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% \Dtitle{$BCO5e@VF;0h$NBg5$GHF0$K4X$9$k8&5f(B}   % $BJQ99IT2D(B
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\def\thechapter{\arabic{chapter}}
\def\thesection{\arabic{chapter}.\arabic{section}}
\def\thesubsection{\arabic{chapter}.\arabic{section}.\arabic{subsection}}
\def\theequation{\arabic{chapter}.\arabic{section}.\arabic{equation}}
\def\thepage{\arabic{page}}
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\def\thetable{\arabic{chapter}.\arabic{section}.\arabic{table}}
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\begin{document}

\title{$BCO5e@VF;0h$NBg5$GHF0$K4X$9$k8&5f(B}
\author{$B?@8MBg3X(B $BM}3XIt(B $BCO5eOG@12J3X2J(B\\
$BCO5e$*$h$SOG@1Bg5$2J3X8&5f<<(B\\
\\
$B?\2l(B $BM'Li(B}
\date{2009$BG/(B2$B7n(B26$BF|(B}
\maketitle

%\chapter{}    % $B>O$N;O$a$+$i$N>l9g$O$3$N%3%^%s%I$r;HMQ$9$k(B
%section{}      % $B@a$N;O$a$+$i$N>l9g$O$3$N%3%^%s%I$r;HMQ$9$k(B
\thispagestyle{empty}
$B!!(B
\clearpage

%%% $BMW;](B
\begin{abstract}
Matsuno(1966) $B$r4p$K@VF;GH$K4X$9$k4pACNO3X$K$D$$$F$N%l%S%e!<$r9T$C$?(B. $B@u?eJ}Dx<0$K@VF;%Y!<%?J?LL6a;w$rMQ$$(B, $B@VF;GH$N3F%b!<%I$rF3$-=P$7$?(B. $B$=$N7k2L(B, $BFnKL$N@a$N?t(B ($B%b!<%I(B) $B$,(B$n \geq 1$ $B$N>l9g$K$O47@-=ENOGH(B ($B@>?J(B, $BEl?J(B), $B%m%9%S!<GH(B ($B@>?J(B) $B$N(B3 $B$D$NGH$,B8:_$9$k(B. $n=0$ $B$G$O(B $B47@-=ENOGH(B ($BEl?J(B) $B$H(B, $B:.9g%m%9%S!<=ENOGH(B ($B@>?J(B) $B$,B8:_$9$k(B. $B0^EYJ}8~$NGH$NB.EY$,(B $v=0$ $B$NFCJL$J>l9g$K$O(B, $B%1%k%S%sGH(B($BEl?J(B) $B$,B8:_$9$k(B. $B0J>e$+$i(B, $B@VF;0h$K$O(B 5 $B<oN`$NGHF0$,B8:_$9$k$3$H$,J,$+$C$?(B.

\end{abstract}

%%% $BL\<!(B
\markright{%\arabic{chapter}%.\arabic{section}
$BL\<!(B} %  
\tableofcontents
%
%
%%%%%% $BBh(B1$B>O(B
\chapter{$B$O$8$a$K(B}
\markright{\arabic{chapter}%.\arabic{section}
 $B$O$8$a$K(B} %  
$B@VF;0h$N1@H/@8$N<~4|@-$OFCD'E*$G$"$j(B, $B$=$N<~4|@-$OB>$NJ*M}NL$N<~4|E*JQF0$H4X78$,$"$k(B. $B6qBNE*$JNc$r5s$2$k$H(B, ENSO($BLs(B 4 $BG/<~4|(B), $B%b%s%9!<%s(B(1 $BG/<~4|(B), MJO($B?t==F|<~4|(B), $B3$N&Iw(B(1$BF|<~4|(B)$B$J$I$,$"$j(B, $B@VF;0h$K$OMM!9$J<~4|$ND9$5$N<~4|E*JQF0$,B8:_$7$F$$$k$3$H$,CN$i$l$F$$$k(B. $B$5$i$K<~4|E*JQF0$K$O%3%j%*%jNO$N8~$-$,$,@VF;$r6-$K5UE>$9$k@VF;Bg5$8GM-$NGHF0(B, $B$9$J$o$A@VF;GH$H8F$P$l$kGHF0$,1F6A$7$F$$$k$3$H$,J,$+$C$F$$$k(B. $B@VF;GH$H$O(B, $B@VF;$G?6I}$,:GBg$H$J$j@VF;IU6a$K%H%i%C%W$5$l$k(B, $BEl@>J}8~$KEAGE$9$kGHF0$G$"$k(B. 

$B@VF;GH$K4X$9$kM}O@E*$J8&5fO@J8$K$O(BMatsuno(1966)
%$B$d(BGill(1980)
$B$,$"$k(B. Matsuno(1966)$B$OCf9b0^EYBS$NBg5$!&3$MN$N1?F0$KMQ$$$k@~7A@u?eJ}Dx<0$r@VF;0h$KE,MQ$7$?>l9g(B, $BCf9b0^EYBS$GI=$l$k$h$&$J47@-=ENOGH$d%m%9%S!<GH$,@VF;0h$K$bI=$l$k$N$+(B, $B$H$$$&F05!$+$i8&5f$r9T$C$?(B. $B$^$?Cf9b0^EYBS$N47@-=ENOGH$O%m%9%S!<GH$KHf$Y?6F0?t$,$O$k$+$KBg$-$$$,(B, $B47@-=ENOGH$N?6F0?t$O%3%j%*%jNO$KHfNc$9$k$?$aDc0^EY$K$J$k$H%m%9%S!<GH$N<~4|$K6a$E$$$F$$$/(B. $B@VF;0h$K(B 2 $B$D$NGH$,B8:_$7$?>l9g(B, $BN><T$NGH$rA4$/0c$&GH$H$7$F07$&$3$H$,=PMh$k$N$+$H$$$&5?Ld$b$"$C$?(B. $B$3$l$K$D$$$F$O(B, $BFnKL$N?6F0%b!<%I$,Dc$$>l9g$K(B, $B@>8~$-$KEAGE$9$kN><T$NGH$r6hJL$G$-$J$/$J$k$3$H$,J,$+$C$?(B. $B8&5f$N7k2L(B, $B@VF;0h$N47@-=ENOGH$H%m%9%S!<GH$K2C$((B, $B@VF;8GM-$NGHF0$G$"$k:.9g%m%9%S!<=ENOGH$H@VF;%1%k%S%sGH$NB8:_$rL@$i$+$K$7$?(B. $B$5$i$K@VF;$K1h$C$F@589E*$KG.8;(B($BNd8;(B)$B$rG[CV$7$?>l9g$NDj>o$JBg5$$NMM;R$K$D$$$F5DO@$7$F$$$k(B. 
% $BG.8;$G$-$?$i(BGill(1980)$B$K$D$$$F$b>/$7=q$/(B

$BK\8&5f$G$O(B Matsuno(1966) $B$r4p$K@VF;GH$K4X$9$k4pACNO3X$K$D$$$F$N%l%S%e!<$r9T$C$?(B. $B@VF;0h$G?6I}$,:GBg$K$J$kEl@>J}8~$NGH$K$D$$$F@~7A2=$5$l$?J}Dx<07O$rB.EY(B, $B05NO$K$D$$$F2r$-(B, $B@VF;0h$N>e5-$NGH$r$=$l$>$lF3=P$7(B, $BGH$4$H$NB.EY>l!&05NO>l$N?^$r<($7$F3NG'$9$k$^$G$r9T$C$?(B. 
%
%
%
%%%%%% $BBh(B2$B@a(B
\chapter{$B4pK\%b%G%k(B}
\markright{\arabic{chapter} %.\arabic{section}
$B4pK\%b%G%k(B} %  
\setcounter{section}{1}    % $B@aHV9f(B
%  $BCf9b0^EYBS$NBg5$3$MN$NBg5,LO1?F0$K$D$$$F9M$($k$H$-(B, $B:G$bC1=c2=$7$?%b%G%k$G$"$k@u?eJ}Dx<0(B($BHsG4@-!&Hs05=LN.BN(B, $B@E?e05J?9U>uBV(B, $B?eJ?%9%1!<%k$,1tD>%9%1!<%k$h$j$O$k$+$KBg$-$$(B)$B$H(B, $\beta$ $BJ?LL6a;w$rMQ$$$F5DO@$9$k(B. 
%$BDc0^EY$G$NBg5,LO1?F0$NJQ2=$N@-<A$r8+$k$N$K$b(B, $B$3$N%b%G%k$r;H$C$F9M$($F$$$/(B. 
  $B@VF;0h$G$NBg5$3$MN$NBg5,LOBg5$$K$D$$$F9M$($k>l9g$N4pK\%b%G%k$H$7$F(B, $BCf9b0^EYBS$G$NBg5,LO1?F0$HF1$8$h$&$K(B,
$BBg5,LO1?F0$r9M$($k>l9g$K:G$bC1=c2=$7$?%b%G%k$G$"$k@u?eJ}Dx<0(B($BHsG4@-!&Hs05=LN.BN(B, $B@E?e05J?9U>uBV(B, $B?eJ?%9%1!<%k$,1tD>%9%1!<%k$h$j$O$k$+$KBg$-$$(B)$B$H(B, $\beta$ $BJ?LL6a;w$rMQ$$$F5DO@$9$k(B. $B$3$N$H$-(B $\beta$ $BJ?LL6a;w$O(B, $B@VF;0h$KBP1~$9$k$h$&$KJQ2=$9$k(B.

%%
%\section{$B@u?eJ}Dx<0!&%Y!<%?J?LL6a;w(B}
%  $BHs05=L(B, $BHsG4@-$N=g05N.BN$K$D$$$F(B, $B@ENO3XJ?9U$r2>Dj$7$?1?F0$r9M$($k(B.
$B$^$:7PEY$r(B $x$ $B<4(B, $B0^EY$r(B $y$ $B<4(B, $B1tD>J}8~$r(B $z$ $B<4$K$H$C$?D>8r:BI87O$r9M$((B,
$B1?F0J}Dx<0$H<ANLJ]B8B'$r=q$/(B
\begin{align}
\label{eq:yuuji}
\frac{\partial u}{\partial t} - fv + g\frac{\partial h}{\partial x} = 0 ,\\
\frac{\partial v}{\partial t} + fu + g\frac{\partial h}{\partial y} = 0 ,\\
\frac{\partial h}{\partial t} + 
H\left(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}\right) = 0.
\end{align}
$B$3$3$G(B, $u,v$ $B$O$=$l$>$l(B $x,y$ $BJ}8~$NB.EY(B, $t$ $B$O;~4V(B, $g$ $B$O=ENO2CB.EY(B, $h$ $B$OGH$N>eLL$N9b$5$NHy>.$JJP:9(B, $H$ $B$OGH$N>eLL$NJ?6Q$7$?9b$5$G$"$k(B.
$f$$B$O%3%j%*%j%Q%i%a!<%?$G(B, 
\begin{align}
f = 2 \Omega \sin \theta
\end{align}
$B$G$"$k(B. $\Omega$ $B$OCO5e$N<+E>B.EY(B, $\theta$ $B$O0^EY$G$"$k(B. $B$"$k0^EY(B $\theta _0$ $B$+$i(B $d\theta$ $B$@$1Hy>.$KFnKL$KJQ2=$9$k>l9g$r9M$($k$H(B, $BCO5eH>7B(B $R$ $B$rMQ$$$F(B
\begin{align}
\theta &= \theta_0 + d\theta \\
&= \theta _0 + \frac{y}{R}
\end{align}
$B$HI=$;$k(B. $B$3$N$H$-$N(B $f$ $B$r%F%$%i!<E83+$9$k$H(B,
\begin{align}
f(\theta) &\simeq 2\Omega \sin \theta_0 + \left( \frac{2\Omega}{R} \cos \theta_0 \right) y \\
&= f(\theta_0) + \beta y 
\end{align}
$B$H$J$k(B. $B$3$l$,%Y!<%?J?LL6a;w$G$"$k(B. $B$3$3$G$O@VF;0h$NOC$J$N$G(B $\theta_0 = 0$ $B$H$7(B,
\begin{align}
f = \beta y ,
\end{align}
$B$H2>Dj$9$k(B. 
$B$3$l$r(B $B<0(B(\ref{eq:yuuji}) $B$KBeF~$9$k$H(B, 
\begin{align}
\frac{\partial u}{\partial t} &- \beta yv + \frac{\partial \phi }{\partial x} = 0 ,\\
\frac{\partial v}{\partial t} &+ \beta yu + \frac{\partial \phi }{\partial v} = 0 ,\\
\frac{\partial \phi }{\partial t} &+ c^2\left ( \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}\right ) = 0 
\end{align}
$B$H$J$k(B. $B$3$3$G(B $ c^2 = \sqrt{gH} $$B$G$"$k(B. $B$5$i$K4JC1$N$?$a$K4pK\J}Dx<0$rL5<!857A$KJQ49$9$k(B. $B;~4V!&D9$5$NC10L$r$=$l$>$l(B
\begin{align}
[T] = \left ( \frac{1}{c\beta } \right )^{1/2} \ , \  [L] = \left ( \frac{c}{\beta }\right ) ^{1/2}
\end{align}
$B$H$9$k$H(B, 
%%% (3)
\begin{align}
\label{eq:3}
\frac{\partial u}{\partial t} &- yv + \frac{\partial \phi }{\partial x} = 0 ,\\
\frac{\partial v}{\partial t} &+ yu + \frac{\partial \phi }{\partial v} = 0 ,\\
\frac{\partial \phi }{\partial t} &+ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0 
\label{eq:33}
\end{align}
$B$H$J$k(B. $B0J9_$G$O$3$NJ}Dx<0$rMQ$$$F@VF;GH$K$D$$$F9M$($k(B.
%
%
%%%%%% $BBh(B3$B@a(B
\chapter{$B8GM-%b!<%I$N@VF;GH(B}
\markright{\arabic{chapter} %.\arabic{section}
$B8GM-%b!<%I$N@VF;GH(B} %  
\section{$BJ,;64X78(B}
  $x$$BJ}8~(B, $B$9$J$o$AEl@>J}8~$KEAGE$7$F$$$/GH$r9M$($k(B. $B$3$3$G$9$Y$F$NJQ2=NL$K(B $e^{i\omega t+ikx}$ $B$H$$$&MWAG$,(B
$B4^$^$l$k$H$9$k$H(B, $u,v,\phi$ $B$O(B
\begin{align}
u=u'(y)e^{i\omega t+ikx} ,\\
v=v'(y)e^{i\omega t+ikx} ,\\
\phi = \phi '(y)e^{i\omega t+ikx}
\end{align}
$B$H$J$j(B, $B<0(B (\ref{eq:3})-(\ref{eq:33})$B$O(B
\begin{align}
i\omega u'e^{i\omega t+ikx} - yv'e^{i\omega t+ikx} + ik\phi 'e^{i\omega t+ikx}= 0 , \\
i\omega v'e^{i\omega t+ikx} + yu'e^{i\omega t+ikx} + \frac{d\phi '}{dy}e^{i\omega t+ikx} = 0 ,\\
i\omega \phi 'e^{i\omega t+ikx}+ iku'e^{i\omega t+ikx} + \frac{dv'}{dy}e^{i\omega t+ikx} = 0
\end{align}
$B$H$J$k(B. $B$3$3$+$i(B$e^{i\omega t+ikx}$$B$r=|$-(B, $y$ $B$N4X?t(B $u'(y),v'(y),\phi'(y)$ $B$r2~$a$F(B $u,v,\phi$ $B$HCV$-$J$*$9$H(B,
\begin{align}
\label{eq:ymuji1}
i\omega u - yv + ik\phi = 0 ,\\
i\omega v + yu + \frac{d\phi }{dy} = 0 ,\\
i\omega \phi + iku + \frac{dv}{dy} = 0
\label{eq:ymuji3}
\end{align}
$B$H$J$k(B.
%
% $B$D$^$j(B$u,v,\phi$ $B$O(By$B$N$_$N4X?t$H$7$F9M$($k(B. 
%
$B$3$l$i$r(B$v$$B$K$D$$$F@0M}$9$k(B. $B<0(B(\ref{eq:ymuji1}) $B$H(B (\ref{eq:ymuji3}) $B$+$i(B$u$$B$H(B$\phi$$B$O(B
%%% u$B$H&5$N<0(B
\begin{align}
\label{eq:u}
u &= \frac{1}{i(\omega ^2 - k^2)}\left(\omega yv + k\frac{dv}{dy}\right) ,\\
\phi &= -\frac{1}{i(\omega^2 - k^2)}\left(kyv + \omega\frac{dv}{dy}\right),
\label{eq:phi}
\end{align}
$B$H$J$j(B, $u$$B$H(B$\phi$$B$r=|$$$F(B$v$$B$N<0$K@0M}$9$k$H(B
\begin{align}
\label{eq:dv}
\frac{d^2v}{dy^2} + \left(\omega^2 - k^2 + \frac{k}{\omega} - y^2\right)v = 0
\end{align}
$B$H$$$&<0$,F@$i$l$k(B. $B$3$3$G9M$($F$$$kLdBj$O@VF;(B($y=0$)$BIU6a$G$N$_B8:_$9$kGH$N1?F0$G$"$k(B. 
$B$9$J$o$A@VF;$+$i1s$/N%$l$?COE@$G$OGH$O8:?j$9$k$H9M$($k(B. $B$h$C$F(B, 
\begin{align}
v \rightarrow 0 \hspace{1em}; \hspace{1em} \textrm{when} \hspace{1em} y \rightarrow \pm \infty
\end{align}
$B$H$$$&6-3&>r7o$rE,MQ$G$-$k$H9M$($i$l$k(B. 
$B<0(B(\ref{eq:dv})$B$rJQ7A$9$k$H(B, 
\begin{align}
\left(-\frac{d^2}{dy^2} + y^2\right)v(y) = \left(\omega^2 - k^2 + \frac{k}{\omega}\right)v(y)
\end{align}
$B$H$J$k(B. $\omega^2 - k^2 + k/\omega$ $B$,Dj?t$H$9$k$H(B, $B$3$N<0$N7A$H6-3&>r7o$O(B
$BD4OB?6F0;R$N(B1$B<!85$N1?F0$r07$C$?%7%e%l%G%#%s%,!<J}Dx<0$HBP1~$7$F$$$k(B. 
$B$3$N%7%e%l%G%#%s%,!<J}Dx<0$N2r$O4{CN$G$"$j(B, $v$$B$K4X$9$kJ}Dx<0$K$D$$$F$b(B
$BF1$82rK!$rMQ$$$F$b$h$$$H9M$($k(B.
$B$^$:(B, 
\begin{align}
\label{bunsan}
\omega^2 - k^2 + \frac{k}{\omega} = 2n + 1 \hspace{1em} (n=0,1,2)
\end{align}
$B$H$$$&J,;64X78$,F@$i$l(B\footnote{$BF3=P$N>\:Y$OIUO?(B A $B$r;2>H(B.}, $v$$B$N2r$O(B
\begin{align}
\label{eq:v}
v(y) = Ce^{-\frac{1}{2}y^2}H_n(y)
\end{align}
$B$G$"$k(B. $B$3$3$G(B$H_n(y)$$B$O(B $n$ $B<!$N%(%k%_!<%HB?9`<0$H8F$P$l$k$b$N$G(B, 
\begin{align}
\label{hermite}
H_n(y) = (-1)^ne^{y^2}\frac{d^n}{dy^n}e^{-y^2}
\end{align}
$B$HDj5A$5$l$k(B. 
$B<0(B(\ref{bunsan})$B$O(B $n$ $B$KBP$7$F?6F0?t$H7PEYJ}8~$NGH?t$H$N4X78$,F@$i$l$k(B. 
$B$3$N(B $n$ $B$OGH$N?6F0$NFnKLJ}8~$N%b!<%I$G$"$j(B, $n$ $B$NCM$N$H$-$N(B $\omega$$B$r0J9_$G9M$($F$$$/(B.

%%%%%%
\section{$B%b!<%IKh$N?6F0?t(B}
$B<0(B(\ref{bunsan})$B$+$i(B, $n$$B$H(B$k$$B$,;XDj$5$l$?$H$-(B $\omega$ $B$N2r$,F@$i$l$k(B.
$B$3$N$H$-(B, $n$ $B$NCM$K$h$C$F(B $\omega$ $B$N2r$,0[$J$k>l9g$,$"$k(B. 

$n \geq$ $B$N>l9g(B, $B<0(B(\ref{bunsan})$B$rJQ7A$9$k$H(B, 
\begin{align}
\frac{\omega^2}{k^2} - 1 + \frac{1}{\omega k} = \frac{2n + 1}{k^2}
\end{align}
$B$H$J$k(B.  $k$$B$KBP$7$F(B $\omega$$B$,Bg$-$$$+>.$5$$$+$G(B
\begin{align}
\frac{\omega^2}{k^2} - 1 \simeq \frac{2n + 1}{k^2} \hspace{1em} (\omega \rightarrow \textrm{$BBg(B}) ,\\
- 1 + \frac{1}{\omega k} \simeq \frac{2n + 1}{k^2} \hspace{1em} (\omega \rightarrow \textrm{$B>.(B}),
\end{align}
$B$H$$$&6a;w$,$G$-$k(B. $B$h$C$F(B $\omega$$B$N2r(B
\begin{align}
\omega _{1,2} \simeq \mp \sqrt{k^2 + 2n + 1} \hspace{2em} (\omega \rightarrow \textrm{$BBg(B}),\\
\omega _3 \simeq \frac{k}{k^2 + 2n + 1} \hspace{2em} (\omega \rightarrow \textrm{$B>.(B})
\end{align}
$B$,F@$i$l$k(B. 
%%%%%%%%%%%
$B$3$l$i$rM-<!85$NGH$NEAGEB.EY$K$9$k$H(B,
\begin{align}
c_{1,2} &\simeq \pm c_g\sqrt{1 + \frac{1}{k^2}\frac{\beta}{c_g}(2n + 1)}, \\
c_3 &\simeq -\frac{\beta}{k^2 + \displaystyle \frac{\beta}{c_g}(2n+1)}
\end{align}
$B$H$J$k(B. $B$3$3$G;R8a@~J}8~$N=ENOGH$NB.EY(B$c$$B$O:.Mp$rHr$1$k$?$a$KBe$o$j$K(B$c_g$$B$GI=$7$F$$$k(B. 
%%%%%%%%%%%%

$BId9f$GI=$5$l$kGH$NEAGEJ}8~(B, $B$^$?47@-=ENOGH$O%m%9%S!<GH$HHf$Y$F?6F0?t$,Bg$-$$$3$H$r9M$($k$H(B,
$BE:;z$N(B1, 2, 3$B$G<($9?6F0?t$dEAGEB.EY$NGH$O(B, $B<!$N(B3$B%?%$%W$NGH(B\begin{enumerate}
\item $BEl?J$9$k47@-=ENOGH(B
\item $B@>?J$9$k47@-=ENOGH(B
\item $B@>?J$9$k%m%9%S!<GH(B
\end{enumerate}
$B$rI=$9(B. $n \geq 1$ $B$N>l9g$O(B, $B>e5-$N(B 3 $B$D$NGH$,B8:_$9$k(B.

$B4pK\E*$KG$0U$N(B$k$$B$KBP$7(B,  $\omega$ $B$N2r$O(B3$B$DF@$i$l$k(B. $ n \geq 1$ $B$N$H$-(B, 
$B$I$N$h$&$J(B$k$$B$KBP$7$F$b$=$l$>$lL@3N$K0c$&(B$\omega$ $B$,F@$i$l$k(B. 
$B$7$+$7(B$\omega$$B$N2r$O(B$n=0$$B$N$H$-FCJL$K$J$k(B. 
(\ref{bunsan}) $B$+$i(B, 
\begin{align}
\omega^2 - k^2 + \frac{k}{\omega} = 1.
\end{align}
$B$3$l$rJQ7A$7$F(B
\begin{align}
(\omega - k)(\omega^2 + k\omega -1) =0
\end{align}
$B$r2r$/$H(B, $B<!$N(B3$B$D$N2r$,F@$i$l$k(B. 
\begin{align}
\label{eq:n0}
\omega = \left\{
\begin{array}{l}
\displaystyle
-\frac{k}{2}-\sqrt{\left(\frac{k}{2}\right)^2+1}, \\
\displaystyle
-\frac{k}{2}+\sqrt{\left(\frac{k}{2}\right)^2+1}, \\
k. \\
\end{array}
\right.
\end{align}
$B$7$+$7(B, $B$3$l$i$N2r$K$=$l$>$l$NGH$N?6F0?t$O(B1$BBP(B1$B$GBP1~$7$F$$$J$$(B. 
$B47@-=ENOGH$O%m%9%S!<GH$HHf$Y$F?6F0?t$,Bg$-$$$H$$$&$3$H(B, 
$B$^$?(B $\omega = k$ $B$r2r$H$9$k$H(B $B<0(B(\ref{eq:u}),(\ref{eq:phi}) $B$h$j(B$u, \phi $ $B$,H/;6$7$F$7$^$$(B, $B6-3&>r7o$rK~$?$J$/$J$k(B
$B$3$H$r9M$($k$H(B, (\ref{eq:n0})$B$N2r$O(B, 
%%%%%%%%%
\begin{align}
\omega _1 &= -\frac{k}{2}-\sqrt{\left(\frac{k}{2}\right)^2+1}\ \hspace{7em} : \text{$BEl?J47@-=ENOGH(B}\ ,\\
\omega _2 &= -\frac{k}{2}+\sqrt{\left(\frac{k}{2}\right)^2+1}\ 
\hspace{1em} \left\{
\begin{array}{ll}
k\leq 1/\sqrt{2} & :\text{$B@>?J47@-=ENOGH(B} \\
k\geq 1/\sqrt{2} & :\text{$B%m%9%S!<GH(B} \\
\end{array},
\right.
\label{kongou}
\end{align}
%%%%%%%%%
$B$N$h$&$K(B 2 $B$D$@$1$H$J$k(B. 
$B$3$N(B 2 $B$DL\$N2r$O(B, $BGH?t(B $k$ $B$NJQ2=$KBP$7$F(B $k = 1/\sqrt{2}$ $B$r6-$K@>?J$9$k=ENOGH$+$i%m%9%S!<GH$XO"B3E*$KJQ2=$9$k(B.
$B$3$l$O(B, $B@>?J$9$k47@-=ENOGH$H%m%9%S!<GH$H$$$&(B 2 $B$D$NGH$N@-<A$r:.9g$7$?(B 1 $B$D$NGH$H$b9M$($i$l$k(B.
$B$3$NGH$O:.9g%m%9%S!<=ENOGH$H8F$P$l$k(B.
$B$9$J$o$A(B, $n=0$ $B$N>l9g$G$O(B
\begin{enumerate}
\item $BEl?J47@-=ENOGH(B
\item $B:.9g%m%9%S!<=ENOGH(B
\end{enumerate}
$B$N(B 2 $B$D$NGH$,B8:_$9$k(B.
%%%%%%%

%%%%%%%
$B:#$^$G$O@VF;IU6a$G$O(B $v\not=0$ $B$G$"$k$H$$$&A0Ds$GJ}Dx<0$r2r$$$F$-$?$,(B, $BFCJL$J>l9g$H$7$F(B $v=0$ $B$N>l9g$r9M$($k(B. 
$B<0(B(\ref{eq:ymuji1}) $B$K(B $v=0$ $B$r$"$F$O$a$k$H(B,
\begin{align}
i\omega u + ik\phi = 0 ,\\
yu + \frac{d\phi}{dy} = 0, \\
i\omega\phi + ihu = 0 
\end{align}
$B$H$J$k(B. $B$3$l$h$j(B
\begin{align}
(\omega - k)(\omega + k) = 0 ,\\
\omega = \pm k
\end{align}
$B$,F@$i$l$k(B. $B$3$N$H$-$N(B $u,\phi$ $B$N2r$O(B,
\begin{align}
\phi = u = Ce^{-\frac{1}{2}y^2} \ , \ \textrm{for} \ \omega = -k, \\
\phi = -u = Ce^{\frac{1}{2}y^2} \ , \ \textrm{for} \ \omega = k,
\end{align}
$B$H$J$k(B. $B$7$+$7(B $n=0$ $B$N$H$-$HF1MM$K(B $\omega = k$ $B$G$O6-3&>r7o$rK~$?$5$J$$(B.
$B$h$C$F(B $v=0$ $B$N2r$O(B $\omega = -k$ $B$H$J$k(B. $B$9$J$o$A(B$v = 0$ $B$N>l9g$K$OEl?J$9$kGH$,(B 1 $B$DB8:_$9$k(B.
$B$3$N2r$O(B, $B2>$KJ,;64X78<0(B(\ref{bunsan})$B$K$*$$$F(B $n=-1$ $B$H$*$$$?>l9g$KF@$i$l$k2r$HEy$7$$(B.
 $n=-1$ $B$N2r$H9M$($k$3$H$,$G$-$k(B. $B$3$NGH$O@VF;%1%k%S%sGH$H8F$P$l$k(B. 
$v=0$ $B$N>l9g$O@VF;%1%k%S%sGH$,B8:_$9$k(B.
%%% 

$n\geq 1$, $n=0$, $v=0(n=-1)$ $B$N?6F0?t$HGH?t$N4X78$r?^<($9$k$H(B, $B?^(B\ref{fig:bunsan}$B$N$h$&$K$J$k(B.
%%%
\begin{figure}[htbp]
\centering
\Depsf[9cm]{./img/bunsan.eps}
\caption{Matsuno(1966) $B$N3F%b!<%I$4$H$NJ,;64X78$rI=$7$??^$K(B, $BGH$NJ,N`$r=q$-2C$($?(B. $B2#<4$,GH?t(B, $B=D<4$,L5<!852=$7$??6F0?t$G$"$k(B.
}\label{fig:bunsan}
\end{figure}
%%%%%%%

\clearpage
\section{$B8GM-4X?t(B}
%%% 3$B$D$NGH$N(Bu,v,$B&5$r$^$H$a$F=q$/(B
$B?6F0?t$N<0$H(B $v$ $B$N<0$K$h$C$F(B, $B$=$l$>$l$NGH$NB.EY$H05NO(B$(u,v,\phi)$ $B$N<0$,F@$i$l$k(B.
$B<0(B(\ref{eq:u}), (\ref{eq:phi}), (\ref{eq:v})$B$+$i(B
\begin{align}
u &= \frac{C}{i(\omega^2+k^2)}\left\{y\omega e^{-\frac{1}{2}y^2}H_n(y) + k\left(-yH_n(y) + \frac{dH_n(y)}{dy}\right)e^{-\frac{1}{2}y^2}\right\} , \\
\phi &= -\frac{C}{i(\omega^2+k^2)}\left\{yk e^{-\frac{1}{2}y^2}H_n(y) + \omega \left(-yH_n(y) + \frac{dH_n(y)}{dy}\right)e^{-\frac{1}{2}y^2}\right\}.
\end{align}
$B$3$l$r@0M}$9$k$H(B,
\begin{align}
u &= \frac{C}{i(\omega^2+k^2)}\left\{(\omega - k)e^{-\frac{1}{2}y^2}H_n(y) + e^{-\frac{1}{2}y^2}\frac{dH_n(y)}{dy} \right\} ,\\
\phi &= -\frac{C}{i(\omega^2+k^2)}\left\{(k-\omega)e^{-\frac{1}{2}y^2}H_n(y) + e^{-\frac{1}{2}y^2}\frac{dH_n(y)}{dy}\right\}
\end{align}
$B$H$J$k(B.
%%% $B%(%k%_!<%HB?9`<0$NA22=<0(B
$B$^$?%(%k%_!<%HB?9`<0(B(\ref{hermite})$B$+$iA22=<0(B
\begin{align}
\frac{dH_n(y)}{dy} &= 2nH_{n-1}(y) ,\\
H_{n+1}(y) &= 2yH_n(y)-2nH_{n-1}(y)
\end{align}
$B$,F@$i$l$k(B. $B$3$l$rMQ$$$F(B$u,v,\phi$$B$N<0$r$^$H$a$k$H(B,
\begin{enumerate}
\item $n \geq 1 $ $B$N>l9g(B
\begin{align}
\left(
\begin{array}{c}
v \\
u \\
\phi \\
\end{array}
\right)_{nl} = \left(
\begin{array}{l}
i(\omega _{nl}^2 - k^2)\psi _n \\
\frac{1}{2}(\omega _{nl} - k)\psi _{n+1} + n(\omega _{nl} + k)\psi _{n-1} \\
\frac{1}{2}(\omega _{nl} - k)\psi _{n+1} - n(\omega _{nl} + k)\psi _{n-1} \\
\end{array}
\right),
\end{align}
%%% n=0$B$N>l9g(B
\item $n=0$ $B$N>l9g(B
\begin{align}
\left(
\begin{array}{c}
v \\
u \\
\phi \\
\end{array}
\right)_{0l} = \left(
\begin{array}{c}
2i(\omega _nl + k)\psi _0 \\
\psi _{1} \\
\psi _{1} \\
\end{array}
\right),
\end{align}
%%% n=-1$B$N>l9g(B
\item $v=0 (n=-1)$ $B$N>l9g(B
\begin{align}
\left(
\begin{array}{c}
v \\
u \\
\phi \\
\end{array}
\right)_{-1} = \left(
\begin{array}{c}
0 \\
\psi _{0} \\
\psi _{0} \\
\end{array}
\right),
\end{align}
\end{enumerate}
$B$G$"$k(B.
$B$3$3$G(B,
\begin{align}
\psi _n = e^{-\frac{1}{2}y^2}H_n(y)
\end{align}
$B$G$"$k(B. $B<0(B(\ref{eq:v}) $B$h$j(B
\begin{align}
v = Ce{-\frac{1}{2}y^2}H_n(y) = C \psi_n
\end{align}
$B$J$N$G(B, $B<0(B(\ref{eq:dv}) $B$+$i(B
\begin{align}
\left(\frac{d^2}{dy^2} + y^2\right)\psi_n = \left(\omega^2 - k^2 + \frac{k}{\omega}\right)\psi_n
\end{align}
$B$G$"$k(B.
$\psi_n$ $B$O8GM-CM(B $\omega$ $B$KBP$9$k8GM-4X?t$H8@$($k(B.

\setcounter{figure}{0}     % $B?^HV9f(B
\section{$B3F%b!<%IKh$NFCD'(B}
$BA0@a$^$G$G8GM-%b!<%IKh$NGHF0$H(B, $B$=$l$i$NB.EY$H05NO$N<0$rF3$-=P$7$?(B. $B$3$3$G$OGHF0$4$H$NB.EY>l(B, $B05NO>l$N?^$r<($7$F$=$NFCD'$r9M$($F$$$/(B.

%%%%%%%%%
\subsection{$n \geq 1$ $B$NGHF0(B}
$n \geq 1$ $B$N>l9g$NGHF0$OEl?J!&@>?J47@-=ENOGH$H%m%9%S!<GH$,B8:_$9$k(B. $B$3$l$i$NGH$O$=$l$>$l$K(B, $BFC$K47@-=ENOGH$H%m%9%S!<GH$N4V$KL@3N$J6hJL$,$D$1$i$l$k(B. 
% $B05NO$N9b$$ItJ,$N%;%k$H05NO$NDc$$ItJ,$N%;%k$N(B

$B?^(B\ref{n1east}, \ref{n1west}$B$N47@-=ENOGH$O@VF;$K6a$$NN0h$G$OFnKLJ}8~$KB.EY$N%Y%/%H%k$,8~$$$F$*$j(B, $BEl@>J}8~$N05NO79EY$,:GBg$G$"$k>l=j$G<}B+$HH/;6$,5/$3$C$F$$$k(B. $B$^$?@VF;$+$iN%$l$k$[$I(B, $B%3%j%*%jNO$N1F6A$r<u$1B.EY%Y%/%H%k$N8~$-$,6J$2$i$l$F$$$k(B.

$B?^(B\ref{n1rossby}$B$N%m%9%S!<GH$OA4BNE*$K5$05>l$HB.EY>l$N4V$G$[$\CO9UN.J?9U$,@.$jN)$C$F$$$k(B. $n=1$ $B$G$OM#0l@VF;IU6a$G%3%j%*%jNO$,F/$+$J$$$?$aCO9UIwJ?9U$,@.$jN)$?$J$/$J$C$F$$$k(B. $n=2$ $B$G$O@VF;$rCf?4$KCO9UN.$H$O5U2sE>$NB.EY>l$,$G$-$F$$$k(B. 

\begin{figure}[htbp!]
\centering
\Depsf[10cm]{./img/n1east.eps}
\Depsf[10cm]{./img/n2east.eps}
\caption{$n\geq1$ $B$NEl?J47@-=ENOGH(B($k=0.5$). $BLp0u$O?eJ?B.EY%Y%/%H%k(B, $BEyCM@~$O05NO$r<($9(B. $B>e$O(B $n=1$, $B2<$O(B$n=2$ $B$N%b!<%I(B.}\label{n1east}
\end{figure}
\begin{figure}[htbp!]
\centering
\Depsf[10cm]{./img/n1west.eps}
\Depsf[10cm]{./img/n2west.eps}
\caption{$n\geq1$ $B$N@>?J47@-=ENOGH(B($k=0.5$). $BLp0u$O?eJ?B.EY%Y%/%H%k(B, $BEyCM@~$O05NO$r<($9(B. $B>e$O(B $n=1$, $B2<$O(B$n=2$ $B$N%b!<%I(B.}\label{n1west}
\end{figure}
\begin{figure}[htbp!]
\centering
\Depsf[10cm]{./img/n1rossby.eps}
\Depsf[10cm]{./img/n2rossby.eps}
\caption{$n\geq1$ $B$N%m%9%S!<GH(B($B@>?J(B, $k=0.5$). $BLp0u$O?eJ?B.EY%Y%/%H%k(B, $BEyCM@~$O05NO$r<($9(B. $B>e$O(B $n=1$, $B2<$O(B$n=2$ $B$N%b!<%I(B.}\label{n1rossby}
\end{figure}

\clearpage
%%%%%%%%%%
\subsection{$n=0$ $B$NGHF0(B}
$n=0$ $B$N>l9g$NGHF0$OEl?J47@-=ENOGH$H:.9g%m%9%S!<=ENOGH$,B8:_$9$k(B.

$B?^(B\ref{n0east}$B$NEl?J47@-=ENOGH$O(B $n \geq 1$ $B$N>l9g$HB.EY%Y%/%H%k$NEl@>J}8~$N8~$-$,5U$K$J$C$F$$$k$,(B, $B$=$l0J30$NJ,I[$NMM;R$O(B$n \geq 1$ $B$NEl?J47@-=ENOGH$NFCD'$HF1$8$G$"$k(B.

$B?^(B\ref{westrossby}$B$N:.9g%m%9%S!<=ENOGH$O(B, $B9b0^EY$G$OB.EY>l$H05NO>l$NCO9UIwJ?9U$,@.$jN)$C$F$$$k(B. $B$3$l$O%m%9%S!<GH$N@-<A$G$"$k(B. $B$7$+$7@VF;IU6a$G$O47@-=ENOGH$N$h$&$JB.EY>l$H$J$C$F$$$k(B. $B@VF;IU6a$G$NB.EY>l!&05NO>l$NJ,I[$,0lHV6a$$$N$,(B, $n=2$ $B$N@>?J47@-=ENOGH$G$"$k(B. $B$3$N$h$&$K(B, $B:.9g%m%9%S!<=ENOGH$O%m%9%S!<GH$H47@-=ENOGH$NN>J}$NFCD'$,:.9g$7$F$$$k(B. $B$^$?(B, (\ref{kongou})$B$G$O(B, $BGH?t(B $k$ $B$,(B $k \leq 1/\sqrt{2}$ $B$N>l9g$O47@-=ENOGH(B, $k \geq 1/\sqrt{2}$ $B$N>l9g$O%m%9%S!<GH$KJ,N`$5$l$k$,(B, $k=0.5 (<1/\sqrt{2})$ $B$H(B $K=1.0 (>1/\sqrt{2}$ $B$G$OL@3N$J0c$$$,8+$F<h$l$J$$(B. $B$3$N$3$H$+$i$b@>?J$9$kGH$O:.9g%m%9%S!<=ENOGH$H$$$&(B 1 $B$D$NGH$G$"$k$H8@$($k(B.

\begin{figure}[htbp!]
\centering
\Depsf[10cm]{./img/n0east.eps}
\Depsf[10cm]{./img/10n0east.eps}
\caption{$n=0$ $B$NEl?J47@-=ENOGH(B. $BLp0u$O?eJ?B.EY%Y%/%H%k(B, $BEyCM@~$O05NO$r<($9(B. $B>e$OGH?t$,(B $k=0.5$, $B2<$O(B$k=1.0$ .
}\label{n0east}
\end{figure}

\begin{figure}[htbp!]
\centering
\Depsf[10cm]{./img/k05westrossby.eps}
\Depsf[10cm]{./img/k10westrossby.eps}
\caption{$n=0$ $B$N:.9g%m%9%S!<=ENOGH(B($B@>?J(B). $BLp0u$O?eJ?B.EY%Y%/%H%k(B, $BEyCM@~$O05NO$r<($9(B. $B>e$OGH?t$,(B $k=0.5$, $B2<$O(B$k=1.0$.}\label{westrossby}
\end{figure}

\clearpage
%%%%%%%%%%
\subsection{$v=0(n=-1)$ $B$NGHF0(B}
$v=0$ $B$N>l9g$NGHF0$O@VF;%1%k%S%sGH$,B8:_$9$k(B. 

$B?^(B\ref{v0kelvin}$B$N%1%k%S%sGH$OB.EY%Y%/%H%k$,NN0hA4BN$G@VF;$KJ?9T$G(B, $B05NO>l$N;3$HC+$,FnKLJ}8~$KBP$7$F@VF;>e$K(B 1 $B$D$7$+$J$$(B.
$B$^$?%1%k%S%sGH$O@VF;$+$iN%$l$k$K$D$l$FB.EY$,>.$5$/$J$C$F$$$k(B. $BEl@>J}8~$K$O@VF;>e$N05NO$N8{G[$,Bg$-$$ItJ,$G<}B+$7(B, $B=ENOGH$NFCD'$,I=$l$F$$$k(B. $B0lJ}FnKLJ}8~$K$O@VF;0h$+$i>/$7N%$l$?ItJ,$G$OEl@>$N5$0579EY$,>.$5$/$J$j(B, $BCO9UN.J?9U$,@.$jN)$D(B. 

\begin{figure}[htbp!]
\centering
\Depsf[10cm]{./img/kelvin.eps}
\caption{$n=0$ $B$N%1%k%S%sGH(B. $BLp0u$O?eJ?B.EY%Y%/%H%k(B, $BEyCM@~$O05NO$r<($9(B. $B>e$OGH?t$,(B $k=0.5$, $B2<$O(B$k=1.0$ .}
\label{v0kelvin}
\end{figure}


%%%%%%%%%%%%
\chapter*{$B$^$H$a(B}
\markright{%\arabic{chapter} %.\arabic{section}
$B$^$H$a(B}
\addcontentsline{toc}{chapter}{$B$^$H$a(B}
$B@VF;GH$K4X$9$k4pACNO3X$K$D$$$F(B, Matsuno(1966)$B$N%l%S%e!<$r9T$C$?(B. 
$B@VF;0h$G$NBg5$!&3$MN$NEl@>J}8~$KEAGE$9$kGHF0$N1?F0$K$D$$$F(B
$B@~7A$N@u?eJ}Dx<0$H@VF;0h$K$D$$$F$N%Y!<%?J?LL6a;w$rMQ$$$F2r$$$?(B. $B$^$:(B $B%b!<%I(B $n \geq 1$ $B$K$D$$$FCf9b0^EYBS$HF1$8$/?6F0?t$NBg$-$$GH$,(B 2 $B$D(B, $B>.$5$$GH$,(B 1 $B$DF@$i$l(B, $B47@-=ENOGH$,%m%9%S!<GH$N?6F0?t$h$j$bBg$-$$>l9g$r9M$($k$H(B, $B$3$l$i$OEl?J!&@>?J47@-=ENOGH$H%m%9%S!<GH$G$"$k(B. $B<!$K(B $n = 0 $ $B$N>l9g$r9M$($k$H(B, $BEl?J47@-=ENOGH$H(B, $B@>?J$9$kGH$N(B 2 $B$D$,F@$i$l$?(B. $B$3$N@>?J$9$kGH$O$"$kGH?t$r6-$K@>?J47@-=ENOGH$H%m%9%S!<GH$H$N@-<A$,F~$lBX$o$k(B. $B$3$NGH$O:.9g%m%9%S!<GH$G$"$k(B. $B$=$7$F(B $v=0$ $B$G$"$kFCJL$J>l9g$K$O(B, $BEl?J$9$kGH$,F@$i$l$?(B. $B$3$NGH$O(B $n=-1$ $B$H$7$?>l9g$KI=$l$kGH$HF1$87A$G$"$j(B, $B$3$NGH$O%1%k%S%sGH$G$"$k(B. $B$3$N$h$&$K(B, $B@VF;0h$G$O(B 5 $B<oN`$NGH$,B8:_$9$k$3$H$,J,$+$C$?(B.

$B:#2s$OFbMF$K4^$a$F$$$J$$$,(B Matsuno(1966) $B$NG.8;1~Ez$K$D$$$F$b%l%S%e!<$r9T$C$F$*$j(B, $B0z$-B3$-?J$a$F$$$/M=Dj$G$"$k(B. $BG.8;1~Ez$K$D$$$F$O(B Matusno(1966) $B$NB>$K(B, Gill(1980) $B$d(B Kosaka and Matsuda(2005), $BNS(B(2006) $B$,$"$k(B. $B$=$l$>$lG.8;$N@_CV$N;EJ}$,0[$J$j(B, Matsuno(1966) $B$G$O@VF;1h$$$KG.8;$HNd8;$r8r8_$K@_CV$7$F$*$j(B, Gill(1980) $B$G$O(B $BG.8;$r@VF;>e$K(B 1 $BE@$N$_@_CV$7$F$$$k(B. Kosaka and Matsuda(2005) $B$dNS(B(2006)$B$G$O(B, $BCO5e$r0O$`%j%s%0>u$NG.8;$r9M$($F$$$k(B. $B$3$l$i$r4p$KIw!&05NO>l$N?^$r:F8=$7(B, $B$^$?B>$NG.8;1~Ez$N7A$b9M$($k$3$H$GG.8;$N1F6A$K$h$kBg5$$N1?F0$K$D$$$FD4$Y$F$$$-$?$$(B. $B$^$?(B, $B3$$KHf$Y$FHfG.$N>.$5$$N&$,F|<M$GCH$a$i$l$k$3$H$G0l;~E*$KG.8;$H$J$j(B, $B$=$l$K$h$kN&>e$G$NF|<~4|$N1@$NH/@8$,Bg5,LO1?F0$KM?$($k1F6A$K$D$$$F9M$($?$$$H;W$C$F$$$k(B.

\chapter*{$B<U<-(B}
\markright{%\arabic{chapter} .\arabic{section}
$B<U<-(B}
\addcontentsline{toc}{chapter}{$B<U<-(B}
$BK\8&5f$r9T$&$K$"$?$j(B, $BB?$/$NJ}!9$K$4;XF3(B, $B$46(NO$rD:$-$^$7$?(B. $B;XF36541$N?@8MBg3X(B $BNS(B $B>M2p(B $B65<x$K$O(B, $BB46H8&5f$NJ}?K$r<($7$FD:$-$^$7$?(B. JAMSTEC($BFHN)9T@/K!?M3$MN8&5f3+H/5!9=(B)/$B?@8MBg3X(B  $B;3Cf(B $BBg3X(B $B65<x$K$O8&5f$NA4BN$K$o$?$C$FHs>o$KB?$/$N$4;XF3$rD:$-$^$7$?(B. $B?@8MBg3X(B $BCO5e$*$h$SOG@1Bg5$2J3X8&5f<<(B $BG<B?(B $BE/;K(B $B;a$K$OO@J8$N:n@.$K$D$$$F$40U8+$rD:$-$^$7$?(B. $B$=$7$F?@8MBg3X(B $BCO5e$*$h$SOG@1Bg5$2J3X8&5f<<$N3'$5$^$K$OB?$/$N=u8@$d7cNe$rD:$-$^$7$?(B. $B$3$3$K?<$/46<U$7(B, $B?4$h$j$*Ni?=$7>e$2$^$9(B.

$B?tCM7W;;$H?^$N:n@.$K$O(B, $BCO5eN.BNEEG>%i%$%V%i%j(B dcl-5.3.3 (http://www.gfd-dennou.org/library/dcl/) $B$r;HMQ$5$;$F$$$?$@$-$^$7$?(B. 

%%%%%%%%%%%%%%%%%
%
\chapter*{$BIUO?(B A $BJ,;64X78<0$NF3=P(B}
\markright{%\arabic{chapter} .\arabic{section}
$BIUO?(B A $BJ,;64X78<0$NF3=P(B}
\addcontentsline{toc}{chapter}{$BIUO?(B A $BJ,;64X78<0$NF3=P(B}
$B<0(B(\ref{eq:dv}) $B$+$i(B $B<0(B(\ref{bunsan})$B$rF3=P$9$k(B.
$B<0(B(\ref{eq:dv})$B$r:F7G$9$k$H(B,
\begin{align}
\frac{d^2v}{dy^2} + \left(\omega^2 - k^2 + \frac{k}{\omega} - y^2\right)v = 0 \ .
\end{align}
$B$3$l$rJQ7A$7$F(B,
\begin{align}
\left( -\frac{d^2}{dy^2} + y^2\right)v = (\omega^2 - k^2 + \frac{\omega}{k})v.
\end{align}
$B$3$N<0$N7A$O(B, 1 $B<!85$ND4OB?6F0;R$KBP$9$k%7%e%l%G%#%s%,!<J}Dx<0(B
\begin{align}
\left( -\frac{d^2}{dy^2} + \xi^2\right)X(\xi) = \lambda X(\xi)
\end{align}
$B$H0lCW$9$k(B.
$B$3$N<0$N2rK!$K$D$$$F=g$K3NG'$7$F$$$/(B.

$B$^$:<0$N7A$+$i2r$r?dDj$9$k$H(B,
\begin{align}
v(y) = f(y) e^{-\frac{t^2}{2}}
\end{align}
$B$H?dDj$G$-$k(B.
$BJX59>e(B $\lambda=\omega^2 - k^2 + k/\omega $ $B$HCV$$$F(B, $B2r$r<0$KEv$F$O$a$k$H(B, 
\begin{align}
-\left\{\frac{d^2f}{dy^2} - 2y\frac{f}{y} - (1-y^2)f\right\}e{-\frac{y^2}{2}} + y^2fe{-\frac{y^2}{2}}
&= \lambda f e{-\frac{y^2}{2}},\\
\frac{d^2f}{dy^2} = 2y\frac{df}{dy} + (1-\lambda)f&
\label{eq:df}
\end{align}
$B$H$J$k(B. $B$3$N$h$&$JJ}Dx<0$N2r(B $f(y)$ $B$r5a$a$k$K$O(B, $f(y)$ $B$r<!$N$h$&$J$Y$-5i?t(B
\begin{align}
f(y) = a_0 + a_1 y + a_2 y^2 + \cdots + a_n y^n + \cdots = \sum_{n=0}^{\infty}a_n y^n
\end{align}
$B$H9M$($F2r$/(B.
$B$3$l$r(B (\ref{eq:df}) $B$KBeF~$7$F(B,
\begin{align}
($B:8JU(B)\hspace{1em} \frac{df^2}{dy^2} &= 1 \cdot 2 a_2 + 2 \cdot 3 a_3 y + 3 \cdot 4 a_4 y^2 + \cdots \notag \\
&= \sum_{n=0}^{\infty}(n+1)(n+2)a_(n+2)y^n ,\\
($B1&JU(B)\hspace{1em} 2y\frac{df}{dy} - (\lambda -1) &= (1-\lambda )a_0 + (1 \cdot 2 + 1 - \lambda )a_1 y 
+(2 \cdot 2 + 1- \lambda)a_2y^2 + \cdots \notag \\
&= \sum_{n=0}^{\infty}(2n+1-\lambda)a_ny^n,
\end{align}
$B$H$J$k(B. $BN>JU$,0lCW$9$k$?$a$K$O(B $y$ $B$NF1$8$Y$-9`$N78?t$,Ey$7$/$J$i$J$/$F$O$J$i$J$$$N$G(B,
\begin{align}
\label{eq:lambda}
(n+1)(n+2)a_{n+2} = (2n + 1 - \lambda)a_n
\end{align}
$B$H$J$k(B. $B$3$l$G(B $ a_2, a_3, a_4, \cdots$ $B$H78?t$r7h$a$k$3$H$,=PMh$k$,(B, $n \rightarrow \infty $ $B$^$G78?t$,$"$k$H$9$k$H(B, $ y \rightarrow \pm \infty$ $B$N$H$-5i?t(B $f(y)$ $B$OH/;6$7$F$7$^$&$N$G(B, $v(y)$ $B$b<}B+$7$J$/$J$k(B.
$B$3$3$G07$C$F$$$kLdBj$dD4OB?6F0;R$K$D$$$F9M$($k$H(B, $y=0$ $B$NIU6a$GM-8B$NCM$r;}$A(B, $y \rightarrow \infty$ $B$G5^B.$K(B $0$ $B$K$J$k$h$&$J>r7o$,$"$k$H9M$($i$l$k(B.
$B$=$N$h$&$J>r7o$,B8:_$9$k$K$O(B, (\ref{eq:lambda}) $B$N1&JU$N3g8L$NCf$,(B 0 $B$K$J$l$P$h$$(B.
$B$9$J$o$A(B,
\begin{align}
\lambda = 2n + 1
\end{align}
$B$rK~$?$;$P(B, $ a_{n+2}, a_{n+4}, \cdots$ $B$O$9$Y$F(B 0 $B$K$J$k(B.
$B$7$+$7(B $\cdots a_{n+1}, a_{n*3} \cdots$ $B$OJL$KB8:_$7$F$$$F(B, $n \rightarrow \infty $ $B$GH/;6$7$F$7$^$&(B. 
$B$3$l$G$O0UL#$,$J$$$N$G(B, $\cdots a_{n+1}, a_{n*3} \cdots$$B$O:G=i$N9`(B ($a_0$ $B$^$?$O(B $a_1$) $B$+$i(B 0 $B$G$J$1$l$P$J$i$J$$(B.
$B$D$^$j5i?t(B $f(y)$ $B$O4q?t$Y$-9`$b$7$/$O6v?t$Y$-9`$@$1$r$b$A(B, (\ref{eq:df})$B$G7h$^$k$h$&$J(B$n$ $B$K$h$C$F(B $a_n$ $B0J9_$,(B 0 $B$K$J$kB?9`<0$G$"$k(B. 
$B$3$3$G(B $\lambda = \omega^2 - k^2 + k/\omega$ $B$h$j(B
\begin{align}
\omega^2 - k^2 + \frac{k}{\omega} = 2n + 1 \hspace{1em} (n = 0,1,2 \cdots)
\end{align}
$B$G$"$j(B, $BJ,;64X78(B (\ref{bunsan}) $B$,F@$i$l$?(B.

\chapter*{$BIUO?(B B  $B9b%b!<%I$NGHF0$NB.EY!&05NOJ,I[?^(B}
\addcontentsline{toc}{chapter}{$BIUO?(B B  $B9b%b!<%I$NGHF0$NB.EY!&05NOJ,I[?^(B}
\markright{%\arabic{chapter}.\arabic{section}
$BIUO?(B B  $B9b%b!<%I$NGHF0$NB.EY!&05NOJ,I[?^(B}
\setcounter{section}{5}
\setcounter{figure}{0}     % $B?^HV9f(B

$n \geq 1$ $B$N>l9g$NGHF0$K$D$$$F(B, $n=3,4,5$ $B$N>l9g$K$D$$$F$bB.EY>l!&05NO>l$N?^$r%W%m%C%H$7$?$N$G(B, 
$B$3$3$G5s$2$F$*$/(B.

$B?^(B \ref{n3east}, \ref{n3west}, \ref{n3rossby} $B$N47@-=ENOGH$b%m%9%S!<GH$b(B, $BFnKL$N@a$,A}$(J#;($K$J$C$F$O$$$k$,(B, $B$=$l$>$l$NGH$N@-<A$,I=$l$F$$$k(B. \ref{n3rossby} $B$N%m%9%S!<GH$K$D$$$F$O(B, $B9b0^EY0h$G$NCO9UN.J?9U$NB.EY>l$,(B, $B5$05>l$+$iDc0^EYB&$K8~$+$C$F$:$l$k798~$,$"$k(B.% $B$I$N$h$&$K(B?%%%%%%%%%
$n$ $B$,Bg$-$/$J$k$[$I(B, $B$:$l$,5$05>l$NCf?4$H$=$N<~0O$N@a$H$N0LCV$KBP$7$FAjBPE*$KBg$-$/$J$k(B.
\begin{figure}[htbp!]
\Depsf[8cm]{./img/n3east.eps}
\Depsf[8cm]{./img/n4east.eps}
\Depsf[8cm]{./img/n5east.eps}
\caption{$n\geq1$ $B$NEl?J47@-=ENOGH(B($k=0.5$). $BLp0u$O?eJ?B.EY%Y%/%H%k(B, $BEyCM@~$O05NO$r<($9(B. $B:8>e$O(B $n=3$, $B1&>e$O(B $n=4$, $B2<$O(B$n=5$ $B$N%b!<%I(B.}\label{n3east}
\end{figure}
\begin{figure}[htbp!]
\Depsf[8cm]{./img/n3west.eps}
\Depsf[8cm]{./img/n4west.eps}
\Depsf[8cm]{./img/n5west.eps}
\caption{$n\geq1$ $B$N@>?J47@-=ENOGH(B($k=0.5$). $BLp0u$O?eJ?B.EY%Y%/%H%k(B, $BEyCM@~$O05NO$r<($9(B. $B:8>e$O(B $n=3$, $B1&>e$O(B $n=4$, $B2<$O(B$n=5$ $B$N%b!<%I(B.}\label{n3west}
\end{figure}
\begin{figure}[htbp!]
\Depsf[8cm]{./img/n3rossby.eps}
\Depsf[8cm]{./img/n4rossby.eps}
\Depsf[8cm]{./img/n5rossby.eps}
\caption{$n\geq1$ $B$N%m%9%S!<GH(B($B@>?J(B, $k=0.5$). $BLp0u$O?eJ?B.EY%Y%/%H%k(B, $BEyCM@~$O05NO$r<($9(B. $B:8>e$O(B $n=3$, $B1&>e$O(B $n=4$, $B2<$O(B$n=5$ $B$N%b!<%I(B.}\label{n3rossby}
\end{figure}

%%%%%%%%%%%%%%%%
\chapter*{$B;29MJ88%(B}
\markright{%\arabic{chapter}.\arabic{section}
$B;29MJ88%(B}
\addcontentsline{toc}{chapter}{$B;29MJ88%(B}
\begin{description}
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\item Matsuno, T., 1966:  Quasi-geostrophic motions in the equatorial area. 
\textit{J. Meteor. Soc. Japan}, \textbf{44}, 25-42.
\item $BLZB<N5<#(B, 1983: $BCO5eN.BNNO3XF~Lg(B, $BEl5~F2=PHG(B, pp.247.
\item $B>.=P(B $B><0lO:(B, 1969: $B4pACJ*M}3XA*=q(B 5A $BNL;RNO3X(B (1), $B>X2ZK<(B, pp.225.
\item $BNS(B $BKcMx;R(B, 2006: $B@VF;IU6a$K$*$1$kG.8;1~Ez$H@VF;GH(B, $BB46HO@J8(B, 1-65.
\item $B>>LnB@O:(B-$BEg:jC#IW(B, 1981:  $BBg5$2J3X9V:B(B 3 $B@.AX7w$HCf4V7w$NBg5$(B, $BEl5~Bg3X=PHG2q(B, pp.279.
\item $B;3CfBg3X(B, 2002: $BElFn%"%8%"$N5$8u!&5$>]M}2r$N$?$a$NNO3XE*4pAC(B, $B5$>]8&5f%N!<%H(B 202 $B9f(B, $BF|K\5$>]3X2q(B, pp1-56.
\end{description}
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