% 表題  2 次元非弾性流体モデル deepconv-mars 解説: 離散モデル編
%       第 4 節  ダスト移流モデルの離散化
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% 履歴  2001/11/01  小高正嗣
%       2007/04/26  小高正嗣: 英語版作成
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\section{\DselJ{ダストの輸送モデル}{Dust transport}} \Dseclab{離散化ダスト}

\DselJ{
移流項 $[\mbox{DQADV}]_{i,j}^{n}$ は熱力学の式と同様に 4 次中央差分, 重
力沈降項 $[\mbox{DQFALL}]_{i,j}^{N}$ は 1 次の上流差分で空間離散化する. 
摩擦項 $[\mbox{DQDIF}]_{i,j}^{N}, [\mbox{DQNLD}]_{i,j}^{N}$ と重力沈降項
の時間積分に対しては常に前進差分を用いる. $[\mbox{DQADV}]_{i,j}^{n}$,
$[\mbox{DQDIF}]_{i,j}^{N}$, $[\mbox{DQNLD}]_{i,j}^{N}$ の表現はそれぞれ
\Deqref{スカラー移流の離散表現}, \Deqref{スカラー拡散の離散表現},
\Deqref{スカラーの非線形数値拡散の離散表現}式と同様なので, ここでは省略
する.
}
{
In the advection diffusion equation of dust (
   equation (20) in Part I), the advection term 
   $[\mbox{DQADV}]_{i,j}^{n}$ is evaluated by the forth order centered
   scheme and the vertical advection term associated with the
   gravitational settling $[\mbox{DQFALL}]_{i,j}^{N}$ is evaluated by the first
   order upstream scheme.  
In time integration, the forward scheme is adapted for the friction
   terms 
   $[\mbox{DQDIF}]_{i,j}^{N}$, $[\mbox{DQNLD}]_{i,j}^{N}$ 
   and the gravitational settling term $[\mbox{DQFALL}]_{i,j}^{N}$.
Representation of 
  $[\mbox{DQADV}]_{i,j}^{n}$, $[\mbox{DQDIF}]_{i,j}^{N}$ and 
  $[\mbox{DQNLD}]_{i,j}^{N}$ are same as those of 
  \Deqref{スカラー移流の離散表現}, \Deqref{スカラー拡散の離散表現} and 
  \Deqref{スカラーの非線形数値拡散の離散表現}.
}
\begin{equation}
 q_{i,j}^{n+1} 
 = q_{i,j}^{N} + dt \left\{
  [\mbox{DQADV}]_{i,j}^{n} +
  [\mbox{DQDIF}]_{i,j}^{N} +
  [\mbox{DQFALL}]_{i,j}^{N}+
  [\mbox{DQNLD}]_{i,j}^{N} \right\}
\end{equation}
\begin{eqnarray}
 \mbox{DQFALL}_{i,j}^{n} &=& - \frac{1}{\rho _{0,j}\Delta z_{j}}
  \left\{ FQf_{z(i,j+\frac{1}{2})}^{n}-FQf_{z(i,j-\frac{1}{2})}^{n}
\right\}, \\
 FQf_{z(i,j-\frac{1}{2})}^{n} &=&
 - \frac{4\rho _{d}gr_{mod}^{2}}{18\eta }\left( 1 + 
         2\frac{\lambda _{r}}{r_{mod}}\frac{p_{r}}{P_{0,j}}\right)
   \rho _{0,j}q_{i,j}^{n},\nonumber 
\end{eqnarray}

