%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% ɽ��  3 ���������ϳإ�ǥ� deepconv -- ������ǥ�
%       ��Ͽ A-1 �వ���������Ϥ�Ƴ�� - ������������- 
% 
% ����  2005/11/15 �����̰�ϯ
%       2006/03/12 �����̰�ϯ
%       2007/06/22 ��������
%       2012/04/18 ��������      3 ������
%
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�ϵ��絤�ˤ����뼾����ή���꼰��Ʊ��, �絤�δ�����ʬ�ȼ�����ʬ��ʬ����
�κ���̩�٤μ��ˤϹ�θ���뤬, Ǯ�μ��ˤϹ�θ���ʤ��褦�ʷϤ�ͤ���. ��
�ηϤǤ��絤��Ǯ���ͥ륮���ϴ����絤��Ǯ���ͥ륮���Ƿ�ޤ뤳�Ȥˤʤ�.
���Τ褦�ʷϤǤϲ��� $\theta$ ����¸�̤Ȥ����Ѥ��뤳�Ȥ��Ǥ���.


\subsection{���� $T$, ̩�� $\rho$, ��® $u, v, w$ ��ͽ���ѿ��Ȥ�����}

��ʿ��ľ 3 �����絤�ξ��֤�, ���� $T$, ���� $p$, ��® $u, v, w$, ̩��
$\rho$ ��ɽ��������, �����������Ϥϰʲ��Τ褦�ˤʤ�.
%
\begin{description}
%
 \item[��ư������]~
%
\begin{eqnarray}
&& \DD{u}{t} + \Dinv{\rho}\DP{p}{x} = Turb.u 
  \Deqlab{equations:u} \\
&& \DD{v}{t} + \Dinv{\rho}\DP{p}{y} = Turb.v
  \Deqlab{equations:v} \\
&& \DD{w}{t} + \Dinv{\rho}\DP{p}{z} = -g + Turb.w
  \Deqlab{equations:w} 
\end{eqnarray}
%
 \item[Ϣ³�μ�]~
%
\begin{eqnarray}
 \DD{\rho}{t} + \rho\left( \DP{u}{x} + \DP{v}{y} + \DP{w}{z}\right) = 0
  \Deqlab{equations:continue}
\end{eqnarray}
%
 \item[̩�٤μ�(����������)]~
%
\begin{eqnarray}
 \rho = \frac{p}{R_{d} T_{v}}
  \Deqlab{equations:rho}
\end{eqnarray}
%
 \item[Ǯ�μ�]~
%
\begin{eqnarray}
 {c_{p}}_{d}\DD{T}{t} - \Dinv{\rho_{d}} \DD{p}{t} = Q + Turb.T
  \Deqlab{equations:T}
\end{eqnarray}
%
 \item[�Ž���ʬ�κ�������¸��]~
%
\begin{eqnarray}
&& \DD{q_{v}}{t} = Src.q_{v} + Turb.q_{v}
  \Deqlab{equations:qv} \\
&& \DD{q_{c}}{t} = Src.q_{c} + Turb.q_{c}
  \Deqlab{equations:qc} \\
&& \DD{q_{r}}{t} = Src.q_{r} + Fall.q_{r} + Turb.q_{r}
  \Deqlab{equations:qr}
\end{eqnarray}
%
\end{description}
%
������ $R_{d}$, ${c_{p}}_{d}$, $\rho_{d}$ ��ñ�̼���������δ�����ʬ��
�������, �갵��Ǯ, ̩�٤Ǥ���, $Q$ ������Ǯ��Ǯ, $q_{v}$ �ϵ�����ʬ��
������, $q_{c}$ �ϱ��庮����, $q_{r}$ �ϱ��庮����Ǥ���.  $q_{v},
q_{r}, q_{c}$ ��, �Ž���ʬ�ο�����¸�ߤ���.  $Turb$, $Src$, $Fall$ ����
������Ϥ��줾��Ȼ���, �������ǹ�, �����̣����.

̩�٤μ��ˤ϶Ž���ʬ�κ����椬��θ����Ƥ���. 
%
\begin{eqnarray}
 \rho &=& \rho_{d} + \sum \rho_{v} + \sum \rho_{c} + \sum \rho_{r}
  \nonumber \\
 &=& \rho_{d} (1 + \sum q_{v} + \sum q_{c} + \sum q_{r} ). 
  \Deqlab{rho:rho}
\end{eqnarray}
%
������, $q_v = \rho_v/\rho_{d}$, $q_{c}$, $q_{r}$ �Ϥ��줾��Ž�������,
����, ����κ�������̣����. �����Ǵ�����ʬ��ʬ�� $p_{d}$ ��.
%
\begin{eqnarray}
 p_{d} 
  &=& p \left( 1 - \frac{\sum p_{v}}{p} \right) \nonumber
  \\
 &=& p \left( 1 - \frac{\sum p_{v}}{p_{d} + \sum p_{v} }\right) \nonumber
  \\
 &=& p \left( 1 - \frac{\sum \rho_{v} R_{v} T}{\rho_{d} R_{d}T + \sum
	\rho_{v} R_{v} T }\right) \nonumber
  \\
 &=& p \left( 1 - \frac{\sum q_{v}/M_{v}}{1/M_{d} + \sum q_{v}/M_{v}
	}\right)
 \nonumber
\end{eqnarray}
%
�Ȥʤ�Τ�, 
%
\begin{eqnarray}
 \rho_{d} 
  = \frac{p_{d}}{R_{d} T}
  = \frac{p}{R_{d} T} \left(\frac{1/M_{d}}{1/M_{d} + \sum q_{v}/M_{v}
	}\right)
  \Deqlab{rho:rho_d}
\end{eqnarray}
%
�Ǥ���. â�� $M$ ��ʬ���̤�ɽ��, �Ž���ʬ�����Ѥ�̵��Ǥ����Τȸ��ʤ���. 
\Deqref{rho:rho}, \Deqref{rho:rho_d} �����, 
%
\begin{eqnarray}
 \rho &=& \frac{p}{R_{d}T}
  \left( \frac{1/M_{d}}{1/M_{d} + \sum q_{v}/M_{v} }\right)
  (1 + \sum q_{v} + \sum q_{c} + \sum q_{r} )
\Deqlab{rho:p-T_1}
\end{eqnarray}
%
�Ȥʤ�. 
%
\begin{eqnarray}
 f \equiv  
  \left(\frac{1/M_{d}}{1/M_{d} + \sum q_{v}/M_{v}	}\right)
  (1 + \sum q_{v} + \sum q_{x} )
\nonumber 
\end{eqnarray}
%
����������, \Deqref{rho:p-T_1} ���ϰʲ��Τ褦�˽񤱤�. 
%
\begin{eqnarray}
 \rho &=& \frac{p}{R_{d} (T/f)}
\Deqlab{rho:p-T}
\end{eqnarray}
%
�ޤ�, ���̤ȥ������ʡ��ؿ����Ѥ���ɽ�������, 
%
\begin{eqnarray}
 \rho 
  &=& \frac{p}{R_{d} \pi (\theta /f )}
  \nonumber \\
  &=& \frac{p_{0} \pi^{{c_{v}}_{d}/R_{d}}}{R_{d} (\theta/f)}
\Deqlab{rho:p-theta}
\end{eqnarray}
%
�Ǥ���. â���������ʡ��ؿ� $\pi$ �� $\pi = T/\theta$ �δط���������. 

���̤ϴ�����Ǯ���֤ˤ�������¸�̤Ǥ���. ������Ǯ���֤�ɽ��Ǯ�ϳؤμ���
%
\begin{eqnarray}
 c_{p}dT - \alpha dp = 0
\Deqlab{theta1}
\end{eqnarray}
% 
�Ǥ���. ������ $T$ �ϲ���, $p$ �ϰ���, $c_{p}$ ��ñ�̼������������Ǯ,
$\alpha$ �����ƤǤ���. \Deqref{theta1} ���� $\alpha$ ��, ���۵��Τξ���
���������Ѥ����,
%
\begin{eqnarray}
 \alpha = \frac{RT}{p}
 \Deqlab{theta2}
\end{eqnarray}
%
�Ƚ񤱤�. ������ $M$ ��ʬ����, $R$ �ϵ�������Ǥ���. \Deqref{theta1} ��
�� \Deqref{theta2} �������������������,
%
\begin{eqnarray}
 \frac{c_{p}}{T}dT - \frac{R}{p} dp = 0
\Deqlab{theta3}
\end{eqnarray}
% 
�Ȥʤ�. �Ž̤������ʤ����ˤϵ������������Ѳ����ʤ��Τ�$c_{p}$ �� $R$
�϶��� $p$ �˰�¸���ʤ�. ���̤� $c_{p}$ �� $T$ �δؿ��Ǥ��뤬,
{\bf \boldmath $c_{p}$ ������Ȥߤʤ���},
%
\begin{eqnarray}
\int^{T_{0}}_{T} \Dinv{T}dT 
 &=& \frac{R}{c_{p}} \int^{p_{0}}_{p} \Dinv{p} dp \nonumber \\
\ln{(T_{0}/T)}
 &=& \frac{R}{c_{p}} \ln{(p_{0}/p)} \nonumber \\
 \theta 
 &=& T \left(\frac{p_{0}}{p}\right)^\frac{R}{c_{p}} 
\Deqlab{theta4}
\end{eqnarray}
% 
�Ȥʤ�, ���̤�������. 



\subsection{���� $\theta$, ���� $p$, ��® $u, v, w$ ��ͽ���ѿ��Ȥ�����}

��ʿ��ľ 3 �����絤�ξ��֤򲹰� $\theta$, ���� $p$, ��® $u, v, w$, ̩
�� $\rho$ ��ɽ��������, �����������Ϥϰʲ��Τ褦�ˤʤ�.  CReSS(���ڤ�
�縶, 2001)�Ǥ�, ���δ������������Ѥ��Ƥ���.
%
\begin{description}
%
 \item[��ư������]~
%
\begin{eqnarray}
&& \DD{u}{t} + \Dinv{\rho}\DP{p}{x} = Turb.u 
  \Deqlab{equations:u-w-p-theta:u} \\
&& \DD{v}{t} + \Dinv{\rho}\DP{p}{y} = Turb.v
  \Deqlab{equations:u-w-p-theta:v} \\
&& \DD{w}{t} + \Dinv{\rho}\DP{p}{z} = - g + Turb.w
  \Deqlab{equations:u-w-p-theta:w} 
\end{eqnarray}
%
 \item[����������]~
%
\begin{eqnarray}
 \DD{p}{t} = \rho {C_{s}}^{2} 
  \left\{
   - \Ddiv{\Dvect{u}}
   + \Dinv{\theta}\DD{\theta}{t} 
   - \Dinv{f} 
   \left( 
     \sum \DP{f}{q_{v}} \DD{q_{v}}{t}
    + \sum  \DP{f}{q_{c}} \DD{q_{c}}{t}
    + \sum  \DP{f}{q_{r}} \DD{q_{r}}{t}
   \right)
  \right\} 
\end{eqnarray}
%
 \item[̩�٤μ�(����������)]~
%
\begin{eqnarray}
 \rho = \frac{p_{0}}{R_{d} \theta_{v}}
  \left( \frac{p}{p_{0}}\right)^{{c_{v}}_{d}/{c_{p}}_{d}}
  \Deqlab{equations:rho}
\end{eqnarray}
%
 \item[Ǯ�μ�]~
%
\begin{eqnarray}
 \DD{\theta}{t} = Q + Turb.\theta
  \Deqlab{equations:u-w-p-theta:theta}
\end{eqnarray}
%
 \item[�Ž���ʬ�κ��������¸��]~
%
\begin{eqnarray}
&& \DD{q_{v}}{t} = Src.q_{v} + Turb.q_{v}
  \Deqlab{equations:u-w-p-theta:qv} \\
&& \DD{q_{c}}{t} = Src.q_{c} + Turb.q_{c}
  \Deqlab{equations:u-w-p-theta:qc} \\
&& \DD{q_{r}}{t} = Src.q_{r} + Fall.q_{r} + Turb.q_{r}
  \Deqlab{equations:u-w-p-theta:qr}
\end{eqnarray}
%
\end{description}
%
���������� $\theta$ ��
%
\begin{eqnarray}
 \theta \equiv T \left(\frac{p_{0}}{p}\right)^{R_{d}/{c_{p}}_{d}}
  \Deqlab{equations:theta}
\end{eqnarray}
�Ǥ���, ������ $\theta_{v}$ ��, 
%
\begin{eqnarray}
 \theta_{v} \equiv \frac{\theta}{f}
\end{eqnarray}
%
�Ǥ���. ��® $C_{s}$ ��
%
\begin{eqnarray}
 {C_{s}}^{2}
  \equiv  \frac{{c_{p}}_{d}}{{c_{v}}_{d}} R_{d} T_{v}
  =   \frac{{c_{p}}_{d}}{{c_{v}}_{d}} R_{d} \frac{T}{f}
\end{eqnarray}
%
�Ǥ���. ${c_{p}}_{d}$ �� ${c_{v}}_{d}$ �Ϥ��줾��ñ�̼���������δ�����
ʬ���갵��Ǯ��������Ǯ�Ǥ���, ${c_{v}}_{d} + R_{d} = {c_{p}}_{d}$ �Ȥ�
���ط��ˤ���.

������������̩�٤μ���Ϣ³�μ����Ȥ߹�碌�뤳�Ȥ�������. �ޤ�̩�٤�
$\rho= \rho(\theta,p,q_{v},q_{x})$ �Ȥ��� $\rho$ ������ʬ�����.
%
\begin{eqnarray}
 d\rho 
  &=&d \left[
	\frac{p_{0}}{R_{d} \theta_{v}} 
	\left( \frac{p}{p_{0}} \right)^{{c_{v}}_{d}/{c_{p}}_{d}}	
     \right]
\nonumber \\
  &=&d \left[
	\frac{p_{0}}{R_{d} (\theta / f) } 
	\left( \frac{p}{p_{0}} \right)^{{c_{v}}_{d}/{c_{p}}_{d}}	
     \right]
\nonumber \\
 &=& 
  \frac{p_{0} }{R_{d} (\theta/f)} 
      \frac{{c_{v}}_{d}}{{c_{p}}_{d}}
      \left( \frac{p}{p_{0}} \right)^{-R_{d}/{c_{p}}_{d}}
      \Dinv{p_{0}}dp
-
      \frac{p_{0} f}{R_{d}}
      \left( \frac{p}{p_{0}} \right)^{{c_{v}}_{d}/{c_{p}}_{d}}
       \frac{d\theta}{\theta^{2}}
\nonumber \\
  &&+ 
   \frac{p_{0}}{R_{d} \theta} 
      \left( \frac{p}{p_{0}} \right)^{{c_{v}}_{d}/{c_{p}}_{d}}
      \left\{
       \sum \DP{f}{q_{v}} dq_{v}
       + \sum \DP{f}{q_{c}} dq_{c}
       + \sum \DP{f}{q_{r}} dq_{r}
      \right\}
\nonumber \\
 &=& 
  \Dinv{{C_{s}}^{2}} dp
  - \frac{\rho}{\theta} d\theta
  + \sum \frac{\rho}{f} \DP{f}{q_{v}} dq_{v}
  + \sum \frac{\rho}{f} \DP{f}{q_{c}} dq_{c}
  + \sum \frac{\rho}{f} \DP{f}{q_{r}} dq_{r}
\Deqlab{pressure:theta-p:drho}
\end{eqnarray}
%
�Ȥʤ�. \Deqref{pressure:theta-p:drho} �����Ϥμ��Ȥ������������, 
%
\begin{eqnarray}
\DD{p}{t}
 = {C_{s}}^{2} 
 \left(
    \DD{\rho}{t}
  + \frac{\rho}{\theta} \DD{\theta}{t} 
  - \sum \frac{\rho}{f} \DP{f}{q_{v}} dq_{v}
  - \sum \frac{\rho}{f} \DP{f}{q_{c}} dq_{c}
  - \sum \frac{\rho}{f} \DP{f}{q_{r}} dq_{r}
 \right)
\nonumber
\end{eqnarray}
%
�Ǥ���, Ϣ³�μ����Ѥ����, 
%
\begin{eqnarray}
\DD{p}{t}
 = \rho {C_{s}}^{2} 
 \left(
  -  \Ddiv \Dvect{u}
  + \Dinv{\theta} \DD{\theta}{t} 
  - \sum \Dinv{f} \DP{f}{q_{v}} dq_{v}
  - \sum \Dinv{f} \DP{f}{q_{c}} dq_{c}
  - \sum \Dinv{f} \DP{f}{q_{r}} dq_{r}
 \right)
\nonumber
\end{eqnarray}
%
�Ȥʤ�, ������������������. 


\subsection{���� $\theta$, ̵�������� $\pi$, ��® $u, v, w$ ��ͽ���ѿ��Ȥ�����}

��ʿ��ľ 2 �����絤�ξ��֤򲹰� $\theta$, ̵�������� $\pi$, ��® $u,
v, w$, ̩�� $\rho$ ��ɽ��������, �����������Ϥϰʲ��Τ褦�ˤʤ�. Ϣ³
�μ� \Deqref{equations:continue} �Ⱦ��������� \Deqref{equations:rho} ��
�Ѥ��뤳�Ȥ������밵�������������Ѥ���.  Klemp and Willhelmson
(1978)�Ǥ�, ���δ������������Ѥ��Ƥ���.
%
\begin{description}
%
 \item[��ư������]~
%
\begin{eqnarray}
&& \DD{u}{t} + {{c_{p}}_{d}} \theta_{v} \DP{\pi}{x} = Turb.u 
  \Deqlab{equations:u-w-pi-theta:u} \\
&& \DD{v}{t} + {{c_{p}}_{d}} \theta_{v} \DP{\pi}{y} = Turb.v
  \Deqlab{equations:u-w-pi-theta:u} \\
&& \DD{w}{t} + {{c_{p}}_{d}} \theta_{v} \DP{\pi}{z} = -g + Turb.w
  \Deqlab{equations:u-w-pi-theta:w} 
\end{eqnarray}
%
 \item[����������]~
%
\begin{eqnarray}
\DD{\pi}{t}
 =  \frac{{C_{s}}^{2}}{ {c_{p}}_{d} \rho \theta_{v} }
 \left\{
  - \Ddiv \Dvect{u}
  + \Dinv{\theta}\DD{\theta}{t} 
  - \Dinv{f}
  \left(
     \sum \DP{f}{q_{v}} \DD{q_{v}}{t}
   + \sum \DP{f}{q_{c}} \DD{q_{c}}{t}
   + \sum \DP{f}{q_{r}} \DD{q_{r}}{t}
  \right)
 \right\}
  \Deqlab{equations:u-w-pi-theta:continue}
\end{eqnarray}
%
 \item[����������]~
%
\begin{eqnarray}
 \rho = \frac{p_{0} \pi^{{c_{v}}_{d}/R_{d}}}{R_{d} \theta_{v}}
  \Deqlab{equations:u-w-pi-theta:rho}
\end{eqnarray}
%
 \item[Ǯ�μ�]~
%
\begin{eqnarray}
 \DD{\theta}{t} = Q + Turb.\theta
  \Deqlab{equations:u-w-pi-theta:theta}
\end{eqnarray}
%
 \item[���������ӿ�ʪ��������μ�]~
%
\begin{eqnarray}
&& \DD{q_{v}}{t} = Src.q_{v} + Turb.q_{v}
  \Deqlab{equations:u-w-pi-theta:qv} \\
&& \DD{q_{x}}{t} = Src.q_{c} + Turb.q_{c}
  \Deqlab{equations:u-w-pi-theta:qc} \\
&& \DD{q_{x}}{t} = Src.q_{r} + Fall.q_{x} + Turb.q_{r}
  \Deqlab{equations:u-w-pi-theta:qr}
\end{eqnarray}
%
\end{description}
%
������, �������ʡ��ؿ� $\pi$ ��, 
%
\begin{eqnarray}
\pi \equiv \frac{T}{\theta} 
 = \left( \frac{p}{p_{0}} \right)^{R_{d}/{c_{p}}_{d}} 
  \Deqlab{equations:pi}
\end{eqnarray}
%
�Ǥ���, ��® $C_{s}$ ��
%
\begin{eqnarray}
 {C_{s}}^{2} 
  \equiv  \frac{{c_{p}}_{d}}{{c_{v}}_{d}} R_{d} \pi \theta_{v}
  =  \frac{{c_{p}}_{d}}{{c_{v}}_{d}} R_{d} \pi 
  \left( \frac{\theta}{f} \right)
\end{eqnarray}
%
�Ǥ���. 

��ư�������ΰ��ϸ��ۤ�, ���̤ȥ������ʡ��ؿ����Ѥ��뤳�Ȥ�������. 
%
\begin{eqnarray}
 \Dinv{\rho} dp 
  &=& \frac{R_{d}  \pi (\theta/f)}{p} 
  d \left( 
     p_{0} \pi^{{c_{p}}_{d}/R_{d}}
    \right)
  \nonumber \\
 &=& \frac{R_{d}  \pi (\theta/f) }{p} 
  \left(
  \frac{p_{0} {c_{p}}_{d}}{R_{d}} \pi^{{c_{p}}_{d}/R - 1} 
  \right)
  d\pi
  \nonumber \\
 &=& \frac{R_{d}  \pi (\theta/f) }{p} 
  \left(
   \frac{{c_{p}}_{d}}{R_{d}} p \pi^{-1} 
  \right)d\pi
  \nonumber \\
 &=& {c_{p}}_{d} (\theta/f)   d\pi
  \nonumber \\
 &=& {c_{p}}_{d} \theta_{v}   d\pi
\end{eqnarray}
%


������������̩�٤μ���Ϣ³�μ����Ȥ߹�碌�뤳�Ȥ�������. 
�ޤ�̩�٤� $\rho= \rho(\theta, \pi, q_{v}, q_{x})$ �Ȥ���  
$\rho$ ������ʬ��׻�����. 
%
\begin{eqnarray}
 d\rho 
  &=&d \left[
      \frac{p_{0} \pi^{c_{v/R_{d}}}}{R_{d} \theta_{v}} 
     \right]
  \nonumber \\
  &=&d \left[
      \frac{p_{0} \pi^{c_{v/R_{d}}}}{R_{d} (\theta/f)} 
     \right]
  \nonumber \\
 &=& 
  \frac{p_{0}}{R_{d} (\theta/f) } 
  \pi^{(c_{v/R_{d}}-1)} 
  \frac{{c_{v}}_{d}}{R_{d}}
  d\pi
  -
      \frac{p_{0} \pi^{c_{v/R_{d}}} f}{R_{d}}
       \frac{d\theta}{\theta^{2}}
\nonumber \\
  &&+ 
      \frac{p_{0} \pi^{c_{v/R_{d}}} }{R_{d} \theta} 
      \left( 
      \sum \DP{f}{q_{v}} dq_{v}
      + \sum \DP{f}{q_{c}} dq_{c}
      + \sum \DP{f}{q_{r}} dq_{r}
      \right)
\nonumber \\
 &=& 
  {c_{p}}_{d} (\theta/f)
  \left( \frac{{c_{v}}_{d}}{{c_{p}}_{d} R_{d} \pi (\theta/f) } \right)
  \left(\frac{p_{0} \pi^{c_{v/R_{d}}} }{R_{d} (\theta/f)} \right)
  d\pi
  -
      \frac{p_{0} \pi^{c_{v/R_{d}}} }{R_{d} (\theta/f)}
       \frac{d\theta}{\theta}
\nonumber \\
  &&+ 
   \Dinv{f}
   \left(
    \frac{p_{0} \pi^{c_{v/R_{d}}} }{R_{d} (\theta / f)} 
   \right)
   \left(
    \sum \DP{f}{q_{v}} dq_{v}
    + \sum \DP{f}{q_{c}} dq_{c}
    + \sum \DP{f}{q_{r}} dq_{r}
   \right)
\nonumber \\
 &=& 
    \frac{ {c_{p}}_{d} \rho (\theta/f)}{{C_{s}}^{2} } d\pi
  - \frac{\rho}{\theta} d\theta
  + \rho \Dinv{f} 
  \left( 
   \sum \DP{f}{q_{v}} dq_{v}
   + \sum \DP{f}{q_{c}} dq_{c}
   + \sum \DP{f}{q_{r}} dq_{r}
  \right)
\nonumber \\
 &=& 
    \frac{ {c_{p}}_{d} \rho \theta_{v}}{{C_{s}}^{2} } d\pi
  - \frac{\rho}{\theta} d\theta
  + \rho \Dinv{f} 
  \left( 
   \sum \DP{f}{q_{v}} dq_{v}
   + \sum \DP{f}{q_{c}} dq_{c}
   + \sum \DP{f}{q_{r}} dq_{r}
  \right)
\Deqlab{pressure:theta-pi:drho}
\end{eqnarray}
%
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%
\begin{eqnarray}
\DD{\pi}{t}
 = 
 \frac{{C_{s}}^{2} }{{c_{p}}_{d} \rho \theta_{v} }
 \left\{
   \Dinv{\rho} \DD{\rho}{t}
  + \Dinv{\theta}\DD{\theta}{t} 
  - \Dinv{f}
  \left(
   \sum \DP{f}{q_{v}} dq_{v}
   + \sum \DP{f}{q_{c}} dq_{c}
   + \sum \DP{f}{q_{r}} dq_{r}
  \right)
\right\}
\Deqlab{pressure:theta-p:p}
\end{eqnarray}
%
�Ȥʤ�, Ϣ³�μ����Ѥ����, 
%
\begin{eqnarray}
\DD{\pi}{t}
 = \frac{{C_{s}}^{2} }{{c_{p}}_{d} \rho \theta_{v} }
 \left\{
  - \Ddiv \Dvect{u} 
  + \Dinv{\theta}\DD{\theta}{t} 
  - \Dinv{f} 
  \left(
   \sum \DP{f}{q_{v}} dq_{v}
   + \sum \DP{f}{q_{c}} dq_{c}
   + \sum \DP{f}{q_{r}} dq_{r}
  \right)
\right\}
\Deqlab{pressure:theta-p:p}
\end{eqnarray}
%
�Ȥʤ�, ������������������. 
