% 表題 準圧縮方程式系のエネルギー方程式の導出
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% 履歴: 2011/12/26  小高正嗣(deepconv/arare5 定式化マニュアルに取り込み)
% 履歴: 2011/12/19  黒田美紀(追記)
% 履歴: 2011/11/30  黒田美紀(追記)
% 履歴: 2011/11/29  黒田美紀(修正 and 追記)
% 履歴: 2011/11/28  黒田美紀(追記)
% 履歴: 2011/11/26  黒田美紀(追記 ((A.9) の変形から))
% 履歴: 2011/11/25  黒田美紀 (新規作成(北守修論の付録のトレース))
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準圧縮方定式におけるエネルギー方程式を導出する. ただしここでは乾燥大気
を想定し, 凝結に関する項を無視する. 

乾燥大気の場合, 準圧縮方程式の連続の式は
\begin{eqnarray}
  \DP{\rho}{t} + \Ddiv{\overline{\rho}\Dvect{u}} = 0
  \Deqlab{乾燥大気の連続の式}
\end{eqnarray}
である. この式より任意のスカラー量 $\phi$ に対し
\begin{eqnarray}
  \overline{\rho} \DD{\phi}{t}
  = \DP{\overline{\rho} \phi}{t}
   + \Ddiv{ (\overline{\rho} \phi \Dvect{u})}
   + \phi \DP{\rho}{t}
  \Deqlab{スカラーの保存則}
\end{eqnarray}
が成り立つ. この式から準圧縮方程式では$\phi$の時間微分を
フラックス形式で書くことができないことがわかる%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\footnote{
  $\partial \phi/\partial t + \Dvect{u} \cdot \nabla \phi = 0$ が
  成り立たないということ. }%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
.

水平方向の運動方程式\Deqref{quasi-equations:u-w-pi-theta:u},
鉛直方向の運動方程式\Deqref{quasi-equations:u-w-pi-theta:w}より, 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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{エネルギー方程式1}
\end{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
となる. 
ここで$K = (u^2 + w^2)/2$, $\Dvect{D} = (D_u, D_w)$とおいた%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\footnote{
%  個人的めも: 導出をどこかに書いておきたいなー. }%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
. 
以降では式\Deqref{エネルギー方程式1}の右辺第一項と右辺第三項を変形していく. 

まず式\Deqref{エネルギー方程式1}の右辺第三項は以下のように変形される. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
式\Deqref{A67}の右辺第二項を変形する. 
熱力学の式\Deqref{quasi-equations:u-w-pi-theta:theta}は, 
凝結がない場合, 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{align}
 \DP{\theta}{t}
  = - \Dvect{u} \cdot \nabla \theta - w \DP{\overline{\theta}}{z}
    + Q + D_{\theta}
  \Deqlab{凝結のない熱力学の式}
\end{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
と表される. 両辺に $\overline{\rho}$ をかけて, 
式\Deqref{乾燥大気の連続の式}を用いて変形すると, 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
となる. また式\Deqref{A67}の右辺第一項は, 式\Deqref{スカラーの保存則}より, 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
となる. よって式\Deqref{エネルギー方程式1}の右辺第三項は, 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
と変形される. 

式\Deqref{エネルギー方程式1}の右辺第一項は以下のように変形される. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
凝結がないときの圧力方程式\Deqref{quasi-equations:u-w-pi-theta:continue}
は, 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
と表される. よって式\Deqref{A71}は, 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
と書き換えられる. 

したがって, 式\Deqref{エネルギー方程式1}は, 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
となり, これをさらに整理すると, 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
となる. この両辺を全領域で積分し, 境界面を出入りする流れはないとすれば, 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
となる. これが準圧縮方程式のエネルギー方程式である.
左辺は全エネルギーの時間変化であり, $\{ \quad \}$ 内の第一項, 第二項, 第三項は
それぞれ運動エネルギー, 浮力による位置エネルギー, 弾性エネルギーである. 

右辺第一項, 第二項, 第三項, 第四項はそれぞれ運動量の拡散, 基本場の鉛直温
位勾配, 非断熱加熱(放射, 散逸), 熱の拡散によるエネルギー変化率である.
右辺最後の項は準圧縮方程式の場合スカラー量の時間微分がフラックス形式で
書けないことによることにより現れる.  この項の存在により準圧縮方程式では
強制項がない場合でもエネルギーが保存しない.


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