%
% Chapter 4   Atmospheric Waves: Part II
%             4-3 CONDITIONS FOR BAROTROPIC INSTABILITY
%
% 2002/11/19  ���� ����   ����
% 2002/11/23  ���� ����   ����
%
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\markboth{5-3 �簵�԰���ξ��}{5-3 �簵�԰���ξ��}
%\markright{4-3 �簵�԰���ξ��}
%\Ddate{2002/11/23}
%\Dfile{hw-04-03.tex}

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\section{�簵�԰���ξ��}

\begin{itembox}[l]{����}
 \Dparskip \vspace{-6mm}
 ������Ǥ�, �簵�԰���ˤĤ��Ƶ�������.
 ����Ū�ˤϽ簵�԰��꤬������Τ�ɬ�פʾ���Ƴ��, �ޤ�,
 ���ξ���ȯã���뤫���ꤹ�뤫���Τ뤿��Υ��ͥ륮���μ������.

 �簵�԰��꤬������Τ�ɬ�פʾ�������ǵ�᤿ή���ؿ���
 �Ρ��ޥ�⡼�ɤβ�˴ؤ��뼰(5-5)�������.
 ����줿������簵�԰���ξ���,
 {\bf �ͤ������������б��٤����ͤ����}���ȤǤ���.

 ���ͥ륮���μ�������ǵ�᤿ή���ؿ��˴ؤ��뼰(5-4)�������.
 ���, �����ȯã�����{\bf ��������������η������������̥����ˤ�äƷ�
 �ޤ�}���Ȥ��狼��.
\end{itembox}

����Ǥ�, �簵�԰��꤬������Τ�ɬ�פʾ���Ƴ��, ���ͥ륮�������������.
$\psi = \Psi (y) e^{i \mu (x - ct)}$�Υ����פξ����԰���ˤʤ�ˤ�,
����®�٤�ʣ�ǿ��Ǥʤ���Фʤ�ʤ�.
���ʤ��$c = c_r + i c_i$�Ǥʤ���Фʤ�ʤ�.
������, �����Ƥ��Ͽ����ؿ�$\Psi$��ʣ�ǿ��Ǥ���.
$\Psi$��ʣ�Ƕ���Ǥ���$\Psi^{*}$��(5-5)�ˤ������,
%
\begin{eqnarray}
 (U - c)
  \left(
   \Psi^{*} \DD[2]{\Psi}{y} - \mu^2 \Psi^{*} \Psi
  \right)
  -
  \left(
   \DD[2]{U}{y} - \beta
  \right) \Psi^{*} \Psi = 0
\end{eqnarray}
%
�Ȥʤ�.
�ǽ�ι�ϰʲ��Τ褦�˽񤱤�.
%
\begin{eqnarray}
 \Psi^{*} \DD[2]{\Psi}{y}
  = \DD{}{y} \left( \Psi^{*} \DD{\Psi}{y} \right)
  - \DD{\Psi^{*}}{y} \DD{\Psi}{y}.
  \nonumber
\end{eqnarray}
%
ʣ�ǿ��Ȥ��ζ�����ѤϤ����ͤ������ͤ�2��ʤΤ�,
%
\begin{eqnarray}
 \Psi \Psi^{*} = |\Psi|^2, \quad
  \DD{\Psi^{*}}{y} \DD{\Psi}{y} = \left| \DD{\Psi}{y} \right|^2
  \nonumber
\end{eqnarray}
%
�Ǥ���.
���η�̤����Ѥ���(5-11)���ѷ���, �����$(U - c)$�Ǥ��μ�����,
���ξ�$\pm d$�֤���ʬ�򤹤�Ȱʲ��μ���������.
%
\begin{eqnarray}
 \int_{-d}^{d}
  \left[
   \DD{}{y} \left( \Psi^{*} \DD{\Psi}{y} \right) - \mu^2 |\Psi|^2
   - \left| \DD{\Psi}{y} \right|^2
  \right] dy
  = \int_{-d}^{d}
  \frac{(d^2U/dy^2 - \beta) |\Psi|^2}{(U - c)} dy.
\end{eqnarray}
%
���������$\Psi$��$\pm d$�ǤϾä��ʤ���Фʤ�ʤ�
\footnote{4-2��, (5-6)����.}.
���Τ���, $\Psi$�Ϥ��μ����ȵ������̡��˾ä��ʤ���Фʤ�ʤ�.
���Τ褦�˹ͤ����, $\Psi^{*}$��$\Psi$��Ʊ�ͤ˶����Ǥϥ����ˤʤ�.
�����, (5-12)�κ��դκǽ�ι����ʬ����ƥ����ˤʤ�.
���,
%
\begin{eqnarray}
 \int_{-d}^{d}
  \left(
   \mu^2 |\Psi|^2 + \left| \DD{\Psi}{y} \right|^2
  \right) dy
  = - \int_{-d}^{d}
  \frac{(d^2U/dy^2 - \beta) (U - c)^{*} |\Psi|^2}{|U - c|^2} dy
  \nonumber
\end{eqnarray}
%
�Ȥʤ�. �ʤ�, ���դ�����ʬ�ؿ��ˤ�, ����ʬ�Ҥ�ʬ���$(U - c)^{*}$��
������.
����, $c$��$c_r + i c_i$���֤�������. ����⤽�Τ褦��ɽ����,
%
\begin{eqnarray}
 \int_{-d}^{d}
  \left(
   \mu^2 |\Psi|^2 + \left| \DD{\Psi}{y} \right|^2
  \right) dy
  = - \int_{-d}^{d}
  \frac{(d^2U/dy^2 - \beta) (U - c_r) |\Psi|^2}{|U - c|^2} dy
  \hspace{20mm}
  \nonumber \\
 - i c_i \int_{-d}^{d}
  \frac{(d^2U/dy^2 - \beta) |\Psi|^2}{|U - c|^2} dy
  \nonumber 
\end{eqnarray}
%
�Ȥʤ�.
���μ��μ����ȵ����򸫤���, 3�Ĥ���2�Ĥ���ʬ�������Ǥ��뤿��,
�Ĥä�$i$�������Ǥ����������ʬ�ι�Ͼä��ʤ���Фʤ�ʤ�.
���Ū��,
%
\begin{eqnarray}
 c_i \int_{-d}^{d}
  \frac{(d^2U/dy^2 - \beta) |\Psi|^2}{|U - c|^2} dy = 0
\end{eqnarray}
%
�Ǥʤ���Фʤ�ʤ��Ȥ������Ȥˤʤ�.
���������Ȥ�¸�ߤ�����, $c_i \neq 0$�Ǥ���.
���Τ���, (5-13)����ʬ�Ͼä���ɬ�פ�����.
���Τ���ˤ�, $d^2U/dy^2 - \beta$��$-d < y < d$��ˤ�����
���ʤ��Ȥ������椬�Ѥ��ʤ��ƤϤʤ�ʤ�.
���ä�, �簵�԰����ɬ�׾���$y$�Τ����Ĥ�����($y_k$�Ȥ���)���ʲ��δط�
�����������ȤǤ���.
%
\begin{eqnarray}
 \left( \DD[2]{U}{y} - \beta \right)_{y_k} = 0, \quad
  -d < y_k < d.
\end{eqnarray}
%

���(5-14)�ϰʲ��Τ褦�ʷ��ǽ񤯤��Ȥ������.
%
\begin{eqnarray}
 \DD{}{y} \left( - \DD{U}{y} + f \right) = 0  \qquad
  \textrm{�ޤ���} \qquad
  \DD{\zeta_a}{y} = 0, \qquad
  y = y_k.
  \nonumber
\end{eqnarray}
%
���μ���, ���ܾ���Τ����Ĥ�����($y = y_k$)�Ǥ����б���
$\zeta_a = \zeta + f$  ($\zeta = - dU/dy$)�϶���, �ޤ��϶˾���
�ʤ�ʤ���Фʤ�ʤ����Ȥ򼨤��Ƥ���.

��������˴ؤ��륨�ͥ륮���μ�����뤳�Ȥ�, �簵�԰���˴ؤ���
���¿���ξ�������뤳�Ȥ������.
(5-4)��$- \psi$�򤫤�, $x$��$y$�˴ؤ�����ʬ�����,
%
\begin{eqnarray}
 \int_{-d}^{d}
  \overline{
   \left[
    - \psi \Ddiv \DP{}{t} \nabla \psi - U \psi \DP{(\Dlapla \psi)}{x}
    - \DP{}{x} \left( \frac{\psi^2}{2} \right)
    \left( \beta - \DD[2]{U}{y} \right)
   \right] dy
  }
  = 0
\end{eqnarray}
%
�Ȥʤ�. �ʤ�, (��) $= L^{-1} \int_{0}^{L} (\quad) dx$�Ǥ���.
�������ˤ��������̤ζ�������,
%
\begin{eqnarray}
 \psi (x + L, y, t) &=& \psi (x,y,t),
  \nonumber \\
 \psi (x, d, t) \hspace{8mm} &=& \psi (x, -d, t) = 0
\end{eqnarray}
%
�Ǥ���.
(5-15)�Τ��줾��ι����ʬ��ʬ��, ���(5-16)��Ȥ���,
�ʲ��Υ��ͥ륮���μ���Ƴ�����
\footnote{
 ���ͥ륮����Ƴ�Ф˴ؤ��Ƥ�{\bfseries 5-3-A}����.
}.
%
\begin{eqnarray}
 \DD{}{t} \int_{-d}^{d} \frac{\overline{{u'}^2 + {v'}^2}}{2} dy
  = - \int_{-d}^{d} \overline{u' v'} \DD{U}{y} dy.
\end{eqnarray}
%
�ʤ�, $u' = - \partial \psi / \partial y$, $v' = \partial \psi /
\partial x$�Ǥ���.
���α��դ�����$\psi$��ʲ��Τ褦�˹ͤ��뤳�ȤǷ��뤳�Ȥ������.
%
\begin{eqnarray}
 \psi (x, y, t) = \Psi (y, t) \cos [\mu x - \theta (y, t)].
\end{eqnarray}
%
$\Psi$�Ͽ�����, $\theta$�ϰ���Ǥ���
\footnote{���Τˤ�$x$�������Ѳ�ʬ�������������Ǥ���.}.
�����{\bfseries ��ư�̥ե�å���}({\itshape momentum flux})
\footnote{
 ��ư�̥ե�å���(�ޤ��ϱ�ư��ή®)�Ȥ�ʸ���ɤ��걿ư�̤�ή��Ǥ���.
 �ܤ����ϥ��������ե��å�(1970)���ȤΤ���.
}
�ϰʲ��Τ褦�˵��ҤǤ���.
%
\begin{eqnarray}
 \overline{u' v'}
  = \overline{- \partial \psi / \partial y \ \partial \psi / \partial x}
  = \Psi^2 \partial \theta / \partial y \ \mu \ 
  \overline{\sin^2 (\mu x - \theta)}
  \hspace{40mm}
  \nonumber \\
  + \mu \ \partial \Psi / \partial y \ \Psi \ 
  \overline{\sin (\mu x - \theta) \ \cos (\mu x - \theta)}.
  \nonumber
\end{eqnarray}
%
����ˡ��Ƴ����, $L$��$2 \pi \mu^{-1}$���ܿ��Ǥ���Ȥ�������
�Ѥ����, ��ư�̥ե�å����ϰʲ��Τ褦�ˤʤ�
\footnote{
 ����ˡ���$\sin^2 \alpha = (1/2) (1 - \cos 2 \alpha)$�ʤΤ�,
 %
 \begin{eqnarray}
  \overline{\sin^2 (\mu x - \theta)}
   &=& \frac{1}{L} \int_{0}^{L} \{ 1/2 - (1/2)\cos (2 \mu x - 2 \theta)\} dx
   \nonumber \\
  &=& 1/2 - (1/4 \mu) [\sin (2 \mu x - 2 \theta)]_{0}^{L}
   \nonumber \\
  &=& 1/2 - (1/4 \mu) \{\sin (2 \mu L - 2 \theta) - \sin (- 2 \theta) \}
   \nonumber
 \end{eqnarray}
 %
 �Ȥʤ�. $L = 2 \pi \mu^{-1} n$ ($n$:����)�ʤΤ�
 $\sin (2 \mu L - 2 \theta) = \sin (-2 \theta)$�Ȥʤ�,
 2���ܤϾ��Ǥ���.
 ���, $\overline{\sin^2 (\mu x - \theta)} = 1/2$�Ȥʤ�.

 ����, ����ˡ���$2 \sin \alpha \cos \alpha = \sin 2 \alpha$�ʤΤ�,
 %
 \begin{eqnarray}
  \overline{\sin (\mu x - \theta) \cos (\mu x - \theta)}
   &=& \frac{1}{L} \int_{0}^{L}  (1/2)\sin (2 \mu x - 2 \theta) dx
   \nonumber \\
  &=& (1/2L) [(- 1/2 \mu) \cos(2\mu x - 2 \theta)]_{0}^{L}
   \nonumber \\
  &=& (- 1/4 \mu L) \{ \cos (2 \mu L - 2 \theta) - \cos 2 \theta\}
   \nonumber \\
  &=& 0
   \nonumber
 \end{eqnarray}
 %
 �Ȥʤ�. �����Ǥ�$L = 2 \pi \mu^{-1} n$ ($n$:����)���Ѥ���.
 ���, ��ư�̥ե�å�����(5-19)�Ȥʤ�.
}.
%
\begin{eqnarray}
 \overline{u' v'} = (\mu/2) \Psi^2 \partial \theta / \partial y.
\end{eqnarray}
%
���μ���, $\overline{u' v'}$����椬�Ȥΰ���ΰ��������η���
$(\partial \theta / \partial y)$�ˤ�äƷ��ꤵ��뤳�Ȥ򼨤��Ƥ���.
(5-19)���, ���ͥ륮���μ�(5-17)�ϰʲ��Τ褦�˵��ҤǤ���.
%
\begin{eqnarray}
 \DD{}{t} \int_{-d}^{d}
  \frac{\overline{{u'}^2 + {v'}^2}}{2} dy
  = - \frac{\mu}{2} \int_{-d}^{d}
  \Psi^2 \DP{\theta}{y} \DD{U}{y} dy.
\end{eqnarray}
%
���μ���, ���������Υ�����ȿ�Фξ��
[$(\partial \theta / \partial y)(dU/dy) < 0$]�ˤϾ���ȯã��,
�դ˷��������Υ�����Ʊ�������ξ��
[$(\partial \theta / \partial y)(dU/dy) > 0$]�ˤϾ���ϼ�ޤ뤳�Ȥ�
�����Ƥ���.





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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%      5-3-A      %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{5-3-A\qquad ���ͥ륮���μ���Ƴ��}
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���ͥ륮���μ�(5-17)��ή���ؿ��μ�(5-15)�ȶ������(5-16)����Ƴ��.
(5-15), (5-16), (5-17)��ʲ��ˤޤȤ�Ƶ��Ҥ���.
%
\begin{eqnarray}
 \frac{1}{L} \int_{-d}^{d} \int_{0}^{L}
  \left[
   - \psi \Ddiv \DP{}{t} \nabla \psi - U \psi \DP{(\Dlapla \psi)}{x}
   - \DP{}{x} \left( \frac{\psi^2}{2} \right)
  \left( \beta - \DD[2]{U}{y} \right)
 \right] \ dx \ dy
  = 0 ,
  \nonumber \\
 \Deqlab{hw-04-03-���ͥ륮���μ���Ƴ��-ή���ؿ��μ�}
\end{eqnarray}
%
\begin{eqnarray}
 \Deqlab{hw-04-03-���ͥ륮���μ���Ƴ��-�������}
  \psi (x + L, y, t) &=& \psi (x,y,t),
  \\
 \psi (x, d, t) \hspace{8mm} &=& \psi (x, -d, t) = 0,
  \nonumber
\end{eqnarray}
%
\begin{eqnarray}
 \Deqlab{hw-04-03-���ͥ륮���μ���Ƴ��-���ͥ륮���μ�}
  \DD{}{t} \int_{-d}^{d} \frac{\overline{{u'}^2 + {v'}^2}}{2} dy
  = - \int_{-d}^{d} \overline{u' v'} \DD{U}{y} dy.
\end{eqnarray}
%


\subsubsection*{5-3-A-a \quad \Deqref{hw-04-03-���ͥ륮���μ���Ƴ��-ή���ؿ��μ�}����1��}

\Deqref{hw-04-03-���ͥ륮���μ���Ƴ��-ή���ؿ��μ�}����1���ܤ�׻�����.
%
\begin{eqnarray}
 \frac{1}{L} \int_{-d}^{d} \int_{0}^{L}
  \left[
   - \psi \Ddiv \DP{}{t} \nabla \psi
  \right] \ dx \ dy 
  \hspace{-50mm} & & \hspace{40mm}
  \nonumber \\
 &=&
  \frac{1}{L} \int_{-d}^{d} \int_{0}^{L}
  \left[
   - \psi \DP{}{t} \Dlapla \psi
  \right] \ dx \ dy
  \nonumber \\  
 &=&
  \frac{1}{L} \int_{-d}^{d} \int_{0}^{L}
  \left[
   - \psi \DP{}{t} \DP[2][]{\psi}{x}
  \right] \ dx \ dy
  + \frac{1}{L} \int_{-d}^{d} \int_{0}^{L}
  \left[
   - \psi \DP{}{t} \DP[2][]{\psi}{y}
  \right] \ dx \ dy.
  \nonumber 
\end{eqnarray}
%
�����Ǻ��ι��$x$��, ���ι��$y$����ʬ��ʬ��Ԥ���,
%
\begin{eqnarray}
 &=&
  \frac{1}{L} \int_{-d}^{d} 
  \left\{
   \left[ - \psi \DP{}{t} \DP[][]{\psi}{x} \right]_{0}^{L}
   - \int_{0}^{L} - \DP{\psi}{x} \DP{}{t} \DP[][]{\psi}{x} dx
  \right\} \ dy
  \nonumber \\
 & & \hspace{10mm}
 + \frac{1}{L} \int_{0}^{L} 
  \left\{
   \left[ - \psi \DP{}{t} \DP[][]{\psi}{y} \right]_{-d}^{d}
   - \int_{-d}^{d} - \DP{\psi}{y} \DP{}{t} \DP[][]{\psi}{y} dy
  \right\} \ dx
  \nonumber 
\end{eqnarray}
%
�Ȥʤ�.
�������\Deqref{hw-04-03-���ͥ륮���μ���Ƴ��-�������}���,
$\psi_{x=L} = \psi_{x=0}, \
(\partial \psi / \partial x)_{x=L} = (\partial \psi / \partial x)_{x=0}, \
\psi_{y=-d} = \psi_{y=d} = 0$
�Ǥ���. �ޤ�$u' = - \partial \psi / \partial y,
v' = \partial \psi / \partial x$�ʤΤ�,
%
\begin{eqnarray}
 &=& \frac{1}{L} \int_{-d}^{d} \int_{0}^{L}
  \left\{
   v' \DP{v'}{t} + u' \DP{u'}{t}
  \right\}
  \ dx \ dy
  \nonumber \\
 &=& \frac{1}{L} \int_{-d}^{d} \int_{0}^{L}
  \left\{
   \frac{1}{2} \DP{{u'}^2}{t} + \frac{1}{2} \DP{{v'}^2}{t}
  \right\}
  \ dx \ dy
  \nonumber \\
 &=& \frac{1}{L} \int_{-d}^{d} \int_{0}^{L}
  \DP{}{t}
  \left\{
   \frac{{u'}^2 + {v'}^2}{2} 
  \right\}
  \ dx \ dy
  \nonumber \\
 \Deqlab{hw-04-03-���ͥ륮���μ���Ƴ��-1}
  &=& \DD{}{t} \int_{-d}^{d}
  \overline{\frac{{u'}^2 + {v'}^2}{2}} 
  \ dy.
\end{eqnarray}
%


\subsubsection*{5-3-A-b \quad \Deqref{hw-04-03-���ͥ륮���μ���Ƴ��-ή���ؿ��μ�}����2��}

\Deqref{hw-04-03-���ͥ륮���μ���Ƴ��-ή���ؿ��μ�}����2���ܤ�׻�����.
%
\begin{eqnarray}
 \frac{1}{L} \int_{-d}^{d} \int_{0}^{L}
  \left[
   - U \psi \DP{(\Dlapla \psi)}{x}
 \right] \ dx \ dy
  \hspace{-50mm} & & \hspace{40mm}  
  \nonumber \\
 &=& 
  \Deqlab{hw-04-03-���ͥ륮���μ���Ƴ��-2}
  \frac{1}{L} \int_{-d}^{d} \int_{0}^{L}
   \left\{
    - U \psi \DP{}{x} \DP[2][]{\psi}{x}
    - U \psi \DP{}{x} \DP[2][]{\psi}{y}
   \right\} \ dx \ dy.
\end{eqnarray}
%
�ޤ��Ϥ���\Deqref{hw-04-03-���ͥ륮���μ���Ƴ��-2}����1���ͤ���.
%
\begin{eqnarray}
 \frac{1}{L} \int_{-d}^{d} \int_{0}^{L}
  \left\{
   - U \psi \DP{}{x} \DP[2][]{\psi}{x}
  \right\} \ dx \ dy
  \hspace{-50mm} & & \hspace{40mm}  
  \nonumber \\
 &=&
  \frac{1}{L} \int_{-d}^{d} 
  \left\{
   \left[ - U \psi \DP[2]{\psi}{x} \right]_{0}^{L}
   - \int_{0}^{L} \left\{ - U  \DP{\psi}{x} \DP[2]{\psi}{x} \right\} \ dx 
  \right\} \ dy
  \nonumber \\
 &=&
  \frac{1}{L} \int_{-d}^{d} U
  \left\{
   \left[ - \psi \DP[2]{\psi}{x} \right]_{0}^{L}
   - \int_{0}^{L}
   \left\{ - \frac{1}{2} \DP{}{x} \DP[][]{\psi}{x}^2 \right\} \ dx 
  \right\} \ dy
  \nonumber \\
 &=&
  \frac{1}{L} \int_{-d}^{d} U
  \left\{
   \left[ - \psi \DP[2]{\psi}{x} \right]_{0}^{L}
   - \left[ - \frac{1}{2} \DP[][]{\psi}{x}^2 \right]_{0}^{L}
  \right\} \ dy
  \nonumber \\
 \Deqlab{hw-04-03-���ͥ륮���μ���Ƴ��-3} 
  &=& 0.
\end{eqnarray}
%
�����ǤϺǸ�˶������\Deqref{hw-04-03-���ͥ륮���μ���Ƴ��-�������}
���Ѥ���.
����, \Deqref{hw-04-03-���ͥ륮���μ���Ƴ��-2}����2���ͤ���.
%
\begin{eqnarray}
 \frac{1}{L} \int_{-d}^{d} \int_{0}^{L}
  \left\{
   - U \psi \DP{}{x} \DP[2][]{\psi}{y}
  \right\} \ dx \ dy
  \hspace{-50mm} & & \hspace{40mm}  
  \nonumber \\
 &=&
  \frac{1}{L} \int_{-d}^{d} \int_{0}^{L}
  \left\{
   - U \DP{}{x} \left\{ \psi \DP[2][]{\psi}{y} \right\}
   + U \DP{\psi}{x} \DP[2]{\psi}{y}
  \right\} \ dx \ dy
  \nonumber \\
 &=&
  \frac{1}{L} \int_{-d}^{d} \int_{0}^{L}
  \bigg\{
   - U \DP{}{x} \left\{ \psi \DP[2][]{\psi}{y} \right\}
   + \DP{}{y} \left\{ U \DP{\psi}{x} \DP{\psi}{y} \right\}
   \nonumber \\
 & & \hspace{50mm}
  - U \frac{\partial^2 \psi}{\partial x \partial y} \DP{\psi}{y}
   - \DP{U}{y} \DP{\psi}{x} \DP{\psi}{y}
  \bigg\} \ dx \ dy
  \nonumber \\
 &=&
  \frac{1}{L} \int_{-d}^{d} \int_{0}^{L}
  \bigg\{
   - U \DP{}{x} \left\{ \psi \DP[2][]{\psi}{y} \right\}
   + \DP{}{y} \left\{ U \DP{\psi}{x} \DP{\psi}{y} \right\}
   \nonumber \\
 & & \hspace{45mm}
  - \frac{U}{2} \DP{}{x}  \DP[][]{\psi}{y}^2
   - \DP{U}{y} \DP{\psi}{x} \DP{\psi}{y}
  \bigg\} \ dx \ dy.
  \nonumber 
\end{eqnarray}
%
��1, 3���$x$����ʬ��, ��2���$y$����ʬ�����, ��������ꤽ���ι��
�����ˤʤ�. ��ä�,
%
\begin{eqnarray}
 \frac{1}{L} \int_{-d}^{d} \int_{0}^{L}
  \left\{
   - U \psi \DP{}{x} \DP[2][]{\psi}{y}
  \right\} \ dx \ dy
  \hspace{-50mm} & & \hspace{40mm}
  \nonumber \\
 &=&
  \frac{1}{L} \int_{-d}^{d} \int_{0}^{L}
  \bigg\{
   - \DP{U}{y} \DP{\psi}{x} \DP{\psi}{y}
  \bigg\} \ dx \ dy
  \nonumber \\
 &=&
  \frac{1}{L} \int_{-d}^{d} \int_{0}^{L}
  \bigg\{
   u' v' \DP{U}{y} 
  \bigg\} \ dx \ dy
  \nonumber \\
 \Deqlab{hw-04-03-���ͥ륮���μ���Ƴ��-4} 
  &=&
  \int_{-d}^{d}
   \overline{u' v'} \DP{U}{y} \ dy.
\end{eqnarray}
%
\Deqref{hw-04-03-���ͥ륮���μ���Ƴ��-3},  
\Deqref{hw-04-03-���ͥ륮���μ���Ƴ��-4}��
\Deqref{hw-04-03-���ͥ륮���μ���Ƴ��-2}���������뤳�Ȥ�
%
\begin{eqnarray}
 \Deqlab{hw-04-03-���ͥ륮���μ���Ƴ��-5}
  \frac{1}{L} \int_{-d}^{d} \int_{0}^{L}
  \left[
   - U \psi \DP{(\Dlapla \psi)}{x}
  \right] \ dx \ dy
  = \int_{-d}^{d}
  \overline{u' v'} \DP{U}{y} \ dy
\end{eqnarray}
%
�Ȥʤ�.


\subsubsection*{5-3-A-c \quad \Deqref{hw-04-03-���ͥ륮���μ���Ƴ��-ή���ؿ��μ�}����3��}

\Deqref{hw-04-03-���ͥ륮���μ���Ƴ��-ή���ؿ��μ�}����3���ܤ�׻�����.
$\beta$��$U$��$x$�ˤϰ�¸���ʤ��Τ�,
%
\begin{eqnarray}
 \frac{1}{L} \int_{-d}^{d} \int_{0}^{L}
  \left[
   - \DP{}{x} \left( \frac{\psi^2}{2} \right)
   \left( \beta - \DD[2]{U}{y} \right)
  \right] \ dx \ dy
  \hspace{-50mm} & & \hspace{40mm}
  \nonumber \\
 &=&
  \frac{1}{L} \int_{-d}^{d} 
  \left( \beta - \DD[2]{U}{y} \right)
  \left\{ \int_{0}^{L}
  \left[
   - \DP{}{x} \left( \frac{\psi^2}{2} \right)
  \right] \ dx
  \right\} \ dy
  \nonumber \\
 &=&
  \frac{1}{L} \int_{-d}^{d} 
  \left( \beta - \DD[2]{U}{y} \right)
  \left[
   - \frac{\psi^2}{2}
  \right]_{0}^{L}
  \ dy
  \nonumber \\
 \Deqlab{hw-04-03-���ͥ륮���μ���Ƴ��-6}
  &=& 0
\end{eqnarray}
%
�Ȥʤ�.


\Deqref{hw-04-03-���ͥ륮���μ���Ƴ��-1},
\Deqref{hw-04-03-���ͥ륮���μ���Ƴ��-5},
\Deqref{hw-04-03-���ͥ륮���μ���Ƴ��-6}��
\Deqref{hw-04-03-���ͥ륮���μ���Ƴ��-ή���ؿ��μ�}
���������뤳�Ȥ�, ��ʸ�Υ��ͥ륮���μ��Ǥ���
\Deqref{hw-04-03-���ͥ륮���μ���Ƴ��-���ͥ륮���μ�}
������줿.



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%\subsection*{Appendix \qquad }

%\subsubsection*{}


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