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% Chapter 4   Atmospheric Waves: Part II
%             4-8 BAROCLINIC INSTABILITY WITH LINEAR SHEAR
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\section{���������ˤ����뷹���԰���}

\begin{itembox}[l]{����}
 \Dparskip \vspace{-6mm}
 ����Ǥ�, �����԰���ΰ���Ū�ʾ���Ƴ�Ф�����ޤǤ����ʤä�.
 (����Ū�ʾ���Ƴ�Ф�ʣ���������񤷤�).
 ������Ǥ�, �������������, ���˴�ñ�ʾ��֤��ꤷ���絤�򰷤�.
 ���Τ褦�ʾ�ﲼ�����������Ѥ�, �ºݤ��԰������Ƴ��.

 �ޤ�, �������������, ��ñ���絤���ꤹ��.
 ���������ǵ�᤿�����Ĥ��μ������˴�ñ�ˤʤ�, ����Ū���԰������
 ���뤳�Ȥ������褦�ˤʤ�.
 ����ݤˤ�, ���ޤǤ�Ʊ�ͤ˸�ͭ��������.
 �ºݤ��԰��������Ƥߤ��, ���礦�����ѵ��Ϥξ��֤��԰������
 ���פ��뤳�Ȥ�ʬ����, ���ѵ��Ϥξ��𤬷����԰���ˤ�äư������������
 ���Ȥ�ʬ����.

 ��Ⱦ�Ǥ�, $\beta$���̤ʤɤ⤦����ʣ�����絤���԰�����򰷤�.
 ������������ϲ���Ū�˲򤯤Τ������񤷤�����, ����Ƴ�Ф�����ޤǤ�,
 �Ǹ����ʬ�ϼºݤο��ͷ׻��ˤ�äƺ������줿�ޤ�Ҳ𤹤��α�ޤ�.
\end{itembox}

Ϣ³Ū�����ץ��ե��������ķ����絤�ˤ�����, �Ǥ��ñ���԰���β��
Eady(1949)
�ˤ�äƵ���줿.
��ϰʲ��Τ褦�ʴ�ñ�ʾ��֤��ꤷ��.
%
\begin{Dlist}[1.]
 \item[1.] $\beta = 0$.
 \item[2.] �֥��ͥ��� ($e^{-Z} \approx $ ����).
 \item[3.] ��Ū�����ٰ��� ($\Gamma =$ ����)
	   \hspace{77mm} (5-40)
 \item[4.] ���$y$���Ф�����Ω ($\partial / \partial y = 0$)
 \item[5.] ����������$Z=1$�ˤ�äƤդ��򤵤�Ƥ���. ($W_{Z=1} = 0$)
\end{Dlist}
\setcounter{equation}{40}
%
���Υץ��ե������$Z$�������δؿ�
%
\begin{eqnarray}
 U = SZ
\end{eqnarray}
%
�Ǥ���. $S$������Ǥ���.
(5-40)���Ѥ����, �ݥƥ󥷥�뱲�٤δ��ܾ�θ���(5-36)�ϰʲ��Τ褦�ˤʤ�.
%
\begin{eqnarray}
 \DP{\overline{q}}{y} = 0.
\end{eqnarray}
%
�ݥƥ󥷥�뱲���������ϰʲ��Τ褦�˴�ñ�ˤʤ�.
%
\begin{eqnarray}
 (U - c)
  \left(
   \frac{{f_0}^2}{\Gamma} \DP[2]{\Psi}{Z} - \mu^2 \Psi
  \right)
  = 0.
\end{eqnarray}
%
���μ��β�ϰʲ��Τ褦�˽񤱤�.
%
\begin{eqnarray}
 \Psi = A \sinh \left( \frac{Z}{\varepsilon^{1/2}} \right)
  + B \cosh \left( \frac{Z}{\varepsilon^{1/2}} \right)
\end{eqnarray}
%
$\varepsilon = {f_0}^2 / \mu^2 \Gamma$�Ǥ���. ((4-48)
\footnote{
 $$
  \varepsilon \equiv f^2 L^2 \Gamma^{-1}.
  \eqno \textrm{(4-48)}
 $$
}����).
�������$W = 0 \ (Z = 0,1)$��, (5-39)����ʲ��Τ褦�˵��ҤǤ���.
%
\begin{eqnarray}
 (U - c) \DD{\Psi}{Z} - S \Psi = 0,
  \qquad \textrm{at} \quad
  Z = 0,1.
\end{eqnarray}
%
(5-44)����ʬ��, (5-45)�����������,
%
\begin{eqnarray}
 - c \varepsilon^{-1/2} A - SB = 0,
  \hspace{35mm}
  \nonumber \\
 \\
 (S - c) \varepsilon^{-1/2}
  \left[
   A \cosh \left( \varepsilon^{-1/2} \right)
   + B \sinh \left( \varepsilon^{-1/2} \right)
  \right]
  \hspace{30mm}
  \nonumber \\
 - S
  \left[
   A \sinh \left( \varepsilon^{-1/2} \right)
   + B \cosh \left( \varepsilon^{-1/2} \right)
  \right] = 0
  \nonumber
\end{eqnarray}
%
�Ȥ�������������.
�����ι��󼰤��ä���ʤ��, $A$��$B$��Ʊ���褦�˥����ˤʤ뤳�Ȥ�̵��.
���Τ��Ȥ���, ����®�٤˴ؤ���ʲ��μ���Ƴ�����
\footnote{
 (5-46)����������
 %
 \begin{eqnarray}
  \left(
   \begin{array}{cc}
    - c \varepsilon^{-1/2} & -S \\
    (S-c) \varepsilon^{-1/2} \cosh \varepsilon^{-1/2} - S \sinh \varepsilon^{-1/2} &
     (S-c) \varepsilon^{-1/2} \sinh \varepsilon^{-1/2} - S \cosh \varepsilon^{-1/2}\\
   \end{array}
  \right)
  \left(
   \begin{array}{c}
    A \\ B
   \end{array}
  \right)
  = 0
  \nonumber 
 \end{eqnarray}
 %
 �ι��󼰤������ˤʤ�褦�˷׻������(5-47)��������.
 �ʤ�, $\coth x = \cosh x / \sinh x$�Ǥ���.
}.
%
\begin{eqnarray}
 c^2 - Sc + S^2
  \left( \varepsilon^{1/2} \coth \varepsilon^{-1/2} - \varepsilon \right)
  = 0.
\end{eqnarray}
%
$c$�˴ؤ���(5-47)��򤤤����󤷤���Τ��ʲ��Ǥ���
\footnote{
 1���ܤ�, ñ��2��������$y = ax^2 + bx + c$�β�θ���
 $$
  x = - \frac{- b \pm \sqrt{b^2 - 4ac}}{2a}
 $$
 ���Ѥ��������Ǥ���.
}.
%
\begin{eqnarray}
 c &=& \frac{S}{2}
  \pm S
  \left\{
   \frac{1}{4}
   - \left( \varepsilon^{1/2} \coth \varepsilon^{-1/2} - \varepsilon \right)
  \right\}^{1/2}
  \nonumber \\
 &=& \frac{S}{2}
  \pm S \varepsilon^{1/2}
  \left\{
   \left[
    \frac{\varepsilon^{-1/2}}{2} - \tanh \left( \frac{\varepsilon^{-1/2}}{2}\right)
   \right]
   \left[
    \frac{\varepsilon^{-1/2}}{2} - \coth \left( \frac{\varepsilon^{-1/2}}{2}\right)
   \right]
  \right\}^{1/2}.
\end{eqnarray}
%
ʿ��������̤���ˤʤ�Τϰʲ��ξ��Ǥ���.
%
\begin{eqnarray}
 \frac{\varepsilon^{-1/2}}{2} < 1.20.
\end{eqnarray}
%
���ξ���$c$��ʣ�ǿ��ˤʤ�, ����԰���ˤʤ�.
(4-48)���$\varepsilon = L^2 / {L_R}^2$ ($L_R = \Gamma^{1/2} / f_0$)
�ʤΤ�, �԰����ɬ�׾��(5-49)�ϰʲ��Τ褦�˽񤱤�
\footnote{
 $\mathcal{L}$����Ĺ�Ǥ���.
}.
%
\begin{eqnarray}
 \frac{\mathcal{L}}{2 \pi} = L > \frac{L_R}{2.40}.
\end{eqnarray}
%
���ξ�ﲼ�Ǥ�, ��ĹΨ�ϰʲ��Τ褦�ˤʤ�
\footnote{
 $L = \mu$�Ȥ��Ƥ���.
}.
%
\begin{eqnarray}
 \mu c_i
  =
  \pm \frac{S}{L_R} \varepsilon^{1/2}
  \left\{
   \left[
    \frac{\varepsilon^{-1/2}}{2} - \tanh \left( \frac{\varepsilon^{-1/2}}{2}\right)
   \right]
   \left[
    \frac{\varepsilon^{-1/2}}{2} - \coth \left( \frac{\varepsilon^{-1/2}}{2}\right)
   \right]
  \right\}^{1/2}. 
\end{eqnarray}
%
���Τ褦��, ��ĹΨ��ľ��Ū�˱�ľ����®����$S$�����㤹��.
�ޤ�, $L$��(5-50)�����������ˤ�, $S \neq 0$�Ǥ���¤�, ���Ƥ��Ȥ�
�԰���ˤʤ�.
��ĹΨ������ˤʤ�Τϰʲ��ξ��Ǥ���.
%
\begin{eqnarray}
 L = \frac{L_R}{1.92}.
\end{eqnarray}
%
���β��Ϥ������Τ�, �����԰��꤬���������륹�������$L \sim L_R$,
�ޤ���$\varepsilon \sim 1$�Ǥ���Ȥ������ȤǤ���.
���ѵ��ϤǤξ����$\varepsilon \sim 1$�Ǥ��뤳�Ȥϴ�¬����Ƥ���Τ�,
���ѵ��Ϥξ���μ��פʥ��ͥ륮�����Ϸ����԰���Ǥ���ȹͤ��뤳�Ȥ�
�����Ǥ���.

�԰�����Ȥΰ���®�٤�
%
\begin{eqnarray}
 c_r = \frac{S}{2}
\end{eqnarray}
%
�Ǥ���.
���Υ�ǥ�Ǥα�ľ���о�������, ���ΰ���®�٤������ʤ�ΤǤ���ȸ�����.
�԰���ʸ�ͭ���$Z=1/2$�˴ؤ����о��������.
�ޤ����α�ľ��¤��, ��¬���줿���ѵ��Ϥξ��������Ū�ˤϻ��Ƥ���.

�����԰���η��֤Ͼ��������������(5-45)��̩�ܤ˴ط����Ƥ���.
�⤷�����ξ���$\Psi = 0$ (at ������������)���֤��������ʤ��,
�������԰�����Ȥ�¸�ߤ����ʤ�.

���������, �����˽簵�԰���ǽҤ٤�������֤ΰ�Ĥ���Ǥ���.
���ξ��֤Ȥ�, Υ���Ρ��ޥ�⡼�ɤϴ����Ϥ�����Ǥ��ʤ��Ȥ���
���֤Τ��Ȥ�
�ؤ��Ƥ���.
����ξ��, �ơ���$\mu$�˴ؤ��Ƥ��礦�ɥڥ��β�¸�ߤ���.
�����������������Ȥ��ư������Ȥˤ�ä�,
������𤬳�餫�˼�ޤ�褦��Ϣ³���ڥ��ȥ�β���դ��뤳�Ȥ�
�Ǥ���.

Charney(1947)��Eady���Ȥä��Τ�Ʊ�����ץ��ե�������Ѥ�����,
Eady���������������Ĥ��θ��̤�ޤ᤿.
Charney��$\beta$���̤�ޤ�, ̵�±�ˤ�����������������ꤷ,
�֥��ͥ�������򤪤��ʤ�ʤ��ä�.
���ξ��, �ݥƥ󥷥�뱲�٤δ��ܾ�θ���(5-36)�ϰʲ��Τ褦�ˤʤ�.
%
\begin{eqnarray}
 \DP{\overline{q}}{y} = \beta + {f_0}^2 S / \Gamma.
\end{eqnarray}
%
$\Gamma$�ϰ���Ǥ���.
���ܾ줬����($S>0$)�ξ��, �ݥƥ󥷥�뱲�٤θ��ۤ���������Ȥʤ�.
����, Eady�Υ�ǥ�ǤϤ��θ��ۤϥ����ˤʤäƤ���.
��ͭ������(5-38)�Ϥ��ξ��, �ʲ��Τ褦�ˤʤ�.
%
\begin{eqnarray}
 (S Z - c)
  \left[
   \left( \frac{{f_0}^2}{\Gamma} \right)
   \left( \DD[2]{\Psi}{Z} - \DD{\Psi}{Z} \right)
   - \mu^2 \Psi
  \right]
  + \left( \beta + {f_0}^2 S / \Gamma \right) \Psi = 0.
\end{eqnarray}
%
���μ���(5-43)�����, �򤯤Τ������񤷤�.
�ʤ��ʤ�, $\partial \overline{q} / \partial y$�ϥ����ǤϤʤ�����Ǥ���.
�����������Ϲ�ή��Ķ�����ؿ����������ѷ��Ǥ���.
�ѷ������������ˤ����ƽ��פ��ѿ���
%
\begin{eqnarray}
 r = \frac{\Gamma \left( \beta + S {f_0}^2 / \Gamma \right)}
  {{f_0}^2 S \left( 1 + 4 \Gamma {f_0}^{-2} \mu^2 \right)^{1/2}}
\end{eqnarray}
%
�Ǥ���.
��ή��Ķ�����ؿ����ѷ��������������������줿���, $c$�˴ؤ���
�򤯤Τ������񤷤�.
��5-7��, $Z=1$�Ǥդ�������Ƥ�����֤ΰ�������
Green��������ʬ�ˤ�äƵ�᤿�ޤǤ���.
%
\begin{figure}[htb]
 \begin{center}
  \Depsf[][]{./eps/hwfig-04-07.eps}
 \end{center}
 \begin{quote}
  \caption{
  {\footnotesize Green(1960)�ˤ�äƷ׻����줿, $Z=1$�Ǥդ������줿���֤Ǥ�
  Charey��ǥ�ΰ������ο�
  (From J.S.A Green, ``A problem in Baroclinic Stability''
  {\itshape Royal Meteorological Society, } 1960.)
  �ʤ�, ���οޤ�
  Haltiner, G. J., and Williams, R. T., 1979:
	Numerical Prediction and Dynamic Meteorology.
	{\itshape John Wiley \& Sons,} 477pp ���ž��.}
  }
  \Dfiglab{hwfig-04-07}
 \end{quote}
\end{figure}
%
����ɸ����Ĺ$1/\mu$, �ĺ�ɸ�����α�ľ����$S$�Ǥ���,
������ĹΨ$\mu c_i$�����Ͷ����Ǥ���.
��������������
\footnote{
 0, 0.1, 0.2 �Ȥ���ź�����դ���������Ӥ����������ʬ�����Τ��ȤǤ���.
}
��$r=1$����Ω�����Ǥ���.
������, ��ĹΨ�κ����ͤ򼨤��Ƥ���.
$r$����Ω�����Ǥ������������ʳ��ʤ��, �԰���ϵ���������.
����ɸ�˶ᤤ������$r=2$, $r=3$���б�����.

�����������������������ȸ�����.
�ʤ��ʤ�, ��Ω�������԰�����ΰ�Ȱ���, �ޤ�����Ω���ΰ��
ʬ���Ƥ��ʤ�����Ǥ���.
������, 0.1 �Ȱ����դ��������⳰����ĹΨ��, ���˾���������,
�໤�Τ����Ǥ��Ȥ�¸�ߤ��ʤ�.
$S$�������礭���ͤ�Ȥä����, �ޤκǤ��԰������ʬ��
Eady�η�̤˶᤯�ʤ�.













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