% 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

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

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

%\Depsfdrafttrue

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

\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 Jovian planet
atmospheres. This method calculates equilibrium composition by
minimizing a thermodynamic function. Its advantage 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 method yields 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 planet atmospheres 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 with many
condensable species, the vertical profiles of temperature, pressure, and
static stability achieved by the adiabatic change of an air parcel are to
be known.
%
Especially in the Jovian planets, 
the released latent heat possibly causes the
atmospheric structure to have complicated static stability, 
because multiple species are condensable.
%
But, just for this reason, the
thermodynamic consideration which is necessary for studying the dynamic
structure of the atmospheres has not been sufficiently given.
%
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
of the atmosphere, with a wide possible range of bulk elemental composition
\bigskip

%木星型惑星大気の対流平衡計算において考慮しなくてはならないことは, (1) 大
%気中で生じる化学反応と凝縮物質の混合による効果, (2) 木星型惑星大気の元素
%組成が明らかにされていないこと, である.  \bigskip

There are two factors which must be considered in convective equilibrium
calculation for the atmosphere of the Jovian planets: (1) the effects of
chemical reactions and mixing of condensed species. 
(2) the fact that the elemental compositions of the atmospheres 
have not yet been determined.
\bigskip


%木星型惑星の大気中では固体の水, 固体のアンモニア, 固体のメタンだけでなく, 
%化学反応によって硫化アンモニウム, 凝縮物の混合によってアンモニア水溶液が
%生成されると考えられている(Atreya and Romeni, 1985). そのため対流平衡状
%態を計算するためには, これらの化学反応と凝縮物質の混合を考慮せねばならな
%い. 水の凝縮だけを考えればよい地球大気の場合に比べ計算は複雑である.
%\bigskip

In the atmospheres of the Jovian planets, it is expected that not only solid
H$_2$O, NH$_3$, CH$_4$, but also NH$_4$SH produced by the chemical
reaction of NH$_3$ and H$_2$S and
aqueous ammonia solution produced by mixing of liquid H$_2$O and NH$_3$
are created (Atreya and Romani, 1985). 
%
Therefore, these phase change, chemical reactions, and mixing must
be considered when the convective equilibrium state is
calculated. 
%
Compared to the case of the earth's atmosphere where water is only the
dominant condensate, the calculation of the atmospheres of the Jovian
planets is more complicated.  \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.

According to the Voyagers' observations, the elemental compositions of 
the surface atmospheres in Jovian planets is different from each other.
%
Also the elemental composition may be laterally* inhomogeneous in a
planet. It probably has a large difference particularly amoung regions
with and without cloud.
%
Therefore, it is necessary to give the elemental composition as a
parameter for calculating the convective equilibrium state.



\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 of the convective equilibrium state in the
atmospheres of the Jovian planets, 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 reaction
(Atreya and Romani, 1985, Atreya {\it et al}, 1999).
%
Each term which contributes entropy change is formulated according to 
individual chemical formula.
%
All the chemical reactions which occured in an air parcel must have
beforehand been known.
%
If the elemental composition changes, concerning chemical reactions
are also changed so that basic part of the numerical code have to be
modified. 
%
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 about chemical
reaction formulae. 
%
In this paper, we develop a thermodynamic calculation method using Gibbs
free energy $G$. This method requires no details of chemical reactions
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 a 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 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, ${n_{i}}^{\phi}$ is the mole number of chemical species $i$ in
phase $\phi$, $R$ is gas constant, $\alpha$ is a constant ($\alpha
= 1$ with gas species, $\alpha = 0$ with condensate species), 
${\mu_{i}^{\circ}}^{\phi}$ is standard chemical potential. 
Here we assume ideal gas and ideal solution. 
The quantity of ${\mu_{i}}^{\phi}$ is found from thermochemical tables.  
The equilibrium
composition ${n_{i}}^{\phi}$ is derived from minimizing $G$ under the
condition of number of elements 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 could be described by conserving whole 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 by \Dfigref{pseudo}. 


\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. 
%
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 $p_1 = p_0 + dp$. Entropy is
calculated as temperature changes so that temperature $T_1$
at which entropy is equal to entropy $S_0$ is obtained.
%
Step3: If any species condensate at the condition ($T_1$, $p_1$),
entropy reserved at next step is entropy of the gas component 
${S^{'}_{4}}_{\rm gas}$.
%
Step4: Adiabatic curve is given by the set of temperature and pressure
derived from Step1 -- Step3.
%
}
\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.

We calculated the convective equilibrium state of the atmosphere of the
Jupiter with above-mentioned method.
%
Because of the verification of our numerical model, the elemental
composition and initial conditions are same as Atreya {\it et al}
(1999).


\subsection{verification 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 with our result and that of Atreya {\it et al}. (1999) are
shown in \Dfigref{kenshou-1}. In \Dfigref{kenshou-1}, The vertical
profile of condensed species, cloud density, and temperature are
plotted.
%
As compared with each other, cloud condensation level, quantity of
cloud density, and distribution of temperature are quite similer. 
%
Partial pressure and saturated pressure of 
H$_2$O and NH$_3$ are plotted in \Dfigref{kenshou-2}. 
%
After condensation, the partial pressure of H$_2$O and NH$_3$
together agrees with saturated pressure of these.
%
It is clear that the phase equilibrium has accurately been reproduced
in the 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 with our result and that of Atreya {\it et al}. (1999). 
 left: The vertical plofile of condensed species and temperature in our model.
 right: The vertical plofile of condensed species and temperature in
 Atreya {\it et al}. (1999). 
%
 }
\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 pressure and saturated pressure.  left:
Partial pressure and saturated pressure of H$_2$O.  right: Partial
pressure and saturated pressure of NH$_3$.  The partial pressure agree
with saturated pressure after condensation. 
%
 }
\Dfiglab{kenshou-2}
\end{figure}



\subsection{Calculation of 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 adiabatic lapse rate and static stability of atosphere are showen in
\Dfigref{keisan}.  The quantity of adiabatic lapse rate are changed by 
latent heat of phase change and heat of reaction, thereby the atmosphere
does the stable layer. The change of the adiabatic lapse rate with
condensation of the water is the most remarkable, and the static
stability with it is $2.5 \times 10^{-5}$ s$^{-2}$. The static stability
of the atmosphere of the earth is $1 \times 10^{-5}$ s$^{-2}$, so the static
stability of the atmosphere of the Jupiter 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 and the static stability of the atmosphere of
 the Jupiter.  left: The adiabatic lapse rate.  The average adiabatic
 lapse rate is $-2.0$ K/km.  Because of condensation, the adiabatic
 lapse rate is decreased.
%
right: The static stability of the atmosphere. the quantity of the
 static stability with H$_2$O, NH$_3$, NH$_4$SH is respectively $2.5
 \times 10^{-5}$ s$^{-2}$, $8.0 \times 10^{-6}$ s$^{-2}$, $4.0 \times
 10^{-6}$ s$^{-2}$.
%
 }
\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


An thermodynamic method for calculating the convective equilibrium
state of the atmosphere of the Jovian planets is newly developed.  
%
The advantage of our method is that we do not have to know the detail of
corresponding chemical reaction formulae.  Therefore, we can say that
our model is more convinent for the conventional study than previoues
studies.
\bigskip
%

The thermodynamic state of the atmosphere is calculated for the
atmosphere of the Jupiter. Here the elemental composition is equal to
Atreya {\it et al}. (1999).
%
The distribution of condensed species obtained from our calculation is
quite similer to Atreya {\it et al}. (1999).  In addition, it is
confirmed that the phase equilibrium is rightly expressed.
%
Therefore our thrmodynamical method is confirmed to work well.
\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. 

As a convective equilibrium state, adiabatic lapse rate and static
stability are also quantitatively calculated.
%
The result showed that the atmosphere does stable layer by 
latent heat of phase change and heat of reaction. 
%
The static stability with the condensation of H$_2$O is shown to be the
biggest than that of 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}