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% ����: 2001-08-19 �����̰�ϯ; �Ȥˤ����񤤤Ƥߤ��С������
% ����: 2001-08-28 �����̰�ϯ
% ����: 2001-09-03 �����̰�ϯ
% ����: 2001-09-06 �����̰�ϯ
% ����: 2001-09-08 �����̰�ϯ

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\begin{document}

   \begin{center}
    {\LARGE {\bf Thermodynamic calculation of the atmospheres of the Jovian planets}}
    \vspace{7mm}

    {\bf 
    Ko-ichiro SUGIYAMA$^{*}$, Masatsugu ODAKA$^{**}$, 
    Kiyoshi KURAMOTO$^{*}$, Yoshi-Yuki HAYASHI$^{*}$ \\
  \vspace{5mm}
  \scriptsize
  ${}^{*}$Division of Earth and Planetary Sciences, Hokkaido
    University, 
  Japan \\
  ${}^{**}$Graduate School of Mathematical Sciences, University of Tokyo, 
  Japan \\ 
     \vspace{3mm} 
    }
   \end{center}

   \begin{abstract}

%�����������絤�β��٤ȶŷ�ʪ���̤α�ľʬ�ۤ���뤿���ʿ��Ǯ�ϳط׻���
%ˡ�򿷤��˳�ȯ����. �Ѥ�����ˡ��, Ǯ�ϳشؿ���Ǿ���������ʿ�դ򤵤����
%ˡ�Ǥ���, ������ħ�ϴ�Ϳ����ȿ������ͤ����˺Ѥळ�Ȥˤ���. �絤��ʬ����
%��Ž���ʬ�����۵���, �����ϱդ�ˡ§�˽����Ȳ��ꤹ�뤳�Ȥˤ��Ǿ����׻�
%�������˴�ά������, �絤�������ñ���ѹ����뤳�Ȥ��Ǥ���褦�ˤʤ�.
%\bigskip
%
%������ˡ���Ѥ��������絤�α�ľ���١�ʪ��ʬ�ۤ�׻���, ���θ���Ȱ���
%���뤳�Ȥ�Τ��᤿. �ܷ׻���ˡ�ˤ��, �������������Ф�����Ǯ���ٸ�Ψ��
%�Ž�ʪ���α�ľʬ�ۤ��ưפ�������褦�ˤʤä�.


An equilibrium thermodynamic calculation method is newly developed to
	investigate the vertical profiles of adiabatic lapse
	rate, gas composition, and amount of condensed species of
	the Jovian planetary atmospheres. 
In this method, the equilibrium composition is calculated by 
	minimizing the thermodynamic function. 
The advantage of this method is that we do not 
	have to know the details of corresponding chemical
	reaction formulae.  
Assuming ideal gas and ideal solution, the calculation scheme 
	is greatly simplified so that the calculations for
	different atmospheric compositions can be carried out
	with ease.
Our new scheme employing this method is confirmed to yield the 
	vertical distributions of condensed species of a model 
	Jovian atmosphere quite similar to those obtained 
	in the previous studies. 
By the use of our new scheme, the equilibrium vertical profiles 
	of the Jovian planetary atmospheres can be obtained more
	easily. 
The complete set of numerical code is presented as 
	Sugiyama {\it et al}. (2001).
\end{abstract}



\section{Introduction}

%����ޤǤ������������絤�˴ؤ��븦��ˤ����Ƥ�, ��ήʿ�վ��֤˴ؤ���Ǯ��
%��Ū�ͻ������ޤꤪ���ʤ�����ʤ��ä�. �ŷ���ʬ��¸�ߤ����絤����ή��¤
%�򸦵椹��˺ݤ��Ƥ�, ����������Ǯ�Ѳ������ꤹ�뤳�Ȥˤ��, ���β��ٰ���
%ʬ�ۤ�ͽ��, ���Ԥ���봥����Ū�����٤��Ѥ�뤳�Ȥ�ɬ�ܤμ��Ǥ���. ��
%��, �����������ˤ����Ƥ�ʣ������ʬ���ŷ�˴�Ϳ��, ������տ魯����Ǯ����
%�Ф�ʣ������Ū�����٤��ä��絤��¤��⤿�餹��ǽ��������. �Ȥ�����, ��
%����ʣ������ʬ���ŷ뤹��Ȥ���������ͳ�ˤ�ä�, �絤���ϳ�Ū�ʹ�¤�򸦵�
%���褦�Ȥ���ݤ�ɬ�פ�Ǯ�ϳ�Ū�ͻ�����ʬ�ˤϹԤ�����ʤ��ä��ΤǤ���. 
%�������ܸ���Ǥ�, �����������絤��ǰƬ��, ����Ǯ�ϳ�Ū�˷�ޤ���ήʿ�վ�
%��, ���ʤ����Ǯ���ٸ��ۤȶŽ̤�ȼ���絤�������Ѳ�, ���ά�˷׻����뤿��
%��Ǯ�ϳإ����ɤ�ȯ����.  �����Ƹ���������Ϳ�����Ȥ��˼¸�������ήʿ��
%���֤򵭽Ҥ���.
%\bigskip

\hspace{5mm}
In the previous studies on the atmospheres of the Jovian planets,
	the thermodynamic consideration on the convective
	equilibrium state was not sufficiently given. 
For studying the convective structure of atmosphere, the possible
	vertical profiles of temperature, pressure, and static
	stability obtained by considering adiabatic change of an
	air parcel have to be studied in advance. 
Especially in the Jovian planets, the latent heat release from many
	condensable species and reaction heat from many chemical 
	processes possibly produce complicated structure of 
	static stability. 
The vertial distribution of condensed species has been calculated 
	in many paper (e.g. Atreya and Romani, 1985, Atreya {\it
	et al}., 1999). 
In those studies, however, not much attention has been paid to 
	the aspect of dynamics, and consequently
	information such as the vertical profile of static
	stability has not been described very intensely. 


According to the Voyagers' observation, 
	the elemental compositions of the surface atmospheres 
	in Jovian planets are different from each other. 
In each planet, the elemental composition may be inhomogeneous.
Probably, the composition differs greatly among the regions 
	with and without cloud. 
In order to consider the possible convective equilibrium states 
	for the variety of elemental composition, 
	it is necessary to give the elemental composition easily
	as a parameter for calculation. 


In this paper, an equilibrium thermodynamic calculation method is 
	newly developed to investigate convective equilibrium 
	state, that is, vertical profiles of adiabatic lapse
	rate, gas composition, and condensed species in the
	atmospheres for a wide range of bulk elemental 
	composition.  


%�����������絤�θ�������������Ʊ���Ȥ����櫓�ǤϤʤ�. �ܥ����㡼�δ�¬��
%����, �������������絤ɽ�ؤθ���������������˰ۤʤ�. �ޤ���������Ǹ�
%��������Ʊ���Ǥ���Ȥ����ݾڤϤʤ�. �ä˱��Τ����ΰ��̵���ΰ�Ǥ϶Ž̤�
%��Ϣ���븵�Ǥ�¸���̤��礭���ۤʤ�Ǥ�����. ���Τ�����ήʿ�վ��֤�׻���
%�뤿��ˤ�, ����������ѥ�᡼���Ȥ���Ϳ����ɬ�פ�����.



\section{Calculation method}

%����������������絤����ήʿ�ո���Ǥ�, ����ȥ��ԡ� $S$ ����¸����, ��
%�۵��Τξ�������������Ǯ��ȿ��Ǯ���Ѥ��Ʋ��١����ϡ������δؿ��Ȥ����꼰
%������(Atreya and Romani, (1985), Atreya {\it et al}., 1999). ����ȥ��ԡ�
%���Ѳ���⤿�餹�ƹ��, �������ˤ�������������ġ��β���ȿ�����б���
%��Ϳ������. ���������ȯ�����뤹�٤Ƥβ���ȿ���򤢤餫���Ᾰ�����Ƥ���
%����Фʤ�ʤ�. ���ͥ����ɤˤ����Ƥ�, ����������ʬ�ۤΰۤ��絤��ͤ��뤿
%�Ӥ��������벽��ȿ�����Ѥ��Τ�, ���ͥ����ɤδ���ʬ��񤭴�����Ȥ���
%��Ȥ�ȯ�����뤳�Ȥˤʤ�. ����Ǥ�, �絤��¤�Υѥ�᥿����ˤ���������
%�Ǥ���.  
%\bigskip

\hspace{5mm}
In the previous studies, entropy $S$ is formulated as a 
	function of temperature, pressure, and composition using
	the equation of state for ideal gas, latent heat of phase
	change, and heat of chemical reaction 
	(Atreya and Romani, 1985, Atreya {\it et al}, 1999). 
Each entropy production term in the equation of entropy change is 
	formulated according to individual chemical formula. 
All of the chemical reactions which occur in an air parcel must 
	be known beforehand. 
If the elemental composition changes, concerning chemical 
	reactions possibly changes, 
	and consequently the related entropy production term 
	should be altered and 
	the basic part of the numerical code have to be 
	modified. 
In this point, the previous method is very inconvenient for 
	parameter study of the atmospheric composition. 


%���⤽��ʿ�վ��֤��ꤷ�Ƥ���Τ�, ����ȿ���˴ؤ����
%�󤬤ʤ��Ȥ���Ǯ���ٸ�Ψ�ȶŷ�ʪ���α�ľʬ�ۤϷ׻��Ǥ���Ϥ��Ǥ���.  ��
%�����ܸ���Ǥ��絤��ʿ�վ��֤򥮥֥��μ�ͳ���ͥ륮�� $G$ ���Ѥ��Ƶ��Ҥ�
%�뤳�Ȥˤ��, �絤���������ȿ������ͤ����˺Ѥ�褦�ˤ���. ���Τ����絤
%�θ����������ưפ��ѹ��Ǥ���褦�ˤʤä�.  
%\bigskip

We have to recall that,
	assuming that the air parcel is in a thermodynamic
	equilibrium state, 
	there is no need of information on the details of 
	chemical reactions to have the equilibrium  
	composition of air parcel;
	one can calculate the adiabatic lapse rate and the
	vertical distributions of condensed species 
	only with the knowledge of chemical potentials of
	the possible components.
Here in this study, we develop a thermodynamic calculation method
	using Gibbs free energy $G$. 
This method requires no information on chemical reaction 
      formulae in the atmosphere, 
      thereby allowing the elemental composition of 
      the atmosphere to be easily changed. 

%����Ū�ʷ׻���ˡ�ϰʲ����̤�Ǥ���: (1) ���١����ϸ���, ���ǿ���¸�ξ��
%�Τ�ȥ��֥��μ�ͳ���ͥ륮�� $G$ ��Ǿ�����, ʿ�����������. (2) ���١�
%���϶��֤Ǥ�ʿ���������Ѥ��ƥ���ȥ��ԡ� $S$ ��׻�����. (3)���١����϶�
%�֤ǤΥ���ȥ��ԡ�$S$ ������Ǯ�� $dS = 0$ ��׻�����. ��, ��Ǯ���η׻���
%�����Ƥϵ�������Ǯ�Ѳ����ꤹ��. ����ϵ��Τ��Ž̤����ݤ˶Ž�ʪ���Ϥ���
%Υæ����ȹͤ������˼¸�������Ǯ��Ψ��, Υæ�����Ž�ʪ����ޤ᤿������
%�ȥ��ԡ�����¸����ȹͤ���.  \bigskip

Our calculation method consists of three procedures as follows: 
(1) equilibrium composition is calculated by minimizing Gibbs
	free energy for given temperature and pressure,
(2) entropy is calculated for the equilibrium composition, and 
(3) adiabatic curve $dS = 0$ is calculated by seeking temperature
	and pressure under which the value of entropy is
	conserved. 
The details of these procedures are explained below. 




%(1)�絤��ʿ�վ��֤ε���: Ǯ�ϳ��ѿ��Ȥ��Ʋ��١����ϡ�ʪ��¸���̤�����
%��. ���ΤȤ��絤�ξ��֤�Ϳ����Ŭ�ڤ�Ǯ�ϳشؿ��ϥ��֥���ͳ���ͥ륮�� $G$ 
%�Ǥ���. �絤��ʿ�վ��֤ϥ��֥���ͳ���ͥ륮�� $G$ ���Ǿ������줿���֤Ǥ�
%��Ȥ���. ���١����Ϥ�Ϳ�����, $G$ �ϰʲ��Τ褦�˽񤱤�. 

(1) Calculation of equilibrium composition: 
Temperature, pressure and composition are chosen as the
	thermodynamic variables which 
	determine the thermodynamic state of the air parcel. 
Correspondingly, the appropriate thermodynamic function which 
	is used in the criteria for the thermodynamic equilibrium
	is Gibbs free energy. 
The thermodynamic equilibrium is defined as the state where Gibbs
	free energy is minimized. 
Gibbs free energy $G$ can be written as follows:

\begin{eqnarray}
G(T, p, n^{\phi}_{i}) 
&=& \sum \mu_{i}^{\phi}(T, p, n^{\phi}_{i}) n_{i}^{\phi} \nonumber \\
&=& \sum 
    \left\{
        {\mu_{i}^{\circ}}^{\phi}(T) 
      + RT \ln \frac{n_{i}^{\phi}}{\sum n_{i}^{\phi}}
      + \alpha_{i} RT \ln{\frac{p}{p_0}} 
    \right\}
    n_{i}^{\phi}, \nonumber
\Deqlab{1}
\end{eqnarray}
%
%â�� ${\mu_{i}^{\circ}}^{\phi}$ �ϴ����֤Ǥβ��إݥƥ󥷥��Ǥ���, ʪ��
%�ͤ����ޤ��̤Ǥ���. ���ǿ���¸�ξ�ﲼ�Ǿ弰��Ǿ�������ʪ��¸���� 
%$n_{i}^{\phi}$ �����.  \bigskip
%
where 
	$T$ is temperature, 
	$p$ is pressure, 
	$p_0$ is pressure at standard state, 
	$R$ is gas constant, 
	${n_{i}}^{\phi}$  is molar number of chemical specie $i$ 
	in phase $\phi$, 
	${\mu_{i}^{\circ}}^{\phi}$ is its reference chemical potential,
	$\alpha_{i}$ is a parameter to distinguish phase 
	($\alpha_{i} = 1$ for gas species, 
	$\alpha_{i} = 0$ for condensed species). 
Here,  we assume ideal gas and ideal solution.  
The values of ${\mu_{i}}^{\phi}$ are found from thermochemical 
	tables.  
The equilibrium composition ${n_{i}}^{\phi}$ is derived from
	minimizing $G$ under the condition 
	that the number of each element is conserved.  



%(2)����ȥ��ԡ��η׻�: ����ȥ��ԡ��� Maxwell �δط���������뤳�Ȥ���
%����. ����, ����, ���إݥƥ󥷥��, ʿ��������Ϳ���뤳�Ȥˤ�äƥ���ȥ�
%�ԡ� $S$ ����ޤ�. 

(2) Calculation of entropy: 
%
Entropy $S$ can obtained by using a Maxwell's relation as follows:

\begin{eqnarray}
S 
  &=& - \left( \DP{G}{T} \right)_{p, n_{i}} 
    \nonumber \\
  &=& - 
    \sum_{i} \left\{ \DP{{\mu_i^{\circ}}^{\phi}(T)}{T}
    + R \ln \left( \frac{n_i^{\phi}}{\sum n_i^{\phi}}\right) 
    + \alpha R \ln p \right\} 
    n_{i}^{\phi}. \nonumber
\end{eqnarray}
%
Entropy of the air parcel is calculated for the equilibrium
	composition determined in procedure (1). 


%(3)�絤����Ǯ�Ѳ��ε���: �絤����Ǯ�Ѳ��ϥ���ȥ��ԡ� S ����¸�Ȥ��Ƶ���
%���뤳�Ȥ��Ǥ���. ���١����϶��֤� dS = 0 �ζ������̤뵰�פ��ˤ��ɤ��, 
%��Ǯ���ٸ�Ψ�ȶŷ�ʪ����¸���̤���뤳�Ȥ��Ǥ���(\Dfigref{pseudo} ����).

(3) Calculation of adiabatic curve $dS = 0$: 
The adiabatic change of the gas parcel is determined by
	conservation of total entropy. 
We can choose pure adiabatic change and pseudo adiabatic change. 
The pseudo adiabatic change is a change in which 
	condensed species are removed from the air parcel as they 
	condense, while total entropy including the removed species 
	conserves. 
The locus where $dS = 0$ in temperature and pressure space is 
	estimated by the steps shown in \Dfigref{pseudo}. 
In the following examples, pseudo adiabatic change is assumed. 



\begin{figure}[h]
\begin{center}
\Depsf[120mm]{../ps/pseudo3.eps}
\end{center}
\caption{
% ��Ǯ���ε����. Step1: ������� $T_0$, ������� $p_0$�Ǥ�ʿ����
% ����׻���, ����ȥ��ԡ� $S_0$ �����. Step2: ���� $p_1 = p_0 + dp$ 
% ���Ѳ�������. ���٤��Ѳ����������Υ���ȥ��ԡ���缡�׻���, ���Υ��ƥ�
% �פǤΥ���ȥ��ԡ� $S_0$ �Ȱ��פ��벹�� $T_1$ �����. Step3: $T_1,
% p_1$ �ˤ����ƶŽ̤����������, ���Υ��ƥåפ���¸�����륨��ȥ��ԡ���,
% $T_1, p_1$ �Ǥε�����ʬ�ΤߤΥ���ȥ��ԡ� ${S_4}_{\rm gas}$. Step4:
% Step 1 ���� Step 3 �ˤ���������줿���١����Ϥ��˷�����Ǯ�������. 
%
Calculation method of adiabatic curve. 
%
Step1: The equilibrium composition is calculated for 
		initial temperature $T_0$ and initial pressure $p_0$. 
       Entropy $S_0$ is obtained from the equilibrium composition.
%
Step2: Pressure is changed to $p_1 = p_0 + dp$.  
       Temperature $T_1$ at which entropy is equal to $S_0$ is
	       recursively sought. 
%
Step3: If a specie condenses at ($T_1$, $p_1$), 
		the value of entropy to be reserved at next step 
		is that of the gaseous component. 
%
Step4: Adiabatic curve is given by the set of temperatures and 
		pressures derived repeatedly from Step 1 -- 3. 
%
}
\Dfiglab{pseudo}
\end{figure}



\section{Result}

%�絤��ʬ���Τ䤽�ζŷ���ʬ�����۵���, �����ϱդ�ˡ§�˽����Ȳ��ꤷ, ����
%�η׻���ˡ��§�ä������絤����ήʿ�վ��֤η׻���Ԥä�. ��ǥ�θ��ڤ��
%�碌�ƹԤ������, �絤�����Ƚ������ Atreya {\it et al.} (1999) ��Ʊ��
%�ˤ���.

\hspace{5mm}
In the followings, we will demonstrate the performance of our
	model by showing some results for a model Jovian 
	atmosphere.
In order to verify our scheme, 
	the elemental composition and initial conditions 
	are chosen as those of Atreya {\it et al} (1999).

\subsection{Performance check of our model}

%��ǥ�θ��ڷ�̤� \Dfigref{kenshou-1}, \Dfigref{kenshou-2} �˼���.
%\Dfigref{kenshou-1} �Ǥ��絤�������Ѳ�����Ž�ʪ���α�ľʬ�ۤ�׻���, ��
%��� Atreya {\it et al.} (1999) ����Ӥ���. ��Ӥη��, �Ž̹���, �Ž���, 
%����ʬ�۶��� Atreya {\it et al}. (1999) �η�̤Ȱ��פ���. 
%\Dfigref{kenshou-2} �� H$_2$O(g), NH$_3$(g) ��ʬ����˰�¾������Ȥ�ץ���
%�Ȥ�����ΤǤ���. �Ž̤������Ƥ����, H$_2$O(g), NH$_3$(g) ����ʬ����˰
%�¾������Ȥ����פ��Ƥ���. ����ˤ���ǥ������ʿ�դ����Τ˺Ƹ�����Ƥ�
%�뤳�Ȥ��狼��.  \bigskip

\hspace{5mm}
Comparison between the result of present model 
	and that of Atreya {\it et al}. (1999) is 
	shown in \Dfigref{kenshou-1}. 
The vertical profiles of condensed species, cloud density, and
	temperature show good agreement between both results. 
In \Dfigref{kenshou-2}, 
	partial and saturated pressures of H$_2$O and NH$_3$ are
	plotted as a function of total pressure 
When condensation occurs, the partial pressures of both species 
	agree with their saturated pressures.  
This indicates that the phase equilibrium is accurately reproduced
	in our model.
\vspace{5mm}

\begin{figure}[h]
\begin{center}
  \Depsf[60mm]{../ps/cloud-t.eps}
  \hspace{15mm}
  \Depsf[60mm]{../ps/AW1999.ps}
\end{center}
\caption{
% ��: Atreya {\it et al.} (1999) �η׻�������̩�١�����ʬ��
%
Comparison between our result and that of Atreya {\it et al}. (1999). 
	Vertical profiles of condensed species, cloud density, and
	temperature in our model (left) and 
	in Atreya {\it et al}. (1999, right). 
%
 }
\Dfiglab{kenshou-1}
\end{figure}

\begin{figure}[h]
\begin{center}
  \Depsf[60mm]{../ps/vaper-H2O-edit.eps}
  \hspace{17mm}
  \Depsf[60mm]{../ps/vaper-NH3-edit.eps}
\end{center}
\caption{
% ʬ����˰�¾������δط�. ��: H$_2$O ��ʬ����˰�¾�����. ��: NH$_3$ ��
%ʬ����˰�¾�����. ξ�Զ��˶Ž̤������Ƥ����ʬ����˰�¾����������פ���. 
%
The relationship between partial (solid line) and saturated (broken
 line) pressures of H$_2$O (left) and NH$_3$ (right).
%
 }
\Dfiglab{kenshou-2}
\end{figure}


\subsection{Calculation of the convective equilibrium state}

%�����Ǥ���Ǯ���ٸ�Ψ���絤����Ū�����٤�\Dfigref{keisan} �˼���. �ŷ��
%��Ǯ�ˤ�ä���Ǯ���ٸ�Ψ���礭�����Ѳ���, �絤���������ؤ��Ƥ��뤳�Ȥ���
%����. ��ζŽ̤�ȼ����Ǯ���ٸ�Ψ���Ѳ����Ǥ⸲����, �����ȼ����Ū������
%�� $2.5 \times 10^{-5}$ s$^{-2}$ �Ǥ���. �ϵ��絤����Ū�����٤� $1
%\times 10^{-5}$ s$^{-2}$ �ʤΤ�, �����絤����Ū�����٤��ϵ����٤ƾ���
%���ȸ�����.
%\bigskip

\hspace{5mm}
Adiabatic lapse rate and static stability of the atmosphere are
	shown in \Dfigref{keisan}.  
The value of adiabatic lapse rate changes because of 
	the latent heat of phase change and the heat of chemical
	reaction. 
In those regions, the atmosphere is stably stratified. 
The change of the adiabatic lapse rate associated with
	condensation of water is the most remarkable, 
	producing a layer with static stability of 
	$2.5 \times 10^{-5}$ s$^{-2}$. 
Static stability of the cloud layer in the Jovian atmosphere is
	smaller than that of the earth, 
	whose value is about $1 \times 10^{-4}$ s$^{-2}$.


\begin{figure}[h]
\begin{center}
  \Depsf[60mm]{../ps/gamma2.eps}
  \hspace{20mm}
  \Depsf[60mm]{../ps/stability2.eps}
\end{center}
\caption{
%��:  ��Ǯ���ٸ�Ψ. ʿ����Ǯ���ٸ�Ψ�� - 2.0 K/km. H$_2$O, NH$_3$ �ζŽ�,
% ����ȿ���ˤ�� NH$_4$SH �������ˤ��, ���ٸ�Ψ���������ʤäƤ���. 
%��: �絤����Ū������. H$_2$O, NH$_3$ �ζŽ�, NH$_4$SH �������ˤ�������
% �Ϥ��줾��, $2.5 \times 10^{-5}$ s$^{-2}$, $8.0 \times 10^{-6}$
% s$^{-2}$, $4.0 \times 10^{-6}$ s$^{-2}$. 
%
Adiabatic lapse rate (left) and static stability (right) of the 
	Jovian atmosphere. 
%
 }
\Dfiglab{keisan}
\end{figure}


\section{Conclusion}

%�����������絤����ήʿ�վ��֤�׻����뤿���Ǯ�ϳإ����ɤ�ȯ��, ������
%���� Atreya {\it et al}. (1999) ��Ʊ�ͤˤ������������絤��Ǯ�ϳؾ��֤��
%������.  �ܸ���η׻���ˡ����ħ���絤����������벽��ȿ�����Τ�ʤ��Ȥ�
%�׻��Ǥ��뤳�ȤǤ���, ����椨����θ���������������⤤���ȤǤ���. 
%�׻���������줿�Ž�ʪʬ�ۤ� Atreya {\it et al}. (1999) ��������, �����
%ʬ����˰�¾������Ȥδط�������ʿ�դ�������ɽ������Ƥ��뤳�Ȥ��Τ�����
%��. ����ˤ����Ǯ�ϳإ����ɤ����������Τ����줿. 
%\bigskip

\hspace{5mm}
A thermodynamic method for calculating convective equilibrium
	state of the atmospheres of the Jovian planets is newly
	developed.  
The advantage of the new method is that we do not have to
	know the details of corresponding chemical reaction
	formulae. 
Because of this, our model is more convenient 
	for the parameter study of atmospheric thermo equilibrium structure
	than those used by previous studies.

%

An example of calculation is shown for a parameter set of 
	a model Jovian atmosphere. 
It is confirmed that the phase equilibrium is accurately 
	reproduced in our new scheme. 
The distribution of condensed species obtained by our model 
	is quite similer to Atreya {\it et al}. (1999).  


%��ήʿ�վ��֤Ȥ���, ��Ǯ���ٸ�Ψ, �����ȼ����Ū�����٤�����Ū�˷׻�����. 
%���η�̶Ž�ʪ������Ǯ�ˤ�ä��絤���������ؤ��뤳�Ȥ������줿. 
%H$_2$O �ζŽ̤�ȼ����Ū�����٤�, ¾��ʪ���ζŽ̤�ȼ���������٤ƺǤ�
%�礭�����Ȥ������줿. 

Adiabatic lapse rate and corresponding static stability of the 
	model Jovian atmosphere are also calculated.
It is shown that the Jovian atmosphere has stably stratified
	layers because of the latent heat of phase change and the
	heat of chemical reaction. 
The static stability formed by the condensation of H$_2$O is 
	larger than those of the other species. 


\section*{Acknowledgement}

\hspace{5mm}
The authors wish to thank Prof. Hashimoto of Hokkaido University 
for many helpful suggestion for the formulation of the method of
minimization of chemical potential. The data are manipulated by
gtool4 (Toyoda 2001) and other softwares are from GFD Dennou Club
(http://www.gfd-dennou.org/). 


\section{Reference}

\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}, 
Cambridge University Press, pp. 17--68. 

\item
Atreya, S. K. and Wong, M. H., Owen, T. C., Mahaffy, P. R., 
Niemann, H. B., de Pater, I., Drossart, P., Encrenaz, Th., 1999:  
A Composition of the atmospheres of Jupiter and Saturn: 
deep atmospheric composition, cloud stracture, vertical mixing, 
and Origin. 
{\it Planetary and Space Science}, {\bf 47}, p1243--1262.

\item
Sugiyama, K., Odaka, M., Kuramoto, K., and Hayashi, Y.-Y., 2001:
Oboro.
http://www.gfd-dennou.org/arch/oboro/ \ \ 
GFD Dennou Club.

\item
Toyoda, E., and Davis Project, 2001: 
gtool4. 
http://www.gfd-dennou.org/arch/gtool4/ \ \ 
GFD Dennou Club.

\end{description}


\end{document}