% 表題  2 次元非弾性流体モデル deepconv-mars 解説: 離散モデル編
%       第 3 節  乱流モデルの離散化
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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{乱流モデル}{Turbulent parameterization}}

\subsection{\DselJ{乱流パラメタリゼーション}{Subgrid turbulent mixing parameterization}}

\DselJ{
移流項 $[\mbox{DKADV}]_{i,j}^{n}$ は熱力学の式の場合と同様に 4 次中央差
分, その他の項は 2 次中央差分で離散化する. 摩擦項
$[\mbox{DKDIF}]_{i,j}^{N}$, $[\mbox{DKNLD}]_{i,j}^{N}$ の時間積分に対し
ては常に前進差分を用いる.  $[\mbox{DKADV}]_{i,j}^{n}$,
$[\mbox{DKDIF}]_{i,j}^{N}$ 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}式と同様なので, ここでは省略する.
}
{
The space differencing in the turbulent kinetic energy equation
   (equation (10) in Part I) is evaluated by the forth
   order centered scheme for advection terms and the second order
   centered scheme for other terms. 
In time integration, the forward scheme is adapted for the
   friction terms.
Representations of $[\mbox{DKADV}]_{i,j}^{n}$ and 
   $[\mbox{DKDIF}]_{i,j}^{N}$ 
   are same as those of 
   \Deqref{スカラー移流の離散表現} and \Deqref{スカラー拡散の離散表現}.
}
\begin{eqnarray}
  \varepsilon _{i,j}^{n+1} &=&
  \varepsilon _{i,j}^{N} + dt \left\{
  [\mbox{DKADV}]_{i,j}^{n} + 
  [\mbox{DKDIF}]_{i,j}^{N} +
  [\mbox{DKNLD}]_{i,j}^{N} \right. \nonumber \\ 
 &&  + \left.
  [\mbox{DKBP}]_{i,j}^{n} +
  [\mbox{DKSP}]_{i,j}^{n} - 
  \frac{C_{\epsilon}}{l}(\varepsilon _{i,j}^{N})^{\frac{3}{2}} \right\}
\end{eqnarray}

\begin{eqnarray}
 \mbox{DKNLD}_{i,j}^{n}
 &=& \frac{1}{(\Delta x)^{2}}
 \left[
  K_{NLD,i+\frac{1}{2},j}^{n}(\varepsilon_{i+1,j}^{n}-\varepsilon_{i,j}^{n})
 -K_{NLD,i-\frac{1}{2},j}^{n}(\varepsilon_{i,j}^{n}  -\varepsilon_{i-1,j}^{n})
 \right] + \frac{1}{\rho _{0,j}\Delta z_{j}}\nonumber \\
 & &
 \left[ \rho _{0,j+\frac{1}{2}}
     K_{NLD,i,j+\frac{1}{2}}^{n}
     \frac{
   (\varepsilon_{i,j+1}^{n}-\varepsilon_{i,j}^{n})}{\Delta z_{j+\frac{1}{2}}}
    - \rho _{0,j-\frac{1}{2}}
     K_{NLD,i,j-\frac{1}{2}}^{n}
     \frac{
   (\varepsilon_{i,j}^{n}-\varepsilon_{*i,j-1}^{n})}{\Delta z_{j-\frac{1}{2}}}
 \right], \Deqlab{スカラーの非線形数値拡散の離散表現}
\end{eqnarray}
\begin{eqnarray*}
 K_{NLD,i+\frac{1}{2},j}^{n} 
 &=& 
 \mbox{MIN}
 \left[ K_{NLD,max}, \quad
        0.01\frac{(\Delta x)^{2}}{\Delta t}
        \frac{{\cal L}(\varepsilon)_{i+1,j}^{n}
              +{\cal L}(\varepsilon)_{i,j}^{n}}{2} \right], \\
 K_{NLD,i,j+\frac{1}{2}}^{n} 
 &=&
 \mbox{MIN}
 \left[ K_{NLD,max}, \quad
        0.01\frac{(\Delta x)^{2}}{\Delta t}
        \frac{{\cal L}(\varepsilon)_{i,j+1}^{n}
              +{\cal L}(\varepsilon)_{i,j}^{n}}{2} \right], \\
 {\cal L}(\varepsilon)_{i,j}^{n}
 &=& \left. \left(
 3\left|\varepsilon_{i+1,j}^{n} + 
        \varepsilon_{i-1,j}^{n} - 2\varepsilon_{i,j}^{n}\right|
 +\left|\varepsilon_{i,j+1}^{n} + 
        \varepsilon_{i,j-1}^{n} - 2\varepsilon_{i,j}^{n}\right|
 \right)\right /2000, \\
 K_{NLD,max} &=& 0.2\frac{(\Delta x)^{2}}{\Delta t}.
\end{eqnarray*}

\begin{eqnarray}
 \mbox{DKBP}_{i,j}^{n} &=& -\frac{g}{\Theta _{0,j}}K_{i,j}^{n}
 \frac{1}{\Delta z_{j}}\left[ 
  (\theta _{i,j+\frac{1}{2}}+\Theta _{0,j+\frac{1}{2}}) - 
  (\theta _{i,j-\frac{1}{2}}+\Theta _{0,j-\frac{1}{2}}) \right], \\
 \mbox{DKSP}_{i,j}^{n} &=& 2K_{i,j}^{n}\left[
 \left(
 \frac{u_{i+\frac{1}{2},j}^{n}-u_{i-\frac{1}{2},j}^{n}}{\Delta x}
 \right)^{2}
 + \left(
 \frac{w_{i,j+\frac{1}{2}}^{n}-w_{i,j-\frac{1}{2}}^{n}}{\Delta z_{j}}
 \right)^{2} 
 \right] \nonumber \\
 &&
 + \frac{2}{3}\varepsilon _{i,j}^{n} \left[
 \frac{u_{i,j+\frac{1}{2}}^{n}-u_{i,j-\frac{1}{2}}^{n}}{\Delta z_{j}} + 
 \frac{w_{i+\frac{1}{2},j}^{n}-w_{i-\frac{1}{2},j}^{n}}{\Delta x}\right]
 \nonumber \\
&& + K_{i,j}^{n}
\left[
 \frac{u_{i,j+\frac{1}{2}}^{n}-u_{i,j-\frac{1}{2}}^{n}}{\Delta z_{j}} + 
 \frac{w_{i+\frac{1}{2},j}^{n}-w_{i-\frac{1}{2},j}^{n}}{\Delta x}\right]^{2}
\end{eqnarray}

\[
 u_{i,j+\frac{1}{2}}^{n} =
 0.5\left(u_{i+\frac{1}{2},j+\frac{1}{2}}^{n} + 
          u_{i-\frac{1}{2},j+\frac{1}{2}}^{n}\right), 
 \quad 
 w_{i+\frac{1}{2},j}^{n} =
 0.5\left(w_{i+\frac{1}{2},j+\frac{1}{2}}^{n} + 
          w_{i+\frac{1}{2},j-\frac{1}{2}}^{n}\right).
\]

\subsection{\DselJ{地表フラックスパラメタリゼーション}{Surface flux parameterization}}

\DselJ{
地表からの運動量と熱の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 surface flux are as follows.
}
\begin{eqnarray}
 F_{u,i} &=& - \rho _{0}C_{D,i}|u_{i,\frac{1}{2}}|u_{i,\frac{1}{2}}, \\
 F_{\theta,i}&=& \rho _{0}C_{D,i}|u_{i,\frac{1}{2}}|(T_{sfc,i}-T_{i,1}).
\end{eqnarray}

\DselJ{
ここで,
}
{where
}

\begin{equation}
  C_{D,i} = \left\{
  \begin{array}{lcl}
  C_{Dn}\left( 1 -
  	\frac{a\mbox{Ri}_{B,i}}{1+c|\mbox{Ri}_{B,i}|^{1/2}} \right) & 
  	for & \mbox{Ri}_{B,i} < 0,  \\
  C_{Dn}\frac{1}{(1+b\mbox{Ri}_{B,i})^{2}}& for &  \mbox{Ri}_{B,i} \geq 0,
  \end{array}\right. 
\end{equation}

\DselJ{
である. バルクリチャードソン数は以下のように評価する.
}
{
The bulk Richardson number is calculated as follows.
}
\begin{equation}
  \mbox{Ri}_{B,i} \equiv \frac{gz_{1}(\Theta _{sfc,i}-\Theta _
  {i,1})}{\overline{\Theta }_{0,1}u_{i,\frac{1}{2}}}.
\end{equation}

