%#! platex master_thesis-M2.tex
\chapter{$B@~7?0BDj@-2r@O(B} \label{chap:baroclinic}
$B7905>lCf$N@~7?@]F0$NH/E8$K$D$$$F9M;!$9$k$?$a$K(B, $B;YG[J}Dx<0(B
(\ref{eq:troeq}), (\ref{eq:clieq}) $B$NHs@~7?9`$H6/@)!&;60o$N9`$r=|$$$?J}(B
$BDx<07O(B
\begin{align}
 &\DP{\bnabla^2 \psi}{t} + U\DP{\bnabla^2\tau}{x} =0, \label{eq:tro_lneq}\\
 &\DP{(\bnabla^2-k_d^2)\tau}{t}+U\DP{\bnabla^2\psi}{x}+U
 k_d^2\DP{\psi}{x}=0 \label{eq:cli_lneq}
\end{align}
$B$rMQ$$$k(B. $B$3$NJ}Dx<07O$NGH7?$N2r(B
\begin{equation*}
 \psi=\hat{\psi}\ee^{i(\bm{k\cdot\bm{x}}-\omega t)}, \tau=\hat{\tau}\ee^{i(\bm{k\cdot\bm{x}}-\omega t)}
\end{equation*}
$B$r2>Dj$7(B, (\ref{eq:tro_lneq}) $B$H(B (\ref{eq:cli_lneq}) $B$K$=$l$>$lBeF~$9$k(B
$B$H(B,
\begin{align}
 &i\{\omega(k_x^2+k_y^2)\hat{\psi}-Uk_x(k_x^2+k_y^2)\hat{\tau}\} = 0,\\
 &i\{-Uk_x(k_x^2+k_y^2-k_d^2)\hat{\psi}+\omega(k_x^2+k_y^2+k_d^2)\hat{\tau}\}
 = 0
\end{align}
$B$H$J$k(B. $B$3$l$r9TNs$GI=<($9$k$H(B,
\begin{align}
 i\underbrace{\left(
 \begin{array}{rr}
  \omega(k_x^2+k_y^2) & -Uk_x(k_x^2+k_y^2) \\
  -Uk_x(k_x^2+k_y^2-k_d^2) & \omega(k_x^2+k_y^2+k_d^2) 
 \end{array}
 \right)}_{=M}
 \left(
 \begin{array}{r}
  \hat{\psi}\\
  \hat{\tau}
 \end{array}
 \right) =0 
\end{align}
$B$H$J$k(B. $BJ}Dx<0$N2r$,<+L@$J2r$H$J$i$J$$$?$a$K$O(B, $M$ $B$N5U9TNs$,B8:_$7$J$$(B,$B$D(B
$B$^$j(B, $\mathrm{det}M=0$ $B$G$"$kI,MW$,$"$k$?$a(B,
\begin{align*}
 &\mathrm{det}M=\omega^2(k_x^2+k_y^2)(k_x^2+k_y^2+k_d^2)-U^2k_x^2(k_x^2+k_y^2)(k_x^2+k_y^2-k_d^2)=0, \\
 \Leftrightarrow\hspace{3pt}& \omega^2=k_x^2U^2\frac{k_x^2+k_y^2-k_d^2}{k_x^2+k_y^2+k_d^2}
\end{align*}
$B$G$J$/$F$O$J$i$J$$(B. $B$D$^$j(B, $B@.D9N($O(B
\begin{align}
 \omega=k_x U\sqrt{\frac{k_d^2-(k_x^2+k_y^2)}{k_x^2+k_y^2+k_d^2}} =
 \frac{k}{\sqrt{2}}U\sqrt{\frac{k_d^2-k^2}{k_d^2+k^2}} \label{eq:omega}
\end{align}
$B$H$J$k(B. 2 $B$DL\$NEy9f$O(B, $x$ $BJ}8~$H(B $y$ $BJ}8~$N@ZCGGH?t$,Ey$7$$$H$-(B,
$k=\sqrt{k_x^2+k_y^2}=\sqrt{2}k_x$ $B$G$"$k$3$H$rMQ$$$?(B. $B$3$N@.D9N($O(B, $B%((B
$B%M%k%.!<%9%Z%/%H%k$N%T!<%/$,GH?t(B $k$ $B$K$"$k$H$-$N@.D9N($G(B, $B:GB.@.D9N($G(B
$B$O$J$$(B.

$B:GB.@.D9N($O(B, $\mathrm{Im}(\omega)$ $B$,B8:_$7(B, $k_y=0$ $B$N$H$-(B
$\partial\omega/\partial k_x=0$ $B$J$k(B $k_x$ $B$N$H$-$N(B
$\mathrm{Im}(\omega)$ $B$,:GB.@.D9N($G$"$k(B. $B0J2<(B, $k_x\rightarrow k$ $B$HCV(B
$B$-49$($k$H(B, $\displaystyle \omega^2=k^2U^2\frac{k^2-k_d^2}{k^2+k_d^2}$$B$h(B
$B$j(B, 
\begin{align*}
 &\DP{\omega^2}{k}=2\omega\DP{\omega}{k},\\
 &\DP{}{k}=\DP{k^2}{k}\frac{\partial}{\partial
 k^2}=2k\frac{\partial}{\partial k^2},\\
 &\frac{\partial\omega^2}{\partial k^2}=\frac{1}{2k}\DP{\omega^2}{k}=\frac{\omega}{k}\DP{\omega}{k}, \\
 &\frac{\partial\omega^2}{\partial k^2}=\frac{\{U^2(k^2-k_d^2)+k^2 U^2 \}\{k^2+k_d^2\}-k^2 U^2(k^2-k_d^2)}{k^2+k_d^2}
\end{align*}
$B$N7W;;$+$i(B, $\partial\omega^2/\partial k^2 = 0$ $B$J$i$P(B,
\begin{align*}
 k^2&=\frac{-2k_d^2\pm\sqrt{4k_d^4+4k_d^4}}{2}\\
 &=\frac{-2\pm 2\sqrt{2}}{2}k_d^2 \\
 &=(-1\pm\sqrt{2})k_d^2 
\end{align*}
$B$H$J$k(B. $k^2>0$$B$h$j(B, $k^2=(\sqrt{2}-1)k_d^2$, $B$D$^$j(B, $k=0.64k_d$ $B$H$J$k(B.
$B$3$l$h$j(B,
\begin{align*}
 \omega^2&=(\sqrt{2}-1)k_d^2
 U^2\frac{(\sqrt{2}-1-1)k_d^2}{(\sqrt{2}-1+1)k_d^2}\\
 &= -\frac{(\sqrt{2}-1)(2-\sqrt{2})}{\sqrt{2}}k_d^2 U^2\\
 &=-(\sqrt{2}-1)^2 U^2 k_d^2
\end{align*}
$B$J$N$G(B, $B:GB.@.D9N($O(B
\begin{align}
 \mathrm{Im}(\omega)=(\sqrt{2}-1)U k_d \label{eq:alpha_max}
\end{align}
$B$G$"$k(B.

$B<B:]$K(B, $BBh(B \ref{chap:result} $B>O$G=R$Y$?=i4|$N%(%M%k%.!<@.D9N($r7W;;$9$k(B.
$BN.@~H!?t(B $\psi$ $B$O(B $\psi\propto \exp(\mathrm{Im}(\omega)t)$ $B$h$j(B, $B%(%M%k(B
$B%.!<(B $E$ $B$O(B $E \propto \exp(2\,\mathrm{Im}(\omega)t)$ $B$G$"$k$+$i(B, $B%(%M%k(B
$B%.!<$N@.D9N((B $\alpha$ $B$O(B $\alpha = 2\,\mathrm{Im}(\omega)$$B$H$J$k(B. $B%(%M%k(B
$B%.!<%9%Z%/%H%k$N%T!<%/$,GH?t(B $\sqrt{2}$ $B$K$"$k$H$-$N@.D9N((B
$\omega_{\mathrm{k=\sqrt{2}}}$ $B$O(B (\ref{eq:alpha_max}) $B$K(B
$U=2.5\times10^{-2}$, $k=\sqrt{2}$ $B$rBeF~$9$k$H(B,
\begin{align*}
 \omega_{\mathrm{k=\sqrt{2}}} &= \frac{\sqrt{2}}{\sqrt{2}}2.5\times
 10^{-2}\sqrt{\frac{10^2-(\sqrt{2})^2}{10^2+(\sqrt{2})^2}}\\ 
 &\approx 0.0245 
\end{align*}
$B$H$J$k$N$G(B, $B%(%M%k%.!<$N@.D9N($O(B,
\[
 \alpha_{\mathrm{k=\sqrt{2}}}=2\,\omega=0.049
\]
$B$H$J$k(B. $B$^$?(B, $B:GB.@.D9N($O(B, (\ref{eq:alpha_max}) $B$K(B $k_d=10$,
$U=2.5\times10^{-2}$ $B$rBeF~$7$F(B, 
\begin{align*}
 \mathrm{Im}(\omega) &= (\sqrt{2}-1)\times 2.5 \times 10^{-2} \times
 10 \\
 &\approx  0.41 \times 2.5 \times 10^{-2} \times 10 \\
 &= 0.1025
\end{align*}
$B$H$J$k$N$G(B, $B%(%M%k%.!<$N:GB.@.D9N((B
$\alpha_{\mathrm{max}}$ $B$O(B $\alpha_{\mathrm{max}}=2\,\mathrm{Im}(\omega)
= 0.205$ $B$H$J$k(B. 

%%% Local Variables:
%%% mode: yatex
%%% TeX-master: "platex master_thesis-M2"
%%% End:
