%#! platex master_thesis-M2.tex

\chapter{$B;YG[J}Dx<0$NF3=P(B} \label{chap:governing_equation}

$B$3$3$G$O(B, $B%b%G%k$GMQ$$$?;YG[J}Dx<0$NF3=P$r9T$&(B.

\section{$BLdBj@_Dj(B}

$B9M$($kN.BN$O(B$f$$BJ?LL>e$NHsG4@-%V%7%M%9%/N.BN$G(B, $B@E;_>uBV$G$O@E?e05J?9U(B$\displaystyle
\DP{\bar{p}}{z}=-\bar{\rho}g$$B$,@.$jN)$C$F$$$k$H$9$k(B. $B$3$N7O$G$N;YG[J}Dx(B
$B<0$O(B,
\begin{align}
 \DDD{u}{t}-fv=-\frac{1}{\bar{\rho}}\DP{p}{x},\\
 \DDD{v}{t}+fu=-\frac{1}{\bar{\rho}}\DP{p}{y},\\
 \DDD{w}{t}=-\frac{1}{\bar{\rho}}\DP{p}{z}-\frac{\rho}{\bar{\rho}}g,\label{eq:em3}\\
 \DP{u}{x}+\DP{v}{y}+\DP{w}{z}=0,\\
 \DDD{\theta}{t}=-w\DD{\bar{\theta}}{z}, \label{eq:em5}
\end{align}
$B$G$"$k(B. $B$3$3$G(B, $\displaystyle
\DDD{}{t}\equiv\DP{}{t}+\bvec{v}\cdot\bnabla$$B$G$"$k(B. $B$^$?(B, $\bar{}$$B$O9b$5(B$z$$B$K$*$1$k(B
$BE,Ev$JD9$5$N;~4VJ?6Q$HN.$l$NNN0h$K$o$?$k?eJ?J}8~$NJ?6Q$r$H$C$?NL$rI=$7$F(B
$B$$$k(B. $B$5$i$K(B, $BCO9UN.$N4X78(B, $($B%3%j%*%jNO(B)=($B05NO79EYNO(B)$, $B$,@.N)$7$F$$$k$H$9$k(B:
\begin{align*}
 \begin{cases}
  \displaystyle -f v_g =-\frac{1}{\bar{\rho}}\DP{p}{x}, \\
  \displaystyle f u_g = -\frac{1}{\bar{\rho}}\DP{p}{y},
 \end{cases}
 \hspace{2pt}\Leftrightarrow\hspace{2pt}
 \begin{cases}
  \displaystyle u_g = -\frac{1}{\bar{\rho}f}\DP{p}{y},\\
  \displaystyle v_g = \frac{1}{\bar{\rho}f}\DP{p}{x},
 \end{cases}\\
 \hspace{2pt}\Leftrightarrow\hspace{2pt}
 \bvec{v}_g=\frac{1}{\bar{\rho}f}\bvec{k}\times\bnabla_h p.
\end{align*}
$B$3$3$G(B,  
$a_{tot}(x,y,z,t)=\bar{a}(z)+a(x,y,z,t)$$B$G$"$k(B\footnote{
$B%V%7%M%9%/N.BN$G$O(B, $BL)EY$NJ?6QNL(B$\bar{\rho}$$B$O(B$z$$B$NH!?t$G$O$J$/(B, $BDj?t$H(B
$B$7$F07$&(B.
}. $B$3$3$G(B, $\displaystyle\bnabla_h\equiv\bvec{i}\DP{}{x}+\bvec{j}\DP{}{y}$ $B$G(B
$B$"$k(B. 

\section{$B=`Hw(B}

$BCO9UN.E*AjBP12EY$O(B
\begin{equation}
 \zeta_g\equiv\DP{v_g}{x}-\DP{u_g}{y}=\frac{1}{\bar{\rho}f}\bnabla_h^2 p
\end{equation}
$B$GDj5A$5$l$k(B. $B<!$K(B, $B%9%1!<%k2r@O$r9T$&(B. $B$3$3$G(B, $B?eJ?J}8~B.EY$H?eJ?J}8~%9(B
$B%1!<%kBeI=E*$JBg$-$5$r$=$l$>$l(B $U, L$ $B$H$*$/(B. $B$3$N$H$-(B, $B;~4V$NBeI=E*$JBg(B
$B$-$5$O(B $L/U$ $B$H$J$k(B. $B$=$3$G(B,
\begin{equation}
 \DP{}{t}+u\DP{}{x}+v\DP{}{y}\sim\frac{U}{L}.
\end{equation}
$B0\N.9`$N1tD>J}8~@.J,$K$D$$$F$O(B, $w\sim Ro\frac{H}{L}U$\footnote{
$B1tD>N.$H1tD>%9%1!<%k$NBeI=E*$JBg$-$5$r$=$l$>$l(B $W, H$ $B$H$9$k(B. $B$3$N$H$-(B,
$BO"B3$N<0(B
\begin{equation*}
 \underbrace{\DP{u}{x}+\DP{v}{y}}_{\sim\frac{U}{L}}+\underbrace{\DP{w}{z}}_{\sim\frac{W}{H}}=0 
\end{equation*}
$B$+$i(B, $BC1=c$K$O(B$\displaystyle W\sim \frac{H}{L}U$ $B$G$"$k$,(B, $B?eJ?N.$OHsH/(B
$B;6$G$"$k$3$H$+$i(B, $\displaystyle \DP{u}{x}+\DP{v}{y}$ $B$O6K$a$F>.$5$$$?$a(B,
$BBh(B 1 $B9`$HBh(B 2 $B9`$NOB$NBg$-$5$O(B $\displaystyle \frac{U}{L}$ $B$K>.$5$JNL$N(B
$B%m%9%S!<?t(B $\displaystyle Ro = \frac{U}{fL}$ $B$r3]$1$k$3$H$K$h$C$FBg$-$5$r%P%i%s%9$5$;$F$$$k(B.
}$B$h$j(B,
\begin{equation}
 w\DP{}{z}\sim\frac{U}{L}Ro.
\end{equation}
$B$5$i$K(B,$B?eJ?J}8~$N1?F0$rCO9UN.$K$h$kItJ,$H$=$l$+$i$N%:%l$NItJ,$H$KJ,$1$k(B: 
\begin{equation}
 \bvec{v}_h = \bvec{v}_g + Ro\,\bvec{v}_a. \label{eq:separation}
\end{equation}
$B$h$C$F(B, $Ro$ $B$N:GDc<!$N%*!<%@!<$G(B Lagrange $BHyJ,$O(B
\begin{equation}
 \DDD{}{t} = \left(\DDD{}{t}\right)_g
  \equiv\DP{}{t}+\bvec{v}_g\cdot\bnabla_h =\DP{}{t}+u_g\DP{}{x}+v_g\DP{}{y} \label{eq:lag}
\end{equation}
$B$H=q$1$k(B.

$B1tD>J}8~$K$O(B, $B4p=`>uBV$+$i$N@]F0$b$^$?@E?e05J?9U$G$"$k$H2>Dj$9$k(B. $B%V%7%M(B
$B%9%/N.BN$G$O(B, $BL)EY$O290L$NLr3d$r2L$?$9$N$G(B, (\ref{eq:em3}) $B$O(B
\begin{equation}
 \DP{p}{z}=\frac{\bar{\rho}\theta}{\bar{\theta}}g \label{eq:bq_onni}
\end{equation}
$B$H$J$k(B\footnote{
$B290L$HL)EY$N4X78$O(B, $BJ?6QNL$KBP$7$F$O(B$\bar{\theta}=c\bar{\rho}$$B$G$"$k$,(B,
$B@]F0$KBP$7$F$O(B$\theta=-c\rho$$B$G$"$k(B. $B$3$3$G(B, $c$$B$O(B$\si{K\per
(kg.m^{-3})}$$B$N<!85$r;}$D@5$NDj?t$G$"$k(B. $B$3$l$i$N4X78<0$NId9f$K$D$$$F$O(B,
$\bar{\theta},\bar{\rho}$$B$O@5DjCM$G$"$k$N$KBP$7$F(B, $BJ?6QNL$K$OId9f$,$D$+(B
$B$J$$(B. $B@]F0$K4X$7$F$O(B, $B<~0O$h$j7Z$$%Q!<%;%k(B($\bar{\rho}<0$)$B$O(B, $B<~0O$h$j29(B
$B$+$$(B($\bar{\theta}>0$)$B$N$G(B, $BIiId9f$,$D$/(B.
}.

$B$^$?(B, $BCO9UN.$N4X78<0$r(B $z$ $B$GHyJ,$7(B, (\ref{eq:bq_onni}) $B$rMQ$$$k$H(B,
\begin{align}
 \begin{split}
  \DP{\bvec{v}_g}{z} &= \frac{1}{\bar{\rho}f}\bvec{k}\times\bnabla_h\DP{p}{z}\\
  &=
  \frac{1}{\bar{\rho}f}\bvec{k}\times\bnabla_h\left(\frac{\bar{\rho}\theta}{\bar{\theta}}g\right) \\
  &= \frac{g}{f\bar{\theta}}\bvec{k}\times\bnabla_h \theta \label{eq:hydro}
 \end{split} 
\end{align}
$B$G$"$k(B.

$B=`CO9U$K$*$1$kG.NO3X%(%M%k%.!<J}Dx<0$O(B, (\ref{eq:em5}) $B$K(B (\ref{eq:lag})
$B$rE,MQ$7$?(B
\begin{equation}
 \left(\DDD{\theta}{t}\right)_g =-w\DD{\bar{\theta}}{z}
\end{equation}
$B$G$"$k(B. $B$3$N<0$O(B
\begin{equation}
 N=\left[\frac{g}{\bar{\theta}}\DD{\bar{\theta}}{z}\right]^{1/2}
\end{equation}
$B$rMQ$$$k$H(B,
\begin{equation}
 \left(\DDD{\theta}{t}\right)_g =-w\frac{\bar{\theta}}{g}N^2
\end{equation}
$B$H=q$1$k(B.

$B0J>e(B, $B$^$H$a$k$H(B,
\begin{align}
 \left(\DDD{u}{t}\right)_g-fv=-\frac{1}{\bar{\rho}}\DP{p}{x}\label{eq:QGeq-1}\\
 \left(\DDD{v}{t}\right)_g+fu=-\frac{1}{\bar{\rho}}\DP{p}{y}\label{eq:QGeq-2}\\
 \DP{p}{z}=\bar{\rho}\frac{\theta}{\bar{\theta}}g \\
 \DP{u}{x}+\DP{v}{y}+\DP{w}{z}=0 \label{eq:QGeq-4}\\
 \left(\DDD{\theta}{t}\right)_g=-w\frac{\bar{\theta}}{g}N^2\label{eq:QGeq-5}
\end{align}
$B$G$"$k(B.

$B$3$3$G(B, (\ref{eq:QGeq-1})$B$H(B(\ref{eq:QGeq-2})$B$r%Y%/%H%k7A<0$G=q$/$H(B,
\begin{align*}
 \left(\DP{}{t}+\bvec{v}_g\cdot\bnabla_h
 \right)\bvec{v}_g+Ro\left(\DP{}{t}+\bvec{v}_g\cdot\bnabla_h \right)\bvec{v}_a +
 f\bvec{k}\times\bvec{v}_g+f\,Ro\,\bvec{k}\times\bvec{v}_a
 =-\frac{1}{\bar{\rho}}\bnabla_h p.
\end{align*}
$BCO9UN.$N4X78$+$i(B, $\displaystyle f\bvec{k}\times\bvec{v}_g =
-\frac{1}{\bar{\rho}}\bnabla_h p$, $B$5$i$K(B, $B>e$N<0$NBh(B 2 $B9`$O(B$O(Ro)$$BDxEY(B
$B$H8+@Q$b$l$k$N$G(B, $O(Ro)$ $B$NHO0OFb$G(B,
\begin{align*}
 \left(\DP{}{t}+\bvec{v}_g\cdot\bnabla_h
 \right)\bvec{v}_g + f\bvec{k}\times\bvec{v}_a=0
\end{align*}
$B$H$J$k(B. $B7k6I(B, (\ref{eq:QGeq-1})$B$H(B(\ref{eq:QGeq-2}) $B$O(B
\begin{align}
 \DP{u_g}{t}+u_g\DP{u_g}{x}+v_g\DP{u_g}{y}-fv_a=0, \\
 \DP{v_g}{t}+u_g\DP{v_g}{x}+v_g\DP{v_g}{y}+fu_a=0
\end{align}
$B$H$J$k(B. $B$^$?(B, (\ref{eq:QGeq-4}) $B$r(B (\ref{eq:separation}) $B$KBeF~$9$k$H(B,
$BCO9UN.$OHsH/;6$G$"$k$+$i(B,
\begin{equation}
 \DP{u_a}{x}+\DP{v_a}{y}+\DP{w}{z}=0 \label{eq:QGeq2-4}
\end{equation}
$B$rF@$k(B.

$B0J>e$h$j(B, $f$ $BJ?LL>e$NHsG4@-%V%7%M%9%/N.BN$K=`CO9U6a;w$rE,MQ$7$?J}Dx<07O(B
$B$O(B, $B$3$3$^$G$GF3=P$7$?<0$HCO9UN.$N<0$r9g$o$;$?$b$N$H$J$k(B:

\begin{shadebox}
 \begin{align}
  \DP{u_g}{t}+u_g\DP{u_g}{x}+v_g\DP{u_g}{y}-fv_a=0, \label{eq:QVeq2-1} \\
  \DP{v_g}{t}+u_g\DP{v_g}{x}+v_g\DP{v_g}{y}+fu_a=0, \label{eq:QVeq2-2} \\
  \DP{p}{z}=\bar{\rho}\frac{\theta}{\bar{\theta}}g, \label{eq:QGeq-3} \\
  \DP{u_a}{x}+\DP{v_a}{y}+\DP{w}{z}=0, \label{eq:QGeq2-4} \\
  \left(\DDD{\theta}{t}\right)_g=-w\frac{\bar{\theta}}{g}N^2, 
  \label{eq:QGeq-5} \\
  u_g = -\frac{1}{\bar{\rho}f}\DP{p}{y}, \\
  v_g = \frac{1}{\bar{\rho}f}\DP{p}{x}, \label{eq:QGeq-7}
 \end{align}
\end{shadebox}

\section{$B=`CO9U120LJ}Dx<0$NF3=P(B}

$B$3$3$+$i(B, $B120LJ}Dx<0$rF3$/$?$a$K(B, (\ref{eq:QVeq2-2})$B$r(B$x$$B$GJPHy(B
$BJ,$7$?<0$+$i(B(\ref{eq:QVeq2-1})$B$r(B$y$$B$GJPHyJ,$7$?<0$r0z$/$H(B,
\begin{align*}
 & \DP{}{t}\DP{v_g}{x}+\DP{u_g}{x}\DP{v_g}{x}+u_g\DP{}{x}\DP{v_g}{x}+\DP{v_g}{x}\DP{v_g}{y}+v_g\DP{}{y}\DP{v_g}{x}+f\DP{u_a}{x}=0\\
 -\bigr) &
 \underline{\DP{}{t}\DP{u_g}{y}+\DP{u_g}{y}\DP{u_g}{x}+u_g\DP{}{x}\DP{u_g}{y}+\DP{v_g}{y}\DP{u_g}{y}+v_g\DP{}{y}\DP{u_g}{y}-f\DP{v_a}{y}=0} \\
 &\DP{}{t}\left(\DP{v_g}{x}-\DP{u_g}{y}\right)+\left(u_g\DP{}{x}+v_g\DP{}{y}\right)\left(\DP{v_g}{x}-\DP{u_g}{y}\right)+\underbrace{\left(\DP{u_g}{x}+\DP{v_g}{y}\right)}_{\bnabla\cdot\bvec{v}_g
 =0}\left(\DP{v_g}{x}-\DP{u_g}{y}\right)\\
 &+ f\left(\DP{u_a}{x}+\DP{v_a}{y}\right)=0 
\end{align*}
$B$H$J$k(B. $\zeta_g$$B$NDj5A$rMQ$$$F=q$-D>$9$H(B,
\begin{align}
 \left(\DDD{\zeta_g}{t}\right)_g &=
 -f\left(\DP{u_a}{x}+\DP{v_a}{y}\right) \nonumber\\
 &= f\DP{w}{z} \label{eq:voreq_qg}\\
 &=
 -\frac{fg}{N^2}\DP{}{z}\left\{\left(\DDD{}{t}\left(\frac{\theta}{\bar{\theta}}\right)\right)_g\right\}
 \nonumber\\ 
 &= -\frac{fg}{N^2}\left[\left(\DDD{}{t}\right)_g
 \left\{\DP{}{z}\left(\frac{\theta}{\bar{\theta}}\right)\right\}+\DP{\bvec{v}_g}{z}\cdot\bnabla_h
 \left(\frac{\theta}{\bar{\theta}}\right)\right] \nonumber\\
 &= -\left(\DDD{}{t}\right)_g
 \left\{\frac{fg}{N^2}\DP{}{z}\left(\frac{\theta}{\bar{\theta}}\right)\right\}
 \nonumber 
\end{align}
$B$H$J$k(B. $B:G8e$NEy<0$O(B, (\ref{eq:hydro}) $B$rMQ$$$?(B. $B$h$C$F(B,
\begin{equation}
 \left(\DDD{}{t}\right)_g \underbrace{\left\{\zeta_g +
  \frac{fg}{N^2}\DP{}{z}\left(\frac{\theta}{\bar{\theta}}\right)\right\}}_{(*)}=0 \label{eq:QGPV}
\end{equation}
$B$H$J$k(B. $B$3$3$G(B, $(*)$$BItJ,$r=`CO9UN.120L(B(QG PV)$B$H8F$V(B. $BCO9UN.$N4X78$+$i$N(B
$B<0(B
\begin{equation*}
 \zeta_g=\frac{1}{\bar{\rho}f}\bnabla_h^2 p
\end{equation*}
$B$H@E?e05$N<0(B
\begin{equation*}
 \frac{\theta}{\bar{\theta}}=\frac{1}{\bar{\rho} g}\DP{p}{z}
\end{equation*}
$B$r(B(\ref{eq:QGPV})$B$KBeF~$7(B, $BN>JU$K(B$\bar{\rho}f$$B$r$+$1$k$H(B,
\begin{screen}
 \begin{equation}
 \left(\DDD{}{t}\right)_g\left\{\bnabla_h^2 p+\frac{f^2}{N^2}\DP[2]{p}{z}\right\}=0
\end{equation}
\end{screen}
$B$H$J$k(B. $B=`CO9UN.6a;w$K$*$1$kJ}Dx<07O(B(\ref{eq:QVeq2-1})$\sim$(\ref{eq:QGeq-7})$B$O(B
$B>e$N<0$K$^$H$a$i$l$k(B. $B$D$^$j(B, $B05NO(B$p$$B$,$o$+$l$P(B, 7 $B$D$NL$CNJQ?t(B
$u_g,v_g,w,u_a,v_a,\theta,p$ $B$N$&$A(B, p $B0J30$NL$CNJQ?t$rCN$k$3$H(B
$B$,=PMh$k(B.

\section{$B=`CO9U(B 2 $BAX120LJ}Dx<0$NF3=P(B}

$B=`CO9U(B 2 $BAX%b%G%k$K$*$1$kJ}Dx<0$rF3=P$9$k(B. 

\begin{figure}[ht] \label{fig:two_layer}
 \centering 
 \includegraphics[scale=.3,bb=0 0 720 540]{./appendix/2layer.png}
 \caption{2 $BAX%b%G%k$N35G0?^(B.}
\end{figure}

$B?^(B \ref{fig:two_layer} $B$N$h$&$K(B, $BBg5$$r8|$5$NEy$7$$(B 2 $BAX$KJ,$1(B, $B>eAX$NJ*(B
$BM}NL$rAX$NCf?4$N9b$5$GBeI=$7(B, $B2<AX$NJ*M}NL$bAX$NCf?4$GBeI=$9$k(B. $B6-3&>r7o(B
$B$H$7$F>e2<$K$O8GBNJI$r2>Dj$9$k(B. $BE:;z$N?t;zIU$NJ*M}NL$O(B, $B$=$N?t;z$N%l%Y%k(B
$B$NJ*M}NL$rI=$9(B. $B12EYJ}Dx<0(B (\ref{eq:voreq_qg}) ($B0J8e(B, $BE:;z$N(B $g$ $B$O>JN,(B
$B$9$k(B) 
\begin{equation}
 \left(\DP{}{t}+u\DP{}{x}+v\DP{}{y}\right)\zeta = f\DP{w}{z}
\end{equation}
$B$r(B 1 $BAXL\(B($B%l%Y%k(B 1)$B$H(B 2 $BAXL\(B($B%l%Y%k(B 3)$B$K$=$l$>$lE,MQ$9$k(B:
\begin{align}
 &\left(\DP{}{t}+u_1\DP{}{x}+v_1\DP{}{y}\right)\zeta_1 =
 f\frac{w_0-w_2}{\Delta z} = -\frac{f}{\Delta z}w_2, \label{eq:layer1}\\
 &\left(\DP{}{t}+u_3\DP{}{x}+v_3\DP{}{y}\right)\zeta_3 =
 f\frac{w_2-w_4}{\Delta z} = \frac{f}{\Delta z}w_2. \label{eq:layer2}
\end{align}
$B$3$3$G(B, $\displaystyle \DP{w}{z}$ $B$N8+@Q$b$j$K$OCf1{:9J,$rMQ$$$?(B. $B$^$?(B,
$BG.NO3XJ}Dx<0$O%l%Y%k(B 2 $B$KE,MQ$7(B, 
\begin{align}
 &\left(\DP{}{t}+u_2\DP{}{t}+v_2\DP{}{y}\right)\theta_2 =
 -\frac{\bar{\theta}_2}{g}N^2 w_2, \label{eq:theq1}\\
 &\frac{\theta_2}{\bar{\theta}_2}=\frac{1}{\bar{\rho}g}\left(\DP{p}{z}\right)_2
 = \frac{1}{\bar{\rho}g}\frac{p_1-p_3}{\Delta z} \label{eq:theq2}
\end{align}
$B$G$"$k(B. (\ref{eq:theq1}) $B$r(B (\ref{eq:layer1}) $B$KMQ$$$k$H(B,
\begin{align}
 ($B1&JU(B) &=
 \left(\DP{}{t}+u_2\DP{}{x}+v_2\DP{}{y}\right)\left(\frac{f}{\Delta
 z}\frac{g}{N^2}\frac{\theta_2}{\bar{\theta}_2}\right) \nonumber\\
 &=
 \left(\DP{}{t}+\bvec{v}_2\cdot\bnabla\right)\left\{\frac{f}{\bar{\rho}}\frac{1}{(\Delta
 z)^2 N^2}(p_1-p_3)\right\} \label{eq:layer1-2}
\end{align}
$B$H$J$k(B. $BF1MM$K(B, (\ref{eq:theq1}) $B$r(B (\ref{eq:layer2}) $B$KMQ$$$k$H(B,
\begin{align}
 ($B1&JU(B) &=
 -\left(\DP{}{t}+u_2\DP{}{x}+v_2\DP{}{y}\right)\left(\frac{f}{\Delta
 z}\frac{g}{N^2}\frac{\theta_2}{\bar{\theta}_2}\right) \nonumber\\
 &=
 \left(\DP{}{t}+\bvec{v}_2\cdot\bnabla\right)\left\{\frac{f}{\bar{\rho}}\frac{1}{(\Delta
 z)^2 N^2}(p_3-p_1)\right\} \label{eq:layer2-2}
\end{align}
$B$H$J$k(B. $B$3$3$G(B, $BCO9UN.$N4X78(B
\begin{align}
 u=-\frac{1}{\bar{\rho} f}\DP{p}{y}, \hspace{3pt} v=\frac{1}{\bar{\rho} f}\DP{p}{x}
\end{align}
$B$+$i(B, $BN.@~H!?t(B $\Psi$ $B$O(B
\begin{align}
 \Psi \equiv \frac{p}{\bar{\rho} f}
\end{align}
$B$HDj5A$G$-(B,
\begin{align}
 \bvec{v}=\bvec{k}\times\bnabla\Psi
\end{align}
$B$H$J$k(B. $B$3$l$rMQ$$$k$H(B (\ref{eq:layer1-2}) $B$O(B
\begin{align}
 \left(\DP{}{t}+\bvec{v}_2\cdot\bnabla\right)\left\{\frac{f^2}{N^2}\frac{1}{(\Delta
 z)^2}(\Psi_1-\Psi_3)\right\} \label{eq:layer1-3}
\end{align}
$B$H$J$j(B, (\ref{eq:layer2-2}) $B$O(B
\begin{align}
 \left(\DP{}{t}+\bvec{v}_2\cdot\bnabla\right)\left\{\frac{f^2}{N^2}\frac{1}{(\Delta
 z)^2}(\Psi_3-\Psi_1)\right\} \label{eq:layer2-3}
\end{align}
$B$H$J$k(B. $B$3$3$G(B,
\begin{align*}
 \bvec{v}_2 =\frac{\bvec{v}_1+\bvec{v}_3}{2} &=
 \bvec{v}_1+\frac{1}{2}(\bvec{v}_3-\bvec{v}_1) \\
 &= \bvec{v}_3+\frac{1}{2}(\bvec{v}_1-\bvec{v}_3) 
\end{align*}
$B$rF'$^$($k$H(B,
\begin{align*}
 &\bvec{v}_2\cdot\bnabla(\Psi_1-\Psi_3) \\
 =&
 \left\{\bvec{v}_1+\frac{1}{2}(\bvec{v}_3-\bvec{v}_1)\right\}\cdot\bnabla(\Psi_1-\Psi_3) \\
 =&
 \left\{\bvec{v}_1+\frac{1}{2}\{\bvec{k}\times\bnabla(\Psi_3-\Psi_1)\}\right\}\cdot\{\bnabla(\Psi_1-\Psi_3)\} \\
 =& \bvec{v_1}\cdot\bnabla(\Psi_1-\Psi_3) 
\end{align*}
$B$H$J$k(B. $B$h$C$F(B, (\ref{eq:layer1-3}) $B$O(B
\begin{align}
 \left(\DP{}{t}+\bvec{v}_1\cdot\bnabla\right)\left\{\frac{f^2}{N^2}\frac{1}{(\Delta
 z)^2}(\Psi_1-\Psi_3)\right\}
\end{align}
$B$H$J$j(B, (\ref{eq:layer2-3}) $B$OF1MM$K$7$F(B, 
\begin{align}
 \left(\DP{}{t}+\bvec{v}_3\cdot\bnabla\right)\left\{\frac{f^2}{N^2}\frac{1}{(\Delta
 z)^2}(\Psi_3-\Psi_1)\right\} 
\end{align}
$B$H$J$k(B. $B7k6I(B, 2 $BAX%b%G%k$K$*$1$k=`CO9U120LJ}Dx<0$O(B
\begin{align}
 &\left(\DP{}{t}+\bvec{v}_1\cdot\bnabla\right)\left\{\bnabla^2
 \Psi_1 + k_d^2\frac{\Psi_3-\Psi_1}{2}\right\} = 0, \label{eq:2layer-1}\\
 &\left(\DP{}{t}+\bvec{v}_3\cdot\bnabla\right)\left\{\bnabla^2
 \Psi_3 + k_d^2\frac{\Psi_1-\Psi_3}{2}\right\} = 0  \label{eq:2layer-2}
\end{align}
$B$G$"$k(B. $B$3$3$G(B, $\displaystyle k_d^2 \equiv\frac{2f^2}{N^2(\Delta z)^2}$ $B$H(B
$B$7$?(B.

\section{$B;YG[J}Dx<07O$NJQ7A(B}

$B1tD>%7%"!<$rH<$&(B $y$ $BJ}8~$K0lMM$JEl@>J?6QN.$,$"$k$H$9$k(B. $BJ?6QN.$+$i$N%:(B
$B%l$N@.J,$K4X$9$kH/E8J}Dx<0$rF3$/(B. $B$3$N$H$-(B, $BN.$lH!?t$O(B
\begin{align}
 &\Psi_1 = -Uy + \psi_1(x,y,t), \\
 &\Psi_3 = Uy + \psi_3(x,y,t),
\end{align}
$B$HI=8=$G$-$k(B. $B$3$3$G(B, $U$ $B$ODj?t(B. $B$3$l$i$r(B (\ref{eq:2layer-1}) $B$d(B
(\ref{eq:2layer-2}) $B$KBeF~$9$k(B;
\begin{align*}
  &\bnabla^2\Psi_1 + k_d^2\frac{\Psi_3-\Psi_1}{2} \\
 =&\bnabla^2\psi_1 + k_d^2\left(Uy+\frac{\psi_3-\psi_1}{2}\right) \\
 =&\left(\bnabla^2\psi_1+k_d^2\frac{\psi_3-\psi_1}{2}\right)+k_d^2 Uy \\
 =& q_1' + Q_1.
\end{align*}
$B:G8e$NEy<0$G$O(B, $\displaystyle q_1' =
\bnabla^2\psi_1+k_d^2\frac{\psi_3-\psi_1}{2},\, Q_1=k_d^2 Uy$ $B$H$7$?(B. $BF1(B
$BMM$K(B,
\begin{align*}
  &\bnabla^2\Psi_3 + k_d^2\frac{\Psi_1-\Psi_3}{2} \\
 =&\left(\bnabla^2\psi_3+k_d^2\frac{\psi_1-\psi_3}{2}\right)-k_d^2 Uy \\
 =& q_3' + Q_3. 
\end{align*}
$B$^$?(B,
\begin{align*}
 \bvec{v}_1=&\bvec{e}_z\times\bnabla\Psi_1 \\
         =&\bvec{e}_z\times\bnabla\psi_1+U\bvec{i}  \\
         =&U\bvec{i}+\bvec{v}_1', \\
 \bvec{v}_3=&\bvec{e}_z\times\bnabla\Psi_3 \\
         =&\bvec{e}_z\times\bnabla\psi_3-U\bvec{i}  \\
         =&-U\bvec{i}+\bvec{v}_3'.
\end{align*}
$B$G$"$k(B. $B0J>e$h$j(B, (\ref{eq:2layer-1}) $B$O(B
\begin{align}
 &\left(\DP{}{t}+\bvec{v}_1\cdot\bnabla\right)\left\{\bnabla^2\Psi_1 +
 k_d^2\frac{\Psi_3-\Psi_1}{2}\right\} = 0 \nonumber\\ 
 \Leftrightarrow&\left\{\DP{}{t}+\left(U\bvec{i}+\bvec{v}_1'\right)\cdot\bnabla\right\}(q_1'+Q_1)=0
 \nonumber\\ 
 \Leftrightarrow&\left(\DP{}{t}+\bvec{v}_1'\cdot\bnabla\right)q_1'+U\DP{}{x}q_1'+v_1'\DP{}{y}Q_1=0
 \nonumber\\
 \Leftrightarrow&\left(\DP{}{t}+\bvec{v}_1'\cdot\bnabla\right)q_1'+U\DP{}{x}q_1'+k_d^2Uv_1'=0 \label{eq:2layer-rev-1}
\end{align}
$B$H$J$k(B. $BF1MM$K(B, (\ref{eq:2layer-2}) $B$O(B
\begin{align}
 \Leftrightarrow&\left(\DP{}{t}+\bvec{v}_3'\cdot\bnabla\right)q_3'-U\DP{}{x}q_3'-k_d^2Uv_3'=0 \label{eq:2layer-rev-2}
\end{align}
$B$H$J$k(B. $B$3$3$G(B, $B=g05(B(barotropic)$B%b!<%I$NN.@~H!?t$H7905(B(baroclinic)$B%b!<%I(B
$B$NN.@~H!?t$r$=$l$>$l0J2<$N$h$&$KDj5A$9$k(B; 
\begin{align}
 \psi=\frac{\psi_1+\psi_3}{2},\,\tau=\frac{\psi_1-\psi_3}{2}.
\end{align}
$B$3$l$i$rMQ$$$k$H(B, $\bvec{v}_1'=\bvec{v}_{\psi}+\bvec{v}_{\tau}$,
$\bvec{v}_3'=\bvec{v}_{\psi}-\bvec{v}_{\tau}$, $q_1'=q_{\psi}+q_{\tau}$, $q_3'=q_{\psi}-q_{\tau}$ $B$H$J$k(B($B2<(B
$BIU$-J8;z(B $\psi$ $B$O=g05%b!<%I(B, $B2<IU$-J8;z(B $\tau$ $B$O7905%b!<%I$NJ*M}(B
$BNL$rI=$9(B). $B$^$?(B,
\begin{align*}
 &q_{\psi}=\frac{q_1'+q_3'}{2}=\bnabla^2\left(\frac{\psi_1'+\psi_3'}{2}\right)\equiv\bnabla^2\psi,\\
 &q_{\tau}=\frac{q_1'-q_3'}{2}=\bnabla^2\left(\frac{\psi_1'-\psi_3'}{2}\right)+k_d^2\frac{\psi_3'-\psi_1'}{2}\equiv\bnabla^2\tau-k_d^2\tau,\\
 &\bvec{v}_{\psi} = \bvec{e}_{z}\times\bnabla\psi,\,\bvec{v}_{\tau}=\bvec{e}_z \times\bnabla\tau
\end{align*}
$B$G$"$k(B. $B$3$l$i$rBeF~$9$k$H(B, (\ref{eq:2layer-rev-1}) $B$O(B
\begin{align}
 \left\{\DP{}{t}+(\bvec{v}_{\psi}+\bvec{v}_{\tau})\cdot\bnabla\right\}(q_{\psi}+q_{\tau})+U\DP{q_1'}{x}+k_d^2
 Uv_1'=0 \label{eq:2layer-rev2-1}
\end{align}
$B$H$J$j(B, (\ref{eq:2layer-rev-2}) $B$O(B
\begin{align}
 \left\{\DP{}{t}+(\bvec{v}_{\psi}-\bvec{v}_{\tau})\cdot\bnabla\right\}(q_{\psi}-q_{\tau})-U\DP{q_3'}{x}-k_d^2
 Uv_3'=0 \label{eq:2layer-rev2-2}
\end{align}
$B$H$J$k(B.$B=g05%b!<%I$@$1$N<0$K$9$k$?$a(B, 
$\{(\mathrm{\ref{eq:2layer-rev2-1}})+(\mathrm{\ref{eq:2layer-rev2-2}})\}\div 2$$B$r7W;;$9$k(B
$B$H(B,
\begin{align}
 &\DP{q_{\psi}}{t}+(\bvec{v}_{\psi}\cdot\bnabla)q_{\psi}+\underbrace{(\bvec{v}_{\tau}\cdot\bnabla)q_{\tau}}_{J(\tau,q_{\tau})=J(\tau,\bnabla^2\tau)-k_d^2J(\tau,\tau)}+U\DP{}{x}q_{\tau}+k_d^2U
 v_{\tau}=0,
 \nonumber\\
 \Leftrightarrow&\DP{q_{\psi}}{t}+J(\psi,q_{\psi})+J(\tau,\bnabla^2\tau)+U\DP{}{x}q_{\tau}+k_d^2U
 v_{\tau}=0 \label{eq:barotro1}
\end{align}
$B$H$J$k(B. $B$^$?(B, $B7905%b!<%I$@$1$N<0$K$9$k$?$a(B,
$\{(\mathrm{\ref{eq:2layer-rev2-1}})-(\mathrm{\ref{eq:2layer-rev2-2}})\}\div
2$ $B$r7W;;$9$k$H(B, 
\begin{align}
 &\underbrace{\DP{q_{\tau}}{t}+(\bvec{v}_{\psi}\cdot\bnabla)q_{\tau}+(\bvec{v}_{\tau}\cdot\bnabla)q_{\psi}}_{\DP{q_{\tau}}{t}+J(\psi,q_{\tau})+J(\tau,q_{\psi})}+U\DP{}{x}q_{\psi}+k_d^2U
 v_{\psi}=0,
 \nonumber\\
 \Leftrightarrow&\DP{q_{\tau}}{t}+J(\psi,q_{\tau})+J(\tau,q_{\psi})+U\DP{}{x}q_{\psi}+k_d^2U
 v_{\psi}=0 \label{eq:barocli1}
\end{align}
$B$H$J$k(B. (\ref{eq:barotro1}) $B$H(B (\ref{eq:barocli1}) $B$r$=$l$>$l(B $\psi$ $B$H(B
$\tau$ $B$rMQ$$$F=q$/$H(B,
\begin{screen}
 \begin{align}
  &\DP{\bnabla^2\psi}{t}+J(\psi,\bnabla^2\psi)+J(\tau,\bnabla^2\tau)+U\DP{\bnabla^2\tau}{x}+k_d^2
  U\DP{\tau}{x}=0,
  \label{eq:barotro2} \\
  &\DP{(\bnabla^2\tau-k_d^2\tau)}{t}+J(\psi,\bnabla^2\tau-k_d^2\tau)+J(\tau,\bnabla^2\psi)+U\DP{\bnabla^2\psi}{x}+k_d^2U\DP{\psi}{x}=0 \label{eq:barocli2}
 \end{align}
\end{screen}
$B$H$J$k(B. 

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