% ɽ�� �వ���������ϤΥ��ͥ륮����������Ƴ��
%
% ����: 2011/12/26  ��������(deepconv/arare5 �꼰���ޥ˥奢��˼�����)
% ����: 2011/12/19  ��������(�ɵ�)
% ����: 2011/11/30  ��������(�ɵ�)
% ����: 2011/11/29  ��������(���� and �ɵ�)
% ����: 2011/11/28  ��������(�ɵ�)
% ����: 2011/11/26  ��������(�ɵ� ((A.9) ���ѷ�����))
% ����: 2011/11/25  �������� (��������(�̼齤������Ͽ�Υȥ졼��))
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%             ��ʸ Start                 %%%%%%%%

�వ�����꼰�ˤ����륨�ͥ륮����������Ƴ�Ф���. �����������Ǥϴ����絤
�����ꤷ, �ŷ�˴ؤ�����̵�뤹��. 

�����絤�ξ��, �వ����������Ϣ³�μ���
\begin{eqnarray}
  \DP{\rho}{t} + \Ddiv{\overline{\rho}\Dvect{u}} = 0
  \Deqlab{�����絤��Ϣ³�μ�}
\end{eqnarray}
�Ǥ���. ���μ����Ǥ�դΥ����顼�� $\phi$ ���Ф�
\begin{eqnarray}
  \overline{\rho} \DD{\phi}{t}
  = \DP{\overline{\rho} \phi}{t}
   + \Ddiv{ (\overline{\rho} \phi \Dvect{u})}
   + \phi \DP{\rho}{t}
  \Deqlab{�����顼����¸§}
\end{eqnarray}
������Ω��. ���μ�����వ���������Ǥ�$\phi$�λ�����ʬ��
�ե�å��������ǽ񤯤��Ȥ��Ǥ��ʤ����Ȥ��狼��%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\footnote{
  $\partial \phi/\partial t + \Dvect{u} \cdot \nabla \phi = 0$ ��
  ����Ω���ʤ��Ȥ�������. }%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
.

��ʿ�����α�ư������\Deqref{quasi-equations:u-w-pi-theta:u},
��ľ�����α�ư������\Deqref{quasi-equations:u-w-pi-theta:w}���, 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{align}
  \DP{\overline{\rho}K}{t} + \Ddiv{(\overline{\rho}K\Dvect{u})}
 = - c_{p} \overline{\theta} \overline{\rho} \Dvect{u} \cdot \Dgrad{\Pi}
   + \overline{\rho}\Dvect{u}\cdot\Dvect{D}
   + \overline{\rho} \frac{\theta}{\overline{\theta}}gw
   - K \DP{\rho}{t}
  \Deqlab{���ͥ륮��������1}
\end{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
�Ȥʤ�. 
������$K = (u^2 + w^2)/2$, $\Dvect{D} = (D_u, D_w)$�Ȥ�����%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\footnote{
%  �Ŀ�Ū���: Ƴ�Ф�ɤ����˽񤤤Ƥ��������ʡ�. }%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
. 
�ʹߤǤϼ�\Deqref{���ͥ륮��������1}�α�������ȱ����軰����ѷ����Ƥ���. 

�ޤ���\Deqref{���ͥ륮��������1}�α����軰��ϰʲ��Τ褦���ѷ������. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{align}
 \overline{\rho} \frac{\theta}{\overline{\theta}}gw
  &= \overline{\rho} \frac{\theta}{\overline{\theta}} g \DD{z}{t}
      \nonumber \\
  &= \overline{\rho} \frac{\theta}{\overline{\theta}} \DD{(gz)}{t}
      \nonumber \\
  &= \overline{\rho}
      \left \{\frac{1}{\overline{\theta}} \DD{(\theta gz)}{t}
       - \frac{gz}{\overline{\theta}} \DD{\theta}{t}
      \right\} \nonumber \\
  &= \frac{\overline{\rho}}{\overline{\theta}} \DD{(\theta gz)}{t}
      - \frac{gz}{\overline{\theta}}
       \left\{ \DP{\overline{\rho}\theta}{t}
               + \nabla \cdot (\overline{\rho}\theta \Dvect{u})
               + \theta \DP{\rho}{t}
        \right\} . 
 \Deqlab{A67} 
\end{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
��\Deqref{A67}�α����������ѷ�����. 
Ǯ�ϳؤμ�\Deqref{quasi-equations:u-w-pi-theta:theta}��, 
�ŷ뤬�ʤ����, 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{align}
 \DP{\theta}{t}
  = - \Dvect{u} \cdot \nabla \theta - w \DP{\overline{\theta}}{z}
    + Q + D_{\theta}
  \Deqlab{�ŷ�Τʤ�Ǯ�ϳؤμ�}
\end{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
��ɽ�����. ξ�դ� $\overline{\rho}$ �򤫤���, 
��\Deqref{�����絤��Ϣ³�μ�}���Ѥ����ѷ������, 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{align}
 \DP{\overline{\rho} \theta}{t}
  &= - \overline{\rho}\Dvect{u} \cdot \nabla \theta - \overline{\rho} w \DP{\overline{\theta}}{z}
    + Q + D_{\theta}
     \nonumber \\
  &= -\nabla \cdot (\overline{\rho} \theta \Dvect{u}) + \theta \nabla \cdot (\overline{\rho}\Dvect{u})
     - \overline{\rho} w \DP{\overline{\theta}}{z}
     + \overline{\rho} (Q + D_{\theta})
     \nonumber��\\
  &= -\nabla \cdot (\overline{\rho} \theta \Dvect{u}) - \theta \DP{\rho}{t}
     - \overline{\rho} w \DP{\overline{\theta}}{z}
     + \overline{\rho} (Q + D_{\theta})
\end{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
�Ȥʤ�. �ޤ���\Deqref{A67}�α��������, ��\Deqref{�����顼����¸§}���, 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{align}
 \overline{\rho} \frac{d}{dt} 
  \left(\frac{\theta gz}{\overline{\theta}}\right)
 = \frac{\partial}{\partial t}\left(\frac {\overline{\rho}\theta gz}{\overline{\theta}}\right)
  + \nabla \cdot \left(\frac{\overline{\rho}\theta gz}{\overline{\theta}}\Dvect{u}\right)
  + \frac{\theta gz}{\overline{\theta}}\DP{\rho}{t}
 \tag{A.9-2}
\end{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
�Ȥʤ�. ��äƼ�\Deqref{���ͥ륮��������1}�α����軰���, 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{align}
 \overline{\rho} \frac{\theta}{\overline{\theta}}gw
  = \frac{\partial}{\partial t}\left(\frac{\overline{\rho}\theta gz}{\overline{\theta}}\right)
   + \nabla \cdot \left(\frac{\overline{\rho}\theta gz}{\overline{\theta}} \Dvect{u}\right)
   + \frac{\theta gz}{\overline{\theta}}\DP{\rho}{t}
   + \frac{\overline{\rho} gz}{\overline{\theta}} \left\{ w\DP{\overline{\theta}}{z}
    - Q -D_{\theta}\right\}
\end{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
���ѷ������. 

��\Deqref{���ͥ륮��������1}�α�������ϰʲ��Τ褦���ѷ������. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{align}
 -c_p \overline{\theta}\overline{\rho}\Dvect{u} \cdot \nabla \Pi
  = -\nabla \cdot (c_p \overline{\theta}\overline{\rho} \Pi \Dvect{u})
   + c_p \Pi \nabla \cdot (\overline{\theta}\overline{\rho}\Dvect{u}) . 
  \Deqlab{A71}
\end{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
�ŷ뤬�ʤ��Ȥ��ΰ���������\Deqref{quasi-equations:u-w-pi-theta:continue}
��, 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}
 \DP{\Pi}{t}
 = -\frac{\overline{C_{s}^{2}}}{c_p \overline{\rho}\overline{\theta}^2}
     \nabla \cdot (\overline{\rho}\overline{\theta}\Dvect{u})
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
��ɽ�����. ��äƼ�\Deqref{A71}��, 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
 -c_p \overline{\theta}\overline{\rho}\Dvect{u} \cdot \nabla \Pi
  &=& -\nabla \cdot (c_p \overline{\theta}\overline{\rho} \Pi \Dvect{u})
      - c_p \Pi \frac{c_p \overline{\rho}\overline{\theta}^2}{\overline{c}^2} \DP{\Pi}{t} \nonumber \\
  &=& -\nabla \cdot (c_p \overline{\theta}\overline{\rho} \Pi \Dvect{u})
       -\frac{\partial}{\partial t} \left\{\frac{\overline{\rho}}{2}
        \left(\frac{c_p \overline{\theta}\Pi}{\overline{c}}\right)^2 \right\}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
�Ƚ񤭴�������. 

�������ä�, ��\Deqref{���ͥ륮��������1}��, 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{align}
 \frac{\partial}{\partial t}&(\overline{\rho}K) + \nabla \cdot (\overline{\rho} K \Dvect{u})
  = -\nabla \cdot (c_p \overline{\theta}\overline{\theta} \Pi \Dvect{u})
     -\frac{\partial}{\partial t} \left\{\frac{\overline{\rho}}{2}
      \left(\frac{c_p \overline{\theta}\Pi}{\overline{c}}\right)^2 \right\}
     + \overline{\rho} \Dvect{u} \cdot \Dvect{D} \nonumber \\
    &+  \frac{\partial}{\partial t}\left(\frac{\overline{\rho}\theta gz}{\overline{\theta}}\right)
      + \nabla \cdot \left(\frac{\overline{\rho}\theta gz}{\overline{\theta}} \Dvect{u}\right)
      + \frac{\theta gz}{\overline{\theta}}\DP{\rho}{t}
      + \frac{\overline{\rho} gz}{\overline{\theta}} \left\{ w\DP{\overline{\theta}}{z}
      - Q - D_{\theta}\right\} - K \DP{\rho}{t}
 \nonumber
\end{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
�Ȥʤ�, ����򤵤�����������, 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{align}
% \split
 \frac{\partial}{\partial t}\left[\overline{\rho}
  \left\{K - \frac{\theta}{\overline{\theta}}gz
         + \frac{1}{2}\left(\frac{c_p \overline{\theta}\Pi}{\overline{c}}\right)^2\right\}\right]
  + \nabla \cdot \left\{\overline{\rho}\left(K - \frac{\theta}{\overline{\theta}}
                                             + c_p \overline{\theta}\Pi \right) \Dvect{u} \right\} \nonumber \\
 = \overline{\rho} \Dvect{u} \cdot \Dvect{D}
      + \frac{\overline{\rho} gz}{\overline{\theta}} \left\{ w\DP{\overline{\theta}}{z}
      - Q -D_{\theta}\right\}
   - \left(K - \frac{\theta}{\overline{\theta}}gz\right) \DP{\rho}{t}
\end{align} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
�Ȥʤ�. ����ξ�դ����ΰ����ʬ��, �����̤�����ꤹ��ή��Ϥʤ��Ȥ����, 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{align}
 &  \DP{}{t} \int \overline{\rho}
     \left\{K - \frac{\theta}{\overline{\theta}} gz 
       + \frac{1}{2} \left( \frac{c_p \overline{\theta}\Pi}
             {\overline{c}} \right)^2\right\} dV
    \nonumber \\
 &= \int \overline{\rho}\Dvect{u}\cdot\Dvect{D} dV 
   + \int \frac{\overline{\rho}wgz}{\overline{\theta}} \DP{\overline{\theta}}{z} dV
   - \int \frac{\overline{\rho}gz}{\overline{\theta}} Q dV
   - \int \frac{\overline{\rho}gz}{\overline{\theta}} D_{\theta} dV
    \nonumber \\
 & \hspace{1em} 
   - \int \left(K - \frac{\theta}{\overline{\theta}} gz \right) \DP{\rho}{t} dV
\end{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
�Ȥʤ�. ���줬�వ���������Υ��ͥ륮���������Ǥ���.
���դ������ͥ륮���λ����Ѳ��Ǥ���, $\{ \quad \}$ �������, �����, �軰���
���줾�챿ư���ͥ륮��, ���Ϥˤ����֥��ͥ륮��, �������ͥ륮���Ǥ���. 

��������, �����, �軰��, ��͹�Ϥ��줾�챿ư�̤γȻ�, ���ܾ�α�ľ��
�̸���, ����Ǯ��Ǯ(����, ����), Ǯ�γȻ��ˤ�륨�ͥ륮���Ѳ�Ψ�Ǥ���.
���պǸ�ι�Ͻవ���������ξ�祹���顼�̤λ�����ʬ���ե�å���������
�񤱤ʤ����Ȥˤ�뤳�Ȥˤ�긽���.  ���ι��¸�ߤˤ��వ���������Ǥ�
�����ब�ʤ����Ǥ⥨�ͥ륮������¸���ʤ�.


% ��������ʸ�����ޤ�
