% 表題: 月惑星シンポジウム 報告書
%
% 履歴: 2001-08-19 杉山耕一朗; とにかく書いてみたバージョン
% 履歴: 2001-08-28 杉山耕一朗
% 履歴: 2001-09-03 杉山耕一朗
% 履歴: 2001-09-06 杉山耕一朗
% 履歴: 2001-09-08 杉山耕一朗

\documentstyle[ascmac,Dmisc,Depsf,Dmath,Dselect]{article}

\setlength{\oddsidemargin}{-0.5cm}
\setlength{\textwidth}{170mm}
\setlength{\topmargin}{-2.4cm}
\setlength{\textheight}{250mm}
\setlength{\columnsep}{0.96cm}

%\Depsfdrafttrue
\Dnewselect{J}
\DoffJ

\pagestyle{empty}
%\setlength{\parindent}{0pt}

\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}

%木星型惑星大気の温度と凝結物質量の鉛直分布を求めるための平衡熱力学計算手
%法を新たに開発した. 用いた方法は, 熱力学関数を最小化して相平衡をさぐる手
%法であり, その特徴は関与する反応式を考えずに済むことにある. 大気成分気体
%や凝縮成分を理想気体, 理想溶液の法則に従うと仮定することにより最小化計算
%は大幅に簡略化され, 大気組成を簡単に変更することができるようになる.
%\bigskip
%
%この方法を用いて木星大気の鉛直温度・物質分布を計算し, 過去の研究と一致
%することを確かめた. 本計算手法により, 木星型惑星に対する断熱温度減率と
%凝縮物質の鉛直分布が容易に得られるようになった.


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}

%これまでの木星型惑星大気に関する研究においては, 対流平衡状態に関する熱力
%学的考察があまりおこなわれて来なかった. 凝結成分の存在する大気の対流構造
%を研究するに際しては, 空気塊の断熱変化を想定することにより, その温度圧力
%分布を予想, 期待される乾燥静的安定度を見積もることは必須の手順である. 特
%に, 木星型惑星においては複数の成分が凝結に関与し, それに付随する潜熱の放
%出は複雑な静的安定度をもった大気構造をもたらす可能性がある. ところが, ま
%さに複数の成分が凝結するというその理由によって, 大気の力学的な構造を研究
%しようとする際に必要な熱力学的考察が十分には行われて来なかったのである. 
%そこで本研究では, 木星型惑星大気を念頭に, その熱力学的に決まる対流平衡状
%態, すなわち断熱温度勾配と凝縮に伴う大気組成の変化, を簡略に計算するため
%の熱力学コードを開発する.  そして元素組成を与えたときに実現する対流平衡
%状態を記述する.
%\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.  


%木星型惑星大気の元素組成は全て同じというわけではない. ボイジャーの観測に
%よれば, 木星型惑星の大気表層の元素組成は惑星毎に異なる. また惑星全球で元
%素組成が同じであるという保証はない. 特に雲のある領域と無い領域では凝縮に
%関連する元素の存在量が大きく異なるであろう. そのため対流平衡状態を計算す
%るためには, 元素組成をパラメータとして与える必要がある.



\section{Calculation method}

%従来の木星型惑星大気の対流平衡研究では, エントロピー $S$ の保存式を, 理
%想気体の状態方程式と潜熱・反応熱を用いて温度・圧力・組成の関数として定式
%化した(Atreya and Romani, (1985), Atreya {\it et al}., 1999). エントロピー
%の変化をもたらす各項は, 空気塊において生じうる個々の化学反応に対応し
%て与えられる. 空気塊内で発生するすべての化学反応をあらかじめ掌握していな
%ければならない. 数値コードにおいては, 元素組成や分布の異る大気を考えるた
%びに生じうる化学反応が変わるので, 数値コードの基幹部分を書き換えるという
%作業が発生することになる. これでは, 大気構造のパラメタ研究には非常に不便
%である.  
%\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. 


%そもそも平衡状態を仮定しているので, 化学反応に関する情
%報がなくとも断熱温度減率と凝結物質の鉛直分布は計算できるはずである.  そ
%こで本研究では大気の平衡状態をギブスの自由エネルギー $G$ を用いて記述す
%ることにより, 大気中で生じる反応式を考えずに済むようにした. そのため大気
%の元素組成を容易に変更できるようになった.  
%\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. 

%具体的な計算方法は以下の通りである: (1) 温度・圧力固定, 元素数保存の条件
%のもとギブスの自由エネルギー $G$ を最小化し, 平衡組成を求める. (2) 温度・
%圧力空間での平衡組成を用いてエントロピー $S$ を計算する. (3)温度・圧力空
%間でのエントロピー$S$ から断熱線 $dS = 0$ を計算する. 尚, 断熱線の計算に
%おいては擬湿潤断熱変化を仮定する. これは気体が凝縮した際に凝縮物が系から
%離脱すると考えた場合に実現する断熱減率で, 離脱した凝縮物質も含めた全エン
%トロピーが保存すると考える.  \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)大気の平衡状態の記述: 熱力学変数として温度・圧力・物質存在量を選択す
%る. このとき大気の状態を与える適切な熱力学関数はギブス自由エネルギー $G$ 
%である. 大気の平衡状態はギブス自由エネルギー $G$ が最小化された状態であ
%るとする. 温度・圧力を与えると, $G$ は以下のように書ける. 

(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}
%
%但し ${\mu_{i}^{\circ}}^{\phi}$ は基準状態での化学ポテンシャルであり, 物性
%値から決まる量である. 元素数保存の条件下で上式を最小化する物質存在量 
%$n_{i}^{\phi}$ を求める.  \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)エントロピーの計算: エントロピーは Maxwell の関係式から求めることがで
%きる. 温度, 圧力, 化学ポテンシャル, 平衡組成を与えることによってエントロ
%ピー $S$ が求まる. 

(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)大気の断熱変化の記述: 大気の断熱変化はエントロピー S の保存として記述
%することができる. 温度・圧力空間で dS = 0 の曲線の通る軌跡を順にたどれば, 
%断熱温度減率と凝結物質の存在量を求めることができる(\Dfigref{pseudo} 参照).

(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{
% 断熱線の求め方. Step1: 初期温度 $T_0$, 初期圧力 $p_0$での平衡組
% 成を計算し, エントロピー $S_0$ を求める. Step2: 圧力 $p_1 = p_0 + dp$ 
% に変化させる. 温度を変化させた時のエントロピーを順次計算し, 前のステッ
% プでのエントロピー $S_0$ と一致する温度 $T_1$ を求める. Step3: $T_1,
% p_1$ において凝縮が生じた場合, 次のステップで保存させるエントロピーは,
% $T_1, p_1$ での気体成分のみのエントロピー ${S_4}_{\rm gas}$. Step4:
% Step 1 から Step 3 において得られた温度・圧力を順に結んで断熱線を引く. 
%
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}

%大気成分気体やその凝結成分を理想気体, 理想溶液の法則に従うと仮定し, 前述
%の計算手法に則って木星大気の対流平衡状態の計算を行った. モデルの検証も合
%わせて行うために, 大気組成と初期条件は Atreya {\it et al.} (1999) と同様
%にした.

\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}

%モデルの検証結果を \Dfigref{kenshou-1}, \Dfigref{kenshou-2} に示す.
%\Dfigref{kenshou-1} では大気組成の変化から凝縮物質の鉛直分布を計算し, そ
%れを Atreya {\it et al.} (1999) と比較した. 比較の結果, 凝縮高度, 凝縮量, 
%温度分布共に Atreya {\it et al}. (1999) の結果と一致した. 
%\Dfigref{kenshou-2} は H$_2$O(g), NH$_3$(g) の分圧と飽和蒸気圧とをプロッ
%トしたものである. 凝縮が生じてからは, H$_2$O(g), NH$_3$(g) 共に分圧と飽
%和蒸気圧とが一致している. これによりモデル中で相平衡が正確に再現されてい
%ることがわかる.  \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{
% 右: Atreya {\it et al.} (1999) の計算した雲密度・温度分布
%
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{
% 分圧と飽和蒸気圧の関係. 左: H$_2$O の分圧と飽和蒸気圧. 右: NH$_3$ の
%分圧と飽和蒸気圧. 両者共に凝縮が生じてからは分圧と飽和蒸気圧が一致する. 
%
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}

%木星での断熱温度減率と大気の静的安定度を\Dfigref{keisan} に示す. 凝結の
%潜熱によって断熱温度減率の大きさが変化し, 大気が安定成層していることがわ
%かる. 水の凝縮に伴う断熱温度減率の変化が最も顕著で, それに伴う静的安定度
%は $2.5 \times 10^{-5}$ s$^{-2}$ である. 地球大気の静的安定度は $1
%\times 10^{-5}$ s$^{-2}$ なので, 木星大気の静的安定度は地球に比べて小さ
%いと言える.
%\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{
%左:  断熱温度減率. 平均断熱温度減率は - 2.0 K/km. H$_2$O, NH$_3$ の凝縮,
% 化学反応による NH$_4$SH の生成により, 温度減率が小さくなっている. 
%右: 大気の静的安定度. H$_2$O, NH$_3$ の凝縮, NH$_4$SH の生成による安定度
% はそれぞれ, $2.5 \times 10^{-5}$ 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}

%木星型惑星大気の対流平衡状態を計算するための熱力学コードを開発し, 元素組
%成を Atreya {\it et al}. (1999) と同様にした時の木星大気の熱力学状態を計
%算した.  本研究の計算手法の特徴は大気中で生じ得る化学反応を知らなくとも
%計算できることであり, それゆえ従来の研究よりも汎用性が高いことである. 
%計算から得られた凝縮物分布は Atreya {\it et al}. (1999) と等しく, さらに
%分圧と飽和蒸気圧との関係から相平衡が正しく表現されていることが確かめられ
%た. これにより本熱力学コードの妥当性が確かめられた. 
%\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).  


%対流平衡状態として, 断熱温度減率, それに伴う静的安定度を定量的に計算した. 
%その結果凝縮物質の潜熱によって大気が安定成層することが示された. 
%H$_2$O の凝縮に伴う静的安定度は, 他の物質の凝縮に伴うそれに比べて最も
%大きいことが示された. 

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}