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\begin{document}
\section{���}
�ܥɥ�����ȤǤ�, NumRu::EP\_Flux ���������� Eliassen-Palm Flux(EP-Flux)
�׻��⥸�塼�뤽��¾�β����Ԥ�. ���ΤȤ���, ���̾�� EP-Flux �Τߤ򥵥ݡ��Ȥ��Ƥ���.
������ǥ�� Andrews et al(1987) ���� 3 �Ϥ򻲹ͤˤ���\cite{AHL1987}.

\subsection{EP-Flux ��TEM �Ϥα�ư������}

Andrews et al(1987)�ˤ�������̾�� EP-Flux �ϼ��Τ褦����������.
\begin{eqnarray}
  \left.
  \begin{array}{cl}
    F_y =& \rho_0 a  \cos \phi \left(\overline{\DP{u}{z}}
    \frac{\overline{v'\theta'}}{\overline{\DP{\theta}{z}}} - \overline{u'v'}\right)\\
    F_z =& \rho_0 a \cos \phi \left(\left[ f - \frac{\DP{\overline{u}\cos \phi}{\phi}}{a\cos\phi} \right]
    \frac{\overline{v'\theta'}}{\overline{\DP{\theta}{z}}} - \overline{u'w'}\right).
  \end{array}
  \right\}
\end{eqnarray}
$F_y, F_z$�����줾�� EP-Flux �ΰ�����ʬ, ����ӱ�ľ��ʬ�Ǥ���. 
$\bar{\star}$������ʿ����, $\star'$������ʿ���̤���ξ�����ʬ��ɽ��.
������, $u, v, w, \theta$�Ϥ��줾��������®, ������®, ��ľ��®, ���̤Ǥ���. �ޤ�$z, \phi$�Ϲ��٤���Ӱ���, $a$������Ⱦ��(���), $\rho_0$�ϻ���̩�٥ץ��ե�����, $f$�ϥ��ꥪ��ѥ�᡼���Ǽ�ž��®��$\Omega$���Ѥ���$2\Omega\sin\phi$���������. 

����, �ܥ⥸�塼��Ǥϼ���������������$\hat{F_y}, \hat{F_z}$�� EP-Flux �ȸƤ�.
\begin{eqnarray}
  \left.
  \begin{array}{cl}
    \hat{F_y} =& \sigma  \cos \phi \left(\overline{\DP{u}{z}}
    \frac{\overline{v'\theta'}}{\overline{\DP{\theta}{z}}} - \overline{u'v'}\right)\\
    \hat{F_z} =& \sigma  \cos \phi \left(\left[ f - \frac{\DP{\overline{u}\cos \phi}{\phi}}{a\cos\phi} \right]
    \frac{\overline{v'\theta'}}{\overline{\DP{\theta}{z}}} - \overline{u'w'}\right)
  \end{array}
  \right\}
\end{eqnarray}
%
������$\sigma$ �ϰ��Ϥ����㤹��̵�����̤�, �絤�λ���̩�٤����������̤Ǥ���. 
%
\begin{align}
  \sigma \equiv \frac{p}{p_{00}} = \exp\left(\frac{-z}{H}\right),
\end{align}
%
$H$�ϰ����п���ɸ�ˤ����륹������ϥ���(���)�Ǥ���. (1.1)���Ȥΰ㤤��
\begin{enumerate}
  \item ����Ⱦ��$a$�������äƤ��ʤ�(����Ⱦ�¤ǵ��ʲ����Ƥ���)
  \item ����̩�٥ץ��ե�����$\rho_0$������ˤ�������㤹��̵������$\sigma$���Ѥ��Ƥ���
\end{enumerate}
�Ǥ���. 

$\sigma$��$\rho_0$�μ����ΰ㤤��̵�뤹���$F_y, F_z$��$\hat{F_y}, \hat{F_z}$�ϰʲ��Τ褦�˴ط��դ�����.
\begin{align}
  (F_y, F_z) = a(\hat{F_y}, \hat{F_z})
\end{align}


���̾��ȯ���ϰʲ��η����������.
%
\begin{align}
  \Ddiv{\hat{\Dvect{F}}} = \Dinv{a \cos \phi} \DP{F_y \cos \phi}{\phi} + \DP{F_z}{z}
\end{align}
%
�ޤ��ĺ��۴�$(0, \bar{v}^*, \bar{w}^*)$��ʲ��η����������.
%
\begin{align}
  \overline{v}^* &\equiv \bar{v} 
   - \Dinv{\sigma}\DP{}{z}\left(\sigma\frac{\overline{v'\theta'}}
                   {\overline{\DP{\theta}{z}}}\right)\\
  \overline{w}^* &\equiv \bar{w} 
   + \Dinv{a \cos\phi}\DP{}{\phi}\left(\cos\phi\frac{\overline{v'\theta'}}
                   {\overline{\DP{\theta}{z}}}\right)
\end{align}
%
�������ѷ������顼ʿ��(TEM)�Ϥ�ή�Τα�ư��������
%
\begin{align}
  \DP{\bar{u}}{t} + \bar{v}^*
          \left[
              \Dinv{a\cos\phi}\DP{(\bar{u}\cos\phi)}{\phi}-f
          \right]
          + \bar{w}^*\DP{\bar{u}}{z} - \bar{X} = \Dinv{\sigma \cos\phi}\Ddiv{\hat{\Dvect{F}}}\label{eq:TEM-ME}
\end{align}
%
�Ƚ񤱤�. ���դˤ�����, ����Ⱦ��$a$�ǳ�äƤ��ʤ��Ȥ������̾�� TEM �Ϥα�ư�������Ȱۤʤ�Ȥ����Ǥ��뤬, $\hat{\Dvect{F}}=(\hat{F_y}, \hat{F_z})$������Ⱦ�¤ǵ��ʲ����� EP-Flux �Ǥ��뤳�Ȥ��θ�����Ǽ������������. ���Τ褦��, ���դ��̤ϰ���Ū���������� TEM �Ϥα�ư������Ʊ�ͤ˲�®���̣���뤳�Ȥˤʤ�.

\subsection{����ή���ؿ�}

�ĺ��۴Ĥ�������$\sigma$�򤫤������ȯ���Ȥʤ�. ������Ѥ����ܥ⥸�塼��Ǥϻĺ��۴Ĥμ���ή���ؿ�$\Psi^*$��
%
\begin{align}
  \sigma\bar{v}^* &= -g\Dinv{2\pi a \cos\phi }\DP{}{p}\Psi^*\\
  \sigma\bar{w}^* &= g\Dinv{2\pi a^2\cos\phi}\DP{}{\phi}\Psi^*
\end{align}
%
���������. ������ z(log-P)�Ϥ��鰵�Ϻ�ɸ�Ϥؤ��Ѵ���(1.3)����
%
\begin{align}
  \DP{}{z}\Psi^* &= -\frac{p}{H}\DP{}{p}\Psi^*
\end{align}
%
���Ѥ���. �ܥ⥸�塼��ǤϾ弰����ʬ����, �絤��ü($p=0$)�ˤ�����$\Psi^* = 0$�Ȥ���, 
%
\begin{align}
  \Psi^*(\theta, p) = \frac{2\pi a \cos\phi}{g} \int_{0}^{p}\bar{v}^*\Dd p
\end{align}
%
�Ȥ��Ƽ���ή���ؿ���Ƴ��.

\subsection{�ѿ��Ѵ�}


���Ϥ����ǡ����α�ľ�����������Ǥ��ä����, �ʲ��δط������Ѥ��ƹ��ټ����Ѵ���, �׻���Ԥ�.
\begin{align}
  z &= -H \log \left( \frac{p}{p_{00}} \right),\\
  p &= p_{00} \exp \left( -\frac{z}{H} \right)
\end{align}
������$p$�ϰ���, $p_{00}$����ɽ�̻��͵���(���)�Ǥ���.

���Ϥ�$\theta$��$w$ �Ǥʤ�, ����$T$, ���ϡ�®�١�$\omega \equiv Dp/Dt$ �ξ��Ϥ��줾��򸵤�$w$, $\theta$�����ɬ�פ�����. �����Ǥϰʲ��μ����Ѥ���$w, \theta$���᤿.
%
\begin{align}
  w &= -\omega H / p\\
  \theta &= T \left(\frac{p_{00}}{p}\right)^\kappa, \kappa = R/C_p
\end{align}
%
������$R$, $C_p$�Ϥ��줾�촥������ε������������갵��Ǯ�Ǥ���. �ޤ�(1.7)����(1.5)�����Ѥ���
%
\begin{align}
  \theta = T\exp \left( \frac{\kappa z}{H} \right )
\end{align}
%
�Ƚ񤯤��Ȥ��Ǥ���.


\clearpage%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{�ƥ��ȥץ������δؿ�}
����Ǥϥƥ��ȥץ������(demo/test.rb)����Ѥ�����ϴؿ��γ��פ�Ҥ٤�. 
�ƥ��ȥץ������Ǥϥ᥽�åɤ�Ϳ��������˴ؤ��� 3 �ĤΥѥ�����ˤĤ���
���줾��ƥ��Ȥ�Ԥ�. �ѥ�����2, �ѥ�����3���Ѥ�����ؿ��ϥѥ�����1����������
��Τ�Ʊ��Ǥ���. 

\subsection{�ѥ����� 1: W, ����, z ��ɸ}
�ƥ����Ѥδؿ���ʲ����������.
\begin{align}
  u(\lambda, \phi, z) &= \sin \lambda \cos \phi + 10 + z,\notag\\
  v(\lambda, \phi, z) &= 2\sin \lambda \cos \phi + 20,\notag\\
  w(\lambda, \phi, z) &= 3\sin \lambda \cos \phi + 30,\notag\\
  T(\lambda, \phi, z) &= 4\sin \lambda \cos \phi + 40 \notag,
\end{align}
������ $u$ ��������®[m/s],
       $v$ ��������®[m/s],
       $w$ �ϱ�ľ®��[m/s],
       $T$ �ϲ���[K],
($\lambda, \phi$)�Ϸ��٤���Ӱ���, ������$z$�Ϲ���[m]�Ȥ���.
$u, v, T, \omega$ ���줾�������ʿ�Ѥ���Ӿ�����ʬ��
\begin{align}
  \overline{u} &= 10 + z,\notag\\
  \overline{v} &= 20,\notag\\
  \overline{w} &= 30,\notag\\
  \overline{\theta} &= 40\left(\frac{p_{00}}{p}\right)^\kappa,\notag\\
                    &= 40\exp\left( \frac{\kappa z}{H} \right),\notag\\
  {u'} &= \sin \lambda \cos \phi,\notag\\
  {v'} &= 2\sin \lambda \cos \phi,\notag\\
  {w'} &= 3\sin \lambda \cos \phi,\notag\\
  {\theta'} &= 4\sin \lambda \cos \phi\exp\left( \frac{\kappa z}{H} \right)\notag
\end{align}
�Ȥ�����. ������걲��ư�̥ե�å���, ��Ǯ�ե�å����Ϥ��줾��
\begin{align*}
  \overline{u'v'} &= \Dinv{2\pi} \int_0^{2\pi} 2 \sin^2 \lambda \cos^2 \phi \Dd \lambda\notag\\
                  &= \cos^2 \phi\notag\\
  \overline{v'\theta'} &= \Dinv{2\pi} \int_0^{2\pi}
                   8\sin^2 \lambda \cos^2 \phi\exp\left( \frac{\kappa z}{H} \right) \Dd \lambda,\\
                   & = 4 \cos^2 \phi\exp\left(\frac{z \kappa}{H}\right),\\
  \overline{u'w'} &= \Dinv{2\pi} \int_0^{2\pi} 3 \sin^2 \lambda \cos^2 \phi \Dd \lambda\\                 &= 1.5 \cos^2 \phi
\end{align*}
�Ȥʤ�. 

�ޤ�$\overline{\DP{u}{z}}$�����$\overline{\DP{\theta}{z}}$��
\begin{align}
\overline{\DP{u}{z}} &= 1 \notag\\
\overline{\DP{\theta}{z}} &= \frac{40 \kappa}{H}\exp\left( \frac{\kappa z}{H} \right) 
\end{align}
�Ȥʤ�.

���ꥪ��ѥ�᡼��$f$��

\begin{align}
  f = 2 \Omega \sin \phi
\end{align}

����������. ������$\Omega$�ϼ�ž��®�٤Ǥ���. ���������б��٤�����ʿ���ͤ�
\begin{align}
  M &= f - \frac{\DP{\overline{u}\cos\phi}{\phi}}{a\cos\phi}\\
    &= f + \frac{10 + z}{a}\tan\phi
\end{align}

�Ȥ�����. �������Ѥ��� Eliassen-Palm Flux ��
\begin{align}
  F_y &= \exp\left(\frac{-z}{H}\right) \cos \phi 
  \left( \frac{
     4 \cos^2\phi \exp\left(\frac{\kappa z}{H}\right) }
    { \frac{40\kappa}{H}\exp\left(\frac{\kappa z}{H}\right) } - \cos^2 \phi \right)\notag\\      &= \exp\left(\frac{-z}{H}\right) \cos^3 \phi \left( \frac{H}{10\kappa} - 1 \right),\\
  F_z &= \exp\left(\frac{-z}{H}\right) \cos \phi \left(
   \left[ f + \frac{10 + z}{a}\tan\phi \right]
  \frac{
     4 \cos^2\phi \exp\left(\frac{\kappa z}{H}\right) }
    { \frac{40\kappa}{H}\exp\left(\frac{\kappa z}{H}\right) } - 1.5 \cos^2 \phi \right)\notag\\
      &= \exp\left(\frac{-z}{H}\right) \cos^3 \phi 
           \left( \left[ f + \frac{10 + z}{a}\tan\phi \right] \frac{H}{10\kappa} - 1.5 \right)
\end{align}

�Ȥ�����. 
%������$H=a=1$�Ȥ���ʤ��
%\begin{align}
%  F_y &= \exp\left(\frac{-z}{H}\right)\cos \phi 
%  \left( \frac{
%    - 4 \cos^2\phi p }
%    { 300*\kappa } - \cos^2 \phi \right),\\
%  F_z &= \exp\left(\frac{-z}{H}\right) \cos \phi \left(
%   \left( f + ({10 + p})\tan\phi \right)
%    \frac{
%    4 \cos^2\phi}
%    {300*\kappa } - 1.5 \cos^2 \phi \right).
%\end{align}
%
�ޤ�(F$_y$, F$_z$)��ȯ����
%
\begin{align}
\Ddiv(F) & = \Dinv{a \cos \phi}\DP{F_y}{\phi} + \DP{F_z}{z}\notag\\
         & = \frac{-4}{3a}\exp\left(\frac{-z}{H}\right)\cos^3 \phi \left\{ \frac{4 p}{300 \kappa} + 1 \right\}\notag\\
         & -\Dinv{H} \exp\left(\frac{-z}{H}\right) \cos \phi \left(
   \left( f + ({10 + p})\tan\phi \right)
    \frac{
    4 \cos^2\phi}
    {300*\kappa } - 1.5 \cos^2 \phi \right)\notag\\
         & -\Dinv{H} \frac{4\cos^3\phi p}{300\kappa} \exp\left(-\frac{z}{H}\right)
\end{align}
�Ȥʤ�.

�ĺ��۴Ĥ�(1.9),(1.10)��ꤽ�줾��
%
\begin{align}
  \overline{v}^* &= 
  20 - \Dinv{\sigma}\DP{}{z}
     \left(
        \sigma\frac{4\cos^2\phi\exp\frac{z\kappa}{H}}
                   {\frac{40 \kappa}{H}\exp\frac{\kappa z}{H}}
     \right)\notag\\
                 &= 
  20 - \Dinv{\exp\left(-\frac{z}{H}\right)}\DP{}{z}
     \left(
        H\exp\left(-\frac{z}{H}\right)\frac{\cos^2\phi}
                   {10 \kappa}
     \right)\notag\\
                 &=
  20 - \Dinv{\exp\left(-\frac{z}{H}\right)}
     \left(
        H\cdot-\Dinv{H}\exp\left(-\frac{z}{H}\right)\frac{\cos^2\phi}
                   {10 \kappa}
     \right)\notag\\
                 &=
  20 + \frac{\cos^2\phi}{10 \kappa},\\
  \overline{w}^* &= 
  30 + \Dinv{a\cos\phi}\DP{}{\phi}
     \left(
        \cos\phi\frac{4\cos^2\phi\exp\frac{z\kappa}{H}}
                   {\frac{40 \kappa}{H}\exp\frac{\kappa z}{H}}
     \right)\notag\\
                 &= 
  30 + \Dinv{a\cos\phi}\DP{}{\phi}
     \left(
        H\frac{\cos^3\phi}
                   {10 \kappa}
     \right)\notag\\
                 &=
  30 + \Dinv{a\cos\phi}
     \left(
        H\frac{-3\cos^2\phi\sin\phi}
                   {10 \kappa}
     \right)\notag\\
                 &=
  30 - \frac{3H\sin\phi\cos\phi}{10 a \kappa}.
\end{align}
%

(1.15)��(2.23)�������ή���ؿ�$\Psi^*$��
%
\begin{align}
  \Psi^* = \frac{2\pi a \cos\phi}{g}
             \left(20 + \frac{\cos^2\phi}{10\kappa}\right)p
\end{align}
�Ƚ񤱤�.

\subsection{�ѥ����� 2: $\Omega$, ����, z ��ɸ}

�ѥ����󣲤ϰ�����Ϳ�����ľ®�٤���Ӳ��٤����줾��$\Omega$(Pa/s), ����$\theta$(K)
�Ǥ������, ��������������������뤫��ƥ��Ȥ��Ƥ���. ���Ѥ���ؿ��η��ϥѥ�����1
��Ʊ�ͤǤ���.

\subsection{�ѥ����� 3: W, ����, p ��ɸ}

�ѥ����� 3 �ϰ�����Ϳ����ʪ���̤����Ϻ�ɸ���������Ƥ�����Υƥ��ȤǤ���.
���Ϻ�ɸ��Ϳ����줿���, �⥸�塼�������ǰ�ö z �Ϥ��Ѵ�������˷׻���Ԥ�, �Ǹ��
����åɤ��Ϻ�ɸ���ᤷ���ͤ��֤��褦�ˤʤäƤ���.

\clearpage

\begin{thebibliography}{1}
\bibitem{AHL1987} D.G. Andrews, J.R. Holton, and C.B. Leovy. 
                  Middle atmosphere dynamics, International Geophysics Series. 
                  Academic Press, 1987
\end{thebibliography}


\end{document}
%%%%%%%%              Text End                  %%%%%%%%
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%%%%%%%%            ���� (��٥��դ�)           %%%%%%%%
%
%\begin{eqnarray}
% \Deqlab{1.1} % ���ʽ�Ǥμ��ֹ�������.
%  \DP{\rho}{t} + \Ddiv (\rho \Dvect{V}) = 0.
%\end{eqnarray}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%  ���� (���ֹ����Ω���ƽ񤭤������)   %%%%%%%%
%
%$$
% \DP{p}{z} = \rho g.
% \eqno \textrm{(1.11)}
%$$
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%         ����ʸ�� (��ʸ�˽񤯾��)      %%%%%%%%
%
%{\bfseries ����ʸ��}
%\vspace{-7mm}
%\begin{description}
% \item	����̾, 2000:
%	����̾, (��, ��).
%	���Ǽ�, 319pp.
%\end{description}
%
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%%%%%%%%             �ޤ�Ž����                 %%%%%%%%
%
%\begin{figure}[hbtp]
% \begin{center}
%  \Depsf[][]{./SEC01/images/fig0101.eps}
% \end{center}
% \caption{
%  ���Ф�
% }
% \Dfiglab{fig0101} % ���ʽ�οޤ��ֹ�������,
%                   % table �οޤξ��� tab0101
%\end{figure}
%
