% 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
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% 履歴: 2001-09-07 杉山耕一朗; 小高さんのチェックを反映

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\begin{document}

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

    {\bf 
    SUGIYAMA Ko-ichiro \\
    (Division of earth and planetary science, Hokkaido University)\\
    ODAKA Masatsugu \\
    (Department of Mathematics, University of Tokyo)\\
    KURAMOTO Kiyoshi \\
    (Division of earth and planetary science, Hokkaido University)\\
    HAYASHI Yoshi-Yuki \\
    (Division of earth and planetary science, Hokkaido University)\\
     \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 atmospheres of the
Jovian planets. This method calculates equilibrium
composition by minimizing a thermodynamic function. Its advantage is
that we do not have to know the detail information 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 method yields the vertical distributions of condensed species of a
    model Jovian atmosphere agree with those obtained in the previous
    studies. By the use of our new scheme, the equilibrium vertical
    profiles of the atmospheres of the Jovian planet can be obtained
    more easily.
    \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


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 vertical
profiles of temperature, pressure, and static stability obtained by the
adiabatic motion of an air parcel are necessary to be known.
% 
Especially in the Jovian planets, the released latent heat from many
condensable species and heat of chemical reactions 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). 
%
Not much attention, however, has been paid to the other aspects
so far in spite of its significance in studying dynamics of the
atmospheres.  \bigskip


According to the Voyagers' observation, the elemental compositions of 
the surface atmospheres in Jovian planets are different from each other.
%
Also the elemental composition may be inhomogeneous in every Jovian
planet, since there is a large compositional difference among cloud
regions and the others. 
%
Therefore, it is necessary to give the elemental composition as a
parameter for calculating the convective equilibrium state.
\bigskip

%
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 with a wide range of bulk elemental
composition.  \bigskip


%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


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 term which contributes entropy change is formulated corresponding to 
individual chemical formula.
%
All chemical reactions which occured in an air parcel must have
been known before the numerical modeling.
%
If the elemental composition changes, basic part of the numerical code
have to be modified since concerning chemical reactions are possibly
changed.
%
Therefore, the previous method is very inconvenient for parameter study of the
atmospheric structure.

\bigskip


%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

Assuming the thermodynamic equilibrium, one can calculate the adiabatic
lapse rate and the vertical distributions of condensed species, even if
there is no information of chemical reaction formulae.
%
In this study, we develop a thermodynamic calculation method using Gibbs
free energy $G$. 
%
This method requires no detail information of chemical reaction formulae
in the atmosphere, thereby allowing the elemental composition of the
atmosphere to be easily changed.  \bigskip


%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 minimizating 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. 
\bigskip



%(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 thermodynamic variables determing the
thermodynamic state of the air parcel. In this case, the appropriate
thermodynamic function which gives criteria for the thermodynamic
equilibrium is Gibbs free energy. The thermodynamic equilibrium is
defined by 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 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, ${\mu_{i}^{\circ}}^{\phi}$ is reference chemical potential of
chemical species $i$ in phase $\phi$, ${n_{i}}^{\phi}$ are the mole
number, $R$ is gas constant, $\alpha$ is a constant ($\alpha = 1$ for
gas species, $\alpha = 0$ for condensed species).  Here we assume ideal
gas and ideal solution.  The each value of ${\mu_{i}}^{\phi}$ is found
from thermochemical tables.  The equilibrium composition
${n_{i}}^{\phi}$ is derived from minimizing $G$ under the condition
where the number of each element is conserved.  \bigskip


%(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 be obtained 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 by procedure (1). 
\bigskip

%(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.  
In this procedure, the pseudo adiabatic change is
assumed. The pseudo adiabatic change means an adiabatic change in which 
condensed species are removed from the air parcel as they condense and 
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}. 


\begin{figure}[h]
\begin{center}
\Depsf[125mm]{../ps/pseudo3.eps}
\end{center}
\caption{
\DselJ{
 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. }
{
The 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 sought. 
%
Step3: If a species condenses at the condition ($T_1$, $p_1$),
entropy reserved at next step is that of the gaseous component. 
%
Step4: Adiabatic curve is given by the set of temperatures and pressures
derived 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.

%
In order to check our model's performance, the elemental
composition and initial conditions are the same as 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

Comparison between our result and that of Atreya {\it et al}. (1999) are
shown in \Dfigref{kenshou-1}. 
%
\Dfigref{kenshou-1} shows the vertical profile of condensed species,
cloud density, and temperature. 
%
The results quite agree with those of Atreya {\it et al}. (1999). 
%
\Dfigref{kenshou-2} shows 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.
%
The results clearly show that the phase equilibrium is accurately represented
in our model.
\bigskip

\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). 
 The vertical distribution of condensed species 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

  The value of adiabatic lapse rate are changed by latent heat of
     phase change and heat of chemical reaction, thereby the atmosphere stably
     stratifies.
  The change of the adiabatic lapse rate associated with condensation
     of water is the most remarkable,
  The static stability produced by condensation of water is $2.5 \times
     10^{-5}$s$^{-2}$.
  The result suggests that the static stability of cloud region in the
     Jovian atmosphere is smaller than that of the earth, since that of
     the earth's atmosphere is $1 \times >10^{-4}$ > s$^{-2}$, so


%\Dfigref{keisan}.  The quantity of adiabatic lapse rate are changed by 
%latent heat of phase change and heat of reaction, thereby the atmosphere
%stably stratifies. 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$^{-2}$. The static stability
%of the earth's atmosphere is $1 \times 10^{-4}$ s$^{-2}$, so the static
%stability of the model Jovian atmosphere is smaller than that of the earth.


\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$^{-2}$. 
%
The adiabatic lapse rate (left) and the stataic stability (right) of the
 atmosphere of the Jupiter. 
%
 }
\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


A thermodynamic method for calculating convective equilibrium
state of the atmospheres of the Jovian planets is newly developed.  
%
The advantage of our method is that we do not have to know detail
information of corresponding chemical reaction formulae.  Therefore, our
model is more convinent for the conventional study than those used by
previous studies.  \bigskip
%

The convective equilibrium state of the atmosphere is calculated for a
model Jovian atmosphere.  The distributions of condensed species
obtained from our calculation are quite agree with those obtained by
Atreya {\it et al}. (1999).  In addition, it is confirmed that the phase
equilibrium is accurately represented in our method.  \bigskip

%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 are also
calculated.
%
The results show that the model Jovian atmosphere has stable layers by 
latent heat of phase change and heat of chemical reaction. 
%
The static stability formed by the condensation of H$_2$O is 
larger than those of the other species.


\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.

\end{description}


\end{document}



 - according -> concerning
 - all the chemical reactions -> all chemical 
 - so that の使い方
   so that ... have to ... は変 consequentry

 - reference chemical potential

 - below