% 表題  2 次元非弾性流体モデル deepconv-mars 解説: 離散モデル編
%       第 5 節  放射モデルの離散化
%
% 履歴  2001/11/01  小高正嗣
%       2007/04/26  小高正嗣: 英語版作成
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\section{\DselJ{放射モデル}{Radiation}} \Dseclab{離散化放射モデル}

\DselJ{
放射加熱項 $Q_{rad,i,j}^{n}$ を正味放射フラックスから計算する際には 2 
次中央差分を用いる.}
{
The radiative heating rate is calculated from the radiative flux by
   using the second order centered scheme.
}
\begin{eqnarray}
  Q_{rad,i,j}^{n} &=& 
  Q_{rad,IR,i,j}^{n} + Q_{rad,NIR,i,j}^{n} + 
  Q_{rad,dust,SR,i,j}^{n} + Q_{rad,dust,IR,i,j}^{n}, \\
  &&\nonumber \\
  Q_{rad,*,i,j}
             &=& - \frac{g}{c_{p}}
                 \frac{F_{*,net,i,j+\frac{1}{2}} - 
                        F_{*,net,i,j-\frac{1}{2}}}
                 {\Delta P _{0,j}},
\end{eqnarray}
\[
  F_{*,net,i,j+\frac{1}{2}} =
                 F_{IR,i,j+\frac{1}{2}}^{\uparrow} - 
                 F_{IR,i,j+\frac{1}{2}}^{\downarrow}, \quad
  \Delta P_{0,j} = P_{0,j+\frac{1}{2}} - P_{0,j-\frac{1}{2}}.
\]

\DselJ{
以下, CO${}_{2}$ およびダストの放射フラックス計算の具体的な表現を示す.
放射フラックスは鉛直方向の半整数格子点上で評価する. 下付き添字 $m$ は波
数方向の差分を表す. 時間方向の差分を示す上付き添字は省略してある.}
{
The radiative flux is evaluated on the half grid point
The subscript $m$ shows grid point in the wave number.
In the following sections, the superscript which shows time step is omitted.
}

\subsection{\DselJ{CO${}_{2}$ の放射}{Radiative transfer of atmospheric CO${}_{2}$}}

\DselJ{
CO${}_{2}$ の放射フラックスは以下のように離散化して計算する. 
}
{
The finite difference form of the infrared radiative flux 
   and the Plank function are represented as follows.
}
\begin{eqnarray}
  F_{IR,i,j+\frac{1}{2}}^{\uparrow} &=& \sum _{m}\Delta \nu _{m}\left\{
                   \pi B_{\nu _{i},T_{sfc,i}}{\cal T}_{i}(0,z_{j+\frac{1}{2}})
  \right. \nonumber \\
  && \left.
  +  \sum _{j'=1}^{j} \pi B_{\nu _{m},T_{i,j'}}
      \frac{ {\cal T }_{m}(z_{j+\frac{1}{2}},z_{j'+\frac{1}{2}}) - 
             {\cal T }_{m}(z_{j+\frac{1}{2}},z_{j'-\frac{1}{2}}) }
           { \Delta z_{j'} } \right\}, \\
  F_{IR,i,j+\frac{1}{2}}^{\downarrow} &=& \sum _{m}\Delta \nu _{m}\left\{
      \sum _{j'=j}^{J}\pi B_{\nu _{m},T_{i,j'}}
      \frac{ {\cal T }_{m}(z_{j+\frac{1}{2}},z_{j'+\frac{3}{2}}) - 
             {\cal T }_{m}(z_{j+\frac{1}{2}},z_{j'+\frac{1}{2}}) }
           { \Delta z_{j'} }
      \right\} , \\
  B_{\nu _{m},T_{i,j}} 
               &=& \frac{1.19\times 10^{-8}\nu _{m}^{3}}
                 {e^{1.4387\nu_{i}/T_{i,j}}-1}, \\
  T_{i,j} &=& \Pi _{0,j}(\theta _{i,j}+\Theta _{0,j})
\end{eqnarray}

\DselJ{
透過関数, 透過幅, 光路長は以下のように離散化する.
}
{
where the averaged transmission function, the equivalent width, and
  the effective path length are represented as follows.
}

\[
  {\cal T}_{m}(z_{j+\frac{1}{2}},z_{j'+\frac{1}{2}}) = 
  \exp ( - W_{m,j,j'}/\Delta \nu _{m}),
  \quad
  W_{m,j,j'} = \frac{S_{m} u(z_{j+\frac{1}{2}},z_{j'+\frac{1}{2}})}
  {\sqrt{ 1 + S_{m}u^{*}(z_{j+\frac{1}{2}},z_{j'+\frac{1}{2}})/
   \alpha ^{*}_{m}}},
\]
\[
  u(z_{j+\frac{1}{2}},z_{j'+\frac{1}{2}}) = 1.67 
  g | P_{0,j+\frac{1}{2}} - P_{0,j'+\frac{1}{2}} |,
  \quad
  \alpha ^{*}_{m}=\alpha _{m} \frac{\overline{p}_{jj'}}{p_{0}}, 
\]
\[
  P_{0,j+\frac{1}{2}} = 0.5(P_{0,j}+P_{0,j+1}),
  \quad
  \overline{p}_{jj'} = 
  \sum _{l=j}^{j'-1} \frac{P_{0,l+1}\cdot 1.67
  g|P_{0,l+\frac{3}{2}}-P_{0,l+\frac{1}{2}}|}
  {u(z_{j+\frac{1}{2}},z_{j'+\frac{1}{2}})},
\]

\DselJ{
CO${}_{2}$ の近赤外放射フラックスは以下のように離散化して計算する.
}
{
The finite form of the near infrared radiative flux 
  and the effective path length are represented as follows.
}
\begin{equation}
  F_{NIR,i,j+\frac{1}{2}}^{\downarrow} = \sum _{m}\Delta \nu _{m}\left\{
      S_{\nu _{m}}{\cal T}_{i}(z_{j+\frac{1}{2}},z_{J+\frac{3}{2}})\cos \zeta 
      \right\}.
\end{equation}
\DselJ{
ここで光路長は以下のように計算される.
}
{}

\[
  u(z_{j+\frac{1}{2}},z_{J+\frac{3}{2}}) = 
  \frac{1.67 g | P_{0,j+\frac{1}{2}} - P_{0,J+\frac{3}{2}} |}
  {\mbox{MAX}(\cos \zeta, \epsilon )}.
\]

\DselJ{
$\epsilon $ は除算例外を防ぐための微小量である. 
}
{
where $\epsilon$ is a small parameter to ensure
  $u = 0$ when $\cos \zeta =0$.
}

\subsection{\DselJ{ダストの放射}{Radiative transfer of dust}}

\DselJ{
ダストの太陽放射フラックスは以下のように離散化して計算する.
}
{
The finite difference form of the solar radiative transfer equation of dust
  are represented as follows.
}
\begin{eqnarray}
  F_{dif,\nu_{m},i,j+\frac{1}{2}}^{\uparrow} &=& 
  F_{dif,\nu_{m},i,j-\frac{1}{2}}^{\uparrow} - 
  \Delta \tau _{\nu_{m},j}\left[
  \gamma _{1,\nu_{m}}\frac{F_{dif,\nu_{m},i,j+\frac{1}{2}}^{\uparrow} + 
                   F_{dif,\nu_{m},i,j-\frac{1}{2}}^{\uparrow}}{2} 
  \right. \nonumber \\
&& - \left.
  \gamma _{2,\nu_{m}}\frac{F_{dif,\nu_{m},i,j+\frac{1}{2}}^{\downarrow} + 
                   F_{dif,\nu_{m},i,j-\frac{1}{2}}^{\downarrow}}{2}
  -\gamma _{3,\nu_{m}}\tilde{\omega}_{\nu_{m}}^{*}
              S_{0}e^{-\tau _{\nu_{m},j+\frac{1}{2}}^{*}
   /\mu_{0}} \right], \Deqlab{離散化されたダストの上向き太陽放射} \\
  F_{dif,\nu_{m},i,j-\frac{1}{2}}^{\downarrow} &=& 
  F_{dif,\nu_{m},i,j+\frac{1}{2}}^{\downarrow} + 
  \Delta \tau _{\nu_{m},j}\left[
      \gamma _{2,\nu_{m}}\frac{F_{dif,\nu_{m},i,j+\frac{1}{2}}^{\uparrow} + 
                       F_{dif,\nu_{m},i,j-\frac{1}{2}}^{\uparrow}}{2} 
  \right. \nonumber \\
&& - \left.
   \gamma _{1,\nu_{m}}\frac{F_{dif,\nu_{m},i,j+\frac{1}{2}}^{\downarrow} + 
                    F_{dif,\nu_{m},i,j-\frac{1}{2}}^{\downarrow}}{2}
   +(1-\gamma _{3,\nu_{m}})\tilde{\omega}_{\nu_{m}}^{*} 
                   S_{0}e^{-\tau _{\nu_{m},j+\frac{1}{2}} ^{*}/\mu_{0}}
  \right], \Deqlab{離散化されたダストの下向き太陽放射}
\end{eqnarray}
\[
 \Delta \tau _{\nu_{m}, j}= \frac{\overline{Q}_{e,\nu_{m}}}{r_{eff}}
     \frac{3\rho _{0,j}q_{i,j}}{4\pi \rho _{s}} \Delta z_{j}, \quad
 \tau _{\nu_{m},j+\frac{1}{2}} = \sum _{j'=j+1}^{J+1}\Delta \tau _{\nu_{m},j'}.
\]

\DselJ{
ここで 
$\Dvect{F}_{dif,\nu_{m}}^{\uparrow}=(
F_{dif,\nu_{m},i,\frac{1}{2}}^{\uparrow},
F_{dif,\nu_{m},i,\frac{3}{2}}^{\uparrow},...,
F_{dif,\nu_{m},i,J+\frac{1}{2}}^{\uparrow})^{T}$ のように表すと\Deqref
{離散化されたダストの上向き太陽放射},\Deqref{離散化されたダストの下向き
太陽放射}は形式的に以下のような行列表現にすることができる.
}
{
\Deqref{離散化されたダストの上向き太陽放射} and 
\Deqref{離散化されたダストの下向き太陽放射}
   can be represented in matrix form as follows.
}
\begin{eqnarray}
 \Dvect{F}_{dif,\nu_{m}}^{\uparrow}&=&
 \Dvect{A}\Dvect{F}_{dif,\nu_{m}}^{\uparrow} + 
 \Dvect{B}\Dvect{F}_{dif,\nu_{m}}^{\downarrow} + \Dvect{R}, 
 \Deqlab{行列形式の上向き太陽放射} \\
 \Dvect{F}_{dif,\nu_{m}}^{\downarrow}&=&
 \Dvect{C}\Dvect{F}_{dif,\nu_{m}}^{\uparrow} + 
 \Dvect{D}\Dvect{F}_{dif,\nu_{m}}^{\downarrow} + \Dvect{S}. 
 \Deqlab{行列形式の下向き太陽放射} 
\end{eqnarray}
\DselJ{
これらの式を反復法で解く. 各係数行列の成分は以下のように表される(ただし 
$0\leq j \leq J$).
}
{
where 
  $\Dvect{F}_{dif,\nu_{m}}^{\uparrow}=(
  F_{dif,\nu_{m},i,\frac{1}{2}}^{\uparrow},
  F_{dif,\nu_{m},i,\frac{3}{2}}^{\uparrow},...,
  F_{dif,\nu_{m},i,J+\frac{1}{2}}^{\uparrow})^{T}$ 
  and so on.
\Deqref{行列形式の上向き太陽放射} and \Deqref{行列形式の下向き太陽放射} 
   are solved by iteration method.
The elements of the matrixes in 
  \Deqref{行列形式の上向き太陽放射} and \Deqref{行列形式の下向き太陽放射} 
  are represented as follows. 
}
\begin{eqnarray*}
 A_{jk} &=& \left\{
 \begin{array}{cl}
 - \frac{1}{2}\Delta \tau _{\nu_{m},j}\gamma_{1,\nu_{m}} & k=j, j\geq 1\\
 1 - \frac{1}{2}\Delta \tau _{\nu_{m},j}\gamma_{1,\nu_{m}} & k=j-1, j\geq 1\\
 0 & \mbox{others}
 \end{array}
 \right. \\
 B_{jk} &=& \left\{
 \begin{array}{cl}
 A & j=k=0 \\
 \frac{1}{2}\Delta \tau _{\nu_{m},j}\gamma_{2,\nu_{m}}& k=j, j-1 ,j\geq 1\\
 0 & \mbox{others}
 \end{array}
 \right. \\
 C_{jk} &=& \left\{
 \begin{array}{cl}
  \frac{1}{2}\Delta \tau _{\nu_{m},j+1}\gamma _{2,\nu_{m}} & k=j,j+1, j\leq J-1
 \\
 0 & \mbox{others}
 \end{array}
 \right. \\
%\]
%\[
 D_{jk} &=& \left\{
 \begin{array}{cl}
  - \frac{1}{2}\Delta \tau _{\nu_{m},j+1}\gamma_{1,\nu_{m}} & k=j,j\leq J-1 \\
 1 - \frac{1}{2}\Delta \tau _{\nu_{m},j+1}\gamma_{1,\nu_{m}} & k=j+1 \\ 
 0 & \mbox{others}
 \end{array}
 \right. \\
%\]
%\[
 R_{j} &=& \left\{
 \begin{array}{cl}
 A S_{0}e^{-\tau _{\nu_{m},\frac{1}{2}}^{*}/\mu_{0}} & j=0 \\
 \Delta \tau _{\nu_{m},j}
 \gamma _{3,\nu_{m}}\tilde{\omega}_{\nu_{m}}^{*}
            S_{0}e^{-\tau _{\nu_{m},j+\frac{1}{2}}^{*}/\mu_{0}}
 & j\geq 1 
 \end{array}
 \right. \\
%\]
%\[
 S_{j} &=& \left\{
 \begin{array}{cl}
 \Delta \tau _{\nu_{m},j+1}(1-\gamma _{3,\nu_{m}})
  \tilde{\omega}_{\nu_{m}}^{*}S_{0}e^{-\tau _{\nu_{m},j+\frac{3}{2}}^{*}
  /\mu_{0}} &
 j\leq J-1 \\
 0 & j=J
 \end{array}
 \right.
%\]
\end{eqnarray*}

\DselJ{
ダストの赤外放射フラックスの離散化はダストの太陽放射フラックスの場合と同
様に行う. 
}
{
The finite difference form of the solar radiative transfer equation of dust
  are represented as follows.
}
\begin{eqnarray}
  F_{IR,\nu_{m},i,j+\frac{1}{2}}^{\uparrow} &=& 
  F_{IR,\nu_{m},i,j-\frac{1}{2}}^{\uparrow} - 
  \Delta \tau _{\nu_{m},j}\left[
  \gamma _{1,\nu_{m}}\frac{F_{IR,\nu_{m},i,j+\frac{1}{2}}^{\uparrow} + 
                   F_{IR,\nu_{m},i,j-\frac{1}{2}}^{\uparrow}}{2} 
  \right. \nonumber \\
&& - \left.
  \gamma _{2,\nu_{m}}\frac{F_{IR,\nu_{m},i,j+\frac{1}{2}}^{\downarrow} + 
                   F_{IR,\nu_{m},i,j-\frac{1}{2}}^{\downarrow}}{2}
  -2\pi (1-\tilde{\omega}_{\nu_{m}}^{*})B_{\nu_{m},T_{i,j}}
\right], \Deqlab{離散化されたダストの上向き赤外放射} \\
  F_{IR,\nu_{m},i,j-\frac{1}{2}}^{\downarrow} &=& 
  F_{IR,\nu_{m},i,j+\frac{1}{2}}^{\downarrow} + 
  \Delta \tau _{\nu_{m},j}\left[
      \gamma _{2,\nu_{m}}\frac{F_{IR,\nu_{m},i,j+\frac{1}{2}}^{\uparrow} + 
                       F_{IR,\nu_{m},i,j-\frac{1}{2}}^{\uparrow}}{2} 
  \right. \nonumber \\
&& - \left.
   \gamma _{1,\nu_{m}}\frac{F_{IR,\nu_{m},i,j+\frac{1}{2}}^{\downarrow} + 
                    F_{IR,\nu_{m},i,j-\frac{1}{2}}^{\downarrow}}{2}
  +2\pi (1-\tilde{\omega}_{\nu_{m}}^{*})B_{\nu_{m},T_{i,j}}
  \right]. \Deqlab{離散化されたダストの下向き赤外放射}
\end{eqnarray}

\DselJ{
ここで
$\Dvect{F}_{IR,\nu_{m}}^{\uparrow}=(F_{IR,\nu_{m},i,\frac{1}{2}}^{\uparrow},
F_{IR,\nu_{m},i,\frac{3}{2}}^{\uparrow},...,
F_{IR,\nu_{m},i,J+\frac{1}{2}}^{\uparrow})^{T}$ 等と表すと, \Deqref{離
散化されたダストの上向き赤外放射}, \Deqref{離散化されたダストの下向き赤
外放射}も行列形式で表現することができる.
}
{
\Deqref{離散化されたダストの上向き赤外放射} and 
\Deqref{離散化されたダストの下向き赤外放射}
   can be represented in matrix form as follows.
}
\begin{eqnarray}
 \Dvect{F}_{IR,\nu_{m}}^{\uparrow}&=&
 \Dvect{A}\Dvect{F}_{IR,\nu_{m}}^{\uparrow} + 
 \Dvect{B}'\Dvect{F}_{IR,\nu_{m}}^{\downarrow} + \Dvect{R}', 
 \Deqlab{行列形式のダストの上向き赤外放射} \\
 \Dvect{F}_{IR,\nu_{m}}^{\downarrow}&=&
 \Dvect{C}\Dvect{F}_{IR,\nu_{m}}^{\uparrow} + 
 \Dvect{D}\Dvect{F}_{IR,\nu_{m}}^{\downarrow} + \Dvect{S}'. 
 \Deqlab{行列形式のダストの下向き赤外放射}
\end{eqnarray}
\DselJ{
ここで
}
{
where 
  $\Dvect{F}_{IR,\nu_{m}}^{\uparrow}=(F_{IR,\nu_{m},i,\frac{1}{2}}^{\uparrow},
  F_{IR,\nu_{m},i,\frac{3}{2}}^{\uparrow},...,
  F_{IR,\nu_{m},i,J+\frac{1}{2}}^{\uparrow})^{T}$ and so on.
\Deqref{行列形式のダストの上向き赤外放射} and 
  \Deqlab{行列形式のダストの下向き赤外放射}
  are solved by iteration method.
The elements of the matrixes in 
  \Deqref{行列形式のダストの上向き赤外放射} and 
  \Deqlab{行列形式のダストの下向き赤外放射}
  are represented as follows. 
}
\begin{eqnarray*}
 B'_{jk} &=& \left\{
 \begin{array}{cl}
 0      & j=k=1 \\
 B_{jk} & \mbox{others}
 \end{array}
 \right. \\
%\]
%\[
 R'_{j} &=& \left\{
 \begin{array}{cl}
 \pi B_{\nu_{m}, T_{sfc,i}} & j=0 \\
  2\pi \Delta \tau _{\nu_{m},j}( 1 - \tilde{\omega }^{*})
  B_{\nu_{m}, T_{i,j}} & j\geq 1 \\
 \end{array}
 \right.\\
%\]
%\[
 S'_{j} &=& \left\{
 \begin{array}{cl}
 2\pi \Delta \tau _{\nu_{m},j}( 1 - \tilde{\omega }^{*})
  B_{\nu_{m}, T_{i,j}} & j\leq J-1 \\
 0 & i=N
 \end{array}
 \right. 
\end{eqnarray*}
\DselJ{
である. 
}
{}
