: 3. $BNO3X2aDx(B : dcpam5 $B;YG[J}Dx<07O$H$=$NN%;62=(B : 1. $B$3$NJ8=q$K$D$$$F(B

2. $B:BI87O!&JQ498x<0(B

2.1 $B$O$8$a$K(B

$B$3$3$G$O(B, $B:BI87O$*$h$S?eJ?3J;RE@(B, $B1tD>%l%Y%k$N 2.2 $B:BI87O(B

$B:BI87O$O(B, $B?eJ?J}8~$K$O0^EY(B $\varphi$ , $B7PEY(B $\lambda$ $B$r(B, $B1tD>J}8~$K$O(B $\sigma \equiv \frac{p}{p_s}$ $B$r$H$k(B. $B$3$3$G(B $B$O5$05(B, $B$OCOI=LL5$05$G$"$k(B.

$B:BI8$N\:Y$OJL;f!X(B $B;YG[J}Dx<07O$NF3=P$K4X$9$k;29M;qNA(B$B!Y(B $B$N!X:BI87O$NH$;$h(B.

2.3 $B?eJ?3J;RE@(B

$B?eJ?J}8~$N3J;RE@$N0LCV$O(B, Gauss $B0^EY(B ($B3J;RE@?t(B $B8D(B ^2.1), $BEy4V3V$N7PEY(B ($BF1(B $B8D(B) $B$G$"$k(B.

Gauss $B0^EY(B
Gauss $B0^EY$r(B $B $P_J(\sin \varphi)$ $B$NNmE@(B $\varphi_j (j=1,2,3,\cdots,J)$ $B$H$7$FDj5A$9$k(B. $B=gHV$H$7$F$O(B, ${\displaystyle \frac{\pi}{2} > \varphi_1 > \varphi_2 > \cdots > \varphi_{J} > -\frac{\pi}{2} }$ $B$H$9$k(B ^2.2. $B$J$*0J8e(B, $\sin \varphi = \mu$ $B$H=q$/$3$H$,$"$k(B.
$B7PEYJ}8~$N3J;RE@(B
$B7PEYJ}8~$N3J;RE@$N0LCV$r(B

$\displaystyle \lambda_i = \frac{2 \pi (i-1)}{I} \ \ (i=1,2,\cdots,I)$ (2.2)

$B$H$H$k(B.

2.4 $B1tD>%l%Y%k(B

Arakawa and Suarez (1983) $B$N%9%-!<%`$rMQ$$$k(B. $B$H$jJ}$O0J2<$N$H$*$j$G$"$k(B ^2.3. $B2<$NAX$+$i>e$X$HAX$NHV9f$r$D$1$k(B. $B@0?t%l%Y%k$HH>@0?t%l%Y%k$rDj5A$9$k(B ^2.4. $BH>@0?t%l%Y%k$G$N(B $\sigma$ $B$NCM(B $\sigma_{k-1/2} \ (k=1,2,\cdots,K)$ $B$rDj5A$9$k(B. $B$?$@$7(B, $B%l%Y%k(B $\frac{1}{2}$ $B$O2), $B%l%Y%k(B $K+\frac{1}{2}$ $B$O>eC<(B( $\sigma=0$ )$B$H$9$k(B. $B@0?t%l%Y%k$N(B $\sigma$ $B$NCM(B $\sigma_k \ (k=1,2,\ldots K)$ $B$O

$\displaystyle \sigma_k = \left\{ \frac{1}{1+\kappa} \left( \frac{ \sigma^{\kapp... ... +1}_{k+1/2} } { \sigma_{k-1/2} - \sigma_{k+1/2} } \right) \right\}^{1/\kappa}.$

(2.3)

$B$?$@$7(B, $\kappa=\frac{R}{C_p}$ $B$G$"$k(B. $B$3$3$G(B,

$B$O4%Ag6u5$$N5$BNDj?t(B,

$B$O4%Ag6u5$$NDj05HfG.$G$"$k(B ^2.5. $B$^$?(B, $B%l%Y%k2C=E(B $\Delta \sigma$ $B$O0J2<$N$h$&$KDj5A$5$l$k(B.

$\begin{align*}\begin{split}\Delta \sigma_k &\equiv \sigma_{k-1/2} - \sigma_{k+1/... ...K+1/2} &\equiv \sigma_{K} - \sigma_{K+1/2} = \sigma_{K}. \end{split}\end{align*}$

(2.4)

$\begin{picture}(300,150)(50,10) \put(50,20){\line(1,0){220}} \put(50,40){\line... ...){\shortstack{$\sigma=1$}} \put(280,136){\shortstack{$\sigma=0$}} \end{picture}$

2.5 $B?eJ?%9%Z%/%H%k(B

$B$3$3$G$O(B, $BNO3X2aDx$N;~4V@QJ,$G$N7W;;$K$*$$$FMQ$$$k%9%Z%/%H%k$rF3F~$7(B, $B3J;RE@$G$NCM$H%9%Z%/%H%k$N78?t$H$N$d$j

2.5.1 $B?eJ?%9%Z%/%H%k$N4pDl$NF3F~(B

$B3J;RE@>e$NE@$GDj5A$5$l$?J*M}NL$O(B, $B3J;RE@>e$G$N$_CM$r;}$D(B ($B0J2<$3$N$3$H$r(B, $B!VN%;62=$7$?!W$H8F$V(B) $B5eLLD4OBH!?t$NOB$N7A$GI=8=$5$l$k(B. $B$^$?(B, $B3F3J;RE@$K$*$1$kJ*M}NL$N?eJ?HyJ,$rI>2A$9$k$?$a$K(B, $(\lambda, \varphi)$ $BLL$GDj5A$5$l$?(B ($B0J2<(B, $B!VO"B37O$N!W$H8F$V(B) $B5eLLD4OBH!?t7O$GFbA^$7$FF@$i$l$k4X?t$rMQ$$$k(B. $B$3$3$G$O$=$N5eLLD4OBH!?t$rF3F~$9$k(B. $B$J$*(B, $B4JC1$N$?$a$K(B, $BO"B37O$N5eLLD4OBH!?t$N$_$rM[$K5-$9(B. $BN%;67O$N5eLLD4OBH!?t$O(B $BO"B37O$N5eLLD4OBH!?t$K3J;RE@$N:BI8$rBeF~$7$?$b$N$+$i9=@.$5$l$k(B.

$(\lambda, \varphi)$ $BLL$K$*$$$F(B, $B5eLLD4OBH!?t(B $Y_n^m(\lambda,\varphi)$ $B$O

$\displaystyle Y_n^m(\lambda,\varphi) \equiv P_n^m(\sin \varphi) \exp(im \lambda),$

(2.5)

$B$?$@$7(B,

$B$O(B $\ 0 \le \vert m\vert \le n$ $B$rK~$?$9@0?t$G$"$j(B, $P_n^m(\sin \varphi)$ $B$O(B 2$B$G5,3J2=$5$l$?(BLegendre$BH!?t!&GfH!?t(B

	$\displaystyle P_n^m(\mu)\equiv \sqrt{\frac{(2n+1)(n-\vert m\vert)!}{(n+\vert m\... ...u^2)^{\frac{\vert m\vert}{2}} }{2^n n!} \DD[n+\vert m\vert]{}{\mu} (\mu^2-1)^n,$	(2.6)
	$\displaystyle \int_{-1}^1 P_n^m(\mu) P_{n'}^m(\mu) d \mu = 2 \delta_{nn'}$	(2.7)

$B$G$"$k(B. $B$J$*(B,

$B$r(B

$B$H$b=q$/(B. $B$^$?(B $\sin \varphi = \mu$ $B$G$"$k$3$H$r:F7G$7$F$*$/(B.

2.5.2 $BGH?t@ZCG(B

$BGH?t@ZCG$O;03Q7A@ZCG(B (T) $B$^$?$OJ?9T;MJU7A@ZCG(B (R) $B$H$9$k(B. , $B$O;03Q7A@ZCG(B, $BJ?9T;MJU7A@ZCG$N$H$-$K$D$$$F(B $B$=$l$>$l0J2<$N$H$*$j$G$"$k(B. $B$?$@$7(B, $B@ZCGGH?t$r(B $N_{tr}$ $B$H$9$k(B.

$B;03Q7A@ZCG$N>l9g(B
$M= N_{tr}, \ N =N_{tr}$ , $I \ge 3N_{tr}+1$ , $B$+$D(B $J \ge \frac{3N_{tr}+1}{2}$ .
$B<+M3EY$O(B, $(N_{tr}+1)^2$ $B$G$"$k(B.
$BJ?9T;MJU7A@ZCG$N>l9g(B
$M= N_{tr}, \ N(m) =N_{tr}+\vert m\vert$ , $I \ge 3N_{tr}+1$ , $B$+$D(B $J \ge 3N_{tr}+1$ .
$B<+M3EY$O(B, $(2N_{tr}+1) (N_{tr}+1)$ $B$G$"$k(B.

$B$h$/MQ$$$i$l$kCM$NNc$H$7$F$O(B, T42 $B$N>l9g(B $I=128,\ J=64$ , R21 $B$N>l9g(B $I=64,\ J=64$ $B$,$"$k(B.

$B5eLLD4OBH!?t$HGH?t@ZCG$K4X$9$k>\:Y$O(B, $BBh(BA.1$B@a$*$h$SBh(BA.8$B@a(B $B$r;2>H$;$h(B.

2.5.3 $BN%;62=$7$?%9%Z%/%H%k$N4pDl$ND>8r@-(B

$BN%;62=$7$?(BLegendre$BH!?t$H;03Q4X?t$O(B $B8r>r7o$rK~$?$9(B ^2.6.

$\displaystyle \sum_{j=1}^{J} P_n^m (\mu_j) P_{n'}^m (\mu_j) w_j$	$\displaystyle = \delta_{nn'},$	(2.8)
$\displaystyle \sum_{i=1}^{I} \exp(im \lambda_i) \exp(-im' \lambda_i)$	$\displaystyle = I \delta_{mm'}.$	(2.9)

$B$3$3$G(B

$B$O(B Gauss $B2Y=E$G(B, ${\displaystyle w_j \equiv \frac{(2J-1)(1-\sin^2 \varphi_j)} {\left\{J P_{J-1}(\sin \varphi_j)\right\}^2 } }$ $B$G$"$k(B.

2.5.4 $B3J;RE@CM$H%9%Z%/%H%k$N78?t$H$NJQ49K!(B

$BJ*M}NL(B $B$N(B $B3J;RE@(B $(\lambda_i,\varphi_j)$ ($B$?$@$7(B $i=1,2,\cdots,I. \quad j=1,2,\cdots,J$ ) $B$G$NCM(B $A_{ij}=A(\lambda_i,\varphi_j)$ $B$H(B $B%9%Z%/%H%k6u4V$G$N(B ($B$?$@$7(B $m=-M,\cdots,M. \quad n=\vert m\vert,\cdots,N(m)$ ) $B$N78?t(B $\tilde{A}_n^m$ $B$H$O$&(B ^2.7.

$\displaystyle A_{ij}$	$\displaystyle \equiv \sum_{m=-M}^{M} \sum_{n=\vert m\vert}^{N} \tilde{A}_n^m Y_n^m (\lambda_i,\varphi_j),$	(2.10)
$\displaystyle \tilde{A}_n^m$	$\displaystyle = \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J} A_{ij} Y_n^{m*} (\lambda_i, \varphi_j) w_j .$	(2.11)

$B$, ${\displaystyle \left\{\tilde{A}^m_n \exp(im\lambda) \right\}^* = \tilde{A}^{-m}_n \exp(-im\lambda) }$ $B$J$N$G(B,

$B$K$D$$$F$OIi$G$J$$@0?t$NHO0O$G(B $BOB$r$H$k$3$H$,$G$-$k(B ^2.8. $B$3$3$G(B, ``

'' $B$OJ#AG6&Lr$rI=$9(B. $B$?$@$7(B,

$B$NDj5A$r0J2<$N$h$&$K=$@5$7$F$$$k$3$H$KCm0U$;$h(B.

$\displaystyle A_{ij}$	$\displaystyle = \sum_{m=0}^{M} \sum_{n=m}^{N} \Re \tilde{A}_n^m Y_n^m(\lambda_i, \varphi_j),$	(2.12)
$\displaystyle \tilde{A}_n^m$	$\displaystyle = \left\{ \begin{array}{ll} {\displaystyle \frac{1}{I} \sum_{i=1}... ...varphi_j) w_j , & \ \ \ 1 \le m \le M, \quad m \le n \le N. \end{array} \right.$	(2.13)

2.5.5 $BFbA^8x<0(B

$(\lambda, \varphi)$ $B6u4V$GDj5A$5$l$kJ*M}NL(B $A(\lambda,\varphi)$ $B$r(B $B3J;RE@CM(B $A_{ij}$ $B$r$b$H$KFbA^$9$k>l9g$K$O(B, $BJQ498x<0$rMQ$$$F(B $A_{ij}$ $B$+$i(B $\tilde{A}_n^m$ $B$r5a$a$?>e$G(B,

$\displaystyle A(\lambda,\varphi)$

$\displaystyle \equiv \sum_{m=-M}^{M} \sum_{n=\vert m\vert}^{N} \tilde{A}_n^m Y_n^m (\lambda, \varphi)$

(2.14)

$B$H$7$FF@$k(B.

2.5.6 $B6u4VHyJ,$NI>2A(B

$B3F3J;RE@$K$*$1$k6u4VHyJ,CM$NI>2A$O(B, $BFbA^8x<0$rMQ$$$FF@$?O"B34X?t$N6u4VHyJ,$N3J;RE@CM$GI>2A$9$k(B.

$\lambda$ $BHyJ,(B

$\displaystyle \left( \DP{f}{\lambda} \right)_{ij}$ $\displaystyle \equiv \sum_{m=-M}^M \sum_{n=\vert m\vert}^N im \tilde{f}_n^m Y_n^m (\lambda_i,\varphi_j) ,$ (2.15)

$\displaystyle \widetilde{ \left( \DP{f}{\lambda} \right) }_n^m$ $\displaystyle = \frac{1}{I} \sum_{i=1}^I \sum_{j=1}^J im f_{ij} Y_n^{m*} (\lambda_i, \varphi_j) w_j .$ (2.16)
$\varphi$ $BHyJ,(B

$\displaystyle \left(\DP{f}{\varphi}\right)_{ij}$ $\displaystyle \equiv \sum_{m=-M}^M \sum_{n=\vert m\vert}^N \tilde{f}_n^m \left. \DD{P_n^m}{\varphi} \right\vert _j \exp(im \lambda_i) ,$ (2.17)

$\displaystyle \widetilde{\left(\DP{f}{\varphi}\right)}_n^m$ $\displaystyle = - \frac{1}{I} \sum_{i=1}^I \sum_{j=1}^J f_{ij} \left. \DD{P_n^m}{\varphi}\right\vert _j \exp(-im \lambda_i) w_j .$ (2.18)

2.6 $B;29MJ88%(B

: Arakawa, A., Suarez, M. J., 1983: Vertical differencing of the primitive equations in sigma coordinates. Mon. Wea. Rev., 111, 34-35.
: $B5$>]D#M=JsIt(B, 1982 : $B%9%Z%/%H%kK!$K$h$k?tCMM=Js!J$=$N86M}$H. $B5$>]D#(B, 111pp.
: Haltiner, G.J., Williams, R.T., 1980: Numerical Prediction and Dynamic Meteorology (2nd ed.). John Wiley & Sons, 477pp.
: $B?98}(B, $B1'ED@n(B, $B0l>>JT(B ,1956 : $B4dGH?t3X8x<0(BI . $B4dGH=qE9(B, 318pp.
: $B?98}(B, $B1'ED@n(B, $B0l>>JT(B ,1960 : $B4dGH?t3X8x<0(BIII . $B4dGH=qE9(B, 310pp.
: $B0l>>(B $B?.(B, 1982 : $B?tCM2r@O(B. $BD+AR=qE9(B, 163pp.
: $B?9(B $B@5Ip(B, 1984 : $B?tCM2r@OK!(B. $BD+AR=qE9(B, 202pp.
: $B;{Bt420l(B, 1983 : $B<+A32J3X. $B4dGH=qE9(B, 711pp.

... $B8D(B ^2.1

$B0J2<(B, $B$O6v?t$H$9$k(B. dcpam5$B$G$O(B, (Gauss $B0^EY$H$7$F$H$k>l9g$K$O(B) $B$O6v?t$G$J$1$l$P$J$i$J$$(B.

... $B$H$9$k(B ^2.2

$B $B$O(B

$\displaystyle \left[ \DD{}{\mu} \left\{ (1-\mu^2) \DD{}{\mu} \right\} + J(J+1) \right] P_J(\mu) = 0$ (2.1)

$B$rK~$?$9(B $B $B$NNmE@$OA4$F(B $-1 < \mu < 1$ $B$K$"$k(B. $B$J$*(B, Gauss $B0^EY$O6a;wE*$K$O(B ${\displaystyle \sin^{-1} \left( \cos \frac{j-1/2}{J}\pi \right) }$ $B$GM?$($i$l$k(B.

... $B$H$jJ}$O0J2<$N$H$*$j$G$"$k(B ^2.3

$B$3$N%9%-!<%`$O
$BA4NN0h@QJ,$7$?
$BA4NN0h@QJ,$7$?%(%M%k%.!<$rJ]B8(B

$BA4NN0h@QJ,$N3Q1?F0NL$rJ]B8(B

$BA4
$B@E?e05$N<0$,(Blocal$B$K$-$^$k(B. ($B2eAX$N29EY$K0MB8$7$J$$(B)

$B?eJ?J}8~$K0lDj$N(B, $B$"$kFCDj$N29EYJ,I[$K$D$$$F(B, $B@E?e05$N<0$,@53N$K$J$j(B, $B5$0579EYNO$,(B0$B$K$J$k(B.

$BEy290LBg5$$O$$$D$^$G$bEy290L$KN1$^$k(B

... $B@0?t%l%Y%k$HH>@0?t%l%Y%k$rDj5A$9$k(B ^2.4

$BJ*M}NL$K$h$j(B, $B@0?t%l%Y%k$GDj5A$5$l$k$b$N$H(B, $BH>@0?t%l%Y%k$GDj5A$5$l$k$b$N$,$"$k(B.

... $B$O4%Ag6u5$$NDj05HfG.$G$"$k(B ^2.5

$B$$$:$l$bDj?t$H$7$F$$$k(B.

... $B8r>r7o$rK~$?$9(B ^2.6

$B>\$7$/$OBh(BA.5$B@a(B $B$r;2>H$;$h(B.

... $B$H$O$&(B ^2.7

$B@5JQ49(B, $B5UJQ49;~$N78?t$O(B consistent $B$KM?$($F$5$($$$l$PLdBj$,$J$$(B.

... $BOB$r$H$k$3$H$,$G$-$k(B ^2.8

$B$5$i$K(B, $B $P_n^m(\sin \varphi)$ $B$,(B, $B$,(B $B6v?t(B (even) $B$N;~(B $\varphi=0$ $B$K$D$$$FBP>N(B, $B$,(B $B4q?t(B (odd) $B$N;~(B $\varphi=0$ $B$K$D$$$FH?BP>N(B $B$G$"$k$3$H$r9MN8$7$F1i;;2s?t$r8:$i$9$3$H$,$G$-$k(B. $B$9$J$o$A(B, $A_{ij}$ $B$N7W;;$G$OKLH>5e$N$_$K$D$$$F(B $BFnKLBP>[email protected],(B $A_{ij}^{even}$ $B$H(B $BH?BP>[email protected],(B $A_{ij}^{odd}$ $B$K$D$$$F(B $B$=$l$>$l7W;;$7(B, $BFnH>5e$K$D$$$F$O(B $A{i,J-j}=A_{ij}^{even}-A_{ij}^{odd}$ $B$H$9$l$P$h$$(B. $B$^$?(B, $B$N7W;;$K$*$$$F$O(B, $B$=$NBP>N@-(B, $BH?BP>N@-$K4p$E$$$F(B $A_{i,j}+A_{i,J-j}$ $B$^$?$O(B $A_{i,j}-A_{i,J-j}$ $B$N0lJ}$r(B $B$K$D$$$F(B 1$B$+$i(B $B$^$G2C$($l$P$h$$(B.

: 3. $BNO3X2aDx(B : dcpam5 $B;YG[J}Dx<07O$H$=$NN%;62=(B : 1. $B$3$NJ8=q$K$D$$$F(B
Yasuhiro MORIKAWA $BJ?@.(B21$BG/(B3$B7n(B6$BF|(B

$\displaystyle \left( \DP{f}{\lambda} \right)_{ij}$	$\displaystyle \equiv \sum_{m=-M}^M \sum_{n=\vert m\vert}^N im \tilde{f}_n^m Y_n^m (\lambda_i,\varphi_j) ,$	(2.15)
$\displaystyle \widetilde{ \left( \DP{f}{\lambda} \right) }_n^m$	$\displaystyle = \frac{1}{I} \sum_{i=1}^I \sum_{j=1}^J im f_{ij} Y_n^{m*} (\lambda_i, \varphi_j) w_j .$	(2.16)

$\displaystyle \left(\DP{f}{\varphi}\right)_{ij}$	$\displaystyle \equiv \sum_{m=-M}^M \sum_{n=\vert m\vert}^N \tilde{f}_n^m \left. \DD{P_n^m}{\varphi} \right\vert _j \exp(im \lambda_i) ,$	(2.17)
$\displaystyle \widetilde{\left(\DP{f}{\varphi}\right)}_n^m$	$\displaystyle = - \frac{1}{I} \sum_{i=1}^I \sum_{j=1}^J f_{ij} \left. \DD{P_n^m}{\varphi}\right\vert _j \exp(-im \lambda_i) w_j .$	(2.18)