% ɽ��   dcpam5-primitive �ϳز����λ�����������Ƴ�� -- ��-��ɸ�ץ�ߥƥ���������
%
% ���� 
%\Drireki{2011/08/11 �ݹ�����}
%\Drireki{2008/06/15 ��������}
%\Drireki{1997/04/15 ���ٹ���}
%\Drireki{1994/04/13 ��������}
%

\section{$\sigma$-��ɸ�ץ�ߥƥ���������}

���ϳ�ʿ�դΤ�ȤǤ�, ���� $p$ �ϱ�ľ��ɸ $z$ ���Ф�ñĴ��������ؿ���
����. ������, ��ľ��ɸ�� $z$ ����, ��ɽ�̵��� $p_s$
�ǵ��ʲ�����������ɸ,
    \begin{align}
      \Deqlab{������}
      \sigma \equiv \frac{p}{p_s}
    \end{align}
���Ѵ�����. $\sigma$ �� $z$ �δط���, ���ϳ�ʿ�դμ�
\Deqref{����������}���ѷ�����������. 
    \begin{align} 
      \Deqlab{���������ϳ�}
      \DP{\sigma}{z} = - \frac{g \sigma}{R T}.
    \end{align} 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{$\sigma$-��ɸ�Ѵ�����}

$z$- ��ɸ���� $\sigma$- ��ɸ�ؤ��Ѵ������򼨤�. 

��ľ��ʬ
    \begin{align}
     \begin{split}
       \DP{\bullet}{z} 
        & = \DP{\sigma}{z} \DP{\bullet}{\sigma} \\
        & = - \frac{g \sigma}{R T} \DP{\bullet}{\sigma}.
     \end{split}    
    \end{align}
��ʿ��ʬ
    \begin{align}
     \Deqlab{������ʬ}
     \begin{split}
        \left( \DP{\bullet}{\lambda} \right)_z
       & =   \left( \DP{\bullet}{\lambda} \right)_{\sigma}
       - \DP{\sigma}{z} \DP{\bullet}{\sigma}
         \left( \DP{z}{\lambda} \right)_{\sigma}   \\
       & =   \left( \DP{\bullet}{\lambda} \right)_{\sigma}
       + \frac{g \sigma}{R T} \DP{\bullet}{\sigma}
         \left( \DP{z}{\lambda} \right)_{\sigma},
     \end{split}     
     \\ \nonumber \\
     \Deqlab{�ե�����ʬ}
     \begin{split}
         \left( \DP{\bullet}{\varphi} \right)_z
       & =   \left( \DP{\bullet}{\varphi} \right)_{\sigma}
       - \DP{\sigma}{z} \DP{\bullet}{\sigma}
         \left( \DP{z}{\varphi} \right)_{\sigma}   \\
       & =   \left( \DP{\bullet}{\varphi} \right)_{\sigma}
       + \frac{g \sigma}{R T} \DP{\bullet}{\sigma}
         \left( \DP{z}{\varphi} \right)_{\sigma}.
     \end{split}
    \end{align}
������ʬ
    \begin{align}
     \begin{split}
         \left( \DP{\bullet}{t} \right)_z
       & =   \left( \DP{\bullet}{t} \right)_{\sigma}
       - \DP{\sigma}{z} \DP{\bullet}{\sigma}
         \left( \DP{z}{t} \right)_{\sigma}  \\
       & =   \left( \DP{\bullet}{t} \right)_{\sigma}
       + \frac{g \sigma}{R T} \DP{\bullet}{\sigma}
         \left( \DP{z}{t} \right)_{\sigma}.
     \end{split}
    \end{align}
�饰��󥸥���ʬ�Ϥ������Ѥ���,
    \begin{align}
     \Deqlab{�饰��󥸥���ʬ�Һ�ɸ}
      \left( \DD{\bullet}{t} \right)_z
     & =   \left( \DP{\bullet}{t} \right)_z
      + \frac{u}{a \cos \varphi} \left( \DP{\bullet}{\lambda} \right)_z
      + \frac{v}{a} \left( \DP{\bullet}{\varphi} \right)_z
      + w \left( \DP{\bullet}{z} \right)_z \nonumber \\
     & =   \left( \DP{\bullet}{t} \right)_{\sigma}
      + \frac{u}{a \cos \varphi} \left( \DP{\bullet}{\lambda} \right)_{\sigma}
      + \frac{v}{a} \left( \DP{\bullet}{\varphi} \right)_{\sigma}
      \nonumber \\
     & \quad
       + \frac{g \sigma}{R T} \left\{ 
        \left( \DP{z}{t} \right)_{\sigma}
       + \frac{u}{a \cos \varphi} \left( \DP{z}{\lambda} \right)_{\sigma}
       + \frac{v}{a} \left( \DP{z}{\varphi} \right)_{\sigma}
       -w \right\} \DP{\bullet}{\sigma} \nonumber \\
     & =   \left( \DD{\bullet}{t} \right)_{\sigma}.
    \end{align}
������, $\sigma$-��ɸ��ľ®�� $\dot{\sigma}$ ���������. 
    \begin{align}
     \Deqlab{�ҥɥå����}
      \dot{\sigma} \equiv
        \frac{g \sigma}{RT} \left\{ 
        \left( \DP{z}{t} \right)_{\sigma}
       + \frac{u}{a \cos \varphi} \left( \DP{z}{\lambda} \right)_{\sigma}
       + \frac{v}{a} \left( \DP{z}{\varphi} \right)_{\sigma}
       -w \right\}.
    \end{align}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{$\sigma$-��ɸ�ץ�ߥƥ�����������}

\subsubsection{���ϳ�ʿ�դμ�}

\Deqref{���������ϳ�}����ϥݥƥ󥷥�� $\Phi=gz$ ���Ѥ��ƽ񤱤�,
    \begin{align}
      \DP{\Phi}{\sigma}=-\frac{R T}{\sigma}.
    \end{align}

\subsubsection{��ư������}

��ʿ�ΰ��ϸ��ۤ�, \Deqref{������ʬ}�����\Deqref{�ե�����ʬ} �� 
$p$ ���Ф���Ŭ�Ѥ�, \Deqref{������} ���Ѥ���м��Τ褦���Ѵ������. 
    \begin{align}
      \frac{1}{\rho} \left( \DP{p}{\lambda} \right)_z
     & =   \frac{1}{\rho} \left\{ \DP[][\sigma]{p}{\lambda}
       + \frac{g \sigma}{R T} \DP{p}{\sigma} \DP[][\sigma]{z}{\lambda}
       \right\} \nonumber \\
     & =   \frac{R T}{p_s} \DP{p_s}{\lambda}
       + \frac{R T}{p} \frac{g \sigma}{R T} p_s \DP[][\sigma]{z}{\lambda}
        \nonumber \\
     & =   R^d T_v \DP[][\sigma]{\pi}{\lambda} + \DP{\Phi}{\lambda},
     \\
      \frac{1}{\rho} \left( \DP{p}{\varphi} \right)_z
     & = R T \DP[][\sigma]{\pi}{\varphi} + \DP{\Phi}{\varphi}.
    \end{align}
������ $\pi \equiv \ln p_s$ �Ǥ���. �������ä�, ��ư�������ο�ʿ��ʬ��,
    \begin{align}
%     \Deqlab{��ư������(����)�ΦҺ�ɸ�ץ�ߥƥ���}
             \DD{u}{t}  -f v - \frac{uv}{a} \tan \varphi
       & = - \frac{1}{a \cos \varphi} \DP{\Phi}{\lambda}
           - \frac{RT}{a \cos \varphi} \DP{\pi}{\lambda}
           + {\cal F}_{\lambda},
     \\
%     \Deqlab{��ư������(����)�ΦҺ�ɸ�ץ�ߥƥ���}
             \DD{v}{t} + fu + \frac{u^2}{a} \tan \varphi
       & = - \frac{1}{a}  \DP{\Phi}{\varphi}
           - \frac{R T}{a}  \DP{\pi}{\varphi}
           + {\cal F}_{\varphi}.
    \end{align}

\subsubsection{Ϣ³�μ�}

®�٤�ȯ����,
    \begin{align}
         \left( \Ddiv \Dvect{v} \right)_z
%       &=    \frac{1}{a \cos \varphi} \DP{u}{\lambda}
%           + \frac{1}{a \cos \varphi} \DP{}{\varphi} (\cos \varphi v)
%           + \DP{w}{z}            \nonumber \\
       &=    \frac{1}{a \cos \varphi} 
             \left[   \DP[][\sigma]{u}{\lambda} 
                    + \frac{g \sigma}{R T} \DP{u}{\sigma}
                    \DP[][\sigma]{z}{\lambda} \right]   \nonumber \\
       & \quad
           + \frac{1}{a \cos \varphi} 
             \left[   \left( \DP{}{\varphi} (v \cos \varphi) \right)_{\sigma}
           + \frac{g \sigma}{R T} \DP{}{\sigma} ( v \cos \varphi) 
                    \DP[][\sigma]{z}{\lambda} \right]
           - \frac{g \sigma}{R T} \DP{}{\sigma} \left( \DD{z}{t}
             \right)_{\sigma}
             \nonumber \\
       &=    \frac{1}{a \cos \varphi} 
             \left[   \DP[][\sigma]{u}{\lambda} 
                    + \frac{g \sigma}{R T} \DP{u}{\sigma} 
                    \DP[][\sigma]{z}{\lambda} \right]           \nonumber \\
       & \quad
           + \frac{1}{a \cos \varphi} 
             \left[  \left( \DP{}{\varphi} (v \cos \varphi) \right)_{\sigma}
                    + \frac{g \sigma}{R T}\DP{}{\sigma} ( v \cos \varphi)
                    \DP[][\sigma]{z}{\lambda} \right]          \nonumber \\
       & \quad
           - \frac{g \sigma}{R^d T_v} \DP{}{\sigma} \left[
              \DP[][\sigma]{z}{t} + \frac{u}{a \cos \varphi}
              \DP[][\sigma]{z}{\lambda}
             + \frac{v}{a} \DP[][\sigma]{z}{\varphi}
             + \dot{\sigma} \DP{z}{\sigma}
             \right]     \nonumber \\
       &=   \frac{1}{a \cos \varphi} \DP[][\sigma]{u}{\lambda}
           + \frac{1}{a \cos \varphi} \left( \DP{}{\varphi} (v \cos
             \varphi) \right)_{\sigma} 
           + \DP{\dot{\sigma}}{\sigma}              \nonumber \\
       & \quad
           - \frac{g \sigma}{R T} \left[ \DP{}{\sigma} \DP[][\sigma]{z}{t}
           + \frac{u}{a \cos \varphi} \DP{}{\sigma}
             \DP[][\sigma]{z}{\lambda}
           + \frac{v}{a} \DP{}{\sigma} \DP[][\sigma]{z}{\varphi} 
           + \dot{\sigma} \DP{}{\sigma} \DP[][]{z}{\sigma} \right]
           \nonumber \\
%       &=   ( \Ddiv{\Dvect{v}})_{\sigma} 
%           + \DP{\sigma}{z}
%             \left[ \left( \DP{}{t}
%           + \frac{u}{a \cos \varphi} \DP{}{\lambda}
%           +   \frac{v}{a} \DP{}{\varphi}
%           + \dot{\sigma} \DP{}{\sigma} \right)_{\sigma} \DP{z}{\sigma} \right]
%           \nonumber \\
       &=   ( \Ddiv{\Dvect{v}_H})_{\sigma} + \DP{\dot{\sigma}}{\sigma} 
           + \DP{\sigma}{z}
             \left( \DD{}{t} \DP{z}{\sigma} \right)_{\sigma}.
    \end{align}
������, 
    \begin{align}
       \Ddiv{\Dvect{v}_H} 
         \equiv \frac{1}{a \cos \varphi}
               \DP[][\sigma]{u}{\lambda} 
             + \frac{1}{a \cos \varphi} \left( \DP{}{\varphi} (v \cos
               \varphi ) \right)_{\sigma}.
    \end{align}
�椨��, $z$- ��ɸϢ³�μ��ϼ��Τ褦���Ѵ������. 
    \begin{align}
     \frac{1}{\rho} \left( \DD{\rho}{t} \right)_z
       + \left( \Ddiv{\Dvect{v}} \right)_z
     & =    \frac{1}{\rho} \left( \DD{\rho}{t} \right)_{\sigma}
       + \left( \Ddiv{\Dvect{v}_H} \right)_{\sigma}
       + \DP{\dot{\sigma}}{\sigma}
       + \DP{\sigma}{z} \left( \DD{}{t} \DP{z}{\sigma} \right)_{\sigma}
       \nonumber \\
     & =   \frac{1}{\rho} \left( \DD{\rho}{t} \right)_{\sigma}
       + \left( \Ddiv{\Dvect{v}_H} \right)_{\sigma}
       + \DP{\dot{\sigma}}{\sigma}
       + \frac{\rho}{p_s} \left( \DD{}{t} \frac{p_s}{\rho} \right)_{\sigma}
       \nonumber \\
     & =   \left( \DD{\ln p_s}{t} \right)_{\sigma} 
       + \left( \Ddiv{\Dvect{v}_H} \right)_{\sigma}
       + \DP{\dot{\sigma}}{\sigma}. 
    \end{align}
�������ä� $\pi \equiv \ln p_s$ ���Ѥ��Ƶ��Ҥ���м��Τ褦�ˤʤ�. 
      \begin{align}
            \DD{\pi}{t}
          + \Ddiv{\Dvect{v}_H} 
          + \DP{\dot{\sigma}}{\sigma}
        = 0.
      \end{align} 

\subsubsection{Ǯ�ϳؤμ�}

\Deqref{zǮ��}�α�����1��ϼ��Τ褦���Ѵ������. 
    \begin{align}
     \frac{1}{C_p \rho} \DD{p}{t} 
        & =     \frac{1}{C_p \rho} \left\{ \DP{p}{t}
              + \Dvect{v}_H \cdot \nabla_{\sigma} p 
              + \dot{\sigma}  \DP{p}{\sigma} \right\}
                                             \nonumber \\
       & =      \frac{1}{C_p \rho} \left\{ \sigma \DP{p_s}{t}
              + \sigma \Dvect{v}_H \cdot \nabla_{\sigma} p_s 
              + \dot{\sigma}  p_s \right\}
                                              \nonumber \\
       & =      \frac{R T}{C_p^d} \left\{   \DP{\pi}{t} 
              + \Dvect{v}_H \cdot \nabla_{\sigma} \pi
              + \frac{\dot{\sigma}}{\sigma} \right\}.
    \end{align}
������,
    \begin{align}
            \Dvect{v}_H \cdot \nabla_{\sigma} 
        =   \frac{u}{a \cos \varphi} \DP{}{\lambda}
          + \frac{v}{a} \DP{}{\varphi}.
    \end{align}
�������ä�, Ǯ�ϳؤμ��ϼ��Τ褦�ˤʤ�. 
    \begin{align}
          \DD{T}{t} 
      =   \frac{R T}{C_p}
            \left\{   \DP{\pi}{t} 
                    + \Dvect{v}_H \cdot \nabla_{\sigma} \pi
                    + \frac{\dot{\sigma}}{\sigma} \right\}
                    + \frac{Q^*}{C_p^d}.
    \end{align}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{�������}

������, $\sigma$ ��ɸ�ˤ����붭�����ˤĤ��ƽҤ٤�. 

\subsubsection{��ɽ�̹���}
\vspace{-7ex}
    \begin{align}
      \Phi = \Phi_s (\lambda, \varphi)   \ \ \ \ {\rm at} \ \ \sigma=1.
    \end{align}
���ʤ��, $\Phi_s$ ��ɽ���Ϸ���ɽ��. ���ζ��������Ѥ���, ���ϳ�ʿ��
�μ����ľ��ʬ���뤳�Ȥ�, Ǥ�դ� $\sigma$ �ˤ�������� $\Phi$ �����
���Ȥ��Ǥ���. 
%
\subsubsection{$\sigma$ ��ɸ��ľ®��}
\vspace{-7ex}
    \begin{align}
      \dot{\sigma} = 0  \ \ \ at \ \ \sigma = 0, \ 1.
    \end{align}
%
\subsubsection{��ʿή�����Ǯ�ϳ��ѿ�}
�����ǤϽҤ٤ʤ�.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{����������}

Ϣ³�μ����ľ������ $\sigma=0$ ���� $\sigma=1$ �ޤ���ʬ��,
$\dot{\sigma}$ �˴ؤ��붭�������Ѥ����, �����������Ȥ�Ф�� $\pi$ 
�λ����Ѳ��˴ؤ��뼰��������.
    \begin{align}
       \frac{\partial \pi}{\partial t}
       = - \int_{0}^{1} \Dvect{v}_{H} \cdot \nabla_{\sigma} \pi d \sigma
         - \int_{0}^{1} D  d \sigma.
    \end{align}
���μ����Ѥ����, $\dot{\sigma}$ �ξ��󤬤ʤ��Ƥ���ɽ�̵����λ����Ѳ�
����뤳�Ȥ��Ǥ���. �ʤ�, �����Ǥϸ�Τ��Ȥ�ͤ��� 
$\Ddiv{\Dvect{v}_H}$ �� $D$ ��ɽ�����Ƥ���. $D$ �ˤĤ��Ƥϼ���ǲ����
�������.

��ľ®��$\dot{\sigma}$��, Ϣ³�μ����ľ������ $\sigma=0$ ���� 
$\sigma=\sigma$ �ޤ���ʬ���뤳�Ȥǿ���Ū��������. 
    \begin{align}
       \dot{\sigma} 
       = - \sigma 
         \frac{\partial \pi}{\partial t}
         - \int_{0}^{\sigma} D d \sigma
         - \int_{0}^{\sigma} 
             \Dvect{v}_{H} \cdot \nabla_{\sigma} \pi d \sigma.
    \end{align}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{�����������κǽ���}
\Dseclab{�ץ�ߥƥ���--����������}

�ʾ���Ѥ����������Ϥ򵭽Ҥ���м��Τ褦�ˤʤ�.

�ſ尵�μ�
    \begin{align}
      \DP{\Phi}{\sigma}=-\frac{R T}{\sigma}.
    \end{align}
Ϣ³�μ�: $\pi \equiv \ln p_s$ ���Ф���, 
      \begin{align}
      \Deqlab{Ϣ³�μ�}
            \DD{\pi}{t}
          + \Ddiv{\Dvect{v}_H} 
          + \DP{\dot{\sigma}}{\sigma}
        = 0.
      \end{align} 
��ư������
    \begin{align}
     \Deqlab{��ư������(����)�ΦҺ�ɸ�ץ�ߥƥ���}
             \DD{u}{t}  -f v - \frac{uv}{a} \tan \varphi
       & = - \frac{1}{a \cos \varphi} \DP{\Phi}{\lambda}
           - \frac{RT}{a \cos \varphi} \DP{\pi}{\lambda}
           + {\cal F}_{\lambda},
     \\
     \Deqlab{��ư������(����)�ΦҺ�ɸ�ץ�ߥƥ���}
             \DD{v}{t} + fu + \frac{u^2}{a} \tan \varphi
       & = - \frac{1}{a}  \DP{\Phi}{\varphi}
           - \frac{R T}{a}  \DP{\pi}{\varphi}
           + {\cal F}_{\varphi}.
    \end{align}
Ǯ�ϳؤμ�
    \begin{align}
     \Deqlab{Ǯ�ϳؤμ��Һ�ɸ}
          \DD{T}{t} 
      =   \frac{R T}{C_p}
            \left\{   \DP{\pi}{t} 
                    + \Dvect{v}_H \cdot \nabla_{\sigma} \pi
                    + \frac{\dot{\sigma}}{\sigma} \right\}
                    + \frac{Q^*}{C_p^d}.
    \end{align}
����������: Ϣ³�μ�������˼��μ����Ѥ���, 
�����餷�����֤� $\pi$ �� $\dot{\sigma}$ ��׻��Ǥ���. 
    \begin{align}
       \frac{\partial \pi}{\partial t}
       = - \int_{0}^{1} \Dvect{v}_{H} \cdot \nabla_{\sigma} \pi d \sigma
         - \int_{0}^{1} D  d \sigma.
    \end{align}
    \begin{align}
       \dot{\sigma} 
       = - \sigma 
         \frac{\partial \pi}{\partial t}
         - \int_{0}^{\sigma} D d \sigma
         - \int_{0}^{\sigma} 
             \Dvect{v}_{H} \cdot \nabla_{\sigma} \pi d \sigma.
    \end{align}
