% ɽ��: ECCM �Ǥ���Ǯ���ٸ�Ψ�η׻���ˡ
%
% ����: 2004/11/06 �����̰�ϯ

\documentclass[a4j,12pt]{jarticle}
\usepackage{ascmac}
\usepackage{Dennou6}

\Dtitle{ECCM}
\Ddate{2004/11/06}
\Dauthor{�����̰�ϯ}

\setlength{\parindent}{0mm}
\setlength{\parskip}{10pt}


\begin{document}

\section{������Ǯ��Ψ�η׻�}


Weidenschilling and Lewis (1973) �μ�����Ǯ��Ψ���꼰����ե��������롥
Ǯ�ϳؤ��� 1 ˡ§��, 
%
\begin{eqnarray}
dU = \delta Q + \delta W + \delta Z ,
\Deqlab{1}
\end{eqnarray}
�Ǥ���. 
%
������ $dU$ ���������ͥ륮����
 $\delta Q$ �ϷϤ˲ä�����Ǯ�̡�
 $\delta W$ �ϷϤ˲ä���Ż���
 $\delta Z$ �ϲ��إ��ͥ륮���Ǥ��롥
�ͤ��Ƥ���Ϥˤ����Ƶ��Τ����۵��ΤȤ��Ƽ�갷�����Ȥ��Ǥ���
�����Ѳ�����ǮŪ�ˤǤ���Ȥ���ȡ�
 \Deqref{3} ���γƹ�ϰʲ��Τ褦�˽񤱤롥
%
\begin{eqnarray}
dU &=& \bar{c_{v}} dT. 
 \Deqlab{2}\\
\delta Q &=& 0. 
 \Deqlab{3}\\
\delta W &=& - p dV, \nonumber \\
&=& - d(pV) + V dp,\nonumber \\
&=& - R dT + V dp, \nonumber \\
&=& - R dT + V \left( \frac{- \bar{M} p g}{R T} \right) dz, 
\; (�� �ſ尵ʿ�դμ�)\nonumber \\
&=&  - R dT - \bar{M} g  dz . 
 \Deqlab{4}\\
\delta Z &=& - \sum \lambda_{k} dX_{k}. 
 \Deqlab{5}
\end{eqnarray}
%
������ $\bar{c_{v}}$ ���絤�����ѥ����Ǯ��ʿ���͡�
$T$ �ϲ���, $p$ �ϰ���, 
$V$ �� ����ʬ�Ҥ� 1 �������������ѡ�
$R$ �ϵ������, $\bar{M}$ ��ʿ��ʬ����, 
$g$ �Ͻ��ϲ�®��, 
 $\lambda_{k}$ �� $k$ ���ܤζŽ���ʬ���ΤΥ���������
�Ž̤Υ��󥿥�ԡ���
 $dX_{k}$ �ϵ��� $k$ �Υ������Ѳ��Ǥ���. 
%
\Deqref{1} ���� \Deqref{2} -- \Deqref{5} �����������뤳�Ȥ�, 
%
\begin{eqnarray}
&& \bar{c_{v}}dT + R dT + \bar{M} g dz   + \sum \lambda_{k} dX_{k} = 0 ,\nonumber \\
&& \bar{c_{p}}dT + \bar{M} g dz   + \sum \lambda_{k} dX_{k} = 0 ,
\Deqlab{dQ}
\end{eqnarray}
%
�Ȥʤ롥â�� $\bar{c_{p}}$ ���絤���갵�����Ǯ��ʿ���ͤǡ�
���۵��Τξ�� $\bar{c_{p}} = \bar{c_{v}} + R$ �Ǥ��롥


\subsection{������Ǯ���ٸ�Ψ}

\Deqref{dQ} ������Ǯ�ˤ����̵�뤹�뤳�ȤǴ�����Ǯ���ٸ�Ψ��
��ޤ롥
%
\begin{eqnarray}
 && \bar{c_{p}}dT + \bar{M} g dz = 0, \nonumber \\
 &&\DD{T}{z} = - \frac{ \bar{M} g }{\bar{c_{p}}}.
\Deqlab{dtdz-dry}
\end{eqnarray}


\subsection{������Ǯ���ٸ�Ψ}

\Deqref{dQ} ����� $dX_{k}$ ����ʬΨ��ʬ���δط�
%
\begin{eqnarray}
dX_{k} &=& \Dinv{p} dp_{k} - \left( \frac{p_{k}}{p^{2}} \right) dp.
 \Deqlab{X}
\end{eqnarray}
%
�ȥ��饦������������ڥ�����μ�
%
\begin{eqnarray}
dp_{k} = \frac{p_{k} \lambda_{k}dT}{R T^{2}}, 
 \Deqlab{CK}
\end{eqnarray}
%
���Ѥ����ѷ������, 
%
\begin{eqnarray}
dX_{k} &=& \Dinv{p} dp_{k} - \frac{p_{k}}{p^{2}} dp, \nonumber \\
&=& \Dinv{p} \left( \frac{p_{k} \lambda_{k} dT}{R T^{2}} \right)
- \frac{p_{k}}{p^{2}} \left( - \frac{ \bar{M} p g}{R T} dz \right) , \nonumber \\ 
&=& \frac{p_{k}}{p} \frac{\lambda_{k}}{ R_{v} T^{2}} dT
+  \frac{p_{k}}{p} \frac{\bar{M} g}{ R T} dz , \nonumber \\ 
&=& \frac{ \lambda_{k} X_{k} }{ R T^{2}} dT
+  \frac{\bar{M} g X_{k}}{ R T} dz .
\Deqlab{Xk}
\end{eqnarray}
%
�Ǥ���. 

\Deqref{dQ} ���� \Deqref{Xk} �����������뤳�ȤǼ�����Ǯ���ٸ�Ψ����ޤ롥
%
\begin{eqnarray}
 &&\bar{c_{p}} dT + \bar{M} g dz + \sum \lambda_{k} dX_{k} = 0, \nonumber \\
 &&\bar{c_{p}} dT + \bar{M} g dz + 
  \sum \lambda_{k} 
  \left(\frac{\lambda_{k} X_{k} }{ R T^{2}} dT
    +  \frac{\bar{M} g X_{k}}{ R T} dz \right)  = 0, \nonumber \\
&&\bar{c_{p}} 
\left( 1+ \frac{\sum \lambda^{2}_{k} X_{k}}{ \bar{c_{p}} R T^{2}} \right) dT 
+ \bar{M} g \left( 1 +  \frac{\sum \lambda_{k} X_{k}}{R T} \right) dz 
= 0, \nonumber \\
 &&\DD{T}{z} = - \frac{\bar{M} g}{\bar{c_{p}}} 
  \left(
 \frac{ 1 +  \frac{ \lambda_{k} X_{k}}{R T}}
{ 1 + \frac{ \lambda^{2}_{k} X_{k}}{ \bar{c_{p}} R T^{2}} }
 \right). 
\Deqlab{dtdz}
\end{eqnarray}
%
â���¤ε�����Ѥ��Ƥ���. 



\Deqref{dtdz} �ˤ����ƽ�ʬ�˶Ž�����ʬ�ξ��ʤ����, �Ĥޤ�
$ \lambda^{2}_{k} X_{k}/ \bar{c_{p}} R T^{2} \ll 1 $ �ξ��ˤ�, 
%
\begin{eqnarray}
\DD{T}{z} 
 &\approx& - \frac{\bar{M} g}{\bar{c_{p}}} 
  \left( 1 +  \frac{ \lambda_{k} X_{k}}{R T} \right)
  \left( 1 - \frac{ \lambda^{2}_{k} X_{k}}{ \bar{c_{p}} R
   T^{2}} \right) ,
  \nonumber \\
 &\approx& - \frac{\bar{M} g}{\bar{c_{p}}} 
  \left\{  1 -  \frac{\lambda_{k} X_{k}}{\bar{c_{p}} T} 
   \left( \frac{ \lambda_{k}}{ R T} 
    - \frac{\bar{c_{p}}}{R} \right) \right\}. 
\Deqlab{dtdz_approx}
\end{eqnarray}
%
�ȶ���Ǥ���. 
â��
$ \lambda_{k} / \bar{c_{p}} T \approx 10 $ �ʤΤ�, 
$ \lambda^{2}_{k} X_{k}/ \bar{c_{p}} R T^{2} \ll 1 $ �ʤ��ɬ��
$ \lambda_{k} X_{k} / R T \ll 1 $ �Ȥʤ�. 
�ѷ��ˤ����� 2 ���������̤Ͻ�ʬ�˾�������ΤȤߤʤ�̵�뤷��. 


\Deqref{dtdz} �ˤ����ƽ�ʬ�˶Ž�����ʬ��¿�����, ���ʤ�� 
$ \lambda_{k} X_{k} / R T \gg 1 $ �ξ��ˤ�, 
%
\begin{eqnarray}
\DD{T}{z} 
 &\approx& - \frac{\bar{M} g}{\bar{c_{p}}} 
  \frac{\frac{ \lambda_{k} X_{k}}{R T}}
  {\frac{ \lambda^{2}_{k} X_{k}}{ \bar{c_{p}} R
   T^{2}} } ,
  \nonumber \\
 &\approx& 
  - \frac{\bar{M} g T}{\lambda_{k}}
\Deqlab{dtdz_approx}
\end{eqnarray}
%
�ȶ���Ǥ���. â��
$ \lambda_{k} / \bar{c_{p}} T \approx 10 $ �ʤΤ�, 
$ \lambda_{k} X_{k} / R T \gg 1 $ �ʤ��ɬ��
$ \lambda^{2}_{k} X_{k}/ \bar{c_{p}} R T^{2} \gg 1 $ �Ȥʤ�. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{�����ٸ�Ψ�η׻�}

��Ǯ���ٸ�Ψ���鲾���ٸ�Ψ��׻�����. 
%
\begin{eqnarray}
 \DD{T_{v}}{z} 
  &=& \DD{}{z}\left( \frac{M_{d}}{\bar{M}} T \right)
  \nonumber \\
 &=& \frac{M_{d}}{\bar{M}} \DD{T}{z} - \frac{M_{d} T}{\bar{M}^{2}} \DD{\bar{M}}{z}
  \nonumber \\
 &=& \frac{M_{d}}{\bar{M}} \DD{T}{z} - \frac{M_{d} T}{\bar{M}^{2}} 
  \DD{}{z}(M_{d}(1 - X) + M_{v} X)
  \nonumber \\
 &=& \frac{M_{d}}{\bar{M}} \DD{T}{z} - \frac{M_{d} T}{\bar{M}^{2}} 
  \DD{}{z}\left\{ (M_{v} - M_{d}) X \right\}
  \nonumber \\
 &=& \frac{M_{d}}{\bar{M}} \DD{T}{z} 
  - \frac{M_{d} (M_{v} - M_{d}) T}{\bar{M}^{2}} 
    \DD{X}{z}
  \nonumber 
\end{eqnarray} 
%
������, $M_{d}$ �ϴ��絤����ʬ����, $M_{v}$ �϶Ž���ʬ��ʬ����, $T_{v} =
M_{d} T / \bar{M}$ �ϲ����٤Ǥ���. ������ \Deqref{Xk} �������������, 
%
\begin{eqnarray}
 \DD{T_{v}}{z} 
  &=& \frac{M_{d}}{\bar{M}} \DD{T}{z} 
  - \frac{M_{d} (M_{v} - M_{d}) T}{\bar{M}^{2}} 
\left\{
  \frac{ \lambda_{k} X_{k} }{ R T^{2}} \DD{T}{z}
  +  \frac{\bar{M} g X_{k}}{ R T} 
\right\} 
\nonumber \\
  &=& \frac{M_{d}}{\bar{M}} \DD{T}{z} 
  - \frac{M_{d} (M_{v} - M_{d}) }{\bar{M}^{2}} 
    \frac{ \lambda_{k} X_{k} }{ R T } \DD{T}{z}
  - \frac{M_{d} (M_{v} - M_{d}) }{\bar{M}} 
    \frac{ g X_{k}}{ R} 
\nonumber \\
  &=& \frac{M_{d}}{\bar{M}}
   \left(  
    1 - \frac{(M_{v} - M_{d}) }{\bar{M}} 
    \frac{ \lambda_{k} X_{k} }{ R T } 
   \right) \DD{T}{z}
  - \frac{M_{d} (M_{v} - M_{d}) }{\bar{M}} 
    \frac{ g X_{k}}{ R} 
\end{eqnarray}
%
����� \Deqref{dtdz} �������������, 
%
\begin{eqnarray}
\DD{T_{v}}{z}
  = 
  - \frac{ M_{d} (M_{v} - M_{d}) }{\bar{M}} 
    \frac{ g X_{k} }{ R} 
  - 
  \frac{M_{d} g}{\bar{c_{p}}}
   \left(  
    1 - \frac{ M_{v} - M_{d} }{\bar{M}} 
    \frac{ \lambda_{k} X_{k} }{ R T } 
   \right) 
 \frac{ 1 +  \frac{ \lambda_{k} X_{k}}{R T}}
{ 1 + \frac{ \lambda^{2}_{k} X_{k}}{ \bar{c_{p}} R T^{2}} }
\Deqlab{vdtdz}
\end{eqnarray}
%
�����ǶŽ���ʬ����ʬ�˾��ʤ��Ȳ��ꤹ���, 
$ \lambda^{2}_{k} X_{k}/ \bar{c_{p}} R T^{2} \ll 1 $ ����
$ \lambda_{k} X_{k} / R T \ll 1 $ �ʤΤ�, 
%
\begin{eqnarray}
\DD{T_{v}}{z}
  = 
  - \frac{ M_{d} (M_{v} - M_{d}) }{\bar{M}} 
    \frac{ g X_{k} }{ R} 
  - 
  \frac{M_{d} g}{\bar{c_{p}}}
  \left(
   1 
   + \frac{ \lambda_{k} X_{k}}{R T}
   - \frac{ \lambda^{2}_{k} X_{k}}{ \bar{c_{p}} R T^{2}} 
   - \frac{ M_{v} - M_{d} }{\bar{M}}  \frac{ \lambda_{k} X_{k}}{R T}
  \right)
\end{eqnarray}
%
�ȶ���Ǥ���. 



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{��Ū�����٤η׻�}

\Deqref{vdtdz} �����Ѥ����, ��Ū������ $N^{2}$ ��, 
%
\begin{eqnarray}
 N^{2} = 
  \frac{g}{T_{v}} \left( \DD{T_{v}}{z} + \frac{M_d g}{c_{p}} \right)
\end{eqnarray}
%
�Ȥʤ�. ������ ${c_{p}}$ �ϴ��絤������Ǯ�Ǥ���. 
���μ���ŷ�ʪ�����ʤ���ΤȤ��ƶ�������, 
%
\begin{eqnarray}
N^{2} &=& 
 \frac{\bar{M} g}{M_{d} T}
 \left( \DD{T_{v}}{z} + \frac{M_d g}{c_{p}} \right)
\nonumber \\
&=&
 \frac{\bar{M} g}{M_{d} T}
\left\{
  - \frac{ M_{d} (M_{v} - M_{d}) }{\bar{M}} 
    \frac{ g X_{k} }{ R} 
  - 
  \frac{M_{d} g}{ c_{p} }
  \left(
    \frac{ \lambda_{k} X_{k}}{R T}
   - \frac{ \lambda^{2}_{k} X_{k}}{ c_{p} R T^{2}} 
   - \frac{ M_{v} - M_{d} }{\bar{M}}  \frac{ \lambda_{k} X_{k}}{R T}
  \right)
\right\}
\nonumber \\
&=&
 \frac{\bar{M} g^{2}}{ c_{p} T}
\left\{
 - \frac{ M_{v} - M_{d} }{\bar{M}} 
    \frac{ c_{p} }{ R} 
  -  \frac{ \lambda_{k}}{R T}
   + \frac{ \lambda^{2}_{k}}{ c_{p} R T^{2}} 
   + \frac{ M_{v} - M_{d} }{\bar{M}}  \frac{ \lambda_{k}}{R T}
\right\} X_{k} 
\nonumber \\
&=&
 \frac{\bar{M} g^{2}}{ c_{p} T}
\left\{
 \frac{ M_{v} - M_{d} }{\bar{M}} 
 \left( 
  \frac{ \lambda_{k}}{R T} - \frac{ c_{p} }{ R} 
 \right)
   + \frac{ \lambda_{k}}{ c_{p} T}
   \left(
     \frac{ \lambda_{k} }{ R T } 
    - \frac{ c_{p} }{ R }
   \right)
\right\} X_{k} 
\nonumber \\
&=&
 \frac{M_{d}  g^{2}}{ c_{p} T}
\left\{
 \left( 
  \frac{ \lambda_{k}}{R T} - \frac{ c_{p} }{ R} 
 \right)
 \left(
  \frac{ \lambda_{k}}{ c_{p} T}
  + \frac{ M_{v} - M_{d} }{M_{d}} 
 \right)
\right\} X_{k} 
\end{eqnarray}
%
������, �Ž�ʪ���̤����ʤ��Ȥ���
$\bar{c_{p}} = c_{p}$, $\bar{M} = M_{d}$ �Ȥߤʤ��Ƥ���. 




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{����ʸ��}
\begin{description}

\item	Atreya, S. K., Romani, P. N., 1985 : 
Photochemistry and clouds of Jupiter, Saturn and Uranus. 
in {\it Recent advances in planetary meteorology}, 
edited by Hunt, G. E., Cambridge University Press, 17-68.


\item	����μ�� ��, 1961 : 
��ر齬 Ǯ�ء������ϳ� ������. �ز�˼.

\item  
Lewis, J. S., 1969a : 
 The clouds of Jupiter and the NH${}_{3}$--H${}_{2}$O and 
NH${}_{3}$--H${}_{2}$S systems. 
 {\it Icarus}, {\bf 10}, 365.

\item	���ع��ض��� ��, 1988 : 
 ���ع������� ������ 5 ��. �����������

\item	���ҵ��� ����1984 : 
���̵��ݳ�. �����ؽ��ǲ�


\item
Weidenschilling, S. J., Lewis, J. S., 1973 : 
 Atmospheric and Cloud Structures of the Jovian Planets.
{\it Icarus}, {\bf 20}, 465-476.

\end{description}






\end{document}


