - \cos \varphi \DP{u}{\varphi} \right) \right\}$$

$$= \DP{}{\varphi} \left( \frac{u}{a} \DP{v}{\lambda} \right) - \DP{}... ...t( u \DP{\cos \varphi}{\varphi} + \cos \varphi \DP{u}{\varphi} \right) \right\}$$

$$= \DP{}{\varphi} \left( \frac{u}{a} \DP{v}{\lambda} \right) - \DP{}... ...rphi} \left\{ \frac{u}{a} \DP{}{\varphi} \left( u \cos \varphi \right) \right\}$$

$$= \DP{}{\varphi} \left\{ u \cos \varphi \left( \frac{1}{a \cos \var... ...}{a \cos \varphi} \DP{}{\varphi} \left( u \cos \varphi \right) \right) \right\}$$

$$= \DP{}{\varphi} \left( \zeta u \cos \varphi \right) .$$ (3.107)

(3.102)$B$r(B (3.103), (3.104), (3.105), (3.106), (3.107) $B$rMQ$$$F@0M}$7(B, $BN>JU$K(B $\frac{1}{a \cos \varphi}$ $B$r3]$1$k$3$H$G(B, (3.99)$B$,F@$i$l$k(B.

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

3.8.3 $BG.NO3X$N<0(B

(3.85)$B$h$j(B

$$\begin{align*}\begin{split}\DP{T}{t} \ &= \ - \Dinv{a \cos \varphi} \DP{(u T)}{\... ...{ \dot{\sigma} }{ \sigma } \right) + \frac{Q^{*}}{C_p} . \end{split}\end{align*}$$ (3.108)

$B$3$3$G(B,

$$\kappa = \frac{R^d}{C_p^d}$$ (3.109)

$B$G$"$k(B.

3.8.4 $B29EY$N4pK\>l$H$:$l$NJ,N%(B

$B2>29EY(B $T_v$ $B$r $B$N$_$K0MB8$9$k>l(B $\overline{T}_v(\sigma)$ $B$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

$$- \frac{1}{a \cos \varphi} \DP{}{\lambda} \left[ \frac{R^d T_v}{a... ...\cos \varphi} \DP{}{\varphi} \left[ \frac{R^d T_v}{a} \DP{\pi}{\lambda} \right]$$

$$= - \frac{1}{a \cos \varphi} \DP{}{\lambda} \left[ \frac{R^d \overl... ...phi} \DP{}{\lambda} \left[ \frac{R^d T_v^{\prime}}{a} \DP{\pi}{\varphi} \right]$$

$$+ \frac{1}{a \cos \varphi} \DP{}{\varphi} \left[ \frac{R^d \overl... ...phi} \DP{}{\varphi} \left[ \frac{R^d T_v^{\prime}}{a} \DP{\pi}{\lambda} \right]$$

$$= - \frac{1}{a \cos \varphi} \left\{ \frac{R^d \overline{T}_v}{a} \... ...}{\varphi} \left[ \frac{R^d T_v^{\prime}}{a} \DP{\pi}{\lambda} \right] \right\}$$

$$= - \frac{1}{a \cos \varphi} \left\{ \DP{}{\lambda} \left[ \frac{R^... ...\varphi} \left[ \frac{R^d T_v^{\prime}}{a} \DP{\pi}{\lambda} \right] \right\} .$$ (3.110)

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

$$- \frac{1}{a \cos \varphi} \DP{}{\lambda} \left[ \frac{R^d T_v}{a... ... \DP{}{\varphi} \left[ \frac{R^d T_v}{a} \cos \varphi \DP{\pi}{\varphi} \right]$$

$$= - \frac{1}{a \cos \varphi} \DP{}{\lambda} \left[ \frac{R^d \overl... ...ambda} \left[ \frac{R^d T_v^{\prime}}{a \cos \varphi} \DP{\pi}{\lambda} \right]$$

$$- \frac{1}{a \cos \varphi} \DP{}{\varphi} \left[ \frac{R^d \overl... ...arphi} \left[ \frac{R^d T_v^{\prime}}{a} \cos \varphi \DP{\pi}{\varphi} \right]$$

$$= - \frac{1}{a^2 \cos^2 \varphi} \DP[2]{}{\lambda} \left( R^d \over... ...ambda} \left[ \frac{R^d T_v^{\prime}}{a \cos \varphi} \DP{\pi}{\lambda} \right]$$

$$- \frac{1}{a^2 \cos \varphi} \DP{}{\varphi} \left[ \cos \varphi \... ...arphi} \left[ \frac{R^d T_v^{\prime}}{a} \cos \varphi \DP{\pi}{\varphi} \right]$$

$$= - \nabla_{\sigma}^2 \left( R^d \overline{T}_v \pi \right) - \frac... ...phi} \left[ \frac{R^d T_v^{\prime}}{a} \cos \varphi \DP{\pi}{\varphi} \right] .$$ (3.111)

$B$3$3$G(B

$$\nabla_{\sigma}^{2} \equiv \frac{1}{a^{2} \cos^2 \varphi} \DP[2]{}{\lambda} + \frac{1}{a^{2} \cos \varphi} \DP{}{\varphi} \left( \cos \varphi \DP{}{\varphi} \right)$$ (3.112)

$B$rMQ$$$?(B.

$BG.NO3X$N<0$G$O(B, $B29EY(B $T$ $B$r(B $\sigma$ $B$N$_$K0MB8$9$k>l(B $\overline{T}(\sigma)$ $B$H(B, $B$=$3$+$i$N$:[email protected],(B $T'$ $B$K$o$1$F5-=R$9$k(B. $B$9$J$o$A(B, $B1&JUBh(B1-3$B9`$O

$$\quad - \Dinv{a \cos \varphi} \DP{(u T)}{\lambda} - \Dinv{a \cos \varphi} \DP{(v T \cos \varphi)}{\varphi} + T D$$

$$= - \Dinv{a \cos \varphi} \left\{ \DP{(u \overline{T})}{\lambda} ... ...{(v T^{\prime} \cos \varphi)}{\varphi} \right\} + \overline{T} D + T^{\prime} D$$

$$= - \Dinv{a \cos \varphi} \left\{ \overline{T} \DP{u}{\lambda} + ... ...(v \cos \varphi)}{\varphi} + \DP{(v T^{\prime} \cos \varphi)}{\varphi} \right\}$$

$$\qquad \qquad + \overline{T} \left[ \frac{1}{a \cos \varphi} \DP{... ...frac{1}{a \cos \varphi} \DP{}{\varphi} ( v \cos \varphi) \right] + T^{\prime} D$$

$$= - \Dinv{a \cos \varphi} \DP{(u T^{\prime})}{\lambda} - \Dinv{a \cos \varphi} \DP{(v T^{\prime} \cos \varphi)}{\varphi} + T^{\prime} D .$$ (3.113)

3.8.5 $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 $U \equiv u \cos \varphi$, $V \equiv \cos \varphi$ $B$KJQ49$9$k(B. $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.

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

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

$$= \frac{1}{a ( 1- \mu^2 )} \DP{V}{\lambda} - \frac{1}{a} \DP{U}{\mu},$$ (3.114)

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

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

$$= \frac{1}{a ( 1-\mu^2)} \DP{U}{\lambda} + \frac{1}{a} \DP{V}{\mu}.$$ (3.115)

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

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

$$\quad + \frac{1}{a \cos \varphi} \left\{ \DP{}{\varphi} (v \cos \varphi \bullet) - \bullet \DP{}{\varphi} ( v \cos \varphi ) \right\}$$

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

$$= \frac{1}{a (1-\mu^2)} \DP{}{\lambda} (U\bullet) +\frac{1}{a} \DP{}{\mu} (V\bullet) -\bullet D.$$ (3.116)

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

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

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

$$\equiv \Dvect{v}_H \cdot \Dgrad_{\sigma} \bullet.$$ (3.117)

3.8.6 $B;YG[J}Dx<0$N:G=*7A(B

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

$$\DP{\pi}{t} + \Dvect{v}_H \cdot \Dgrad_{\sigma} \pi = - D - \DP{\dot{\sigma}}{\sigma}.$$ (3.118)

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

$$\DP{\Phi}{\sigma}=-\frac{R^d T_v}{\sigma}.$$ (3.119)

$B1?F0J}Dx<0(B

$$\DP{\zeta}{t} = \ \Dinv{a} \left( \Dinv{1-\mu^2} \DP{V_A}{\lambda} - \DP{U_A}{\mu} \right),$$ (3.120)

$$\DP{D}{t} = \ \Dinv{a} \left( \Dinv{1-\mu^2} \DP{U_A}{\lambda} + \DP{V_A}{\mu} \right) - \nabla^{2}_{\sigma} ( \Phi + R \overline{T} \pi + ) .$$ (3.121)

$B$3$3$G(B,

$$U_A\ (\varphi, \lambda, \sigma) \equiv ( \zeta + f ) V - \dot{\sigma} \DP{U}{\sigma} - \frac{R T_v^{\prime}}{a} \DP{\pi}{\lambda} + {\cal F}_{\lambda} \cos \varphi,$$ (3.122)

$$V_A\ (\varphi, \lambda, \sigma) \equiv - ( \zeta + f ) U - \dot{\sigma} \DP{V}{\sigma} - \frac{R T_v^{\prime}}{a} (1-\mu^2) \DP{\pi}{\mu} + {\cal F}_{\varphi} \cos \varphi .$$ (3.123)

$BG.NO3X$N<0(B

$$\begin{align*}\begin{split}\DP{T}{t} \ &= \ - \Dinv{a} \left( \Dinv{1-\mu^2} \DP... ...{ \dot{\sigma} }{ \sigma } \right) + \frac{Q^{*}}{C_p} . \end{split}\end{align*}$$ (3.124)

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

$$\DP{q}{t} \ = \ - \Dinv{a \cos \varphi} \left( \Dinv{1-\mu^2} \DP{U q}{\lambda} + \DP{V q}{\mu} \right) + q D - \dot{\sigma} \DP{q}{\sigma} + S_{q} .$$ (3.125)

(3.16)$B$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, $C_p^d$ $B$r$=$l$>$l(B $R$, $C_p$ $B$N$h$&$K$"$i$?$a$FCV$-$J$*$;$P(B, dcpam5$B$NNO3X2aDx$N;YG[J}Dx<07O$rF@$k(B.

3.9 $B;29MJ88%(B

Haltiner, G.J., Williams, R.T., 1980: Numerical Prediction and Dynamic Meteorology (2nd ed.). John Wiley & Sons, 477pp.

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

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

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

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

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

$B$^$?(B, $B:8JUBh(B5$B9`$O $\DP{\Phi^*}{t}=0$ $B$G$"$k$H$7$F$$$k(B.

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

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

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$$\rho \varepsilon = \rho^d \varepsilon^d + \rho^v \varepsilon^v$$

$$= \rho^d C_v^d T + \rho^v C_v^v T$$

$$= \rho \left( \frac{ \rho^d C_v^d + \rho^v C_v^v}{\rho} \right) T$$

$$= \rho \left( \frac{ C_v^d + r C_v^v }{ 1+r } \right) T,$$

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$$C_v \equiv \frac{ C_v^d + r C_v^v }{ 1+r },$$

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$$R \equiv \frac{R^d + r R^v}{1+r} \left( = R^d \frac{ 1 + r/\epsilon}{1+r} \right) ,$$

$B$G$"$k$+$i(B,

$$C_p = C_v + R$$

$$= \frac{ C_v^d + r C_v^v + R^d + r R^v }{1+r}$$

$$= \frac{ C_p^d + r C_p^v}{1+r}$$

$$= C_p^d \frac{ 1+rC_p^v/C_p^d}{1+r}$$

$$\sim C_p^d \frac{ 1+ 8 r/7 \epsilon}{1+r},$$

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