% 表題: 基本場と擾乱場の分離
%
% 履歴: 2005-11-26 杉山耕一朗
%       2007/06/22 小高正嗣
%       2009/09/11 山下達也
%       2009/11/12 山下達也
%       2009/11/24 山下達也
%       2009/12/01 山下達也
%       2009/12/08 山下達也
%       2010/03/17 山下達也

準圧縮方程式系では, 変数を基本場と擾乱場に分離し, 線形化を行う. 


\subsection{基本場と擾乱場の分離}


変数を基本場と擾乱場に分離し, 基本場は静水圧平衡にあると仮定する. 
この時, 変数は以下のように書ける. 
%
\begin{eqnarray}
u (x,z,t) &=& u^{\prime} (x,z,t), \nonumber \\
v (x,z,t) &=& v^{\prime} (x,z,t), \nonumber \\
w (x,z,t) &=& w^{\prime} (x,z,t), \nonumber \\
\theta (x,z,t) &=& \overline{\theta} (z) + \theta^{\prime} (x,z,t),
 \nonumber \\
\Pi (x,z,t) &=& \overline{\Pi} (z) + \Pi^{\prime} (x,z,t), 
 \nonumber \\
q_d (x,z,t) &=& \overline{q_d} (z) +
 {q_d}^{\prime} (x,z,t), \nonumber \\
q_v (x,z,t) &=& \overline{q_v} (z) +
 {q_v}^{\prime} (x,z,t), \nonumber \\
q_c (x,z,t) &=& {q_c}^{\prime} (x,z,t), \nonumber \\
q_r (x,z,t) &=& {q_r}^{\prime} (x,z,t). \nonumber
\end{eqnarray}
% 
ここで基本場の風速 $w$ と雲水比湿と雨水比湿はゼロとみなした. 
そして基本場には静水圧平衡
%
\begin{eqnarray}
 \DP{\bar{\Pi}}{z} 
  = - \frac{g}{\langle c_p \rangle_0 \bar{\theta_{v}}} 
\end{eqnarray}
%
の関係が成り立つものとする. 



\subsection{水平方向の運動方程式の線形化}

水平方向の運動方程式を基本場と擾乱場に分離する. 
%
\begin{eqnarray}
 \DP{u^{'}}{t} 
  &=& 
  - \left[
      u^{'}  \DP{u^{'}}{x} 
     + v^{'}  \DP{u^{'}}{y} 
     + w^{'} \DP{u^{'}}{z} 
    \right] 
    \nonumber \\
  & &
  - \langle c_p \rangle_0
  \left(
     \bar{\theta_{v}} \DP{\bar{\Pi}}{x}
   + \bar{\theta_{v}} \DP{\Pi^{'}}{x}
   + {\theta_{v}}^{'} \DP{\bar{\Pi}}{x}
   + {\theta_{v}}^{'} \DP{\Pi^{'}}{x}
  \right)
  + D_{u} \nonumber \\
 \DP{v^{'}}{t} 
  &=& 
  - \left[
      u^{'}  \DP{v^{'}}{x} 
     + v^{'}  \DP{v^{'}}{y} 
     + w^{'} \DP{v^{'}}{z} 
    \right] 
    \nonumber \\
  & &
  - \langle c_p \rangle_0
  \left(
     \bar{\theta_{v}} \DP{\bar{\Pi}}{y}
   + \bar{\theta_{v}} \DP{\Pi^{'}}{y}
   + {\theta_{v}}^{'} \DP{\bar{\Pi}}{y}
   + {\theta_{v}}^{'} \DP{\Pi^{'}}{y}
  \right)
  + D_{v}. \nonumber
\end{eqnarray}
%
上式において移流項以外の 2 次の微小項を消去し, さらに基本場は $x$ 方向に
は変化しないことを利用すると, 以下の擾乱成分の式が得られる.
%
\begin{eqnarray}
 \DP{u^{'}}{t} 
  &=&   
- u^{\prime} \DP{u^{\prime}}{x}
- v^{\prime} \DP{u^{\prime}}{y} 
- w^{\prime} \DP{u^{\prime}}{z}
 \nonumber \\
  & &
  - \langle c_p \rangle_0 \overline{\theta_v} \DP{\Pi^{\prime}}{x}
  + D_{u}
  \Deqlab{bunri:moist:dudt} \\
 \DP{v^{'}}{t} 
  &=&   
- u^{\prime} \DP{v^{\prime}}{x}
- v^{\prime} \DP{v^{\prime}}{y} 
- w^{\prime} \DP{v^{\prime}}{z}
\nonumber \\
  & &
  - \langle c_p \rangle_0 \overline{\theta_v} \DP{\Pi^{\prime}}{y}
  + D_{v}.
  \Deqlab{bunri:moist:dvdt}
\end{eqnarray}
%


\subsection{鉛直方向の運動方程式の線形化}

鉛直方向の運動方程式を基本場と擾乱場に分離する. 
%
\begin{eqnarray}
 \DP{w^{'}}{t} 
 &=&
  - \left[
       u^{'}  \DP{w^{'}}{x} 
     + v^{'} \DP{w^{'}}{y} 
     + w^{'} \DP{w^{'}}{z} 
    \right] \nonumber \\
 & &
  - \langle c_p \rangle_0
  \left(
     \bar{\theta_{v}} \DP{\bar{\Pi}}{z}
   + \bar{\theta_{v}} \DP{\Pi^{'}}{z}
   + {\theta_{v}}^{'} \DP{\bar{\Pi}}{z}
   + {\theta_{v}}^{'} \DP{\Pi^{'}}{z}
  \right) \nonumber \\
 & &
  - g
  + D_{w^{'}}
\nonumber 
\end{eqnarray}
%
上式において移流項以外の 2 次の微小項を消去すると以下となる. 
%
\begin{eqnarray}
 \DP{w^{'}}{t} 
  &=& 
    - u^{'} \DP{w^{'}}{x} 
    - v^{'} \DP{w^{'}}{y} 
    - w^{'} \DP{w^{'}}{z} 
      \nonumber \\
  & &
  - \langle c_p \rangle_0
  \left(
     \bar{\theta_{v}} \DP{\bar{\Pi}}{z}
   + \bar{\theta_{v}} \DP{\Pi^{'}}{z}
   + {\theta_{v}}^{'} \DP{\bar{\Pi}}{z}
  \right)
    - g
    + D_{w^{'}} . \nonumber 
\end{eqnarray}
%
さらに静水圧の式
\begin{eqnarray}
\DP{\overline{\Pi}}{z} = - \frac{g}{\langle c_p \rangle_0
 \overline{\theta_v}} 
\end{eqnarray}
を利用すると以下のようになる. 
%
\begin{eqnarray}
 \DP{w^{'}}{t} 
  &=& 
     - u^{'} \DP{w^{'}}{x} 
     - v^{'} \DP{w^{'}}{y} 
     - w^{'} \DP{w^{'}}{z} 
    \nonumber \\
  & &
  + \langle c_p \rangle_0 \bar{\theta_{v}} 
  \left( \frac{g}{\langle c_p \rangle_0 \bar{\theta_{v}}} \right)
  -   \langle c_p \rangle_0 \bar{\theta_{v}} \DP{\Pi^{'}}{z}
  +   \langle c_p \rangle_0 {\theta_{v}}^{'} 
  \left( \frac{g}{\langle c_p \rangle_0 \bar{\theta_{v}}} \right)
  - g + D_{w^{\prime}}  \nonumber \\
  &=& 
  - u^{'} \DP{w^{'}}{x} 
  - v^{'} \DP{w^{'}}{y} 
  - w^{'} \DP{w^{'}}{z} 
  -   \langle c_p \rangle_0 \bar{\theta_{v}} \DP{\Pi^{'}}{z}
  +   \frac{{\theta_{v}}^{'}}{\bar{\theta_{v}}} g
  + D_{w^{\prime}}  \nonumber 
\end{eqnarray}
%
仮温位
\begin{eqnarray}
\theta_v
  \equiv  \frac{T_v}{\Pi}
  =  \left( \frac{\langle M \rangle_0}{M_d} q_d
     + \sum \frac{\langle M \rangle_0}{ M_v } q_v
         \right) \theta
\end{eqnarray}
を基本場成分と擾乱成分に分けると, 
%
\begin{eqnarray}
\bar{\theta_v}
&=&  
\left( \frac{\langle M \rangle_0}{M_d} \bar{q_d}
     + \sum \frac{\langle M \rangle_0}{ M_v } \bar{q_v}
         \right) \bar{\theta}
 \nonumber \\
&=&
 \left[
  1 + \sum \left( -1 + \frac{R_v}{R_d} \right) \overline{q_v}
 \right] \bar{\theta}
 \\
\theta_v^{\prime}
&=&
 \left( \frac{\langle M \rangle_0}{M_d} \overline{q_d}
     + \sum \frac{\langle M \rangle_0}{ M_v } \overline{q_v}
         \right) \theta^{\prime}
+  \left( \frac{\langle M \rangle_0}{M_d} q_d^{\prime}
     + \sum \frac{\langle M \rangle_0}{ M_v } q_v^{\prime}
         \right) \overline{\theta} 
 \nonumber \\
&=&
 \left[
  1 + \sum \left( -1 + \frac{R_v}{R_d} \right) \overline{q_v}
 \right] \theta^{\prime}
  \nonumber \\
& &
 + \left[
    \sum \left( -1 + \frac{R_v}{R_d} \right) q_v^{\prime}
    - \sum \left( q_c^{\prime} + q_r^{\prime} \right)
 \right] \overline{\theta} 
  \Deqlab{DTheta}
\end{eqnarray}
%
となるので, 浮力項は
\begin{eqnarray}
\frac{\theta_v^{\prime}}{\overline{\theta_v}} g
 =
\frac{\theta^{\prime}}{\overline{\theta}} g
 + 
  \frac{ 
    \sum \left( -1 + \frac{R_v}{R_d} \right) q_v^{\prime}
    - \sum \left( q_c^{\prime} + q_r^{\prime} \right) }
   {  1 + \sum \left( -1 + \frac{R_v}{R_d} \right) \overline{q_v} } g
\end{eqnarray}
と書き換えられる.
従って線形化された鉛直方向の運動方程式は
\begin{eqnarray}
\DP{w^{\prime}}{t}
&=& 
  - u^{\prime} \DP{w^{\prime}}{x} 
  - v^{\prime} \DP{w^{\prime}}{y} 
  - w^{\prime} \DP{w^{\prime}}{z}
  \nonumber \\
& &
  - \langle c_p \rangle_0 \overline{\theta_v} \DP{\Pi^{\prime}}{z}
  + \frac{\theta^{\prime}}{\overline{\theta}} g
  + \frac{ 
    \sum \left( \frac{R_v}{R_d} -1 \right) q_v^{\prime}
    - \sum \left( q_c^{\prime} + q_r^{\prime} \right) }
   {  1 + \sum \left( \frac{R_v}{R_d} -1 \right) \overline{q_v} } g
  + D_{w^{\prime}} 
  \Deqlab{bunri:moist:dwdt}
\end{eqnarray}
となる.
%


\subsection{圧力方程式の線形化}

Klemp and Wilhelmson (1978) では, 非断熱的な加熱による熱膨張と
凝結に伴う圧力変化を無視しているが, 本モデルではこれを無視しない. 
\Deqref{pressure:theta-p:p} に \Deqref{sphm6}, \Deqref{sphm7} を代入する
と, 
\begin{eqnarray}
\DD{\Pi}{t}
&=&
 \frac{\langle c_s \rangle^2}{\langle c_p \rangle_0 \theta_v}
\Bigg[ - \DP{u_j}{x_j} + \Dinv{\rho} \sum M_{fall} (\rho_r) +
\Dinv{\theta }  
 \left\{ \Dinv{\Pi} ( Q_{cond} + Q_{rad} + Q_{dis} ) 
  + D_{\theta} \right\} 
 \nonumber \\
& &
+ \Dinv{q_d  \langle M \rangle_0 / M_d
    + \sum q_v \langle M \rangle_0 / M_v} 
  \Bigg\{ \frac{\langle M \rangle_0}{M_d} 
 \left( - \frac{q_d}{\rho} \sum M_{fall} (\rho_r) + D_{q_d} \right) 
 \nonumber \\
& &    
 + \sum \frac{\langle M \rangle_0}{M_v} 
 \left( - \frac{q_v}{\rho} \sum M_{fall} (\rho_r) + M_{src} (q_v) +
  D_{q_v} \right) 
\Bigg\}
\Bigg]
 \nonumber \\
&=&
 \frac{\langle c_s \rangle^2}{\langle c_p \rangle_0 \theta_v}
\Bigg[ - \DP{u_j}{x_j}  +
\Dinv{\theta }  
 \left\{ \Dinv{\Pi} ( Q_{cond} + Q_{rad} + Q_{dis} ) 
  + D_{\theta} \right\} 
 \nonumber \\
& &
+ \Dinv{ q_d 
    + M_d \sum q_v  / M_v} 
  \Bigg\{ 
   D_{q_d} + M_d \sum \Dinv{M_v} 
    \left[  M_{src} (q_v) + D_{q_v} \right] 
  \Bigg\}
\Bigg]
\Deqlab{pressEQ1} 
\end{eqnarray}
%
となる. 
%圧力方程式に関して, 平均成分と擾乱成分に分ける. ただし, 擾乱成分は平均成
%分よりも十分小さいという仮定を用い, 
%$1/\theta = 1/\bar{\theta}$, $1/f = 1/\bar{f}$ とする. 
\Deqref{pressEQ1} の左辺を線形化すると, 
%
\begin{eqnarray}
\DD{\Pi}{t}
  &=& 
\DP{\bar{\Pi} + \Pi^{'}}{t} 
 + u^{'} \DP{\bar{\Pi}+\Pi^{'}}{x} 
 + v^{'} \DP{\bar{\Pi}+\Pi^{'}}{y} 
 + w^{'} \DP{\bar{\Pi}+\Pi^{'}}{z}
  \nonumber \\
  &\approx&
\DP{\Pi^{'}}{t}
 + w^{'} \DP{\bar{\Pi}}{z}
%  =   - \frac{\overline{{C_{s}}^{2} }}{ \langle c_p \rangle_0
%  (\bar{\theta}/\bar{f})} 
%   \Ddiv \Dvect{u^{'}}
\nonumber 
\end{eqnarray}
%
%
となる. 
乱流拡散項・生成項・雲粒落下項は擾乱成分とみなすと, 
圧力方程式を線形化したときの $\langle c_{s} \rangle^2$, 及び 
$\theta_v$ からの寄与は基本場成分のみとなる. 
従って \Deqref{pressEQ1} の右辺を線形化すると, 
%
\begin{eqnarray}
& &
 \frac{\langle c_s \rangle^2}{\langle c_p \rangle_0 \theta_v}
\Bigg[ - \DP{u_j}{x_j}  +
\Dinv{\theta }  
 \left\{ \Dinv{\Pi} ( Q_{cond} + Q_{rad} + Q_{dis} ) 
  + D_{\theta} \right\} 
 \nonumber \\
& &
+ \Dinv{ q_d 
    + M_d \sum q_v  / M_v} 
  \Bigg\{ 
   D_{q_d} + M_d \sum \Dinv{M_v} 
    \left[  M_{src} (q_v) + D_{q_v} \right] 
  \Bigg\}
\Bigg]
 \nonumber \\
&\approx&
 \frac{ \overline{\langle c_s \rangle^2} }
      {\langle c_p \rangle_0 \overline{\theta_v} }
\Bigg[ - \DP{u_j^{'}}{x_j}  +
\Dinv{ \overline{\theta} }  
 \left\{ \Dinv{\overline{\Pi}} ( Q_{cond} + Q_{rad} + Q_{dis} ) 
  + D_{\theta} \right\} 
 \nonumber \\
& &
+ \Dinv{ 
    \overline{q_d} + M_d \sum \overline{q_v}  / M_v } 
  \Bigg\{ 
   D_{q_d} + M_d \sum \Dinv{M_v} 
    \left[  M_{src} (q_v) + D_{q_v} \right] 
  \Bigg\}
\Bigg]
 \nonumber \\
&=&
 \frac{ \overline{\langle c_s \rangle^2} }
      {\langle c_p \rangle_0 \overline{\theta_v} }
\Bigg[ - \DP{u_j^{'}}{x_j}  +
\Dinv{ \overline{\theta} }  
 \left\{ \Dinv{\overline{\Pi}} ( Q_{cond} + Q_{rad} + Q_{dis} ) 
  + D_{\theta} \right\} 
 \nonumber \\
& &
+ \Dinv{ 1 + \sum \left( \frac{R_v}{R_d} - 1 \right) \overline{q_v} } 
  \Bigg\{ 
   - \sum ( D_{q_v} + D_{q_c} + D_{q_r} ) 
    + \sum \frac{R_v}{R_d} 
      \left[  M_{src} (q_v) + D_{q_v} \right] 
  \Bigg\}
\Bigg]
 \nonumber
\end{eqnarray}
%
となる. 
ここで $\overline{q_d} + \overline{q_v} = 1$, $D_{q_d} + D_{q_v} +
D_{q_c} + D_{q_r} = 0$ となることを用いた. 
また
\begin{eqnarray}
\overline{\langle c_s \rangle^2} 
 = 
 \frac{\langle c_p \rangle_0 \langle R \rangle_0
 \bar{\Pi} \bar{\theta_v}}{\langle c_v \rangle_0} \nonumber
\end{eqnarray}
である. 
従って線形化された圧力方程式は
\begin{eqnarray}
\DP{ \Pi^{'}}{t}
 &=& 
 - w^{'} \DP{\bar{\Pi}}{z} 
 - \frac{ \overline{\langle c_s \rangle^2} }{\langle c_p \rangle_0
 \bar{\theta_v} } \DP{u_j^{'}}{x_j}
 \nonumber \\
 & &
 + \frac{ \overline{\langle c_s \rangle^2} }
      {\langle c_p \rangle_0 \overline{\theta_v} }
\Bigg[ 
 \Dinv{ \overline{\theta} }  
 \left\{ \Dinv{\overline{\Pi}} ( Q_{cond} + Q_{rad} + Q_{dis} ) 
  + D_{\theta} \right\} 
 \nonumber \\
& &
+ \Dinv{ 1 + \sum \left( \frac{R_v}{R_d} - 1 \right) \overline{q_v} } 
  \Bigg\{ 
   - \sum ( D_{q_v} + D_{q_c} + D_{q_r} ) 
    + \sum \frac{R_v}{R_d} 
      \left[  M_{src} (q_v) + D_{q_v} \right] 
  \Bigg\}
\Bigg]
 \nonumber \\
\Deqlab{pressEQ2}
\end{eqnarray}
と表される. 
\Deqref{pressEQ2} の右辺第 1 項, 第 2 項をまとめると, 
%さらに $\Pi$ を理想気体の状態方程式で変形してまとめると, 
%圧力の擾乱成分の時間発展方程式が得られる. 
%
\begin{eqnarray}
 & &
 - w^{'} \DP{\bar{\Pi}}{z} 
 - \frac{ \overline{\langle c_s \rangle^2} }{\langle c_p \rangle_0
 \bar{\theta_v} } \DP{u_j^{'}}{x_j} \nonumber \\
 &=&  
 - w^{'} \DP{}{z} 
 \left( \frac{\bar{\rho} \langle R \rangle_0 \bar{\theta_v}
  }{p_{0}}\right)^{ \langle R \rangle_0 / \langle c_v \rangle_0 }  
 - \frac{\overline{{ \langle c_{s} \rangle }^{2}} }{ \langle c_p \rangle_0
 \bar{\theta_v} }
   \left( \DP{ u^{'}}{x} + \DP{ w^{'}}{z} \right)
\nonumber \\
%
 &=& 
- w^{'} 
 \frac{\langle R \rangle_0}{ \langle c_v \rangle}_0 \bar{\Pi}
 \Dinv{ \left( \frac{\bar{\rho}
	\langle R \rangle_0 \bar{\theta_v} }{p_{0}}\right)}
 \frac{ \langle R \rangle_0}{p_{0}}
 \DP{ ( \bar{\rho} \bar{\theta_v} )}{z} 
 - \frac{\overline{{\langle c_{s} \rangle}^{2}} }{ \langle c_p \rangle_0
 \bar{\theta_v} }
   \left( \DP{u^{'}}{x} + \DP{w^{'}}{z} \right)
\nonumber \\
%
 &=&
 - \frac{\overline{ {\langle c_{s} \rangle}^{2} }}{\langle c_p \rangle_0
 \bar{\rho} 
 \bar{\theta_v}^{2}}
 \left\{
 w^{'} 
 \DP{ ( \bar{\rho} \bar{\theta_v} ) }{z} 
 + \bar{\rho} \bar{\theta_v}
   \left( \DP{u^{'}}{x} + \DP{w^{'}}{z} \right)
\right\}
\nonumber \\
%
 &=&
- \frac{\overline{{c_{s}}^{2}}}{\langle c_p \rangle_0  \bar{\rho}
 \bar{\theta_v}^{2}}
 \Ddiv \left(
	\bar{\rho} \bar{\theta_v} \Dvect{u^{'}}
       \right)
\nonumber 
\end{eqnarray}
%
以上より, 
\begin{eqnarray}
\DP{ \Pi^{'}}{t}
 &=& 
- \frac{\overline{{c_{s}}^{2}}}{\langle c_p \rangle_0  \bar{\rho}
 \bar{\theta_v}^{2}}
 \Ddiv \left(
	\bar{\rho} \bar{\theta_v} \Dvect{u^{'}}
       \right)
 \nonumber \\
 & &
 + \frac{ \overline{\langle c_s \rangle^2} }
      {\langle c_p \rangle_0 \overline{\theta_v} }
\Bigg[ 
 \Dinv{ \overline{\theta} }  
 \left\{ \Dinv{\overline{\Pi}} ( Q_{cond} + Q_{rad} + Q_{dis} ) 
  + D_{\theta} \right\} 
 \nonumber \\
& &
+ \Dinv{ 1 + \sum \left( \frac{R_v}{R_d} - 1 \right) \overline{q_v} } 
  \Bigg\{ 
   - \sum ( D_{q_v} + D_{q_c} + D_{q_r} ) 
    + \sum \frac{R_v}{R_d} 
      \left[  M_{src} (q_v) + D_{q_v} \right] 
  \Bigg\}
\Bigg]
 \nonumber \\
\Deqlab{pressEQ3}
\end{eqnarray}
%
である. 


\subsection{熱の式の線形化}

熱の式を平均成分と擾乱成分に分離する. 
%
\begin{eqnarray}
 \DP{(\bar{\theta} + \theta^{'})}{t} 
 &=& 
 - u^{'} \DP{(\bar{\theta} + \theta^{'})}{x} 
 - v^{'} \DP{(\bar{\theta} + \theta^{'})}{y} 
 - w^{'}\DP{(\bar{\theta} + \theta^{'})}{z} 
  \nonumber \\
 & &
 + \Dinv{\bar{\Pi} + \Pi^{'}} (Q_{cond} + Q_{rad} + Q_{dis}) 
 + D_{\bar{\theta}} + D_{\theta^{'}}
\nonumber 
\end{eqnarray}
%
ここで平均場の量は $z$ の関数であることを用いると, 
%
\begin{eqnarray}
 \DP{\theta^{'}}{t} 
 &=& 
 - \left(
    u^{'}\DP{\theta^{'}}{x} 
    + v^{'}\DP{\theta^{'}}{y} 
    + w^{'}\DP{\theta^{'}}{z} 
   \right)
 - w^{'}\DP{\bar{\theta}}{z} 
  \nonumber \\
 & &
 + \Dinv{\bar{\Pi}} (Q_{cond} + Q_{rad} + Q_{dis}) 
 + D_{\bar{\theta}}   + D_{\theta^{'}}
 \Deqlab{Thermeq}
\end{eqnarray}
%
となる. 


\subsection{比湿の保存式の線形化}

凝結成分の比湿の保存式についても, 変数を平均成分と擾乱成分に分離する. 
熱の式と同様に, 以下のように書ける. 但し, 生成項, 落下項は擾乱成分のみ
存在すると仮定する. この仮定は平均場では凝結は生じていないと考えることに
等しい. 
%
\begin{eqnarray}
%%
%\DP{q_d^{\prime}}{t}
% &=& - u^{\prime} \DP{q_d^{\prime}}{x}
%     - v^{\prime} \DP{q_d^{\prime}}{y}
%     - w^{\prime} \DP{q_d^{\prime}}{z}
%     - \overline{u} \DP{q_d^{\prime}}{x}
%     - \overline{v} \DP{q_d^{\prime}}{y}
%     - w^{\prime} \DP{\overline{q_d}}{z} 
%      \nonumber \\
% & &
%     - \frac{\overline{q_d}}{\overline{\rho}} \sum M_{fall}
%     (\rho_r^{\prime})  + D_{\overline{q_d}}
%     + D_{q_d^{\prime}}, \\
%%
\DP{q_v^{\prime}}{t}
 &=& - u^{\prime} \DP{q_v^{\prime}}{x}
     - v^{\prime} \DP{q_v^{\prime}}{y}
     - w^{\prime} \DP{q_v^{\prime}}{z}
     - w^{\prime} \DP{\overline{q_v}}{z} 
       \nonumber \\
 & &
     - \frac{\overline{q_v}}{\overline{\rho}} \sum M_{fall}
     (\rho_r^{\prime}) + \Dinv{\overline{\rho}} M_{src} (\rho_v^{\prime})
     + D_{\overline{q_v}}
     + D_{q_v^{\prime}}, \\
%
\DP{q_c^{\prime}}{t}
 &=& - u^{\prime} \DP{q_c^{\prime}}{x}
     - v^{\prime} \DP{q_c^{\prime}}{y}
     - w^{\prime} \DP{q_c^{\prime}}{z}
       \nonumber \\
 & &
     + \Dinv{\overline{\rho}} M_{src} (\rho_c^{\prime})
     + D_{q_c^{\prime}}, \\
%
\DP{q_r^{\prime}}{t}
 &=& - u^{\prime} \DP{q_r^{\prime}}{x}
     - v^{\prime} \DP{q_r^{\prime}}{y}
     - w^{\prime} \DP{q_r^{\prime}}{z}
       \nonumber \\
 & &
     + \Dinv{\overline{\rho}} M_{fall} (\rho_r^{\prime})
     + \Dinv{\overline{\rho}} M_{src} (\rho_r^{\prime})
     + D_{q_r^{\prime}}.
%
\end{eqnarray}
%
但し雲水量と雨水量は擾乱成分のみの量である. 


\subsection{エネルギー方程式の導出}

準圧縮方程式系におけるエネルギー方程式を導出する. 

\Deqref{bunri:moist:dudt}, \Deqref{bunri:moist:dvdt},
\Deqref{bunri:moist:dwdt} にそれぞれ 
$\overline{\rho} u^{\prime}$, 
$\overline{\rho} v^{\prime}$, 
$\overline{\rho} w^{\prime}$ を掛けて足し合わせると
\begin{eqnarray}
\DP{ (\overline{\rho} K ) }{t} 
 &=&
 - \overline{\rho} \Dvect{u} \cdot \nabla K 
 - \langle c_p \rangle_0
  \overline{\rho} \overline{\theta_v} \Dvect{u} \cdot \nabla \Pi^{\prime}
 + \overline{\rho} \Dvect{u} \cdot \Dvect{D} 
 + g 
    \frac{\theta_v^{\prime}}
         {\overline{\theta_v}} \overline{\rho} w^{\prime}
\Deqlab{AAA}
\end{eqnarray}
となる.
但し $\Dvect{u} = (u^{\prime}, v^{\prime},
w^{\prime})$, $\Dvect{D} = (D_u, D_v, D_w)$, 
$K = [{u^{\prime}}^2 + {v^{\prime}}^2 + {w^{\prime}}^2 ]/2$ と置いた.
連続の式
\begin{equation}
\DP{\rho_d^{\prime}}{t}
 + \DP{}{x} \left( u^{\prime} \overline{\rho_d} \right)
 + \DP{}{z} \left( w^{\prime} \overline{\rho_d} \right)
 = 0, 
 \Deqlab{AAA-2}
\end{equation}
\begin{equation}
\DP{\rho_v^{\prime}}{t}
 + \DP{}{x} \left( u^{\prime} \overline{\rho_v} \right)
 + \DP{}{z} \left( w^{\prime} \overline{\rho_v} \right)
 = M_{src} (\rho_v^{\prime}) 
 \Deqlab{AAA-3}
\end{equation}
より
\begin{equation}
\DP{}{t} \left( \rho_d^{\prime} + \sum \rho_v^{\prime} \right)
+ \nabla \cdot (\overline{\rho} \Dvect{u})
 = \sum M_{src} (\rho_v^{\prime})
 \Deqlab{AAB}
\end{equation}
となる. 
但し $\overline{\rho} = \overline{\rho_d} + \sum \overline{\rho_v}$ であ
ることを用いた. 
\Deqref{AAB} を用いると, \Deqref{AAA} の右辺第 1 項は
\begin{eqnarray}
- \overline{\rho} \Dvect{u} \cdot \nabla K
&=&
 - \nabla \cdot ( \overline{\rho} K \Dvect{u} )
 + K \nabla \cdot (\overline{\rho} \Dvect{u}) \nonumber \\
&=&
 - \nabla \cdot ( \overline{\rho} K \Dvect{u} )
 - K \DP{}{t} \left( \rho_d^{\prime} + \rho_v^{\prime} \right) 
 + K M_{src} (\rho_v^{\prime})
\Deqlab{AAC}
\end{eqnarray}
となる.
また \Deqref{pressEQ3} を用いて \Deqref{AAA} の右辺第 2 項を書き換えると
\begin{eqnarray}
 - \langle c_p \rangle_0 \overline{\rho} \overline{\theta_v} \Dvect{u}
  \cdot \nabla \Pi^{\prime} 
&=&
 - \nabla \cdot [ \langle c_p \rangle_0 \overline{\rho} \overline{\theta_v}
 \Pi^{\prime} \Dvect{u} ]
 + \langle c_p \rangle_0 \Pi^{\prime} \nabla \cdot (\overline{\rho}
 \overline{\theta_v} \Dvect{u} ) 
  \nonumber \\
&=&
 - \nabla \cdot [ \langle c_p \rangle_0 \overline{\rho} \overline{\theta_v}
 \Pi^{\prime} \Dvect{u} ] 
  \nonumber \\
 & &
 + \Pi^{\prime}
   \frac{\langle c_p \rangle_0^2 \overline{\rho} \overline{\theta_v}^2}
        {\overline{\langle c_s \rangle}^2}
   \nonumber \\
 & &
  \Bigg[
   - \DP{\Pi^{\prime}}{t}
   + \frac{ \overline{\langle c_s \rangle^2} }
     {\langle c_p \rangle_0 \overline{\theta_v} }
   \Bigg[ 
    \Dinv{ \overline{\theta} }  
     \left\{ \Dinv{\overline{\Pi}} ( Q_{cond} + Q_{rad} + Q_{dis} ) 
     + D_{\theta} \right\} 
   \nonumber \\
 & &
  + \Dinv{ 1 + \sum \left( \frac{R_v}{R_d} - 1 \right) \overline{q_v} } 
  \Bigg\{ 
   - \sum ( D_{q_v} + D_{q_c} + D_{q_r} ) 
   \nonumber \\
  & &
    + \sum \frac{R_v}{R_d} 
      \left[  M_{src} (q_v) + D_{q_v} \right] 
  \Bigg\}
  \Bigg] \Bigg]
   \nonumber \\
&=&
 - \nabla \cdot [ \langle c_p \rangle_0 \overline{\rho} \overline{\theta_v}
 \Pi^{\prime} \Dvect{u} ] 
 - \DP{}{t} 
    \left[ 
     \frac{\overline{\rho}}{2} 
     \frac{\langle c_p \rangle_0^2 \overline{\theta_v}^2 {\Pi^{\prime}}^2}
          {\overline{\langle c_s \rangle}^2}
    \right]
  \nonumber \\
 & &
 + \langle c_p \rangle_0 \overline{\rho} \overline{\theta_v} \Pi^{\prime}
   \nonumber \\
 & &
   \Bigg[ 
    \Dinv{ \overline{\theta} }  
     \left\{ \Dinv{\overline{\Pi}} ( Q_{cond} + Q_{rad} + Q_{dis} ) 
     + D_{\theta} \right\} 
   \nonumber \\
 & &
  + \Dinv{ 1 + \sum \left( \frac{R_v}{R_d} - 1 \right) \overline{q_v} } 
  \Bigg\{ 
   - \sum ( D_{q_v} + D_{q_c} + D_{q_r} ) 
   \nonumber \\
  & &
    + \sum \frac{R_v}{R_d} 
      \left[  M_{src} (q_v) + D_{q_v} \right] 
  \Bigg\}
  \Bigg]
\Deqlab{AAD}
\end{eqnarray}
となる. 
%但し
%\begin{eqnarray}
%\Phi 
% &=&
% \Dinv{\bar{\rho}} \sum M_{fall} (\rho_r^{'}) +
% \Dinv{ \bar{\theta} }
%  \left\{ \Dinv{\bar{\Pi}} ( Q_{cond} + Q_{rad} + Q_{dis} )
%   + D_{\theta^{'}} \right\}
%  \nonumber \\
% & &
%  + \Dinv{ \bar{q_d}  \langle M \rangle_0 / M_d
%    + \sum \bar{q_v} \langle M \rangle_0 / M_v}
%   \Bigg\{ \frac{\langle M \rangle_0}{M_d}
%  \left( - \frac{\bar{q_d}}{\bar{\rho}} \sum M_{fall} (\rho_r^{'}) +
%   D_{q_d^{'}} \right)
% \nonumber \\
% & &
%  + \sum \frac{\langle M \rangle_0}{M_v}
%  \left( - \frac{\bar{q_v}}{\bar{\rho}} \sum M_{fall} (\rho_r^{'}) +
%   M_{src} (q_v^{'}) +  D_{q_v^{'}} \right)
% \Bigg\}
%\Deqlab{AAD-2}
%\end{eqnarray}
%である. 
\Deqref{AAB} より任意のスカラー量 $\phi$ について
\begin{equation}
\overline{\rho} \DD{\phi}{t}
= \DP{}{t} \left( \overline{\rho} \phi \right)
  + \nabla \cdot \left( \overline{\rho} \phi \Dvect{u} \right)
  + \phi \DP{}{t} \left( \rho_d^{\prime} + \sum \rho_v^{\prime} \right)
  - \phi \sum M_{src} (\rho_v^{\prime})
\Deqlab{AAE}
\end{equation}
が成り立つ.
\Deqref{Thermeq} 及び \Deqref{AAE} を用いて \Deqref{AAA} の右辺第 4 項
を書き換えると,
\begin{eqnarray}
\frac{\theta^{\prime}}{\overline{\theta}} \overline{\rho} w^{\prime}
 &=&
 \frac{\theta^{\prime}}{\overline{\theta}} \overline{\rho} \DD{z}{t}
  \nonumber \\
 &=&
 \frac{\overline{\rho}}{\overline{\theta}} \theta^{\prime} \DD{}{t} (g z)
+ \frac{\overline{\rho} g z}{\overline{\theta}}
  \left[ \DD{\theta^{\prime}}{t} + w^{\prime} \DP{\overline{\theta}}{z}
   - \Dinv{\overline{\Pi}}
     \left( Q_{cond} + Q_{rad} + Q_{dis} \right) - D_{\theta} \right]
  \nonumber \\
 &=&
\frac{\overline{\rho}}{\overline{\theta}} \DD{}{t}
(\theta^{\prime} g z)
+ \frac{\overline{\rho} g z}{\overline{\theta}}
  \left[ w^{\prime} \DP{\overline{\theta}}{z}
   - \Dinv{\overline{\Pi}}
     \left( Q_{cond} + Q_{rad} + Q_{dis} \right) - D_{\theta} \right]
  \nonumber \\
 &=&
\Dinv{\overline{\theta}} \DP{}{t} \left( \overline{\rho} \theta^{\prime}
 g z \right) + \Dinv{\overline{\theta}}
 \nabla \cdot \left( \overline{\rho} \theta^{\prime} g z \Dvect{u} \right)
 + \frac{\theta^{\prime} g z}{\overline{\theta}}
  \left[ 
   \DP{}{t} \left( \rho_d^{\prime} + \rho_v^{\prime} \right) 
   - \sum M_{src} (\rho_v^{\prime})
  \right] \nonumber \\
& &
+ \frac{\overline{\rho} g z}{\overline{\theta}}
  \left[ w^{\prime} \DP{\overline{\theta}}{z}
   - \Dinv{\overline{\Pi}}
     \left( Q_{cond} + Q_{rad} + Q_{dis} \right) - D_{\theta} \right]
  \nonumber \\
&=&
\Dinv{\overline{\theta}} \DP{}{t} \left( \overline{\rho} \theta^{\prime}
 g z \right) +
 \nabla \cdot \left( \frac{ \overline{\rho} \theta^{\prime} g z \Dvect{u}
   }{\overline{\theta}} \right)
 + \frac{ \overline{\rho} \theta^{\prime} g z w^{\prime}}
        { {\overline{\theta}}^2}
 \DP{\overline{\theta}}{z} 
  \nonumber \\
& &
 + \frac{\theta^{\prime} g z}{\overline{\theta}}
  \left[ 
    \DP{}{t} \left( \rho_d^{\prime} + \rho_v^{\prime} \right) 
    - \sum M_{src} (\rho_v^{\prime})
  \right]
\nonumber \\
& &
+ \frac{\overline{\rho} g z}{\overline{\theta}}
  \left[ w^{\prime} \DP{\overline{\theta}}{z}
   - \Dinv{\overline{\Pi}}
     \left( Q_{cond} + Q_{rad} + Q_{dis} \right) 
   - D_{\theta} 
  \right], 
\Deqlab{AAF}
\end{eqnarray}
%
\begin{eqnarray}
\frac{\overline{\rho} g}
     { 1 + \sum \left( \frac{R_v}{R_d} - 1 \right) \overline{q_v} } 
  q_v^{\prime}
 &=& 
  \DP{}{t}
   \left[
    \frac{ q_v^{\prime} \overline{\rho} g z}
     { 1 + \sum \left( \frac{R_v}{R_d} - 1 \right) \overline{q_v} } 
   \right]
  + 
  \nabla \cdot
   \left[
    \frac{ q_v^{\prime} \overline{\rho} g z \Dvect{u} }
     { 1 + \sum \left( \frac{R_v}{R_d} - 1 \right) \overline{q_v} } 
   \right]
  \nonumber \\
& &
 - q_v^{\prime} g z \nabla \cdot
   \left[
    \frac{ \overline{\rho} \Dvect{u} }
     { 1 + \sum \left( \frac{R_v}{R_d} - 1 \right) \overline{q_v} } 
   \right]
  \nonumber \\
& &
 + g z 
  \left[
   w^{\prime} \DP{\overline{q_v}}{z}
   + \frac{\overline{q_v}}{\overline{\rho}} \sum M_{fall} (\rho_r)
   - \Dinv{\overline{\rho}} M_{src} (q_v)
   - D_{q_v}
  \right]
  \nonumber \\
\Deqlab{AAF-2}
\end{eqnarray}
%
\begin{eqnarray}
\frac{\overline{\rho} g}
     { 1 + \sum \left( \frac{R_v}{R_d} - 1 \right) \overline{q_v} } 
  q_c^{\prime}
 &=& 
  \DP{}{t}
   \left[
    \frac{ q_c^{\prime} \overline{\rho} g z}
     { 1 + \sum \left( \frac{R_v}{R_d} - 1 \right) \overline{q_v} } 
   \right]
  + 
  \nabla \cdot
   \left[
    \frac{ q_c^{\prime} \overline{\rho} g z \Dvect{u} }
     { 1 + \sum \left( \frac{R_v}{R_d} - 1 \right) \overline{q_v} } 
   \right]
  \nonumber \\
& &
 - q_c^{\prime} g z \nabla \cdot
   \left[
    \frac{ \overline{\rho} \Dvect{u} }
     { 1 + \sum \left( \frac{R_v}{R_d} - 1 \right) \overline{q_v} } 
   \right]
  \nonumber \\
& &
 + g z 
  \left[
   - \Dinv{\overline{\rho}} M_{src} (q_c)
   - D_{q_c}
  \right]
  \nonumber \\
\Deqlab{AAF-3}
\end{eqnarray}
%
\begin{eqnarray}
\frac{\overline{\rho} g}
     { 1 + \sum \left( \frac{R_v}{R_d} - 1 \right) \overline{q_v} } 
  q_r^{\prime}
 &=& 
  \DP{}{t}
   \left[
    \frac{ q_r^{\prime} \overline{\rho} g z}
     { 1 + \sum \left( \frac{R_v}{R_d} - 1 \right) \overline{q_v} } 
   \right]
  + 
  \nabla \cdot
   \left[
    \frac{ q_r^{\prime} \overline{\rho} g z \Dvect{u} }
     { 1 + \sum \left( \frac{R_v}{R_d} - 1 \right) \overline{q_v} } 
   \right]
  \nonumber \\
& &
 - q_r^{\prime} g z \nabla \cdot
   \left[
    \frac{ \overline{\rho} \Dvect{u} }
     { 1 + \sum \left( \frac{R_v}{R_d} - 1 \right) \overline{q_v} } 
   \right]
  \nonumber \\
& &
 + g z 
  \left[
   - \Dinv{\overline{\rho}} M_{fall} (\rho_r)
   - \Dinv{\overline{\rho}} M_{src} (q_r)
   - D_{q_r}
  \right]
  \nonumber \\
\Deqlab{AAF-3A}
\end{eqnarray}
%但し
%\begin{eqnarray}
%\overline{\Omega_d}
% &=&
% \frac{\langle M \rangle_0 /M_d  }{\overline{q_d} \langle M \rangle_0 /M_d +
%    \sum \overline{q_v} \langle M \rangle_0 / M_v}, 
%\Deqlab{AAF-4} 
% \\
%\overline{\Omega_v}
% &=&
% \frac{
%    \langle M \rangle_0 / M_v }{\overline{q_d} \langle M \rangle_0 /M_d +
%    \sum \overline{q_v} \langle M \rangle_0 / M_v}
%\Deqlab{AAF-5} 
%\end{eqnarray}
%である. 
\Deqref{AAC}, \Deqref{AAD}, \Deqref{AAF}, \Deqref{AAF-2},
\Deqref{AAF-3}, \Deqref{AAF-3A}
より \Deqref{AAA} は以下のように書き換えられる.
\begin{eqnarray}
& &
\DP{}{t}
\left\{ 
 \overline{\rho} 
  \left[ 
   K 
   - \frac{\theta_v^{\prime}}{\overline{\theta_v}} g z 
   + \Dinv{2}  
     \left( 
       \frac{\langle c_p \rangle_0 \overline{\theta_v} \Pi^{\prime}}
            {\overline{\langle c_s \rangle}}
     \right)^2
  \right] \right\}
+ \nabla \cdot \left[
  \overline{\rho} \left( K - \frac{\theta_v^{\prime}}{\overline{\theta_v}}
  g z + \langle c_p \rangle_0 \overline{\theta_v} \Pi^{\prime} \right)
  \Dvect{u} \right]  
   \nonumber \\
&=&
   - K \DP{}{t} \left( \rho_d^{\prime} + \rho_v^{\prime} \right) 
%
 + \overline{\rho} \Dvect{u} \cdot \Dvect{D} 
 + K M_{src} (\rho_v^{\prime})
  \nonumber \\
& &
 + \langle c_p \rangle_0 \overline{\rho} \overline{\theta_v} \Pi^{\prime}
   \nonumber \\
 & &
   \Bigg[ 
    \Dinv{ \overline{\theta} }  
     \left\{ \Dinv{\overline{\Pi}} ( Q_{cond} + Q_{rad} + Q_{dis} ) 
     + D_{\theta} \right\} 
   \nonumber \\
 & &
  + \Dinv{ 1 + \sum \left( \frac{R_v}{R_d} - 1 \right) \overline{q_v} } 
  \Bigg\{ 
   - \sum ( D_{q_v} + D_{q_c} + D_{q_r} ) 
    + \sum \frac{R_v}{R_d} 
      \left[  M_{src} (q_v) + D_{q_v} \right] 
  \Bigg\}
  \Bigg]
 \nonumber \\
 & &
  + \left[ 
    - \sum (q_v^{\prime} + q_c^{\prime} + q_r^{\prime} ) 
    + \sum (R_v / R_d) q_v^{\prime}
    \right]
   \nabla \cdot
   \left[
    \frac{ \overline{\rho} \Dvect{u} }
     { 1 + \sum \left( \frac{R_v}{R_d} - 1 \right) \overline{q_v} } 
   \right]
  \nonumber \\  
 & &
  - \theta^{\prime} g z 
    \nabla \cdot 
    \left[ \frac{\overline{\rho} \Dvect{u}}{\overline{\theta}} \right]
  \nonumber \\
 & &
  + \frac{\overline{\rho} g z}
       { 1 + \sum \left( \frac{R_v}{R_d} - 1 \right) \overline{q_v} } 
  \nonumber \\
 & &
   \Bigg[
    \left\{ w^{\prime} \sum \left( -1 + R_v/R_d \right) \DP{q_v}{z} \right\}
    + \left( \sum \frac{\overline{q_v} R_v}{\overline{\rho} R_d} \right)
      \left( \sum M_{fall} (\rho_r) \right)
  \nonumber \\
 & &
    + \sum (D_{q_v} + D_{q_c} + D_{q_r})
    - \sum R_v D_{q_v} / R_d
   \Bigg]
  \nonumber \\
 & &
  + \frac{\overline{\rho} g z}{\overline{\theta}}
    \left[ 
     w^{\prime} \DP{\theta}{z}
     - \Dinv{\overline{\Pi}}
       \left( Q_{cond} + Q_{rad} + Q_{dis} \right) - D_{\theta} \right]
\Deqlab{AAG}
\end{eqnarray}
計算領域として矩形領域を想定し, 鉛直方向の境界からの流出は無く, 水平境界
の両端では周期的であるとすると, 計算領域境界面でのフラックスはゼロとなる.
従って \Deqref{AAG} を全計算領域にわたって積分すると,
\begin{eqnarray}
& &
\DP{}{t} \int
\left\{ 
 \overline{\rho} 
  \left[ 
   K 
   - \frac{\theta_v^{\prime}}{\overline{\theta_v}} g z 
   + \Dinv{2}  
     \left( 
       \frac{\langle c_p \rangle_0 \overline{\theta_v} \Pi^{\prime}}
            {\overline{\langle c_s \rangle}}
     \right)^2
  \right] \right\} dV
   \nonumber \\
&=&
   - \int K \DP{}{t} \left( \rho_d^{\prime} + \rho_v^{\prime} \right) dV
%
 + \int \overline{\rho} \Dvect{u} \cdot \Dvect{D} dV
 + \int K M_{src} (\rho_v^{\prime}) dV
  \nonumber \\
& &
 + \int \langle c_p \rangle_0 \overline{\rho} \overline{\theta_v} \Pi^{\prime}
   \nonumber \\
 & &
   \Bigg[ 
    \Dinv{ \overline{\theta} }  
     \left\{ \Dinv{\overline{\Pi}} ( Q_{cond} + Q_{rad} + Q_{dis} ) 
     + D_{\theta} \right\} 
   \nonumber \\
 & &
  + \Dinv{ 1 + \sum \left( \frac{R_v}{R_d} - 1 \right) \overline{q_v} } 
  \Bigg\{ 
   - \sum ( D_{q_v} + D_{q_c} + D_{q_r} ) 
    + \sum \frac{R_v}{R_d} 
      \left[  M_{src} (q_v) + D_{q_v} \right] 
  \Bigg\}
  \Bigg] dV
 \nonumber \\
 & &
  + \int \left[ 
    - \sum (q_v^{\prime} + q_c^{\prime} + q_r^{\prime} ) 
    + \sum (R_v / R_d) q_v^{\prime}
    \right]
   \nabla \cdot
   \left[
    \frac{ \overline{\rho} \Dvect{u} }
     { 1 + \sum \left( \frac{R_v}{R_d} - 1 \right) \overline{q_v} } 
   \right] dV
  \nonumber \\  
 & &
  - \int \theta^{\prime} g z 
    \nabla \cdot 
    \left[ \frac{\overline{\rho} \Dvect{u}}{\overline{\theta}} \right]
    dV
  \nonumber \\
 & &
  + \int \frac{\overline{\rho} g z}
       { 1 + \sum \left( \frac{R_v}{R_d} - 1 \right) \overline{q_v} } 
  \nonumber \\
 & &
   \Bigg[
    \left\{ w^{\prime} \sum \left( -1 + R_v/R_d \right) \DP{q_v}{z} \right\}
    + \left( \sum \frac{\overline{q_v} R_v}{\overline{\rho} R_d} \right)
      \left( \sum M_{fall} (\rho_r) \right)
  \nonumber \\
 & &
    + \sum (D_{q_v} + D_{q_c} + D_{q_r})
    - \sum R_v D_{q_v} / R_d
   \Bigg] dV
  \nonumber \\
 & &
  + \int \frac{\overline{\rho} g z}{\overline{\theta}}
    \left[ 
     w^{\prime} \DP{\theta}{z}
     - \Dinv{\overline{\Pi}}
       \left( Q_{cond} + Q_{rad} + Q_{dis} \right) - D_{\theta} \right]
     dV
\Deqlab{AAH}
\end{eqnarray}
となり, 準圧縮方程式に関するエネルギー方程式が得られる.

\Deqref{AAH} の左辺は全エネルギーの時間変化を表している.
左辺の被積分関数の第 1 項, 第 2 項, 第 3 項はそれぞれ運動エネルギー, 浮
力による位置エネルギー, 弾性エネルギー(熱エネルギー)を表す.
右辺第 1 項, 第 5 項, 第 6 項は準圧縮近似によって現れる項であり, 一般に
ゼロとなることはない. 
右辺第 2 項は運動量の乱流拡散, 第 3 項は凝結, 第 4 項は非断熱加熱と乱流
拡散, 第 7 項は蒸気の勾配・雲粒落下・水蒸気拡散, 第 8 項は温位勾配・非断
熱加熱を表す. 
非断熱加熱や乱流拡散や基本場の空間変化が存在しなかったとしても, 右辺がゼ
ロとなることは無い. 
即ち, 準圧縮方程式では全エネルギーが保存されることはない.



\section{まとめ}

準圧縮方程式系は以下のようにまとめられる. 
%
\begin{description}
%
 \item[運動方程式]~
%
\begin{eqnarray}
 \DP{u^{'}}{t}
  &=&
- u^{\prime} \DP{u^{\prime}}{x}
- v^{\prime} \DP{u^{\prime}}{y}
- w^{\prime} \DP{u^{\prime}}{z}
\nonumber \\
  & &
  - \langle c_p \rangle_0 \overline{\theta_v} \DP{\Pi^{\prime}}{x}
  + D_{u}
  \Deqlab{quasi-equations:u-w-pi-theta:u} \\
%
 \DP{v^{'}}{t}
  &=&
- u^{\prime} \DP{v^{\prime}}{x}
- v^{\prime} \DP{v^{\prime}}{y}
- w^{\prime} \DP{v^{\prime}}{z}
\nonumber \\
  & &
  - \langle c_p \rangle_0 \overline{\theta_v} \DP{\Pi^{\prime}}{y}
  + D_{v}.
  \Deqlab{quasi-equations:u-w-pi-theta:v} \\
%
\DP{w^{\prime}}{t}
&=& 
  - u^{\prime} \DP{w^{\prime}}{x} 
  - v^{\prime} \DP{w^{\prime}}{y} 
  - w^{\prime} \DP{w^{\prime}}{z}
  \nonumber \\
& &
  - \langle c_p \rangle_0 \overline{\theta_v} \DP{\Pi^{\prime}}{z}
  + \frac{\theta^{\prime}}{\overline{\theta}} g
  + \frac{ 
    \sum \left( -1 + \frac{R_v}{R_d} \right) q_v^{\prime}
    - \sum \left( q_c^{\prime} + q_r^{\prime} \right) }
   {  1 + \sum \left( -1 + \frac{R_v}{R_d} \right) \overline{q_v} } g
  + D_{w^{\prime}} 
\nonumber \\
  \Deqlab{quasi-equations:u-w-pi-theta:w} 
\end{eqnarray}
%
 \item[圧力方程式]~
%
\begin{eqnarray}
\DP{ \Pi^{'}}{t}
 &=& 
- \frac{\overline{{ \langle c_{s} \rangle}^{2}}}{\langle c_p \rangle_0
\bar{\rho} \bar{\theta_v}^{2}}
 \Ddiv \left(
	\bar{\rho} \bar{\theta_v} \Dvect{u^{'}}
       \right)
 \nonumber \\
 & &
+ \Dinv{ 1 + \sum \left( \frac{R_v}{R_d} - 1 \right) \overline{q_v} } 
 \nonumber \\
 & &
  \Bigg\{ 
   - \sum ( D_{q_v} + D_{q_c} + D_{q_r} ) 
    + \sum \frac{R_v}{R_d} 
      \left[  M_{src} (q_v) + D_{q_v} \right] 
  \Bigg\}
\Bigg]
 \nonumber \\
  \Deqlab{quasi-equations:u-w-pi-theta:continue}
\end{eqnarray}
%
 \item[熱の式]~
%
\begin{eqnarray}
 \DP{\theta^{'}}{t}
 &=&
 - u^{'}\DP{\theta^{'}}{x}
 - v^{'}\DP{\theta^{'}}{y}
 - w^{'}\DP{\theta^{'}}{z}
 - w^{'}\DP{\bar{\theta}}{z}
  \nonumber \\
 & &
 + \Dinv{\bar{\Pi}} (Q_{cond} + Q_{rad} + Q_{dis})
 + D_{\bar{\theta}}   + D_{\theta^{'}}
  \Deqlab{quasi-equations:u-w-pi-theta:theta}
\end{eqnarray}
%
 \item[比湿の保存式]~
%
\begin{eqnarray}
%\DP{q_d^{\prime}}{t}
% &=& - u^{\prime} \DP{q_d^{\prime}}{x}
%     - w^{\prime} \DP{q_d^{\prime}}{z}
%     - \overline{u} \DP{q_d^{\prime}}{x}
%     - w^{\prime} \DP{\overline{q_d}}{z}
%     - \frac{\overline{q_d}}{\overline{\rho}} \sum M_{fall}
%     (\rho_r^{\prime}) \nonumber \\
% & & + D_{\overline{q_d}}
%     + D_{q_d^{\prime}},
%  \Deqlab{quasi-equations:u-w-pi-theta:qd} \\
%
\DP{q_v^{\prime}}{t}
 &=& - u^{\prime} \DP{q_v^{\prime}}{x}
     - v^{\prime} \DP{q_v^{\prime}}{y}
     - w^{\prime} \DP{q_v^{\prime}}{z}
     - w^{\prime} \DP{\overline{q_v}}{z}
       \nonumber \\
 & &
     - \frac{\overline{q_v}}{\overline{\rho}} \sum M_{fall}
     (\rho_r^{\prime}) + \Dinv{\overline{\rho}} M_{src}
     (\rho_v^{\prime})
     + D_{\overline{q_v}}
     + D_{q_v^{\prime}},
  \Deqlab{quasi-equations:u-w-pi-theta:qv} \\
%
\DP{q_c^{\prime}}{t}
 &=& - u^{\prime} \DP{q_c^{\prime}}{x}
     - v^{\prime} \DP{q_c^{\prime}}{y}
     - w^{\prime} \DP{q_c^{\prime}}{z}
       \nonumber \\
 & &
     + \Dinv{\overline{\rho}} M_{src} (\rho_c^{\prime})
     + D_{q_c^{\prime}}, 
  \Deqlab{quasi-equations:u-w-pi-theta:qc} \\
%
\DP{q_r^{\prime}}{t}
 &=& - u^{\prime} \DP{q_r^{\prime}}{x}
     - v^{\prime} \DP{q_r^{\prime}}{y}
     - w^{\prime} \DP{q_r^{\prime}}{z}
       \nonumber \\
 & &
     + \Dinv{\overline{\rho}} M_{fall} (\rho_r^{\prime})
     + \Dinv{\overline{\rho}} M_{src} (\rho_r^{\prime})
     + D_{q_r^{\prime}}. 
  \Deqlab{quasi-equations:u-w-pi-theta:qr}
\end{eqnarray}
%
 \item[エネルギー方程式]~
%
\begin{eqnarray}
& &
\DP{}{t} \int
\left\{ 
 \overline{\rho} 
  \left[ 
   K 
   - \frac{\theta_v^{\prime}}{\overline{\theta_v}} g z 
   + \Dinv{2}  
     \left( 
       \frac{\langle c_p \rangle_0 \overline{\theta_v} \Pi^{\prime}}
            {\overline{\langle c_s \rangle}}
     \right)^2
  \right] \right\} dV
   \nonumber \\
&=&
   - \int K \DP{}{t} \left( \rho_d^{\prime} + \rho_v^{\prime} \right) dV
%
 + \int \overline{\rho} \Dvect{u} \cdot \Dvect{D} dV
 + \int K M_{src} (\rho_v^{\prime}) dV
  \nonumber \\
& &
 + \int \langle c_p \rangle_0 \overline{\rho} \overline{\theta_v} \Pi^{\prime}
   \nonumber \\
 & &
   \Bigg[ 
    \Dinv{ \overline{\theta} }  
     \left\{ \Dinv{\overline{\Pi}} ( Q_{cond} + Q_{rad} + Q_{dis} ) 
     + D_{\theta} \right\} 
   \nonumber \\
 & &
  + \Dinv{ 1 + \sum \left( \frac{R_v}{R_d} - 1 \right) \overline{q_v} } 
  \Bigg\{ 
   - \sum ( D_{q_v} + D_{q_c} + D_{q_r} ) 
    + \sum \frac{R_v}{R_d} 
      \left[  M_{src} (q_v) + D_{q_v} \right] 
  \Bigg\}
  \Bigg] dV
 \nonumber \\
 & &
  + \int \left[ 
    - \sum (q_v^{\prime} + q_c^{\prime} + q_r^{\prime} ) 
    + \sum (R_v / R_d) q_v^{\prime}
    \right]
   \nabla \cdot
   \left[
    \frac{ \overline{\rho} \Dvect{u} }
     { 1 + \sum \left( \frac{R_v}{R_d} - 1 \right) \overline{q_v} } 
   \right] dV
  \nonumber \\  
 & &
  - \int \theta^{\prime} g z 
    \nabla \cdot 
    \left[ \frac{\overline{\rho} \Dvect{u}}{\overline{\theta}} \right]
    dV
  \nonumber \\
 & &
  + \int \frac{\overline{\rho} g z}
       { 1 + \sum \left( \frac{R_v}{R_d} - 1 \right) \overline{q_v} } 
  \nonumber \\
 & &
   \Bigg[
    \left\{ w^{\prime} \sum \left( -1 + R_v/R_d \right) \DP{q_v}{z} \right\}
    + \left( \sum \frac{\overline{q_v} R_v}{\overline{\rho} R_d} \right)
      \left( \sum M_{fall} (\rho_r) \right)
  \nonumber \\
 & &
    + \sum (D_{q_v} + D_{q_c} + D_{q_r})
    - \sum R_v D_{q_v} / R_d
   \Bigg] dV
  \nonumber \\
 & &
  + \int \frac{\overline{\rho} g z}{\overline{\theta}}
    \left[ 
     w^{\prime} \DP{\theta}{z}
     - \Dinv{\overline{\Pi}}
       \left( Q_{cond} + Q_{rad} + Q_{dis} \right) - D_{\theta} \right]
     dV
  \Deqlab{quasi-equations:u-w-pi-theta:energy}
\end{eqnarray}
%
\end{description}
