\section{��ľ��ɸ�Ѵ�}

��ľ��ɸ�Ȥ���
\begin{equation}
  \sigma \equiv \frac{p}{p_s}
\end{equation}
��Ȥ�. $p_s$ ����ɽ�̵���.
$\sigma$ ��������
\begin{equation}
  \DP{p}{\sigma} = p_s
\end{equation}
������Ω��.

\vspace{5mm}

��ľ��ɸ�Ѵ��Ρ���\footnotemark[4]
\footnotetext[4]{kiso.ref/gfd/formula/vertical ��ľ��ɸ�Ѵ�( $\sigma$ 
  ��ɸ�� ����}
���, ���Τ褦���Ѵ�§��������.
\begin{eqnarray}
    \left( \DD{}{t} \right)_z
  & = & \left( \DD{}{t} \right)_{\sigma}
  = \left( \DP{}{t} \right)_{\sigma}
    + \Dvect{v}_H \cdot \nabla_{\sigma}
    + \dot{\sigma} \DP{}{\sigma} \nonumber \\
\label{����ʬ�Ѵ�§}
  & = & \left( \DP{}{t} \right)_{\sigma}
    + \frac{u}{a \cos \varphi} \left( \DP{}{\lambda} \right)_{\sigma}
    + \frac{v}{a} \left( \DP{}{\varphi} \right)_{\sigma}
    + \dot{\sigma} \DP{}{\sigma} \\
\label{lambda ��ʬ�Ѵ�§}
          \DP[][z]{f}{\lambda}
  & = &   \DP[][\sigma]{f}{\lambda} 
        - \DP{f}{\sigma} \DP{\sigma}{z} \DP[][\sigma]{z}{\lambda}  \\
\label{phi ��ʬ�Ѵ�§}
          \DP[][z]{f}{\varphi}
  & = &   \DP[][\sigma]{f}{\varphi} 
        - \DP{f}{\sigma} \DP{\sigma}{z} \DP[][\sigma]{z}{\varphi} \\
\label{ȯ���Ѵ�§}
     \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} 
                - \DP{u}{\sigma} \DP{\sigma}{z}
                \DP[][\sigma]{z}{\lambda} \right]   \nonumber \\
   &  &+ \frac{1}{a \cos \varphi} 
         \left[   \DP{}{\varphi} (v \cos \varphi)
                - \DP{}{\sigma} ( v \cos \varphi) \DP{\sigma}{z}
                \DP[][\sigma]{z}{\lambda} \right]
       + \DP{w}{\sigma} \DP{\sigma}{z}           \nonumber \\
   &=&   \frac{1}{a \cos \varphi} 
         \left[   \DP[][\sigma]{u}{\lambda} 
                - \DP{u}{\sigma} \DP{\sigma}{z}
                \DP[][\sigma]{z}{\lambda} \right]           \nonumber \\
   & & + \frac{1}{a \cos \varphi} 
         \left[   \DP{}{\varphi} (v \cos \varphi)
                - \DP{}{\sigma} ( v \cos \varphi) \DP{\sigma}{z}
                \DP[][\sigma]{z}{\lambda} \right]
       + \DP{}{\sigma} \left( \DD{z}{t} \right) \DP{\sigma}{z}
                                 \nonumber \\
   &=&    \frac{1}{a \cos \varphi} \DP[][\sigma]{u}{\lambda} 
        - \frac{1}{a \cos \varphi} \DP{u}{\sigma} \DP{\sigma}{z}
          \DP[][\sigma]{z}{\lambda}            \nonumber \\
   &  & + \frac{1}{a \cos \varphi} \DP{}{\varphi} (v \cos \varphi)
        - \frac{1}{a \cos \varphi} \DP{}{\sigma} ( v \cos \varphi)
          \DP{\sigma}{z} \DP[][\sigma]{z}{\lambda}       \nonumber \\
    & & + \DP{\sigma}{z} \DP{}{\sigma} 
          \left[ \left( \DD{z}{t} \right)_{\sigma}
    + \frac{u}{a \cos \varphi} \left( \DP{z}{\lambda} \right)_{\sigma}
    + \frac{v}{a} \left( \DP{z}{\varphi} \right)_{\sigma}
    + \dot{\sigma} \DP{z}{\sigma} \right]           \nonumber \\
   &=&   \frac{1}{a \cos \varphi} \DP{u}{\lambda}
       + \frac{1}{a \cos \varphi} \DP{}{\varphi} ( v \cos \varphi)
       + \DP{\sigma}{z} \DP{}{\sigma} \DD{z}{t}
       + \DP{\dot{\sigma}}{\sigma} 
\end{eqnarray}

\vspace{5mm}

�ſ尵ʿ��, ��ư������, Ϣ³�μ�, ���٤μ��ϼ��Τ褦�ˤʤ�.

\begin{itemize}
  \item �ſ尵ʿ�ռ� 
        \begin{equation}
        \label{�ſ尵ʿ��}
            p_{s} \DP{\sigma}{z} = - \rho g . 
        \end{equation}
        ���ݥƥ󥷥�� $\Phi \equiv gz$ ���Ѥ����
        \begin{equation}
           \DP{\Phi}{\sigma} = - \frac{R T}{\sigma}
        \end{equation}
        �Ȥʤ�.
  \item ��ư������ \\
        ���ϸ��ۤι��, (\ref{lambda ��ʬ�Ѵ�§}), 
        (\ref{phi ��ʬ�Ѵ�§}) �ˤ���Ѵ�����.
        \begin{eqnarray}
        \label{��ư������lambda��ʬ}
                   \left( \DD{u}{t} \right)_{\sigma} - f v 
          & = &    \frac{u v \tan \varphi}{z} 
                 - \frac{1}{\rho a \cos \varphi} 
                   \left\{ \DP[][\sigma]{p}{\lambda}
                 - p_{s} \DP{\sigma}{z} \DP[][\sigma]{z}{\lambda} \right\}
                 + F_{\lambda}, \\
        \label{��ư������varphi��ʬ}
                   \left( \DD{v}{t} \right)_{\sigma} + f u 
          & = &  - \frac{u^2 \tan \varphi}{a} 
                 - \frac{1}{\rho a} \left\{ \DP[][\sigma]{p}{\varphi}
                 - p_{s} \DP{\sigma}{z} \DP[][\sigma]{z}{\varphi} \right\}
                 + F_{\varphi}. 
        \end{eqnarray}
        (\ref{�ſ尵ʿ��})���Ѥ����
        \begin{eqnarray}
        \label{u �μ� sigma}
                \left( \DD{u}{t} \right)_{\sigma} - f v 
          &=& - \frac{1}{\rho a \cos \varphi} 
                \left\{   \DP[][\sigma]{p}{\lambda} 
                       + \rho g \DP[][\sigma]{z}{\lambda} \right\}
              + F_{\lambda},
                                                              \\
        \label{v �μ� sigma}
                \left( \DD{v}{t} \right)_{\sigma} + f u 
          &=& - \frac{1}{\rho a} \left\{ \DP[][\sigma]{p}{\varphi}
              + \rho g \DP[][\sigma]{z}{\varphi} \right\} + F_{\varphi}
        \end{eqnarray}
        �Ȥʤ�.
        $p=p_s \sigma$ ���
        \begin{equation}
            \DP[][\sigma]{p}{\lambda} 
          = \sigma \DP[][\sigma]{p_s}{\lambda}, \ \ \ 
            \DP[][\sigma]{p}{\varphi} 
          = \sigma \DP[][\sigma]{p_s}{\varphi}, 
        \end{equation}
        $\Phi = gz$ ���
      \begin{equation}
          -g \DP[][\sigma]{z}{\lambda} = - \DP{\Phi}{\lambda}, \ \ \ 
          -g \DP[][\sigma]{z}{\varphi} = - \DP{\Phi}{\varphi},
      \end{equation}
      �ʤɤ��Ѥ���� (\ref{u �μ� sigma}), 
      (\ref{v �μ� sigma}) ��
      \begin{equation}
               \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}
             + F_{\lambda},
      \end{equation}
      \begin{equation}
               \DD{v}{t} + fu + \frac{u^2}{a} \tan \varphi
           = - \frac{1}{a}  \DP{\Phi}{\varphi}
             - \frac{RT}{a}  \DP{\pi}{\varphi}
             + F_{\varphi}.
      \end{equation}
      ������, $\pi \equiv \ln p_s$ �Ȥ�����.
\item Ϣ³�μ� \\
      (\ref{����ʬ�Ѵ�§}), (\ref{ȯ���Ѵ�§}) ���
      \begin{equation}
            \frac{1}{\rho} \left( \DD{\rho}{t} \right)_{\sigma}
          + \frac{1}{a \cos \varphi} \DP{u}{\lambda}
          + \frac{1}{a \cos \varphi} \DP{}{\varphi}(v \cos \varphi)
          + \DP{\dot{\sigma}}{\sigma}
          + \DP{\sigma}{z} 
            \left( \DD{}{t} \DP{z}{\sigma} \right)_{\sigma}
        = - \frac{S[Pr]}{\rho}.
      \end{equation}
      (\ref{�ſ尵ʿ��})���
      \begin{eqnarray}
            \frac{1}{\rho} \left( \DD{\rho}{t} \right)_{\sigma}
          + \DP{\sigma}{z} 
            \left( \DD{}{t} \DP{z}{\sigma} \right)_{\sigma}
        =   \frac{1}{\rho} \left( \DD{\rho}{t} \right)_{\sigma}
          + \frac{\rho}{p_s} \DD{}{t} \frac{p_s}{\rho}
        =   \DD{}{t} \ln p_s
      \end{eqnarray}
      �ʤΤ�
      \begin{equation}
      \label{Ϣ³�μ�}
            \DD{\pi}{t}
          + \frac{1}{a \cos \varphi} \DP{u}{\lambda}
          + \frac{1}{a \cos \varphi} \DP{}{\varphi}(v \cos \varphi)
          + \DP[]{\dot{\sigma}}{\sigma}
        = - \frac{S[Pr]}{\rho}. 
      \end{equation} 
\item Ǯ͢���μ�  \\
      $c_p$ ������ξ��ˤϼ��Τ褦�ˤʤ�.
      \begin{eqnarray}
         c_p \DD{T}{t} & = & \frac{1}{\rho} \DD{p}{t} + Q
                                                            \\
                \DP{T}{t} 
              + \Dvect{v} \nabla_{\sigma} T
              + \dot{\sigma}  \DP{T}{\sigma}
        & = &   \frac{1}{c_p \rho}
                \left\{ \DP{p}{t} + \Dvect{v} \nabla_{\sigma} p 
              + \dot{\sigma}  \DP{p}{\sigma} \right\}
              + \frac{Q}{c_p}
                                             \nonumber \\
       & = &    \frac{1}{c_p \rho}
                \left\{ \sigma \DP{p_s}{t}
              + \sigma \Dvect{v} \nabla_{\sigma} p_s 
              + \dot{\sigma}  p_s \right\}
              + \frac{Q}{c_p}
                                              \nonumber \\
       & = &    \frac{R T}{c_p}
                \left\{   \DP{\pi}{t} 
                        + \Dvect{v} \nabla_{\sigma} \pi
                        + \frac{\dot{\sigma}}{\sigma} \right\}
                        + \frac{Q}{c_p}
      \end{eqnarray}
      ������, 
      \begin{equation}
            \Dvect{v} \nabla_{\sigma} 
        =   \frac{u}{a \cos \varphi} \DP{}{\lambda}
          + \frac{v}{a} \DP{}{\varphi}.
      \end{equation}
\end{itemize}
