\chapter{���Ϻ�ɸ�Ϥˤ�����ץ�ߥƥ�����������Ƴ��}
������Ͽ�Ǥ�, $z$�Ϥǵ��Ҥ��줿�ץ�ߥƥ����������Ϥ�$p$�Ϥǵ��Ҥ��뤿��˱�ľ��ɸ�Ѵ���Ԥ�.

�����絤�ˤ�����ץ�ߥƥ�����������, ���ˤ����뱿ư������, �ſ尵ʿ�դμ�, Ϣ³�μ�,
���۵��Τξ���������, Ǯ�ϳ�Ū���ͥ륮����¸�μ��ˤ�äƹ��������.
%
\begin{eqnarray}
\Deqlab{motion_eq_zcd}
 \DD{\Dvect{v}}{t} &=& - \dfrac{1}{\rho} \nabla p - f \Dvect{k} \times \Dvect{v} + \Dvect{F}  \\
\Deqlab{hydrodyn_eq}
 \DP{p}{z} &=& - \rho g  \\
\Deqlab{continious_eq_zcd}
 \DD{\rho}{t} &=& \rho \left(\nabla \cdot \Dvect{v} + \DP{w}{z} \right) \\
\Deqlab{ideal_gas_law_eq}
 p &=& \rho RT \\
\Deqlab{thermodyn_energy_conserve_eq}
 c_{v} \DD{T}{t} + p \DD{\alpha}{t} &=& \dot{Q}
\end{eqnarray}
%
������, $\Dvect{v}=(u,v)$�Ͽ�ʿ®�٥٥��ȥ�, $w$�ϱ�ľ®��, $\nabla=(\partial/\partial x, \partial/\partial y)$, 
$\rho$��̩��, $p$�ϰ���, $T$ �ϲ���, $\alpha(=\rho^{-1})$������,
$f(=2 \Omega \sin \theta)$�ϥ��ꥪ��ѥ�᡼��, $R$�ϴ����絤�ε������, $c_v$�ϴ����絤���갵��Ǯ, 
$\Dvect{F}$���໤��, $\dot{Q}$������Ǯ��Ǯ�ˤ���ǮΨ, $\Dvect{k}$�ϱ�ľ�������ñ�̥٥��ȥ�Ǥ���.
�ޤ�, ����ʬ�ȶɽ���ʬ�ˤ�,
%
\begin{equation}
\DD{}{t} = \DP{}{t} + u\DP{}{x} + v\DP{}{y} + w\DP{}{z}
\end{equation}
%
�δط�������. 

�ޤ�, Ǯ�ϳ�Ū���ͥ륮����¸�μ�\Deqref{thermodyn_energy_conserve_eq}��, ����
%
\begin{equation}
 \Deqlab{potential_temperture}
 \theta = T \left( \dfrac{p_{s}}{p} \right)^\kappa
\end{equation}
%
��Ȥä�,
%
\begin{equation}
\Deqlab{thermodyn_energy_conserve_eq_theta}
 \DD{ \textrm{ln} \; \theta}{t} = \dfrac{1}{c_{p} T} \dot{Q}
\end{equation}
%
�Ƚ�ľ����.
������, $p_{s}$�ϴ��Ȥʤ�����ΰ��Ϥ�Ǥ�դ����٤Ф褤��, �Ͼ嵤��(1000 hPa)���뤳�Ȥ�¿��.
�ޤ�, $\kappa=R/c_{p}$�Ǥ���.

����, $z$ �ϤΥץ�ߥƥ���������\Deqref{motion_eq_zcd} $\sim$ \Deqref{thermodyn_energy_conserve_eq}�ˤ�����,
��ľ��ɸ�����Ū�ʹ��٤��鰵�Ϥ˼��ľ�����Ȥ�ͤ���.
$z$��ɸ�Ϥ���Ǥ�դα�ľ��ɸ�Ϥؤ��Ѵ���ȼ����ʬ�黻�Ҥ��Ѵ�§�Ĥ��Ƥ�, ��Ͽ A �ˤޤȤ᤿.
�����Ǥ�, ��Ͽ A �ˤ�����Ǥ�դα�ľ��������Ω�ѿ�$\zeta$������$p$�Ǥ���Ȥ���, ���η�̤��Ѥ��뤳�Ȥˤ���.
��Ͽ A �� $\dot{\zeta}$ �Ϥ��ξ��$ dp/dt $�Ȥʤ�, �����$\omega$���������.
����$\omega$��, $p$��ɸ�ϤǤα�ľ®�٤�ɽ�����Ȥˤʤ�.
$p$��ɸ�Ϥˤ�����饰��󥸥���ʬ��, ��Ͽ A ��\Deqref{lagrange_derivative_zeta}���鼡�Τ褦�˽񤱤�.
%
\begin{equation}
 \Deqlab{lagrange_derivative_p}
 \DD{}{t} = 
  \left( \DP{}{t} \right)_{p} + \tilde{u} \left( \DP{}{x} \right)_{p} + \tilde{v} \left( \DP{}{y} \right)_{p} + \tilde{\omega} \DP{}{p}
\end{equation}
%
������, ź������$p$���������̾�Ǥζɽ���ʬ��ɽ�路�Ƥ���.
�ޤ�, ������ΤĤ��Ƥ����ѿ���, ��Ω�ѿ�$(x,y,p,t)$�ν�°�ѿ��Ǥ��뤳�Ȥ�ɽ���Ȥ���.

\paragraph*{��ư������\Deqref{motion_eq_zcd}�α�ľ��ɸ�Ѵ�}
��®�ٹ�Υ饰��󥸥���ʬ��, \Deqref{lagrange_derivative_p}�ˤ�ä��Ѵ������.
���Ϸ����Ϥι��, ��Ͽ A ��\Deqref{rule_delx_z_to_zeta}, �ſ尵ʿ�դμ�\Deqref{hydrodyn_eq}���Ѥ����
���Τ褦���Ѵ������.
%
\begin{equation}
 \begin{split}
  - \dfrac{1}{\rho} \DP[][z]{\,p}{x} 
   &= - \dfrac{1}{ \tilde{\rho} } \left[\DP[][p]{\,\tilde{p}}{x} - \DP{p}{p} \DP{p}{z} \DP[][p]{\,\tilde{z}}{x}\right] \\
   &= - g \DP[][p]{\,\tilde{z}}{x} = - \DP[][p]{\,\tilde{\Phi}}{x}
 \end{split}
\end{equation}
%
������, $\tilde{\Phi}(=gz)$�ϥ����ݥƥ󥷥��Ǥ���.

�������ä�, $p$�Ϥˤ����뱿ư��������,
%
\begin{equation}
\Deqlab{motion_eq_pcd0}
\DD{ \tilde{\Dvect{v}} }{t} = - \nabla_{p} \tilde{\Phi} - f \Dvect{k} \times \Dvect{\tilde{v}} + \Dvect{\tilde{F}}  \\
\end{equation}
%
�Ȥʤ�.
������,
%
\begin{equation*}
\nabla_{p}= \left( \DP[][p]{}{x}, \DP[][p]{}{y} \right).
\end{equation*}


\paragraph*{�ſ尵ʿ�դμ�\Deqref{hydrodyn_eq}�α�ľ��ɸ�Ѵ�}
��Ͽ A ��\Deqref{rule_delz_z_to_zeta}�ˤ�����, $\psi=z$, $\tilde{\psi}=\tilde{z}$�Ȥ����,
%
\begin{equation*}
 \begin{split}
  \DP{z}{z} &= \DP{\tilde{z}}{p} \DP{p}{z} \\
            &= - \tilde{\rho} g \DP{\tilde{\Phi}}{p} = - \tilde{\rho} \DP{\tilde{\Phi}}{p} = 1
 \end{split}
\end{equation*}
%
�Ȥʤ�.
��ä�, ���۵��Τξ���������\Deqref{ideal_gas_law_eq}���Ѥ����, $p$ �Ϥˤ������ſ尵ʿ�դμ�
%
\begin{equation}
\Deqlab{hydrodyn_eq_pcd0}
 \DP{\tilde{\Phi}}{p} = - \dfrac{R\tilde{T}}{p}
\end{equation}
������.

\paragraph*{Ϣ³�μ�\Deqref{continious_eq_zcd}�α�ľ��ɸ�Ѵ�}
Ϣ³�μ�\Deqref{continious_eq_zcd}
%
\begin{equation*}
\Deqlab{continious_eq_zcd1}
 - \dfrac{1}{\rho} \left(\DD{\rho}{t}\right)_z 
  = \DP[][z]{\, u}{x} + \DP[][z]{\, v}{y} + \DP{w}{z}
\end{equation*}
%
��$z$�ϤǤγ���ʬ���Ф���, \Deqref{lagrange_derivative_p}, ��Ͽ A ��\Deqref{rule_delx_z_to_zeta},
\Deqref{rule_dely_z_to_zeta}, \Deqref{rule_delz_z_to_zeta}��Ŭ�Ѥ���.
�ޤ�, ��Ͽ A ��\Deqref{zeta_dot_relation}���,
%
\begin{equation*}
\tilde{\omega} =
 - \DP{p}{z} \biggl \{ \DP[][p]{\, \tilde{z} }{t} + \tilde{u} \DP[][p]{\, \tilde{z} }{x} + \tilde{v} \DP[][p]{\, \tilde{z}}{y} - \tilde{w} \biggr \}
\end{equation*}
%
�Ǥ���Τ�, �ſ尵ʿ�դμ����Ѥ����$w$ ��$\omega$�δط���
\begin{equation*}
\tilde{w} = - \dfrac{\tilde{\omega}}{\rho g} 
 + \DP[][p]{\,\tilde{z}}{t} + \tilde{u} \DP[][p]{\,\tilde{z}}{x} + \tilde{v} \DP[][p]{\,\tilde{z}}{y}
\end{equation*}
%
������.
�������ä�, Ϣ³�μ���,
%
\begin{equation}
 \begin{split}
  - \dfrac{1}{\tilde{\rho}} & \left[ \DP[][p]{\,\tilde{\rho}}{t} + \tilde{u} \DP[][p]{\,\tilde{\rho}}{x} + \tilde{v} \DP[][p]{\, \tilde{\rho} }{y} + \tilde{\omega} \DP{ \tilde{\rho} }{p} \right] \\
     = & \DP[][p]{\tilde{u}}{x} + \DP[][p]{\tilde{v}}{y} + \tilde{\rho} g \left[ \DP[][p]{\,\tilde{z}}{x}  \DP{\tilde{u}}{p} + \DP[][p]{\,\tilde{z}}{y} \DP{\tilde{v}}{p} \right] \\
     &+ \DP{\tilde{\omega}}{p} - \dfrac{\tilde{\omega}}{\tilde{\rho}} \DP{\tilde{\rho}}{p} - \tilde{\rho} g \DP{}{p} \left[ \DP[][p]{\,\tilde{z}}{t} + \tilde{u} \DP[][p]{\,\tilde{z}}{x} + \tilde{v} \DP[][p]{\,\tilde{z}}{y} \right]
 \end{split}
\end{equation}
%
�Ȥʤ�. �����, �弰�α�����$6$��򤵤��Ÿ�������,
%
\begin{equation*}
- \dfrac{1}{\tilde{\rho}} \left[ \DP[][p]{\,\tilde{\rho}}{t} + \tilde{u} \DP[][p]{\, \tilde{\rho}}{x} + \tilde{v} \DP[][p]{\,\tilde{\rho}}{y} \right] 
 - \tilde{\rho} g \left[ \DP{\tilde{u}}{p} \DP[][p]{\,\tilde{z}}{x} + \DP{\tilde{v}}{p} \DP[][p]{\,\tilde{z}}{y} \right]
\end{equation*}
%
�Ȥʤ�Τ�, ¿���ι���Ǥ��ä��������Ȥ�ʬ����.

�ǽ�Ū��, $p$�Ϥˤ�����Ϣ³�μ��ϼ��Τ褦�ʴʷ�ʷ���Ϳ������.
%
\begin{equation}
\Deqlab{continious_eq_pcd0}
 \DP[][p]{\,\tilde{u}}{x} + \DP[][p]{\,\tilde{v}}{y} + \DP{\tilde{\omega}}{p} = 0 .
\end{equation}
%
��������$z$�ϤǤ��󰵽�ή�Τ��Ф���Ϣ³�μ��Τ褦�Ǥ��뤬, �����Ǥ�, �󰵽�ή�Τβ�����֤��Ƥ��ʤ����Ȥ����դ��줿��.

\paragraph{Ǯ�ϳ�Ū���ͥ륮����¸�μ�\Deqref{thermodyn_energy_conserve_eq}�α�ľ��ɸ�Ѵ�}
�饰��󥸥���ʬ��\Deqref{lagrange_derivative_p}�ˤ�äƽ񤫤�뤬, $p$�Ϥˤ�����
Ǯ�ϳ�Ū���ͥ륮����¸�μ���, ��������$z$ �ϤΤ�Τ��Ѥ��ʤ�.
%
\begin{equation}
 \Deqlab{thermodyn_energy_conserve_eq_pcd0}
 c_{v} \DD{\tilde{T}}{t} + p \DD{\tilde{\alpha}}{t} = \dot{\tilde{Q}}
\end{equation}
