%  $BI=Bj(B  $B2F$N(B GFD $B%;%_%J!<(B 2006 $B9V1i%N!<%H(B
% 
%  $BMzNr(B  2006/09/07 $B>.9b@5;L(B
%
\documentclass[a4j,12pt]{jarticle}

\usepackage{Dennou6}

\Dtitle[GFD $B%;%_%J!<(B 2006 $B2F(B]{$B0LAj0MB8@-$N$J$$(B \\ $BGH$N3hF0EY%U%i%C%/%9(B}
\Ddate{2006 $BG/(B 9 $B7n(B 5 $BF|(B}

\Dauthor[$B9bC+(B $B9/B@O:(B]{$B9bC+(B $B9/B@O:(B \\ $B=q5-(B: $B>.9b@5;L(B, $B:4!9LZMNJ?(B}

\setlength{\parskip}{1em}

\begin{document}


% $B%?%$%H%k$NI=<((B
\maketitle

\section{$B?eJ?(B 2 $B<!85(B, $\beta$ $BJ?LL(B, $BHsH/;6$N>l9g(B}

\subsection{$B>qMp>l$N<0(B}

$B4pACJ}Dx<0$O(B
\begin{eqnarray}
  && \DP{u}{t} + u\DP{u}{x} + v\DP{u}{y} - fv = -\frac{1}{\rho _{0}}\DP{p}{x},
     \Deqlab{2D QG $B&BLL(B u $B$N<0(B}
     \\
  && \DP{v}{t} + u\DP{v}{x} + v\DP{v}{y} + fu = -\frac{1}{\rho _{0}}\DP{p}{y},
     \Deqlab{2D QG $B&BLL(B v $B$N<0(B}
     \\
  && \DP{u}{x} + \DP{v}{y} =0
     \Deqlab{2D QG $BO"B3$N<0(B}
\end{eqnarray}
$B$G$"$k(B. $B$3$3$G(B $f=f_{0} + \beta y$, $u, v$ $B$ON.@~4X?t(B $\psi$ $B$rMQ$$$F(B
\[
   u = - \DP{\psi}{y}, \quad v = \DP{\psi}{x}
\]
$B$HI=$5$l$k(B. 

$B12EYJ}Dx<0$O(B $\displaystyle \DP{}{x}\Deqref{2D QG $B&BLL(B v $B$N<0(B}-
\DP{}{y}\Deqref{2D QG $B&BLL(B u $B$N<0(B}$ $B$r7W;;$9$k$3$H$GF@$i$l$k(B.
\begin{equation}
  \DP{\zeta}{t} + u\DP{\zeta}{x} + v\DP{\zeta}{y} + \beta v = 0.
    \Deqlab{2D QG $B12EYJ}Dx<0(B}
\end{equation}

$B$3$3$GJ*M}NL$rDj>o$J4pK\>l$H$=$l$h$j$b?6I}$N>.$5$$;~4VJQ2=$9$k>qMp>l$K(B
$BJ,$1$k(B. $B4pK\>l$O(B $y$ $B$K$N$_0MB8$9$k$H2>Dj$9$k(B.
\begin{eqnarray}
  u &=& \overline{u}(y) + u' + u^{(2)} + \cdots , \\
  v &=&                   v' + v^{(2)} + \cdots , \\
  \psi &=& \Psi(y) + \psi' + \psi ^{(2)} + \cdots, \\
  \zeta &=& \overline{\zeta}(y) + \zeta' + \zeta^{(2)} + \cdots 
\end{eqnarray}
$B$3$3$G(B $O(a') = \varepsilon$, $O(a^{n}) = \varepsilon ^{n}$,
$|\varepsilon|\ll 1$ $B$G$"$k(B.

$B$3$l$i$r(B\Deqref{2D QG $B12EYJ}Dx<0(B}$B$KBeF~$7(B, $O(\varepsilon)$ $B$N9`$@$1$r4^$`(B
$B<0$rN)$F$k$H(B, $B0J2<$N<0$,F@$i$l$k(B.
\begin{equation}
  \DP{\zeta'}{t} + \overline{U}\DP{\zeta'}{x} + \beta_{e}v' = 0
    \Deqlab{2D QG $B>qMp>l$N12EYJ}Dx<0(B}
\end{equation}
$B$3$3$G(B, $\displaystyle \beta_{e} = \beta - \DP[2]{\overline{U}}{y}$ $B$G$"$k(B.

\subsection{$B%m%9%S!<GH(B}

$B>qMp>l$N2r$H$7$F(B
\[
  \psi ' = A\sin (kx + ly - \omega t) 
\]
$B$N$h$&$JJ?LLGH2r$r2>Dj$9$k$H(B, $k, l, \omega$ $B$NK~$?$9<0$H$7$F(B,
\begin{equation}
  \omega = \overline{U}k - \frac{\beta _{e}k}{k^{2}+l^{2}}
\end{equation}
$B$,F@$i$l$k(B. $B$3$N$H$-0LAjB.EY$O(B
\begin{eqnarray}
  C_{x} &=& \frac{\omega}{k} \nonumber \\
        &=& \overline{U} 
              - \frac{\beta_{e}}{k^{2}+l^{2}}, \\
  C_{y} &=& \frac{\omega}{l} \nonumber \\
        &=& \left(\overline{U}
              - \frac{\beta _{e}}{k^{2}+l^{2}}\right)\frac{k}{l} 
\end{eqnarray}
$B72B.EY$O(B
\begin{eqnarray}
  C_{gx} &=& \DP{\omega}{k} \nonumber \\
         &=& \overline{U} 
               + \frac{\beta_{e}(k^{2}-l^{2})}{(k^{2}+l^{2})}, \\
  C_{gy} &=& \DP{\omega}{l} \nonumber \\
         &=&  \frac{2\beta_{e} kl}{(k^{2}+l^{2})^{2}}
\end{eqnarray}
$B$HI=$5$l$k(B.  $\beta_{e} >0$ $B$N>l9g(B, $B0LAjB.EY(B $C_{x}$ $B$O4pK\>l$NN.B.(B 
$\overline{U}$ $B$h$j$b>o$KCY$$$N$KBP$7(B, $B72B.EY(B $C_{gx}$ $B$H(B 
$\overline{U}$ $B$N4X78$O(B $k$ $B$H(B $l$ $B$NBg>.4X78$K0MB8$9$k(B.


\subsection{$B%(%s%9%H%m%U%#!<(B}

$B12EY(B $\zeta$ $B$N<+>h$K(B $1/2$ $B$r$+$1$?$b$N$r%(%s%9%H%m%U%#!<$H$$$&(B. $B$3$3(B
$B$G$O>qMp>l$N%(%s%9%H%m%U%#!<$N<0$r5a$a$k(B.

\Deqref{2D QG $B>qMp>l$N12EYJ}Dx<0(B}$B$K(B $\zeta'$ $B$r$+$1$k$H(B,
\[
  \DP{}{t}\frac{\zeta ^{'2}}{2} 
    + \overline{U}\DP{}{x}\frac{\zeta ^{'2}}{2}
    + \beta_{e}v'\zeta' = 0.
\]
$\beta_{e}$ $B$O0lDj$H2>Dj$9$k$H(B, 
\[
  \DP{}{t}\frac{\zeta ^{'2}}{2\beta_{e}} 
    + \overline{U}\DP{}{x}\frac{\zeta ^{'2}}{2\beta_{e}}
    + v'\zeta' = 0.
\]
$B$5$i$KEl@>J?6Q(B
\[
   [A] = \frac{1}{\alpha _{0}}\int _{0}^{\alpha _{0}}A \Dd x
\]
$B$r$H$k$H(B, 
\begin{equation}
  \DP{}{t}\left[\frac{\zeta ^{'2}}{2\beta _{e}}\right] + [v'\zeta'] = 0
    \Deqlab{$BEl@>J?6Q$7$?>qMp$N%(%s%9%H%m%U%#!<$N<0(B}
\end{equation}
$B$,F@$i$l$k(B.

$B>qMp>l$KBP$9$kO"B3$N<0$rMQ$$$k$H(B
\begin{eqnarray*}
  v'\zeta' &=& v\left(\DP{v'}{x} - \DP{u'}{y}\right) \\
           &=& v'\DP{v'}{x} - u'\DP{u'}{x} - u'\DP{v'}{y} - v'\DP{u'}{y} \\
           &=& \DP{}{x}\left(\frac{v^{'2} - u^{'2}}{2}\right)
                 + \DP{}{y}(-u'v')
\end{eqnarray*}
$B$G$"$k(B. $B$7$?$,$C$F(B
\[
  [v'\zeta'] = \DP{}{y}(-[u'v'])
\]
$B$HJQ7A$5$l$k(B. $B$3$l$rMQ$$$k$H(B\Deqref{$BEl@>J?6Q$7$?>qMp$N%(%s%9%H%m%U%#!<(B
$B$N<0(B}$B$O(B
\begin{equation}
  \DP{}{t}\left[\frac{\zeta ^{'2}}{2\beta _{e}}\right] 
    + \DP{}{y}(-[u'v']) = 0
      \Deqlab{$BEl@>J?6Q$7$?>qMp$N%(%s%9%H%m%U%#!<$N<0(B(2)}
\end{equation}
$B$H$J$k(B. $B$3$N<0$O(B $\displaystyle A \equiv [\frac{\zeta ^{'2}}{2\beta
_{e}}]$, $\displaystyle \Dvect{F} \equiv (0, \quad [-u'v'])$ $B$H$*$/$H(B,
\begin{equation}
  \DP{A}{t} + \Ddiv \Dvect{F} = 0
    \Deqlab{$BEl@>J?6Q$7$?>qMp$N%(%s%9%H%m%U%#!<$N<0(B(3)}
\end{equation}
$B$HI=$9$3$H$,$G$-$k(B. $A$ $B$OGH$N3hF0EY(B (wave activity), $\Dvect{F}$ $B$O(B
$BGH$N3hF0EY%U%i%C%/%9(B (wave activity flux) $B$H8F$P$l$k(B.


\subsection{$BJ?LLGH$N>l9g$NGH$N3hF0EY(B}

$B2r$H$7$FJ?LLGH(B
\[
  \psi ' = \psi _{0}\sin (kx + ly - \omega t) 
\]
$B$rM?$($k$H(B, 
\begin{eqnarray}
  A &=& \left[\frac{(k^{2}+l^{2})^{2}\psi _{0}^{2}}{2\beta _{e}}\right] 
        \nonumber \\
    &=& \frac{(k^{2}+l^{2})^{2}\psi _{0}^{2}}{4\beta _{e}}.
\end{eqnarray}
$B$^$?(B, 
\begin{eqnarray}
  [-u'v'] &=& \left[\DP{\psi'}{y}\DP{\psi'}{x}\right] \nonumber \\
          &=& \frac{kl\psi _{0}^{2}}{2} \nonumber \\
          &=& \frac{2\beta _{e}kl}{(k^{2}+l^{2})^{2}}
              \frac{(k^{2}+l^{2})^{2}\psi _{0}^{2}}{4\beta _{e}} \nonumber \\
          &=& C_{gy}A
\end{eqnarray}
$B$HI=$5$l$k(B. $B$7$?$,$C$F(B
\begin{equation}
  \Dvect{F} = (0, \quad C_{gy}A).
\end{equation}


\subsection{2 $B<!$N>qMp>l$N<0(B}

1 $B<!$N>qMp>l$N<0$r5a$a$?>l9g$HF1MM$N<j=g$K1h$C$F(B, 2 $B<!$N>qMp>l$N<0$r5a(B
$B$a$k$H(B,
\begin{equation}
  \DP{u^{(2)}}{t} + \overline{u}\DP{u^{(2)}}{x} 
    + u'\DP{u'}{x} + u^{(2)}\DP{\overline{u}}{x} 
    + v'\DP{u'}{y} + v^{(2)}\DP{\overline{u}}{y} - fv^{(2)} 
    = -\frac{1}{\rho _{0}}\DP{p^{2}}{x}.
\end{equation}
$B$3$l$rEl@>J?6Q$9$k$H(B, 
\begin{equation}
  \DP{[u^{(2)}]}{t} = - \DP{}{y}[u'v'] = \Ddiv \Dvect{F} = [v'\zeta'].
  \Deqlab{$BEl@>J?6Q$7$?(B 2 $B<!$N>qMp$N<0(B}
\end{equation}
$B$3$N<0$H(B\Deqref{$BEl@>J?6Q$7$?>qMp$N%(%s%9%H%m%U%#!<$N<0(B(3)}$B$rAH$_9g$o$;$k$H(B,
\begin{equation}
  \DP{}{t}[u^{(2)} + A ] = 0,
\end{equation}
$B$H$J$k(B. $B$3$l$O(B $A$ $B$,8:>/$9$k$H@>Iw$O2CB.$5$l(B, $A$ $B$,A}2C$9$k$H@>Iw$O(B
$B8:B.$5$l$k$3$H$rI=$7$F$$$k(B. $B$^$?(B, $B$=$l$>$l$N<0$OJ]B87A$K$J$C$F$$$k$N$G(B, 
$B$?$H$($P$"$k(B $y$ $B$K$*$$$F(B $A<0, u^{(2)}>0$ $B$J$i$P(B, $BJL$N>l=j$G$O(B $A>0,
u^{(2)}<0$ $B$H$J$C$F$$$J$1$l$P$J$i$J$$(B. $B$3$N$H$-(B $A<0$ $B$NNN0h$+$i(B $A>0$ 
$B$NNN0h$X(B $A$ $B$,(B $\Dvect{F}$ $B$K$h$C$FM"Aw$5$l$k(B, $B$H9M$($k$3$H$,$G$-$k(B
(\Dfigref{$B?^(B1}).

\begin{figure}[bth]
\begin{center}
\Depsf[][]{figs/Fig1-1.ps}
\caption{}
\Dfiglab{$B?^(B1}
\end{center}
\end{figure}


\subsection{$BJ*M}E*9M;!(B}

\Deqref{$BEl@>J?6Q$7$?>qMp$N%(%s%9%H%m%U%#!<$N<0(B(2)},
\Deqref{$BEl@>J?6Q$7$?(B 2 $B<!$N>qMp$N<0(B}$B$NJ*M}E*$J0UL#$r9M;!$9$k(B.

\Deqref{$BEl@>J?6Q$7$?(B 2 $B<!$N>qMp$N<0(B}$B$O@>Iw1?F0NL%U%i%C%/%9$N<}B+$,$"$l(B
$B$P@>Iw$r2CB.$7(B, $BH/;6$,$"$l$P@>Iw$r8:B.$9$k$3$H$rI=$7$F$$$k(B(\Dfigref{$B?^(B
3}).

\begin{figure}[hbt]
\begin{center}
\Depsf[][]{figs/Fig1-3.ps}
\caption{}
\Dfiglab{$B?^(B3}
\end{center}
\end{figure}

\newpage
\section{$B@.AXN.BN(B}

\subsection{$B=`CO9UN.12EYJ}Dx<0$NF3=P(B}

$B4pACJ}Dx<0$O(B
\begin{eqnarray}
  && \DP{u_g}{t}+u_g\DP{u_g}{x}+v_g\DP{u_g}{y}-f(v_g+v_a)=-\DP{\phi}{x}, 
       \Deqlab{3D QG u $B$N<0(B}
     \\
  && \DP{v_g}{t}+u_g\DP{v_g}{x}+v_g\DP{v_g}{y}+f(u_g+u_a)=-\DP{\phi}{y}, 
       \Deqlab{3D QG v $B$N<0(B}
     \\
  && \DP{\theta}{t}+u_g\DP{\theta}{x}+v_g\DP{\theta}{y}+w_a\DP{\theta_0}{z}=0,
       \Deqlab{3D QG $B290L$N<0(B}
     \\
  && \DP{u_a}{x}+\DP{v_a}{y}+\frac{1}{\rho_0}\DP{\rho_0w_a}{z}=0
       \Deqlab{3D QG $BO"B3$N<0(B}
\end{eqnarray}
$B$G$"$k(B. $B$3$3$G(B $\displaystyle z=H\ln (p/p_s), u_g = - \DP{\phi}{y},
v_g = \DP{\phi}{x}, \theta _0=\theta _0(z), \rho_0 = \rho_s e^{-z/H}$ 
$B$G$"$k(B. $(u_a, v_a)$ $B$OIwB.$NHsCO9UN.@.J,$G(B, $u_g \gg u_a$ $B$G$"$k(B.

$\displaystyle \DP{}{x}\Deqref{3D QG v $B$N<0(B}-
\DP{}{y}\Deqref{3D QG u $B$N<0(B}$ $B$h$j(B, 
\begin{equation}
  \DP{}{t}\zeta _{g} 
    + u_g\DP{\zeta_g}{x}+v_g\DP{\zeta_g}{y}  
    + \beta (v_g+v_a) 
  = \frac{f_0}{\rho_0}(\rho_0w_a)_{z}
\end{equation}
\Deqref{3D QG $B290L$N<0(B}$B$h$j(B
\begin{equation}
  \left(\DP{}{t}+u_g\DP{}{x}+v_g\DP{}{y}\right)
    \left(\rho_0\frac{\theta}{{\theta _0}_{z}}\right)+\rho_0w_a=0
\end{equation}
$B$3$3$G(B
\[
  \theta/{\theta_0}_{z} 
    = (HR^{-1}f_0e^{\kappa z/H}\psi _{z})/{\theta _0}_{z}
    = f_0 \psi _{z}/N^{2}
\]
$B$H$*$/(B($N^{2}=H^{-1}Re^{-\kappa z/H}{\theta_0}_{z}$). $B$3$l$h$j(B, 
\begin{eqnarray}
  && \left(\DP{}{t}+u_g\DP{}{x}+v_g\DP{}{y}\right)q = 0 \\
  && q \equiv f_0 + \beta y + \psi _{xx} + \psi _{yy} + \rho_0^{-1}
       \left(\rho _0\frac{f_0^{2}}{N^{2}}\psi_{z}\right)_{z}
\end{eqnarray}
$B$,F@$i$l$k(B. 

\subsection{$B=`CO9UN.12EYJ}Dx<0$N@~7A2=(B}

$BJ*M}NL$rDj>o$J4pK\>l$H$=$l$h$j$b?6I}$N>.$5$$;~4VJQ2=$9$k>qMp>l$K(B
$BJ,$1$k(B. $B4pK\>l$O(B $x$ $B$K$O0MB8$7$J$$$H2>Dj$9$k(B.
\begin{eqnarray}
  q &=& \overline{Q}(y,z,t) + q' , \\
  u &=& \overline{u}(y,z,t) + u' , \\
  \phi &=& \Phi + \psi '
\end{eqnarray}
$B@~7A2=$5$l$?>qMp$N<0$O(B
\begin{equation}
  \DP{q'}{t} + \overline{U}\DP{q'}{x} + \DP{\overline{Q}}{y}v' = 0
\end{equation}
$B$H$J$k(B. $B$3$l$K(B $q'$ $B$r$+$1$k$H(B
\[
  \DP{}{t}\frac{q^{'2}}{2} 
     + \overline{U}\DP{}{x}\frac{q^{'2}}{2} 
     + \overline{Q}_{y}v'q' = 0.
\]
$B$3$3$GGH$N3hF0EY(B $A$ $B$r(B $A\equiv q^{'2}/2\overline{Q}_{y}$ $B$HDj5A$9$k$H(B,
$B>e<0$O(B,
\begin{equation}
  \DP{}{t}A + \DP{}{x}\overline{U}A + v'q' = 0
\end{equation}
$B$H$J$k(B. $v'q'$ $B$O(B
\begin{eqnarray*}
  v'q' &=& v'\left\{\DP{v'}{x}-\DP{u'}{y}+\frac{1}{\rho_0}\DP{}{z}\left(
             \rho_0\frac{f_0^{2}}{N^{2}}\DP{\psi}{z}\right)\right\} \\
       &=& \DP{}{x}\left(\frac{v^{'2}-u^{'2}}{2}\right) + \DP{}{y}(-u'v')
             + \frac{\psi '_{x}}{\rho_0}\DP{}{z}\left(
             \rho_0\frac{f_0^{2}}{N^{2}}\DP{\psi}{z}\right) \\
       &=& \DP{}{x}\left(\frac{v^{'2}-u^{'2}}{2}\right) 
             + \DP{}{y}(-u'v')
             + \left\{
	         \frac{1}{\rho_0}\DP{}{z}
                 \left(
                   \rho_0\frac{f_0^{2}}{N^{2}}\DP{\psi'}{z}\DP{\psi'}{x}\right)
	              - \DP{}{x}\left(\frac{1}{2}\frac{f_0^{2}}{N^{2}}
                      \left(\DP{\psi '}{z}\right)^{2}
	         \right)
	       \right\} \\
       &=& \DP{}{x}
           \left\{
             \frac{1}{2}
             \left(
               v^{'2}-u^{'2} 
               - \frac{1}{2}\frac{f_0^{2}}{N^{2}}
                 \left(
                   \DP{\psi '}{z}
                 \right)^{2}
             \right)
           \right\} 
           + \DP{}{y}(-u'v')
           + \frac{1}{\rho_0}\DP{}{z}
               \left(
                 \rho_0\frac{f_0^{2}}{N^{2}}\DP{\psi'}{z}\DP{\psi'}{x}
               \right) \\
       &=& \DP{}{x}
           \left\{
             \frac{1}{2}
               \left(
                 v^{'2}-u^{'2} 
                   - \frac{1}{2}\frac{f_0^{2}}{N^{2}}
                     \left(
                       \DP{\psi '}{z}
                     \right)^{2}
               \right)
           \right\} 
           + \DP{}{y}(-u'v')
           + \frac{1}{\rho_0}\DP{}{z}
               \left(
                 \rho_0f_0\frac{v'\theta'}{{\theta_0}_{z}}
               \right)
\end{eqnarray*}
$B$HJQ7A$5$l$k(B. $B1&JU$NBh(B 3 $B9`$N:G8e$NJQ7A$K$O(B, 
\[
  N^{2}=\frac{g}{\theta _{0}}\DD{\theta_0}{z}, \quad
\]
$B$rMQ$$$?(B. $B1&JUBh(B 1 $B9`$N(B $\displaystyle \left(\DP{\psi'}{z}\right)^{2}$ 
$B$N9`$O(B$\displaystyle \DP{\psi'}{z}\propto \theta '$ $B$G$"$k$3$H$+$i(B,
$BM-8z0LCV%(%M%k%.!<$KHfNc$9$kNL$G$"$k$3$H$,$o$+$k(B. $B$=$3$G(B,
$BGH$N%(%M%k%.!<$r(B
\begin{equation}
  e = \frac{1}{2}
        \left( 
          {\psi^{'}}_{x}^{2} + {\psi^{'}}_{y}^{2}
          + \frac{f_0^{2}}{N^{2}}{\psi^{'}}_{z}^{2}
        \right)
\end{equation}
$B$HDj5A$9$k$H(B, 
\begin{equation}
  v'q' = \frac{1}{\rho_0}\Ddiv \rho_0 
    \left(
      \begin{array}{c}
        v^{'2} - e  \\
        - u'v'      \\
        f_0\frac{v'\theta'}{{\theta_0}_{z}}
      \end{array}
    \right).
\end{equation}
$B$h$C$F(B, $BGH$N3hF0EY$NJ]B8$N<0$O(B,
\begin{equation}
  \DP{}{t}A + \frac{1}{\rho_0}\Ddiv \rho_0 
    \left(
      \begin{array}{c}
        \overline{u}A + v^{'2} - e  \\
        - u'v'      \\
        f_0 v'\theta'/{\theta_0}_{z}
      \end{array}
    \right) = 0.
\end{equation}
$B$3$l$rEl@>J?6Q$9$k$H(B, 
\begin{equation}
  \DP{}{t}[A] + \frac{1}{\rho_0}\Ddiv \Dvect{F} = 0
\end{equation}
$B$,F@$i$l$k(B. $B$3$3$G(B $\Dvect{F}$ $B$OGH$N3hF0EY%U%i%C%/%9$G(B
\begin{equation}
  \Dvect{F} \equiv \rho_0 
    \left(
      \begin{array}{c}
        0             \\
        - [u'v']      \\
        f_0[v'\theta']/{\theta_0}_{z}
      \end{array}
    \right)
\end{equation}
$B$HM?$($i$l$k(B.

\subsection{TEM $BJ}Dx<0(B}

$B85$NJ}Dx<0$rEl@>J?6Q$9$k$H(B,
\begin{eqnarray}
  && \DP{[u_g]}{t}+\DP{}{y}[u'v'] - f[v_a]= 0 
     \\
  && \DP{[\theta]}{t}+\DP{}{y}[v'\theta']+\left[w_a\DP{\theta_0}{z}\right]=0,
     \\
  && \DP{v_a}{y}+\frac{1}{\rho_0}\DP{\rho_0w_a}{z}=0.
\end{eqnarray}
$B$3$l$rJQ7A$9$k$H(B Tronsformed Eulerian Mean (TEM) $BJ}Dx<0$,F@$i$l$k(B.
\begin{eqnarray}
  && \DP{[u_g]}{t} - f[v_a^{*}]= \Ddiv \Dvect{F} 
     \\
  && \DP{[\theta]}{t}+\left[w_a^{*}\DP{\theta_0}{z}\right]=0,
     \\
  && \DP{v_a^{*}}{y}+\frac{1}{\rho_0}\DP{\rho_0w_a^{*}}{z}=0.
\end{eqnarray}
$B$3$3$G(B,
\begin{eqnarray}
  v_a^{*} &\equiv & [v_a] - \frac{1}{\rho_0}\DP{}{z}
                    \left(
                      \rho_0\frac{v'\theta'}{{\theta _0}_{z}}
                    \right), \\
  w_a^{*} &\equiv & [w_a] + \DP{}{y}
                    \left[
                       \frac{v'\theta'}{{\theta _0}_{z}}
                    \right]
\end{eqnarray}


\section{$B$5$^$6$^$JGH$N3hF0EY%U%i%C%/%9(B}

\subsection{Trenberth (1986) $B$N%U%i%C%/%9(B}

\subsection{Plum (1986) $B$N%U%i%C%/%9(B}

$\Dvect{F}=\Dvect{C}_{g}A$ $B$H$J$k$3$H$r<($9(B. $B=PH/E@$O(B
\[
  \DP{q'}{t} + \overline{u}\DP{q'}{x} + v'\DP{\overline{Q}}{y} = 0 
\]
$B$G$"$k(B. $B$3$3$G(B,
\begin{eqnarray*}
  q'      &=& \psi'_{xx} 
                + \psi'_{yy} 
                + \frac{1}{\rho _0}\DP{}{z}
                  \left(
                     \rho_0\frac{f_0^{2}}{N^{2}}\DP{\psi}{z}
                  \right), \\
  \rho _0 &=& \rho _s\exp (-z/H)
\end{eqnarray*}
$B$G$"$k(B. $\psi'$ $B$H$7$F1tD>J}8~$N@.AX$N9=B$$rJ?LLGH$N2r(B
\begin{equation}
  \psi ' = \psi _0\sin (kx+ly-\omega t)\exp(-z/H)
  \Deqlab{Plum $B$N%U%i%C%/%9$K$*$1$kJ?LLGH2r(B}
\end{equation}
$B$rM?$($k(B. $B$3$N$H$-(B
\[
  q' = - \left\{
           k^{2}+l^{2}
             + \frac{f_0^{2}}{N^{2}}\left(m^{2}+\frac{1}{4H^{2}}\right)
	 \right\}
\]
$B$HI=$5$l$k(B. $B0J2<$G$O(B $m^{2} \gg 1/4H^{2}$ $B$H$9$k(B. $B$3$l$O1tD>J}8~$K==J,(B
$B$J?t$N@a$,B8:_$9$k$3$H$r0UL#$9$k(B. $BJ,;64X78$O(B,
\[
  \omega = k\overline{u} 
             - \frac{\overline{Q}_{y}k}
                                {k^{2}+l^{2}+\frac{f_0^{2}}{N^{2}}m^{2}}
\]
$B$H$J$k(B. $x$ $BJ}8~$N0LAjB.EY$O(B $\Dvect{k} = (k, l, (f_0/N)m)$ $B$H$9$k$H(B,
\[
  C_x = \overline{u} - \frac{\overline{Q}_{y}k}{\Dvect{k}^{2}}
\]
$B72B.EY$O(B
\begin{eqnarray*}
  C_{gx} &=& \overline{u} + \frac{k^{2}-l^{2}-\frac{f_0^{2}}{N^{2}}m^{2}}
                                  {\Dvect{k}^{4}}\overline{Q}_{y}, \\
  C_{gy} &=& \frac{2kl}{\Dvect{k}^{4}}\overline{Q}_{y}, \\
  C_{gz} &=& \frac{2km\frac{f_0^{2}}{N^{2}}}{\Dvect{k}^{2}}\overline{Q}_{y}
\end{eqnarray*}
$B$H$J$k(B. 

$B$3$l$i$N72B.EY$NI=8=$rMQ$$$F(B Plumb (1986) $B$N%U%i%C%/%9$N3F@.J,$rI=$9(B. 
\begin{eqnarray*}
F_x &=& \overline{u}\overline{A} + \overline{v^{'2}} - \overline{e} \\
    &=& \overline{u}\overline{A} +
         \frac{1}{2}\overline{v^{'2}} - \frac{1}{2}\overline{u^{'2}}-
         \frac{1}{2}\frac{f_0^{2}}{N^{2}}\overline{m^{2}} 
\end{eqnarray*}
\Deqref{Plum $B$N%U%i%C%/%9$K$*$1$kJ?LLGH2r(B}$B$N$h$&$K(B $\psi'$ $B$r$*$/$H(B
$\displaystyle \overline{v^{'2}} = k^{2}\psi_0^{2}/4$ $B$H$J$k$3$H$J$I$r(B
$BMQ$$$k$H(B, $B1&JUBh(B 2 $B9`$O(B
\begin{eqnarray*}
  \frac{1}{2}\overline{v^{'2}} 
    - \frac{1}{2}\overline{u^{'2}}
    - \frac{1}{2}\frac{f_0^{2}}{N^{2}}\overline{m^{2}} 
  &=& 
    \frac{\psi _0^{2}}{4}\left(k^2-l^2-\frac{f_0^{2}}{N^{2}}m^2\right) \\
  &=& 
    \frac{(k^2-l^2-\frac{f_0^{2}}{N^{2}}m^2)\overline{Q}_{y}}{\Dvect{k}^{4}}
      \frac{\Dvect{k}^{4}}{4}\frac{\psi _0^{2}}{\overline{Q}_{y}} \\
  &=& 
    \frac{(k^2-l^2-\frac{f_0^{2}}{N^{2}}m^2)\overline{Q}_{y}}{\Dvect{k}^{4}}
      \overline{A}
\end{eqnarray*}
$B$H$J$k(B. $B$3$l$h$j(B, 
\begin{eqnarray}
F_x &=& \left\{\overline{u}\overline{A} 
          + \frac{(k^2-l^2-\frac{f_0^{2}}{N^{2}}m^2)\overline{Q}_{y}}{\Dvect{k}^{4}}
      \right\}\overline{A} \nonumber \\
    &=& C_{gx}\overline{A}
\end{eqnarray}
$B;D$j$N@.J,$K$D$$$F$bF1MM$K5a$a$k$3$H$,$G$-$k(B.


\subsection{Plumb (1985) $B$N%U%i%C%/%9(B}

$B0LAjB.EY$,Hs>o$K$f$C$/$j$H$7$?DdBZ@->qMp$N>l9g(B, Trenberth (1986) $B$d(B 
Plumb (1986) $B$N%U%i%C%/%9$G$O0LAj0MB8@-$r>C5n$9$k$3$H$,$G$-$J$$(B. $BDdBZ(B
$B@->qMp$X$NE,MQ$rA[Dj$79M$($i$l$?GH$N3hF0EY%U%i%C%/%9$,(B, Plumb (1985) 
$B$N%U%i%C%/%9$G$"$k(B.

Plumb (1985) $B$O(B Plumb (1986) $B$NGH$N3hF0EY$NJ]B8$N<0(B
\[
  \DP{}{t}A + \Ddiv \Dvect{F} = 0 
\]
$B$KBP$7(B, $\Ddiv \Dvect{G}=0$ $B$rK~$?$9$h$&$J%Y%/%H%k(B $\Dvect{G}$ $B$r9M$($k(B.
$B$3$N$H$-(B
\[
  \DP{}{t}A + \Ddiv (\Dvect{F}+\Dvect{G}) = 0 
\]
$B$,$J$j$?$D(B. $\Dvect{G}$ $B$O(B
\begin{equation}
 \Dvect{G} = 
   \left(
     \begin{array}{c}
       \DP[2]{\psi ^{'2}}{y} + \DP{}{z}\frac{f_0^{2}}{N^{2}}\DP{\psi ^{'2}}{z} \\
       -\DP{}{x}\DP{}{y}\psi ^{'2} \\
       -\frac{f_0^{2}}{N^{2}}\DP{}{x}\DP{}{z}\psi ^{'2}
     \end{array}
   \right)
\end{equation}
$B$H$9$k(B. $B$3$N$H$-(B
\begin{equation}
  \Dvect{F} + \Dvect{G} = \frac{1}{2}
    \left(
      \begin{array}{c}
        \psi^{'2}_{x} 
          - \psi'\psi'_{xx}+\psi'q'+\overline{u}q^{'2}/\overline{Q}_{y} \\
        \psi '_{z}\psi'_{y}-\psi'\psi'_{xy} \\
        \frac{f_0^{2}}{N^{2}}(\psi '_{x}\psi '_{z} - \psi '\psi'_{xz})
      \end{array}
   \right)
\end{equation}
$BDdBZ@->qMp$N>l9g$O(B
\[
  \overline{u}\DP{q'}{x} + v'\overline{Q}_{y} = 0
\]
$B$,$[$\@.$jN)$D$H9M$($k(B. $B$5$i$K(B $\overline{u}$ $B$O(B $x$ $B$K0MB8$7$J$$$H$7(B
$B$F(B,
\[
  \DP{}{x}\left(\overline{u}q' + \psi'\overline{Q}_{y}\right) = 0
\]
$B$H$9$k(B. $(\Dvect{F}+\Dvect{G})_{x}$ $B$NBh(B 3 $B9`$HBh(B 4 $B9`$O(B,
\[
  \psi'q'+\overline{u}q^{'2}/\overline{Q}_{y} 
    = \frac{q'}{\overline{Q}_{y}}
        \left(
          \psi'\overline{Q}_{y}+\overline{u}q'
        \right)
\]
$B$HI=$5$l(B, $B$3$N(B $x$ $BHyJ,$O(B 0 $B$J$N$GGH$N3hF0EY$NA}8:$K$O4sM?$7$J$$(B.  $B$=(B
$B$3$G$3$l$i$r(B 0 $B$H$*$-(B, $B$=$N>l9g$N(B $\Dvect{F}+\Dvect{G}$ $B$r(B 
$\Dvect{F}_{s}$ $B$H$*$/$H(B,
\begin{equation}
 \Dvect{F}_{s} = \frac{1}{2}
   \left(
     \begin{array}{c}
       \psi^{'2}_{x} - \psi'\psi'_{xx} \\
       \psi '_{z}\psi'_{y}-\psi'\psi'_{xy} \\
       \frac{f_0^{2}}{N^{2}}(\psi '_{x}\psi '_{z} - \psi '\psi'_{xz})
     \end{array}
   \right)
\end{equation}
$B$HI=$5$l$k(B. $\psi ' \propto \sin (kx+ly+(f_0/N)mz)$ $B$H$9$k$H(B, 
\[
  \Dvect{F}_{s} \approx \frac{\psi_0^{2}}{2}
    \left(
      \begin{array}{c}
        k^{2} \\
        kl \\
        \frac{f_0^{2}}{N^{2}}km
      \end{array}
    \right)
\]
$B$H$J$C$F(B, $B0LAj0MB8@-$r>C5n$9$k$3$H$,$G$-$k(B. 

$B%m%9%S!<GH$N0LAjB.EY$H72B.EY(B
\begin{eqnarray*}
  C_{x} &=& \overline{u} - \frac{\overline{Q}_{y}}{\Dvect{k}^{2}}, \\
  C_{gx}&=& \overline{u} + \frac{k^{2}-l^{2}-\frac{f_0^{2}}{N^{2}}m^{2}}
                                  {\Dvect{k}^{4}}\overline{Q}_{y}
\end{eqnarray*}
$B$rMQ$$$k$H(B, $BDdBZ@->qMp$N>l9g(B $C_{x}\approx 0$ $B$J$N$G(B, 
\[
  \overline{u} = \frac{\overline{Q}_{y}}{\Dvect{k}^{2}}
\]
$B$3$N$H$-(B
\[
  C_{gx} = \frac{2k^{2}\overline{Q}_{y}}{\Dvect{k}^{4}}
\]
$B$H$J$N$G(B, $\Dvect{F}_{s} \neq \Dvect{C}_{g}A$ $B$G$"$k(B. $B$5$i$K(B
$BGH$N3hF0EY(B $A$ $B$O(B
\[
  A = \frac{q^{'2}}{2\overline{Q}_{y}} \propto \sin ^{2}x
\]
$B$H$J$C$F(B, $B0LAj0MB8@-$,>C5n$G$-$J$$(B. $B$=$3$G(B, $B0LAj0MB8@-$,>C5n$G$-$k$h$&(B
\[
  A_{s} \propto A + \frac{e}{\overline{u}} \propto \sin ^{2}x + \cos ^{2}y
\]
$B$H$J$k$h$&$J(B $A_{s}$ $B$rDj5A$9$k$H(B $\Dvect{F}_{s} = \Dvect{C}_{g}A_{s}$ 
$B$,@.$jN)$D(B.

$\psi ' \propto \sin \chi$ $B$H$9$k$H(B, 
\begin{eqnarray*}
  A &\sim& \frac{\Dvect{k}^{4}\psi _0^{2}}{2\overline{Q}_{y}}\sin ^{2}\chi, \\
  e &\sim& \frac{\Dvect{k}^{2}\psi _0^{2}}{2}\cos ^{2}\chi \\
\end{eqnarray*}
$B$G$"$k(B. $B%m%9%S!<GH$N0LAjB.EY(B
\[
  C_{x} = \overline{u} - \frac{\overline{Q}_{y}}{\Dvect{k}^{2}}
\]
$B$rMQ$$$k$H(B, 
\[
  \frac{e}{\overline{u}-C_{x}} \sim  \frac{\Dvect{k}^{4}\psi _0^{2}}
                                     {2\overline{Q}_{y}}\cos ^{2}\chi
\]
$B$HI=$5$l$k(B. $B$h$C$F(B
\[
  \frac{1}{2}\left[A - \frac{e}{\overline{u}-C_{x}}\right] =
  \frac{\Dvect{k}^{4}\psi _0^{2}}{4\overline{Q}_{y}}.
\]


\subsection{$BJ]B8B'$rMQ$$$?(B Plum (1985) $B$N%U%i%C%/%9F3=P(B}

Plum (1985) $B$O(B
\[
   A + \frac{e}{\overline{u}}
\]
$B$J$kNL$rDj5A$9$k$3$H$G(B, $B0LAj0MB8@-$N$J$$GH$N3hF0EY$NJ]B8B'$r5a$a$?(B.
$B$3$3$G$O(B $A$ $B$H(B $e$ $B$NJ]B8$rMQ$$$F(B Plumb (1985) $B$N%U%i%C%/%9$NF3=P$r(B
$B9T$&(B

$B$^$:(B $e/(\overline{u}-C_{x}$ $B$NJ]B8B'$r(B, $B>qMp$N12EYJ}Dx<0$+$iF3$/(B.
\[
  \DP{q'}{t} + \overline{u}\DP{q'}{x} + \overline{Q}_{y}v' = 0
\]
$B$K(B $\psi '$ $B$r$+$1$k$H(B
\begin{eqnarray*}
  \psi' \DP{q'}{t} 
    &=& \psi '\left\{
          \psi '_{xx} 
            + \ psi '_{yy} 
            + \frac{1}{\rho _0}\DP{}{z}
                \left(
                  \frac{f_0^{2}}{N^{2}}\DP{\psi '}{z}
                \right)
             \right\} \\
    &=& \DP{}{x}(\psi '\psi'_{xt}) 
          - \DP{}{t}\left(\frac{\psi ^{'2}_{x}}{2}\right) \\
    & & + \DP{}{y}(\psi '\psi'_{yt}) 
        - \DP{}{t}\left(\frac{\psi ^{'2}_{y}}{2}\right) \\
    & & + \frac{1}{\rho_{0}}
            \DP{}{z}\left(\rho_{0}\frac{f_0^{2}}{N^{2}}\psi'\psi'_{zt}\right)
        - \DP{}{t}\left(\frac{1}{2}\frac{f_0^{2}}{N^{2}}\psi^{'2}_{z}\right) \\
    &=& \frac{1}{\rho_{0}}\Ddiv
          \left(
             \begin{array}{c}
                \rho_0 \psi '\psi'_{xt} \\
                \rho_0 \psi '\psi'_{yt} \\
                \rho_{0}\frac{f_0^{2}}{N^{2}}\psi'\psi'_{zt}
             \end{array}
          \right) 
        - \DP{}{t}e
\end{eqnarray*}
$B$H$J$k$3$H$J$I$rMQ$$$k$H(B, 
\[
  \frac{1}{\rho_0}\Ddiv \rho_0 
    \left(
      \begin{array}{c}
        \psi '\psi'_{xt} \\
        \psi '\psi'_{yt} \\
        \frac{f_0^{2}}{N^{2}}\psi'\psi'_{zt}
      \end{array}
    \right) 
  - \DP{}{t}e
  + \frac{\overline{u}}{\rho_0}\Ddiv \rho_0 
    \left(
      \begin{array}{c}
         \psi '\psi'_{xx} \\
         \psi '\psi'_{xy} \\
	 \frac{f_0^{2}}{N^{2}}\psi'\psi'_{xz}
      \end{array}
    \right)
  -  \overline{u}\DP{}{x}e 
  + \frac{1}{2}\DP{}{x}(\overline{Q}_{y}\psi ^{'2}) = 0.
\]
$B:8JUBh(B 1 $B9`$HBh(B 3 $B9`$NH/;6$r$H$k%Y%/%H%k$r$=$l$>$l(B $\Dvect{R}_{1},
\Dvect{R}_{2}$ $B$H$*$-(B, $ - 1/(\overline{u}-C_x)$ $B$r$+$1$k$H(B,
\[
  \DP{}{t}\frac{e}{\overline{u}-C_x} 
    + \overline{u}\DP{}{x}\frac{e}{\overline{u}-C_x} 
    - \frac{1}{\rho_0}\frac{1}{\overline{u}-C_x}\Ddiv \Dvect{R}_{1}
    - \frac{\overline{u}}{\rho_0(\overline{u}-C_x)}\Ddiv \Dvect{R}_{2}
    - \DP{}{x}\left(\frac{\overline{Q}_{y}}{2(\overline{u}-C_x)}\psi ^{'2}\right)
   =0.
\]
$B$5$i$KJQ7A$7$F(B
\[
  \DP{}{t}\frac{e}{\overline{u}-C_x} 
    + C_{x}\DP{}{x}\frac{e}{\overline{u}-C_x} 
    + \DP{}{x}e
    - \frac{1}{\rho_0}\frac{1}{\overline{u}-C_x}
        \Ddiv (\Dvect{R}_{1} + C_x \Dvect{R}_{2})
    - \frac{1}{\rho_0}\Ddiv \Dvect{R}_{2}
    - \DP{}{x}\left(\frac{\overline{Q}_{y}}{2(\overline{u}-C_x)}\psi ^{'2}\right)
    = 0.
\]

$B0lJ}$G(B, $A$ $B$NJ]B8B'$O(B
\[
  \DP{A}{t} 
    + C_{x}\DP{A}{x} 
    + \DP{}{x}\frac{(\overline{u}-C_{x})q^{'2}}{2\overline{Q}_y} 
    + \frac{1}{\rho_0}\Ddiv \rho_0 
      \left(
         \begin{array}{c}
	   v^{'2} - e \\
	   -u'v' \\
	   \frac{f_0^{2}}{N^{2}}\psi'_{x}\psi'_{z}
	 \end{array}
      \right)
    = 0.
\]

$BDdBZ@->qMp$K$D$$$F$O(B,
\begin{eqnarray*}
 && \DP{A}{t} 
      + C_{x}\DP{A}{x} 
      + \DP{}{x}\frac{(\overline{u}-C_{x})q^{'2}}{2\overline{Q}_y} 
      + \frac{1}{\rho_0}\Ddiv \rho_0 
        \left(
	  \begin{array}{c}
	    v^{'2} - e \\
	    -u'v' \\
	    \frac{f_0^{2}}{N^{2}}\psi'_{x}\psi'_{z}
	  \end{array}
	\right)
      = 0 , \\
 &&
    \DP{}{t}\frac{e}{\overline{u}-C_x} 
      + C_{x}\DP{}{x}\frac{e}{\overline{u}-C_x} 
      + \frac{1}{\rho_0}\Ddiv \rho _0
      \left(
        \begin{array}{c}
	  - \psi '\psi'_{xx} \\
	  - \psi '\psi'_{xy} \\
	  - \frac{f_0^{2}}{N^{2}}\psi'\psi'_{xz}
	\end{array}
      \right)
      + \DP{}{x}e
      - \DP{}{x}\left(\frac{\overline{Q}_{y}}{2(\overline{u}-C_x)}\psi ^{'2}\right) \\
 &&
    - \frac{1}{\rho_0}\frac{1}{\overline{u}-C_x}
      \Ddiv (\Dvect{R}_{1} + C_x \Dvect{R}_{2})
    = 0
\end{eqnarray*}
$B$H$J$k(B. $B$3$3$G(B
\[
  M \equiv \frac{1}{2}\left(A + \frac{e}{\overline{u}-C_x}\right)
\]
$B$J$kNL$rDj5A$9$k$H(B, 
\begin{eqnarray*}
  && \DP{M}{t} 
       + C_x\DP{M}{x}
       + \frac{1}{\rho_0}\Ddiv \rho _0
       \left(
         \begin{array}{c}
	   \psi ^{'2}_{x}- \psi '\psi'_{xx} \\
	   \psi '_{x}\psi '_{y}- \psi '\psi'_{xy} \\
	   \frac{f_0^{2}}{N^{2}}(\psi '_{x}\psi'_{z}-\psi'\psi'_{xz})
	 \end{array}
       \right)
       + \DP{}{x}
         \left(
           \frac{(\overline{u}-C_{x})q^{'2}}{4\overline{Q}_y} 
	   - \frac{\overline{Q}_{y}}{4(\overline{u}-C_x)}\psi ^{'2}
	 \right) \\
  &&   - \frac{1}{\rho_0}\frac{1}{\overline{u}-C_x}\Ddiv (\Dvect{R}_{1} + C_x
     \Dvect{R}_{2})
= 0
\end{eqnarray*}
$B$3$3$GDdBZ@->qMp$G$"$k$3$H$+$i(B, 
\[
  (\overline{u}-C_x)\DP{q'}{x} + v'\overline{Q}_{y} = 0.
\]
\[
  \DP{}{x}\left\{(\overline{u}-C_x)q' + \psi'\overline{Q}_{y}\right\} = 0
\]
$B$rMQ$$$F(B, 
\begin{eqnarray*}
 \DP{}{x}\left(
  \frac{(\overline{u}-C_{x})q^{'2}}{4\overline{Q}_y} 
- \frac{\overline{Q}_{y}}{4(\overline{u}-C_x)}\psi ^{'2}\right)
 &=& 
 \DP{}{x}\left\{
  \frac{\overline{Q}_{y}}{4(\overline{u}-C_{x})} 
  \left(\frac{(\overline{u}-C_{x})^{2}}{(\overline{Q}_{y})^{2}}q^{'2}
  - \psi ^{'2}\right)
  \right\} \\
 &=&
 \DP{}{x}\left\{
  \frac{\overline{Q}_{y}}{4(\overline{u}-C_{x})} 
  \left(\frac{\overline{u}-C_{x}}{\overline{Q}_{y}}q' + \psi '\right)
  \left(\frac{\overline{u}-C_{x}}{\overline{Q}_{y}}q' - \psi '\right)
  \right\} \\
 &=& 0.
\end{eqnarray*}
$B$H$J$k(B
\footnote{
$BK\Ev$O(B
\[
  (\overline{u}-C_x)q' + \psi'\overline{Q}_{y} = Const. 
\]
$B$G$"$k(B. $BE,Ev$J6-3&>r7o$r$H$l$P$3$NCM$r(B 0 $B$H$9$k$3$H$,$G$-$k$@$m$&(B.
}
. $B$5$i$K(B
\[
  \frac{1}{\rho_0}\frac{1}{\overline{u}-C_x}\Ddiv (\Dvect{R}_{1} + C_x
    \Dvect{R}_{2}) = 
  \left(\DP{}{t}+C_x\DP{}{x}\right)e 
  + \psi '\left(\DP{}{t}+C_x\DP{}{x}\right)q'
\]
$B$G$"$j(B, $\psi '=\psi _0\sin \chi$ $B$J>qMp$K$D$$$F$O(B 
$\displaystyle \DP{}{t} \sim -i\omega \sim -iC_{x}k \sim -C_{x}\DP{}{x}$
$B$H9M$($k$3$H$,$G$-$k$N$G(B, $B$3$NItJ,$O(B 0 $B$H$9$k(B.
$B0J>e$h$j(B
\begin{equation}
  \DP{M}{t} 
+ C_x\DP{M}{x}
+ \frac{1}{\rho_0}\Ddiv \rho _0
 \left(
 \begin{array}{c}
  \psi ^{'2}_{x}- \psi '\psi'_{xx} \\
  \psi '_{x}\psi '_{y}- \psi '\psi'_{xy} \\
  \frac{f_0^{2}}{N^{2}}(\psi '_{x}\psi'_{z}-\psi'\psi'_{xz})
 \end{array}
 \right)
= 0
\end{equation}
$B$H$J$C$F(B, Plumb (1985) $B$NGH$N3hF0EY$NJ]B8$N<0$rF@$k$3$H$,$G$-$k(B.

\[
\Dvect{W} \equiv \left(
 \begin{array}{c}
  \psi ^{'2}_{x}- \psi '\psi'_{xx} \\
  \psi '_{x}\psi '_{y}- \psi '\psi'_{xy} \\
  \frac{f_0^{2}}{N^{2}}(\psi '_{x}\psi'_{z}-\psi'\psi'_{xz})
 \end{array}
 \right)
\]
$B$H$*$/$H(B,
\begin{equation}
  \DP{M}{t} + \DP{}{x}C_{x}M + \Ddiv \Dvect{W}=0
  \Deqlab{$BJ]B8B'$+$i$b$H$a$?(B Plumb (1985) $B$NGH$N3hF0EYJ]B8$N<0(B}
\end{equation}
$B$^$?(B
\[
\Dvect{W}_{T} \equiv \left(
 \begin{array}{c}
  CM + \psi ^{'2}_{x}- \psi '\psi'_{xx} \\
  \psi '_{x}\psi '_{y}- \psi '\psi'_{xy} \\
  \frac{f_0^{2}}{N^{2}}(\psi '_{x}\psi'_{z}-\psi'\psi'_{xz})
 \end{array}
 \right)
= \Dvect{C}_{g}M
\]
$B$H$J$k(B.

Plumb (1985) $B$NF3=P$h$j$b$h$$E@$O(B, 2 $B$D$NJ]B8B'$rAH$_9g$o$;$F5a$a$?$3(B
$B$H$K$h$jF3=P$N8+DL$7$,$h$$$3$H$H(B, $C_x\neq 0$ $B$G$J$$>qMp$KBP$7$F$bMQ$$(B
$B$k$3$H$,$G$-$k$3$H$G$"$k(B.


\newpage
\section{$BEl@>Hs0lMM$J>l$KBP$9$kGH$N3hF0EY%U%i%C%/%9(B}

\Deqref{$BJ]B8B'$+$i$b$H$a$?(B Plumb (1985) $B$NGH$N3hF0EYJ]B8$N<0(B}$B$rEl@>Hs0lMM(B
$B$J4pK\>l$r;}$D>l9g$KBP$73HD%$9$k(B. $B4pK\N.$r(B $\overline{\Dvect{u}}=(U, V, 0)$
$B$H$7(B, $B4pK\>l$OHs>o$K$f$C$/$j$HJQ2=$9$k$H2>Dj$9$k(B. $B4pK\>l$NN.B.$K1h$C$?(B
$B0LAjB.EY$r(B $C_p$ $B$H$9$k(B.

\begin{figure}[h]
\begin{center}
\Depsf[][]{figs/Fig4-1.ps}
\caption{$\overline{u}$ $B$H(B $C_p$}
\Dfiglab{$B?^(B1}
\end{center}
\end{figure}

$B4pK\>l$O(B
\[
  \overline{u}\DP{\overline{q}}{x} + \overline{v}\DP{\overline{q}}{y}=0
\]
$B$rK~$?$9$H$9$k(B.

\begin{figure}[h]
\begin{center}
\Depsf[][]{figs/Fig4-2.ps}
\caption{$B4pK\>l$N%]%F%s%7%c%k12EY$HN.$l$N>l(B}
\Dfiglab{$B?^(B1}
\end{center}
\end{figure}

$B@~7A2=$5$l$?%]%F%s%7%c%k12EYJ]B8B'$O(B
\begin{equation}
  \DP{q'}{t}+U\DP{q'}{x}+V\DP{q'}{y}+u'\DP{\overline{Q}}{x}+v'\DP{\overline{Q}}{y}=0
  \Deqlab{$BEl@>Hs0lMM4pK\>l$N@~7A2=$5$l$?%]%F%s%7%c%k12EYJ]B8B'(B}
\end{equation}
$B$G$"$k(B. $BGH$N3hF0EY(B $A$ $B$O(B
\[
  A = \frac{q^{'2}}{2|\Dgrad _h\overline{Q}|} 
\]
$B$HDj5A$9$k(B. $B$3$3$G(B
\[
  \Dgrad _h\overline{Q}| = \sqrt{\left(\DP{\overline{Q}}{x}\right)^{2} + 
                           \left(\DP{\overline{Q}}{y}\right)^{2}}
\]
$B$G$"$k(B. $B$^$?(B $e/(\overline{u}-C_x)$ $B$KBP1~$9$kNL$H$7$F(B
\[
  {\cal E} = \frac{e}{|\overline{u}|- C_p}
\]
$B$rMQ0U$7(B,
\[
  M =\frac{1}{2}(A + {\cal E})
\]
$B$N6a;wE*$JJ]B8B'$r5a$a$k(B.

\subsection{$A$ $B$N<0(B}

\Deqref{$BEl@>Hs0lMM4pK\>l$N@~7A2=$5$l$?%]%F%s%7%c%k12EYJ]B8B'(B}$B$K(B
$\displaystyle \frac{q^{'2}}{|\Dgrad _h\overline{Q}|}$ $B$r$+$1$k$H(B
\[
  \frac{1}{|\Dgrad _h\overline{Q}|}\left(
  \DP{}{t}+U\DP{}{x}+V\DP{}{y}\right)\frac{q^{'2}}{2}
  + 
  \frac{1}{|\Dgrad _h\overline{Q}|}\left(
   +u'q'\DP{\overline{Q}}{x}+v'q'\DP{\overline{Q}}{y}  
  \right)
  =0.
\]
$B:8JUBh(B 1 $B9`$K$D$$$F$O(B
\begin{eqnarray*}
  \DP{}{t}\frac{q^{'2}}{2|\Dgrad _h\overline{Q}|} &=& \DP{}{t}A, \\
  \frac{1}{|\Dgrad _h\overline{Q}|}\left(
  U\DP{}{x}+V\DP{}{y}\right)\frac{q^{'2}}{2} &=&
  \frac{1}{|\Dgrad _h\overline{Q}|}\left(
  \DP{}{x}(Uq^{'2})+\DP{}{y}(Vq^{'2}\right) \\
  &\sim& \DP{}{x}(UA) + \DP{}{y}(VA)
\end{eqnarray*}
$BBh(B 2, $BBh(B 3 $B9`$K$D$$$F$O(B, $B$3$l$i$N6a;w$G$O4pK\>l$NJQ2=$OHs>o$K$f$C$/$j(B
$B$H$7$F$$$k$3$H$r2>Dj$7(B,
\begin{eqnarray*}
  \frac{1}{|\Dgrad _h\overline{Q}|}\left(
   +u'q'\DP{\overline{Q}}{x}+v'q'\DP{\overline{Q}}{y}  
  \right)
  &=&
  \frac{\overline{Q}_x}{\rho_0|\Dgrad _h \overline{Q}|}  
    \Ddiv \rho _0 \left(
  \begin{array}{c}
    u'v' \\
    e - u ^{'2} \\
    - \frac{f_0^{2}}{N^{2}}\psi '_{y}\psi '_{z}
  \end{array}
  \right) \\
  &&
  + 
  \frac{\overline{Q}_y}{\rho_0|\Dgrad _h \overline{Q}|}  
    \Ddiv \rho _0 \left(
  \begin{array}{c}
    v^{'2} - e \\
    - u'v' \\
    \frac{f_0^{2}}{N^{2}}\psi '_{x}\psi '_{z}
  \end{array}
  \right)
  \\ 
  &\approx&
  \frac{1}{\rho_0}\Ddiv \frac{\rho _0}{|\Dgrad _h\overline{Q}|}
    \left(
  \begin{array}{c}
    \overline{Q}_{y}(v^{'2}-e) + \overline{Q}_{x}u'v' \\
    \overline{Q}_{y}(-u'v') + \overline{Q}_{x}(e - u ^{'2}) \\
    \overline{Q}_{y} \frac{f_0^{2}}{N^{2}}\psi '_{x}\psi '_{z}
    - \overline{Q}_{x}\frac{f_0^{2}}{N^{2}}\psi '_{y}\psi '_{z}
  \end{array}
  \right) \\
  &\approx&
  \frac{1}{\rho_0}\Ddiv \frac{\rho_0}{|\overline{\Dvect{u}}|}
    \left(
  \begin{array}{c}
    U(v^{'2}-e) + Vu'v' \\
    Uu'v' + V(e - u ^{'2}) \\
    U\frac{f_0^{2}}{N^{2}}\psi '_{x}\psi '_{z}
    + V\frac{f_0^{2}}{N^{2}}\psi '_{y}\psi '_{z}
  \end{array}
  \right).
\end{eqnarray*}
$B$H$9$k(B. $B:G8e$N6a;w$G$O(B
\[
 \frac{\overline{Q}_y}{|\Dgrad _h\overline{Q}|} \approx \frac{U}{|\overline{\Dvect{u}}|}, 
    \quad
 \frac{\overline{Q}_x}{|\Dgrad _h\overline{Q}|} \approx - \frac{V}{|\overline{\Dvect{u}}|}
\]
$B$H$7$?(B.  

$B0J>e$h$j(B
\begin{equation}
  \DP{}{t}A + \Ddiv \left(
  \begin{array}{c}
    UA \\
    VA \\
    0
  \end{array}
  \right) 
  +
  \frac{1}{\rho_0}\Ddiv \frac{\rho_0}{|\overline{\Dvect{u}}|}
    \left(
  \begin{array}{c}
    U(v^{'2}-e) + Vu'v' \\
    Uu'v' + V(e - u ^{'2}) \\
    U\frac{f_0^{2}}{N^{2}}\psi '_{x}\psi '_{z}
    - V\frac{f_0^{2}}{N^{2}}\psi '_{y}\psi '_{z}
  \end{array}
  \right).
\end{equation}

$B$5$i$K(B
\begin{eqnarray*}
  \Ddiv \left(
  \begin{array}{c}
    UA \\
    VA \\
    0
  \end{array}
  \right) 
  &=& \DP{}{x}\left(\frac{|\overline{\Dvect{u}}|-C_p}{|\overline{\Dvect{u}}|}U +
                  \frac{C_pU}{|\overline{\Dvect{u}}|}A
    \right) +
   \DP{}{y}\left(\frac{|\overline{\Dvect{u}}|-C_p}{|\overline{\Dvect{u}}|}V +
                  \frac{C_pV}{|\overline{\Dvect{u}}|}A
    \right) \\
  &=&
   \Ddiv
   \left(
  \begin{array}{c}
    \frac{|\overline{\Dvect{u}}|-C_p}{|\overline{\Dvect{u}}|}UA \\
    \frac{|\overline{\Dvect{u}}|-C_p}{|\overline{\Dvect{u}}|}VA \\
    0
  \end{array}
  \right)
  + 
   \Ddiv
   \left(
  \begin{array}{c} 
    \frac{C_pU}{|\overline{\Dvect{u}}|} \\
    \frac{C_pV}{|\overline{\Dvect{u}}|} \\
    0
  \end{array}
  \right)A \\
  &=& 
   \Ddiv \Dvect{N}^{(1)} + \Ddiv \Dvect{C}_{v}A, \\
\Dvect{E} &=& \frac{\rho_0}{|\overline{\Dvect{u}}|}\left(
  \begin{array}{c}
    U(v^{'2}-e) + Vu'v' \\
    Uu'v' + V(e - u ^{'2}) \\
    U\frac{f_0^{2}}{N^{2}}\psi '_{x}\psi '_{z}
    + V\frac{f_0^{2}}{N^{2}}\psi '_{y}\psi '_{z}
  \end{array}
  \right)
\end{eqnarray*}
$B$H$9$k$H(B, 
\begin{equation}
  \DP{}{t}A + \Ddiv \Dvect{C}_{v}A + \frac{1}{\rho_0}\Ddiv \Dvect{E} + 
  \frac{1}{\rho_0}\Ddiv \Dvect{N}^{(1)} = 0
  \Deqlab{A $B$N<0(B}
\end{equation}
$B$H$J$k(B. $B$3$l$OEl@>0lMM$J>l$KBP$9$kGH$N3hF0EY$NJ]B8$N<0(B
\[
  \DP{}{t}A 
    + \DP{}{x}C_{x}A + \Ddiv \Dvect{E} 
    + \DP{}{x}\frac{(\overline{u}-C_x)}{2\overline{Q}_{y}}q^{'2} = 0
\]
$B$KBP1~$7$F$$$k(B.

\subsection{{\cal E} $B$N<0(B}

\Deqref{$BEl@>Hs0lMM4pK\>l$N@~7A2=$5$l$?%]%F%s%7%c%k12EYJ]B8B'(B}$B$K(B
$\displaystyle -\frac{\psi '}{|\overline{\Dvect{u}}|-C_p}$ $B$r$+$1$k$H(B
\begin{equation}
  - \frac{1}{|\overline{\Dvect{u}} -C_p|}\left(
  \psi ' \DP{q'}{t} 
     + \overline{\Dvect{u}}\psi '\Dgrad q'  
     + \overline{\Dvect{u}}\psi '\Dgrad \overline{Q}
  \right)
  =0.
\end{equation}
$B:8JUBh(B 1 $B9`$O(B
\begin{eqnarray*}
  \psi '\DP{q'}{t} &=&
  \DP{}{t}{\cal E} - \frac{1}{\overline{\Dvect{u}}-C_p}
  \Ddiv \left(
  \begin{array}{c}
    \psi ' \psi '_{xt}  \\
    \psi ' \psi '_{yt}  \\ 
    \frac{f_0^{2}}{N_{2}}\psi '\psi '_{zt}
  \end{array}
  \right) \\
  &=&
  \DP{}{t}{\cal E} - \frac{1}{\overline{\Dvect{u}} - C_p}\Ddiv \Dvect{R}_{1}
\end{eqnarray*}
$B$H$J$k(B.

$BBh(B 2 $B9`$O0J2<$N$h$&$KJQ7A$9$k(B.
\begin{eqnarray*}
  \overline{\Dvect{u}}\psi '\Dgrad q'  
  &=&
    - \frac{U}{|\overline{\Dvect{u}}|-C_p}\psi '\DP{q'}{x} 
    - \frac{V}{|\overline{\Dvect{u}}|-C_p}\psi '\DP{q'}{y}  \\
  &=&
    - \left(
        \frac{U}{|\overline{\Dvect{u}}|} 
          + \frac{C_p}{|\overline{\Dvect{u}}|-C_p}
              \frac{U}{|\overline{\Dvect{u}}|} 
      \right) 
      \psi '\DP{q'}{x} 
    - \left(
        \frac{V}{|\overline{\Dvect{u}}|} 
          + \frac{C_p}{|\overline{\Dvect{u}}|-C_p}
              \frac{V}{|\overline{\Dvect{u}}|} 
      \right) 
      \psi '\DP{q'}{y} 
\end{eqnarray*}
$B$5$i$K(B
\[
  \psi '\DP{q'}{x} = \Ddiv \left(
    \begin{array}{c}
      \psi '\psi'_{xx} \\
      \psi '\psi'_{yx} \\
      \frac{f_0^{2}}{N^{2}}\psi '\psi'_{zx}
    \end{array}
    \right) 
    - \DP{e}{x} 
\]
$B$G$"$k$+$i(B, 
\begin{eqnarray*}
  - \frac{U}{|\overline{\Dvect{u}}|}\psi '\DP{q'}{x} 
     - \frac{V}{|\overline{\Dvect{u}}|}\psi '\DP{q'}{y} 
  &=& 
    \frac{U}{|\overline{\Dvect{u}}|}
    \left\{
      \frac{1}{\rho_0}\Ddiv \left(
        \begin{array}{c}
          -\psi '\psi'_{xx} \\
          -\psi '\psi'_{xy} \\
          -\frac{f_0^{2}}{N^{2}}\psi '\psi'_{xz}
        \end{array}
      \right) 
      + \DP{e}{x}
    \right\} \\
 &  &
  +    
    \frac{V}{|\overline{\Dvect{u}}|}
    \left\{
      \frac{1}{\rho_0}\Ddiv \left(
        \begin{array}{c}
          -\psi '\psi'_{xy} \\
          -\psi '\psi'_{yy} \\
          -\frac{f_0^{2}}{N^{2}}\psi '\psi'_{yz}
        \end{array}
      \right) 
      + \DP{e}{y}
    \right\} \\
  &\approx&
    \frac{1}{\rho_0}\Ddiv \frac{\rho_0}{|\overline{\Dvect{u}}|}
    \left(
      \begin{array}{c}
        eU - U\psi '\psi'_{xx} - V\psi '\psi_{xy} \\
        eV - U\psi '\psi'_{xy} - V\psi '\psi_{yy} \\
        - \frac{f_0^{2}}{N^{2}}(U\psi '\psi'_{xz} + V\psi '\psi _{yz})
      \end{array}
    \right) \\
  &\equiv& \Dvect{H}a
\end{eqnarray*}
$B$H$J$k(B
\footnote{$B$3$l$O(B Takaya and Nakamura (2001)$B$N(B (19) $B<0$NJQ7A$G$"$k(B.}.
$B$^$?(B,
\begin{eqnarray*}
 - \frac{C_p}{|\overline{\Dvect{u}}|-C_p}
     \frac{U}{|\overline{\Dvect{u}}|}\psi '\DP{q'}{x} 
 - \frac{C_p}{|\overline{\Dvect{u}}|-C_p}
     \frac{V}{|\overline{\Dvect{u}}|}\psi '\DP{q'}{y} 
 &\approx& 
    \DP{}{x}
      \left(
        \frac{UC_p}{|\overline{\Dvect{u}}|}
        \frac{e}{|\overline{\Dvect{u}}|-C_p} 
      \right)  \\
  &  & 
    - \frac{C_p}{\rho_0(|\overline{\Dvect{u}}|-C_p)}
        \frac{U}{|\overline{\Dvect{u}}|}\Ddiv \rho_0 \left(
        \begin{array}{c}
	  -\psi '\psi '_{xx} \\
	  -\psi '\psi '_{xy} \\
	  -\frac{f_0^{2}}{N^{2}}\psi'\psi'_{xz} 
        \end{array}
	\right) \\
  &  & 
  - \DP{}{y}
      \left(
        \frac{VC_p}{|\overline{\Dvect{u}}|}
        \frac{e}{|\overline{\Dvect{u}}|-C_p} 
      \right) \\
  &  &
   - \frac{C_p}{\rho_0(|\overline{\Dvect{u}}|-C_p)}
        \frac{V}{|\overline{\Dvect{u}}|}\Ddiv \rho_0 \left(
        \begin{array}{c}
	  -\psi '\psi '_{xy} \\
	  -\psi '\psi '_{yy} \\
	  -\frac{f_0^{2}}{N^{2}}\psi'\psi'_{yz} 
        \end{array}
	\right) \\
   &=& 
     \Ddiv \Dvect{C}_{v}{\cal E}  
       - \frac{C_p}{\rho_0(|\overline{\Dvect{u}}|-C_p)}
         \left(
            \frac{U}{|\overline{\Dvect{u}}|}\Ddiv \Dvect{R}_{2}
	  + \frac{V}{|\overline{\Dvect{u}}|}\Ddiv \Dvect{R}_{3}
         \right)
\end{eqnarray*}
$B$H$J$k(B
\footnote{$B$3$l$O(B Takaya and Nakamura (2001) $B$N(B (17) $B<0$NJQ7A(B.}.
$BBh(B 3 $B9`$O(B
\begin{eqnarray*}
  - \frac{1}{|\overline{\Dvect{u}} -C_p|}
    \overline{\Dvect{u}}\psi '\Dgrad \overline{Q}
  &=& 
  - \frac{1}{|\overline{\Dvect{u}} -C_p|}
     \left(u'\psi'\DP{\overline{Q}}{x} + v'\psi'\DP{\overline{Q}}{y}\right)
    \\
  &=& 
  - \frac{1}{|\overline{\Dvect{u}} -C_p|}
     \left\{
       - \DP{}{y}\left(\frac{1}{2}\overline{Q}_{x}\psi ^{'2}\right)
       + \DP{}{x}\left(\frac{1}{2}\overline{Q}_{y}\psi ^{'2}\right)
     \right\} \\
  &\approx&
    \Ddiv \left(
      \begin{array}{c}
      \frac{1}{2}\frac{\overline{Q}_{y}}{|\overline{\Dvect{u}}|-C_p}\psi^{'2}\\
    - \frac{1}{2}\frac{\overline{Q}_{x}}{|\overline{\Dvect{u}}|-C_p}\psi^{'2}\\
      0
      \end{array}
    \right) \\
  &\equiv&
     \Ddiv \Dvect{N}^{(2)}
\end{eqnarray*}
$B$HJQ7A$9$k(B\footnote{$B$3$l$O(B Takaya and Nakamura (2001) $B$N(B (21), (22) $B<0$NJQ7A(B.}

$B0J>e$r$^$H$a$k$H(B,

\begin{equation}
  \DP{{\cal E}}{t} 
    + \Ddiv \Dvect{C}_{v}{\cal E} 
    + \frac{1}{\rho_0}\Ddiv \Dvect{H}
    + \frac{1}{\rho_0}\Ddiv \Dvect{N}^{(2)}
    - \frac{1}{\rho _{0}(|\overline{\Dvect{u}}|-C_p)}
        \left\{
	  \Ddiv \Dvect{R}_{1} 
             + \frac{C_{p}U}{|\overline{\Dvect{u}}|}\Ddiv \Dvect{R}_{2}
             + \frac{C_{p}V}{|\overline{\Dvect{u}}|}\Ddiv \Dvect{R}_{3}
        \right\}
    = 0
  \Deqlab{E $B$N<0(B}
\end{equation}


\subsection{$BEl@>Hs0lMM4pK\>l$NGH$N3hF0EY$NJ]B8B'(B}

\Deqref{A $B$N<0(B}$B$H(B\Deqref{E $B$N<0(B}$B$NOB$r$H$k$3$H$G(B,
\begin{eqnarray}
  &&\DP{{\cal M}}{t} 
    + \Ddiv \Dvect{C}_{v}{\cal M} 
    + \frac{1}{\rho_0}\Ddiv \frac{1}{2}(\Dvect{E} + \Dvect{H})
    + \frac{1}{\rho_0}\Ddiv \frac{1}{2}(\Dvect{N}^{(1)} + \Dvect{N}^{(2)})
  \nonumber \\
  &&
    - \frac{1}{2\rho _{0}(|\overline{\Dvect{u}}|-C_p)}
        \left\{
	  \Ddiv \Dvect{R}_{1} 
             + \frac{C_{p}U}{|\overline{\Dvect{u}}|}\Ddiv \Dvect{R}_{2}
             + \frac{C_{p}V}{|\overline{\Dvect{u}}|}\Ddiv \Dvect{R}_{3}
        \right\}
    = 0
  \Deqlab{M $B$N<0(B}
\end{eqnarray}
$BBh(B 4 $B9`$O(B,
\begin{eqnarray*}
  \frac{1}{\rho_0}\Ddiv \frac{1}{2}(\Dvect{N}^{(1)} + \Dvect{N}^{(2)})
  &=&
    \frac{1}{2\rho_0}
    \left(
    \begin{array}{c}
    \frac{|\overline{\Dvect{u}}|-C_p}{|\overline{\Dvect{u}}|}UA 
    - \frac{1}{2}\frac{\overline{Q}_{y}}{|\overline{\Dvect{u}}|-C_p}\psi^{'2}\\
    \frac{|\overline{\Dvect{u}}|-C_p}{|\overline{\Dvect{u}}|}VA 
    + \frac{1}{2}\frac{\overline{Q}_{x}}{|\overline{\Dvect{u}}|-C_p}\psi^{'2}\\
      0
    \end{array}
    \right)  \\
  &\approx&
    \frac{1}{2\rho_0}
    \left(
    \begin{array}{c}
    \frac{|\overline{\Dvect{u}}|-C_p}{|\overline{\Dvect{u}}|}UA 
    - \frac{1}{2}
        \frac{U|\Dgrad _h\overline{Q}|}
           {|\overline{\Dvect{u}}|(|\overline{\Dvect{u}}|-C_p)}\psi^{'2}\\
    \frac{|\overline{\Dvect{u}}|-C_p}{|\overline{\Dvect{u}}|}VA 
    - \frac{1}{2}
        \frac{V|\Dgrad _h\overline{Q}|}
           {|\overline{\Dvect{u}}|(|\overline{\Dvect{u}}|-C_p)}\psi^{'2}\\
      0
    \end{array}
    \right) 
\end{eqnarray*}
$BESCf$G(B
\[
 \frac{\overline{Q}_y}{|\Dgrad _h\overline{Q}|} \approx \frac{U}{|\overline{\Dvect{u}}|}, 
    \quad
 \frac{\overline{Q}_x}{|\Dgrad _h\overline{Q}|} \approx - \frac{V}{|\overline{\Dvect{u}}|}
\]
$B$rMQ$$$?(B. $x$ $B@.J,$O(B
\begin{eqnarray*}
  \frac{|\overline{\Dvect{u}}|-C_p}{|\overline{\Dvect{u}}|}UA 
   - \frac{1}{2}
      \frac{U|\Dgrad _h\overline{Q}|}
         {|\overline{\Dvect{u}}|(|\overline{\Dvect{u}}|-C_p)}\psi^{'2}
  &=& ...
\end{eqnarray*}
$B$3$3$G(B, $B4pK\>l$NN.$l$K1h$C$?J}8~$K(B $X$ $B<4(B, $B$=$l$HD>8r$9$kJ}8~$K(B $Y$ $B<4(B
$B$r$H$k$H(B, $B$b$H$N@~7A2=$5$l$?>qMp$N%]%F%s%7%c%k12EY$N<0(B,
\[
  \DP{q'}{t} + \overline{\Dvect{u}}\Dgrad q' + \Dvect{u}'\Dgrad \overline{Q}
  = 0
\]
$B$O(B
\[
  \DP{q'}{t} 
    + $B!C(B\overline{\Dvect{u}}|\DP{q'}{X} 
    + \DP{\psi '}{X}\DP{\overline{q}}{Y} 
  = 0
\]
$B$H$J$k(B. $q'\approx \sin \chi$ $B$N$h$&$JJ?LLGH2r$r2>Dj$9$k$H(B,
\[
  \DP{}{t} \sim -C_p\DP{}{x}
\]
$B$H9M$($F$h$$$N$G(B, $B>e5-$N<0$O(B
\[
  \DP{}{x}\left\{ (\overline{\Dvect{u}}|-C_p)q' + |\Dgrad _h \overline{Q}|\psi '\right\}
  = 0
\]
$BE,Ev$J6-3&>r7o$N$b$H$G$O(B
\[
  (\overline{\Dvect{u}}|-C_p)q' + |\Dgrad _h \overline{Q}|\psi ' = 0
\]
$B$H$9$k$3$H$,$G$-$k(B. $B$7$?$,$C$F(B, 
\[
  \frac{1}{\rho_0}\Ddiv \frac{1}{2}(\Dvect{N}^{(1)} + \Dvect{N}^{(2)}) =0
\]
$B$H$9$k$3$H$,$G$-$k(B. 

$B<!$KBh(B 5 $B9`$O(B
\begin{eqnarray*}
  \Ddiv \Dvect{R}_{1} 
    + \frac{C_{p}U}{|\overline{\Dvect{u}}|}\Ddiv \Dvect{R}_{2}
    + \frac{C_{p}V}{|\overline{\Dvect{u}}|}\Ddiv \Dvect{R}_{3}
  &=& 
    \left(\DP{}{t}+C_p\DP{}{X}\right)e 
    + \psi '\left(\DP{}{t}+C_p\DP{}{X}\right)q' \\
  &\sim& 0
\end{eqnarray*}

$BBh(B 3 $B9`$O(B
\begin{eqnarray*}
 \frac{1}{2}(\Dvect{E} + \Dvect{H}) 
   &=& 
   \left(
  \begin{array}{c}
    U(v^{'2}-e) + Vu'v' + eU - U\psi '\psi'_{xx} - V\psi '\psi_{xy} \\
    Uu'v' + V(e - u ^{'2}) + eV - U\psi '\psi'_{xy} - V\psi '\psi_{yy} \\
    U\frac{f_0^{2}}{N^{2}}\psi '_{x}\psi '_{z}
    + V\frac{f_0^{2}}{N^{2}}\psi '_{y}\psi '_{z}
     - \frac{f_0^{2}}{N^{2}}(U\psi '\psi'_{xz} + V\psi '\psi _{yz})
  \end{array}
  \right) \\
  &=&
   \left(
  \begin{array}{c}
    U((\psi ^{'}_{x})^{2} - \psi '\psi'_{xx}) + 
      V(\psi '_{x}\psi '_{y} - \psi '\psi'_{xy}) \\
    U(\psi '_{x}\psi '_{y} - \psi '\psi '_{xy}) +
      V((\psi ^{'}_{y})^{2} - \psi '\psi '_{yy}) \\
    \frac{f_0^{2}}{N^{2}}\{
    U(\psi '_{x}\psi '_{z}-\psi '\psi'_{xz})
    + V(\psi '_{y}\psi '_{z} - \psi '\psi _{yz})\}
  \end{array}
  \right) \\
 &\equiv& \Dvect{W}_{s}
\end{eqnarray*}

$B0J>e$r$^$H$a$k$H(B, \Deqref{M $B$N<0(B}$B$O(B
\begin{eqnarray}
  \DP{{\cal M}}{t} 
    + \Ddiv \Dvect{C}_{v}{\cal M} 
    + \Ddiv \Dvect{W}_{s}
  = 0
\end{eqnarray}
$\Dvect{W} = \Dvect{C}_{p}{\cal M} + \Dvect{W}_{s}$ $B$H$9$k$H(B,
\begin{eqnarray}
  \DP{{\cal M}}{t} + \Ddiv \Dvect{W} = 0
\end{eqnarray}
($B7W;;$O>JN,$9$k$,(B) $\Dvect{W} = \Dvect{C}_{g}{\cal M}$ $B$H$J$C$F$$$k(B.

$\Dvect{W}_{s}$ $B$N3F@.J,$O(B $\psi'$ $B$N(B 1 $B3,HyJ,$N<+>h$H(B, $\psi'$ $B$H(B 
$\psi '$ $B$N(B 2 $B3,HyJ,$N@Q$H$NOB$+$i$J$k(B. $B$7$?$,$C$F(B, $\psi '\propto
\sin \chi$ $B$J>qMp$KBP$7$F$O(B, $BJ?6QA`:n$r9T$&$3$H$J$/0LAj0MB8@-$r>C5n(B
$B$9$k$3$H$,$G$-$k(B.

\newpage

\section{3 $B<!85(B TEM $BJ}Dx<07O(B}

TEM $BJ}Dx<07O(B

\begin{eqnarray}
  && \DP{[A]}{t} + \Ddiv \Dvect{F} = 0,          \\
  && \DP{[u]}{t} - fv^{*}_{a} = \Ddiv \Dvect{F}, \\
  && \Ddiv \Dvect{F} = \Ddiv
       \left(
       \begin{array}{c}
	 0        \\
	 - [u'v'] \\
	 \frac{f_{0}}{{\theta _{0}}_{z}}[v'\theta']
       \end{array}
       \right)
      = [v'q']
\end{eqnarray}
$B$r(B 3 $B<!85$K3HD%$9$k$3$H$r9M$($k(B.
\[
  \psi = \Psi (y,z) = \psi '
\]
$B$HJ*M}NL$rJ?6Q>l$H>qMp$K$o$1$?$H$-$K(B, $B>qMp>l$K$D$$$F$O0J2<$N$h$&$JJ?6Q(B
$B$,$H$l$kM}A[E*$J7O$r9M$($k(B.
\[
  \frac{1}{T}\int _{0}^{T}\psi \Dd t =0.
\]

\subsection{Plumb (1986) $B$N>l9g(B}

3 $B<!85$NGH$N3hF0EY$NJ]B8$N<0$O(B
\[
  \DP{\overline{A}}{t} 
    + \DP{}{x}(\overline{u}\overline{A})
    + \Ddiv \left(
      \begin{array}{c}
	\overline{v^{'2}} - \overline{e} \\
	- \overline{u'v'} \\
	\frac{f_0}{{\theta _0}_z}\overline{v'\theta '}
      \end{array}
      \right) 
      = 0
\]
$B$3$l$O(B Plumb (1986) $B$NGH$N3hF0EY$NJ]B8$N<0$G(B $\overline{v}=0$ $B$H$7$?<0(B
$B$KEy$7$$(B.

$BJ?6QN.$N<0$O1&JU$K(B
\[
  \overline{v'q'} = \Ddiv \left(
       \begin{array}{c}
	 \overline{v^{'2}}-\overline{e}        \\
	 - [u'v'] \\
	 \frac{f_{0}}{{\theta _{0}}_{z}}[v'\theta']
       \end{array}
  \right)
\]
$B$,8=$l$k$h$&(B, $B0J2<$N$h$&$KJQ7A$r9T$&(B.
\begin{eqnarray*}
  \DP{}{t}\overline{u} - f \overline{v}_{a} &=&
    - \DP{}{x}u^{'2}
    - \DP{}{y}u'v'
    + \DP{}{x}v^{'2}
    - \DP{}{x}v^{'2}
    + \DP{}{x}e
    - \DP{}{x}e
    + \DP{}{z}\frac{f_0}{{\theta _0}_z}v'\theta' 
    - \DP{}{z}\frac{f_0}{{\theta _0}_z}v'\theta'  \\
  &=&
    \DP{}{x}(v^{2}-e) 
    + \DP{}{y}(-u'v') 
    + \DP{}{z}\frac{f_0}{{\theta _0}_z}v'\theta' \\
  &=&
    + \left\{
      \DP{}{x}(e - u^{'2}) - \DP{}{z}\frac{f_0}{{\theta _0}_z}v'\theta' 
      \right\}
\end{eqnarray*}
\[
  v^{*}_{a} \equiv v_{a} - \frac{1}{f_0}
    \left\{
      \DP{}{x}\left(
        \frac{u^{'2}}{2} 
          + \frac{v^{'2}}{2} 
          - \frac{1}{2}\frac{f_0^{2}}{N^{2}}\psi ^{'2}_{z}
      \right) 
      + \frac{1}{\rho_0}\DP{}{z}
	\left(\rho _{0}\frac{f_{0}^{2}}{N^{2}}\overline{\psi'_{x}\psi'_{z}}\right)
    \right\}
\]
$B$HDj5A$9$k$H(B,
\[
 \DP{}{t}\overline{u} - fv_{a}^{*} = \overline{v'q'} = \Ddiv \left(
       \begin{array}{c}
	 \overline{v^{'2}}-\overline{e}        \\
	 - [u'v'] \\
	 \frac{f_{0}}{{\theta _{0}}_{z}}[v'\theta']
       \end{array}
 \right)
\]
$B$3$N$H$-GH$N3hF0EY$N<0$O(B
\[
  \DP{\overline{A}}{t} 
  +\DP{}{x}(\overline{u}\overline{A}) 
  +
    \Ddiv
       \left(
       \begin{array}{c}
	 \overline{v^{'2}}-\overline{e}        \\
	 - \overline{u'v'} \\
	 \frac{f_{0}}{{\theta _{0}}_{z}}\overline{v'\theta'}
       \end{array}
       \right)
      = 0
\]
$B$H$J$k(B. $B$3$3$GCm0U$9$Y$-$O(B, $BGH$N3hF0EY$NJQ2=(B $B$O(B $\Ddiv \Dvect{F}$ $B$@$1$G(B
$B$J$/(B, $B0\N.9`$K$b0MB8$9$k$3$H$G$"$k(B. 

$B7A<0E*$K%U%i%C%/%9$N<}B+$K$h$C$FGH$N3hF0EY$,JQ2=$9$k(B, $B$H$$$&<0$K$9$k$H(B,
\[
  \Dvect{M}_T \equiv = 
       \left(
       \begin{array}{c}
	 \overline{u}A + \overline{v^{'2}}-\overline{e}        \\
	 - \overline{u'v'} \\
	 \frac{f_{0}}{{\theta _{0}}_{z}}\overline{v'\theta'}
       \end{array}
       \right)
\]
$B$rDj5A$7(B, 
\begin{eqnarray*}
  && \DP{}{t}{\overline{u}} - fv_{a}^{**} = \Ddiv \Dvect{M}_{T} \\
  && v_{a}^{**} = v_{a} - 
     \frac{1}{f_{0}}\left\{
      \DP{}{x}\left(
        \frac{u^{'2}}{2} 
          + \frac{v^{'2}}{2} 
          - \frac{1}{2}\frac{f_0^{2}}{N^{2}}\psi ^{'2}_{z}
	  + \overline{u}A
      \right) 
      + \frac{1}{\rho_0}\DP{}{z}
	\left(\rho _{0}\frac{f_{0}^{2}}{N^{2}}\overline{\psi'_{x}\psi'_{z}}\right)
     \right\}
\end{eqnarray*}
$B$H$J$k(B. $B$?$@$7(B, $B%U%i%C%/%9$N(B $x$ $B@.J,$K(B $\overline{u}A$ $B$NG$0U@-$,8=$l(B
$B$k$3$H$KCm0U$,I,MW$G$"$k(B. $B$3$l$OJ?6QN.$K>h$C$?:BI8$+$i8+$?>l9g$HCOLL$K(B
$B8GDj$5$l$?:BI8$+$i8+$?>l9g$G%U%i%C%/%9$NBg$-$5$,0[$J$k$3$H$r0UL#$9$k(B.

Plumb (1986) $B$G$OJ?6QN.$NJ}Dx<0$rF3$/:]$K$O(B $\beta $ $BLL$G$O$J$/(B $f $BLL(B$
$B$rMQ$$$F$$$k(B.

$B$I$N$h$&$J%U%i%C%/%9$NI=8=$rMQ$$$k$N$,$h$$$+$O(B, $B7k6I<B:]$K8=<BBg5$$KE,(B
$BMQ$7?^$r:n@.$7$F$+$i9M$($J$$$H$o$+$i$J$$$+$b$7$l$J$$(B.

%  $B9bC+%N!<%H$r$b$H$KJdB-$9$k$3$H(B
% 
%

\subsection{Trenberth (1986) $B$N>l9g(B}

f $BLL$N;~4VJ?6Q$7$?J?6Q>l$N<0$H$7$F(B
\[
  \overline{u}\DP{\overline{u}}{x} 
      + \overline{v}\DP{\overline{u}}{x} 
      - f\overline{v}
      = -\DP{\overline{\phi}}{x} 
      - \DP{}{x}\overline{u^{'2}}-\DP{}{y}\overline{u'v'} 
\]
$B$rF@$k(B. $B$3$l$rJQ7A$7$F(B
\[
  \overline{u}\DP{\overline{u}}{x} 
      + \overline{v}\DP{\overline{u}}{x} 
      - fv_{T}^{*}
      = \frac{1}{\rho_0}\Ddiv \rho_0
        \left(
	\begin{array}{c}
	  \frac{1}{2}(\overline{v^{'2}-u^{'2}}) \\
	  - \overline{u'v'} \\
	  \frac{f_{0}}{{\theta _{0}}_{z}}\overline{v'\theta'}
	\end{array}
	\right)  
\]
\[
v_{T}^{*} \equiv \overline{v} - \frac{1}{f}\DP{}{x}(\overline{\phi}+K)
 - \frac{1}{\rho _0}\DP{}{z}\left(\rho _{0}\frac{f_0}{{\theta _0}_z}
 \overline{v'\theta'}\right)
\]
$B$3$3$G(B $K$ $B$O1?F0%(%M%k%.!<$G$"$k(B. $B>e5-$N<0$G(B
\[
  \rho_0
        \left(
	\begin{array}{c}
	  \frac{1}{2}(\overline{v^{'2}-u^{'2}}) \\
	  - \overline{u'v'} \\
	  \frac{f_{0}}{{\theta _{0}}_{z}}\overline{v'\theta'}
	\end{array}
	\right) 
 \neq \Dvect{C}_{g}A
\]
$B$G$"$k$3$H$KCm0U$9$kI,MW$,$"$k(B. 

\subsection{Plumb (1985) $B$N>l9g(B}

Plumb (1985) $B$G$O(B TEM $B7O$NJ}Dx<0$NF3=P$O9T$C$F$$$J$$(B. $BJ?6Q>l$N<0(B
\[
  \DP{u^{(2)}}{t} + U\DP{u^{(2)}}{x} + v^{(2)}\DP{U}{y} - fv_{a}
   = -\DP{}{x}u^{'2} - \DP{}{y}u'v'
\]
$B$N1&JU$N9`$N0LAj0MB8@-$N>C5n$,LLE]$G$"$C$?$?$a$HA[A|$5$l$k(B. 

$B1&JU$NJQ7A$O0J2<$N$h$&$K9T$&$H$h$$(B.
\begin{eqnarray*}
 -\DP{}{x}u^{'2} - \DP{}{y}u'v' 
 &=&
    - \psi '_{y}\psi '_{xy} + \psi '_{x}\psi '_{yy} \\
 &=&
    - \psi '_{y}\psi '_{xy} -\psi'\psi'_{xyy} + \psi'\psi'_{xyy} + \psi '_{x}\psi '_{yy} \\
  &=&
    - \DP{}{y}(\psi'\psi'_{xy})  + \DP{}{x}(\psi'\psi'_{yy})
\end{eqnarray*}
$B$3$N$H$-(B
\[
  \DP{u^{(2)}}{t} + U\DP{u^{(2)}}{x} + v^{(2)}\DP{U}{y} - fv_{a}
   = - \frac{1}{2}\DP{}{x}
      \left(
        \psi ^{'2}_{y} - \psi'\psi'_{yy}
      \right)
     + \frac{1}{2}\DP{}{y}
      \left(
        \psi '_{x}\psi '_{y} - \psi'\psi'_{xy}
      \right)
\]
$B$H$J$C$F(B, $B1&JU$N0LAj0MB8@-$r>C5n$9$k$3$H$,$G$-$k(B.
\[
  Du^{(2)} - fv_{a}^{***} = \Ddiv \Dvect{W} \\
\]
\[
  v_{a}^{***} = v_{a} -\frac{1}{f}\left(\cdots \right)
\]
$B$N$h$&$KI=$9$3$H$,$G$-$k(B.

\end{document}