% 表題   AGCM5 第1部 数理モデル 雲
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% 履歴 
\Drireki{2006/10/27 石渡正樹}
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\section{湿潤対流調節}

\subsection{はじめに}

連続した 2 つのレベルの間の層において, 次の条件が
満たされる時調節を行う.
\begin{enumerate}
 \item 温度減率が湿潤断熱減率よりも大きい
 \item 飽和もしくは過飽和.
\end{enumerate}

\subsection{水蒸気が少ないという近似を行う場合}

上記の条件 (1) に関して, 
\begin{eqnarray}
 \DD{s}{z} < 0
\end{eqnarray}
の条件を, 「水蒸気が少ない」という近似をふんだんに
用いて書きかえると
\begin{eqnarray}
 \DP{T}{z} - \frac{RT}{c_p p} \DP{p}{z} + \frac{L}{c_p} \DP{q^*}{z} < 0
\end{eqnarray}
となる.

上記の条件 (2) に関しては, そのまま使う.

これらを用いて温度と比湿を調節するのが dcpam のデフォルトの
湿潤対流調節スキームである.

以下, スキームの定式化の説明を行う.
(差分法と混ざった話になってしまっているので, あとでちゃんと整理が
必要だとおもう).

\subsection{温度と比湿の調節量の計算方法}

比湿と温度を, $(\hat{q}, \hat{T})$ から $(q, T)$ へ
調節するものとする.

条件式は以下の通りである.
\begin{eqnarray}
 q_{k-1} &=& q^* (T_{k-1}, p_{k-1}), \\
 q_{k} &=& q^{*} (T_{k}, p_{k})
\end{eqnarray}
\begin{eqnarray}
  T_{k-1} - T_{k} 
+ \frac{L}{C_p} 
  \left\{ q^{*} (T_{k-1}, p_{k-1})  - q^{*} (T_{k}, p_{k}) \right\}
- \frac{R}{C_p} \frac{\Delta p_{k-1/2}}{p_{k-1/2}}
  \frac{T_{k-1} + T_{k}}{2}
=0
\end{eqnarray}
\begin{eqnarray}
    (c_p T_{k} + L q_{k}) \Delta p_{k}
  + (c_p T_{k-1} + L q_{k-1}) \Delta p_{k-1}
=   (c_p \hat{T}_{k} + L \hat{q}_{k}) \Delta p_{k}
  + (c_p \hat{T}_{k-1} + L \hat{q}_{k-1}) \Delta p_{k-1}
\end{eqnarray}

解は以下のようになる(で, 良いんだっけかな?)

\begin{itemize}
\item  $q_{k}$, $q_{k-1}$ ($q \ll 1$ は使っていない)

$\Delta T_{k}$, $\Delta T_{k-1}$ が求められたとすると, 
「飽和するべし」という条件から, 
\begin{eqnarray}
 q_{k} &=& q^{*} (\hat{T}_{k},p_{k}) 
  + \left. \DP{q^{*}}{T} \right|_{\hat{T}_{k}}
    \cdot \Delta T_{k}, \\
 q_{k-1} &=& q^{*} (\hat{T}_{k-1},p_{k-1}) 
  + \left. \DP{q^{*}}{T} \right|_{\hat{T}_{k-1}}
    \cdot \Delta T_{k-1}
\end{eqnarray}
によって計算できる.

\item $T_{k-1}$ ($q \ll 1$ は使っていない)

$\Delta T_{k}$ がわかったとすると, 
\begin{eqnarray}
 q_{k} &=& q^{*} (\hat{T}_{k},p_{k}) 
  + \left. \DP{q^{*}}{T} \right|_{\hat{T}_{k}}
    \cdot \Delta T_{k}
\end{eqnarray}
もわかる.

$\sum h =0$ より
\begin{eqnarray}
      (c_p T_{k-1} + L q_{k-1}) \Delta p_{k-1}
    - (c_p \hat{T}_{k-1} + L \hat{q}_{k-1}) \Delta p_{k-1}
&=& - (c_p T_{k} + L q_{k}) \Delta p_{k}
    + (c_p \hat{T}_{k} + L \hat{q}_{k}) \Delta p_{k}, \nonumber \\
%
    \left\{ \Delta T_{k-1} 
      + \frac{L}{c_p} (q_{k-1} - \hat{q}_{k-1} ) \right\}
    \Delta p_{k-1}
&=& - \left\{ \Delta T_{k} + \frac{L}{c_p} (q_{k} - \hat{q}_{k}) \right\}
    \Delta p_{k}, \nonumber
\end{eqnarray}
\begin{eqnarray}
&&    \left[   \Delta T_{k-1} 
           + \frac{L}{c_p} 
             \left\{   q^{*} (\hat{T}_{k-1} ) 
                     + \left. \DP{q^{*}}{T} \right|_{k-1}
                       \Delta T_{k-1}
                     - \hat{q}_{k-1}           
             \right\}
    \right] \Delta p_{k-1} \nonumber \\
&& = - \left[   \Delta T_{k} 
           + \frac{L}{c_p} 
             \left\{   q^{*} (T_{k}) 
                     + \left. \DP{q^{*}}{T} \right|_{k}
                       \Delta T_{k}
                     - \hat{q}_{k}
             \right\}
    \right] \Delta p_{k}, \nonumber \\
%
&&     \left\{  1+ 
                    \frac{L}{c_p}   \left. \DP{q^{*}}{T} \right|_{k-1}
       \right\} 
       \Delta T_{k-1} \Delta p_{k-1}
     + \frac{L}{c_p} 
             \left\{   q^{*} (\hat{T}_{k-1} ) 
                     - \hat{q}_{k-1}           
             \right\} 
      \Delta p_{k-1}
     \nonumber \\
&& = - \left[   1
           + \frac{L}{c_p} 
             \left. \DP{q^{*}}{T} \right|_{k}
       \right] \Delta T_{k}  \Delta p_{k}
     - \frac{L}{c_p}
       \left\{  q^{*} (\hat{T}_{k}) 
                     - \hat{q}_{k}
       \right\}
       \Delta p_{k}, \nonumber
\end{eqnarray}
ここで
\begin{eqnarray}
 \gamma_{k} \equiv \frac{L}{c_p} \left. \DP{q^{*}}{T} \right|_{k}
\end{eqnarray}
とおくと, 
\begin{eqnarray}
 &&    \left\{  1 + \gamma_{k-1} \right\} 
       \Delta T_{k-1} \Delta p_{k-1}
     \nonumber \\
&& = 
     - \left(  1
           + \gamma_{k}
       \right) \Delta T_{k}  \Delta p_{k}
     + \frac{L}{c_p} 
       \left[
           \left\{ \hat{q}_{k-1} - q^{*} (\hat{T}_{k-1} ) \right\} 
           \Delta p_{k-1}
         + \left\{ \hat{q}_{k} - q^{*} (\hat{T}_{k}) \right\}
           \Delta p_{k}
       \right]
\end{eqnarray}
となる.
ここで, 
\begin{eqnarray}
 \Delta \hat{Q} \equiv 
           \left\{ \hat{q}_{k-1} - q^{*} (\hat{T}_{k-1} ) \right\} 
           \Delta p_{k-1}
         + \left\{ \hat{q}_{k} - q^{*} (\hat{T}_{k}) \right\}
           \Delta p_{k}
\end{eqnarray}
とおき, $\Delta T_{k-1}$ について解けば, 
\begin{eqnarray}
 &&    \left\{  1 + \gamma_{k-1} \right\} 
       \Delta T_{k-1} \Delta p_{k-1}
 = 
     - \left(  1
           + \gamma_{k}
       \right) \Delta T_{k}  \Delta p_{k}
     + \frac{L}{c_p} \Delta \hat{Q}, \nonumber \\
%
 &&  \Delta T_{k-1} 
 = 
     - \frac{ 1 + \gamma_{k} }{ 1 + \gamma_{k-1}}
       \frac{\Delta p_{k}}{\Delta p_{k-1}}
       \Delta T_{k}
     + \frac{1}{ 1 + \gamma_{k-1}}
       \frac{L}{c_p} \Delta \hat{Q} \frac{1}{\Delta p_{k-1}}
\end{eqnarray}
となる.

\item $T_{k}$

断熱条件
\begin{eqnarray}
 \DP{S}{z} = 0
\end{eqnarray}
の式より (最初の $q (T_{k-1}, p_{k-1})$ って
$q^* (T_{k-1}, p_{k-1})$ とするのが正しい???), 
\begin{eqnarray}
&&   T_{k-1} - T_{k} 
 + \frac{L}{c_p}
   \left\{ q (T_{k-1}, p_{k-1}) - q^{*} (T_{k}, p_{k}) \right\}
 - \frac{R}{c_p}
   \frac{\Delta p_{k-1/2}}{p_{k-1/2}}
   \frac{T_{k-1} + T_{k}}{2}
= 0,  \nonumber \\
%
&&   \hat{T}_{k-1} + \Delta T_{k-1} 
 - \left( \hat{T}_{k}  + \Delta T_{k} \right)
 + \frac{L}{c_p}
   \left\{   q^{*} (\hat{T}_{k-1})  
           + \left. \DP{q^{*}}{T} \right|_{k-1} \Delta T_{k-1}
           - q^{*} (\hat{T}_{k})  
           - \left. \DP{q^{*}}{T} \right|_{k} \Delta T_{k}
    \right\}
     \nonumber \\
&& \qquad
 - \frac{R}{c_p}
   \frac{\Delta p_{k-1/2}}{p_{k-1/2}}
   \frac{\hat{T}_{k-1} + \Delta T_{k-1} + \hat{T}_{k} + \Delta T_{k}}{2}
= 0,  \nonumber \\
%
&&   \hat{T}_{k-1} - \hat{T}_{k}
 + \frac{L}{c_p}
   \left\{   q^{*} (\hat{T}_{k-1})  
           - q^{*} (\hat{T}_{k})  
    \right\}
 - \frac{R}{c_p}
   \frac{\Delta p_{k-1/2}}{p_{k-1/2}}
   \frac{\hat{T}_{k-1} + \hat{T}_{k} }{2}
     \nonumber \\
&& \qquad
 + \Delta T_{k-1} - \Delta T_{k}
 +  \frac{L}{c_p} \left. \DP{q^{*}}{T} \right|_{k-1} \Delta T_{k-1}
 - \frac{L}{c_p} \left. \DP{q^{*}}{T} \right|_{k} \Delta T_{k}
 - \frac{R}{c_p}
   \frac{\Delta p_{k-1/2}}{p_{k-1/2}}
   \frac{\Delta T_{k-1} + \Delta T_{k}}{2}
= 0.
\end{eqnarray}
ここで, 
\begin{eqnarray}
 St \equiv 
   \hat{T}_{k-1} - \hat{T}_{k}
 + \frac{L}{c_p}
   \left\{   q^{*} (\hat{T}_{k-1})  
           - q^{*} (\hat{T}_{k})  
    \right\}
 - \frac{R}{c_p}
   \frac{\Delta p_{k-1/2}}{p_{k-1/2}}
   \frac{\hat{T}_{k-1} + \hat{T}_{k} }{2}
\end{eqnarray}
とおくと, 
\begin{eqnarray}
&& \Delta T_{k-1} - \Delta T_{k}
 + \gamma_{k-1} \Delta T_{k-1}
 - \gamma_{k} \Delta T_{k}
 - \frac{R}{c_p}
   \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}}
   \Delta T_{k-1} 
 - \frac{R}{c_p}
   \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}}
   \Delta T_{k}
           \nonumber \\
&& \qquad = - St, 
           \nonumber \\
%
&& 
  \left( - 1 - \gamma_{k}  
         - \frac{R}{c_p}
           \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}}
  \right)
  \Delta T_{k}
 + \left(
       1 + \gamma_{k-1}
     - \frac{R}{c_p}
       \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}}
   \right)
   \Delta T_{k-1} 
= - St 
\end{eqnarray}
げー, ノートが間違っているような気がする.
分母に入っている $p_{k-1/2}$ が $p_{k-1}$ になってしまっている....

ここで $\Delta T_{k-1}$ の表式を代入する.
\begin{eqnarray}
&& 
  \left( - 1 - \gamma_{k}  
         - \frac{R}{c_p}
           \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}}
  \right)
  \Delta T_{k}
                 \nonumber \\
&&
 + \left(
       1 + \gamma_{k-1}
     - \frac{R}{c_p}
       \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}}
   \right)
\left( 
     - \frac{ 1 + \gamma_{k} }{ 1 + \gamma_{k-1}}
       \frac{\Delta p_{k}}{\Delta p_{k-1}}
       \Delta T_{k}
     + \frac{1}{ 1 + \gamma_{k-1}}
       \frac{L}{c_p} \Delta \hat{Q} \frac{1}{\Delta p_{k-1}}
\right)
                 \nonumber \\
&&
= - St 
\end{eqnarray}
$\kappa = \frac{R}{c_p}$ を使うと (もっと前から使おうよ...)
\begin{eqnarray}
&& 
  \left( - 1 - \gamma_{k}  
         - \kappa
           \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}}
  \right)
  \Delta T_{k}
                 \nonumber \\
&&
 + \left(
       1 + \gamma_{k-1}
     - \kappa
       \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}}
   \right)
\left( 
     - \frac{ 1 + \gamma_{k} }{ 1 + \gamma_{k-1}}
       \frac{\Delta p_{k}}{\Delta p_{k-1}}
       \Delta T_{k}
     + \frac{1}{ 1 + \gamma_{k-1}}
       \frac{L}{c_p} \Delta \hat{Q} \frac{1}{\Delta p_{k-1}}
\right)
                 \nonumber \\
&&
= - St 
\end{eqnarray}
この式を $\Delta T_{k}$ について解く.
\begin{eqnarray}
&& 
  \left[
      \left( - 1 - \gamma_{k}  \right)
         - \kappa
           \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}}
     + 
       \left(
            1 + \gamma_{k-1}
          - \kappa
            \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}}
       \right)
       \left( - \frac{ 1 + \gamma_{k} }{ 1 + \gamma_{k-1}} 
       \frac{\Delta p_{k}}{\Delta p_{k-1}}
       \right) 
   \right]
   \Delta T_{k}
                 \nonumber \\
&&
= - St 
  - \left(
            1 + \gamma_{k-1}
          - \kappa
            \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}}
       \right)
       \frac{1}{ 1 + \gamma_{k-1}}
       \frac{L}{c_p} \Delta \hat{Q} \frac{1}{\Delta p_{k-1}}
\end{eqnarray}
\begin{eqnarray}
&& 
   \Delta T_{k}
                 \nonumber \\
&&
= \frac{
  - St 
  - \left(
            1 + \gamma_{k-1}
          - \kappa
            \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}}
       \right)
       \frac{1}{ 1 + \gamma_{k-1}}
       \frac{L}{c_p} \Delta \hat{Q} \frac{1}{\Delta p_{k-1}}
  }{
      \left( - 1 - \gamma_{k}  \right)
         - \kappa
           \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}}
     + 
       \left(
            1 + \gamma_{k-1}
          - \kappa
            \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}}
       \right)
       \left( - \frac{ 1 + \gamma_{k} }{ 1 + \gamma_{k-1}} 
       \frac{\Delta p_{k}}{\Delta p_{k-1}}
       \right) 
  }
            \nonumber \\
%
&&
= \frac{
   St 
  + \left(
            1 + \gamma_{k-1}
          - \kappa
            \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}}
       \right)
       \frac{1}{ 1 + \gamma_{k-1}}
       \frac{L}{c_p} \Delta \hat{Q} \frac{1}{\Delta p_{k-1}}
  }{
       \left( 1 + \gamma_{k}  \right)
     + \kappa
       \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}}
     + 
       \left(
            1 + \gamma_{k-1}
          - \kappa
            \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}}
       \right)
       \frac{ 1 + \gamma_{k} }{ 1 + \gamma_{k-1}} 
       \frac{\Delta p_{k}}{\Delta p_{k-1}}
  }
            \nonumber \\
%
&&
= \frac{
    (1 + \gamma_{k-1}) \Delta p_{k-1} St 
  + \left(
            1 + \gamma_{k-1}
          - \kappa
            \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}}
       \right)
       \frac{L}{c_p} \Delta \hat{Q} 
  }{
       \left( 1 + \gamma_{k}  \right) (1 + \gamma_{k-1}) \Delta p_{k-1}
     + \kappa
       \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}}
       (1 + \gamma_{k-1}) \Delta p_{k-1}
     + 
       \left(
            1 + \gamma_{k-1}
          - \kappa
            \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}}
       \right)
       (1 + \gamma_{k})
       \Delta p_{k}
  }
            \nonumber \\
%
&&
= \frac{
    (1 + \gamma_{k-1}) \Delta p_{k-1} St 
  + \left(
            1 + \gamma_{k-1}
          - \kappa
            \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}}
       \right)
       \frac{L}{c_p} \Delta \hat{Q} 
  }{
       \left( 1 + \gamma_{k}  \right) (1 + \gamma_{k-1}) 
       (\Delta p_{k-1} + \Delta p_{k})
     + \kappa
       \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}}
       \left\{
          (1 + \gamma_{k-1}) \Delta p_{k-1}
        -  (1 + \gamma_{k}) \Delta p_{k}
       \right\}
  }
\end{eqnarray}
\end{itemize}


\subsection{水蒸気が少ないという近似をしない場合}

$l$ が一定の場合,
\begin{eqnarray}
     \DD{s}{t}
&=& - R_d \DD{}{t} (\ln p_d) 
  + \left(1 + \frac{q}{1-q} \right) c_p
    \DD{}{t} (\ln T) 
  + l \DD{}{t} \frac{r}{T}
                   \nonumber \\
&=& - R_d \DD{}{t} \left\{ \ln (p (1-q)) \right\}
  + \frac{1}{1-q} c_p
    \DD{}{t} (\ln T) 
  + l \DD{}{t} \left( \frac{1}{T} \frac{q}{1-q} \right)
                   \nonumber \\
\end{eqnarray}
全部 $q$ を使って書き換えた.
更に変形すると
\begin{eqnarray}
     \DD{s}{t}
&=& - R_d   \frac{1}{p (1-q)} \DD{}{t} \left\{ p (1-q) \right\}
  + \frac{1}{1-q} \frac{c_p}{T}
    \DD{T}{t}
  + l 
    \left[
           - \frac{1}{T^2} \DD{T}{t} \frac{q}{1-q}
        +    \frac{1}{T} \DD{}{t} \left(  \frac{q}{1-q} \right)
    \right]
\end{eqnarray}
これより, $\DD{S}{z} =0$ となる条件は
\begin{eqnarray}
  - \frac{R_d}{p (1-q)} \DD{}{z} \left\{ p (1-q) \right\}
  + \frac{1}{1-q} \frac{c_p}{T}
    \DD{T}{z}
  - \frac{l}{T^2} \frac{q}{1-q} \DD{T}{z} 
  + \frac{l}{T} \DD{}{z} \left(  \frac{q}{1-q} \right)
= 0
\end{eqnarray}

\vspace{5mm}

$dS=0$ の式をどのような形にするのが best なのかは
よくわからない.
とりあえず, 扱いが容易かなと思った分母を全部払った
形にしてみる.
\begin{eqnarray}
  - \frac{R_d}{p (1-q)} \DD{}{z} \left\{ p (1-q) \right\}
  + \frac{1}{1-q} \frac{c_p}{T}
    \DD{T}{z}
  - \frac{l}{T^2} \frac{q}{1-q} \DD{T}{z} 
  + \frac{l}{T}
    \frac{\DD{q}{z} (1-q) - q \DD{}{z}(1-q)}
         {(1-q)^2} 
= 0,
                \nonumber \\
  - \frac{R_d}{p (1-q)} \DD{}{z} \left\{ p (1-q) \right\}
  + \frac{1}{1-q} \frac{c_p}{T}
    \DD{T}{z}
  - \frac{l}{T^2} \frac{q}{1-q} \DD{T}{z} 
  + \frac{l}{T} \frac{1}{(1-q)} \DD{q}{z}
  + \frac{l}{T}  \frac{q}{(1-q)^2}  \DD{q}{z}
= 0.
\end{eqnarray}
この式の両辺に $T^2(1-q)^2$ をかける.
\begin{eqnarray}
  - \frac{R_d}{p} T^2 (1-q)  \DD{}{z} \left\{ p (1-q) \right\} 
  + c_p T (1-q)
    \DD{T}{z}
  - l q (1-q)  \DD{T}{z} 
  + l T (1-q) \DD{q}{z}
  + l T q \DD{q}{z}
= 0, 
    \nonumber \\
  - \frac{R_d}{p} T^2 (1-q)  \DD{}{z} \left\{ p (1-q) \right\} 
  + c_p T (1-q)
    \DD{T}{z}
  - l q (1-q)  \DD{T}{z} 
  + l T \DD{q}{z}
= 0, 
\end{eqnarray}
更に $dz$ をかければ
\begin{eqnarray}
  - \frac{R_d}{p} T^2 (1-q)  d \left\{ p (1-q) \right\} 
  + c_p T (1-q)
    dT
  - l q (1-q)  dT
  + l T dq
= 0, 
\end{eqnarray}

\vspace{5mm}

近似をせずに, 分母を払った形の式をそのまま離散化する.

\begin{eqnarray}
&&  - \frac{R_d}{p_{k-1/2}} T_{k-1/2}^2 (1-q_{k-1/2}) 
    \left[ p_{k-1} (1-q_{k-1}) - p_{k} (1-q_{k}) \right]
  + c_p (1-q_{k-1/2}) T_{k-1/2} 
    (T_{k-1} - T_{k})
                 \nonumber \\
&& \qquad
  - l q_{k-1/2} (1-q_{k-1/2})  (T_{k-1} - T_{k})
  + l T_{k-1/2} (q_{k-1} - q_{k})
= 0, 
\end{eqnarray}
ここで, 
\begin{eqnarray}
  T_{k-1/2}  &=& \frac{T_{k-1} + T_{k}}{2}, \\
  q_{k-1/2}  &=& \frac{q_{k-1} + q_{k}}{2}
\end{eqnarray}
とすると(これ, 本当は良くないのだろう. $T_{k-1/2}$ については
Arakawa and Suarez (1983) の正しい補間式を使うべきなような気がする.
しかし, agcm5 時代に, サブルーチンの引数を変えるのが嫌だったので
こうしている. dcpam ではサブルーチン内で $T_{k-1/2}$ を作る
のでも良いかもしれない), 
\begin{eqnarray}
&&  - \frac{R_d}{p_{k-1/2}} 
      \left( \frac{T_{k-1} + T_{k}}{2} \right)^2 
      \left( 1 - \frac{q_{k-1} + q_{k}}{2} \right) 
      \left[ p_{k-1} (1-q_{k-1}) - p_{k} (1-q_{k}) \right]
                 \nonumber \\
&& \qquad
  + c_p \left( 1 - \frac{q_{k-1} + q_{k}}{2} \right)
    \frac{T_{k-1} + T_{k}}{2}
    (T_{k-1} - T_{k})
                 \nonumber \\
&& \qquad
  - l \frac{q_{k-1} + q_{k}}{2} 
    \left( 1 - \frac{q_{k-1} + q_{k}}{2} \right)  (T_{k-1} - T_{k})
                 \nonumber \\
&& \qquad
  + l \frac{T_{k-1} + T_{k}}{2} (q_{k-1} - q_{k})
= 0, 
                 \nonumber \\
%
&&  - \frac{R_d}{c_p}  
      \left( \frac{T_{k-1} + T_{k}}{2} \right)^2 
      \left( 1 - \frac{q_{k-1} + q_{k}}{2} \right) 
      \left[   \frac{p_{k-1}}{p_{k-1/2}}  (1-q_{k-1}) 
             - \frac{p_{k}}{p_{k-1/2}}  (1-q_{k})
      \right]
                 \nonumber \\
&& \qquad
  + \left( 1 - \frac{q_{k-1} + q_{k}}{2} \right)
    \frac{T_{k-1} + T_{k}}{2}
    (T_{k-1} - T_{k})
                 \nonumber \\
&& \qquad
  - \frac{L}{c_p} \frac{q_{k-1} + q_{k}}{2} 
    \left( 1 - \frac{q_{k-1} + q_{k}}{2} \right)  (T_{k-1} - T_{k})
                 \nonumber \\
&& \qquad
  + \frac{L}{c_p} \frac{T_{k-1} + T_{k}}{2} (q_{k-1} - q_{k})
= 0
\end{eqnarray}
潜熱が大文字になっちゃった...
最初から $L$ にしておくべき.

$\Delta T_{k-1}$ などを使って書き換える.
\begin{eqnarray}
&&  - \frac{R_d}{c_p}  \frac{1}{4}
      \left( 
         \hat{T}_{k-1} + \Delta T_{k-1} + \hat{T}_{k} + \Delta T_{k} 
      \right)^2 
                 \nonumber \\
&& \qquad
      \times
      \left\{   1 
         - \frac{1}{2} 
           \left(   q^{*} (\hat{T}_{k-1}) 
                  + \left. \DP{q^{*}}{T} \right|_{k-1} \Delta T_{k-1} 
                  + q^{*} (\hat{T}_{k}) 
                  + \left. \DP{q^{*}}{T} \right|_{k} \Delta T_{k} \right)
           \right\}
                 \nonumber \\
&& \qquad
      \times
      \left[ \frac{p_{k-1}}{p_{k-1/2}} 
             \left( 1 - q^{*} (\hat{T}_{k-1}) 
                   - \left. \DP{q^{*}}{T} \right|_{k-1} \Delta T_{k-1} \right) 
             - \frac{p_{k}}{p_{k-1/2}}  \left( 1-q^{*} (\hat{T}_{k}) - \left. \DP{q^{*}}{T} \right|_{k} \Delta T_{k} 
              \right)
      \right]
                 \nonumber \\
&& \quad
  + \left\{ 1 - \frac{1}{2} \left( q^{*} (\hat{T}_{k-1}) + \left. \DP{q^{*}}{T} \right|_{k-1} \Delta T_{k-1} + q^{*} (\hat{T}_{k}) + \left. \DP{q^{*}}{T} \right|_{k} \Delta T_{k} \right) \right\}
                 \nonumber \\
&& \qquad
    \times
    \frac{1}{2} \left( \hat{T}_{k-1} + \Delta T_{k-1} + \hat{T}_{k} + \Delta T_{k} \right)
    (\hat{T}_{k-1} + \Delta T_{k-1} - \hat{T}_{k} - \Delta T_{k})
                 \nonumber \\
&& \quad
  - \frac{L}{c_p} \frac{1}{2} \left( q^{*} (\hat{T}_{k-1}) + \left. \DP{q^{*}}{T} \right|_{k-1} \Delta T_{k-1} + q^{*} (\hat{T}_{k}) + \left. \DP{q^{*}}{T} \right|_{k} \Delta T_{k} \right)
                 \nonumber \\
&& \qquad
    \times
    \left\{  1 - \frac{1}{2} \left( q^{*} (\hat{T}_{k-1}) + \left. \DP{q^{*}}{T} \right|_{k-1} \Delta T_{k-1} + q^{*} (\hat{T}_{k}) + \left. \DP{q^{*}}{T} \right|_{k} \Delta T_{k} \right) \right\}  
                 \nonumber \\
&& \qquad
    \times
    (\hat{T}_{k-1} + \Delta T_{k-1} - \hat{T}_{k} - \Delta T_{k})
                 \nonumber \\
&& \quad
  + \frac{L}{c_p} 
    \frac{1}{2}
    \left( \hat{T}_{k-1} + \Delta T_{k-1} + \hat{T}_{k} + \Delta T_{k} \right)
                 \nonumber \\
&& \qquad
    \times
    \left( q^{*} (\hat{T}_{k-1}) + \left. \DP{q^{*}}{T} \right|_{k-1} \Delta T_{k-1} - q^{*} (\hat{T}_{k}) - \left. \DP{q^{*}}{T} \right|_{k} \Delta T_{k} \right)
= 0
\end{eqnarray}

ここで, 以下の変数達を導入する.
\begin{eqnarray}
MM &\equiv& 
  1  - \frac{1}{2} q^{*} (\hat{T}_{k-1}) 
     - \frac{1}{2} q^{*} (\hat{T}_{k}), \\
D_{k-1} &\equiv& \left. \DP{q^{*}}{T} \right|_{k-1}, \\ 
D_{k-1} &\equiv& \left. \DP{q^{*}}{T} \right|_{k}, \\
TP &\equiv& \hat{T}_{k-1} + \hat{T}_{k}, \\
TM &\equiv& \hat{T}_{k-1} - \hat{T}_{k}, \\
M_{k-1} &\equiv& 1 - q^{*} (\hat{T}_{k-1}), \\ 
M_{k} &\equiv& 1 - q^{*} (\hat{T}_{k}), \\ 
F &\equiv& \frac{R_d}{c_p}, \\
E &\equiv& \frac{L}{c_p}, \\
QP &\equiv& q^{*} (\hat{T}_{k-1}) + q^{*} (\hat{T}_{k}), \\ 
QM &\equiv& q^{*} (\hat{T}_{k-1}) - q^{*} (\hat{T}_{k}), \\ 
P_{k-1} &\equiv& \frac{p_{k-1}}{p_{k-1/2}}, \\
P_{k} &\equiv& \frac{p_{k}}{p_{k-1/2}}
\end{eqnarray}

これらの記号を用いて, 先程の式を書き換えると以下のように
なる.
\begin{eqnarray}
&&  - \frac{F}{4}
      \left( TP + \Delta T_{k-1} + \Delta T_{k} 
      \right)^2 
      \left\{   MM 
              - \frac{1}{2} D_{k-1} \Delta T_{k-1} 
              - \frac{1}{2} D_{k} \Delta T_{k}
      \right\}
                 \nonumber \\
&& \qquad
      \times
      \left[   P_{k-1} \left( M_{k-1}  - D_{k-1} \Delta T_{k-1}  \right) 
             - P_{k} \left( M_{k} - D_{k} \Delta T_{k} \right)
      \right]
                 \nonumber \\
&& \quad
  + \frac{1}{2}
    \left\{ MM - \frac{1}{2} \left( D_{k-1} \Delta T_{k-1} 
               + D_{k} \Delta T_{k} \right) 
    \right\}
    \left( TP + \Delta T_{k-1} + \Delta T_{k} \right)
    (TM + \Delta T_{k-1} - \Delta T_{k})
                 \nonumber \\
&& \quad
  - \frac{E}{2} 
    \left( QP + D_{k-1} \Delta T_{k-1} + D_{k} \Delta T_{k} \right)
    \left\{  MM - \frac{1}{2} \left( D_{k-1} \Delta T_{k-1} + D_{k} \Delta T_{k} \right) \right\}  
                 \nonumber \\
&& \qquad
    \times
    (TM + \Delta T_{k-1} - \Delta T_{k})
                 \nonumber \\
&& \quad
  + \frac{E}{2}
    \left( TP + \Delta T_{k-1} + \Delta T_{k} \right)
    \left( QM + D_{k-1} \Delta T_{k-1} - D_{k} \Delta T_{k} \right)
= 0
\end{eqnarray}
この式をまともに解くことは大変なので, やむをえず近似する.
$\Delta$ が 2 つ以上かかった項を無視することにする.
おそらく「1 次近似」と言って良いのだろう, とは
思っているが, この近似の妥当性に関して現段階ではまったく
検討していない.

式を展開しつつ「2 次以上の項」を順次無視していくと, 以下のようになる.
\begin{eqnarray}
&&  - \frac{F}{4}
      \left\{ TP^2  + 2 TP \cdot (\Delta T_{k-1} + \Delta T_{k}) \right\}
      \left(   MM 
              - \frac{1}{2} D_{k-1} \Delta T_{k-1} 
              - \frac{1}{2} D_{k} \Delta T_{k}
      \right)
                 \nonumber \\
&& \qquad
      \times
      \left(   P_{k-1} M_{k-1}  - P_{k-1} D_{k-1} \Delta T_{k-1} 
             - P_{k} M_{k} + P_{k} D_{k} \Delta T_{k}
      \right)
                 \nonumber \\
&& \quad
  + \frac{1}{2}
    \left( MM  - \frac{1}{2} D_{k-1} \Delta T_{k-1} 
                - \frac{1}{2} D_{k} \Delta T_{k}
    \right)
    \left\{ TP \cdot TM  + (\Delta T_{k-1} + \Delta T_{k}) TM 
           + ( \Delta T_{k-1} - \Delta T_{k}) TP 
    \right\}
                 \nonumber \\
&& \quad
  - \frac{E}{2} 
    \left( QP + D_{k-1} \Delta T_{k-1} + D_{k} \Delta T_{k} \right)
                 \nonumber \\
&& \qquad
    \times
    \left\{   MM \cdot TM 
            + ( \Delta T_{k-1} - \Delta T_{k}) MM
            + \left(- \frac{1}{2} D_{k-1} \Delta T_{k-1} - \frac{1}{2} D_{k} \Delta T_{k}\right)TM  \right\}  
                 \nonumber \\
&& \quad
  + \frac{E}{2}
    \left\{   TP \cdot QM 
           +  \left(  D_{k-1} \Delta T_{k-1} - D_{k} \Delta T_{k} \right) TP
           + (\Delta T_{k-1} + \Delta T_{k}) QM 
    \right\}
= 0
\end{eqnarray}
\begin{eqnarray}
&&  - \frac{F}{4}
      \left\{   TP^2 \cdot MM 
              - \frac{1}{2} TP^2 
                \left(    
                     D_{k-1} \Delta T_{k-1} 
                  +  D_{k} \Delta T_{k}
                \right) 
              + 2 TP \cdot MM (\Delta T_{k-1} + \Delta T_{k}) 
      \right\}
                 \nonumber \\
&& \quad
      \times
      \left\{  \left( P_{k-1} M_{k-1} - P_{k} M_{k} \right)  
              - P_{k-1} D_{k-1} \Delta T_{k-1} 
              + P_{k} D_{k} \Delta T_{k}
      \right\}
                 \nonumber \\
&&
  + \frac{1}{2}
    \left\{  
         MM \cdot TP \cdot TM  
      + \left\{ 
            (\Delta T_{k-1} + \Delta T_{k}) TM 
          + ( \Delta T_{k-1} - \Delta T_{k}) TP 
        \right\} MM
                  \right.     \nonumber \\
&& \quad \left.
      - TP \cdot TM
        \left(   \frac{1}{2} D_{k-1} \Delta T_{k-1} 
               + \frac{1}{2} D_{k} \Delta T_{k} 
        \right) 
    \right\} 
                 \nonumber \\
&& 
  - \frac{E}{2} 
    \left\{   QP  \cdot MM \cdot TM 
            + MM \cdot QP ( \Delta T_{k-1} - \Delta T_{k}) 
            + TM \cdot  QP
              \left( - \frac{1}{2} D_{k-1} \Delta T_{k-1} 
                     - \frac{1}{2} D_{k} \Delta T_{k}
              \right)
            \right.     \nonumber \\
&& \quad \left.
            + MM \cdot TM (D_{k-1} \Delta T_{k-1} + D_{k} \Delta T_{k} )
    \right\}
                 \nonumber \\
&& 
  + \frac{E}{2}
    \left\{   TP \cdot QM 
           + (  TP \cdot D_{k-1} + QM ) \Delta T_{k-1}
           + ( - TP \cdot D_{k}  + QM )  \Delta T_{k}
    \right\}
= 0
\end{eqnarray}
更に第 1 項を展開する.
\begin{eqnarray}
&&  - \frac{F}{4}
      \left[   TP^2 \cdot MM \left( P_{k-1} M_{k-1} - P_{k} M_{k} \right)  
              + TP^2 \cdot MM
                \left\{  
                   - P_{k-1} D_{k-1} \Delta T_{k-1} 
                   + P_{k} D_{k} \Delta T_{k}
                \right\}
            \right.     \nonumber \\
&& \qquad \left.
              + \left( P_{k-1} M_{k-1} - P_{k} M_{k} \right)  
                \left\{   - \frac{1}{2} TP^2 
                \left(    
                     D_{k-1} \Delta T_{k-1} 
                  +  D_{k} \Delta T_{k}
                \right) 
              + 2 TP \cdot MM (\Delta T_{k-1} + \Delta T_{k}) 
                \right\}
      \right]
                 \nonumber \\
&&
  + \frac{1}{2}
    \left\{  MM \cdot TP \cdot TM  
           + \left\{ 
                  (\Delta T_{k-1} + \Delta T_{k}) TM 
                + ( \Delta T_{k-1} - \Delta T_{k}) TP 
             \right\} MM
            \right.     \nonumber \\
&& \qquad \left.
           - TP \cdot TM
             \left(   \frac{1}{2} D_{k-1} \Delta T_{k-1} 
               + \frac{1}{2} D_{k} \Delta T_{k} \right) 
    \right\} 
                 \nonumber \\
&&
  - \frac{E}{2} 
    \left\{   QP  \cdot MM \cdot TM 
            + MM \cdot QP ( \Delta T_{k-1} - \Delta T_{k}) 
            + TM \cdot  QP
              \left( - \frac{1}{2} D_{k-1} \Delta T_{k-1} 
                     - \frac{1}{2} D_{k} \Delta T_{k}
              \right)
            \right.     \nonumber \\
&& \qquad \left.
            + MM \cdot TM (D_{k-1} \Delta T_{k-1} + D_{k} \Delta T_{k} )
    \right\}
                 \nonumber \\
&&
  + \frac{E}{2}
    \left\{   TP \cdot QM 
           + (  TP \cdot D_{k-1} + QM ) \Delta T_{k-1}
           + ( - TP \cdot D_{k}  + QM )  \Delta T_{k}
    \right\}
= 0
\end{eqnarray}
この式を $\Delta T_{k-1}$ の項と $\Delta T_{k}$ の項に
まとめていく. まずばらす.
\begin{eqnarray}
&&  - \frac{F}{4}
      \left[   TP^2 \cdot MM \left( P_{k-1} M_{k-1} - P_{k} M_{k} \right)  
             - TP^2 \cdot MM P_{k-1} D_{k-1} \Delta T_{k-1} 
             + TP^2 \cdot MM P_{k} D_{k} \Delta T_{k}
            \right.     \nonumber \\
&& \qquad \left.
             - \frac{1}{2} \left( P_{k-1} M_{k-1} - P_{k} M_{k} \right)  TP^2 D_{k-1} \Delta T_{k-1} 
             - \frac{1}{2} \left( P_{k-1} M_{k-1} - P_{k} M_{k} \right)  TP^2  D_{k} \Delta T_{k}
            \right.     \nonumber \\
&& \qquad \left.
             + 2 \left( P_{k-1} M_{k-1} - P_{k} M_{k} \right)   TP \cdot MM \Delta T_{k-1} 
             + 2 \left( P_{k-1} M_{k-1} - P_{k} M_{k} \right)   TP \cdot MM \Delta T_{k} 
      \right]
                 \nonumber \\
&&
  + \frac{1}{2}
    \left\{   MM \cdot TP \cdot TM  
            + TM \cdot MM \Delta T_{k-1} + TM \cdot MM \Delta T_{k} 
            + TP \cdot MM \Delta T_{k-1} - TP \cdot MM \Delta T_{k} 
            \right.     \nonumber \\
&& \qquad \left.
            - TP \cdot TM  \frac{1}{2} D_{k-1} \Delta T_{k-1} 
            - TP \cdot TM \frac{1}{2} D_{k} \Delta T_{k}
    \right\} 
                 \nonumber \\
&&
  - \frac{E}{2} 
    \left\{   QP  \cdot MM \cdot TM 
            + MM \cdot QP \Delta T_{k-1} 
            - MM \cdot QP \Delta T_{k} 
            - TM \cdot  QP \frac{1}{2} D_{k-1} \Delta T_{k-1} 
            - TM \cdot  QP\frac{1}{2} D_{k} \Delta T_{k}
            \right.     \nonumber \\
&& \qquad \left.
            + MM \cdot TM D_{k-1} \Delta T_{k-1} 
            + MM \cdot TM D_{k} \Delta T_{k} 
    \right\}
                 \nonumber \\
&&
  + \frac{E}{2}
    \left\{   TP \cdot QM 
           + (  TP \cdot D_{k-1} + QM ) \Delta T_{k-1}
           + ( - TP \cdot D_{k}  + QM )  \Delta T_{k}
    \right\}
= 0
\end{eqnarray}
ついで, まとめる.
\begin{eqnarray}
&& - \frac{F}{4} TP^2 \cdot MM \left( P_{k-1} M_{k-1} - P_{k} M_{k} \right)  
              \nonumber \\
&& - \frac{F}{4}
     \left\{ 
       - TP^2 \cdot MM \cdot P_{k-1} \cdot D_{k-1} 
       + \left( P_{k-1} M_{k-1} - P_{k} M_{k} \right) 
         \left(- \frac{1}{2} TP^2 \cdot D_{k-1} + 2 TP \cdot MM \right) 
     \right\} \Delta T_{k-1} 
               \nonumber \\
&& - \frac{F}{4}
     \left\{
         TP^2 \cdot MM \cdot P_{k} \cdot D_{k} 
       + \left( P_{k-1} M_{k-1} - P_{k} M_{k} \right)
         \left( - \frac{1}{2} TP^2 \cdot  D_{k} + 2 TP \cdot MM \right)
      \right\} \Delta T_{k}
                 \nonumber \\
&& + \frac{1}{2} MM \cdot TP \cdot TM  
              \nonumber \\
&& + \frac{1}{2} 
     \left\{ 
         TM \cdot MM + TP \cdot MM 
       - \frac{1}{2} TP \cdot TM \cdot D_{k-1} 
     \right\} \Delta T_{k-1} 
              \nonumber \\
&& + \frac{1}{2} 
     \left\{ 
         TM \cdot MM - TP \cdot MM
       - \frac{1}{2} TP \cdot TM  \cdot D_{k}
     \right\} \Delta T_{k} 
                 \nonumber \\
&& - \frac{E}{2} QP  \cdot MM \cdot TM 
                 \nonumber \\
&& - \frac{E}{2}
     \left( 
         MM \cdot QP - \frac{1}{2} TM \cdot QP \cdot D_{k-1}
       + MM \cdot TM \cdot D_{k-1}
     \right) \Delta T_{k-1} 
                 \nonumber \\
&& - \frac{E}{2}
     \left( 
       - MM \cdot QP - \frac{1}{2} TM \cdot  QP \cdot D_{k}
       + MM \cdot TM \cdot D_{k}
     \right) \Delta T_{k} 
                 \nonumber \\
&& + \frac{E}{2} TP \cdot QM 
   + \frac{E}{2} ( TP \cdot D_{k-1} + QM ) \Delta T_{k-1}
   + \frac{E}{2} ( - TP \cdot D_{k} + QM ) \Delta T_{k}
= 0
\end{eqnarray}
ここで, 以下のように変数をまとめる (前の $St$ とはちゃんと対応
しているんだろうね???).
\begin{eqnarray}
St &\equiv& 
      \frac{F}{4} TP^2 \cdot MM \left( P_{k-1} M_{k-1} - P_{k} M_{k} \right)  
    - \frac{1}{2} MM \cdot TP \cdot TM  \nonumber \\
&&    + \frac{E}{2} QP  \cdot MM \cdot TM 
    - \frac{E}{2} TP \cdot QM                \\
%
B &\equiv& - \frac{F}{4}
     \left\{ 
       - TP^2 \cdot MM \cdot P_{k-1} \cdot D_{k-1} 
       + \left( P_{k-1} M_{k-1} - P_{k} M_{k} \right) 
         \left(- \frac{1}{2} TP^2 \cdot D_{k-1} + 2 TP \cdot MM \right) 
     \right\}
               \nonumber \\
&& + \frac{1}{2} 
     \left\{ 
         TM \cdot MM + TP \cdot MM 
       - \frac{1}{2} TP \cdot TM \cdot D_{k-1} 
     \right\}
              \nonumber \\
&& - \frac{E}{2}
     \left( 
         MM \cdot QP - \frac{1}{2} TM \cdot QP \cdot D_{k-1}
       + MM \cdot TM \cdot D_{k-1}
     \right)
   + \frac{E}{2} ( TP \cdot D_{k-1} + QM )
                  \\
C &\equiv& - \frac{F}{4}
     \left\{
         TP^2 \cdot MM \cdot P_{k} \cdot D_{k} 
       + \left( P_{k-1} M_{k-1} - P_{k} M_{k} \right)
         \left( - \frac{1}{2} TP^2 \cdot  D_{k} + 2 TP \cdot MM \right)
      \right\}
                 \nonumber \\
&& + \frac{1}{2} 
     \left\{ 
         TM \cdot MM - TP \cdot MM
       - \frac{1}{2} TP \cdot TM  \cdot D_{k}
     \right\}
                 \nonumber \\
&& - \frac{E}{2}
     \left( 
       - MM \cdot QP - \frac{1}{2} TM \cdot  QP \cdot D_{k}
       + MM \cdot TM \cdot D_{k}
     \right)
   + \frac{E}{2} ( - TP \cdot D_{k} + QM )
\end{eqnarray}
これより, 
\begin{eqnarray}
 B \Delta T_{k-1} + C \Delta T_{k} = St
\end{eqnarray}
となる.
ここで, $\sum h =0$ より得られる
\begin{eqnarray}
      \Delta T_{k-1} 
&=& - \frac{1 + \gamma_{k}}{1+\gamma_{k-1}} 
    \frac{\Delta p_{k}}{\Delta p_{k-1}} \Delta T_{k}
  + \frac{1}{1+\gamma_{k-1}} 
    \frac{L}{c_p}
    \Delta \hat{Q}
    \frac{1}{\Delta p_{k-1}} \\
&\equiv& \alpha \Delta T_{k} + \beta
\end{eqnarray}
を代入すると
\begin{eqnarray}
 B \left( \alpha \Delta T_{k} + \beta \right)  + C T_{k} = St
\end{eqnarray}
これを, $\Delta T_{k}$ について解けば
\begin{eqnarray}
   T_{k} = \frac{St - \beta B }{C + \alpha B}
\end{eqnarray}
