%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% 表題: 2 次元非静力学モデル -- 離散モデル  乱流エネルギーの時間方向の離散化
%
% 履歴: 2005-02-04 杉山耕一朗
%       2005/04/13 小高正嗣
%       2006/03/13 杉山耕一朗
%       2009/10/09 山下達也
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{乱流運動エネルギーの式}

Klemp and Wilhelmson (1978) および CReSS (坪木と榊原篤志, 2001) と同様
に, 1.5 次のクロージャーを用いる. 乱流エネルギーの時間発展方程式
をリープフロッグ法を用いて時間方向に離散化すると, 以下のようになる. 
%
\begin{eqnarray}
 [E_{turb}]_{i,k}^{t + \Delta t} = [E_{turb}]_{i,k}^{t - \Delta t}  
  + 2 \Delta t [F_{E_{turb}}]_{i,k}^{t}
\end{eqnarray}
%
ここで, 
%
\begin{eqnarray}
 [F_{E_{turb}}]_{i,k}^{t} &=& 
  - [{\rm Adv}.E_{turb}]_{i,k}^{t} 
  + [{\rm Buoy}.E_{turb}]_{i,k}^{t - \Delta t}  
  + [{\rm Shear}.E_{turb}]_{i,k}^{t - \Delta t}  
\nonumber \\
&&
  + [{\rm Turb}.E_{turb}]_{i,k}^{t - \Delta t}  
  + [{\rm Disp}.E_{turb}]_{i,k}^{t - \Delta t}  
\end{eqnarray}
%
である. CReSS にならい, 移流項を $t$ で, 
移流項以外を $t - \Delta t$ で評価した. 
%

$F_{E_{turb}}$ に含まれる各項は以下のように書き下すことができる. 
%
\begin{eqnarray}
[{\rm Adv}.E_{turb}]_{i,k}^{t} 
  &=& 
    \left\{
        u_{i(u),k} \left( \DP{E_{turb}}{x} \right)_{i(u), k} 
     \right\}_{i,k}^{t} 
    + 
    \left\{
        w_{i,k(w)} \left( \DP{E_{turb}}{z} \right)_{i, k(w)} 
     \right\}_{i,k}^{t}
\\ 
\left[{\rm Buoy}.E_{turb}\right]_{i,k}^{t - \Delta t}   &=& 
   - \left\{
      \frac{3 g C_{m} l}{ 2 \overline{\theta_{v}}} 
       E_{turb_{i,k}}
      \left(\DP{\theta_{el}}{z} \right)_{i,k(w)}
     \right\}_{i,k}^{t-\Delta t}
\\
 \left[{\rm Shear}.K_m\right]_{i,k}^{t - \Delta t}   
  &=& 
   2 C_m l [E_{turb}^{1/2}]_{i,k}^{t - \Delta t} 
    \left\{ 
      \left[
       \left( \DP{u}{x} \right)^{2} 
      \right]_{i,k}^{t - \Delta t}
      +
      \left[
       \left( \DP{w}{z} \right)^{2}
      \right]_{i,k}^{t - \Delta t}
    \right\}
\nonumber \\
 &&
  + C_m l [E_{turb}^{1/2}]_{i,k}^{t - \Delta t} \nonumber \\
 &&
        \left[
     \left\{ 
       \left( \DP{u}{z} \right)_{i(u),k(w)}
     \right\}_{i,k}^{t-\Delta t}     
    +
     \left\{ 
       \left( \DP{w}{x} \right)_{i(u),k(w)} 
     \right\}_{i,k}^{t - \Delta t}
    \right]^{2}
\nonumber \\
&&
  - \frac{2}{3} [E_{turb}]_{i,k}^{t - \Delta t}
    \left\{
     \left( \DP{u}{x} \right)_{i,k}^{t-\Delta t}
  + 
     \left( \DP{w}{z} \right)_{i,k}^{t-\Delta t}
    \right\}
\\
 \left[{\rm Turb}.E_{turb}\right]_{i,k}^{t - \Delta t}   
  &=& 
   C_{m} l 
    \left\{ \DP{}{x} \left[ [E_{turb}^{1/2}]_{i,k} \left( \DP{E_{turb}}{x}
     \right)_{i(u),k} \right] \right\}_{i,k}^{t - \Delta t} \nonumber \\
  & &
   + C_{m} l 
    \left\{ \DP{}{z} \left[ [E_{turb}^{1/2}]_{i,k} \left( \DP{E_{turb}}{z}
     \right)_{i,k(w)} \right] \right\}_{i,k}^{t - \Delta t} 
\\
 \left[{\rm Disp}.E_{turb}\right]_{i,k}^{t - \Delta t}   &=& 
   - \frac{C_{\varepsilon}}{l} 
 \left( E_{turb}^{3/2} \right)_{i,k}^{t - \Delta t}
\Deqlab{TurbE_risan}
\end{eqnarray}
%
ここで $C_{\varepsilon} = C_{m} = 0.2$, 
混合距離 $l = \left(\Delta x \Delta z \right)^{1/2}$ とする.
また $\theta_{el}$ は以下で与えられる. 
%
\begin{eqnarray}
 \theta_{el} &=& \overline{ \theta_{v}} + \theta_{v}^{'} \;\;\; (for \;\; q_{c} = 0)\\
%
 \theta_{el} 
  &=& \overline{\theta_{v}} + \theta_{v}^{'} + \frac{ \sum L
  q_{v}}{{c_p}_{d} \bar{\pi}}  
   \;\;\; (for \;\; q_{c} > 0)
\end{eqnarray}
%
ただし, 
%
\begin{eqnarray}
 \overline{\theta_{v}} + \theta_{v}^{'}   &=& 
  \left( 
   \frac{\langle M \rangle}{M_d} \overline{q_d} 
   + \sum \frac{\langle M \rangle}{M_v} \overline{q_v} 
  \right) ( \overline{\theta} + \theta )
   + 
   \left( 
   \frac{\langle M \rangle}{M_d} q_d 
   + \sum \frac{\langle M \rangle}{M_v} q_v
  \right) \overline{\theta}
\end{eqnarray}
%
である.
%
%
%\begin{eqnarray}
%&& \theta_{el} = \theta + \frac{L q_{v}}{{c_p}_{d} \bar{\pi}}
%  + \bar{\theta} \left\{
%    \frac{\sum q_{v}/M_{v}}{1/M_{d} + \sum  \bar{q_{v}}/M_{v}}
%   - \frac{\sum q_{v} + \sum q_{c} + \sum q_{r}}
%          {1 + \sum \bar{q_{v}}}
%\right\}
%\end{eqnarray}
