%  ɽ��  �Ƥ� GFD ���ߥʡ� 2006 �ֱ�Ρ���
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%  ����  2006/09/07 ��������
%        2006/09/10 ��������ʿ
%        2006/09/12 ��������ʿ
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\Dtitle[GFD ���ߥʡ� 2006 ��]{%
GFD ���ߥʡ� 2006 ��: \\
�����¸���Τʤ� \\ �Ȥγ�ư�٥ե�å���}
\Dauthor[�ֱ�:��ë����Ϻ]{%
�ֱ��: ��ë����Ϻ\\ ��: ��������, ��������ʿ}
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\begin{document}
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\section{�Ȥγ�ư�٤Ȥ��Υե�å��� -- ������}

\subsection{��ʿ�󼡸�$\beta$ʿ�̾����ȯ��ή�Τξ��}

��ñ�Τ���
��ʿ�󼡸� $\beta$ �̾����ȯ��ή�Τ�ͤ���.
�����������Ϥϰʲ����̤�Ǥ���:
\begin{eqnarray}
&&\DP{u}{t} + u\DP{u}{x} + v\DP{u}{y} - fv = -\frac{1}{\rho}\DP{p}{x}, 
    \Deqlab{2D������ȯ��u} \\
&&\DP{v}{t} + u\DP{v}{x} + v\DP{v}{y} - fv = -\frac{1}{\rho}\DP{p}{y}, 
    \Deqlab{2D������ȯ��v} \\
&&\DP{u}{x} + \DP{v}{y} = 0.
    \Deqlab{2D������ȯ��Ϣ³}
\end{eqnarray}
������ $f=f_{0} + \beta y$ �Ǥ���.
��ȯ��ή�ΤǤ��뤫��, $u, v$ ��ή���ؿ� $\psi$ ��
�Ѥ���
\[
    u = -\DP{\psi}{y}, \quad v = \DP{\psi}{x}
\]
��ɽ�蘆���.

������������
$\displaystyle\DP{}{x} \Deqref{2D������ȯ��v}- \DP{}{y}\Deqref{2D������ȯ��u}$ 
����������:
\begin{equation}
    \DP{\zeta}{t} + u \DP{\zeta}{x} + v\DP{\zeta}{y} + \beta v = 0.
	\Deqlab{2D������ȯ������������}
\end{equation}

����������ͤʴ��ܾ���ξ����ͤ�, 
ʪ���̤����ʴ��ܾ��, ������⿶���ξ����������Ѳ���������ʬ����.
����Ÿ���η����� $\varepsilon (\ll 1)$ �Ȥ���, 
\begin{eqnarray}
    \begin{array}{ccccc}
	\mathrm{order:} &  1 & \varepsilon & \varepsilon^{2}  & \cdots\\
	u =&  \overline{u}(y) &+ u' &+ u^{(2)}  &+\cdots \\
	v =&  & + v' & + v^{(2)}  &+\cdots \\
	\psi =& \overline{\psi}& +\psi' &+\psi^{(2)} &+ \cdots \\
	\zeta =& \overline{\zeta}(y)& + \zeta' &+ \zeta^{(2)} & +\cdots .
    \end{array}
\end{eqnarray}
�����Ǳ�����ź���Ϥ��줾��, Ǥ�դ�ʪ���� $a$ �ˤĤ���
$\mathcal{O}(a') =\varepsilon$,~
$\mathcal{O}(a^{n}) =\varepsilon^{n}$ ��ɽ�魯�Ȥ���.

��������������\Deqref{2D������ȯ������������}��
������
$\mathcal{O}(\varepsilon)$ ����������Ȥ�,
���������줿������������������:
\begin{eqnarray}
    \DP{\zeta'}{t} + \overline{u} \DP{\zeta'}{x} + \beta_{e} v' =0.
    \Deqlab{2D������������������}
\end{eqnarray}
������ $\beta_{e}$ ��
\begin{equation}
        \beta_{e} \equiv \beta - \DP[2]{\overline{u}}{y}
	    \Deqlab{��e �����}
\end{equation}
�Ȥ���.
$\beta_{e}$ �� $y$ �ˤ�����¸���ʤ��������դ��줿��.

\subsection{Rossby �ȤΤ����餤}

����β�Ȥ���ʿ���Ȳ���ꤹ��. $\psi_{0}$ ������Ȥ���, 
\begin{equation}
    \psi'  = \psi_{0}\exp\{i (kx + ly - \omega t)\}
	\Deqlab{ʿ���Ȳ�}
\end{equation}
����������������������\Deqref{2D������������������}����������:
\begin{eqnarray}
    &&- i\omega \zeta' + ik\overline{u}\zeta' + \beta_{e}\DP{\psi'}{x}= 0
	\nonumber \\
    &&-i\omega\{-(k^{2}+l^{2})\psi'\}
	+ ik\overline{u}\{-(k^{2}+l^{2})\psi'\} + \beta_{e} ik\psi' = 0,
	\Deqlab{�����ӡ���ʬ���ط���Ƴ��}
\end{eqnarray}
������ή���ؿ��ȱ��٤�������
\begin{eqnarray*}
    \zeta'
	= \Dlapla \psi'
	= -(k^{2}+l^{2})\psi'
\end{eqnarray*}
�Ǥ��뤳�Ȥ��Ѥ���.
\Deqref{�����ӡ���ʬ���ط���Ƴ��}�� $\omega$ �ˤĤ��Ʋ򤯤��Ȥ�,
Rossby �Ȥ�ʬ���ط�����������:
\begin{equation}
    \omega = k \overline{u} - \frac{\beta_{e} k}{k^{2}+ l^{2}}.
	\Deqlab{2D������ȯ�� Rossby �Ȥ�ʬ���ط���}
\end{equation}
������1��ϴ��ܾ�ΰ�ή��, ������2�ब�Ȥι�¤��ɽ���Ƥ���.

\Deqref{2D������ȯ�� Rossby �Ȥ�ʬ���ط���}���
����®�٤�
\begin{eqnarray}
    &&C_{x}
	= \frac{\omega}{k}
	= \overline{u} - \frac{\beta_{e}}{k^{2}+l^{2}}, \\
    &&C_{y}
	= \frac{\omega}{l}
	= \left(\overline{U} - \frac{\beta _{e}}{k^{2}+l^{2}}\right)\frac{k}{l},
\end{eqnarray}
��®�٤�
\begin{eqnarray}
    &&C_{gx} = \DP{\omega}{k}
	= \overline{u} + \frac{\beta_{e}(k^{2}-l^{2})}{k^{2}+l^{2}}, \\
    &&C_{gy} = \DP{\omega}{l}
	= \frac{2\beta_{e} kl}{k^{2}+l^{2}}
	\Deqlab{2D������ȯ�������ӡ�Cgy}
\end{eqnarray}
�Ȥʤ�\footnote{%
�Ȥγ�ư�٥ե�å��� $\Dvect{F}$ ��
$\Dvect{F} \equiv \Dvect{C}_{g} A$����������.
����ȼ���Ǥ�,
����ʿ�Ѥ�����ǤΥե�å�����ͤ���Τ� $C_{gy}$ (�� $C_{gz}$) �����ܤ��줿��.
}.
$\beta_{e} >0$ �ξ��ˤ�, 
����®�� $C_{x}$ �ϴ��ܾ��ή® $\overline{u}$ �������٤�.
������Ф��Ʒ�®�� $C_{gx}$ �� $\overline{u}$ �δط���
 $k$ �� $l$ ���羮�ط��˰�¸����.

\subsection{�Ȥγ�ư�٤Ȥ��Υե�å�����Ƴ��}

�����Ǿ����Υ��󥹥ȥ��ե�����ͤ��褦.
��������������������\Deqref{2D������������������}��ξ�դ�
$\zeta'$�򤫤���
\begin{equation}
    \DP{}{t}\frac{\zeta'^{2}}{2}
	+ \overline{u}\DP{}{x}\frac{\zeta'^{2}}{2}
	+ \beta_{e} v'\zeta' = 0.
\end{equation}
\Deqref{��e �����} ���
$\beta_{e}$ �� $y$ �ˤΤ߰�¸����Τ�,
ξ�դ� $\beta_{e}$�ǳ�ä���ʬ�����������
\begin{equation}
    \DP{}{t}\frac{\zeta'^{2}}{2\beta_{e}}
	+ \overline{u}\DP{}{x}\frac{\zeta'^{2}}{2\beta_{e}}
	+  v'\zeta' = 0.
\end{equation}
���μ�������ʿ�Ѥ���.
Ǥ�դ�ʪ���� a ������ʿ�Ѥ� $[a]$ ��ɽ�魯�Ȥ���:
\begin{equation}
    [a] \equiv \frac{1}{x_{0}}\int_{0}^{x_{0}} a \Dd x.
\end{equation}
�󼡸� $\beta$�̤ȤϤ���,
��Ȥ�ȵ��̤�ͤ������Τ�ʿ�Ѥ���ȸ�����äƤ���, �Ȥ���
(���ʤ������������ͤ���).
��ä� $[\overline{u}] = 0$ �Ǥ��뤫��,
\begin{equation}
    \DP{}{t}
	\biggl[
	\frac{\zeta'}{2\beta_{e}}
	\biggr]
	+
	[v'\zeta'] = 0.
\end{equation}
����� $v'\zeta'$ �� Ϣ³�μ��� $u'$ ����($=0$) ����Ϳ���ѷ������
\begin{eqnarray*}
    v'\zeta' - u'\bigl(\DP{u'}{x} + \DP{v'}{y}\bigr)
	&=& v' \bigl(
	\DP{v'}{x} - \DP{u'}{y}
	\bigr)
	- u'\bigl(\DP{u'}{x} + \DP{v'}{y}\bigr) \\
    &=&
	\DP{}{x}\bigl(\frac{1}{2}v'^{2}\bigr)
	- v'\DP{u'}{y} - \DP{}{x}\bigl(\frac{1}{2}u'^{2}\bigr) -u'{\DP{v'}{y}}
	\\
    &=&
	\DP{}{x}\biggl(\frac{v'^{2}-u^{2}}{2}\biggr)
	+\DP{}{y}(-u'v')
\end{eqnarray*}
�Ǥ��뤫��, ���
\begin{equation}
    [v'\zeta'] = \DP{}{y}[-u'v'].
\end{equation}
��ä�
\begin{equation}
    \DP{}{t}
	\biggl[
	\frac{\zeta'}{2\beta_{e}}
	\biggr]
	+
	\DP{}{y}[-u'v']
	= 0
	\Deqlab{zonal_mean_enstropy}
\end{equation}
������줿. ������
\begin{eqnarray}
    A &\equiv& \bigl[\frac{1}{2}\zeta'^{2}\bigr] \\
    \Deqlab{2D����ȯ��wave activity}
    \Dvect{F} &\equiv& 
	\left(
	 \begin{array}{c}
	     0 \\
	     -[u'v']
	 \end{array}
	      \right)
	\Deqlab{2D����ȯ��wave activitiy flux}
\end{eqnarray}
����������,
\begin{equation}
    \DP{A}{t} + \Ddiv\Dvect{F} = 0
	\Deqlab{2D��wave activity ��¸§}
\end{equation}
��ɽ�碌��.
$A$ ��{\bf �Ȥγ�ư��}(wave activity),
$\Dvect{F}$ ��{\bf �Ȥγ�ư�٥ե�å���}(wave activity flux)
�ȸƤФ��.

\subsection{\Dvect{F} ���Ȥˤ��$A$ �ΰ�ή�Ǥ��뤳�Ȥξ���}

$[-u'v']$ ��, �����ӡ��Ȥˤ�� $A$ �ΰ�ή�ȤʤäƤ�������������.
����β�Ȥ���ʿ���Ȳ�\Deqref{ʿ���Ȳ�}���ꤷ�Ƥ���Τ�,
\begin{eqnarray}
    A
	&=&
	\biggl[
	\frac{\zeta'}{2\beta_{e}}
	\biggr]
	=
	\frac{(k^{2}+l^{2})\psi_{0}^{2}}{4\beta_{e}}.
\end{eqnarray}
�ޤ�, 
\begin{eqnarray}
    \left[ u'v' \right]
	&=&
	\biggl[
	\DP{\psi'}{x}\DP{\psi'}{y}
	\biggr]
	=
	[k\psi_{0}\cos(\cdots)\cdot l\psi_{0}\cos(\cdots)]
	= \frac{kl\psi'^{2}}{2}  \nonumber \\
    &=& \frac{2\beta_{e}kl}{k^{2}+l^{2}}\cdot\frac{(k^{2}+l^{2})\psi^{2}}{4\beta_{e}} \nonumber\\
    &=& C_{gy}[A].
\end{eqnarray}
��ä�
\begin{equation}
    \Dvect{F} =
	\left(
	 \begin{array}{c}
	     0\\
	     C_{gy}A
	 \end{array}
	\right)
\end{equation}
�Ǥ���, ��®�٤ˤ�ä� $A$ ����ή����Ƥ�����������줿.


\subsection{�Ȥγ�ư�٤ˤ����������®(��®)}

�Ȥγ�ư�� $A$�λ����Ѳ���,
�������λ����Ѳ��Ȥδط��ˤĤ��ƹͤ���.
������ $u$ �� $\mathcal{O}(\varepsilon^{2})$ �μ���
\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}
���μ�������ʿ�Ѥ���. ���λ�
\begin{eqnarray}
    &&[v^{(2)}] =
	\biggl[
	\DP{\psi^{(2)}}{x}
	\biggr] = 0 \\
    &&\biggl[
	\overline{u}\DP{u^{(2)}}{x}
	\biggr]
	=
    \biggl[
	\DP{}{x}\bigl\{
	\overline{u} u^{(2)}
	\bigr\}
	\biggr]
	=0
\end{eqnarray}
�Ǥ��뤫��, ��̤Ȥ���
\begin{equation}
    \DP{}{t}[u^{(2)}]
	= - \DP{}{y}[u'v'] = \Ddiv\Dvect{F} = [v'\zeta']
	\Deqlab{������®(2)}
\end{equation}
�������, 
\Deqref{������®(2)}���Ȥγ�ư�٤���¸§\Deqref{2D��wave activity ��¸§}
���碌����ǰʲ��μ���������:
\begin{equation}
    \DP{}{t} [u^{(2)} + A] = 0
	\Deqlab{������®(1)}
\end{equation}

\subsection{�Ȥγ�ư�٤���������®\Deqref{������®(1)}�β��}

��������®�μ�\Deqref{������®(1)}��,
�Ȥγ�ư�� $A$ ������(����)�����,
��������®(��®)���뤳�Ȥ�ɽ�路�Ƥ���.

�ޤ�, �Ȥγ�ư����¸§\Deqref{2D��wave activity ��¸§} ��
��������®\Deqref{������®(1)} ��,
���줾����Ω����¸���ȤʤäƤ���:
\begin{eqnarray}
    &&\DP{[A]}{t} + \Ddiv \Dvect{F} = 0 , \\
    &&\DP{[u^{(2)}]}{t}  - \Ddiv \Dvect{F} = 0.
\end{eqnarray}
��ä�
���� $y$ �ˤ����� $A$ ������($u^{(2)}$ ������)���Ƥ���ʤ��, 
�̤� $y$ �ˤ����� $A$ ������($u^{(2)}$ ������)���Ƥ�����,
���λ�,
$A$ �θ����褫�� $A$ �����ð�ؤ� $A$ ��͢������Ƥ���.
���Υե�å����� $\Dvect{F}$ �ˤ�ä�ɽ������Ƥ���(\Dfigref{��1})
\begin{figure}[bth]
    \begin{center}
	\Depsf[][1.5cm]{figs/Fig1-1.ps}
	\caption{$\Dvect{F}$ �� $A$ �ΰ�ή��ɽ�路�Ƥ����.\newline%
	$y$����¦�� $A$ �θ�����($u^{(2)}$ �����ð�), %
	$y$����¦�� $A$ �����ð�($u^{(2)}$ �θ�����) �򼨤�.
	$A$ ������(����)�� $\Dvect{F}$ �μ�«(ȯ��)�Ǥ��뤫��, 
	�ޤκ����鱦��, Rossby �Ȥˤ�ä� $A$ ����ή����Ƥ���.}
	\Dfiglab{��1}
    \end{center}
\end{figure}

�ޤ� 
$\displaystyle{\Dvect{F} = [v'\zeta'] = -\DP{}{y}[u'v'].}$
�Ǥ��뤫��, 
\[
    \DP{}{t}[u^{(2)}] + \DP{}{y}[u'v'] = 0
\]
�Ȥʤ�.
$\displaystyle{\DP{}{y}[u'v']}$ ���������ˤ��������ư�̤ΰ�ή�Ǥ��뤫��,
������ư�̥ե�å����μ�«(ȯ��)�ˤ�ä�������®(��®)�������Ƥ���,
�Ȥ��ᤵ���(\Dfigref{��3}).
\begin{figure}[hbt]
    \begin{center}
	\Depsf[][2.5cm]{figs/Fig1-3.ps}
	\caption{
	������®(��®)��
	$\displaystyle{\frac{\partial}{\partial y}[u'v']}$ �ˤ����.
	\newline %
	$[u'v']$ �����̤���Ӥ����,
	$[u'v']$ �θ�����Ǥ�������®($\Dd u^{(2)} >0$)�������Ƥ���(��),
	$[u'v']$ �����ð�Ǥ�������®($\Dd u^{(2)}<0$)�������Ƥ���(��).
	}
	\Dfiglab{��3}
    \end{center}
\end{figure}

\newpage%----------------------------------------
\subsection{���õ���: ������®(��®)�� $[v'\zeta']$ ��ľ�ܸ����Τϲ��Τ�?}

����äȹͤ�����Ի׵ĤǤ���. ���ǲ��Ǥ��ޤ���? �ȸ���������ʶ�ꤷ��.
�ʲ������, �������о줷������, ���������ƤǤ���.
�ե������Ǥ��Ƥ��ʤ����¿������Τ�, �ӥǥ������ȤȤ��į���줿��.

\subsubsection{��ë�����������}

������į����
\begin{eqnarray}
    &&\DP{u}{t} + u\DP{u}{x} + v \DP{u}{y} + fv
	= -\DP{}{x}\bigl(\frac{p}{\rho_{0}}\bigr) \nonumber, \\
    &&\quad\to
	\DP{u}{t}
	- (f+\zeta) v
	= -\DP{}{x}\biggl\{ \frac{p}{\rho_{0}} + K\biggr\},
	\quad K = \frac{1}{2}(u^{2}+v^{2})
\end{eqnarray}
�ȤʤäƤ���.
�ǽ�������Ȥ����, �٥�̥��ؿ��θ��ۤ� $-(f+ \zeta)v$ ���������ʤäƤ�.
������ $\zeta v$ ����®(��®)�˸���(��������?).
�ޤ�, ���λ��򳨤ǲ��Ǥ����������.

\begin{itemize}
    \item ��߷������: ������ή�������ˤ����������®
    \item ���礵����: ���Υڥ����̤���˰�ư���뤳�Ȥˤ����������®
    \item �в�������(1): 
    \item �в�������(2):����ӥ󥤥�ѥ륹�ǹͤ����
	  \[
	      \int_{a}^{b}\Delta \overline{u}\Dd y
		  = [y\Delta\overline{u}]_{a}^{b}
		  + \int_{a}^{b}y\Delta\overline{\zeta}\Dd y
	  \]
\end{itemize}

\subsubsection{�Ӥ���β���}

�Ȥγ�ư�٥ե�å����θ�����:
���� $\zeta$ ���ۤ˽ФƤ��ʤ��Τ�, ��������®������.

$u^{(2)}$ ��������ͤ�����ˤ�,
$\zeta^{(2)}$ ����ʬ�����������Ȱ��ˤ��ƹͤ�, 
$\zeta^{(2)}$ ���� $u^{(2)}$ ��ղ򤭤����Ȥ�ɬ�פˤʤ�.
������®�پ��ղ򤭤�����ˤ�, ���������ΤäƤ��ʤ���Фʤ�ʤ�.
�������ˤĤ��Ƥϰ��ۤ�λ��(������ $0$ �Ȥ�)��¸�ߤ���.
�����ռ�������, ���ǲ�ᤷ�褦�Ȥ����,
����Ⱥ��������붲�줬����Τ����դ��褦(\Dfigref{��1-4}).
\begin{figure}[bth]
    \begin{center}
	\Depsf[][4.5cm]{figs/Fig1-4.eps}
	\caption{����������, �ۥ磻�ȥܡ��ɾ�Ǵ���������.\newline %
	(��)��«���̤ذ�ư���Ƥ�������, $[u'v']$ ��ʬ��(��)��
	(��)��ư�̥ե�å���(��������®).
	}
	\Dfiglab{��1-4}
    \end{center}
\end{figure}



\newpage %----------------------------------------
\section{�Ҹ��̽۴ľ�Ǥ��Ȥγ�ư�٥ե�å���. }

����Ǥ�, ��ʿ�󼡸����Ȥγ�ư�٤�ͤ���.
�ʲ��Ǥ�����ή�Τξ���ͤ�,
����ʿ�Ѥ��뤳�ȤǻҸ��̽۴ľ�Ǥ��Ȥγ�ư�٥ե�å�����Ƴ�Ф���.

\subsection{ $\beta$ ʿ�̾������ή�Τξ��}

$\beta$ʿ�̾������ή�Τ�ͤ���.
��ľ��ɸ�Ȥ����п����Ϻ�ɸ
\begin{equation}
    z \equiv - H\ln\bigl(\frac{P}{P_{s}}\bigr)
\end{equation}
��Ȥ�, 
���Ϲ�ή��������(�ʲ�, QG ��������)�����ä�Ϥ��:
\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{QGu}\\
    && \DP{v_{g}}{t}
	+ u_{g}\DP{v_{g}}{x}
	+ v_{g}\DP{v_{g}}{y}
	+ f(u_{g} + v_{a}) = -\DP{\phi}{y}, \Deqlab{QGv}\\
    && \DP{\theta}{t}
	+ u_{g}\DP{\theta}{x}
	+ v_{g}\DP{\theta}{y}
	+ w_{a}\DP{\theta_{0}}{z} = 0 \Deqlab{QGT}, \\
    && \DP{u_{a}}{x} + \DP{v_{a}}{y}
	+\frac{1}{\rho_{0}}\DP{}{z}(\rho_{0}w_{a}) = 0. \Deqlab{QGϢ³}
\end{eqnarray}
������ $(u_{g}, v_{g})$ ���Ϲ�ή��ʬ, 
$(u_{a}, v_{a})$ �����Ϲ�ή��ʬ�Ǥ���,
$(u_{g},v_{g}) \gg (u_{a}, v_{a})$ �Ǥ���. �ޤ�,
\begin{eqnarray}
    && u_{g} = -\DP{\psi}{y}, 
	v_{g} = \DP{\psi}{x},  \\
    && \psi \equiv \frac{\phi - \phi_{0}}{f_{0}}, \psi_{0} = \psi_{0}(z)\\
    && \theta = \frac{H}{R}f_{0}e^{\kappa z/H}\DP{\psi}{z} \\
    && \theta_{0} = \theta_{0}(z), \\
    && \rho_{0} = \rho_{s} e^{-z/H}
\end{eqnarray}
�Ǥ���.

������������
$\displaystyle{\DP{}{x}\Deqref{QGv} - \DP{}{y}\Deqref{QGu}}$ ���
������:
\begin{equation}
    \DP{\zeta_{g}}{t}
	+ u_{g}\DP{\zeta_{g}}{x}
	+ v_{g}\DP{\zeta_{g}}{y}
	+ \beta(v_{g} + v_{a})
	=
	\frac{f}{\rho_{0}}\DP{}{z}(\rho_{0}w_{a}).
	\Deqlab{QG����}
\end{equation}
�ݥƥ󥷥�뱲��(�ʲ�, PV)��¸§��,
���̤μ�\Deqref{QGT} ���ѷ���
\footnote{
�褺���Ϲ����β��٤μ�\Deqref{QGT} ��ξ�դ�
$\theta_{0z} \equiv \partial \theta_{0}/\partial z$�ǳ��, 
$\rho_{0}$ �򤫤��뤳�Ȥ�,
\[
  \left(\DP{}{t}+u_g\DP{}{x}+v_g\DP{}{y}\right)
    \left(\rho_0\frac{\theta}{\theta_{0z}}\right)+\rho_0w_a=0.
\]
���μ��� $z$ ����ʬ��, �ʲ��δط���
\begin{eqnarray*}
    &&\DP{u_{g}}{z} = -\DP{}{y}\DP{\psi}{z}
	= - \frac{Re^{-\kappa z/H}}{Hf_{0}}\DP{\theta}{y}, \\
    &&\DP{u_{g}}{z} = \DP{}{x}\DP{\psi}{z}
	=  \frac{Re^{-\kappa z/H}}{Hf_{0}}\DP{\theta}{x}, \\
    && \frac{\theta}{\theta_{0z}}
	= \frac{(HR^{-1}f_0e^{\kappa z/H}\psi _{z})}{\theta_{0z}}
	= \frac{f_{0}}{N^{2}}\DP{\psi}{z}
\end{eqnarray*}
���Ѥ����ѷ����뤳�Ȥ�,
\[
\DP{}{z}(\rho_{0}w_{a}) 
	= 
	-\left(
	  \DP{}{t}+u_{g}\DP{}{x} + v_{g}\DP{}{y}
		\right)
	\DP{}{z}
	\left(
	 \rho_{0}\frac{f_{0}}{N^{2}}\DP{\psi}{z}
	\right).
\]
��������������α��դ�������, ��������.}, �����������ȹ�碌�뤳�Ȥ�������:
\begin{eqnarray}
  && \left(\DP{}{t}+u_g\DP{}{x}+v_g\DP{}{y}\right)q = 0 \\
    \Deqlab{QGq}
  && q \equiv f_0 + \beta y + \psi _{xx} + \psi _{yy} + \frac{1}{\rho_0}
      \DP{}{z}\left(\rho _0\frac{f_0^{2}}{N^{2}}\DP{\psi}{z}\right)
      \Deqlab{qdef}
\end{eqnarray}

�ʹߤ��Ϲ�ή��ʬ��ɽ�魯ź��$~_{g}$���ά����. 
�ޤ�, ���Ĥ�ź�� $_x, _y, _z$  �ϳơ� $x$ ��ʬ, $y$ ��ʬ, $z$ ��ʬ��ɽ�魯���ȤȤ���.

\subsection{�Ȥγ�ư�٤Ȥ��� flux ��Ƴ��}

�����Ʊ�ͤ�, ���̤�ʿ��ή����Ÿ���������������, 
PV ��¸§\Deqref{QGq} ������������.
���ܾ�� $x$ �˰�¸���ʤ��Ȥ���:
\begin{eqnarray}
    \begin{array}{ccccc}
	\mathrm{order:} &  1 & \varepsilon & \varepsilon^{2}  & \cdots\\
	q =&  \overline{q}(y,z, t) &+ q' &+ q^{(2)}  &+\cdots \\
	u =&  \overline{u}(y,z, t) & + u' & + u^{(2)}  &+\cdots \\
	\psi =& \overline{\psi}& +\psi' &+\psi^{(2)} &+ \cdots \\
    \end{array}
\end{eqnarray}

���������줿 PV �ξ���μ���
\begin{eqnarray}
    \DP{q'}{t} 
	+ \overline{u}\DP{q'}{x} + \DP{\overline{q}}{y}v'=0
\end{eqnarray}
�Ȥʤ�. ����� $q'$ �򤫤������������,
\begin{equation}
    \DP{}{t}\frac{q'^{2}}{2} 
	+ \overline{u}\DP{}{x}\frac{q'^{2}}{2} 
	+ \overline{q}_{y}v'q'=0  
\end{equation}
��������.

�Ȥγ�ư�� $A$ ��
\begin{equation}
    A \equiv \frac{q'^{2}}{2\overline{q}_{y}}
\end{equation}
�Ȥ��뤳�Ȥ�,
\begin{equation}
    \DP{}{t}A
	+ \DP{}{x}(\overline{u}A)
	+ v'q'
	= 0
\end{equation}
��������. ����� $v'q'$ ���ѷ������,
\begin{eqnarray}
    v'q'
	&=& 
	v'\biggl\{
	\DP{v'}{x} - \DP{u'}{y} 
	+ \frac{1}{\rho_{0}}
	\biggl(
	\rho_{0}\frac{f_{0}^{2}}{N^{2}} \DP{\psi'}{z}
	\biggr)
	\biggr\} \nonumber \\
    &=&
	\DP{}{x}\biggl(\frac{(v'^{2}-u'^{2})}{2}\biggr)
	+\DP{}{y}(-u'v')
	+\frac{\psi'_{x}}{\rho_{0}}
	\DP{}{z}
	\biggl\{
	\rho_{0}\frac{f_{0}^{2}}{N^{2}}\DP{\psi'}{z}
	\biggr\} \nonumber \\
    &=&
	\DP{}{x}\biggl(\frac{(v'^{2}-u'^{2})}{2}\biggr)
	+\DP{}{y}(-u'v')
	+
	\biggl\{
	\frac{1}{\rho_{0}}
	\DP{}{z}
	\left(
	\rho_{0}\frac{f_{0}^{2}}{N^{2}}\psi'_{x}\psi'_{z}
       \right)
	-
	\DP{}{x}
	\left(
	 \frac{1}{2}
	 \frac{f_{0}^{2}}{N^{2}}\psi'^{2}_{z}
	\right)
	\biggr\} \nonumber \\
    &=&
	\DP{}{x}
	\biggl\{
	\frac{1}{2}
	\left(
	 v^{2}-u^{2}- \frac{1}{2}\frac{f^{2}}{N^{2}}\psi'^{2}_{z}
	\right)
	\biggr\}
	+
	\DP{}{y}(-u'v')
	+
	\frac{1}{\rho_{0}}\DP{}{z}
	\biggl\{
	\rho\frac{f^{2}}{N^{2}}\psi'_{x}\psi'_{z}
	\biggr\}  \nonumber \\
    &=&
	\DP{}{x}
	\biggl\{
	\frac{1}{2}
	\left(
	 v^{2}-u^{2}- \frac{1}{2}\frac{f^{2}}{N^{2}}\psi'^{2}_{z}
	\right)
	\biggr\}
	+
	\DP{}{y}(-u'v')
	+
	\frac{1}{\rho_{0}}\DP{}{z}
	\biggl\{
	\rho_{0}f_{0}\frac{v'\theta'}{\theta_{0z}}
	\biggr\}	
\end{eqnarray}
�Ȥʤ�. �Ǹ�μ��ѷ��Ǥ�
\[
    N^{2} = \frac{g}{\theta}\theta_{0z}
\]
���Ѥ���.  ���μ��ϥե�å��������ȤʤäƤ���.

������ 1 ��� $\psi'^{2}_{z}$ ��, $\psi'_{z} \propto \theta$  ���
ͭ�����֥��ͥ륮�������㤹���̤Ǥ���.
��ä��ȤΥ��ͥ륮��
\begin{equation}
    e = \frac{1}{2}\bigl\{
	\psi_{x}'^{2}
	+ \psi_{y}'^{2}
	+ \frac{f_{0}^{2}}{N^{2}}\psi_{x}'^{2}
	\bigr\}
\end{equation}
���Ѥ��Ƽ��ѷ���Ԥʤ���,
\begin{eqnarray}
    \frac{1}{2}(v'^{2}-u'^{2}- \frac{f^{2}}{N^{2}}\psi'_{z})
	&=&
	v'^{2} - \frac{1}{2}(v'^{2}+u'^{2}+\frac{f^{2}}{N^{2}}\psi'_{z}) 
	\nonumber \\
    &=& v'^{2} -e
\end{eqnarray}
�Ȥʤ�.

�ʾ�λ������Ȥγ�ư�� $A$ ����¸§������줿
\footnote{%
$A$ ��
{\bf pesudo momentum}(����ư��)�ȸƤФ��. 
���̤� pesudo momentum �Υե�å�����
{\bf ���ꥢ�å���--�ѡ���ե�å���}(Eliassen--Palm flux)�ȸ���.
}:
\begin{equation}
    \DP{}{t}A
	+
	\frac{1}{\rho_{0}}\Ddiv \rho_{0}
	\left(
	 \begin{array}{c}
	     v'^{2} - e \\
	     - u'v' \\
	     \rho_{0}f_{0}
		 (v'\theta'/\theta_{0z})
	 \end{array}
	\right) = 0.
	\Deqlab{QGA��¸}
\end{equation}

����� \Deqref{QGA��¸} �� zonal mean ���뤳�Ȥ�,
\begin{equation}
    \DP{}{t}[A]
	+
	\frac{1}{\rho_{0}}\Ddiv\Dvect{F} = 0.
\end{equation}
������ $\Dvect{F}$ �ϻҸ��̽۴ľ�Ǥ��Ȥγ�ư�٥ե�å����Ǥ���,
��������������:
\begin{equation}
\Dvect{F}  \equiv
    \rho_{0}\left(
     \begin{array}{c}
	 0, \\
	 - [u'v'],\\
	 f_{0}
	     [v'\theta']/\theta_{0z}
     \end{array}
    \right)
    \Deqlab{QGF}
\end{equation}

\subsection{TEM(Transform Eulerian Mean)��������}

�ʲ��Ǥ� Transformed Eulerian Mean (�ʲ�, TEM)�������Ϥ�Ƴ�Ф���.
TEM �������ϤˤĤ��Ƥ�
�������������������뤬, 
�ִ��ܾ����������®���Ф����Ȥαƶ����꼰�������ϡפȲ�᤹����ɤ�.

���ܾ�μ��� zonal mean ��
\begin{eqnarray}
    &&\DP{[u]}{t} + \DP{}{y}[u'v'] - f [v_{a}] = 0,  \\
    &&\DP{[\theta]}{t} + \DP{}{y}[v'\theta'] 
	+ \biggl[w_{a}\DP{\Theta_{0}}{z} \biggr] = 0, \\
    &&\DP{}{y}v_{a} + \frac{1}{\rho_{0}}\DP{}{z}\rho_{0}w_{a} = 0
\end{eqnarray}
�Ȥʤ�. �������ѷ����뤳�Ȥ� TEM ����������������:
\begin{eqnarray}
    &&\DP{[u]}{t} - fv_{a}^{*}  = \Ddiv \Dvect{F} \\
    &&\DP{[\theta]}{t} 
	+
	\biggl[w_{a}^{*}\DP{\Theta_{0}}{z}\biggr] = 0 \\
    &&\DP{}{y}v_{a}^{*} + \frac{1}{\rho_{0}}\DP{}{z}w_{a}^{*} = 0
\end{eqnarray}
������ $v_{a}^{*}, w_{a}^{*}$ �ϻĺ��۴ĤȸƤФ�, ��������������:
\begin{eqnarray}
    &&v_{a}^{*} \equiv [v_{a}] 
	- \frac{1}{\rho_{0}}\DP{}{z}
	\biggl( \rho_{0}\frac{v'\theta'}{\Theta_{0z}}\biggr) \\
    &&w_{a}^{*} \equiv [w_{a}]
	+ \DP{}{y}\biggl[ \frac{v'\theta'}{\Theta_{0z}}\biggr]
\end{eqnarray}

\subsection{�Ҹ��̽۴ľ�Ǥ� flux �ˤ����������®(��®)}

��������®����®�� EP flux �μ�«��ȯ���Ȥδط���
\begin{eqnarray*}
    \DP{}{t}[A] + \Ddiv \Dvect{F} = 0 \\
    \DP{}{t}[u] - f v_{a}^{*} = \Ddiv \Dvect{F} 
\end{eqnarray*}
�Ȥʤ�. ���ξ�����������®�ˤĤ��Ƥ�ʪ��Ū���ϰʲ����̤�Ǥ���.

�ե�å����� $y$ ��ʬ�ˤĤ��Ƥβ���
�󼡸��ξ����Ѥ��ʤ�(\Dfigref{��1}����).
���ʤ���ե�å������������ˤ��������ư�̤�͢����ɽ�����Ƥ���.

�ե�å����� $z$ ��ʬ�β��ϲ������δط������Ѥ���.
\Dfigref{��2-1}�ˤ����� $[v'\theta'] > 0$ �ʤ��,
\[
    \DP{[u']}{z} \propto -\DP{\theta'}{y}
\]
�Ǥ��뤫��, ���ٷ��٤���ä���,
���ؤ�������®������, ���ؤ�������®��������.
���λ� $\Ddiv{\Dvect{F}}$ �Ͼ��ؤǼ�«, ���ؤ�ȯ�����Ƥ���,
$A$ �����ؤ�����ؤ�͢������Ƥ��뤳�ȤȤʤ�.
\begin{figure}[bth]
    \begin{center}
	\Depsf[][]{figs/Fig2-1.eps}
	\caption{}
	\Dfiglab{��2-1}
    \end{center}
\end{figure}

\newpage %----------------------------------------
\section{����ʿ�Ѥ���ʤ��ե�å���}

���� 
\begin{equation}
    \psi' = \psi_{0} \sin(kx + ly - \omega t) = \psi_{0}\sin\chi
\end{equation}
��ɽ�蘆���Ȥ���(����� $\chi$ �Ȥ���). ���λ�
\begin{eqnarray}
    u' &=& -\DP{\psi'}{y} = -l \psi_{0}\cos\chi \\
    v' &=&  k\psi_{0}\cos\chi
\end{eqnarray}
���
\begin{equation}
    -u'v' = kl \psi_{0}^{2}\cos^{2}\chi
\end{equation}
�Ȥʤ�. 

��«�����Ť򵭽Ҥ��뤿��ˤ�,
ή��ξ�����Ф��Ƥʤ�餫�ΰ���ʿ����ɬ�פˤʤ�.
����ޤǤΤ褦�� zonal mean ��Ȥ�ʤ��,
\begin{equation}
    \frac{1}{2\pi}\int_{0}^{2\pi}(-u'v')\Dd x \simeq \frac{1}{2}kl \psi_{0}^{2}
\end{equation}
�Ȥʤ�.
�������ʤ��� zonal mean ��Ȥä�
�Ȥγ�ư�٥ե�å����Ǥ�, �ե�å����� $x$ ��ʬ���ä��뤿��,
��®�ο�ʿ���Ť򵭽ҤǤ��ʤ�.

�ʲ��Ǥ�, ����ʿ�Ѥ�Ȥ�ʤ��ե�å������������Ĥ���ӥ塼����
\footnote{%
�ֱ�Ԥ� Trenberth(1986) �Υ�ӥ塼��ͽ�ꤷ�Ƥ�����, ���֤��Թ�ǳ䰦���줿.}.


\subsection{Plum(1986) �Υե�å���}

ʿ�����Ȥ���, �Ȥμ��������Τʾ��ˤ�, ���μ�����ʿ������������Ǥ���
\begin{equation}
    \frac{1}{T}\int_{0}^{T}(-u'v')\Dd t
	\simeq kl \psi_{0}^{2}
	\frac{1}{T}\int_{0}^{T}\cos^{2}\chi\Dd t
	\to_{T\to \infty} \frac{1}{2}kl \psi_{0}^{2}
\end{equation}
�Ȥʤ�. 
����ʿ�����ˤĤ���, ���η׻��ξܺ٤Ͼʤ�, ��ά�ˤȤɤ��.
����ʿ�ѤˤĤ���, 
\begin{equation}
    \int_{0}^{T} u' \Dd t = 0
\end{equation}
�Ȳ��ꤹ��. 
�ʲ��Ǥϻ���ʿ�Ѥ����̤� $[a]$ ��ɽ�魯�Ȥ���.
��̤Ȥ����Ȥγ�ư�٥ե�å�����
\begin{eqnarray}
    &&\DP{}{t} [A] + \frac{1}{\rho_{0}}\Ddiv \Dvect{F} = 0 \\
    &&\Dvect{F}
	= \rho_{0}
	\left(
	 \begin{array}{c}
	     [u][A] + [v'^{2}] -[e] \\
	     -[u'v'], \\
	     \frac{f^{2}}{N^{2}}[\psi'_{x}\psi'_{z}]
	 \end{array}
	\right)
\end{eqnarray}
�Ȥʤ�. 

���줬 Plumb(1986) �δ���Ū�ʥ����ǥ��Ǥ���.
��ư����������ˤ�����ˤ�,
��ʬĹ�����֤�ʿ�Ѥ������«�����Ť򵭽ҤǤ���.

\iffalse
;�Ĥ��󤫤�ֳ����ߤ����פȤ�������������. 
�Ȥ�������, ���ǲ�ᤷ�Ƥߤ��.
\begin{eqnarray*}
    && \overline{V} \neq 0,  \\
    && \overline{u} > 0, |v'| > |u'| \\
    && F_{x}
	\sim \overline{u} A 
	+  \frac{(v'^{2} - u'^{2})}{2} > 0
\end{eqnarray*}
\fi


\subsubsection{$\Dvect{F} = \Dvect{C}_{g} A$ �ξ���}

Plumb(1986) ��Ƴ�Ф��줿 
�Ȥγ�ư�٥ե�å����ȷ�®�٥٥��ȥ뤬ʿ�ԤǤ��뤳�Ȥ��ǧ����.
���������� PV ��¸§
\begin{eqnarray}
    &&\DP{q'}{t} + \overline{u}\DP{q'}{x}+ v'\DP{\overline{q}}{y} = 0, \\
    && q' = \psi'_{xx} + \psi'_{yy} 
	+ \frac{1}{\rho_{0}}\DP{}{z}\biggl\{
	\rho_{0}\frac{f_{0}^{2}}{N^{2}}\DP{\psi'}{z}
	\biggr\}, \quad
	\rho_{0} = \rho_{s}\exp\bigl(-\frac{z}{H}\bigr)
	\Deqlab{���̾���}
\end{eqnarray}
�ˤ����� $\psi'$ �Ȥ���ʿ���Ȥ��ꤹ��:
\begin{equation}
    \psi' = \psi_{0}\sin(kx+ly+mz - \omega t)\exp\bigl(\frac{z}{2H}\bigr)
\end{equation}
����򱲰̾���\Deqref{���̾���}���������뤳�Ȥ�
\begin{equation}
    q' = - \biggl\{k^{2}+l^{2}+\frac{f_{0}^{2}}{N^{2}}\left(m^{2}+\frac{1}{4H^{2}}\right)\biggr\}\psi'.
\end{equation}
��ä�ʬ���ط���
\begin{equation}
    \omega
	=
	k \overline{u}
	-
	\frac{k\overline{q}_{y}}{%
	k^{2} + l^{2} + (f_{0}^{2}/N^{2})m^{2}
	}.
\end{equation}
����®�٤ȷ�®�٤�, 
�ȿ��٥��ȥ�$\Dvect{k}$��
\[
\Dvect{k} = 
\left(
\begin{array}{c}
    k\\
    l\\
    mf_{0}/N
\end{array}
\right)
\]
�Ȥ���
\begin{eqnarray}
    C_{x} &=& \overline{u} - \frac{\overline{q}_{y}}{\Dvect{k}^{2}}, \\
    C_{gx} &=& \overline{u}
	+
	\frac{k^{2}-l^{2}-(f^{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}{\Dvect{k}^{4}}\biggl(\frac{f^{2}}{N^{2}}\biggr)\overline{q}_{y}.
\end{eqnarray}

Plumb(1986) �ˤ������Ȥγ�ư�٥ե�å����� x ��ʬ$F_{(x)}$ ��
\footnote{���դ�ź�� $_x, _y, _z$ ����ʬ��ɽ�路�Ƥ���Τ�,
��ʬɽ���� $_{(x)}, _{(y)}, _{(z)}$ ��ɽ��. }
\begin{eqnarray}
    F_{(x)} 
	&=&
	[u][A] + [v'^{2}] -[e]
\end{eqnarray}
�Ǥ���,
\[
    v'^{2} = \frac{k^{2}}{2}\psi_{0}\cos^{2}\chi \quad \to \quad
    [v'^{2}] = \frac{1}{4}\psi_{0}^{2}, 
\]
��
\[
        [A]
	=
	\left[\frac{q'^{2}}{2\overline{q}_{y}}\right]
	=
	\frac{\Dvect{k}^{4}\psi_{0}^{2}}{4\overline{q}_{y}}
\]
�Ǥ��뤫��,
\begin{eqnarray*}
    [v'^{2}] - [e]
	&=&
	\frac{1}{2}[v'^{2}]
	-
	\frac{1}{2}[u'^{2}]
	-
	\frac{1}{2}\frac{f_{0}^{2}}{N^{2}}m'^{2} \\
    &=&
	\frac{(k^{2}-l^{2}-\frac{f^{2}}{N^{2}}m^{2})\overline{q}_{y}}{\Dvect{k}^{4}} \cdot \frac{\Dvect{k}^{4}}{4}\frac{\psi^{2}}{\overline{q}_{y}}
\end{eqnarray*}
��ä�, 
\begin{equation}
    F_{x} =
	\biggl(
	\overline{u} + \frac{k^{2}-l^{2}-\frac{f^{2}}{N^{2}}m^{2}}{\Dvect{k}^{4}}\overline{q}_{y}
\biggr)\cdot \frac{\Dvect{k}^{4}}{4}\frac{\psi_{0}^{2}}{\overline{q}_{y}}
= C_{gx}\cdot[A]
\end{equation}
�Ȥʤ�. $y, z$ ��ʬ�ˤĤ��Ƥ�Ʊ�ͤǤ���.

���ξ�� $\overline{v} = 0$ �ǵ���������,
$\overline{v}\neq 0$�Ǥ�ե�å�������®�٤�����(ʿ��)�Ǥ��뤳�Ȥ�������.

�ʾ�,
Plumb �Υե�å����ϰ�ư������(���ȡ���ȥ�å�)�ε��Ҥˤ����������Ǥ���.
�������ʤ�������������ε��Ҥˤϸ����ʤ�.
����ʿ�Ѥ�ȤäƤ�, ���������𤬶����ФƤ��뤿��, ��«�����Ť򵭽ҤǤ��ʤ�����.

\subsection{����������:Plumb(1985)�Υե�å���}

����®�٤����ˤ�ä���Ȥ�������������ξ��, 
Trenberth(1986) �� Plumb (1986) �Υե�å����Ǥ�
�����¸����õ�뤳�Ȥ��Ǥ��ʤ�. 
����������ؤ�Ŭ�Ѥ����ꤷ�ͤ���줿�Ȥγ�ư�٥ե�å�����, Plumb (1985) �Υե�å����Ǥ���.

�Ȥγ�ư����¸§�����ä�Ϥ��:
\begin{equation}
    \DP{}{t}A + \Ddiv \Dvect{F} = 0.
\end{equation}
������, ���̶�����¸�ߤ��ʤ�������ȯ���Ȥʤ�٥��ȥ� $\Dvect{G}$ ��
Ƴ����, �Ȥγ�ư����¸§�κ����������ղä���:
\begin{eqnarray}
    \DP{}{t}A + \Ddiv(\Dvect{F}+ \Dvect{G}) = 0.
\end{eqnarray}
$\Dvect{G}$ �ζ���Ū��ɽ���Ȥ��� Plumb �ϰʲ���Ϳ����
\footnote{Plumb �����Τ��Τ褦�˻פ��Ĥ����Τ�������}:
\begin{eqnarray}
    \Dvect{G} 
	=
	\frac{1}{4}
	\left(
	 \begin{array}{c}
	     \DP[2]{}{y}\psi'^{2}
		 +
		 \DP{}{z}\frac{f^{2}}{N^{2}}\DP{\psi'^{2}}{z}
		 \\
	     -\DP{}{x}\DP{}{y}\psi'^{2} \\
	     -\frac{f^{2}}{N^{2}}\DP{}{x}\DP{}{z}\psi'^{2}
	 \end{array}
       \right).
\end{eqnarray}
���λ� $\Dvect{F}+\Dvect{G}$ ��
\begin{eqnarray}
    \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'_{x}\psi'_{y} - \psi'\psi'_{xy}
		    \\
	     \frac{f^{2}}{N^{2}}(\psi'_{x}\psi'_{z} - \psi'\psi'_{xz})
	 \end{array}
	\right)	.
\end{eqnarray}
��������������𤬥������åȤʤΤ�
\[
    \overline{u}\DP{q'}{x}+v'\DP{\overline{q}}{y}=0
\]
�Ȥ�, ����� $\overline{u}$ �� $x$ �˰�¸���ʤ��Ȥ���,
\begin{eqnarray*}
    \DP{}{x}\bigl(\overline{u}q' + \DP{\overline{q}}{y}\psi'\bigr) = 0
\end{eqnarray*}
�Ȥ���. 
$\Dvect{F}+\Dvect{G}$ �� $x$ ��ʬ����3, 4 ���
\begin{eqnarray}
    \psi' q' +\overline{u}q'^{2}/\overline{q}_{y}
	&=&
	\frac{q'}{\overline{q}_{y}} (\psi' \overline{q}_{y} + \overline{u}q')
\end{eqnarray}
�Ȥʤ�, ���� $x$ ��ʬ�� $0$ �ʤΤ��Ȥγ�ư�٤˴�Ϳ���ʤ�.
��äƤ����̵�뤹��.
ʿ���Ȳ���ꤷ�Ƥ��뤳�Ȥ���
\begin{eqnarray*}
    &&\psi' \sim \sin \chi \\
    &&\psi'_{xx} \sim -k^{2}\sin^{2}\chi, \\
    &&\psi'^{2}_{x} \sim k^{2}\cos^{2}\chi, \\
    &&-\psi'\psi'_{xx} \sim k^{2}\sin^{2}\chi
\end{eqnarray*}
�Ǥ��뤫��, $\Dvect{F}+\Dvect{G}$ ��$x$ ��ʬ��
\[
    \frac{\psi'_{0}}{2}k^{2}
\]
�Ȥʤ�.$y, z$ ��ʬ�ˤĤ��Ƥ�Ʊ�ͤ˷׻������, ���
\begin{equation}
    \Dvect{F}+\Dvect{G}
	\equiv \Dvect{F}_{s}
	= \frac{\psi_{0}}{2}
	\left(
	 \begin{array}{c}
	     k^{2}
		 \\
	     kl
		 \\
	     \frac{f^{2}}{N^{2}}km
	 \end{array}
       \right)
\end{equation}
�Ȥʤ�, �����¸����õ�Ǥ���.

�Ȥγ�ư�٥ե�å��� $\Dvect{F}$ ����ȯ���� $\Dvect{G}$ ��ä��Ƥ��뤿��,
���ޤޤǤȰۤʤ� $\Dvect{F}_{s} \neq \Dvect{C}_{g} A$ �ȤʤäƤ���.
������ǧ����.

Rossby �Ȥΰ���®�٤ȷ�®�٤ˤĤ���
�����������ͤ��Ƥ���Ȥ����, ����®�� $C_{x} = 0$ �Ȥʤ�Τ�,
\begin{eqnarray}
    C_{x} = \overline{u} -\frac{\overline{q}_{y}}{\Dvect{k}^{2}} = 0 \quad
    \to \overline{u} = \frac{\overline{q}_{y}}{\Dvect{k}^{2}}.
\end{eqnarray}
��äƷ�®�� $C_{gx}$ ��
\begin{equation}
    C_{gx} = \frac{2k^{2}\overline{q}_{y}}{\Dvect{k}^{4}}
\end{equation}
�Ȥʤ�, $\Dvect{F}_{s} \neq \Dvect{C}_{g} A$ �ȤʤäƤ���.

�����, �Ȥγ�ư�� $A$ ��
\begin{equation}
    A = \frac{q'^{2}}{2\overline{q}_{y}} \propto \sin^{2}\chi
\end{equation}
�ΤޤޤǤ���, �����¸����õ�Ǥ��Ƥ��ʤ�.

������ Plumb(1985) �Ǥ�, �ȤΥ��ͥ륮�� $e$ ��ʿ��ή $\Dvect{u}$ �ǳ�ä��̤�
�Ȥγ�ư�٤��ղä���
\begin{equation}
	A_{s} \equiv A + \frac{e}{\overline{u}}
\end{equation}
�Ȥ���. ���λ�
\begin{equation}
    \Dvect{F}_{s} \propto \Dvect{C}_{g} \cdot A_{s}
\end{equation}
�Ȥʤ뤳�Ȥ�, ľ�ܷ׻��ǳΤ���뤳�Ȥ��Ǥ���
\footnote{
�ܺ٤ϳ䰦��, ��ά�����Ҥ٤��α���:
\begin{eqnarray*}
    &&A \sim \frac{\Dvect{k}^{4}\psi_{0}^{2}}{2\overline{q}_{y}}\sin^{2}\chi
	, \quad (\psi = \psi_{0} \sin \chi) \\
    && e \sim \frac{\Dvect{k}^{2}\psi_{0}^{2}}{2}\cos^{2}\chi ,\\
    && C_{x} = \overline{u} - \frac{\overline{q}_{y}}{\Dvect{k}^{2}} ,\\
    && \frac{e}{\overline{u} - c} \sim
	\frac{\Dvect{k}^{4}\psi_{0}^{2}}{2\overline{q}_{y}}\cos^{2}\chi, \\
    && \to 
	\frac{1}{2}\biggl[
	A + \frac{e}{\overline{u}-c}
	\biggr]
	=
	\frac{\Dvect{k}^{4}\psi_{0}^{2}}{4\overline{q}_{y}}.
\end{eqnarray*}
�ɤ���ȤǤ�ա��ꥨŸ������� $\sin$ or $\cos$ �ʤ��,
��η�̤�Ŭ����ǽ��, �Ȥ����Ф����Ǥ�̵��.
WKBJ ��Ŭ���Ǥ������, �Ȥ�����郎�Ĥ�����.
}.
\begin{itemize}
    \item ����������β��Ϥ��ɤ���: �ȿ���ʬ�� 1��2 �ξ��
    \item ���ܤ���: �ȿ���ʬ�� 3��\\
	  Plumb(1985)�Ǥ�
	  ���ܾ�Ȥ����Ӿ�ʿ�Ѿ���Ѥ��Ƥ���. �ȿ���ʬ�� 3 �ʾ�ˤʤ��,
	  �Ӿ�ʿ�Ѿ줬�������ͤǤ���, �Ȥ������꤬����Ω���ʤ�.
\end{itemize}


\subsection{��¸§���Ѥ��� Plumb(1985) �Υե�å�����Ƴ��}

Plum (1985) ��
\[
   A_{s} \equiv A + \frac{e}{\overline{u}}
\]
�ʤ��̤�������뤳�Ȥ�, �����¸���Τʤ��Ȥγ�ư�٤���¸§���᤿.
������, ���μ�ˡ��ȯ��Ū�Ǥ���, ʪ��Ū�ʰ�̣���ĤߤŤ餤.
�����Ǥ� $A$ �� $e$ ����¸§���Ѥ��� Plumb (1985) �Υե�å�����
����Ū��Ƴ�Ф���.

��¸§�����Ū�˵��Ƥߤ�.
���������줿����������
\begin{equation}
    \DP{q'}{t} + \overline{u}\DP{q'}{x} + \overline{q}_{y}v' = 0
\end{equation}
��ή���ؿ�$\psi'$ ���Ѥ���.
������1��, ��2��Ϥ��줾��
\begin{eqnarray*}
    \mathrm{��1��}:~~
	\psi'\DP{q'}{t}
	&=& \psi'\DP{}{t}\biggl\{
	\psi'_{xx}
	+ \psi'_{yy}
	+ \frac{1}{\rho}\DP{}{z}\biggl(
	\rho_{0}\frac{f^{2}}{N^{2}}\DP{\psi}{z}
	\biggr)
	\biggr\} \\
    &=&
	\frac{1}{\rho_{0}}\Ddiv\rho_{0}
	\left(
	 \begin{array}{c}
	     \psi'\psi'_{xt}     \\
	     \psi'\psi'_{yt}   \\
	     \psi'\psi'_{zt}
	 \end{array}
	\right)
	- \DP{e}{t} \\
    \mathrm{��2��}:~~
	\overline{u}\DP{q'}{x}
	&=&
	\overline{u}\cdot
	\frac{1}{\rho}
	\Ddiv
	\rho_{0}
	\left(
	\begin{array}{c}
	    \psi'\psi'_{xx}\\
	    \psi\psi'_{xy} \\
	    \frac{f^{2}}{N^{2}}\psi'\psi'_{xz}
	\end{array}
	\right)
	- \overline{u}\DP{e}{x}
\end{eqnarray*}
�Ȥʤ�.

������
\begin{equation}
    \Dvect{R}_{1}
	\equiv
	\rho_{0}
	\left(
	\begin{array}{c}
	    \psi'\psi'_{xt} \\
	    \psi'\psi'_{yt} \\
	    \frac{f^{2}}{N^{2}}\psi'\psi'_{zt} \\
	\end{array}
	\right), \quad
    \Dvect{R}_{2}
    \equiv
	\rho_{0}
	\left(
	\begin{array}{c}
	    \psi'\psi'_{xx}\\
	    \psi\psi'_{xy} \\
	    \frac{f^{2}}{N^{2}}\psi'\psi'_{xz}
	\end{array}
	\right)
\end{equation}
�Ȥ���, ����� $1/(\overline{u}-c)$ �򤫤������������
\begin{equation}
    \DP{}{t}\frac{e}{\overline{u}-c}
	+ \overline{u}\DP{}{x}\frac{e}{\overline{u}-c}
	- 
	\frac{1}{\rho_{0}}\frac{1}{\overline{u}-c}\Ddiv\Dvect{R}_{1}
	-
	\frac{\overline{u}}{\rho_{0}(\overline{u}-c)}\Ddiv\Dvect{R}_{2}
	-
	\DP{}{x}
	\biggl(
	\frac{\overline{q}_{y}}{2(\overline{u}-c)}\psi'^{2}
	\biggr)
	=
	0
\end{equation}
��������.

�ޤ�, 
\begin{eqnarray*}
    \overline{u}\DP{}{x}\frac{e}{\overline{u}-c}
	&=&
	\overline{u}\DP{}{x}\frac{e}{\overline{u}-c} \\
	&=&
	c\DP{}{x}\frac{e}{\overline{u}-c} + \DP{}{x}e ,\\
    \frac{\overline{u}}{\rho_{0}(\overline{u}-c)}\Ddiv\Dvect{R}_{2}
	&=&
	-
	\frac{1}{\rho_{0}}\frac{c}{\overline{u}-c}\Ddiv\Dvect{R}_{2}
	-
	\frac{1}{\rho_{0}}\Ddiv\Dvect{R}_{2}
\end{eqnarray*}
�Ǥ��뤫��
\begin{equation}
    \DP{}{t}\frac{e}{\overline{u}-c}
	+
	c\DP{}{x}\frac{e}{\overline{u}-c}
	+
	\frac{1}{\rho_{0}}\Ddiv\rho_{0}
	\left(
	 \begin{array}{c}
	     -\psi'\psi'_{xx}\\
	     -\psi'\psi'_{xy}\\
	     -\frac{f^{2}}{N^{2}}\psi'\psi'_{xz}
	 \end{array}
       \right)
	+
	\DP{}{x}e
	-
	\DP{}{x}\frac{\overline{q}_{y}\psi'^{2}}{2(\overline{u}-c)}
	-
	\frac{1}{\rho_{0}}
	\frac{1}{\overline{u}-c}(\Ddiv\Dvect{R}_{1}+ c\Ddiv\Dvect{R}_{2})
	=0
\end{equation}
�Ȥʤ�.

�ޤ� A �ˤĤ��Ƥ�
\begin{equation}
    \DP{}{t}A
	+
	C\DP{A}{x}
	+
	\frac{1}{\rho_{0}}\Ddiv\rho_{0}
	\left(
	\begin{array}{c}
	    v'^{2}-e\\
	    -u'v' \\
	    \frac{f^{2}}{N^{2}}\psi'_{x}\psi'_{z} 
	\end{array}
	\right)
	+
	\DP{}{x}\biggl(
	\frac{\overline{u}-c}{2}\DP{q'^{2}}{\overline{q}_{y}}
	\biggr)
	=0
\end{equation}
�Ȥʤ�Τ�, ξ�դ�­�����碌��
\begin{equation}
    M \equiv \frac{1}{2}\biggl(A + \frac{e}{\overline{u}-c}\biggr)
\end{equation}
����������,
\begin{eqnarray}
    \DP{}{t}M + c\DP{}{x}M
	&+& \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^{2}}{N^{2}}(\psi'_{x}\psi'_{z} -\psi'\psi'_{xz})
	 \end{array}
       \right) \nonumber \\
    &+&
	\DP{}{x}
	\biggl\{
	\frac{(\overline{u}-c)q'^{2}}{4\overline{q}_{y}}
	-
	\frac{\overline{q}_{y}\psi'^{2}}{4(\overline{u}-c)}
	\biggr\}
	-
	\frac{1}{2\rho_{0}}\frac{1}{\overline{u}-c}
	(\Ddiv\Dvect{R}_{1} + c\Ddiv\Dvect{R}_{2}) = 0.
\end{eqnarray}

�����Ǥ����, �����������ͤ��Ƥ���Ȥ���
\[
    \DP{}{t} \simeq -i \omega \simeq -ick \simeq -c\DP{}{x}    
\]
������������Ŭ�����뤳�Ȥ�,
\begin{eqnarray*}
   && (\overline{u}-c)\DP{}{x}q' + \overline{q}_{y}v' = 0 \\
    && \to
	\DP{}{x}\biggl\{
	(\overline{u}-c)q' + \overline{q}_{y} \psi'
	\biggr\}
	\simeq 0.
\end{eqnarray*}
��ä�
\begin{eqnarray*}
	\DP{}{x}
	\biggl\{
	\frac{(\overline{u}-c)q'^{2}}{4\overline{q}_{y}}
	-
	\frac{\overline{q}_{y}\psi'^{2}}{4(\overline{u}-c)}
	\biggr\}
	=
	\DP{}{x}
	\biggl\{
	\frac{\overline{q}_{y}}{4(\overline{u}-c)}
	\bigl(
	\frac{\overline{u}-c}{\overline{q}_{y}}q' + \psi'
	\bigr)
	\bigl(
	\frac{\overline{u}-c}{\overline{q}_{y}}q' - \psi'
	\bigr)
	\biggr\} = 0
\end{eqnarray*}
�Ǥ���\footnote{%
��������\[
	    (\overline{u}-c)q' + \psi'\overline{q}_{y} = const.
	\]
	�Ǥ���. Ŭ�ڤʶ����������, �����ͤ� $0$ �ˤ��뤳�Ȥ���ǽ�Ǥ���.
}, Ʊ�ͤ�
\[
(\Ddiv\Dvect{R}_{1} + c\Ddiv\Dvect{R}_{2})
    =
    \bigl(
    \DP{}{t} + c\DP{}{x}
    \bigr)e
    +
    \psi'\bigl(
    \DP{}{t} + c\DP{}{x}
    \bigr)q'
    =0
\]
�Ȥʤ�.

�ǽ�Ū�˰ʲ��� Plumb(1985) �Υե�å��� $\Dvect{W}$ ��������:
\begin{eqnarray}
    &&
	\DP{}{t} M 
	+ 
	c\DP{}{x} M
	+
	\Ddiv \Dvect{W} = 0 \\
    &&
	M
	\equiv
	\frac{1}{2}\biggl(A + \frac{e}{\overline{u}-c}\biggr) \\
    &&
	\Dvect{W}
	\equiv
	\frac{1}{2}
	\left(
	 \begin{array}{c}
	     \psi'^{2}_{x} - \psi'\psi'_{xx} \\
	     \psi'_{x}\psi'_{y} - \psi'\psi'_{xy} \\
	     \frac{f^{2}}{N^{2}}
		 (\psi'_{x}\psi'_{z} - \psi'\psi'_{xz})
	 \end{array}
	\right)
\end{eqnarray}

���ξ���ǤΥե�å�����Ƴ�Ф� Plumb(1985) �����ɤ�����,
\begin{itemize}
    \item 2 �Ĥ���¸§���Ȥ߹�碌������Ƴ�Фθ��̤����ɤ�����
    \item $c\neq 0$ �ξ���ˤĤ��Ƥ�Ŭ�ѤǤ��뤳��
\end{itemize}
�Ǥ���.

%\fi

\subsection{�����ޤǤΤޤȤ�.}

�����ޤǤ��ä�ή���ޤȤ��. 

Ƴ�Ф���Ƥ����ե�å����ˤĤ��Ƥ�
\begin{itemize}
    \vspace{-1em}
    \item ���ܾ����������
    \item $c = 0$ �ξ��: Plumb(1985) ��������줿�ե�å���
	  \begin{itemize}
	      \item �����¸����Τ�, ����������ˤϻȤ��ʤ�.
	  \end{itemize}
    \item $c \neq 0$ �ξ��: �������������� $M$ ����¸§�ȥե�å���
	  ���Ѥ���.
	  \begin{eqnarray}
	      &&M \equiv \frac{1}{2}
		  \bigl(
		  A + \frac{e}{\overline{u}-c}
		  \bigr) \\
	      &&\DP{}{t} M
		  + 
		  \DP{}{x}cM
		  +
		  \Ddiv \Dvect{W} = 0 \\
	      &&\Dvect{W}
		  \equiv
		  \left(
		   \begin{array}{c}
		       \psi'^{2}_{x} - \psi'\psi'_{xx}\\
		       \psi'_{x}\psi'_{y} - \psi'\psi'_{xy}\\
		       \frac{f^{2}}{N^{2}}
			   (\psi'_{x}\psi'_{z} - \psi'\psi'_{xz})
		   \end{array}
		  \right)
	  \end{eqnarray}
	  \begin{itemize}
	      \item  �����¸���ʤ��Τ�
		     $\sin^{2}\chi + \cos^{2}\chi = 1$ �Ȥʤ뤫��.
	      \item  ���μ��ϰ���ʿ�������Ѥ��Ƥ��ʤ�.
	      \item  ��̤Ȥ��� flux �ϰ����¸���ʤ��Τ�,
		     ����������β��Ϥˤ�Ȥ���.
	  \end{itemize}
\end{itemize}

\begin{itemize}
    \item �Ӥ���μ���:\\
	  WKBJ ���Ѥ���, 
	  $u$ �λ����Ѳ���ޤ���ˤ�ä����ޤ�������? \\
	  $\to$ �����Ǥ�̵��, �餷��(by ��ë����)
    \item ������μ���:\\
	  $c\neq0$ �Ǥ�ե�å����η����Ѥ�äƤ��ʤ��ΤǤ�? \\
	  $\to $
	  $c$ �η�����ޤäƤ���Τʤ��, 
	  \begin{equation}
	      \Dvect{W}_{T}
		  \equiv
		  \left(
		   \begin{array}{c}
		       cM + \psi'^{2}_{x} - \psi'\psi'_{xx}\\
		       \psi'_{x}\psi'_{y} - \psi'\psi'_{xy}\\
		       \frac{f^{2}}{N^{2}}(\psi'_{x}\psi'_{z}-\psi'\psi'_{xz})
		   \end{array}
		      \right)
		  = \Dvect{C}_{g}\cdot M
	  \end{equation}
	  �Ȥʤ���������Ǥ���.
	  ������, �ºݤˤ� $c$ �η������Τ�����.
	  �ץ��åȤ�����ˤ�, �㤨�� $x$ �����ˤĤ��Ƥ�
	  \begin{equation}
	      cM + \psi'^{2}_{x} - \psi'\psi'_{xx}
	  \end{equation}
	  ���������Ȥˤʤ�.
    \item �������ε���(�ޥ˥��å��ͥ�?) \\
%% ����ä�, ��ë����Υͥ������. �ܤ����ɤ��Τ�����.
	  \begin{eqnarray*}
	      &&M
		  = 
		  \frac{1}{2}
		  \biggl(
		  \frac{q'^{2}}{2\overline{q}_{y}} + \frac{e}{\overline{u}-c}
		  \biggr), \\
	      &&\psi = A(X,Y,Z)\sin(kx+ly+mz-\omega t),  \\
	      &&M 
		  =
		  \frac{\Dvect{k}^{4}}{4}
		  \frac{\psi^{2}_{0}}{\overline{q}_{y}} 
		  + \mathcal{O}(\varepsilon).
	  \end{eqnarray*}
	  ��̤Ȥ��� $\displaystyle{\DP{M}{t}}$ �ΰ���ϻ��֤˰�¸����.
	  �������ʤ���
	  \begin{eqnarray*}
	      \psi'^{2}_{x} - \psi'\psi'_{xx} 
		  = k^{2}\psi_{0}^{2} + \mathcal{O}(\varepsilon^{2})
	  \end{eqnarray*}
	  �ʤΤ�, �ե�å�����ȯ���ΰ���ϻ��֤˰�¸���ʤ�.
	  ������ $M$ �Τ�����,
	  \[
	  \DP{\Gamma}{x} \equiv \zeta'
	  \]
	  �Ȥʤ� $\Gamma$ ��Ƴ������ �������������� $\tilde{M}$ ��ͤ����
	  \[
	  \tilde{M}
	  \equiv
	  \biggl(
	  \frac{\zeta'^{2}}{2\beta}
	  -
	  \frac{\Gamma\zeta' x}{2\beta}
	  \biggr)
	  =
	  \frac{\Dvect{k}^{4}}{4}
	  \frac{\psi^{2}_{0}}{\overline{q}_{y}} + \mathcal{O}(\varepsilon^{2})
	  \]
	  �Ȥʤä�, �����ˤʤ��. 
	  �������ʤ���, ����ʪ��Ū�ʰ�̣���狼��ʤ�.
\end{itemize}





\newpage%----------------------------------------
\section{��������ʹ��ܾ�ؤγ�ĥ:Takaya and Nakamura(2001)}

�ͤ��Ƥ�������ˤ�äƤ�, ���ܾ����������ͤȤ��������Թ礬�ɤ�.
������, 
��������ͤʴ��ܾ�Ǥ�,
��¸§����Ω���ʤ��Τ�, ���Ū����¸§����뤳�ȤȤ���.

�ͤ��Ƥ�����ܾ�����:
\begin{equation}
    \overline{\Dvect{u}} = (U, V, 0).
\end{equation}
������,
���ܾ�����Ǥ���, 
����˴��ܾ�ζ����Ѳ������ˤ�ä���Ȥ����Ѳ�(�����)���Ȳ��ꤹ��.
�ޤ�, ���ܾ�˱�ä������ΰ���®�٤� $C_{p}$ �Ȥ���(\Dfigref{��4-1}).

\begin{figure}[h]
    \begin{center}
	\Depsf[][]{figs/Fig4-1.ps}
	\caption{$\overline{u}$ �� $C_p$}
	\Dfiglab{��4-1}
    \end{center}
\end{figure}


�ޤ�, ���ܾ�϶����ब¸�ߤ��ʤ�, �Ȳ��ꤹ��:
\begin{equation}
    \overline{u}\DP{\overline{q}}{x}
	+
	\overline{v}\DP{\overline{q}}{y} = 0.
\end{equation}
����ϴ��ܾ줬����Ǥ���, �Ȥ������ο����Ǥ�ɽ���Ǥ���(\Dfigref{��4-2}).
\begin{figure}[ht]
    \begin{center}
	\Depsf[][]{figs/Fig4-2.ps}
	\caption{���ܾ�Υݥƥ󥷥�뱲�٤�ή��ξ�}
	\Dfiglab{��4-2}
    \end{center}
\end{figure}

���������줿 PV ��¸§�����ä�Ϥ��:
\begin{equation}
    \DP{q'}{t}
	+
	\overline{U}\DP{q'}{x}
	+
	\overline{V}\DP{q'}{y}
	+
	\overline{u'}\DP{\overline{q}}{x}
	+
	\overline{v'}\DP{\overline{q}}{y}
	=
	0.
\end{equation}

�������ͤξ����Ȥγ�ư�٤�
\[
    A \equiv \frac{q'^{2}}{2\overline{q}_{y}}
\]
�Ǥ��ä�. ��������ͤʴ��ܾ�Ǥ��Ȥγ�ư�� �� Plumb(1985)�˽���,
�������������:
\begin{equation}
    A \equiv \frac{q'^{2}}{2|\Dgrad_{h}\overline{q}|}.
\end{equation}
������ $\Dgrad_{h}$ �Ͽ�ʿ���ۤǤ���.
�ޤ�, �������ͤʾ����ȤΥ��ͥ륮����
\begin{equation}
    \epsilon = \frac{e}{\overline{u}-c}
\end{equation}
�Ǥ��ä�. ��������ͤʾ��ˤ�, �弰��
\begin{equation}
    \epsilon \equiv \frac{e}{|\Dvect{u}|-C_{p}}
\end{equation}
�ȳ�ĥ����.

������­����碌��
\begin{equation}
    M \equiv \frac{1}{2}(A + \epsilon)
\end{equation}
��(���Ū��)��¸§��, �ʲ��Ǥ�Ƴ�Ф���.

\subsection{A ����¸§}

���������줿 PV ��¸§�� $q'/|\Dgrad_{h} \overline{q}|$ ���Ѥ���:
\begin{eqnarray}
    \frac{1}{|\Dgrad_{h} \overline{q}|}
	\bigl(
	\DP{}{t} + U \DP{}{x} + V\DP{}{y}
	\bigr)\frac{q'^{2}}{2}
	+
	\frac{1}{|\Dgrad_{h} \overline{q}|}
	u'q'\DP{\overline{q}}{x}
	+
	\frac{1}{|\Dgrad_{h} \overline{q}|}
	v'q'\DP{\overline{q}}{y} = 0.
\end{eqnarray}
������,
\begin{eqnarray}
    \frac{1}{|\Dgrad_{h} \overline{q}|}\DP{}{t}\frac{q'^{2}}{2}
	&=&
	\DP{}{t}\frac{q'^{2}}{2|\Dgrad_{h}\overline{q}|} = \DP{}{t}A \\
    \frac{1}{|\Dgrad_{h} \overline{q}|}
	\bigl(U\DP{}{x}+V\DP{}{y}(\bigr)\frac{q'^{2}}{2}
	&=&
	\frac{1}{2|\Dgrad_{h}\overline{q}|}\biggl\{
	u'q'\DP{\overline{q}}{x}
	+
	v'q'\DP{\overline{q}}{y}
	\biggr\} \nonumber \\
    &\simeq& \DP{}{x}(UA)+ \DP{}{y}(VA)
\end{eqnarray}
�Ǹ�μ��ѷ��Ǥ�,
���ܾ줬���ˤ�ä�����Ѳ�����Τ���ʬ���������뤳�Ȥ��Ǥ���, �ȶ������.

�ޤ�, 
\begin{eqnarray}
    &&u'q' 
	=
	\Ddiv \rho_{0}
	\left(
	\begin{array}{c}
	    u'v' \\
	    e - u'^{2} \\
	    -\frac{f^{2}}{N^{2}}
	\end{array}
	\right) \quad, 
    v'q' 
	=
	\Ddiv \rho_{0}
	\left(
	\begin{array}{c}
	    v'^{2}-e \\
	    u'v' \\
	    \frac{f^{2}}{N^{2}}\psi'_{x}\psi'_{z}
	\end{array}
	\right) \\
    &&\to
    \frac{1}{|\Dgrad_{h} \overline{q}|}
	u'q'\DP{\overline{q}}{x}
	+
	\frac{1}{|\Dgrad_{h} \overline{q}|}
	v'q'\DP{\overline{q}}{y}
	\simeq
	\frac{1}{\rho_{0}}\Ddiv\frac{1}{|\Dgrad\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^{2}}{N^{2}}\psi'_{x}\psi'_{z}
		-
		\overline{q}_{x}
		\frac{f^{2}}{N^{2}}\psi'_{y}\psi'_{z}
	\end{array}
	\right)
\end{eqnarray}
�����Ǵ��ܾ���Ѳ�����ä���Ǥ���Ȥ���,
$|\Dgrad\overline{q}|$ ����ʬ��������줿.

�ޤ����ܾ줬���Ǥ���Ȥ�����Ｐ���,
\begin{eqnarray}
    \DP{\overline{q}}{x}
	=
	-\frac{V}{U}\DP{\overline{q}}{y}
\end{eqnarray}
�Ǥ��뤫��, ���Ū��
\begin{equation}
    \frac{\overline{q}_{y}}{|\Dgrad_{h}\overline{q}|}
	\simeq
	\frac{U}{|\overline{\Dvect{u}}|}, \quad
	\frac{\overline{q}_{x}}{|\Dgrad_{h}\overline{q}|}
	\simeq
	-\frac{V}{|\overline{\Dvect{u}}|}
\end{equation}
�Ȥʤ�Τ�, ���
\begin{eqnarray}
    \frac{1}{|\Dgrad_{h} \overline{q}|}
	u'q'\DP{\overline{q}}{x}
	+
	\frac{1}{|\Dgrad_{h} \overline{q}|}
	v'q'\DP{\overline{q}}{y}
	&=&
	\frac{1}{\rho_{0}}\Ddiv\frac{1}{|\Dgrad\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^{2}}{N^{2}}\psi'_{x}\psi'_{z}
		 -
		 \overline{q}_{x}
		 \frac{f^{2}}{N^{2}}\psi'_{y}\psi'_{z}
	 \end{array}
	\right)
	\nonumber \\
    &=&
	\frac{1}{\rho_{0}}\Ddiv\frac{\rho_{0}}{|\overline{\Dvect{u}}|}
	\left(
	 \begin{array}{c}
	     U(v'^{2}-e) + V(-u'v') \\
	     U(-u'v') + V(u'^{2}-e) \\
	     \frac{f^{2}}{N^{2}}
		 \{
		 U\psi'_{x}\psi'_{z}
		 +
		 V\psi'_{y}\psi'_{z}		
		 \}
	 \end{array}
	\right)	.
\end{eqnarray}

�ʾ��ޤȤ��
\begin{equation}
    \DP{}{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) + V(-u'v') \\
	     U(-u'v') + V(u'^{2}-e) \\
	     \frac{f^{2}}{N^{2}}
		 \{
		 U\psi'_{x}\psi'_{z}
		 +
		 V\psi'_{y}\psi'_{z}		
		 \}
	 \end{array}
	      \right)
\end{equation}

������, ��������Τ����
\begin{eqnarray}
    	\Ddiv
	\left(
	 \begin{array}{c}
	     UA\\
	     VA\\
	     0
	 \end{array}
	\right)
	&=&
	\DP{}{x}UA + \DP{}{y} VA \nonumber \\
	&=&
	\DP{}{x}
	\biggl\{
	\frac{|\overline{\Dvect{u}}|-C_{p}}{|\overline{\Dvect{u}}|}U
	+
	\frac{C_{p}U}{|\overline{\Dvect{u}}|}A
	\biggr\}
	+
	\DP{}{y}
	\biggl\{
	\frac{|\overline{\Dvect{u}}|-C_{p}}{|\overline{\Dvect{u}}|}V
	+
	\frac{C_{p}V}{|\overline{\Dvect{u}}|}A
	\biggr\} \nonumber \\
    &=&
	\Ddiv \Dvect{N} + \Ddiv\Dvect{C}_{v}A \\
\end{eqnarray}
�Ȥ���.
$\Dvect{N}^{(1)}$, $\Dvect{C}_{U}$, $\Dvect{E}$ �ϰʲ����������:
\begin{eqnarray}
    \Dvect{N}^{(1)}
	&\equiv&
	\left(
	 \begin{array}{c}
	     \frac{|\overline{\Dvect{u}}|-C_{p}}{|\overline{\Dvect{u}}|}U\\
	     \frac{|\overline{\Dvect{u}}|-C_{p}}{|\overline{\Dvect{u}}|}V \\
	     0
	 \end{array}
       \right), 
	\quad
	\Dvect{C}_{U}
	\equiv
	\left(
	 \begin{array}{c}
	     \frac{U}{|\overline{\Dvect{u}}|}C_{p} \\
	     \frac{V}{|\overline{\Dvect{u}}|}C_{p} \\
	     0
	 \end{array}
       \right) , \quad
    \Dvect{E}
    \equiv
    \frac{\rho_{0}}{|\overline{\Dvect{u}}|}
    \left(
     \begin{array}{c}
	 U(v'^{2}-e) + V(-u'v') \\
	 U(-u'v') + V(u'^{2}-e) \\
	 \frac{f^{2}}{N^{2}}
	     \{
	     U\psi'_{x}\psi'_{z}
	     +
	     V\psi'_{y}\psi'_{z}		
	     \}
     \end{array}
    \right)
\end{eqnarray}
$\Dvect{N}^{(1)}, \Dvect{C}_{U}, \Dvect{E}$���Ѥ����,
�������ʹ��ܾ�ˤ������Ȥγ�ư����¸§���б����Ĥ��䤹��:
\begin{eqnarray}
    \mathrm{���������:~~}&&\DP{}{t} A
	+
	\Ddiv\Dvect{C}_{U} A
	+
	\frac{1}{\rho_{0}}\Ddiv\Dvect{E}
	+
	\Ddiv\Dvect{N}^{(1)} 
	= 0,\\
    \mathrm{��������:~~}&&\DP{}{t}A
	+ \DP{}{x}cA
	+ \Ddiv\Dvect{E}
	+ \DP{}{x}(\overline{u}-c)\frac{q'^{2}}{2\overline{q}_{y}}
	=0 .\nonumber
\end{eqnarray}

\subsection{$\epsilon$ ����¸§}

����
\[
    \epsilon \equiv  \frac{e}{|\overline{\Dvect{u}}|-C_{p}}
\]
����¸§�����.
���������줿 PV ��¸§�� $-\psi'/(\overline{\Dvect{u}}-C_{p})$ ���Ѥ�Ȥä�
\begin{equation}
    -\frac{1}{\overline{\Dvect{u}}-C_{p}}
	\bigl(
	\psi'\DP{q'}{t}
	+ \overline{\Dvect{u}}\psi'\Dgrad{q'}
	+ \Dvect{u}'\psi'\Dgrad\overline{q}
	\bigr)
	=0
\end{equation}
������
\[
    \psi'\DP{q'}{t}
    =
    -\Ddiv
    \left(
    \begin{array}{c}
	\psi'\psi'_{xt}\\
	\psi'\psi'_{yt} \\
	0
    \end{array}
    \right)
    -\DP{e}{t}
\]
�Ǥ��뤫��, ������1���
\begin{equation}
    \DP{}{t}\epsilon
    -
    \frac{1}{|\overline{\Dvect{u}}| - C_{p}}
    -\Ddiv
    \left(
    \begin{array}{c}
	\psi'\psi'_{xt}\\
	\psi'\psi'_{yt} \\
	0
    \end{array}
    \right) 
    \equiv
    \DP{}{t}\epsilon
    -
    \frac{1}{|\overline{\Dvect{u}}| - C_{p}}\Ddiv\Dvect{R}_{1}
\end{equation}
�ޤ���2���
\begin{eqnarray}
    -
	\frac{U}{|\overline{\Dvect{u}}| - C_{p}}
	\psi\DP{q'}{x}
	&&-
	\frac{V}{|\overline{\Dvect{u}}| - C_{p}}
	\psi\DP{q'}{y}  \nonumber \\
    &&=
	-
	\biggl(
	\frac{U}{|\overline{\Dvect{u}}|}
	\frac{C_{p}}{|\overline{\Dvect{u}}| - C_{p}}
	\frac{U}{|\overline{\Dvect{u}}|}
	\biggr)
	\psi\DP{q'}{x}
	-
	\biggl(
	\frac{V}{|\overline{\Dvect{u}}|}
	\frac{C_{p}}{|\overline{\Dvect{u}}| - C_{p}}
	\frac{V}{|\overline{\Dvect{u}}|}
	\biggr)
	\psi\DP{q'}{y}.
\end{eqnarray}
������
\begin{equation}
    \psi'\DP{q'}{x}
	=
	-
	\frac{1}{\rho_{0}}
	\Ddiv
	\rho_{0}
	\left(
	\begin{array}{c}
	    \psi'\psi'_{xx}\\
	    \psi'\psi'_{xy}\\
	    -\frac{f^{2}}{N^{2}}\psi'_{xz}
	\end{array}
	\right)
	-\DP{e}{x}
\end{equation}
�Ǥ��뤫��,
\begin{eqnarray}
    -\frac{U}{|\overline{\Dvect{u}}|}
	\psi\DP{q'}{x}
	&&-
	\frac{V}{|\overline{\Dvect{u}}|}
	\psi\DP{q'}{y} \nonumber \\
	&&=
	\frac{U}{|\overline{\Dvect{u}}|}
	\biggl(
	\frac{1}{\rho_{0}}
	\Ddiv
	\rho_{0}
	\left(
	\begin{array}{c}
	    -\psi'\psi'_{xx}\\
	    -\psi'\psi'_{xy}\\
	    -\frac{f^{2}}{N^{2}}\psi'_{xz}
	\end{array}
	\right)
	+\DP{e}{x}
	\biggr)
	+
	\frac{V}{|\overline{\Dvect{u}}|}
	\biggl(
	\frac{1}{\rho_{0}}
	\Ddiv
	\rho_{0}
	\left(
	 \begin{array}{c}
	    -\psi'\psi'_{xy}\\
	    -\psi'\psi'_{yy}\\
	    -\frac{f^{2}}{N^{2}}\psi'_{yz}
	 \end{array}
	\right)
	+\DP{e}{x}
	\biggr) \nonumber \\
    &&\simeq
	\frac{1}{\rho_{0}}\Ddiv
	\frac{\rho_{0}}{|\overline{\Dvect{u}}|}
	\left(
	\begin{array}{c}
	    eU - U\psi'\psi'_{xx} -V\psi'\psi'_{xx}\\
	    eV - U\psi'\psi'_{xy} -V\psi'\psi'_{yy}\\
	    -\frac{f^{2}}{N^{2}}
		(U\psi'\psi'_{xz} + V\psi'\psi'_{yz})
	\end{array}
	\right) \equiv \frac{1}{\rho}\Ddiv\Dvect{H}
\end{eqnarray}
�Ǹ�Ǥ�, ���ܾ���Ѳ�����ä���Ǥ���ȶ������ʬ��������줿
\footnote{%
Takaya and Nakamura(2001), (19) �����ѷ��Ǥ���.
}.

�ޤ�
\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}\nonumber \\
    \simeq &&
	\DP{}{x}
	\biggl(
	\frac{UC_{p}}{|\overline{\Dvect{u}}|}
	\frac{e}{|\overline{\Dvect{u}}| -C_{p}}
	\biggr)
	\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^{2}}{N^{2}}\psi'\psi'_{xz}
	 \end{array}
\right) \nonumber\\
    &&+
	\DP{}{y}
	\biggl(
	\frac{VC_{p}}{|\overline{\Dvect{u}}|}
	\frac{e}{|\overline{\Dvect{u}}| -C_{p}}
	\biggr)
	\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^{2}}{N^{2}}\psi'\psi'_{yz}
	 \end{array}
\right) \nonumber \\
	=&&
	\Ddiv\Dvect{C}_{U}\epsilon
	-
	\frac{C_{p}}{\rho_{0}|\overline{\Dvect{u}}| -C_{p}}
	\biggl\{
	\frac{U}{|\overline{\Dvect{u}}|}\Ddiv\Dvect{R}_{2}
	+
	\frac{V}{|\overline{\Dvect{u}}|}\Ddiv\Dvect{R}_{3}
	\biggr\}.
\end{eqnarray}
������
\begin{eqnarray}
    \Dvect{R}_{2}
	\equiv
	\rho_{0}
	\left(
	 \begin{array}{c}
	     -\psi'\psi'_{xx}\\
	     -\psi'\psi'_{xy}\\
	     -\frac{f^{2}}{N^{2}}\psi'\psi'_{xz}
	 \end{array}
	\right), \quad
	\Dvect{R}_{3}
	\equiv
	\rho_{0}
	\left(
	 \begin{array}{c}
	     -\psi'\psi'_{xy}\\
	     -\psi'\psi'_{yy}\\
	     -\frac{f^{2}}{N^{2}}\psi'\psi'_{yz}
	 \end{array}
       \right)
\end{eqnarray}
���������\footnote{%
Takaya and Nakamura(2001), (17) �����ѷ��Ǥ���.
}

�Ǹ��
\begin{eqnarray}
    -
	\frac{1}{|\overline{\Dvect{u}}| -C_{p}}
	\biggl\{
	u'\psi'\DP{\overline{q}}{x}
	+
	v'\psi'\DP{\overline{q}}{y}
	\biggr\}
	&=&
    -
	\frac{1}{|\overline{\Dvect{u}}| -C_{p}}
	\biggl\{
	-\DP{}{y}\bigl(
	\frac{1}{2}\DP{\overline{q}}{x}\psi'^{2}
	\bigr)
	+\DP{}{x}\bigl(
	\frac{1}{2}\DP{\overline{q}}{y}\psi'^{2}
	\bigr)
	\biggr\} \nonumber \\
    &\simeq&
	\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) \nonumber \\
	&=&
	\Ddiv\Dvect{N}^{(2)}.
\end{eqnarray}
$\Dvect{N}^{(2)}$ �ϼ�������������
\footnote{%
Takaya and Nakamura(2001), (21), (22)�����ѷ��Ǥ���.
}.
\begin{equation}
    \Dvect{N}^{(2)}
	\equiv
	\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) 
\end{equation}

�����μ���ޤȤ�뤳�Ȥ� $\epsilon$ ����¸§������줿:
\begin{equation}
    \DP{}{t}\epsilon
	+ \Ddiv\Dvect{C}_{U}\epsilon
	+\frac{1}{\rho_{0}}\Ddiv\Dvect{H}
	+\frac{1}{\rho_{0}}\Ddiv\Dvect{N}^{(2)}
	-\frac{1}{\rho_{0}|\overline{\Dvect{u}}| - C_{p}}
	\biggl\{
	\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}
	\biggr\}
	=
	0
\end{equation}

\subsection{�Ȥγ�ư����¸§��Ƴ��}

Takaya \& Nakamura(2001) �Ǥ�, �Ȥγ�ư�٤Ȥ���
\begin{equation}
    M = \frac{1}{2}(A+\epsilon)
\end{equation}
���Ѥ��Ƥ���. $M$ �ˤĤ��Ƥ���¸§��,
$A$ �ˤĤ��Ƥ���¸§�� $\epsilon$ �ˤĤ��Ƥ���¸§��­����碌�뤳�Ȥ�
\begin{eqnarray}
    &&\DP{}{t}M
	+
	\Ddiv\Dvect{C}_{U}M \nonumber \\
    &&\quad +
    	\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 \\
    &&\quad -
	\frac{1}{2\rho_{0}(|\overline{\Dvect{u}}| - C_{p})}
	\biggl\{
	\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}
	\biggr\}
	= 0 .
\end{eqnarray}
$\Dvect{N}^{(1)} + \Dvect{N}^{(2)}$ ��ȯ���ι��
\begin{eqnarray}
    \frac{1}{\rho_{0}}\Ddiv\frac{1}{2}(\Dvect{N}^{(1)}+\Dvect{N}^{(2)})
	&=&
	\frac{1}{2\rho_{0}}\Ddiv
	\left(
	 \begin{array}{c}
	     \frac{|\overline{\Dvect{u}}|-C_{p}}{|\overline{u}|} UA
		 -
		 \frac{\overline{q}_{y}}{2|\overline{\Dvect{u}}|-C_{p}} \psi'^{2}
		 \\
	     \frac{|\overline{\Dvect{u}}|-C_{p}}{|\overline{u}|} VA
		 -
		 \frac{\overline{q}_{x}}{2|\overline{\Dvect{u}}|-C_{p}} \psi'^{2}
		 \\
	     0
	 \end{array}
\right) \nonumber\\
    x\mathrm{��ʬ}\quad
	&=&
	\frac{U}{2(|\overline{\Dvect{u}}|-C_{p})|\Dgrad\overline{q}|}
	\biggl(
	(|\overline{\Dvect{u}}|-C_{p})^{2}q'^{2}
	-
	|\Dgrad\overline{q}|^{2}\psi^{2}
	\biggr)
	\nonumber \\
	&=&
	\frac{U}{2(|\overline{\Dvect{u}}|-C_{p})|\Dgrad\overline{q}|}
	\biggl(
	(|\overline{\Dvect{u}}|-C_{p})q'
	+
	|\Dgrad\overline{q}|\psi
	\biggr)
	\biggl(
	(|\overline{\Dvect{u}}|-C_{p})q'
	-
	|\Dgrad\overline{q}|\psi
	\biggr).
\end{eqnarray}
������, ����� PV �����������ͻ������\Dfigref{��5-1}���
\begin{figure}[ht]
    \begin{center}
	\Depsf[][]{figs/Fig5-1.eps}
	\caption{$C_p$ �� $(X, Y)$}
	\Dfiglab{��5-1}
    \end{center}
\end{figure}
\begin{eqnarray}
    &&\DP{}{t}q' + U\cdot\Dgrad{q'} + \Dvect{u}'\Dgrad\overline{q} = 0 \\
    &&\to 
	\DP{}{t}q'
	+
	U \DP{}{X}q'
	+
	\DP{\psi'}{X}\DP{\overline{q}}{Y} = 0 
	\quad (\mathrm{��ɸ�β�ž, ή���˱�ä���ɸ��}) \nonumber \\
    &&\to\DP{}{X}
	\{(|\overline{\Dvect{u}} - C_{p}|)q' 
	+ |\Dgrad_{h}\overline{q}|\psi'\} = 0 
	\quad\mathrm{(��ή��®�٤�} C_{p}\mathrm{������)}
\end{eqnarray}
������������뤳�Ȥ�, ���
\begin{equation}
    \frac{1}{\rho_{0}}\Ddiv\frac{1}{2}(\Dvect{N}^{(1)}+\Dvect{N}^{(2)}) = 0
\end{equation}
�Ȥʤ�.

�ޤ�,
\begin{eqnarray}
    \Ddiv\Dvect{R}_{1} 
	+
	\frac{UC_{p}}{|\overline{\Dvect{u}}|}\Ddiv\Dvect{R}_{2}
	+
	\frac{VC_{p}}{|\overline{\Dvect{u}}|}\Ddiv\Dvect{R}_{2}
	&=&
	\bigl(
	\DP{e}{t}+ C_{p}\DP{e}{X}
	\bigr)
	+ 
	\psi'
	\bigl(
	\DP{q'}{t}+ C_{p}\DP{q'}{X}
	\bigr)
	\nonumber \\
    && \simeq 0 \qquad \because  \DP{}{t}\simeq -C_{p}\DP{}{X}
\end{eqnarray}

�Ǹ��
\begin{eqnarray}
    &&
	\Dvect{E}
	\equiv
	\frac{\rho_{0}}{|\overline{\Dvect{u}}|}
	\left(
	 \begin{array}{c}
	     U(v'^{2}-e) + V(-u'v') \\
	     U(-u'v') + V(u'^{2}-e) \\
	     \frac{f^{2}}{N^{2}}
		 \{
		 U\psi'_{x}\psi'_{z}
		 +
		 V\psi'_{y}\psi'_{z}		
		 \}
	 \end{array}
	\right),  \quad
	\Dvect{H} \equiv
	\frac{\rho_{0}}{|\overline{\Dvect{u}}|}
	\left(
	 \begin{array}{c}
	     eU - U\psi'\psi'_{xx} -V\psi'\psi'_{xx}\\
	     eV - U\psi'\psi'_{xy} -V\psi'\psi'_{yy}\\
	     -\frac{f^{2}}{N^{2}}
		 (U\psi'\psi'_{xz} + V\psi'\psi'_{yz})
	 \end{array}
	\right) \nonumber \\
    &&
	\to
	\frac{1}{2}(\Dvect{E}+\Dvect{H})
	=
	\frac{\rho_{0}}{|\overline{\Dvect{u}}|}
	\left(
	\begin{array}{c}
	    U(\psi'^{2}_{x}-\psi'\psi'_{xx}) + V(\psi'_{x}\psi'_{y}-\psi'\psi'_{xy})\\
	    V(\psi'_{x}\psi'_{y} - \psi'\psi'_{xy}) + V(\psi'^{2}_{y}-\psi'\psi'_{yy}) \\
	    \frac{f^{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}
$\Dvect{W}_{s}$ �γ���ʬ�ι��,
$\psi'$ �ΰ켡��ʬ����, �⤷����$\psi'$ �� $\psi'$ ������ʬ���ѤʤΤ�,
$\psi \sim \psi_{0}\sin\chi$ �Ȥ����, �����¸�����ä���.

��̤�ޤȤ���,
\begin{eqnarray}
    &&M = \frac{1}{2}(A+\epsilon) \\
    &&\DP{}{t}M
	+
	\Ddiv \Dvect{C}_{U}M
	+
	\Ddiv \Dvect{W}_{s}
	= 0
\end{eqnarray}
�Ȥʤ�. $\Dvect{W}$ ��
\begin{equation}
    \Dvect{W} = \Dvect{C}_{U} M + \Dvect{W}_{S}
\end{equation}
����������, $M$ ����¸§�Ȥ���
\begin{equation}
    \DP{}{t}M + \Ddiv\Dvect{W} = 0
\end{equation}
�Ȥʤ�.
�ޤ� $\Dvect{W} = \Dvect{C}_{g}\cdot M$ 
�Ȥʤ뤳�Ȥ�������Ƥ���(�׻��ξܺ٤ϳ䰦����).

\subsection{$\Dvect{W}_{s}$ �θ����פˤĤ���}

\begin{itemize}
    \item ���ܾ�ι�¤��������(�ȿ����礭��)���β��ϤǤ��ɤ�����.
    \item �ץ��åȤ��Ƥ���Τ� $\Dvect{W}_{s}$.
	  $\Dvect{C}_{U}$ �������Τ��񤷤�.
    \item �ɤ��ǻȤ��뤫, �Ȥ��ʤ���?
	  \begin{itemize}
	      \item �����ʤ餽�Τޤ޻Ȥ��������.
	      \item Ĺ���֤θ��ݤ���Ϥ���ʤ��,
		    Plumb(1986)
	      \item ���ʥåץ���åȤ���Ϥ���ʤ��, Takaya \& Nakamura(2001).
		    ������ $C_{p}$ ���Ѥ��Τ��ưפǤ�̵��.
		    �ƥåɥ����ե����ɤ���۩��
		    �����ͥ륮��$\tilde{E}$��Ȥ���, �㤨��
		    \begin{equation}
			\tilde{E} = e - U A = c A 
			\quad \mathrm{(ʿ�����򤹤��)} \quad
			\to \frac{[E]}{[A]} = c
		    \end{equation}
		    ��, ����®�٤������뤫��(��ë����ϻ�Ƥ��ʤ�).
	      \item �������ͤʾ�ǹͤ����,
		    \begin{equation}
			\DP{}{t}M
			    +
			    \DP{}{x}
			    cM
			    +
			    \Ddiv\Dvect{W}_{s} = 0
		    \end{equation}
		    ���ξ�� $cM$ �������ͥ륮���ȤʤäƤ���.
	      \item Takaya \& Nakamura Flux ��ȤäƲ��Ϥ������������.
		    Takaya \& Nakamura Flux ������
		    ή���ؿ� $\psi'$ �ǵ��Ҥ���Ƥ���Τ�,
		    ���ǡ�������®��򤽤Τޤ޻Ȥ��Τ�
		    �����꤬������⤢��(���������ɤ������).
		    $\psi'$ ��,
		    �Ĥޤ���Ϲ�ή��ʬ $v_{g}$ �ʤΤ�,
		    ���Ϲ�ή��ʬ���礭��������.��������դ��ޤ��礦.
	  \end{itemize}
\end{itemize}

\section{������ TEM ��������}

TEM ��������
\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}
�� 3 �����˳�ĥ���뤳�Ȥ�ͤ���
\footnote{�ʲ�����ϥ��ߥʡ���λ���,
ͭ�֤����ǹԤʤ�줿���ƤǤ���}.
\[
  \psi = \Psi (y,z) = \psi '
\]
��ʪ���̤�ʿ�Ѿ�Ⱦ���ˤ櫓���Ȥ���, �����ˤĤ��Ƥϰʲ��Τ褦��ʿ��
���Ȥ������Ū�ʷϤ�ͤ���.
\[
  \frac{1}{T}\int _{0}^{T}\psi \Dd t =0.
\]

\subsection{Plumb (1986) �ξ��}

3 �������Ȥγ�ư�٤���¸�μ���
\[
  \DP{\overline{A}}{t} 
    + \DP{}{x}(\overline{u}\overline{A})
    + \Ddiv \left(
      \begin{array}{c}
	\overline{v^{'2}} - \overline{e} \\
	- \overline{u'v'} \\
	\frac{f_0}{{\theta _0}_z}\overline{v'\theta '}
      \end{array}
      \right) 
      = 0
\]
����� Plumb (1986) ���Ȥγ�ư�٤���¸�μ��� $\overline{v}=0$ �Ȥ�����
��������.

ʿ��ή�μ��ϱ��դ�
\[
  \overline{v'q'} = \Ddiv \left(
       \begin{array}{c}
	 \overline{v^{'2}}-\overline{e}        \\
	 - [u'v'] \\
	 \frac{f_{0}}{{\theta _{0}}_{z}}[v'\theta']
       \end{array}
  \right)
\]
�������褦, �ʲ��Τ褦���ѷ���Ԥ�.
\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*}
\[
  v^{*}_{a} \equiv v_{a} - \frac{1}{f_0}
    \left\{
      \DP{}{x}\left(
        \frac{u^{'2}}{2} 
          + \frac{v^{'2}}{2} 
          - \frac{1}{2}\frac{f_0^{2}}{N^{2}}\psi ^{'2}_{z}
      \right) 
      + \frac{1}{\rho_0}\DP{}{z}
	\left(\rho _{0}\frac{f_{0}^{2}}{N^{2}}\overline{\psi'_{x}\psi'_{z}}\right)
    \right\}
\]
����������,
\[
 \DP{}{t}\overline{u} - fv_{a}^{*} = \overline{v'q'} = \Ddiv \left(
       \begin{array}{c}
	 \overline{v^{'2}}-\overline{e}        \\
	 - [u'v'] \\
	 \frac{f_{0}}{{\theta _{0}}_{z}}[v'\theta']
       \end{array}
 \right)
\]
���ΤȤ��Ȥγ�ư�٤μ���
\[
  \DP{\overline{A}}{t} 
  +\DP{}{x}(\overline{u}\overline{A}) 
  +
    \Ddiv
       \left(
       \begin{array}{c}
	 \overline{v^{'2}}-\overline{e}        \\
	 - \overline{u'v'} \\
	 \frac{f_{0}}{{\theta _{0}}_{z}}\overline{v'\theta'}
       \end{array}
       \right)
      = 0
\]
�Ȥʤ�. ���������դ��٤���, �Ȥγ�ư�٤��Ѳ� �� $\Ddiv \Dvect{F}$ ������
�ʤ�, ��ή��ˤ��¸���뤳�ȤǤ���. 

����Ū�˥ե�å����μ�«�ˤ�ä��Ȥγ�ư�٤��Ѳ�����, �Ȥ������ˤ����,
\[
  \Dvect{M}_T \equiv = 
       \left(
       \begin{array}{c}
	 \overline{u}A + \overline{v^{'2}}-\overline{e}        \\
	 - \overline{u'v'} \\
	 \frac{f_{0}}{{\theta _{0}}_{z}}\overline{v'\theta'}
       \end{array}
       \right)
\]
�������, 
\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*}
�Ȥʤ�. ������, �ե�å����� $x$ ��ʬ�� $\overline{u}A$ ��Ǥ����������
�뤳�Ȥ����դ�ɬ�פǤ���. �����ʿ��ή�˾�ä���ɸ���鸫���������̤�
���ꤵ�줿��ɸ���鸫�����ǥե�å������礭�����ۤʤ뤳�Ȥ��̣����.

Plumb (1986) �Ǥ�ʿ��ή����������Ƴ���ݤˤ� $\beta $ �̤ǤϤʤ� $f ��$
���Ѥ��Ƥ���.

�ɤΤ褦�ʥե�å�����ɽ�����Ѥ���Τ��褤����, ��ɼºݤ˸����絤��Ŭ
�Ѥ��ޤ�������Ƥ���ͤ��ʤ��Ȥ狼��ʤ����⤷��ʤ�.

%  ��ë�Ρ��Ȥ��Ȥ���­���뤳��
% 
%

\subsection{Trenberth (1986) �ξ��}

f �̤λ���ʿ�Ѥ���ʿ�Ѿ�μ��Ȥ���
\[
  \overline{u}\DP{\overline{u}}{x} 
      + \overline{v}\DP{\overline{u}}{x} 
      - f\overline{v}
      = -\DP{\overline{\phi}}{x} 
      - \DP{}{x}\overline{u^{'2}}-\DP{}{y}\overline{u'v'} 
\]
������. ������ѷ�����
\[
  \overline{u}\DP{\overline{u}}{x} 
      + \overline{v}\DP{\overline{u}}{x} 
      - fv_{T}^{*}
      = \frac{1}{\rho_0}\Ddiv \rho_0
        \left(
	\begin{array}{c}
	  \frac{1}{2}(\overline{v^{'2}-u^{'2}}) \\
	  - \overline{u'v'} \\
	  \frac{f_{0}}{{\theta _{0}}_{z}}\overline{v'\theta'}
	\end{array}
	\right)  
\]
\[
v_{T}^{*} \equiv \overline{v} - \frac{1}{f}\DP{}{x}(\overline{\phi}+K)
 - \frac{1}{\rho _0}\DP{}{z}\left(\rho _{0}\frac{f_0}{{\theta _0}_z}
 \overline{v'\theta'}\right)
\]
������ $K$ �ϱ�ư���ͥ륮���Ǥ���. �嵭�μ���
\[
  \rho_0
        \left(
	\begin{array}{c}
	  \frac{1}{2}(\overline{v^{'2}-u^{'2}}) \\
	  - \overline{u'v'} \\
	  \frac{f_{0}}{{\theta _{0}}_{z}}\overline{v'\theta'}
	\end{array}
	\right) 
 \neq \Dvect{C}_{g}A
\]
�Ǥ��뤳�Ȥ����դ���ɬ�פ�����. 

\subsection{Plumb (1985) �ξ��}

Plumb (1985) �Ǥ� TEM �Ϥ���������Ƴ�ФϹԤäƤ��ʤ�. ʿ�Ѿ�μ�
\[
  \DP{u^{(2)}}{t} + U\DP{u^{(2)}}{x} + v^{(2)}\DP{U}{y} - fv_{a}
   = -\DP{}{x}u^{'2} - \DP{}{y}u'v'
\]
�α��դι�ΰ����¸���ξõ���ݤǤ��ä���������������. 

���դ��ѷ��ϰʲ��Τ褦�˹Ԥ��Ȥ褤.
\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*}
���ΤȤ�
\[
  \DP{u^{(2)}}{t} + U\DP{u^{(2)}}{x} + v^{(2)}\DP{U}{y} - fv_{a}
   = - \frac{1}{2}\DP{}{x}
      \left(
        \psi ^{'2}_{y} - \psi'\psi'_{yy}
      \right)
     + \frac{1}{2}\DP{}{y}
      \left(
        \psi '_{x}\psi '_{y} - \psi'\psi'_{xy}
      \right)
\]
�Ȥʤä�, ���դΰ����¸����õ�뤳�Ȥ��Ǥ���.
\[
  Du^{(2)} - fv_{a}^{***} = \Ddiv \Dvect{W} \\
\]
\[
  v_{a}^{***} = v_{a} -\frac{1}{f}\left(\cdots \right)
\]
�Τ褦��ɽ�����Ȥ��Ǥ���.


\end{document}