% ɽ��   dcpam4  �ϳز��� -- ������������
%
% ���� 
%\Drireki{1993/06/16  �¸��ء��ݺ�����}
%\Drireki{2005/04/04  ��������}
%\Drireki{2008/03/25  �������硦��������}
%\Drireki{2008/06/15  ��������}
%

\section{����������}

�����Ǥ��ϳز����λ����������Ϥ򼨤�. 
�����������Ϥξܺ٤˴ؤ��Ƥ�, Haltiner and Williams (1980) �⤷����
�̻��
\htmladdnormallinkfoot{�����������Ϥ�Ƴ�Ф˴ؤ��뻲�ͻ���}
{http://www.gfd-dennou.org/library/dcpam/dcpam4/dcpam4\_current/doc/physical\_model/htm/physical\_model.htm}��
�Ρ��ϳز����λ����������Ϥ�Ƴ�С٤򻲾Ȥ���.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Ϣ³�μ�}
%
\vspace*{-5ex}
\begin{align}
 \Deqlab{Ϣ³�μ�}
    \DP{\pi}{t}
  + \frac{u}{a \cos \varphi} \DP{\pi}{\lambda}
  + \frac{v}{a} \DP{\pi}{\varphi}
 &=
  - D
  - \DP{\dot{\sigma}}{\sigma}.
\end{align}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{�ſ尵�μ�}

\vspace*{-5ex}
\begin{align}
\Deqlab{�ſ尵�μ�}
  \DP{\Phi}{\sigma} = - \frac{RT_v}{\sigma}.
\end{align}

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

\vspace*{-5ex}
\begin{align}
  \Deqlab{����������}
  \DP{\zeta}{t} 
     \ &= \ \Dinv{a \cos \varphi}
           \left\{ \DP{v_A}{\lambda} - \DP{(u_A \cos \varphi)}{\varphi} \right\}
          + {\cal D}(\zeta), \\
  \Deqlab{ȯ��������}
  \DP{D}{t} 
     \ &= \ \Dinv{a \cos \varphi}
           \left\{ \DP{u_A}{\lambda} + \DP{(v_A \cos \varphi)}{\varphi} \right\}
          - \nabla^{2}_{\sigma}
           ( \Phi + R \overline{T} \pi + \mbox{\sl KE} ) 
          + {\cal D}(D).
\end{align}
%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Ǯ�ϳؤμ�}

\vspace*{-5ex}
\begin{align}
 \begin{split}
  \Deqlab{Ǯ�ϳؤμ�}
   \DP{T}{t}
    \ &= \
          - \Dinv{a \cos \varphi}
             \left\{  \DP{(u T^{\prime})}{\lambda}
                    + \DP{(v T^{\prime} \cos \varphi)}{\varphi}
             \right\}
          + T^{\prime} D \\
    \ & \ \quad
       - \dot{\sigma} 
              \DP{T}{\sigma}
       + \kappa T_v
           \left(
                   \DP{\pi}{t}
                 + \frac{u}{a \cos \varphi} \DP{\pi}{\lambda}
                 + \frac{v}{a} \DP{\pi}{\varphi}
                 + \frac{ \dot{\sigma} }{ \sigma } 
           \right)
       + \frac{Q}{C_p}
       + {\cal D}(T) 
       + {\cal D}^{\prime}(\Dvect{v}).
 \end{split}
\end{align}

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

\vspace*{-5ex}
\begin{align}
 \begin{split}
  \Deqlab{������μ�}
  \DP{q}{t}
    \ &= \
          - \Dinv{a \cos \varphi}
             \left\{  \DP{(u q)}{\lambda}
                    + \DP{(v q \cos \varphi)}{\varphi}
             \right\}
          + q D  \\
    \ & \quad
          - \dot{\sigma} \DP{q}{\sigma}
          + S_{q}
          + {\cal D}(q).
 \end{split}
\end{align}

������, ��Ω�ѿ��ϰʲ����̤�Ǥ���. 
%
\begin{align}
 \varphi & : \quad ���� [\mathrm{deg.}], \\ 
 \lambda & : \quad ���� [\mathrm{deg.}], \\ 
 \sigma  & \equiv p/p_s, \\ 
 t       & : \quad ���� [\mathrm{s}]. 
\end{align}
%
������, $p$�ϵ���, $p_s$ ����ɽ�̵����Ǥ���. 
��ǥ�ǻ���ȯŸ��׻����뤳�ȤȤʤ�ͽ���ѿ��ϰʲ����̤�Ǥ���. 
%
\begin{align}
 \pi\ (\varphi, \lambda)
                    & \equiv \ln p_s, \\
 T\ (\varphi, \lambda, \sigma)
                    & : \quad ���� [\mathrm{K}], \\
 q\ (\varphi, \lambda, \sigma)
                    & : \quad �漾 [\mathrm{kg}\ \mathrm{kg}^{-1}], \\
\Deqlab{�������}
 \zeta\ (\varphi, \lambda, \sigma)
                    & \equiv \Dinv{a \cos \varphi}
                        \left\{   \DP{v}{\lambda}
                                - \DP{(u \cos \varphi)}{\varphi} \right\}
                      : \quad ���� [\mathrm{s}^{-1}], \\
\Deqlab{ȯ�����}
 D\ (\varphi, \lambda, \sigma)
                    & \equiv  \Dinv{a \cos \varphi}
                        \left\{   \DP{u}{\lambda}
                                + \DP{(v \cos \varphi)}{\varphi} \right\}
                      : \quad ȯ�� [\mathrm{s}^{-1}].
% u\ (\varphi, \lambda, \sigma)
%                    & : \quad ������® [\mathrm{m}\ \mathrm{s}^{-1}], \\
% v\ (\varphi, \lambda, \sigma)
%                    & : \quad ������® [\mathrm{m}\ \mathrm{s}^{-1}].
\end{align}
%
������ $u$ ��������®, $v$ ��������®�Ǥ���, ���줾��
$(\varphi, \lambda, \sigma)$ �δؿ��Ǥ���. 
%
�ƻ��֥��ƥåפǿ���Ū�˵������ѿ��ϰʲ����̤�Ǥ���. 
%
\begin{align}
 \Phi               & \equiv gz
                      : \quad �����ݥƥ󥷥�����
                              [\mathrm{m}^{2}\  \mathrm{s}^{-2}], \\
 \dot{\sigma}       & \equiv \DD{\sigma}{t}
                      \ \equiv \ 
                         \DP{\sigma}{t}
                       + \frac{u}{a \cos \varphi} \DP{\sigma}{\lambda}
                       + \frac{v}{a} \DP{\sigma}{\varphi}
                       + \DP{\sigma}{\sigma}, \\
 \overline{T}\ (\sigma)
                    & : \quad ��ʿʿ�ѵ��� [\mathrm{K}], \\
 T^{\prime}\ (\varphi, \lambda, \sigma)
                    & \equiv T - \overline{T}, \\
 T_v\ (\varphi, \lambda, \sigma)
                    & \equiv T
                          \left\{
                            1 + \left(\epsilon_v^{-1} - 1\right) q
                          \right\}, \\
 T_v^{\prime}\ (\varphi, \lambda, \sigma)
                    & \equiv T_v - \overline{T}, \\
% \theta       & \equiv T  \left( p/p_{s} \right)^{- \kappa}, \\
\Deqlab{������ư�̰�ή��}
 u_A\ (\varphi, \lambda, \sigma)
                    & \equiv
                          ( \zeta + f ) v 
                        - \dot{\sigma} \DP{u}{\sigma} 
                        - \frac{R T_v^{\prime}}{a \cos \varphi} \DP{\pi}{\lambda} 
                        + {\cal F}_{\lambda}, \\
%
\Deqlab{���̱�ư�̰�ή��}
 v_A\ (\varphi, \lambda, \sigma)
                    & \equiv
                        - ( \zeta + f ) u 
                        - \dot{\sigma} \DP{v}{\sigma} 
                        - \frac{R T_v^{\prime}}{a} \DP{\pi}{\varphi} 
                        + {\cal F}_{\varphi}, \\
%
\Deqlab{��ץ饷����}
 \nabla^{2}_{\sigma}  
       & \equiv 
               \frac{1}{a^{2} \cos^2 \varphi} \DP[2]{}{\lambda} 
             + \frac{1}{a^{2} \cos \varphi}
                \DP{}{\varphi}
                  \left( \cos \varphi \DP{}{\varphi} \right),  \\
%
\Deqlab{��ư���ͥ륮����}
 \mbox{\sl KE}\ (\varphi, \lambda, \sigma)
                    & \equiv  \frac{u^{2}+v^{2}}{2 } \\
%
 {\cal D}(\zeta)  & : \qquad ���ٿ�ʿ�Ȼ�, \Deqlab{���ٿ�ʿ�Ȼ�} \\
 {\cal D}(D)      & : \qquad ȯ����ʿ�Ȼ�, \Deqlab{ȯ����ʿ�Ȼ�} \\
 {\cal D}(T)      & : \qquad ���ٿ�ʿ�Ȼ�, \Deqlab{���ٿ�ʿ�Ȼ�} \\
 {\cal D}(q)      & : \qquad �漾��ʿ�Ȼ�, \Deqlab{�漾��ʿ�Ȼ�} \\
 {\cal F}_\lambda \ (\varphi, \lambda, \sigma)
                  & : \qquad �����ϱ�ư���� (��������),
                     \Deqlab{�����ϱ�ư���� (��������)} \\
 {\cal F}_\varphi \ (\varphi, \lambda, \sigma)
                  & : \qquad �����ϱ�ư���� (��������),
                     \Deqlab{�����ϱ�ư���� (��������)} \\
 Q \ (\varphi, \lambda, \sigma) 
                  & : \qquad ����, �ŷ�, �����ϱ�ư�������ˤ���Ǯ�������Ѳ�, 
      \Deqlab{����, �ŷ�, �����ϱ�ư�������ˤ���Ǯ�������Ѳ�} \\
 S_q \ (\varphi, \lambda, \sigma) 
                  & : \qquad �ŷ�, �����ϱ�ư�������ˤ������������, 
        \Deqlab{�ŷ�, �����ϱ�ư�������ˤ������������} \\
 {\cal D}' \ (\Dvect{v})
                  & : \qquad �໤Ǯ \Deqlab{�໤Ǯ}. 
\end{align}
%
�ƿ�ʿ�Ȼ�\Deqref{���ٿ�ʿ�Ȼ�}��\Deqref{�漾��ʿ�Ȼ�}
�˴ؤ��Ƥ�\ref{�ȿ���¸���ο�ʿ�Ȼ���}������������.
%
%   �໤Ǯ������. ʪ�������˰ܤ����.
%\begin{align}
%  {\cal D}^{\prime} (\Dvect{v}) 
% = - \frac{1}{C_p}
%     \Dvect{v} \cdot  ( \DP{\Dvect{v}}{t} )_{diff} .
%\end{align}
%
%$( \DP{\Dvect{v}}{t} )_{diff} $ ��,
%��ʿ����ӱ�ľ�γȻ��ˤ�� $u,v$ �λ����Ѳ���Ǥ���.
%
����ϰʲ����̤�Ǥ���. 
%
\begin{align}
 a           & : \quad ����Ⱦ�� [\mathrm{m}], \\
 R           & : \quad �����絤�ε������ [\mathrm{J\  kg}^{-1}\  \mathrm{K}^{-1}], \\
 C_p         & : \quad �����絤���絤�갵��Ǯ [\mathrm{J\ kg}^{-1}\ \mathrm{K}^{-1}], \\
 f           & : \quad ���ꥪ��ѥ�᡼�� [\mathrm{s}^{-1}], \\
 \kappa      & \equiv R/C_p, \\
 \epsilon_v  & : \quad �����ʬ������.
%
\end{align}
%

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

��ľή�˴ؤ��붭������
%
\begin{align}
  \dot{\sigma} = 0  \ \ \ at \ \ \sigma = 0 , \ 1 .
\end{align}
%
�Ǥ���. ��ä�\Deqref{Ϣ³�μ�} ����,
��ɽ�����λ����Ѳ�����
$\sigma$�ϤǤα�ľ®��$\dot{\sigma}$�������Ǽ�
%
\begin{align}
   \Deqlab{��������}
   \DP{\pi}{t}
   = - \int_{0}^{1} \Dvect{v}_{H} \cdot \nabla_{\sigma} \pi d \sigma
     - \int_{0}^{1} D  d \sigma ,
\end{align}
%
\begin{align}
   \Deqlab{��ľ®��}
   \dot{\sigma} 
   = - \sigma 
     \DP{\pi}{t}
     - \int_{0}^{\sigma} D d \sigma
     - \int_{0}^{\sigma} 
         \Dvect{v}_{H} \cdot \nabla_{\sigma} \pi d \sigma ,
\end{align}
%
��Ƴ�����.

������ǮŪ�������ˤĤ��Ƥ�
%�̻����ɽ�̲�����
\Dchapref{��ɽ�̲���}
�ˤ����Ƶ��Ҥ���. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{��ʿ�Ȼ���}
%\label{��ʿ�Ȼ���}
\subsection{�ȿ���¸���ο�ʿ�Ȼ���}
\label{�ȿ���¸���ο�ʿ�Ȼ���}

��ʿ�Ȼ����, ���Τ褦��$\nabla^{N_D}$�η��Ƿ׻�����. 
%
\begin{equation}
  \label{��ʿ�Ȼ�}
  {\cal D}(\zeta) = - K_{HD} 
                      \left[ (-1)^{N_D/2} \nabla^{N_D}
                              - \left( \frac{2}{a^2} \right)^{N_D/2} 
                      \right]
                    \zeta ,
\end{equation}
%
\begin{equation}
  {\cal D}(D) = - K_{HD} 
                      \left[ (-1)^{N_D/2} \nabla^{N_D}
                              - \left( \frac{2}{a^2} \right)^{N_D/2} 
                      \right]
                    D ,
\end{equation}
%
\begin{equation}
  {\cal D}(T) = - (-1)^{N_D/2} K_{HD} \nabla^{N_D} T ,
\end{equation}
%
\begin{equation}
  {\cal D}(q) = - (-1)^{N_D/2} K_{HD} \nabla^{N_D} q .
\end{equation}
%
���ο�ʿ�Ȼ���Ϸ׻��ΰ��경�Τ���ΰ�̣�礤������.
�����ʥ������������Ū�ʿ�ʿ�Ȼ���ɽ������,
$N_D$�Ȥ��Ƥ�, 4$\sim$16���Ѥ���.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{�ȿ����¸���ο�ʿ�Ȼ���}
%
% ̤Ƴ���Τ���, Ƴ�������ޤǥ����ȥ�����. 
%
%��ʿ�Ȼ����ȿ��˰�¸���ʤ����ͤ��ͤˤ��뤳�Ȥ�Ǥ���. 
%�ܺپ�ά. 
