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%\markright{1-9 ���Ȥ��䤹����ľ��ɸ��}
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\section{���Ȥ��䤹����ľ��ɸ��}

�⤵$z$(�ޤ���Ⱦ��$r$)���Ѥ�����ɸ�Ϥ�
¿������Ū��������Ǥ�äȤ������ʺ�ɸ�ϤȤϸ����ʤ�.
��������������Ȥ��Ƥ���¾�α�ľ��ɸ�ϤȤ��Ƥ�,
����$p, \ \ln p/p_0$, �Ͼ尵�Ϥǵ��ʲ����줿����
$\sigma = p/p_s$,
�����Ʋ���$\theta$�κ�ɸ�Ϥ�����.

���̲�������ľ��ɸ��$\zeta$ [Kasahara (1974)
]��ͤ���.
����$\zeta$�Ϲ⤵$z$�˴ؤ�����
\footnote{
��Ĥ�$z$���Ф��ư�Ĥ�$\zeta$����ޤ뤳��.
}
��ñĴ�ؿ�
\footnote{
 {\bfseries ñĴ�ؿ�}�Ȥ�, ñĴ���äޤ���ñĴ��������褦�ʴؿ��Τ���.
}
�ˤ�ä�����Ǥ���.
$z$��ɸ�Ϥˤ����Ƥ�, $\zeta$��$x, y, z$������$t$�δؿ��Ǥ���.
�Ĥޤ�, $\zeta=\zeta (x,y,z,t)$�Ǥ���.
¾�θ������򤹤��, $\zeta$�α�ľ��ɸ�ϤǤ�,
$z$�Ͻ�°�ѿ�, �Ĥޤ�$z=z(x,y,\zeta,t)$�Ȥʤ�.
�����륹���顼(�ޤ��ϥ٥��ȥ�)�ν�°�ѿ�
(�����Ǥϲ���$A$�Ȥ���)�Ϥɤ���κ�ɸ�Ϥˤ����Ƥ�
$A(x,y,z,t)$�ޤ���$A(x,y,\zeta,t)$��ɽ���Ǥ���.
�ʤ��ʤ��, �����δؿ���$z$�ޤ���$\zeta$��
����Ȥ��̤κ�ɸ�Ϥδؿ������֤��������
Ʊ���ˤʤ뤫��Ǥ���.
�Ĥޤ�, �ʲ��μ�������Ω�ĤȤ������ȤǤ���.
%
\begin{eqnarray*}
 A(x,y,\zeta ,t) \equiv A(x,y,z(x,y,\zeta ,t),t).
\end{eqnarray*}
%
������$s$�Ǥ�����ʬ����夲��.
������$s$�Ȥ�$s=x,$��$y,$, $t$�Ǥ���.
�����,
%
\begin{eqnarray}
 \left(
  \DP{A}{s}
 \right)_{\zeta}
 =
 \left(
  \DP{A}{s}
 \right)_{z}
 +
 \DP{A}{z}
 \left(
  \DP{z}{s}
 \right)_{\zeta}
\end{eqnarray}
%
�Ȥʤ�
.
�ʤ�, ź����$z$��$\zeta$�Ϥ��줾��κ�ɸ�ϤǤ�
��ΤǤ��뤳�Ȥ򼨤�
\footnote{
���μ��˴ؤ��Ƥ�{\bf 2-9-A}�򻲾�.
}.
Ʊ�ͤ˱�ľ����������ʬ�ϰʲ��Τ褦�ʴط��ˤ���.
%
\begin{eqnarray}
 \DP{A}{\zeta}=\DP{A}{z}\DP{z}{\zeta}
  \quad {\rm  �ޤ�, �̤ʷ��Ǥ�, } \quad
 \DP{A}{z}=\DP{A}{\zeta}\DP{\zeta}{z}.
\end{eqnarray}
%
(2-42)��(2-41)�����������,
���η�̤ϰʲ��Τ褦�ˤʤ�.
%
\begin{eqnarray}
 \left(
  \DP{A}{s}
 \right)_{\zeta}
 =
  \DP[][z]{A}{s}
 +
  \DP{A}{\zeta}\DP{\zeta}{z}
  \DP[][\zeta]{z}{s}.
\end{eqnarray}
%
���κǸ�μ���$s=x$��������������$s=y$�������������Ȥ�
­����碌�뤳�Ȥ�$A$�θ���(����ǥ������)�Ȥʤ�.
Ʊ�ͤ�, $A$���Ѥ��˥٥��ȥ���ʬ$\Dvect{B}$���Ѥ����
2����ȯ���μ��Ȥʤ�.
���ǽ񤯤Ȱʲ��Τ褦�ˤʤ�.
%
\begin{eqnarray}
 \nabla_{\zeta}A
  &=& \nabla_{z}A+\DP{A}{\zeta}\DP{\zeta}{z}\nabla_{\zeta}z ,\\
 \nabla_{\zeta}\cdot \Dvect{B} 
  &=& \nabla_{z} \cdot \Dvect{B}
    + \DP{\Dvect{B}}{\zeta}\DP{\zeta}{z} \cdot \nabla_{\zeta}z.
\end{eqnarray}
%
$s=t$�Ȥ������ϰʲ��Τ褦�ˤʤ�.
%
\begin{eqnarray}
 \left(
  \DP{A}{t}
 \right)_{\zeta}
  =
 \left(
  \DP{A}{t}
 \right)_{z}
 +
 \DP{A}{\zeta}\DP{\zeta}{z}
 \left(
  \DP{z}{t}
 \right)_{\zeta}.
\end{eqnarray}
%
�ʤ�, $\zeta$��ɸ�ϤǤ�����ʬ�ϰʲ��Τ褦�ˤʤ�.
%
\begin{eqnarray}
 \DD{A}{t} =
  \left(
   \DP{A}{t}
  \right)_{\zeta}
 + \Dvect{V} \cdot \nabla_{\zeta}A
 + \dot{\zeta}\DP{A}{\zeta}.
\end{eqnarray}
%
���������$\Dvect{V}$��$\zeta$�̤Ǥ�2����®��,
$\dot{\zeta}=d \zeta / dt$��$\zeta$��ɸ�Ϥˤ������ľ®��,
������$A$�ϥ����顼�ޤ��ϥ٥��ȥ�Ǥ���
(�٥��ȥ�ξ��ϥ�������˴ؤ��ƹ�θ�ѤߤǤ���Ȳ��ꤷ�Ƥ���
\footnote{
����Ū�ˤ�, ����μ�(2-27)�Τ褦��®����ʬ�����
����������Ҥ�ޤ��褦�����Τ��ȤǤ���.
}
).

���Ҥμ��ϱ�ư��������$z$��ɸ��ɽ������$\zeta$��ɸ��ɽ����
�ѷ�����Τ����ѤǤ���.
��(2-44)�����Ѥ���ȿ�ʿ�̰��Ϥ�ɽ���ϰʲ��Τ褦���ѷ��Ǥ���.
%
\begin{eqnarray}
 - \alpha \nabla_{z} p
  =
 - \alpha \nabla_{\zeta} p + \alpha \DP{p}{z} \nabla_{\zeta} z
  =
 - \alpha \nabla_{\zeta} p - \nabla_{\zeta} \phi.
\end{eqnarray}
%
������$\phi$��{\bf �����ݥƥ󥷥��}({\itshape geopotential})
\footnote{
ñ�̼��̤�ʪ�Τ��̤����ϵ�ν��Ϥ˹����Ƥ��ι���
�ˤޤǻ����夲��Τ��פ��륨�ͥ륮��[$N \cdot s / kg$].
}
�Ǥ���, $\phi = gz$�Ǥ���
\footnote{
�ſ尵ʿ�դμ�����ʲ��δط�������Ω��.
\begin{eqnarray*}
 \alpha \ \partial p / \partial z = -g.
\end{eqnarray*}
}.

���Τ褦�ˤ���, �໤̵���Ǥο�ʿ�̤α�ư��������
$\zeta$��ɸ�ϤǤϰʲ��Τ褦�ˤʤ�.
%
\begin{eqnarray}
 \DD{\Dvect{V}}{t}
  = - \alpha \nabla _{\zeta} p - \nabla _{\zeta} \phi
    - f \Dvect{k} \times \Dvect{V} + \Dvect{F}.
\end{eqnarray}
%
(2-42)���ſ尵ʿ�դμ�$\alpha(\partial p /\partial z) + g = 0$
�����������,
%
\begin{eqnarray}
 \alpha \DP{p}{\zeta}\DP{\zeta}{z} + g = 0 \qquad
 \textrm{�ޤ���} \qquad
 \alpha \DP{p}{\zeta} + \DP{\phi}{\zeta} = 0
\end{eqnarray}
%
�Ȥʤ�
\footnote{
���μ���ξ�դ�$\partial z / \partial \zeta$��ݤ����
�ʲ��Τ褦�ˤʤäƱ��μ���Ƴ�Ф����.
\begin{eqnarray*}
 \alpha \DP{p}{\zeta} \DP{z}{\zeta}\DP{\zeta}{z} + g \DP{z}{\zeta}
  &=& 0, \\
 \alpha \DP{p}{\zeta} + \DP{\phi}{\zeta}
  &=& 0.
\end{eqnarray*}
}.

����, Ϣ³�μ��κ�ɸ���Ѵ��ˤĤ��ƹͤ���.
Ϣ³�μ����ѷ������,
%
\begin{eqnarray}
 d(\ln \rho) / dt + \nabla_{z} \cdot \Dvect{V} +
  \partial w/ \partial z = 0
\end{eqnarray}
%
�Ȥʤ�
\footnote{
�����Ǥ�, $\nabla_{z}$, $\Dvect{V}$��2����(��ʿ��)�Υ٥��ȥ�Ǥ���,
$d(\ln \rho)/dt = (1/\rho)d \rho /dt$�Ǥ��뤳�Ȥ����Ѥ���
�ѷ���ԤäƤ���.
}.
�ޤ��Ϥ�˼�(2-42), (2-45)��ȯ���ι�
((2-51)�α�����2, 3��)�����Ѥ����,
%
\begin{eqnarray}
 \nabla_{z} \cdot \Dvect{V} + \DP{w}{z}
  =
 \nabla_{\zeta} \cdot \Dvect{V}
 - \DP{\zeta}{z}\DP{\Dvect{V}}{\zeta} \cdot \nabla_{\zeta} z
 + \DP{w}{\zeta}\DP{\zeta}{z}
\end{eqnarray}
%
�Ȥʤ�. �����,
%
\begin{eqnarray}
 w = dz/dt = (\partial z / \partial t)_{\zeta}
   + \Dvect{V} \cdot \nabla _{\zeta}z
   + \dot{\zeta} \partial z / \partial \zeta
\end{eqnarray}
%
�ʤΤ�
\footnote{
��(2-47)�����Ѥ��Ƥ���.
}
, ���μ���$\zeta$������ʬ����Ȱʲ��Τ褦�ˤʤ�.
%
\begin{eqnarray}
 \DP{w}{\zeta}
  =
 \DP{}{t} \left(
	   \DP{z}{\zeta}
	  \right)
 + \Dvect{V} \cdot \nabla _{\zeta} \DP{z}{\zeta}
 + \DP{\Dvect{V}}{\zeta} \cdot \nabla_{\zeta} z
 + \DP{\dot{\zeta}}{\zeta}\DP{z}{\zeta}
 + \dot{\zeta}\DP{}{\zeta}
   \left(
    \DP{z}{\zeta}
   \right).
\end{eqnarray}
%
��(2-52)�˼�(2-54)����������Ȱʲ��μ���Ƴ�����.
%
\begin{eqnarray}
 \nabla_{z} \cdot \Dvect{V} + \DP{w}{z}
  =
 \nabla_{\zeta} \cdot \Dvect{V}
 + \DP{\zeta}{z}
    \left(
     \DP{}{t} + \Dvect{V} \cdot \nabla_{\zeta} + \dot{\zeta}\DP{}{\zeta}
    \right) \DP{z}{\zeta}
 + \DP{\dot{\zeta}}{\zeta}.
\end{eqnarray}
%
���ξ�, $\partial \zeta / \partial z = (\partial z /\partial \zeta)^{-1}$
�Ǥ��뤳�Ȥ���, ���դο���ι�Ϥ��礦��
$d[\ln (\partial z / \partial \zeta)]/dt$�Ȥʤ�
\footnote{
$d/dt = \partial/ \partial t + \Dvect{V} \cdot \nabla_{\zeta}
+ \dot{\zeta} (\partial / \partial \zeta)$
�Ǥ��뤳�Ȥ�����.
}.
���ι��(2-51)��$d(\ln \rho) /dt$���Ȥ߹�碌�뤳�Ȥ��Ǥ���
�ʲ��Τ褦�ˤʤ�
%
\footnote{
�ſ尵�μ�$dp = - \rho g dz$����
%
\begin{eqnarray*}
 \partial p / \partial \zeta = - \rho g \ \partial z / \partial \zeta.
\end{eqnarray*}
%
���μ��μ����п���Ȥ��
%
\begin{eqnarray*}
 \ln(\partial p / \partial \zeta)
  = \ln(\rho \ \partial z / \partial \zeta)
    + \ln(-  g ).
\end{eqnarray*}
%
���μ��������ʬ�����, ���ϲ�®��$g$�����Ǥ��뤳�Ȥ���
%
\begin{eqnarray*}
 d[\ln(\partial p / \partial \zeta)]/dt
  = d[\ln(\rho \ \partial z / \partial \zeta)] /dt.
\end{eqnarray*}
%
}.
\begin{eqnarray*}
 d [\ln (\rho \ \partial z / \partial \zeta)]/dt
  = d(\ln \partial p/\partial \zeta)/dt.
\end{eqnarray*}
%
���Τ褦�ˤ���, �ѷ����줿Ϣ³�μ��ϰʲ��Τ褦�ˤʤ�.
%
\begin{eqnarray}
 \DD{}{t}
  \left(
   \ln \DP{p}{\zeta}
  \right)
 + \nabla _{\zeta} \cdot \Dvect{V} + \DP{\dot{\zeta}}{\zeta} = 0.
\end{eqnarray}
%
�Ǹ��Ǯ�ϳؤμ��˴ؤ��ƤǤ���.
Ǯ�ϳؤμ��ϱ�ľ��ɸ�Ϥ��Ѵ���ԤäƤ⸫���ܤ�
�Ѥ��ʤ�.
�����Ǯ�ϳؤμ����ʲ��Τ褦�˲���
$\theta = T(p_0/p)^{\kappa}$������ʬ��$T$
��ɽ������뤫��Ǥ���.
%
\def\theequation{1-57\alph{equation}}
\setcounter{equation}{0}
%
\begin{eqnarray}
 c_p T (d \ln \theta)/dt = Q, 
\end{eqnarray}
%
�ޤ���
%
\begin{eqnarray}
 c_p \DD{T}{t}- \alpha \DD{p}{t} = Q.
\end{eqnarray}
%
\def\theequation{\arabic{chapter}-\arabic{equation}}
\setcounter{equation}{57}
%
���ǤϤ����ΰ��̲������������Ϥ�
����ξ��˱��Ѥ��Ƥߤ�.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% 2-9-1 Pressure Vertical Coordinate %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{���ϱ�ľ��ɸ��}

$\zeta = p$�Ǥ������ͤ���.
���ξ��, $\nabla_{\zeta} p = \nabla_{p} p = 0$�Ǥ���
, $\partial p / \partial p = 1$�Ǥ��뤿��,
��ư������(2-49)���ſ尵�μ�(2-50)�ϰʲ��Τ褦�ˤʤ�.
%
\begin{eqnarray}
 \DD{\Dvect{V}}{t}
  &=& - \nabla_{p} \phi
      - f \Dvect{k} \times \Dvect{V} + \Dvect{F}, \\
 \DP{\phi}{p}
  &=& - \alpha.
\end{eqnarray}
%
Ϣ³�μ�(2-56)�Ǥ�, $\zeta = p$�ˤ��
$\ln (\partial \zeta/ \partial p) = \ln (\partial p / \partial p)
= 0$�Ȥʤ뤿����1��Ͼä�,
%
\begin{eqnarray}
 \nabla_p \cdot \Dvect{V} + \partial \omega / \partial p =0 
\end{eqnarray}
%
�Ȥʤ�. ������
$\omega = dp/dt = \dot{p}$��$p$-��ɸ�ϤǤα�ľ®�٤Ǥ���.
�ʤ�, ������������{\bf ����}({\it linear\/})�Ǥ��뤳�Ȥ�����.
���μ���$p=0$(�����Ǥ�$\omega =0$
)
����Ǥ�դΰ���$p$����Ĺ⤵�ޤǤ��ϰϤ���ʬ�����
$\omega$����뼰�Ȥʤ�.
%
\begin{eqnarray}
 \omega = - \int^{p}_{0} \nabla_{p} \cdot \Dvect{V} dp.
\end{eqnarray}
%
��ɽ�̤Ǥϱ�ľ������ή���̵���Ȥ���
{\bf ��ư��Ū�������}({\it kinematic boundary condition\/})���ꤹ���,
%
\begin{eqnarray}
 \omega_{s} = dp_{s}/dt = \partial p_{s} / \partial t
              + \Dvect{V}_{s} \cdot \nabla p_{s}
\end{eqnarray}
%
�Ȥʤ�
\footnote{
���μ���Ƴ�Ф˴ؤ��Ƥ�{\bf 2-9-B}����.
}.
���Τ��Ȥ���, $p_{s}(x,y,t)$�ޤǤ���ʬ��Ԥ���,
��ɽ�̰��Ϥ˴ؤ���ʲ�����������������.
%
\begin{eqnarray}
 \DP{p_{s}}{t} + \Dvect{V}_{s} \cdot \nabla  p_{s}
  = - \int ^{p_{s}} _{0} \nabla _{p} \cdot \Dvect{V} dp.
\end{eqnarray}
%
�Ǹ��Ǯ�ϳؤμ��˴ؤ��ƤǤ���.
Ǯ�ϳؤμ�(2-57b)��$dp/dt$��$\omega$���֤������뤳�Ȥ��Ǥ�,
¾���Ѳ������Τ�̵��.

$p$-��ɸ�Ϥ�$z$�Τ�Τ��ͥ��Ƥ���Ȥ�����
Ϣ³�μ�(2-60)�������Ǥ��뤳�Ȥ�,
��(2-58)�ˤ����ư��Ϲब����������Ƥ���Ȥ����Ǥ���
\footnote{
����ϤĤޤ�,
$- \alpha \nabla_{z}p = - \alpha \nabla_{p} - \nabla_{p}\phi
\simeq - \nabla_{p}\phi$
�ȤʤäƤ��뤳�Ȥ�ؤ��Ƥ���.
}.
��Ԥ�̩�٤����Ǥ���Τ�
\footnote{
$p$-��ɸ�Ϥ�$z$-��ɸ�ϤǤ��줾�첹�����μ�����Ƥߤ��,
�����Ǹ�����̩�٤����Ǥ���٤ΰ�̣���狼�������.
$p$-��ɸ�ϤǤβ������μ���Ƴ�Ф�{\bfseries 2-9-C}��,
$z$-��ɸ�ϤǤβ������μ���Ƴ�Ф�{\bfseries 2-9-D}��
���ȤΤ���.
},
�Ϲ����Ȳ������δط���
��ñ��ɽ���Ǥ���.
�㤨�аʲ��Τ褦�ʼ��������Ǥ���
\footnote{
$p$-��ɸ�ϤǤα�ư������(2-58)�����Ϲ����δط��������
�������δط�����Ƴ�Ф�����ˡ��{\bf 2-9-C}�򻲾�.
}.
%
\begin{eqnarray*}
 \DP{\Dvect{V}_g}{p} = f^{-1} \Dvect{k} \times \nabla \DP{\phi}{p} \qquad
 {\rm and} \qquad
 \Delta \Dvect{V}_g = f^{-1} \Dvect{k} \times \nabla \ \Delta \phi.
\end{eqnarray*}
%
���ʤߤ�, 2���ܤμ���$\Delta$�Ϻ�ʬ���̣����.

\medskip

\subsection{���ʲ����Ϻ�ɸ�� $\sigma = p / p_s$ (Phillips, 1957)}

���α�ľ��ɸ�Ϥ�ͤ�����, ��ɽ�̰���$p_s = p_s (x,y,t)$
��פ��Ф����ߤ���.
��(2-48)��$\zeta = \sigma$�Ȥ����,
%
\begin{eqnarray*}
 - \alpha \nabla_{z} p
 = - \alpha \nabla_{\sigma} p
   - \alpha (\partial p / \partial z) \nabla_{\sigma} z
 = - \alpha \nabla_{\sigma} (\sigma p_s) - \nabla_{\sigma}\phi
\end{eqnarray*}
%
�Ȥʤ�.
$\alpha \nabla_{\sigma}(\sigma p_s) = \alpha \sigma \nabla p_s$
�Ȥʤ뤳�Ȥ���
, ��ư������(2-49)�ϰʲ��μ��Ȥʤ�
\footnote{
�������������,
$\alpha = RT/p$�ʤΤ�, $\alpha \sigma = RT/p_s$
}.
%
\begin{eqnarray}
 \DP{\Dvect{V}}{t}
 + \Dvect{V} \cdot \nabla_{\sigma} \Dvect{V}
 + \dot{\sigma} \DP{\Dvect{V}}{\sigma}
 =
 - \nabla_{\sigma} \phi
 - (RT/p_s) \nabla p_s - f \Dvect{k} \times \Dvect{V} + \Dvect{F}.
\end{eqnarray}
%
�ſ尵�μ�(2-50)�Ǥ�
$\partial p/ \partial \sigma = \partial (\sigma p_s) / \partial \sigma
= p_s$�Ȥʤ�Τ�
\footnote{
�����Ǥ�$p_s= p_s (x,y,t)$�Ǥ��뤳�Ȥ���
$\partial p_s / \partial \sigma = 0$�Ǥ���.
},
%
\begin{eqnarray}
 \partial \phi / \partial \sigma + \alpha p_s = 0
\end{eqnarray}
%
�Ȥʤ�.

Ϣ³�μ�(2-56)����1���$d(\ln \partial p/ \partial \sigma)/dt$
�Ȥʤ�,
$\partial p/ \partial \sigma = p_s$�Ȥʤ뤿��
$\sigma$-��ɸ�Ϥ�Ϣ³�μ��ϰʲ��Τ褦�ˤʤ�.
%
\begin{eqnarray}
 d(\ln p_s)/dt + \nabla_{\sigma} \cdot \Dvect{V}
 + \partial \dot{\sigma} /\partial \sigma = 0 .
\end{eqnarray}
%
��1��$\dot{p}_s/p_s$��Ÿ�����������˰ʲ��μ���������.
%
\begin{eqnarray*}
 \partial p_s / \partial t
 + \Dvect{V} \cdot \nabla p_s
 + \dot{\sigma} \partial p_s / \partial \sigma
 + p_s \nabla_{\sigma} \cdot \Dvect{V}
 + p_s \partial \dot{\sigma} / \partial \sigma
 = 0 .
\end{eqnarray*}
%
$p_s$��$\sigma$�˴ؤ��Ƥ��Ѳ����ʤ��Τ�,
$\partial p_s / \partial \sigma = 0$�Ǥ���.
���Τ��Ȥ����μ���
%
\begin{eqnarray}
 \partial p_s / \partial t
  =
 - \nabla \cdot (p_s \Dvect{V})
 - \partial (p_s \dot{\sigma})/\partial \sigma
\end{eqnarray}
%
�Ȥʤ�.
``��ľ®��''$\dot{\sigma}=d \sigma/dt$�ζ�������
%
\begin{eqnarray}
 ��ɽ��    \hspace{6mm} ��\qquad
  p = p_s �� \qquad
  ��ä� \qquad
  \sigma = 1, \quad \dot{\sigma}=0,
 \nonumber \\
 �絤��ü  \hspace{3mm} ��\qquad
  p = 0 \ �� \qquad
  ��ä� \qquad
  \sigma = 0, \quad \dot{\sigma}=0
\end{eqnarray}
%
�Ȥʤ�.
���ζ�����ﲼ��(2-67)���絤����($0 \leq \sigma \leq 1$)
����ʬ�����,
%
\begin{eqnarray}
 \DP{p_s}{t} = - \int^{1}_{0} \nabla \cdot (p_s \Dvect{V}) d \sigma
\end{eqnarray}
%
�Ȥʤ�
\footnote{
(2-67)��$0 \leq \sigma \leq 1$����ʬ�����,
%
\begin{eqnarray*}
 \int^{1}_{0} \DP{p_s}{t} d \sigma
  &=&
  - \int^{1}_{0} \nabla \cdot (p_s \Dvect{V}) d \sigma
  - \int^{1}_{0} \DP{}{\sigma} (p_s \dot{\sigma}) d \sigma \\
  &=&
  - \int^{1}_{0} \nabla \cdot (p_s \Dvect{V}) d \sigma
  - \left[
     p_s \dot{\sigma}
    \right]^{1}_{0} \\
\end{eqnarray*}
%
�Ȥʤ�,
(2-68)��� $\dot{\sigma}=0$ at $\sigma=0, \ 1$�ʤΤ�
��(2-69)�Ȥʤ�.
}.
�������ɽ�̰��Ϥ˴ؤ����������Ǥ���.
$\sigma=0$����Ǥ�դ���$\sigma$�ޤ���ʬ��������
$\sigma$�ˤ������ľ®��$\dot{\sigma}$��ޤ��������Ȥʤ�.
%
\begin{eqnarray}
 \sigma \DP{p_s}{t}
  + \int^{\sigma}_{0} \nabla \cdot (p_s \Dvect{V}) d \sigma
  = - p_s \dot{\sigma}.
\end{eqnarray}
%
��(2-69)����$\partial p_s/\partial t$���ͤ�����줿��,
�����Ȥä�(2-70)����$\dot{\sigma}$����뤳�Ȥ��Ǥ���.
���٤Ϥ���$\dot{\sigma}$��ȤäƱ�ư������(2-64)��
$\dot{\sigma} \partial \Dvect{V} / \partial \sigma$��
�ͤ���뤳�Ȥ��Ǥ���.
Ǯ�ϳؤμ�
\footnote{
(2-57b)���,
\begin{eqnarray*}
 c_p \DD{T}{t} - \alpha \DD{(\sigma p_s)}{t} = Q.
\end{eqnarray*}
}
�ˤĤ��Ƥ�Ʊ�ͤǤ���.
�����κ�Ȥ�, ���͵���ͽ���ǥ�ˤ�����,
$\Dvect{V}, T$�ʤɤ��ͤζɽ�Ǥλ����Ѳ���
����Τ�ɬ�פʼ��ΰ�ĤǤ���.

��ɽ�̤ˤ�����$\dot{\sigma}=0$�Ȥʤ�Τ�,
��ɽ�̤Ǥ�ή�����ɽ�˱�äƤ��ʤ���Фʤ�ʤ�
(�Ĥޤ�, ��ľ��ʬ�ϥ����Ǥʤ���Фʤ�ʤ�)
����Ǥ���.
���δ�ñ�ʲ���������郎
$\sigma$-��ɸ�Ϥ�$p$-��ɸ�Ϥ���ͥ��Ƥ������Ǥ���
\footnote{
����Ū�ʤ��Ȥ˴ؤ��Ƥ�{\bfseries 2-9-E}�򻲾�.
}.
���δ�ñ�ʲ����������η��, �ʷ�ʼ�(2-69)��¸�ߤ�,
��ɽ�̰��Ϥλ����Ѳ���Ƴ�Ф����Ѥ����.





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% 2-9-2 Isentropic Vertical Coordinate %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{������ȥ��ԡ���ľ��ɸ��}

������ȥ��ԡ���(����ȥ��ԡ����������),
�ޤ���(Ʊ����̣����)�����̤���ľ��ɸ�Ȥ��ư���Ū��
�ʤä��Τ�1930��1940ǯ��Τ��ȤǤ���.
Ĺ���֤ۤȤ�ɻȤ��Ƥ��ʤ��ä���
1970ǯ��Ⱦ�Ф���$\theta$-��ɸ�Ϥ������ܤ���Ӥ뤳�Ȥˤʤä�.
$\theta$-��ɸ�Ϥ�̥��Ū�ʤȤ�����,
������Ǯ({\it dry adiabatic\/})��ư�ˤ����Ƥ�
���줾���{\bfseries �ѡ�����}({\itshape parcel})
\footnote{
{\bfseries �ѡ�����}({\itshape parcel})�Ȥ�
����Ū�ʵ����Τ��ȤǤ���.
���Υѡ�����Ȥ����ͤ������Ѥ����, �絤�ϥѡ�����ν����Τ���
�������Ȥ��Ǥ���.
}
�ˤ����Ʋ��̤���¸�����Ȥ����Ǥ���.
�Ĥޤ�$d \theta /dt =0$�Ȥʤ�.
�ޤ�, ������ȥ��ԡ��̤����Τޤ�
{\bfseries ʪ����}({\itshape material surface})�Ȥʤ� 
\footnote{
��ʪ���̡פ˴ؤ��Ƥ�{\bfseries 2-9-F}����.
}.
������, ���������Ǯ({\itshape diabatic})�ʾ�ﲼ�Ǥ�����Ω���ʤ�.
���κ�ɸ�Ϥ�¾�ˤ�ͥ�줿��ħ������,
��Ǯ��ư�θ���˸¤餺���������Ѥ���Ƥ���.
�㤨��, �����ն���ͻҤʤɤ�$\theta$-��ɸ�Ϥ�
�ɤ�ɽ�����Ȥ��Ǥ���.

(2-44)���Ѥ���
��ʿ�̰��Ϥ����٤Τ褦���ѷ������,
%
\begin{eqnarray}
 - \alpha \nabla_{z} p
 =
 - \alpha \nabla_{\theta} p
 - \alpha \DP{p}{z} \nabla_{\theta} z
 =
 - \alpha \nabla_{\theta} p
 + g \nabla_{\theta} z
\end{eqnarray}
%
�Ȥʤ�.
$\theta = T (p_0 / p)^{\kappa}$�Ǥ��뤳�Ȥ���,
$\nabla_{\theta} \ln T - (R/c_p) \nabla_{\theta} \ln p
= \nabla_{\theta} \ln \theta = 0$�Ȥʤ�.
���Τ��Ȥ���
$- \nabla_{\theta} p = - (p c_p / RT) \nabla_{\theta} T$
������Ω��.
��äƿ�ʿ�̰���(2-71)�ϰʲ��Τ褦�˽�ɽ����.
%
\begin{eqnarray*}
 - \alpha \nabla_{z} p = - \nabla_{\theta}(c_p T)
  - \nabla_{\theta} \phi.
\end{eqnarray*}
%
���μ����ʿ�̱�ư����������������Ȱʲ��μ���������.
%
\begin{eqnarray}
 \DP{\Dvect{V}}{t} + \Dvect{V} \cdot \nabla_{\theta} \Dvect{V}
  + \dot{\theta} \DP{\Dvect{V}}{\theta}
  =
 - \nabla_{\theta} (c_p T + \phi) - f \Dvect{k} \times \Dvect{V}.
\end{eqnarray}
%
ʪ����$c_p T + \phi = M$������
{\bf ��󥴥�꡼ή���ؿ�}({\it Montgomery stream-function\/})
�ȸƤФ��.
���̤����������$p$��$T$����Ф����п�����,
���θ���ʬ������ſ尵�μ������ѤǤ���.
�ޤ�, ���̤������$\theta = T(p_0/p)^{\kappa}$���п���Ȥä�
��ʬ�����,
%
\begin{eqnarray*}
 \frac{1}{\theta}
  = \frac{1}{T} \DP{T}{\theta}
    - \frac{R}{p \ c_p} \DP{p}{\theta}
\end{eqnarray*}
%
�Ȥʤ�
\footnote{
�⤦�����ܺ٤˽�.
�ޤ�, ���̤���������п���Ȥ�.
%
\begin{eqnarray*}
 \ln \theta = \ln T + \kappa (\ln p_0 - \ln p).
\end{eqnarray*}
%
������$\theta$����ʬ�����,
%
\begin{eqnarray*}
 \DP{(\ln \theta)}{\theta}
 &=&
 \DP{(\ln T)}{\theta}
 + \kappa
    \left\{
     \DP{(\ln p_0)}{\theta} - \DP{(\ln p)}{\theta}
    \right\}, \\
 \frac{1}{\theta}
 &=&
 \frac{1}{T} \DP{T}{\theta}
 + \frac{\kappa}{p} \DP{p}{\theta}
\end{eqnarray*}
%
�Ȥʤ�, ��ʸ�μ��Ȥʤ�.
}.
�����ſ尵�μ�(2-50)�����Ѥ���
$\partial p / \partial \theta$��
$- \rho \partial \phi / \partial \theta$���֤��������,
%
\begin{eqnarray}
 \DP{}{\theta} (c_p T + \phi) = \frac{c_p T}{\theta}
\end{eqnarray} 
%
�Ȥʤ�.
�Ĥޤ�, ��󥴥�꡼ή���ؿ�$M$��$\theta$�˴ؤ��Ƥ�
$c_p T / \theta$���Ѳ�����.

����Ϣ³�μ��˴ؤ��ƤǤ���.
Ϣ³�μ�(2-56)�ϰʲ��μ��ˤʤ�.
%
\begin{eqnarray}
 \DD{}{t}
  \left(
   \ln \DP{p}{\theta}
  \right)
 + \nabla_{\theta} \cdot \Dvect{V}
 + \DP{\dot{\theta}}{\theta}
 = 0.
\end{eqnarray}
%
���μ���Ÿ������$\theta$����ʬ�����
��ɽ�̰��Ϥλ����Ѳ�$\partial p_s / \partial t$
�˴ؤ��뼰��������
\footnote{
���μ���Ƴ�Ф˴ؤ��Ƥ� {\bfseries 2-9-G}�򻲾ȤΤ���.
}.
Ǯ�ϳؤμ�(2-57a)�ˤ��Ѵ���ɬ��̵��.
$\dot{\theta}$�ζ�������(2-68)��������Ƥ���,
�ʲ��Τ褦�ˤʤ�
\footnote{
$\theta_{{\rm top}}$�ϲ��̥�ǥ��ĺ���β���
(Kasahara(1974)����).
}.
%
\begin{eqnarray}
 \dot{\theta} = 0 \ \hspace{31mm} &{\rm at}& \ \theta = \theta_{{\rm top}}
 \nonumber ,\\
 \dot{\theta} = \partial \theta_{s} / \partial t
  + \Dvect{V}_s \cdot \nabla \theta_s  \qquad  &{\rm at}& \
  \theta = \theta_s
\end{eqnarray}
%




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%             �ʲ���Appendix             %%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%           ���ֹ�Υ������饤��         %%%%%%%% 
\setcounter{equation}{0}
\setcounter{figure}{0}
\def\theequation{A-\arabic{equation}}
\def\thefigure{A-\arabic{figure}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\begin{center}
% {\bf 
% --------------------------------------
% ��ʸ�����ޤ�
% --------------------------------------
% } 
%\end{center}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%      2-9-A      %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{2-9-A\qquad �ۤʤ��ɸ�ϤǤ�����ʬ�δط���}
\setcounter{equation}{0}
\setcounter{figure}{0}
\def\theequation{2-9-A-\arabic{equation}}
\def\thefigure{2-9-A-\arabic{figure}}


(2-41)��ͤ���.
\Dfigref{hw-01-09-z��ɸ�Ϥ�zeta��ɸ�Ϥ���ʬ�μ����}
�ǹͤ��Ƥߤ褦.
���줾���� a, b, c �Ǥ�$A$��$A_a$, $A_b$, $A_c$�Ȥ���,
a �� c, a �� b, b �� c �򤽤줾����ʬ���ǽ�ɽ���Ƥߤ��,
%
\begin{eqnarray}
 {\rm a �� c:} \hspace{10mm} & &
  \frac{A_c - A_a}{dx} \rightarrow
  \DP[][\zeta]{A}{x}
  \nonumber \\
 {\rm a �� b:} \hspace{10mm} & &
  \frac{A_b - A_a}{dx} \rightarrow
  \DP[][z]{A}{x}
  \nonumber \\
 {\rm b �� c:} \hspace{10mm} & &
  \frac{A_c - A_b}{dz} \DD{z}{x}
  \rightarrow
  \DP{A}{z} \DP[][\zeta]{z}{x}
  \nonumber
\end{eqnarray}
%
�Ȥʤ�. �������Ȥ߹�碌���,
%
\begin{eqnarray}
 \DP[][\zeta]{A}{x} = \DP[][z]{A}{x} + \DP{A}{z} \DP[][\zeta]{z}{x}
  \nonumber
\end{eqnarray}
%
�Ȥ������Ȥʤ�. $x$��$s$���֤������ƹͤ����(2-41)�Ȥʤ�.
�ܤ�������ˤĤ��ƤϾ���(2000)
�򻲾Ȥ��Ƥ�餤����.
%
\begin{figure}[htb]
 \begin{center}
  \Depsf[][]{./eps/alternate_coordinate.eps}
 \end{center}
 \begin{quote}
  \caption{
  {\footnotesize $z$-��ɸ�Ϥ�$\zeta$-��ɸ�Ϥ���ʬ�μ����}
  }
  \Dfiglab{hw-01-09-z��ɸ�Ϥ�zeta��ɸ�Ϥ���ʬ�μ����}
 \end{quote}
\end{figure}
%



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%      2-9-B      %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{2-9-B\qquad ��ɽ�̰���$p_s$������ʬ��Ƴ��}
\setcounter{equation}{0}
\setcounter{figure}{0}
\def\theequation{2-9-B-\arabic{equation}}
\def\thefigure{2-9-B-\arabic{figure}}


(2-62)�˴ؤ��Ƶ�������.
���Ϻ�ɸ�ϤǤ�$p = p(x,y,z,t)$�Ǥ���.
��ɽ�̤ι��٤�$z = H(x,y,t)$�ȹͤ����,
��ɽ�̤ΰ��Ϥ�$p_s = p_s(x,y,H,t) = p_s(x,y,H(x,y,t),t) = p_s(x,y,t)$
�Ȥʤ�.
($p_s$��$p$�Ȥϰۤʤ�, ��ɸ�ǤϤʤ��ؿ��Ǥ��뤳�Ȥ�����).
�����, $p_s$���ѿ�����ͤ���,
%
\begin{eqnarray*}
 \DD{p_s}{t} = \DP{p_s}{t} + u \DP{p_s}{x} + v \DP{p_s}{y}
\end{eqnarray*}
%
�Ȥʤ�. ��̤Ȥ���(2-62)������줿.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%      2-9-C      %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{2-9-C\qquad �Ϲ���, �������δط�����Ƴ��}
\setcounter{equation}{0}
\setcounter{figure}{0}
\def\theequation{2-9-C-\arabic{equation}}
\def\thefigure{2-9-C-\arabic{figure}}


\subsubsection*{2-9-C-a \quad �Ϲ����δط�����Ƴ��}

$p$-��ɸ�ϤǤα�ư������(2-58), �ſ尵�μ�(2-59), �����
����������(2-8)�����Ϲ����δط��������.
�ʤ�, �Ϲ���������Ω�ĤΤϥ��ꥪ���Ϥȵ��������Ϥ��Ĥ��äƤ���
���ʤΤ�,
1) �����������Ǥ�($d\Dvect{V}/dt = \partial \Dvect{V}/\partial t$),
2) �໤�Ϥ�̵��($\Dvect{F}=0$),
3) ��®�����ǻ����Ѳ����ʤ�($\partial \Dvect{V} / \partial t$=0)
�Ȥ�����ﲼ�ǹͤ���.

�ʲ������Ѥ���3���Ǥ���.
%
\begin{eqnarray}
 \Deqlab{hw-01-09-��ư������}
  \DD{\Dvect{V}}{t}
 &=&
  - \nabla_p \phi - f \Dvect{k} \times \Dvect{V} + \Dvect{F},
 \\
 \Deqlab{hw-01-09-�ſ尵�μ�}
  \DP{\phi}{p}
 &=&
  - \alpha,
 \\
 \Deqlab{hw-01-09-����������}
  p
 &=&
  \frac{RT}{\alpha}.
\end{eqnarray}
%
�������Ǥ��뤳�Ȥ��໤�Ϥ�̵��Ǥ��뤳��,
��������®�����Ǥ��뤳�Ȥ��θ��,
\Deqref{hw-01-09-��ư������}���ѷ������,
%
\begin{eqnarray}
 \Deqlab{hw-01-09-�Ϲ���ʿ��}
  - f \Dvect{k} \times \Dvect{V} = \nabla_p \phi
\end{eqnarray} 
%
%\begin{itembox}[l]{�Ϲ��������礤�μ�}
% \begin{eqnarray}
%  \Deqlab{hw-01-09-�Ϲ���ʿ��}
%   - f \Dvect{k} \times \Dvect{V} = \nabla_p \phi
% \end{eqnarray} 
%\end{itembox}
%
�Ȥ�����������Ω��.
���դ����ꥪ���ϤǤ���, ���դ����������ϤǤ���.
���줬�Ϲ��������礤�μ��Ǥ���.
�ʲ��ǤϤ��δط�����������®$\Dvect{V}$��
�Ϲ���$\Dvect{V}_g$��ɽ������.
���μ���ξ�դ�$(1/f) \Dvect{k}$��ݤ����,
%
\begin{eqnarray}
  - \Dvect{k} \times (\Dvect{k} \times \Dvect{V}_g)
 &=&
  f^{-1} \Dvect{k} \times \nabla_p \phi
 \nonumber \\
  - (\Dvect{k} \cdot \Dvect{V}_g) \Dvect{k}
  + (\Dvect{k} \cdot \Dvect{k}) \Dvect{V}_g
 &=&
  f^{-1} \Dvect{k} \times \nabla_p \phi
 \nonumber \\
 \Deqlab{hw-01-09-�Ϲ�����Ƴ�Ф��뼰}
  \Dvect{V}_g
 &=&
  f^{-1} \Dvect{k} \times \nabla_p \phi
\end{eqnarray}
%
�Ȥʤ�.
���μ������Ϲ�������뤳�Ȥ��Ǥ���.
�ʤ�, �׻��������, ����ʸ�κǸ�˷Ǻܤ���Ƥ�����ظ���(A-1)����Ѥ���.
�ޤ�, $\Dvect{V}_g$�Ͽ�ʿ�̤���ʬ�����ʤ�����
$\Dvect{k} \cdot \Dvect{V}_g = 0$�Ȥʤ뤳�Ȥˤ����դ��뤳��.
���ʤߤ�, \Deqref{hw-01-09-�Ϲ�����Ƴ�Ф��뼰}�����ʬ��
ʬ���Ƥߤ��,
%
\begin{eqnarray}
 \Deqlab{hw-01-09-�Ϲ����γ���ʬ}
  u_g = - \frac{1}{f} \DP{\phi}{y}, \qquad
  v_g =   \frac{1}{f} \DP{\phi}{x}
\end{eqnarray}
%
�Ȥʤ�.
�Ĥޤ�, �����̤ˤ������̾夹��ˤĤ���٤������äƤ�����
(�����ؤ�����̾夹��ˤĤ쵤�����������Ƥ�����)��
������������᤯.
�ޤ�, �����̤ˤ�����������ۤɹ��٤������äƤ�����
(�����ؤ������˹Ԥ��ˤĤ쵤�������ä��Ƥ�����)��
�̸����������᤯.

\subsubsection*{2-9-C-b \quad �������δط�����Ƴ��}

����
\Deqref{hw-01-09-�Ϲ�����Ƴ�Ф��뼰}���ѷ����Ʋ������δط��������.
\Deqref{hw-01-09-�Ϲ�����Ƴ�Ф��뼰}
��$p$������ʬ�����,
%
\begin{eqnarray}
 \Deqlab{hw-01-09-�Ϲ����μ���p��ʬ}
  \DP{\Dvect{V}_g}{p} =
  \frac{1}{f} \Dvect{k} \times \nabla_p \DP{\phi}{p}
\end{eqnarray}
%
�Ȥʤ�.
�������ʸ�ˤ��ä����Ǥ���.
³����
\Deqref{hw-01-09-�Ϲ����μ���p��ʬ}
��׻�����.
�ſ尵�μ� \Deqref{hw-01-09-�ſ尵�μ�}��
���������� \Deqref{hw-01-09-����������}�����Ѥ���.
%
\begin{eqnarray}
  \DP{\Dvect{V}_g}{p}
 &=&
  \frac{1}{f} \Dvect{k} \times \nabla_p (- \alpha)
 \nonumber \\
 &=&
  \frac{1}{f} \Dvect{k} \times \nabla_p (- \frac{RT}{p})
 \nonumber \\
 \Deqlab{hw-01-09-�������δط���}
 &=&
  - \frac{R}{fp}
   \left( \Dvect{k} \times \nabla_p T \right).
\end{eqnarray}
%
\Deqref{hw-01-09-�������δط���}���������δط����Ǥ���.
��������ʬ��ʬ����Ȱʲ��Τ褦�ˤʤ�.
%
\begin{eqnarray}
 \Deqlab{hw-01-09-�������γ���ʬ}
  \DP{u}{p} = \frac{R}{fp} \DP{T}{y} ,
  \qquad
  \DP{v}{p} = - \frac{R}{fp} \DP{T}{x}.
\end{eqnarray}
%
���μ���ʪ��Ū�ʰ�̣��ͤ��Ƥߤ�.
�㤨��, ���������̾���̤˹Ԥ��ۤɲ��٤������äƤ����
($\partial T/\partial y < 0$),
���������̾�Ƕ���̩��$\rho$���̤ۤ��礭��.
�������ä�������$p$�ˤ���٤�������$p - \delta p$��
��ä��̤˸����äƷ����Ƥ���(���޻���).
�������ä�������$p- \delta p$����Ϲ���$u$��������$p$���
��������礭��(���ʤ��$\partial u /\partial p < 0$)�ʤ����
�ʤ�ʤ����Ǥ���.
%
\begin{figure}[htb]
 \begin{center}
  \Depsf[][]{./eps/thermal_wind.eps}
 \end{center}
 \begin{quote}
  \caption{
  {\footnotesize �������γ�ǰ��}
  }
  \Dfiglab{hwfig-01-09-������}
 \end{quote}
\end{figure}
%
�ʤ�, �����Ǥ����������ǹͤ�����,
���������Ǥ�Ʊ�ͤ˹ͤ����$v$�μ��˴ؤ��Ƥ�����Ǥ��������.
%



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%      2-9-D      %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{2-9-D\qquad $z$-��ɸ�Ϥ�����벹�����δط���}
\setcounter{equation}{0}
\setcounter{figure}{0}
\def\theequation{2-9-D-\arabic{equation}}
\def\thefigure{2-9-D-\arabic{figure}}


{\bfseries 2-9-C}�Ǥ�$p$-��ɸ�ϤǤα�ư����������ʷ�ʷ���
�������δط��� \Deqref{hw-01-09-�������γ���ʬ}��Ƴ�����Ȥ��Ǥ���.
�Ǥ�, $z$-��ɸ�ϤǤα�ư������(������ſ尵�μ�)���鲹�����μ�
��Ƴ���Ȥɤ��ʤ�Τ�������.
�ʲ��Ǥ�$z$-��ɸ�ϤǤα�ư���������鲹�����μ�����Ƥߤ�.

$z$-��ɸ�ϤǤα�ư������, �ſ尵�μ�,
����Ӿ����������Ϥ��줾��,
%
\begin{eqnarray}
 \Deqlab{hw-01-09-z��ɸ�ϱ�ư������}
  \DD{\Dvect{V}}{t}
 &=&
  - \alpha \nabla_z p - f \Dvect{k} \times \Dvect{V} + \Dvect{F},
 \\
 \Deqlab{hw-01-09-z��ɸ���ſ尵�μ�}
  \DP{p}{z} 
 &=&
  - \frac{g}{\alpha},
 \\
 \Deqlab{hw-01-09-z��ɸ�Ͼ���������}
  p
 &=&
  \frac{RT}{\alpha}
\end{eqnarray}
%
�ȤʤäƤ���.
�������Ǥ�, �໤�Ϥ�̵��Ǥ�, ��®�����Ǥ���Ȥ���
��ﲼ�Ǥ�
\Deqref{hw-01-09-z��ɸ�ϱ�ư������}�ϰʲ��Τ褦�ˤʤ�.
%
\begin{eqnarray}
 \Deqlab{hw-01-09-z��ɸ���Ϲ���ʿ��}
 - f \Dvect{k} \times \Dvect{V} = \alpha \nabla_z p.
\end{eqnarray}
%
�����$p$-��ɸ�ϤǤ�
\Deqref{hw-01-09-�Ϲ���ʿ��}�μ���Ʊ�ͤ��Ϲ��������礤�μ��Ǥ���.
$p$-��ɸ�Ϥλ�������Ʊ�ͤ�, ξ�դ�$(1/f)\Dvect{k}$��ݤ����,
%
\begin{eqnarray}
 \Deqlab{hw-01-09-z��ɸ���Ϲ�����Ƴ�Ф��뼰} 
 \Dvect{V}_g = \frac{\alpha}{f} (\Dvect{k} \times \nabla_z p)
\end{eqnarray}
%
�Ȥ�������Ƴ�ФǤ���.
����ʬ��ʬ���Ƥߤ��,
%
\begin{eqnarray}
 \Deqlab{hw-01-09-z��ɸ�Ǥ��Ϲ����γ���ʬ}
  u_g = - \frac{\alpha}{f} \DP{p}{y}, \qquad
  v_g =   \frac{\alpha}{f} \DP{p}{x}
\end{eqnarray}
%
�Ȥʤ�.
\Deqref{hw-01-09-z��ɸ���Ϲ�����Ƴ�Ф��뼰}
���鲹��������Ƥߤ褦.

\Deqref{hw-01-09-z��ɸ���Ϲ�����Ƴ�Ф��뼰}��ξ�դ�$z$����ʬ����
�׻���Ԥ�.
�ʤ�, �׻��������
\Deqref{hw-01-09-z��ɸ���ſ尵�μ�}��
\Deqref{hw-01-09-z��ɸ�Ͼ���������}�����Ѥ���.
%
\begin{eqnarray}
  f \DP{\Dvect{V}_g}{z}
 &=&
  \DP{}{z}
   \left\{
    \alpha \left( \Dvect{k} \times \nabla_z p \right)
   \right\}
 \nonumber \\
 &=&
  \alpha \left( \Dvect{k} \times \nabla_z \DP{p}{z} \right)
  + \DP{\alpha}{z}
    \left( \Dvect{k} \times \nabla_z p \right)
 \nonumber \\
 &=&
  - \alpha \left( \Dvect{k} \times \nabla_z \frac{g}{\alpha} \right)
  + \DP{\alpha}{z}
    \left( \Dvect{k} \times \nabla_z p \right)
 \nonumber \\
 &=&
  - \alpha \left( \Dvect{k} \times \nabla_z \frac{pg}{RT} \right)
  + \DP{\alpha}{z}
    \left( \Dvect{k} \times \nabla_z p \right)
 \nonumber \\
 &=&
  - \frac{\alpha g p}{R} \left( \Dvect{k} \times \nabla_z \frac{1}{T} \right)
  - \frac{\alpha g}{RT} \left( \Dvect{k} \times \nabla_z p \right)
  + \DP{\alpha}{z}
    \left( \Dvect{k} \times \nabla_z p \right)
 \nonumber \\
 &=&
  \frac{\alpha g p}{R T^2} \left( \Dvect{k} \times \nabla_z T \right)
  + \left( \DP{\alpha}{z} - \frac{\alpha g}{RT} \right)
    \left( \Dvect{k} \times \nabla_z p \right)
 \nonumber \\
 &=&
  \frac{g}{T} \left( \Dvect{k} \times \nabla_z T \right)
  + \left( \DP{\alpha}{z} - \frac{g}{p} \right)
    \left( \Dvect{k} \times \nabla_z p \right)
 \nonumber \\
 &=&
  \frac{g}{T} \left( \Dvect{k} \times \nabla_z T \right)
  + \left( \DP{\alpha}{z} - \frac{g}{\alpha} \frac{\alpha}{p} \right)
    \left( \Dvect{k} \times \nabla_z p \right)
 \nonumber \\
 &=&
  \frac{g}{T} \left( \Dvect{k} \times \nabla_z T \right)
  + \left( \DP{\alpha}{z} + \frac{\alpha}{p} \DP{p}{z} \right)
    \left( \Dvect{k} \times \nabla_z p \right)
 \nonumber \\
 &=&
 \Deqlab{hw-01-09-z��ɸ�Ϥβ������δط���}
  \frac{g}{T} \left( \Dvect{k} \times \nabla_z T \right)
  + \frac{1}{p} \left( \DP{}{z} (p \alpha) \right)
    \left( \Dvect{k} \times \nabla_z p \right).
\end{eqnarray}
%
����
\Deqref{hw-01-09-z��ɸ�Ϥβ������δط���}��
$z$-��ɸ�Ϥμ�����Ƴ���줿�������δط����Ȼ������Ǥ���.
������, $p$-��ɸ�Ϥλ��Ȱ㤤���ޤ�$\alpha$��������ä��ʤ�����,
���ä���Ȥϲ������δط��򼨤��Ƥ���Ȥϸ����ʤ�.
���μ��Ϥ����ޤǤ������٤��ä���Ȥ��������Ǥ����ʤ�.
���Τ��Ȥ���ʸ�Ρ�$p$-��ɸ�Ϥ�̩�٤����Ǥ���ΤǡġפȤ������Ȥ�
��̣�Ǥ���.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%      2-9-E      %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{2-9-E\qquad $\sigma$-��ɸ�Ϥ�����}
\setcounter{equation}{0}
\setcounter{figure}{0}
\def\theequation{2-9-E-\arabic{equation}}
\def\thefigure{2-9-E-\arabic{figure}}


$\sigma$-��ɸ�Ϥ�$p$-��ɸ�Ϥ��ͥ��Ƥ�������
��ñ�ʲ����������Ǥ������ʸ�ǽҤ٤��Ƥ��뤬,
���줬����Ū�ˤϤɤ��������ȤǤ��뤫��������.
�ʲ���\Dfigref{hwfig-01-09-�Һ�ɸ�Ϥ�p��ɸ��}�򸫤Ƥۤ���.
\Dfigref{hwfig-01-09-�Һ�ɸ�Ϥ�p��ɸ��}
�ˤ�����, ���������Ͼ���, ��������$\sigma$��, ��������$p$�̤Ǥ���.
�ޤ��狼��褦��, ��$p$�̤��Ϸ��ε����ˤ��ʬ�Ǥ���Ƥ���.
������Ф�, ��$\sigma$�̤��Ϸ���Ʊ�����ˤʤ�
(���礦��$\sigma=1$���̤��Ͼ�η��Ȱ��פ���),
�������Ϸ��Ǥ��äƤ�ʬ�Ǥ���뤳�Ȥ�̵��.
�㤨��, ���ͷ׻��ǤϤ����ΰ�Υѥ�᡼�������Τ�
���ܤ����ΰ�Υѥ�᡼����ɬ�פˤʤ�.
$p$-��ɸ�ϤǤ��Ϸ��θ��̤����٤Υѥ�᡼���������ʤ����⤢�ꤦ�뤬,
$\sigma$-��ɸ�ϤǤϤ��Τ褦�ʤ��ȤϤʤ�
(\Dfigref{hwfig-01-09-�Һ�ɸ�Ϥ�����}����).
���줬$\sigma$-��ɸ�Ϥ�$p$-��ɸ�Ϥ��ͥ�줿���Ǥ���.
%
\begin{figure}[htb]
 \begin{center}
  \Depsf[][]{./eps/sigma_area.eps}
 \end{center}
 \begin{quote}
  \caption{
  {\footnotesize $\sigma$-��ɸ�Ϥ�����}
  }
  \Dfiglab{hwfig-01-09-�Һ�ɸ�Ϥ�����}
 \end{quote}
\end{figure}
%
%
\begin{figure}[htb]
 \begin{center}
  \Depsf[][]{./eps/sigma_coordinate.eps}
 \end{center}
 \begin{quote}
  \caption{
  {\footnotesize $\sigma$-��ɸ�Ϥ�$p$-��ɸ��}
  }
  \Dfiglab{hwfig-01-09-�Һ�ɸ�Ϥ�p��ɸ��}
 \end{quote}
\end{figure}
%




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%      2-9-F      %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{2-9-F\qquad ʪ����}
\setcounter{equation}{0}
\setcounter{figure}{0}
\def\theequation{2-9-F-\arabic{equation}}
\def\thefigure{2-9-F-\arabic{figure}}


����γ�ҷ���ͤ�����, ����γ�ҷ�����ư�����ϰϤ�{\bfseries ʪ����}
�ȸƤФ���ΤǤ���.
�㤨��, ��$\sigma$�̤ʤɤ�ͤ���Ȥ�����¸�ߤ���γ�ҤϤ�����������
��ư����Τ�$\sigma$�̤�ʪ���̤Ȥϸ����ʤ�
(\Dfigref{hwfig-01-09-ʪ����}����).
��������$\theta$�̤�ͤ��Ƥߤ�. ��ʸ�ˤ�$d\theta /dt = 0$�Ƚ񤤤Ƥ����̤�,
$\theta$���Ѳ����ʤ�
(\Dfigref{hwfig-01-09-ʪ����}����).
�Ĥޤ�γ�Ҥ���$\theta$���⤷����ư���ʤ�.
���Τ��Ȥ�$\theta$�̤�ʪ���̤Ǥ��뤳�Ȥ򼨤��Ƥ���.
���줬��ʸ����̣���뤳�ȤǤ���.
%
\begin{figure}[htb]
 \begin{center}
  \Depsf[][]{./eps/material_surface.eps}
 \end{center}
 \begin{quote}
  \caption{
  {\footnotesize ʪ����}
  }
  \Dfiglab{hwfig-01-09-ʪ����}
 \end{quote}
\end{figure}
%



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%      2-9-G      %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{2-9-G\qquad Ϣ³�μ��������ɽ�̰��Ϥλ����Ѳ�
$\partial p_s/ \partial t$��Ƴ��}
\setcounter{equation}{0}
\setcounter{figure}{0}
\def\theequation{2-9-G-\arabic{equation}}
\def\thefigure{2-9-G-\arabic{figure}}



$\theta$-��ɸ�ϤǤ�Ϣ³�μ�(2-74)������ɽ�̰���$p_s$��
�����Ѳ�$\partial p_s/\partial t$�μ�����뤳�Ȥ��Ǥ���.
�ʤ�, ����Ƴ�Ф˴ؤ��Ƥ�Kasahara(1974)��
���ͤˤ���.

$\theta$-��ɸ�ϤǤ�Ϣ³�μ�(2-74)�ϰʲ��μ��Ǥ���.
%
\begin{eqnarray}
 \Deqlab{hw-01-09-���̺�ɸ��Ϣ³�μ�}
  \DD{}{t} \left( \ln \DP{p}{\theta} \right)
  + \nabla_{\theta} \cdot \Dvect{V}
  + \DP{\dot{\theta}}{\theta}
  = 0.
\end{eqnarray}
%
�ʤ�, $\dot{\theta} = d \theta / dt$�Ǥ���.
���μ����ѷ�����.
%
\begin{eqnarray*}
  \DP{\theta}{p} \DD{}{t} \left(\DP{p}{\theta} \right)
  &+& \nabla_{\theta} \cdot \Dvect{V}
  + \DP{\dot{\theta}}{\theta}
  = 0,
 \\ 
  \DP{}{t} \left(\DP{p}{\theta} \right)
  &+& \Dvect{V} \cdot \nabla_{\theta} \DP{p}{\theta}
  + \dot{\theta} \DP{}{\theta} \left( \DP{p}{\theta} \right)
   + \DP{p}{\theta} \left(
    \nabla_{\theta} \cdot \Dvect{V}
    + \DP{\dot{\theta}}{\theta}
   \right)
  = 0,
 \\ 
  \DP{}{t} \left(\DP{p}{\theta} \right)
  &+& \nabla_{\theta} \cdot \left(\Dvect{V} \DP{p}{\theta} \right)
  + \DP{}{\theta} \left( \dot{\theta} \DP{p}{\theta} \right)
  = 0.
\end{eqnarray*}
%
���μ��򤢤�Ǥ�դ�$\theta$����$\theta_t$
\footnote{
$\theta_t$�ϲ��̤Υ�ǥ��ĺ���β���.
}
�ޤ���ʬ�����,
%
\begin{eqnarray}
  \int_{\theta}^{\theta_t} \DP{}{t} \left( \DP{p}{\theta} \right) d \theta
 &=&
  - \int_{\theta}^{\theta_t}
   \left\{
    \nabla_{\theta} \cdot \left( \Dvect{V} \DP{p}{\theta} \right)
    + \DP{}{\theta} \left( \dot{\theta} \DP{p}{\theta} \right)
   \right\} d \theta,
 \nonumber \\
  \int_{\theta}^{\theta_t} \DP{}{\theta} \left( \DP{p}{t} \right) d \theta
 &=&
  - \int_{\theta}^{\theta_t}
   \left\{
    \nabla_{\theta} \cdot \left( \Dvect{V} \DP{p}{\theta} \right)
    + \DP{}{\theta} \left( \dot{\theta} \DP{p}{\theta} \right)
   \right\} d \theta,
 \nonumber \\
  \DP{p}{t} \bigg| _{\theta = \theta_t} - \DP{p}{t} \bigg| _{\theta = \theta} 
 &=&
  - \int_{\theta}^{\theta_t}
   \left\{
    \nabla_{\theta} \cdot \left( \Dvect{V} \DP{p}{\theta} \right)
    + \DP{}{\theta} \left( \dot{\theta} \DP{p}{\theta} \right)
   \right\} d \theta,
 \nonumber \\
  \DP{p}{t} \bigg| _{\theta} 
 &=&
  \DP{p}{t} \bigg| _{\theta_t}
  + \int_{\theta}^{\theta_t}
   \left\{
    \nabla_{\theta} \cdot \left( \Dvect{V} \DP{p}{\theta} \right)
    + \DP{}{\theta} \left( \dot{\theta} \DP{p}{\theta} \right)
   \right\} d \theta  
 \nonumber \\
 \Deqlab{hw-01-09-��ɽ�̰��Ϥλ����Ѳ�Ƴ�Фμ�1}
\end{eqnarray}
%
�Ȥʤ�.
�����ǰʲ��ζ�������Ƴ������.
�������ʸ��(2-75)��Ʊ�ͤΤ�ΤǤ���.
%
\begin{eqnarray}
 \dot{\theta} = 0 \ \hspace{31mm} &{\rm at}& \ \theta = \theta_{t} ,
 \nonumber \\
 \Deqlab{hw-01-09-���̺�ɸ�ζ������}
 \dot{\theta} = \partial \theta_{s} / \partial t
  + \Dvect{V} \cdot \nabla_{\theta} \theta_s  \qquad &{\rm at}& \
  \theta = \theta_s .
\end{eqnarray}
%
���������,
\Deqref{hw-01-09-��ɽ�̰��Ϥλ����Ѳ�Ƴ�Фμ�1}
��
%
\begin{eqnarray}
  \DP{p}{t} \bigg| _{\theta} 
 &=&
  \DP{p}{t} \bigg| _{\theta_t}
  + \int_{\theta}^{\theta_t}
    \nabla_{\theta} \cdot \left( \Dvect{V} \DP{p}{\theta} \right) d \theta
    + \left( \dot{\theta} \DP{p}{\theta} \right) \bigg|_{\theta_t}
    - \left( \dot{\theta} \DP{p}{\theta} \right) \bigg|_{\theta}   
 \nonumber \\ 
 \Deqlab{hw-01-09-��ɽ�̰��Ϥλ����Ѳ�Ƴ�Фμ�2}
 &=&
  \DP{p}{t} \bigg| _{\theta_t}
  + \int_{\theta}^{\theta_t}
    \nabla_{\theta} \cdot \left( \Dvect{V} \DP{p}{\theta} \right) d \theta
    - \left( \dot{\theta} \DP{p}{\theta} \right) \bigg|_{\theta}   
\end{eqnarray}
%
�Ȥʤ�.
$\theta = \theta_s$�Ȥ����,
%
\begin{eqnarray}
  \DP{p}{t} \bigg| _{\theta_s} 
 &=&
  \DP{p}{t} \bigg| _{\theta_t}
  + \int_{\theta_s}^{\theta_t}
    \nabla_{\theta} \cdot \left( \Dvect{V} \DP{p}{\theta} \right) d \theta
    - \left( \dot{\theta} \DP{p}{\theta} \right) \bigg|_{\theta_s}
 \nonumber \\
 &=&
  \DP{p}{t} \bigg| _{\theta_t}
  + \int_{\theta_s}^{\theta_t}
    \nabla_{\theta} \cdot \left( \Dvect{V} \DP{p}{\theta} \right) d \theta
    - \DP{p}{\theta} \bigg|_{\theta_s} \DP{\theta_s}{t}
    - \left(
       \DP{p}{\theta} 
       \Dvect{V} \cdot \nabla_{\theta} \theta_s
      \right) _{\theta = \theta_s}
 \nonumber \\
 \Deqlab{hw-01-09-��ɽ�̰��Ϥλ����Ѳ�Ƴ�Фμ�3}
\end{eqnarray}
%
�Ȥʤ�.
������, �ʲ���4�Ĥμ������Ѥ���
\footnote{
\Deqref{hw-01-09-ĺ�����Ϥλ����Ѳ��δط���},
\Deqref{hw-01-09-��ɽ�̰��Ϥλ����Ѳ��δط���}
\Deqref{hw-01-09-ĺ�����Ϥζ�����狼��δط���}�����
\Deqref{hw-01-09-��ʬ�μ�}��Ƴ�Ф˴ؤ��Ƥ�
���줾��{\bfseries 2-9-G-a} {\bfseries 2-9-G-b},
{\bfseries 2-9-G-c} ,{\bfseries 2-9-G-d}
�򻲾ȤΤ���.
}.
%
\begin{eqnarray}
 \Deqlab{hw-01-09-ĺ�����Ϥλ����Ѳ��δط���}
  \DP{p_t}{t}
 &=&
  \DP{p}{t} \bigg|_{\theta_t},
 \\
 \Deqlab{hw-01-09-��ɽ�̰��Ϥλ����Ѳ��δط���}
  \DP{p_s}{t}
 &=&
  \DP{p}{t} \bigg|_{\theta_s} + \DP{p}{\theta} \bigg|_{\theta_s}
  \DP{\theta_s}{t},
 \\
 \Deqlab{hw-01-09-ĺ�����Ϥζ�����狼��δط���}
  - \left( \Dvect{V} \cdot \nabla_{\theta} \theta \right)_{\theta = \theta_t}
 &=&
  0,
 \\
  \nabla_{\theta} \cdot \int_{\theta_s}^{\theta_t} \Dvect{V}
  \DP{p}{\theta} d \theta
 &=&
  \int_{\theta_s}^{\theta_t} \nabla_{\theta} \cdot
  \left( \Dvect{V} \DP{p}{\theta} \right) d \theta
 \nonumber \\
 \Deqlab{hw-01-09-��ʬ�μ�}
 & & \hspace{10mm}
  + \left(
     \DP{p}{\theta}
     \Dvect{V} \cdot \nabla_{\theta} \theta_t
    \right) _{\theta = \theta_t}
  - \left(
     \DP{p}{\theta}
     \Dvect{V} \cdot \nabla_{\theta} \theta_s
    \right) _{\theta = \theta_s}.
\end{eqnarray}
%
\Deqref{hw-01-09-ĺ�����Ϥλ����Ѳ��δط���}��
\Deqref{hw-01-09-��ɽ�̰��Ϥλ����Ѳ��δط���}
���
\Deqref{hw-01-09-��ɽ�̰��Ϥλ����Ѳ�Ƴ�Фμ�3}��
%
\begin{eqnarray}
 \Deqlab{hw-01-09-��ɽ�̰��Ϥλ����Ѳ�Ƴ�Фμ�4} 
  \DP{p_s}{t} 
 &=&
  \DP{p_t}{t} 
  + \int_{\theta_s}^{\theta_t}
    \nabla_{\theta} \cdot \left( \Dvect{V} \DP{p}{\theta} \right) d \theta
    - \left(
       \DP{p}{\theta} 
       \Dvect{V} \cdot \nabla_{\theta} \theta_s
      \right) _{\theta = \theta_s}
\end{eqnarray}
%
�Ȥʤ�, ³����
\Deqref{hw-01-09-��ʬ�μ�}�����Ѥ����,
%
\begin{eqnarray*}
  \DP{p_s}{t} 
 &=&
  \DP{p_t}{t}
   + \nabla_{\theta} \cdot \int_{\theta_s}^{\theta_t} \Dvect{V} \DP{p}{\theta}
   d \theta  
    - \left(
       \DP{p}{\theta} 
       \Dvect{V} \cdot \nabla_{\theta} \theta_t
      \right) _{\theta = \theta_t}
\end{eqnarray*}
%
�Ȥʤ�.
������
\Deqref{hw-01-09-ĺ�����Ϥζ�����狼��δط���}
��걦�դ���3��Ͼä���Τ�,
%
\begin{eqnarray}
 \Deqlab{hw-01-09-��ɽ�̰��Ϥλ����Ѳ��μ�}
  \DP{p_s}{t} 
 &=&
  \DP{p_t}{t}
   + \nabla_{\theta} \cdot \int_{\theta_s}^{\theta_t} \Dvect{V} \DP{p}{\theta}
   d \theta  
\end{eqnarray}
%
�Ȥʤ�.
���줬��ʸ�Ǹ��������ɽ�̰��Ϥλ����Ѳ�
$\partial p_s / \partial t$�˴ؤ��뼰�פǤ���.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%      2-9-G-a    %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection*{2-9-G-a \qquad $\theta_t$�Ǥΰ��Ϥ�
�����Ѳ��δط���}

$\displaystyle
  \DP{p_t}{t}
 =
  \DP{p}{t} \bigg|_{\theta_t}
$
��Ƴ�Ф���.
������, 
���Ϥ�$p = p(x,y,\theta,t)$,
���̥�ǥ��ĺ���Ǥΰ��Ϥ�$p_t = p (x,y,\theta_t,t)$,
���̤�$\theta_t = \theta_t (x,y,t)$�Ǥ���Ȥ���.
{\bfseries 2-9-G-b}��Ʊ�ͤ˹ͤ����
%
\begin{eqnarray}
 \Deqlab{hw-01-09-ĺ���ΰ��Ϥδط���1}
  \DP{p_t}{t}
 =
  \DP{p}{t} \bigg|_{\theta_t} + \DP{p}{\theta} \bigg|_{\theta_t}
  \DP{\theta_t}{t}
\end{eqnarray}
%
�Ǥ���.
������, $\theta_t$�ϲ��̥�ǥ��ĺ���ǻ����Ѳ�, �����Ѳ��Ȥ��̵���Τ�
$\partial \theta_t / \partial t = 0$�Ǥ���.
��ä�,
%
\begin{eqnarray}
 \Deqlab{hw-01-09-ĺ���ΰ��Ϥδط���2}
  \DP{p_t}{t}
 =
  \DP{p}{t} \bigg|_{\theta_t}
\end{eqnarray}
%
�Ȥʤ�. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%      2-9-G-b    %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection*{2-9-G-b \qquad $\theta_s$(��ɽ��)�Ǥΰ��Ϥ�
�����Ѳ��δط���}

$\displaystyle
  \DP{p_s}{t}
 =
  \DP{p}{t} \bigg|_{\theta_s} + \DP{p}{\theta} \bigg|_{\theta_s}
  \DP{\theta_s}{t}
$
��Ƴ�Ф���. 
���Ϥ�$p = p(x,y,\theta,t)$,
��ɽ�̰��Ϥ�$p_s = p (x,y,\theta_s,t)$,
��ɽ�̲��̤�$\theta_s = \theta_s (x,y,t)$�Ǥ���.
$p$������ʬ��
%
\begin{eqnarray}
 \Deqlab{hw-01-09-p������ʬ2}
 dp = \DP{p}{x} dx + \DP{p}{y} dy + \DP{p}{\theta} d \theta + \DP{p}{t} dt
\end{eqnarray}
%
�Ǥ���.
������$\theta = \theta_s$�Ȥ����, $d \theta = d \theta_s$�Ǥ���.
�ޤ�, $p_s = p (x,y,\theta_s,t)$�Ǥ��뤳�Ȥ���
$d p = d p_s $�Ǥ���.
��ä�, \Deqref{hw-01-09-p������ʬ2}��
%
\begin{eqnarray}
 \Deqlab{hw-01-09-��ɽ�̰��Ϥ�����ʬ}
 dp_s = \DP{p}{x} \bigg|_{\theta = \theta_s} dx
      + \DP{p}{y} \bigg|_{\theta = \theta_s} dy
      + \DP{p}{\theta} \bigg|_{\theta = \theta_s} d \theta_s
      + \DP{p}{t} \bigg|_{\theta = \theta_s} dt
\end{eqnarray}
%
�Ȥʤ�.
$\theta_s = \theta_s(x,y,t)$�ʤΤ�
$d \theta_s = (\partial \theta_s / \partial x) dx
              + (\partial \theta_s / \partial y) dy
	      + (\partial \theta_s / \partial t) dt
$
�Ǥ��뤿��,
\Deqref{hw-01-09-��ɽ�̰��Ϥ�����ʬ}��
%
\begin{eqnarray}
  dp_s
 &=&
  \DP{p}{x} \bigg|_{\theta_s} dx
  + \DP{p}{y} \bigg|_{\theta_s} dy
  + \DP{p}{\theta} \bigg|_{\theta_s}
   \left\{
    \DP{\theta_s}{x} dx
    + \DP{\theta_s}{y} dy
    + \DP{\theta_s}{t} dt
   \right\}
  + \DP{p}{t} \bigg|_{\theta_s} dt
 \nonumber \\
 &=&
  \left\{
   \DP{p}{x} \bigg|_{\theta_s}
   + \DP{p}{\theta} \bigg|_{\theta_s} \DP{\theta_s}{x}
  \right\} dx
 +\left\{
   \DP{p}{y} \bigg|_{\theta_s}
   + \DP{p}{\theta} \bigg|_{\theta_s} \DP{\theta_s}{y}
  \right\} dy
 \nonumber \\
 \Deqlab{hw-01-09-��ɽ�̰��Ϥ�����ʬ2}
 & &
 +\left\{
   \DP{p}{t} \bigg|_{\theta_s}
   + \DP{p}{\theta} \bigg|_{\theta_s} \DP{\theta_s}{t}
  \right\} dt
\end{eqnarray}
%
�Ȥʤ�.
������ʬ��ͤ����, ���դ���1��, ��2���̵�뤷�ƹͤ�����Τ�,
$p_s$�λ����Ѳ���
%
\begin{eqnarray}
 \Deqlab{hw-01-09-���̺�ɸ�Ϥ����᤿��ɽ�̰��Ϥλ����Ѳ��δط���}
 \DP{p_s}{t} = \DP{p}{t} \bigg|_{\theta_s}
               + \DP{p}{\theta} \bigg|_{\theta_s} \DP{\theta_s}{t}
\end{eqnarray}
%
�Ȥ���������.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%      2-9-G-c    %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection*{2-9-G-c \qquad
\Deqref{hw-01-09-ĺ�����Ϥζ�����狼��δط���} ��Ƴ��}

�����Ǥ� \Deqref{hw-01-09-ĺ�����Ϥζ�����狼��δط���} ��Ƴ�Ф�
�Ԥ�.
\Deqref{hw-01-09-���̺�ɸ�ζ������}���,
%
\begin{eqnarray*}
 \left( \DD{\theta}{t} \right)_{\theta = \theta_t} = 0
\end{eqnarray*}
%
�Ǥ���.
���μ����ѷ����뤳�Ȥ�,
%
\begin{eqnarray*}
 \DP[][\theta_t]{\theta}{t}
 = - \left( \Dvect{V} \cdot \nabla_{\theta} \theta \right)_{\theta_t}
\end{eqnarray*}
%
�Ȥʤ�.
$\theta_t$�϶����Ѳ�, �����Ѳ����ʤ��Τ�,
$(\partial \theta / \partial t)_{\theta_t} =
(\partial \theta_t /\partial t) = 0$
�Ȥʤ�,
\Deqref{hw-01-09-ĺ�����Ϥζ�����狼��δط���} ������Ω��. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%      2-9-G-d    %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection*{2-9-G-d \qquad \Deqref{hw-01-09-��ʬ�μ�}��Ƴ��}

�����Ǥ� \Deqref{hw-01-09-��ʬ�μ�}��Ƴ�ФΤ���η׻���Ԥ�.
�ޤ�, �����ܤ��ñ�ˤ��뤿��$x$��ʬ������ͤ���Ȥ���,
\Deqref{hw-01-09-��ʬ�μ�} �κ���(��$x$��ʬ)�ϰʲ��Τ褦�ˤʤ�
\footnote{
��ʬ�θ���,
\begin{eqnarray*}
 \DP{}{x} \int_{a(x)}^{b(x)} f(x,\theta) d \theta
 =
 \int_{a(x)}^{b(x)} \DP{f(x,\theta)}{x} d \theta
 + f(b) \DP{b}{x} - f(a) \DP{a}{x}
\end{eqnarray*}
�����Ѥ���.
}.
%
\begin{eqnarray}
 \DP{}{x} \int_{\theta_s(x)}^{\theta_t(x)} u \DP{p}{\theta} d \theta
 &=&
 \int_{\theta_s(x)}^{\theta_t(x)}
  \DP{}{x} \left( u \DP{p}{\theta} \right) d \theta
 \nonumber \\
 \Deqlab{hw-01-09-�ִ���ʬ}
 & & \hspace{10mm}
 + \left( u \DP{p}{\theta} \right)_{\theta = \theta_t} \DP{\theta_t}{x}
 - \left( u \DP{p}{\theta} \right)_{\theta = \theta_s} \DP{\theta_s}{x} .
\end{eqnarray}
%
$y$��ʬ�ˤĤ��Ƥ�ͤ����,
%
\begin{eqnarray}
  \nabla_{\theta} \cdot \int_{\theta_s}^{\theta_t} \Dvect{V}
  \DP{p}{\theta} d \theta
 &=&
  \int_{\theta_s}^{\theta_t} \nabla_{\theta} \cdot
  \left( \Dvect{V} \DP{p}{\theta} \right) d \theta
  \nonumber \\
 & &
  + \left(
     \DP{p}{\theta} \Dvect{V} \cdot \nabla_{\theta} \theta_t
    \right)_{\theta = \theta_t}
  - \left(
     \DP{p}{\theta} \Dvect{V} \cdot \nabla_{\theta} \theta_s
    \right)_{\theta = \theta_s}
\end{eqnarray}
%
�Ȥʤ�.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%      2-9-A      %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\subsection*{Appendix \qquad }

%\subsubsection*{}

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\setcounter{equation}{79}
\def\theequation{\arabic{chapter}-\arabic{equation}}
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