% ɽ��   dcpam5  �ѱ��ѥ�᥿�ꥼ�������  ������ήĴ��
%
% ���� 
%\Drireki{2006/10/27 ��������}
%\Drireki{2008/03/27 ��������}
%\Drireki{2010/06/10 �ⶶ˧��}
%
%

\section{������ήĴ��}

\subsection{Υ��ɽ��}

�����Ǥ�, ������ήĴ�� (e.g., Manabe et al., 1965) ���꼰���ˤĤ��Ʋ��⤹��. 
�ʤ�, ������ήĴ����꼰����, ��������ʤ��Ȥ������β���, ������ήĴ���
�������ưפ�Ƴ�ФǤ��뤿��, �����˼������ϴ�����ήĴ��β���ˤ�ʤäƤ���. 

��ήĴ��Ǥ�, Ϣ³���� 2 �Ĥ��ؤˤ�����, ���ξ�郎
������������Ĵ���Ԥ�.
%
\begin{enumerate}
 \item ���ؤȾ��ؤμ�����Ū���ͥ륮���κ������ͤ���礭�� (���ؤμ�����Ū���ͥ륮�������ؤΤ�������礭�� (���ٸ�Ψ��������Ǯ��Ψ�����礭��)), 
\footnote
{
  ñ��ˤ�, �������ͤϥ����Ǥ���. 
  ������, �ºݤˤϥ�ǥ�ʻҴֳ���ǲ��١����٤�ʬ�ۤ����뤳�Ȥ��ͤ����, 
  �ʻҤ�ʿ�ѥ��ͥ륮�����������ʾ�Ǥ��äƤ�, �ʻ���Ǻ��礬������
  ���Ȥ����������. 
}. 
 \item ���м��٤����Ͱʾ�
\footnote
{
  ñ��ˤ�, �ŷ뤬���������м��٤����ͤ� 1 �Ǥ���. 
  ������, �ºݤˤϥ�ǥ�ʻҴֳ���Ǽ��٤�ʬ�ۤ����뤳�Ȥ��ͤ����, 
  �ʻҤ�ʿ�����м��٤� 1 �ʲ��Ǥ��äƤ�, �ʻ���Ƕŷ뤬������
  ���Ȥ����������. 
}. 
\end{enumerate}
%
������, Υ������������ɽ������Ȳ��Τ褦��ɽ�蘆���. 
%
\begin{eqnarray}
  C_p \hat{T}_{k} + L q^*(\hat{T}_{k}) + g \hat{z}_{k}
   - \left( C_p \hat{T}_{k+1} + L q^*(\hat{T}_{k+1}) + g \hat{z}_{k+1} \right) &>& C_p \Delta T_c,
\\
  \frac{ \hat{q}_{k  } }{ q^*(\hat{T}_{k  }, p_{k  }) } &\ge& r_c,
\\
  \frac{ \hat{q}_{k+1} }{ q^*(\hat{T}_{k+1}, p_{k+1}) } &\ge& r_c
\end{eqnarray}
%
������, $\hat{}$ ��Ĵ�������ͤ�ɽ��. 
�ޤ�, $C_p \Delta T_c$ ���԰��꤬�����뼾����Ū���ͥ륮���������ͤǤ���, 
$r_c$ �϶ŷ뤬���������м��٤����ͤǤ���. 

Ĵ���������������, 
%
\begin{eqnarray}
  \left\{ C_p \hat{T}_{k} + L \hat{q}_{k} \right\} 
    \frac{p_{k-\frac{1}{2}} - p_{k+\frac{1}{2}}}{g}
   + \left\{ C_p \hat{T}_{k+1} + L \hat{q}_{k+1} \right\} 
    \frac{p_{k+\frac{1}{2}} - p_{k+\frac{3}{2}}}{g} \nonumber
\\
  = 
  \left\{ C_p T_{k} + L q_{k} \right\} 
    \frac{p_{k-\frac{1}{2}} - p_{k+\frac{1}{2}}}{g}
   + \left\{ C_p T_{k+1} + L q_{k+1} \right\} 
    \frac{p_{k+\frac{1}{2}} - p_{k+\frac{3}{2}}}{g}
\end{eqnarray}
%
\begin{eqnarray}
  C_p T_{k} + L q_{k} + g z_{k}
   = C_p T_{k+1} + L q_{k+1} + g z_{k+1}
  \label{eq:mca:neutralstrat}
\end{eqnarray}
%
\begin{eqnarray}
  q_{k  } &=& q^*(T_{k  }, p_{k  })
\\
  q_{k+1} &=& q^*(T_{k+1}, p_{k+1})
\end{eqnarray}
%
�Ǥ���. 

������, (\ref{eq:mca:neutralstrat}) ���ſ尵ʿ�դμ����Ѥ������������, 
%
\begin{eqnarray}
  C_p ( T_{k} - T_{k+1} ) 
    + L ( q^*(T_{k}) - q^*(T_{k+1}) )
    - \frac{ R T_{k+\frac{1}{2}} } { p_{k+\frac{1}{2}} } \left( p_{k} - p_{k+1} \right)
    = 0
\end{eqnarray}
%
�Ȥʤ�. 
�������ä�, ... �Τ���ʤ�ϢΩ�켡��������򤱤��ɤ�. 
%
�ʤ�, $T_{k+\frac{1}{2}}$ ��
%
\begin{eqnarray}
  T_{k+\frac{1}{2}} = \frac{T_{k} + T_{k+1}}{2}
\end{eqnarray}
%
%�⤷����, 
%%
%\begin{eqnarray}
%  T_{k+\frac{1}{2}} &=& \frac{\Delta p_{k} T_{k} + \Delta p_{k+1} T_{k+1}}{ \Delta p_{k} + \Delta p_{k+1} }
%\\
%  \Delta p_{k} &=& p_{k-\frac{1}{2}} - p_{k+\frac{1}{2}}
%\end{eqnarray}
%%
��ɽ�����뤳�Ȥˤ���.

������, $q_{k}$, $q_{k+1}$ ��ƥ��顼Ÿ����, 
%
\begin{eqnarray}
  q_{k} &=& q^*(T_{k}, p_{k})
%
      = q^*(\hat{T}_{k}, p_{k}) 
         + \left. \frac{\partial q^*}{\partial T}\right|_{T=\hat{T}_{k}} \Delta T_{k}
\\
  q_{k+1}   &=& q^*(T_{k+1}, p_{k+1})
%
      = q^*(\hat{T}_{k+1}, p_{k+1}) 
         + \left. \frac{\partial q^*}{\partial T}\right|_{T=\hat{T}_{k+1}} \Delta T_{k+1}
\\
  \Delta T_{k  } &=& T_{k  } - \hat{T}_{k  }
\\
  \Delta T_{k+1} &=& T_{k+1} - \hat{T}_{k+1}
\end{eqnarray}
%
�Ȥ���ϢΩ�켡��������򤯤�, ���β�������. 
%
\begin{eqnarray}
  \Delta T_{k} 
    &=&   \left\{ \Delta p_{k} \left( 1 + \gamma_{k} \right) \right\}^{-1}
          \left\{
            \frac{L}{C_p} \Delta Q
            - \Delta p_{k+1} \left( 1 + \gamma_{k+1} \right) \Delta T_{k+1}
          \right\}
\\
  \Delta T_{k+1} 
    &=& \left[
          F_{k+\frac{1}{2}}
            \left\{
                \Delta p_{k  } \left( 1 + \gamma_{k  } \right)
              - \Delta p_{k+1} \left( 1 + \gamma_{k+1} \right)
            \right\} 
         + \left( 1 + \gamma_{k} \right) \left( 1 + \gamma_{k+1} \right) 
              \left( \Delta p_{k} + \Delta p_{k+1} \right)
        \right]^{-1} \nonumber
\\
    & &
        \left[
          \Delta p_{k} \left( 1 + \gamma_{k} \right) S_{k+\frac{1}{2}}
          + \left\{
              1 + \gamma_{k} - F_{k+\frac{1}{2}}
            \right\}
            \frac{L}{C_p} \Delta Q
        \right]
\\
  \Delta p_k &=& p_{k-\frac{1}{2}} - p_{k+\frac{1}{2}}
\\
  F_{k+\frac{1}{2}}
    &=&   \frac{R}{C_p}\frac{p_{k} - p_{k+1}}{2 p_{k+\frac{1}{2}}}
\\
  S_{k+\frac{1}{2}} 
    &=&   \hat{T}_{k} - \hat{T}_{k+1}
        + \frac{L}{C_p} \left\{ q^*(\hat{T}_{k},p_{k}) - q^*(\hat{T}_{k+1},p_{k+1}) \right\}
        - F_{k+\frac{1}{2}} \left( \hat{T}_{k} + \hat{T}_{k+1} \right)
\\
  \Delta Q
    &=&       \Delta p_{k}
                \left\{ \hat{q}_{k  } - q^*(\hat{T}_{k  },p_{k  }) \right\} 
            + \Delta p_{k+1}
                \left\{ \hat{q}_{k+1} - q^*(\hat{T}_{k+1},p_{k+1}) \right\} 
\\
  \gamma_{k} &=& \frac{L}{C_p} \left.\frac{\partial q^*}{\partial T}\right|_{T=\hat{T}_{k}}
\end{eqnarray}
%

�ºݤˤ�, �嵭�β�� $q_{k}$, $q_{k+1}$ ��ƥ��顼Ÿ�����Ƶ�᤿�����Ǥ����ʤ�, 
���Τˤ� ... ���������Ƥ��ʤ�. 
�����, �嵭���꼰����, $k$ ���ܤ��ؤ� $k+1$ ���ܤ��ؤκ����ɽ�����Ƥ������
�Ǥ��뤬, �ºݤˤ� 3 �ذʾ���ؤˤ錄�뺮��ⵯ���ꤨ��. 
������, �嵭��Ĵ����٤������֤��Ԥ����Ȥ�, ������Ĵ�ᤷ�Ƥ���. 


�ʤ�, �߿��̤�, 
%
\begin{eqnarray}
  P &=& - \frac{1}{2 \Delta t}
            \sum_{k=k_{max}}^{1} 
              \frac{ p_{k-\frac{1}{2}} - p_{k+\frac{1}{2}} }{ g } 
                \Delta q_{k} \nonumber
\\
    &=& - \frac{1}{2 \Delta t}
            \sum_{k=k_{max}}^{1} 
              \frac{ p_{k-\frac{1}{2}} - p_{k+\frac{1}{2}} }{ g } 
                \left\{ \left( q_{k} \right)_{l_{max}+1} - \hat{q}_{k} \right\}
\end{eqnarray}
%
�Ǥ���
\footnote
{
  ������, ��ľ�������¤Ͼ��ؤ��鲼�ؤ˸������¤��뤳�Ȥˤ��Ƥ���. 
  �����, ���ؤ������ŷ��̤����ʤ�����Ǥ���. 
}.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% �ʲ����Τ�ʸ��
% ����λ��ͤΤ���˻Ĥ��Ƥ���.

\iffalse

\subsection{�Ϥ����}

Ϣ³���� 2 �ĤΥ�٥�δ֤��ؤˤ�����, ���ξ�郎
������������Ĵ���Ԥ�.
%
\begin{enumerate}
 \item ���ٸ�Ψ��������Ǯ��Ψ�����礭��. 
 \item ˰�¤⤷���ϲ�˰��. 
\end{enumerate}
%

\subsection{����������ʤ��Ȥ��������Ԥ����}

�嵭�ξ�� (1) �˴ؤ���, ����ȥ��ԡ� $S$ �ι����Ѳ��˴ؤ�����
%
\begin{align}
 \DD{S}{z} < 0
\end{align}
%
��, �ֿ���������ʤ��פȤ���������Ѥ��ƽ񤭤������
%
\begin{align}
 \DP{T}{z} - \frac{RT}{C_p p} \DP{p}{z} + \frac{L}{C_p} \DP{q^*}{z} < 0
\end{align}
%
�Ȥʤ�. 
������, $C_p$ �ϴ����絤���갵��Ǯ, $L$����Ǯ, $q^{*}$��
˰���漾�Ǥ���. 

�嵭�ξ�� (2) �Ϥ��Τޤ��Ѥ���. 

�������Ѥ��Ʋ��٤��漾��Ĵ�᤹��Τ����Υ�ǥ�ǤΥǥե���Ȥ�
������ήĴ�᥹������Ǥ���.
%
�ʲ�, ����������꼰����������Ԥ�
\footnote{
 �ʲ��Ϻ�ʬˡ�Ⱥ����ä��äˤʤäƤ��ޤäƤ���Τ�, 
 ���Ȥ�Ϣ³�Ϥ��ä�Υ���Ϥ��äȤ�ʬ����ɬ�פ�����. 
}.

\subsection{���٤��漾��Ĵ���̤η׻���ˡ}

�漾�Ȳ��٤�, $(\hat{q}, \hat{T})$ ���� $(q, T)$ ��
Ĵ�᤹���ΤȤ���.

��Ｐ�ϰʲ����̤�Ǥ���.
\begin{align}
 q_{k-1} & = q^* (T_{k-1}, p_{k-1}), \\
 q_{k} & = q^{*} (T_{k}, p_{k}), 
\end{align}
%
\vspace*{-5ex}
%
\begin{align}
 \Deqlab{���ٸ�Ψ�˴ؤ����Ｐ}
     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{align}
%
\vspace*{-5ex}
%
\begin{align}
 \Deqlab{Ĵ������Ǥ���¸��}
    (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{align}
%
��ϰʲ��Τ褦�ˤʤ�. 
%
\begin{itemize}
\item  $q_{k}$, $q_{k-1}$ ($q \ll 1$ �ϻȤäƤ��ʤ�)

�漾�˴ؤ��Ƥ�, ��˰�¤ξ��ˤϰʲ��Τ褦��˰�¾��֤�Ĵ�᤹��.
%
\begin{align}
 \Deqlab{�ؾ�ü���漾��Ĵ��}
 q_{k} &= q^{*} (\hat{T}_{k},p_{k}) 
  + \left. \DP{q^{*}}{T} \right|_{\hat{T}_{k}}
    \Delta T_{k}, \\
 \Deqlab{�ز�ü���漾��Ĵ��}
 q_{k-1} &= q^{*} (\hat{T}_{k-1},p_{k-1}) 
  + \left. \DP{q^{*}}{T} \right|_{\hat{T}_{k-1}}
    \Delta T_{k-1}. 
\end{align}
%
������,
%
\begin{align}
 \Deqlab{�ؾ�ü�Φ�T�����}
  \Delta T_{k}   &= T_{k}   - \hat{T}_{k}   , \\
 \Deqlab{�ز�ü�Φ�T�����}
  \Delta T_{k-1} &= T_{k-1} - \hat{T}_{k-1} 
\end{align}
%
�Ǥ���. 
������Ƴ�Ф˴ؤ��Ƥϰʲ��򻲾ȤΤ���. 

\item $\Delta T_{k-1}$ ($q \ll 1$ �ϻȤäƤ��ʤ�)

\Deqref{Ĵ������Ǥ���¸��}, \Deqref{�ؾ�ü���漾��Ĵ��}, 
\Deqref{�ز�ü���漾��Ĵ��}���,
%
\begin{align}
 \Deqlab{�ز�ü�β����Ѳ�����뤿��η׻�1}
      (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 \\
%
    \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} \hspace{-7em} & \nonumber \\
 & \hspace{-7em} 
   = - \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}
      \hspace{-7em} & \nonumber \\
 & \hspace{-7em}
   = - \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}
\end{align}
%
�Ǥ���. ������
%
\begin{align}
 \gamma_{k} \equiv \frac{L}{C_p} \left. \DP{q^{*}}{T} \right|_{k}
\end{align}
%
�Ȥ�����, 
%
\begin{align}
 & \left\{  1 + \gamma_{k-1} \right\} 
    \Delta T_{k-1} \Delta p_{k-1}
     \nonumber \\
 & \hspace{3em} = 
     - \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{align}
%
�Ȥʤ�.
������,
%
\begin{align}
 \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{align}
%
�Ȥ���, $\Delta T_{k-1}$ �ˤĤ��Ʋ򤱤�, 
%
\begin{align}
 \Deqlab{�ز�ü�Φ�T�η��}
       \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{align}
%
�Ȥʤ�. 
$\Delta T_{k}$ ��Ƴ�ФˤĤ��Ƥϰʲ��򻲾ȤΤ���. 

\item $\Delta T_{k}$

%��Ǯ���
%\begin{eqnarray}
% \DP{S}{z} = 0
%\end{eqnarray}
%�μ���� (�ǽ�� $q (T_{k-1}, p_{k-1})$ �ä�
%$q^* (T_{k-1}, p_{k-1})$ �Ȥ���Τ�������???),

\Deqref{���ٸ�Ψ�˴ؤ����Ｐ}, 
\Deqref{�ؾ�ü���漾��Ĵ��}, \Deqref{�ز�ü���漾��Ĵ��}, 
\Deqref{�ؾ�ü�Φ�T�����}, \Deqref{�ز�ü�Φ�T�����} ���, 
%
\begin{align}
 \Deqlab{���ٸ�Ψ�˴ؤ����Ｐ���ѷ�1}
&   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{align}
%
������, 
%
\begin{align}
 \Deqlab{�����԰���ξ�Ｐ1}
 S_{k-1/2} \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{align}
%
�Ȥ���. ����ȼ����԰���ξ���
%
\begin{align}
 \Deqlab{�����԰���ξ�Ｐ2}
 S_{k-1/2} > 0 
\end{align}
%
��ɽ�����.
\Deqref{�����԰���ξ�Ｐ1} ���Ѥ���
\Deqref{���ٸ�Ψ�˴ؤ����Ｐ���ѷ�1} ���ѷ������, 
%
\begin{align}
& \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} = - S_{k-1/2}, 
           \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} 
= - S_{k-1/2}
\end{align}
%
�Ȥʤ�. ������\Deqref{�ز�ü�Φ�T�η��}���Ѥ����, 
%
\begin{align}
& 
  \left( - 1 - \gamma_{k}  
         - \frac{R}{C_p}
           \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}}
  \right)
  \Delta T_{k}
                 \nonumber \\
& \hspace{2em}
 + \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 \\
& \hspace{4em}
= - S_{k-1/2} 
\end{align}
%
$\kappa = R/C_p$ ��Ȥ���
\footnote{��ä�������Ȥ��٤�??  (�����ɤȤ��б��ˤ�äƤ�$\kappa$��
       �Ȥ�ʤ������褤�Τ���)},
%
\begin{align}
& 
  \left( - 1 - \gamma_{k}  
         - \kappa
           \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}}
  \right)
  \Delta T_{k}
                 \nonumber \\
& \hspace{2em}
 + \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 \\
& \hspace{4em}
= - S_{k-1/2} 
\end{align}
%
�Ȥʤ�. $\Delta T_{k}$�˴ؤ������դˤޤȤ���, 
%
\begin{align}
& 
  \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 \\
& \hspace{2em}
= - S_{k-1/2} 
  - \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{align}
%
�Ȥʤ�. $\Delta T_{k}$ �ˤĤ��Ʋ򤯤�, 
%
\begin{align}
& 
   \Delta T_{k}
                 \nonumber \\
&
= \frac{ \displaystyle
  - S_{k-1/2} 
  - \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}}
  }{ \displaystyle
      \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{ \displaystyle
   S_{k-1/2} 
  + \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}}
  }{ \displaystyle
       \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{ \displaystyle
    (1 + \gamma_{k-1}) \Delta p_{k-1} S_{k-1/2} 
  + \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} 
  }{ \displaystyle
       \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{ \displaystyle
    (1 + \gamma_{k-1}) \Delta p_{k-1} S_{k-1/2} 
  + \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} 
  }{ \displaystyle
       \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{align}
%
��������. 
\end{itemize}

% dcpam5 �Ǥ�̤Ƴ���Τ���, Ƴ�������ޤǥ����ȥ����� 2008/03/25  ��������
%
%
%\subsection{����������ʤ��Ȥ�������򤷤ʤ����}
%
%$l$ ������ξ��,
%\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}
%���� $q$ ��Ȥäƽ񤭴�����.
%�����ѷ������
%\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}
%������, $\DD{S}{z} =0$ �Ȥʤ����
%\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}
%
%$dS=0$ �μ���ɤΤ褦�ʷ��ˤ���Τ� best �ʤΤ���
%�褯�狼��ʤ�.
%�Ȥꤢ����, �������ưפ��ʤȻפä�ʬ�������ʧ�ä�
%���ˤ��Ƥߤ�.
%\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}
%���μ���ξ�դ� $T^2(1-q)^2$ �򤫤���.
%\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}
%���� $dz$ �򤫤����
%\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}
%
%����򤻤���, ʬ���ʧ�ä����μ��򤽤Τޤ�Υ��������.
%
%\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}
%������, 
%\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}
%�Ȥ����(����, �������ɤ��ʤ��Τ�����. $T_{k-1/2}$ �ˤĤ��Ƥ�
%Arakawa and Suarez (1983) ����������ּ���Ȥ��٤��ʤ褦�ʵ�������.
%������, agcm5 �����, ���֥롼����ΰ������Ѥ���Τ������ä��Τ�
%�������Ƥ���. dcpam �Ǥϥ��֥롼������� $T_{k-1/2}$ ����
%�ΤǤ��ɤ����⤷��ʤ�), 
%\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}
%��Ǯ����ʸ���ˤʤä���ä�...
%�ǽ餫�� $L$ �ˤ��Ƥ����٤�.
%
%$\Delta T_{k-1}$ �ʤɤ�Ȥäƽ񤭴�����.
%\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}
%
%������, �ʲ����ѿ�ã��Ƴ������.
%\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}
%
%�����ε�����Ѥ���, �����μ���񤭴�����Ȱʲ��Τ褦��
%�ʤ�.
%\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}
%���μ���ޤȤ�˲򤯤��Ȥ����ѤʤΤ�, ���򤨤��������.
%$\Delta$ �� 2 �İʾ夫���ä����̵�뤹�뤳�Ȥˤ���.
%�����餯��1 ������פȸ��ä��ɤ��Τ�����, �Ȥ�
%�פäƤ��뤬, ���ζ�����������˴ؤ��Ƹ��ʳ��ǤϤޤä���
%��Ƥ���Ƥ��ʤ�.
%
%����Ÿ�����Ĥġ�2 ���ʾ�ι�פ�缡̵�뤷�Ƥ�����, �ʲ��Τ褦�ˤʤ�.
%\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}
%������ 1 ���Ÿ������.
%\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}
%���μ��� $\Delta T_{k-1}$ �ι�� $\Delta T_{k}$ �ι��
%�ޤȤ�Ƥ���. �ޤ��Ф餹.
%\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}
%�Ĥ���, �ޤȤ��.
%\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}
%������, �ʲ��Τ褦���ѿ���ޤȤ�� (���� $S_t$ �ȤϤ������б�
%���Ƥ�����������???).
%\begin{eqnarray}
%S_t &\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}
%������, 
%\begin{eqnarray}
% B \Delta T_{k-1} + C \Delta T_{k} = S_t
%\end{eqnarray}
%�Ȥʤ�.
%������, $\sum h =0$ ���������
%\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}
%�����������
%\begin{eqnarray}
% B \left( \alpha \Delta T_{k} + \beta \right)  + C T_{k} = S_t
%\end{eqnarray}
%�����, $\Delta T_{k}$ �ˤĤ��Ʋ򤱤�
%\begin{eqnarray}
%   T_{k} = \frac{S_t - \beta B }{C + \alpha B}
%\end{eqnarray}

\fi
