%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% 表題: 2001 年地球惑星関連学会 合同大会 ポスター資料 %%% %%% 履歴: 2001/05/03 杉山耕一朗 %%% 2001/05/08 杉山耕一朗 %%% 2001/06/03 杉山耕一朗 %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %%deffont "standard" xfont "comic sans ms-medium-r" %%deffont "thick" xfont "arial black-medium-r" %%deffont "typewriter" xfont "courier new-bold-r" %%deffont "type2writer" xfont "arial narrow-bold-r" %% %% Default settings per each line numbers. %% %%%default 1 leftfill, size 2, fore "darkgreen", back "lemonchiffon" %default 1 leftfill, size 2, fore "darkgreen", back "white" %default 2 size 7, vgap 10, prefix " " %%%default 3 size 2, image "brush-lemonchiffon.jpg", vgap 20 %default 3 size 2, bar "grey70", vgap 20 %default 4 size 5, fore "navyblue", vgap 30, prefix " " %% %% Default settings that are applied to TAB-indented lines. %% %tab 1 size 4, vgap 35, prefix " ", icon arc "tomato" 40 %tab 2 size 4, vgap 20, prefix " ", icon box "seagreen" 40 %tab 3 size 3, vgap 20, prefix " ", icon delta3 "pink" 40 %% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %page %nodefault %%%size 7, fore "darkblue", vgap 20, back "lemonchiffon" %size 7, fore "darkblue", vgap 20, back "white" %center, size 7 木星型惑星大気の熱力学計算 %size 4 杉山耕一朗 (北大・理) 小高正嗣 (東大・数理) 倉本圭 (北大・理) 林祥介 (北大・理) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %page はじめに 本研究の目標: %fore "red" 木星型惑星大気の熱力学状態を記述するための熱力学コードの開発 %mark, fore "navyblue" 熱力学コード作成にあたって 多様な木星型惑星全てに対して応用可能 様々な温度・圧力条件下で計算可能 複数の物質の相変化, 化学反応, 混合(混合気体, 溶液) 計算する物理量 断熱温度減率 凝縮物の鉛直分布 気体成分の組成の鉛直変化 %again %right, image "ps/radiative-conv.eps" 0 180 180 1 %cont %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %page 木星型惑星と一言で言っても... 木星 表層雲層温度・圧力: 120 K 〜, 1 bar 〜 主成分気体: 水素, ヘリウム 凝結物質: 固体のアンモニア, 硫化アンモニウム, 氷 NH3(g) + H2S(g) → NH4SH(s) 天王星 表層雲層温度・圧力: 60 K 〜 , 0.5 bar 〜 凝結物質: 固体のメタン, 固体のアンモニア, 硫化アンモニウム, アンモニア水溶液, 氷 H2O(g) + NH3(g) → NH3(aq) [アンモニア水溶液] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %page 従来の熱力学計算 エントロピーの保存式 %fore "orange" → 断熱線に沿った温度・化学組成を計算 %filter "platex2eps.sh gamma" \begin{eqnarray} dS = \frac{c_{p}}{T} dT - \frac{V}{T} dp + \frac{L_{s}}{T} dw_{s} + \frac{L_{r}}{T} dw_{r} + \frac{L_{aq}}{T} dw_{aq} = 0 \nonumber \end{eqnarray} %endfilter %cont, fore "red", image "gamma.eps" 0 250 250 1 %fore "navyblue" 凝結物質の混合比の変化は経験式から定式化 %filter "platex2eps.sh gamma2" \begin{eqnarray} dw_{s}: 飽和蒸気圧, \hspace{5mm} dw_{r}: 圧平衡定数, \hspace{5mm} dw_{aq}: 溶液濃度と分圧の経験式 \nonumber \end{eqnarray} %endfilter %cont, fore "red", image "gamma2.eps" 0 200 200 1 %fore "navyblue" ポテンシャル温度, 物質の種類の変更が困難 化学反応式を全て把握する必要がある 新しい物質の追加 %fore "red", cont → %fore "navyblue", cont 必要となる化学反応式の再考 %fore "red", cont → %fore "navyblue", cont アルゴリズムの変更 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %page 本研究の計算手法 ギブスの自由エネルギー G 最小化法 与えられた温度, 圧力に対して化学平衡組成を計算 相変化, 化学反応, 混合(溶液) %fore "red", cont → %fore "navyblue", cont 「化学ポテンシャル」に一括 断熱線に沿った計算をするために 平衡組成から, エントロピー S, 断熱曲線(dS = 0)を計算 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %page 計算手順 1: 平衡状態の求め方 考え方 条件付き極値問題 T, P 固定, 元素数保存の条件のもと G を最小化 %fore "red" → %fore "navyblue", cont 物質のモル数(平衡化学組成) 定式化 G(T, p, 組成) の式 %filter "platex2eps.sh gibbs" \begin{eqnarray} G(n^{\phi}_{i}) = \sum \mu_{i}^{\phi} n_{i}^{\phi} \nonumber \end{eqnarray} %endfilter %cont, fore "red", image "gibbs.eps" 0 250 250 1 %fore "navyblue" 化学ポテンシャル(物質の理想性を仮定) %filter "platex2eps.sh mu" \begin{eqnarray} \mu_{i}^{\phi}(T,p) = \underbrace{ {\mu_{i}^{\circ}}^{\phi}(T) }_{標準化学ポテンシャル} + \underbrace{ RT \ln \frac{n_{i}^{\phi}}{\sum n_{i}^{\phi}} + \alpha RT \ln{\frac{p}{p_0}} }_{混合によるエントロピー変化(気相; \alpha = 1, それ以外; \alpha = 0)} \nonumber \end{eqnarray} %endfilter %cont, fore "red", image "mu.eps" 0 180 180 1 %fore "navyblue" 標準化学ポテンシャル %filter "platex2eps.sh mu2" \begin{eqnarray} {\mu_{i}^{\circ}}^{\phi}(T) = \underbrace{h(T_{0}) - T s(T_0, p_0)}_{標準状態での \mu} + \int^{T}_{T_0} c_p dT - T \int^{T}_{T_0} \frac{c_p}{T}dT \nonumber \end{eqnarray} %endfilter %cont, fore "red", image "mu2.eps" 0 180 180 1 %cont, size 3, fore "navyblue" (標準状態でのμ, Cp ← 物性値) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %page 計算手順 2: エントロピーの求め方 考え方 温度・圧力空間での平衡組成から S を計算 定式化 Maxwell の関係式 %filter "platex2eps.sh entropy" \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} %endfilter %cont, fore "red", image "entropy.eps" 0 200 200 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %page 計算手順 3: 断熱曲線の求め方 考え方 大気の擬湿潤断熱(pseudo moist adiabat)変化 凝結性成分気体が凝結 %fore "red", cont → %fore "navyblue", cont 空気塊から離脱(降水) 計算手順 %center, image "ps/pseudo3.eps" 0 180 180 1 %left %size 5 %cont,size 2 初期温度・圧力での平衡組成を計算し, %cont, size 5 %cont,size 2 圧力を dp だけ変化させる. %cont, size 5 %cont,size 2 (T1,P1)において凝縮物質が生じた場合, %cont, size 5 %cont,size 2 Step 1 〜 3 で得られた温度・圧力を %size 5 %cont,size 2 エントロピー S0 を求める. %cont, size 5 %cont,size 2 温度を変化させた時のエントロピーを %cont, size 5 %cont,size 2 (T1,p1) の気相のみのエントロピー %cont, size 5 %cont,size 2 結んで断熱線を引く. %size 5 %cont,size 2 順次計算し, エントロピーが前のステップ %cont, size 5 %cont,size 2 (S'4)gas を保存させる. %size 5 %cont,size 2 でのエントロピー S0 と一致する温度を %size 5 %cont,size 2 求める. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %page モデル計算 計算に必要な物理量 基準面での温度・圧力 (Atreya et al 1999 から推測) 初期温度: 345 K 初期圧力: 12 bar 元素存在度と物質の種類(下表) 太陽系元素存在度 (Anders and Grevesse, 1989) 物質の標準化学ポテンシャル %filter "platex2eps.sh genso" \begin{tabular}{|r|r|}\hline 元素 & 存在比 \\ \hline \hline He/H & 0.095 \\ O/H & 8.51 $\times 10^{-4}$ \\ C/H & 3.62 $\times 10^{-4}$ \\ N/H & 1.12 $\times 10^{-4}$ \\ S/H & 1.62 $\times 10^{-5}$ \\ \hline \end{tabular} %endfilter %filter "platex2eps.sh sosei" \begin{tabular}{|l|}\hline 物質名 \\ \hline \hline H$_2$(g), He(g), H$_2$O(g), CH$_4$(g), NH$_3$(g), H$_{2}$S(g) \\ H$_2$O(l), CH$_4$(l), NH$_3$(l), H$_{2}$S(l) \\ H$_2$O(s), NH$_3$(s), NH$_{4}$SH(s) \\ \hline \end{tabular} %endfilter %size 5, fore "black" %cont, size 3 表1: 太陽系元素存在度 表2: 計算に利用した物質の種類 %center %image "genso.eps" 0 200 200 1 %cont %cont, image "sosei.eps" 0 200 200 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %page 計算結果の検証 (その 1) 鉛直雲密度・温度分布 過去の大気研究と整合的 %fore "black" %mark %cont, image "ps/cloud-t.eps" 0 160 160 1 %size 3 本研究での雲密度・温度分布 %again, right %image "ps/AW1999.ps" 0 80 80 1 %cont %size 2 %size 3 過去の研究での雲密度・温度分布 (Atreya et al 1999) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %page 計算結果の検証 (その 2) 鉛直エントロピーの分布 擬断熱過程が正しく表現されている %fore "black" %center, image "ps/entropy.eps" 0 100 100 1 %size 3 エントロピーの鉛直分布 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %page 計算結果の検証 (その 3) 蒸気圧と飽和蒸気圧の関係 持ち上げ凝結高度より上で, 飽和蒸気圧と蒸気圧が一致 %fore "black" %center %image "ps/vaper-H2O-edit.eps" 0 120 120 1 %cont %cont, image "ps/vaper-NH4SH-edit.eps" 0 120 120 1 %cont, image "ps/vaper-NH3-edit.eps" 0 120 120 1 %left %size 3 左: H2O の蒸気圧と飽和蒸気圧の関係 真中: log(P_NH3 ・ P_H2S) と NH4SH 生成反応の圧平衡定数の関係 %filter "platex2eps.sh Kp" $K_p = \ln(p_{NH3} \cdot p_{H2S}) = 61.781 - \frac{10834}{T}$ %endfilter 相平衡状態にあるならば, %cont, image "Kp.eps" 0 150 150 1 %cont を満たす. 右: NH3 の蒸気圧と飽和蒸気圧の関係 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %page 計算結果の検証 (その 4) 凝縮成分気体のモル比 持ち上げ凝結高度より上で減少. %fore "black" %center %image "ps/ratio-H2O.eps" 0 65 65 1 %cont %cont, image "ps/ratio-H2S.eps" 0 65 65 1 %cont, image "ps/ratio-NH3.eps" 0 65 65 1 %size 3 H2O のモル比 H2S のモル比, NH3 のモル比 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%page %% %%計算結果の検証 (その 4) %% %% 鉛直温度分布 %% 過去の大気研究に同じ %% %%%fore "black" %%%center, image "ps/therm.eps" 0 100 100 1 %% %%%size 3 %%鉛直温度分布 %% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %page まとめ 木星型惑星大気の熱力学状態を計算するための熱力学コードを開発した 本研究の計算手法の利点 反応式を把握する必要がない. 大気中で生成される物質を予想できさえすれば良い 汎用性に富む. どのような惑星大気にも簡単に応用可能 物質の化学ポテンシャルさえわかれば OK 大気中の物質の種類を変えても数値コード自体を変更する必要がない 但し理想気体の仮定が成り立たないとダメ モデルの検証 従来の研究でのモデル計算結果と整合的 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %page 今後の課題 他の木星型惑星大気への応用 現段階では 100 K 以下の化学ポテンシャルの表を持ってない 物性値を集める/作る必要有り 溶液モデル 非理想溶液への拡張 現在は理想溶液を仮定 アンモニア水溶液は非理想的 対流モデルとのカップリング 木星型惑星大気の対流計算に利用 水以外の凝結物質を考慮した木星型惑星大気対流モデルの開発