\section{離散化}

\subsection{鉛直方向の離散化}

鉛直方向の離散化にローレンツ格子を用いる. すなわち鉛直座標, 熱フラック
スを整数格子点におき, 密度, 温位, 温度, 拡散係数(乱流エネルギー)は半整
数格子点に置く. 整数格子点間隔を $\Delta z_{i+\frac{1}{2}} \equiv
z_{i} - z_{i-1}$, 半整数格子点間隔を $\Delta z_{i} \equiv
z_{i+\frac{1}{2}} - z_{i-\frac{1}{2}}$ と定義する.
\begin{eqnarray}
  \DP{\theta _{i+\frac{1}{2}}}{t} 
  &=& \frac{1}{\rho _{i+\frac{1}{2}}c_{p}}
      \frac{\theta _{i+\frac{1}{2}}}{T_{i+\frac{1}{2}}}Q_{i+\frac{1}{2}} 
    + \frac{1}{\rho _{i+\frac{1}{2}}}
      \frac{F_{turb,i+1} - F_{turb,i}}{\Delta z_{i+\frac{1}{2}}}, \\
  F_{turb,i} 
  &=& \rho _{i}K_{h,i}\frac{\theta _{i+\frac{1}{2}}-\theta _{i-\frac{1}{2}}}
    {\Delta z_{i}},\\
  K_{h,i}
  &=& \frac{K_{h,i+\frac{1}{2}}+K_{h,i-\frac{1}{2}}}{2}, \\
  && \nonumber \\
  \DP{\varepsilon _{i+\frac{1}{2}}}{t} 
  &=& -\frac{g}{\theta _{i+\frac{1}{2}}}K_{h,i+\frac{1}{2}}
       \frac{\theta _{i+\frac{3}{2}}-\theta _{i-\frac{1}{2}}}
            {2\Delta z_{i+\frac{1}{2}}}
      + \frac{F_{\varepsilon ,i+1} - F_{\varepsilon ,i}}
             {\Delta z_{i+\frac{1}{2}}}
      - \frac{0.2}{l}\varepsilon ^{3/2}_{i+\frac{1}{2}}, \\
  F_{\varepsilon ,i} 
  &=& K_{h,i}\frac{\varepsilon _{i+\frac{1}{2}}
                   -\varepsilon _{i-\frac{1}{2}}}{\Delta z_{i}}, \\
  K_{h,i+\frac{1}{2}}
  &=& 0.2 \sqrt{\varepsilon _{1+\frac{1}{2}}}\mbox{MAX}(\Delta z, z), \\
  && \nonumber \\
  Q_{i+\frac{1}{2}} 
  &=& -\frac{1}{\Delta z}\left[ 
      F_{net,IR,i+1}-F_{net,IR,i}\right]  
      -\frac{1}{\Delta z}\left[ 
      F_{net,S,i+1}-F_{net,S,i}\right],\\
  F_{net,IR,i} 
  &=& F^{\uparrow}_{IR,i} - F^{\downarrow}_{IR,i}, \\
  F^{\uparrow}_{IR,i}
  &=& 2\pi \left\{\frac{\sigma T_{g}^{4}}{2\pi }
      - 0.488 \sum _{n=1}^{i}E(p_{i},p_{n-1})
        [B(p_{n})-B(p_{n-1})]\right\}, \\
  F^{\downarrow}_{IR,i}
  &=& 0.488 \cdot 2\pi \cdot
  \left\{B(p_{N})E(p_{N},p_{i}) - 
         \sum_{n=i+1}^{N}E(p_{n},p_{i})[B(p_{n})-B(p_{n-1})] \right\},  \\
  B(p_{n})
  &=& \sigma T_{n}^{4}, \\
  E(p_{i},p_{j}) 
  &=& 54.4\left(\overline{T}_{ij}/T_{0}\right)^{0.879} \cdot
      \ln \left[ 1 + 0.153\left(\overline{T}/T_{0}\right)^{-0.256}
         u(p_{i},p_{j})^{0.566}\overline{p}_{ij}^{0.323}\right], \\ 
  u(p_{i}, p_{j})
  &=& [p_{j}-p_{i}]/g, \\
  \overline{T}_{ij}
  &=& \left.\sum _{n=i+\frac{1}{2}}^{j-\frac{1}{2}}T_{n}\Delta z_{n}
      \right/|z_{j}-z_{i}|, \\
  \overline{p}_{ij}
  &=& \left.\sum _{n=i+\frac{1}{2}}^{j-\frac{1}{2}}p_{n}\Delta z_{n}
      \right/|z_{j}-z_{i}|, \\
  && \nonumber \\
  H
  &=& \rho _{\frac{1}{2}}c_{p}C_{d}|U|(T_{g} - \theta _{\frac{1}{2}})
\end{eqnarray} 

\subsection{時間方向の離散化}

時間方向の離散化は前進差分で行なう. ここで時刻 $n\Delta t$ における物
理量 $A$ の値を $A^{n}$ と表す. 放射項については陽解法で, 拡散項につい
ては陰解法(Crank-Nicolson 法)で解く\footnote{Crank-Nicolson 法について
は別紙解説を参照のこと}.
\begin{eqnarray}
  \theta _{i+\frac{1}{2}}^{n+1}
  &=& \theta _{i+\frac{1}{2}}^{n}+\Delta t \cdot
      \left\{
      \frac{1}{\rho _{i+\frac{1}{2}}^{n}c_{p}}
      \frac{\theta _{i+\frac{1}{2}}^{n}}
           {T_{i+\frac{1}{2}}^{n}Q_{i+\frac{1}{2}}^{n}} \right. \nonumber \\ 
  && \left.+\frac{1}{2}\left[
      \frac{1}{\rho _{i+\frac{1}{2}}^{n+1}}
      \frac{F_{turb,i+1}^{n+1} - F_{turb,i}^{n+1}}
           {\Delta z_{i+\frac{1}{2}}}\right]
     +\frac{1}{2}\left[
      \frac{1}{\rho _{i+\frac{1}{2}}^{n}}
      \frac{F_{turb,i+1}^{n} - F_{turb,i}^{n}}
           {\Delta z_{i+\frac{1}{2}}}\right]
      \right\}. 
%  \varepsilon _{i+\frac{1}{2}}^{n+1}
%  &=& \varepsilon _{i+\frac{1}{2}}^{n}+\Delta t \cdot 
%      \left\{
%      -\frac{g}{\theta _{i+\frac{1}{2}}^{n}}K_{h,i+\frac{1}{2}}^{n}
%       \frac{\theta _{i+\frac{3}{2}}^{n}-\theta _{i-\frac{1}{2}}^{n}}
%            {2\Delta z_{i+\frac{1}{2}}}
%      + \frac{F_{\varepsilon ,i+1}^{n} - F_{\varepsilon ,i}^{n}}
%             {\Delta z_{i+\frac{1}{2}}}
%      - \frac{0.2}{l}(\varepsilon ^{n}_{i+\frac{1}{2}})^{3/2}
%      \right\}, 
\end{eqnarray}     

