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%   2017/03/04  付録 1 --- 暴走温室状態
%               当番 : 松田 幸樹
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\Dtitle{Energy Balance Model}   % 変更不可
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\begin{document}

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\chapter{Nakajima et al. (1992) による暴走温室状態の解釈の一部}

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\section{暴走温室状態}

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\begin{wrapfigure}{r}{50mm}
  \centering
  \includegraphics[scale=0.5]{./Introduction/fig/Nakajima1992-1.jpg}
  \caption{対流圏界面温度$T_{tp}$と対流圏における厚さ$\tau_{tp}$の関係
 (Nakajima et al. 1992, Fig 1). 太線は\eqref{bousou3}, 細線は
 \eqref{bousou4}を表す. } \label{tau-T}
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放射伝達方程式
\begin{align}
 \frac{2}{3}\dd{F^{\uparrow}(\tau)}{\tau} &= F^{\uparrow}(\tau) - \pi B(T), \\
 -\frac{2}{3}\dd{F^{\downarrow}(\tau)}{\tau} &= F^{\downarrow}(\tau) - \pi B(T)
\end{align}
を考える. 上 2 式の和と差を取ると,
\begin{align}
 \frac{2}{3}\dd{}{\tau}(F^{\uparrow} + F^{\downarrow}) &= F^{\uparrow} -
 F^{\downarrow}, \label{wa} \\
 \frac{2}{3}\dd{}{\tau}(F^{\uparrow} - F^{\downarrow}) &= F^{\uparrow} +
 F^{\downarrow} - 2\pi B(T) \label{sa}
\end{align}
を得る. 平衡状態において大気には放射エネルギーの収束がないので,
\begin{align}
 \dd{F^{\mathrm{net}}}{\tau} &= \dd{}{\tau}(F^{\uparrow} -
 F^{\downarrow}) = 0,  \label{tamarinasi} \\ 
 F^{\mathrm{net}} &= \mathrm{const}
\end{align}
である. また, 大気上端における熱収支は,
\begin{equation}
 F^{\uparrow}(0) + 0 =  F^{\mathrm{net}} = F^{IR}
\end{equation}
となる. ここで, $F^{IR}$は太陽放射である. \eqref{wa}を光学的厚さ$\tau$に
ついて大気上端($\tau=0$)から$\tau$まで積分すると,
\begin{align}
 F^{\uparrow} + F^{\downarrow} -
 &F^{\uparrow}(0) = \frac{2}{3}F^{\mathrm{net}}\tau, \\
  F^{\uparrow} + F^{\downarrow} &=
 F^{IR}\left(\frac{3}{2}\tau + 1 \right)
\end{align}
を得る. また, \eqref{sa}と\eqref{tamarinasi}より,
\begin{align}
 &F^{\uparrow} + F^{\downarrow} - 2\pi B(T) = 0, \\
 &F^{\uparrow} + F^{\downarrow} = 2\pi B(T) = 2\sigma T^{4}
\end{align}
となる. 以上より,
\begin{equation}
 \sigma T^{4} = \frac{F^{IR}}{2}\left(\frac{3}{2}\tau + 1 \right) \label{bousou1}
\end{equation}
を得る.

次に, 光学的厚さ$\tau$を考える. ここでは, 簡単のため水蒸気のみが長波放射を吸
収すると考えると,
$\tau$は以下のように定義される:
\begin{equation}
 d\tau = \kappa_{v}m_{v}\chi_{v}\frac{dp}{\bar{m}g}. 
\end{equation}
ここで, $p$は全圧, $\kappa_{v}$は水蒸気の吸収係数, $p^{\ast}(T)$は水蒸
気の圧力, $m_{v}$は水蒸気の分子量, $\bar{m}$は大気の分子量,
$\chi_{v}\equiv p^{\ast}(T)/p$は
飽和状態の水蒸気のモル分率である. 大気上
端($\tau=0$)から$\tau$まで積分すると,
\begin{equation}
 \tau =  \kappa_{v}m_{v}\chi_{v} \frac{p}{\bar{m}g} \label{bousou2}
\end{equation}
を得る.

今, 対流圏界面を考えると, \eqref{bousou1}と\eqref{bousou2}は,
\begin{align}
 \sigma T^{4}_{tp} &= \frac{F^{IR}}{2}\left(\frac{3}{2}\tau_{tp} + 1
 \right), \label{bousou3} \\
 \tau_{tp} &=  \kappa_{v}m_{v}\chi_{v}\frac{p_{tp}}{\bar{m}g}
\end{align}
である. また, 対流圏界面ではモル分率は定数であり, $\chi_{v}\equiv
p^{\ast}(T_{tp})/p_{tp}$とかけるので,
\begin{equation}
 \tau_{tp} =  \kappa_{v}p^{\ast}(T_{tp})\frac{m_{v}}{\bar{m}g} \label{bousou4}
\end{equation}
となる. \eqref{bousou3}と\eqref{bousou4}の解を求めるために, 2 式を
$\tau_{tp}-T_{tp}$平面上に描画すると図\ref{tau-T}となる. 

図\ref{tau-T}より, 太陽放射$F^{IR}$がある値を超えると, \eqref{bousou3}と
\eqref{bousou4}は交わらない. このように, 惑星放射の値には上限がある(Komabayashi 1967; Ingersoll 1969;
Nakajima et al. 1992). この惑星放射の上限値を超える太陽放射が入射すると平衡
にならない. この状態を暴走温室状態という.

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