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%ɽ��   2 ���������ϳإ�ǥ� -- ��Ͽ A ������������Ƴ��
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%       2003-09-08 �ⶶ ������   ����ʸ���ɲ�
%       2003-09-09 �ⶶ ������   Appendix �ɲ�
%       2003-11-17 �ⶶ ������   ����
%       2003-11-22 �ⶶ ������   ��ư���ͥ륮���μ�����
%       2004-07-25 ��������      2 ���������絤�Ǥغƹ���
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\chapter{�����������Ϥ�Ƴ��}

\section{���ܾ���к���ʬΥ}

 �ѿ�����ܾ�Ȥ���������к���ʬ����. ���ܾ���ѿ��Ͼ��դ��С� 
 $(\bar{~})$ ��ɽ��, �к�����դ��ץ饤�� $( ^{\prime})$ ��ɽ��.  ������
 �� $\phi$ �ϰʲ��Τ褦��ʬΥ�����.
 \begin{eqnarray}
  \phi = \overline{\phi} + \phi^{\prime}.
 \end{eqnarray}
 ʬΥ������ѿ��� $u, w$, ���� $\theta$, ���� $p$ �ޤ��� $\Pi$,
 ̩�� $\rho$�Ǥ���.

 ���ܾ�Ͽ�ʿ���� ($\overline{\phi} = \overline{\phi}(z)$) ��, ����Τ�
 ���Ż߾��� ($\overline{u} = \overline{v} = \overline{w} = 0$) �Ǥ����
 ����. ���ܾ��̵�������ϴؿ��Ȳ��̤�, ���줾��,
 \begin{eqnarray}
  \overline{\Pi} &=& \left(\frac{\overline{p}}{p_{0}}\right)^{R_{d}/c_{p}}
  , \\
  \overline{\theta} &=& \frac{\overline{T}}{\overline\Pi}
 \end{eqnarray}
 �Ǥ���. ���ܾ��̩�� $\overline{\rho}$ �Ͼ�������������
 \begin{equation}
   \overline{\rho} = \frac{p_{0}}{R_{d}}
                    \frac{\overline{\Pi}^{c_{v}/R_{d}}}{\overline{\theta}}
 \end{equation}
 ��Ϳ������. ������ $c_{v}$ ��������Ǯ�Ǥ���.

 �ʲ��Ǥ� $\Pi^{\prime} = \pi$ �Ȥ�,
 ���Ҥ���ؤˤ��뤿�᤽��¾���ѿ����к��̤��դ����դ��ץ饤�� $(
 ^{\prime})$ �Ͼ�ά����.
 
%--------------------------------------------------------------------
\pagebreak

\section{��ư��������Ƴ��}

 �����Ǥϥ������ʡ��ؿ����Ѥ�����ư��������Ƴ�Ф�, ��ʿ�����Żߴ��ܾ��
 ���������к��Ȥ�ʬΥ��Ԥ�. 

\subsection{�������ʡ��ؿ���Ƴ��}

 ��ư��������
 \begin{eqnarray}
  \DD{u_{i}}{t} = - \frac{1}{\rho} \DP{p}{x_{i}}
                  - \delta_{i3} g
		  + D_{u_{i}}
 \end{eqnarray}
 �Ǥ���.
 ���ϸ��۹�� �������ʡ��ؿ����Ѥ��ƽ�ľ����, 
 \begin{eqnarray}
  \DD{u_{i}}{t} = - \frac{c_{p} p}{\rho R_{d} \Pi}
                    \DP{\Pi}{x_{i}}
                  - \delta_{i3} g
		  + D_{u_{i}}
 \end{eqnarray}
 ������, ����������
 \begin{eqnarray}
  p &=& \rho R_{d} \Pi \theta
 \end{eqnarray}
 ���Ѥ��Ƥ���˽�ľ����, 
 \begin{eqnarray}
  \DD{u_{i}}{t} = - c_{p} \theta \DP{\Pi}{x_{i}}
                  - \delta_{i3} g
		  + D_{u_{i}}
 \end{eqnarray}
 ������.

\subsection{���ܾ���к���ʬΥ}

 ��ư���������ʿ�����Żߴ��ܾ�μ��Ȥ���������к��μ�(�����μ�)
 ��ʬΥ����. ���ܾ�����������ſ尵�μ�
 \begin{equation}
  \DP{\overline{\Pi}}{z} = - \frac{g}{c_{p} \overline{\theta}}
 \end{equation}
 �Ǥ���.

 �����ο�ʿ�����α�ư������ $(i = 1)$ ��
 \begin{eqnarray}
  \DD{u}{t} &=& - c_{p}
                  \left(\overline{\theta}
	        + \theta^{\prime}
	          \right)
		  \DP{\left(\overline{\Pi} + \pi\right)}{x}
		+ D_{u}
  \nonumber \\
             &\simeq& - c_{p}
		     \left(\overline{\theta}_{v}
		           \DP{\overline{\Pi}}{x}
			   + \overline{\theta}_{v}
		             \DP{\pi}{x}
			   + \theta_{v}^{\prime}
		             \DP{\overline{\Pi}}{x}
		     \right)
		    + D_{u}
 \end{eqnarray}
 �Ȥʤ�. �����α�ľ�����α�ư������ $(i = 3)$��
 \begin{eqnarray}
  \DD{w}{t} &=& - c_{p}
                  \left(\overline{\theta}
		        + \theta^{\prime}
		  \right)
		  \DP{\left(\overline{\Pi} + \pi\right)}{z}
                -  g 
		+ D_{w}
  \nonumber \\
             &\simeq& - c_{p}
		     \left(\overline{\theta}_{v}
		           \DP{\overline{\Pi}}{z}
			   + \overline{\theta}_{v}
		             \DP{\pi}{z}
			   + \theta_{v}^{\prime}
		             \DP{\overline{\Pi}}{z}
		     \right)
                    - g
		    + D_{u}
  \Deqlab{���ܾ�ޤ౿ư������}
 \end{eqnarray}
 �Ȥʤ�. ���줾��μ��ǰ��ϸ��۹�ˤ����� 2 ���̤Ͼ������Ȳ��ꤷ̵�뤹��.

 ���ܾ�Ͽ�ʿ���ͤǤ��뤳�Ȥ�, ��ľ�����α�ư�����������ſ尵�μ��򺹤�
 �������Ȥˤ��, �ʲ��ξ����α�ư������������.
 \begin{eqnarray}
  \DD{u}{t} &=&  - c_{p}\overline{\theta}_{v}
		         \DP{\pi}{x}
		+ D_{u}
  , \\
  \DD{w}{t} &=&  - c_{p}
                  \overline{\theta}_{v} \DP{\pi}{z}
		+ g\frac{\theta}{\overline{\theta}}
		+ D_{w}
  .
 \end{eqnarray}

 �ޤȤ��ɽ����, 
 \begin{eqnarray}
  \DD{u_{i}}{t} = - c_{p} \overline{\theta}_{v} \DP{\pi}{x_{i}}
      + \delta_{i3} g\frac{\theta}{\overline{\theta}}
      + D_{u_{i}}
 \end{eqnarray}
 �Ȥʤ�.

%--------------------------------------------------------------------

\clearpage
\section{������������Ƴ��}

\subsection{����������}

 ������������, Ϣ³�μ�
 \begin{eqnarray}
  \DD{\rho}{t}
   + \rho \DP{u_{j}}{x_{j}}
   = 0
  \Deqlab{Ϣ³�μ�}
 \end{eqnarray}
 �ȥ������ʡ��ؿ���ɽ��������������
 \begin{eqnarray}
  \Pi = \left(\frac{R_{d}}{p_{0}}
              \rho \theta_{v}\right)^{R_{d}/c_{v}}
  \Deqlab{�������ʡ��ؿ����Ѥ�������������}
 \end{eqnarray}
 ����Ƴ�Ф���.

 ����������\Deqref{�������ʡ��ؿ����Ѥ�������������}�� $t$ ����ʬ����:
 \begin{eqnarray}
  \DD{\Pi}{t} &=& \left(\frac{R_{d}}{p_{0}}\right)^{R_{d}/c_{v}}
                  \left\{\frac{R_{d}}{c_{v}}
		         \theta^{R_{d}/c_{v}}
		         \rho^{R_{d}/c_{v} - 1}
		         \DD{\rho}{t}
			 + \frac{R_{d}}{c_{v}}
		           \rho^{R_{d}/c_{v}}
		           \theta^{R_{d}/c_{v} - 1}
			   \DD{\theta}{t}
                  \right\}
  \nonumber \\
	      &=& \frac{R_{d}}{c_{v}}
	          \left(\frac{R_{d}}{p_{0}}
		        \rho \theta
		  \right)^{R_{d}/c_{v}}
                  \left\{\frac{1}{\rho}
		         \DD{\rho}{t}
			 + \frac{1}{\theta}
			   \DD{\theta}{t}
                  \right\}
  \nonumber \\
              &=& \frac{R_{d}}{c_{v}} \Pi
                  \left\{\frac{1}{\rho}
		         \DD{\rho}{t}
			 + \frac{1}{\theta}
			   \DD{\theta}{t}
                  \right\}
  .
 \end{eqnarray}
 Ϣ³�μ�\Deqref{Ϣ³�μ�}���դ����������,
 \begin{eqnarray}
  \DD{\Pi}{t}  &=& \frac{R_{d}}{c_{v}} \Pi
                  \left\{- \DP{u_{j}}{x_{j}}
			 + \frac{1}{\theta}
			   \DD{\theta}{t}
                  \right\}
 \end{eqnarray}
 ������.

\subsection{�����к���������}

 �������ʡ��ؿ����ʿ���ʹ��ܾ� $\overline{\Pi}$ �Ȥ���������к� $\pi$ 
 ��ʬΥ��, $\pi$ �������������. 
 \begin{eqnarray}
  \DP{\pi}{t} + u_{j} \DP{\pi}{x_{j}}
              + w \DP{\overline{\Pi}}{z}
	      = - \frac{R_{d}}{c_{v}}
	            \overline{\Pi} \DP{u_{j}}{x_{j}}
		  - \frac{R_{d}}{c_{v}}
		    \pi \DP{u_{j}}{x_{j}}
		  + \frac{R_{d}}{c_{v}}
		    \frac{\Pi}{\overline{\theta} +\theta}
		    \DD{(\overline{\theta}+\theta)}{t}
 .
 \end{eqnarray}
 ��®
 \begin{eqnarray}
  c^{2} &=& \frac{c_{p} R_{d}}{c_{v}} \Pi (\overline{\theta}+\theta)
  , \\
  \overline{c^{2}} &=& \frac{c_{p} R_{d}}{c_{v}}
                  \overline{\Pi} \overline{\theta}
 \end{eqnarray}
 ���Ѥ��ƽ�ľ����,
 \begin{eqnarray}
  \DP{\pi}{t} + \frac{\overline{c}^{2}}{c_{p} \overline{\theta}}
	        \DP{u_{j}}{x_{j}}
              + w \DP{\overline{\Pi}}{z}
	      &=& - u_{j} \DP{\pi}{x_{j}}
		  + \frac{R_{d}}{c_{v}}\pi
		    \DP{u_{j}}{x_{j}}
		  + \frac{c^{2}}{c_{p} (\overline{\theta} +\theta)^{2}}
		    \DD{(\overline{\theta}+\theta)}{t}
 .
 \end{eqnarray}
 �����Ǿ���������\Deqref{�������ʡ��ؿ����Ѥ�������������}���Ѥ���
 ������ 3 ����ѷ�����.
 \begin{eqnarray*}
  w \DP{\overline{\Pi}}{z}
      &=& w \left(\frac{R_{d}}{p_{0}}\right)^{R_{d}/c_{v}}
          \DP{\left(\overline{\rho}
	            \overline{\theta}
	      \right)^{R_{d}/c_{v}}}
	     {z}
  \nonumber \\
      &=& w \left(\frac{R_{d}}{p_{0}}\right)^{R_{d}/c_{v}}
          \frac{R_{d}}{c_{v}}
	  \left(\overline{\rho}
	            \overline{\theta}
	      \right)^{R_{d}/c_{v} - 1}
          \DP{\overline{\rho} \overline{\theta}}{z}
  \nonumber \\
      &=& w \frac{R_{d}}{c_{v}}
          \overline{\Pi}
	  \frac{1}{\overline{\rho} \overline{\theta}}
          \DP{\left(\overline{\rho} \overline{\theta}_{v}\right)}{z}
  \nonumber \\
      &=& \frac{\overline{c}^{2}}
               {c_{p} \overline{\rho} \overline{\theta}_{v}^{2}}
	  w
          \DP{\left(\overline{\rho} \overline{\theta}_{v}\right)}{z}
  \nonumber \\
      &=& \frac{\overline{c}^{2}}
               {c_{p} \overline{\rho} \overline{\theta}^{2}}
	  u_{j}
          \DP{\left(\overline{\rho} \overline{\theta}\right)}{x_{j}}
  .
 \end{eqnarray*}
 �ޤ�, ������ 2 ���
 \begin{eqnarray*}
   \frac{\overline{c}^{2}}{c_{p} \overline{\theta}}
    \DP{u_{j}}{x_{j}}
      = \frac{\overline{c}^{2}}
             {c_{p} \overline{\rho} \overline{\theta}^{2}}
	 \overline{\rho} \overline{\theta}
	\DP{u_{j}}{x_{j}}
 \end{eqnarray*}
 �Ƚ񤭴����뤳�Ȥ��Ǥ���.
 ������ 2, 3 ���ޤȤ��Ȱ����������ϰʲ��Τ褦��ɽ�����.
 \begin{equation}
  \DP{\pi}{t}
     + \frac{\overline{c}^{2}}{c_{p}
         \overline{\rho} \overline{\theta}^{2}}
         \DP{}{x_{j}}
	 \left(\overline{\rho} \overline{\theta} u_{j}\right)
     = f_{\pi}
 \end{equation}
 \begin{equation}
  f_{\pi} = - u_{j} \DP{\pi}{x_{j}}
            + \frac{R_{d} \pi}{c_{v}} \DP{u_{j}}{x_{j}}
  	    + \frac{c^{2}}{c_{p} (\overline{\theta} +\theta)^{2}}
	   \DD{(\overline{\theta}+\theta)}{t}
  .
 \Deqlab{�����������ζ�����}
 \end{equation}
% ���ΰ����������� Tapp and White (1976) �������ϥ᥽��ǥ��
% �Ѥ����Τ�Ʊ���������Ǥ���.

 Klemp and Wilhelmson (1978) �ˤ�������, �ܥ�ǥ�Ǥ� $f_{\pi} = 0$ �Ȥ�
 ��. �������������Ԥ�, ����Ǯ��(\Deqref{�����������ζ�����}
 ������ 3 ��)��̵�뤹�뤳�Ȥ��̣����.


