% ɽ��   dcpam5-primitivve  �ϳز��� -- ������������
%
% ���� 
%\Drireki{2010/09/01 �ݹ�����}
%\Drireki{2010/04/15 �ⶶ˧��}
%\Drireki{2009/10/05 �ⶶ˧��}
%\Drireki{2009/03/03 ��������}
%\Drireki{2008/06/15 ��������}
%\Drireki{2008/03/25 �������硦��������}
%\Drireki{2005/04/04 ��������}
%\Drireki{1993/06/16 �¸��ء��ݺ�����}
%

%\section{����������}
\section{����ɽ��}

�����Ǥ��ϳز����λ����������Ϥο���ɽ���򼨤�. 
�����������Ϥξܺ٤˴ؤ��Ƥ�, Haltiner and Williams (1980) �⤷����
�̻��
\htmladdnormallinkfoot{�����������Ϥ�Ƴ�Ф˴ؤ��뻲�ͻ���}
{http://www.gfd-dennou.org/library/dcpam/dcpam5/dcpam5\_current/doc/derivation/htm/derivation.htm}��
�Ρ��ϳز����λ����������Ϥ�Ƴ�С٤򻲾Ȥ���.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Ϣ³�μ�}
%
\vspace*{-5ex}
\begin{align}
 \Deqlab{Ϣ³�μ�}
    \DP{\pi}{t} + \Dvect{v}_H \cdot \nabla_{\sigma} \pi 
 &=
  - 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}
           \left( \Dinv{1 - \mu^2} \DP{V_A}{\lambda} - \DP{U_A}{\mu} \right)
          + {\cal D}(\zeta), \\
  \Deqlab{ȯ��������}
  \DP{D}{t} 
     \ &= \ \Dinv{a}
           \left( \Dinv{1 - \mu^2} \DP{U_A}{\lambda} + \DP{V_A}{\mu} \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}
             \left(   \Dinv{1 - \mu^2} \DP{U T^{\prime}}{\lambda}
                    + \DP{V T^{\prime}}{\mu}
             \right)
          + T^{\prime} D \\
    \ & \ \quad
       - \dot{\sigma} 
              \DP{T}{\sigma}
       + \kappa T
           \left(
                   \DP{\pi}{t}
                 + \Dvect{v}_H \cdot \Dgrad_{\sigma} \pi 
                 + \frac{ \dot{\sigma} }{ \sigma } 
           \right)
       + \frac{Q}{C_p}
       + {\cal D}(T) 
       + {\cal D}^{\prime}(\Dvect{v}).
 \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���.
�ޤ� $\mu \equiv \sin \varphi$ �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}], \\
\Deqlab{�������}
 \zeta\ (\varphi, \lambda, \sigma)
                    & \equiv \Dinv{a}
                        \left(    \Dinv{1 - \mu^2} \DP{V}{\lambda}
                                - \DP{U}{\mu} \right)
                      : \quad ���� [\mathrm{s}^{-1}], \\
\Deqlab{ȯ�����}
 D\ (\varphi, \lambda, \sigma)
                    & \equiv  \Dinv{a}
                        \left(    \Dinv{1 - \mu^2} \DP{U}{\lambda}
                                + \DP{V}{\mu} \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}
%
������, 
%
\begin{align}
  U(\varphi, \lambda, \sigma) &\equiv u(\varphi, \lambda, \sigma) \cos \varphi, \\
  V(\varphi, \lambda, \sigma) &\equiv v(\varphi, \lambda, \sigma) \cos \varphi, \\
  u & : \mbox{������®}, \\
  v & : \mbox{������®}
\end{align}
%
�Ǥ���. 
ή���ؿ�$\psi$��®�٥ݥƥ󥷥��$\chi$��Ƴ�������,
$U$, $V$, $\zeta$, $D$�Ϥ��줾��ʲ��Τ褦��ɽ�蘆���. 
%
\begin{align}
 \Deqlab{�פȦ֤ˤ��������®��ɽ��}
  U &= \Dinv{a} \left(    \DP{\chi}{\lambda}
                        - (1-\mu^2) \DP{\psi}{\mu} \right), \\
 \Deqlab{�פȦ֤ˤ��������®��ɽ��}
  V &= \Dinv{a} \left(    \DP{\psi}{\lambda}
                        + (1-\mu^2) \DP{\chi}{\mu} \right), \\
 \Deqlab{�פȦ֤ˤ�뱲�٤�ɽ��}
  \zeta &= \Dlapla \psi, \\
 \Deqlab{�פȦ֤ˤ��ȯ����ɽ��}
  D     &= \Dlapla \chi.
\end{align}
%
�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 ��ಹ�� [\mathrm{K}], \\
 T^{\prime}\ (\varphi, \lambda, \sigma)
                    & \equiv T - \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^{\prime}}{a} \DP{\pi}{\lambda} 
                        + {\cal F}_{\lambda} \cos \varphi, \\
%
\Deqlab{���̱�ư�̰�ή��}
 V_A\ (\varphi, \lambda, \sigma)
                    & \equiv
                        - ( \zeta + f ) U 
                        - \dot{\sigma} \DP{V}{\sigma} 
                        - \frac{R T^{\prime}}{a} (1-\mu^2) \DP{\pi}{\mu} 
                        + {\cal F}_{\varphi} \cos \varphi, \\
%
\Deqlab{�Фΰ�ή}
 \Dvect{v}_H \cdot \nabla_{\sigma} \pi
   &\equiv
       \frac{U}{a (1 - \mu^2)} \DP{\pi}{\lambda} 
     + \frac{V}{a} \DP{\pi}{\mu} \\
%
\Deqlab{��ץ饷����}
 \nabla^{2}_{\sigma}  
       & \equiv 
               \frac{1}{a^{2} (1-\mu^2)} \DP[2]{}{\lambda} 
             + \frac{1}{a^{2}}
                \DP{}{\mu}
                  \left[ (1-\mu^2) \DP{}{\mu} \right],  \\
%
\Deqlab{��ư���ͥ륮����}
 \mbox{\sl KE}\ (\varphi, \lambda, \sigma)
                    & \equiv  \frac{U^{2}+V^{2}}{2 (1-\mu^2) } \\
%
 {\cal D}(\zeta)  & : \qquad ���٤ο�ʿ�Ȼ��ȥ��ݥ��ؤˤ����뻶��, \Deqlab{���ٿ�ʿ�Ȼ�} \\
 {\cal D}(D)      & : \qquad ȯ���ο�ʿ�Ȼ��ȥ��ݥ��ؤˤ����뻶��, \Deqlab{ȯ����ʿ�Ȼ�} \\
 {\cal D}(T)      & : \qquad Ǯ�ο�ʿ�Ȼ�, \Deqlab{���ٿ�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�} \\
 {\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, \\
%
\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���}
%\subsection{��ʿ�Ȼ��ȥ��ݥ���}
\subsection{��ʿ�Ȼ���}
\label{�ȿ���¸���ο�ʿ�Ȼ���}

%��ʿ�Ȼ��ȥ��ݥ��ؤˤ����뱲�٤�ȯ���λ���ϼ��Τ褦��ɽ������. 
%
% % \begin{align}
% %   {\cal D}(\zeta) &= {\cal D_{HD}}(\zeta) + {\cal D_{SL}}(\zeta)
% % \\
% %   {\cal D}(D)     &= {\cal D_{HD}}(D)     + {\cal D_{SL}}(D)
% % \\
% %   {\cal D}(T)     &= {\cal D_{HD}}(T)     + {\cal D_{SL}}(T)
% % \\
% %   {\cal D}(q)     &= {\cal D_{HD}}(q)
% % \end{align}
% % %
% % ������, ${\cal D_{HD}}$, ${\cal D_{SL}}$ �Ϥ��줾���ʿ�Ȼ��ȥ��ݥ��ؤˤ�����
% �����ɽ��. 

��ʿ�Ȼ��ˤ����뱲�٤�ȯ���λ���ϼ��Τ褦��ɽ������. 
%
\begin{align}
  {\cal D}(\zeta) &= {\cal D_{HD}}(\zeta) 
\\
  {\cal D}(D)     &= {\cal D_{HD}}(D)     
\\
  {\cal D}(T)     &= {\cal D_{HD}}(T)     
\end{align}
%
������, ${\cal D_{HD}}$ �Ͽ�ʿ�Ȼ��ˤ�뻶���ɽ��. 

��ʿ�Ȼ����, ���Τ褦�� $\nabla^{N_D}$ �η��Ƿ׻�����. 
%
\begin{equation}
  \label{��ʿ�Ȼ�}
  {\cal D_{HD}}(\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_{HD}}(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_{HD}}(T) = - (-1)^{N_D/2} K_{HD} \nabla^{N_D} T ,
\end{equation}
%
% \begin{equation}
%   {\cal D_{HD}}(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������,
����Ȥ��� $N_D$ �ˤ� 4$\sim$16 ���Ѥ��뤳�Ȥ�¿��.

% ���ݥ��ؤˤ����뱿ư�̤λ�����, ����ʿ����ʬ�򸺿ꤵ������Ȥ����ʤ����� 2 �̤��
% �׻�ˡ��Ƴ������. 
% %
% ����ʿ����ʬ�⸺�ꤵ������ˤ�, 
% %
% \begin{align}
%   {\cal D_{SL}}(\zeta) &= - \gamma_M \zeta, 
% \\
%   {\cal D_{SL}}(D)     &= - \gamma_M D, 
% \end{align}
% %
% �Ȥʤ�. ������, $\gamma_M$ �ϥ��ݥ��ؤˤ����뱿ư�̤θ��그���Ǥ���. 
% ����ʿ����ʬ�򸺿ꤵ���ʤ����ˤ�, 
% %
% \begin{align}
%   {\cal D_{SL}}(\zeta) &= - \gamma_M ( \zeta - \bar{\zeta} ), 
% \\
%   {\cal D_{SL}}(D)     &= - \gamma_M ( D - \bar{D} ), 
% \end{align}
% %
% �Ȥʤ�. ������, $\bar{}$ ��, ����ʿ�Ѥ�ɽ��. 

% ���ݥ�����β��پ���θ���ˤϰʲ��ι��Ƴ������. 
% %
% \begin{align}
%   {\cal D_{SL}}(T) &= - \gamma_H ( T - \bar{T} ), 
% \end{align}
% %
% %
% ������, $\gamma_H$ �ϥ��ݥ��ؤˤ����벹�پ���θ��그���Ǥ���. 

% ���그�� $\gamma_M$, $\gamma_H$ �� $\sigma$ ��¸���˰��̷��Ϥʤ���, dcpam �Ǥ�
% ���Τ褦�� $\sigma$ ��¸�����θ����. 
% %
% \begin{align}
%  \gamma_M = \left\{
%  \begin{array}{ll}
%    \gamma_{M,0} \left( \frac{\sigma_0}{\sigma} \right)^{N_{SL}}, & \text{($\sigma \le \sigma_{lim}$)} \\
%    0 . & \text{($\sigma > \sigma_{lim}$)}
%  \end{array} \right. 
% \\
%  \gamma_H = \left\{
%  \begin{array}{ll}
%    \gamma_{H,0} \left( \frac{\sigma_0}{\sigma} \right)^{N_{SL}}, & \text{($\sigma \le \sigma_{lim}$)} \\
%    0 . & \text{($\sigma > \sigma_{lim}$)}
%  \end{array} \right. 
% \end{align}
% %
% ������, $\gamma_{M,0}$, $\gamma_{H,0}$, $N_{SL}$, $\sigma_{lim}$ �Ϥ��줾��, 
% $\sigma = \sigma_0$ �ˤ����븺�그��, $\sigma$ ��¸���λؿ�, ���ݥ��ؤ�
% ���¤� $\sigma$ �Ǥ���. 
% dcpam �Ǥ�, $\sigma_0$ �ϥ�ǥ�Ǿ��ؤ� $\sigma$ �Ȥ��Ƥ���. 


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