%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% 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
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\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}
 [K_{m}]_{i,k}^{t + \Delta t} = [K_{m}]_{i,k}^{t - \Delta t}  
  + 2 \Delta t [F_{K_m}]_{i,k}^{t}
\end{eqnarray}
%
ここで, 
%
\begin{eqnarray}
 [F_{K_m}]_{i,k}^{t} &=& 
  - [{\rm Adv}.K_m]_{i,k}^{t} 
  + [{\rm Buoy}.K_m]_{i,k}^{t - \Delta t}  
  + [{\rm Shear}.K_m]_{i,k}^{t - \Delta t}  
\nonumber \\
&&
  + [{\rm Turb}.K_m]_{i,k}^{t - \Delta t}  
  + [{\rm Disp}.K_m]_{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. 
%

$F_{K_m}$ 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}.K_m]_{i,k}^{t} 
  &=& 
    \left\{
        u_{i(u),k} \left( \DP{K_{m}}{x} \right)_{i(u), k} 
     \right\}_{i,k}^{t} 
    + 
    \left\{
        w_{i,k(w)} \left( \DP{K_{m}}{z} \right)_{i, k(w)} 
     \right\}_{i,k}^{t}
\\ 
\left[{\rm Buoy}.K_m\right]_{i,k}^{t - \Delta t}   &=& 
   - \left\{
      \frac{3 g C_{m}^{2} l^{2}}{ 2 \overline{\theta}} 
      \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}   &=& 
   \left( C_{m}^{2} l^{2} \right)_{i,k}
    \left[
    \left\{ 
       \left( \DP{u}{x} \right)^{2} 
    \right\}_{i(u),k}
    \right]_{i,k}^{t - \Delta t}
  + \left( C_{m}^{2} l^{2} \right)_{i,k}
    \left[
    \left\{ 
       \left( \DP{w}{z} \right)^{2}
    \right\}_{i,k(w)}
    \right]_{i,k}^{t - \Delta t}
\nonumber \\
 &&
  + \left(  \frac{ C_{m}^{2} l^{2} }{2} \right)_{i,k}
    \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 \\
&&
  - \left( \frac{K_{m}}{3} \right)_{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}.K_m\right]_{i,k}^{t - \Delta t}   &=& 
   \Dinv{2}
   \left[
    \left\{
     \DP{}{x}
     \left( 
      \DP{K_{m}^{2}}{x}
     \right)_{i(u),k}
    \right\}_{i,k}^{t-\Delta t}
   +
   \left\{
    \DP{}{z}
       \left(
          \DP{K_{m}^{2}}{z}
       \right)_{i,k(w)}
   \right\}_{i,k}^{t-\Delta t}
   \right]
\nonumber \\
   && 
   + 
   \left[
   \left\{
    \left( \DP{K_{m}}{x}\right)^{2}
   \right\}_{i(u),k}
   \right]_{i,k}^{t - \Delta t}
   + 
   \left[
   \left\{
     \left(\DP{K_{m}}{z}\right)^{2}
   \right\}_{i,k(w)}
   \right]_{i,k}^{t - \Delta t}
\\
 \left[{\rm Disp}.K_m\right]_{i,k}^{t - \Delta t}   &=& 
   - \Dinv{2 l^{2}} \left( K_{m}^{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}^{'}   &=& 
    \bar{\theta_{v}} 
    \left\{
     1 + \frac{\theta}{\bar{\theta}}
     + \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}
%
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}
