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%ɽ��   2 ���������ϳإ�ǥ� -- ��Ͽ B ��ή�ѥ�᥿�ꥼ�������
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%����   2003-07-25 �ⶶ ������   ��������
%       2003-09-03 �ⶶ ������   ���Ҵ���
%       2003-09-08 �ⶶ ������   ����ʸ���ɲ�
%       2003-09-09 �ⶶ ������   Appendix �ɲ�
%       2003-11-17 �ⶶ ������   ����
%       2003-11-22 �ⶶ ������   ��ư���ͥ륮���μ�����
%       2004-07-25 ��������      2 ���������絤�Ǥغƹ���
%       2007-06-05 ��������      ��ή��ư���ͥ륮����������Ƴ�Ф��ɲ�
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\chapter{��ή�ѥ�᥿�ꥼ�������}

%----------------------------------------------------------------------
%\newpage

%\subsection{�쥤�Υ륺����������}


%----------------------------------------------------------------------
%\newpage

\section{��ή�ѥ�᥿�ꥼ�������}

Klemp and Wilhelmson (1978) ����� CReSS ���Ѥ����Ƥ��� 1.5 ���Υ�����
���㡼���Ѥ���. ���ΤȤ���ή��ư���ͥ륮���λ���ȯŸ��������, 
%
\begin{eqnarray}
 \DD{E}{t} &=& B + S + D_{E}  
  - \left(\frac{C_{\varepsilon}}{l}\right)
    E^{\frac{3}{2}}
\Deqlab{B:dEdt}	  
\end{eqnarray} 
%
��Ϳ������. $l$ �Ϻ����Υ��, $l = \left(\Delta x \Delta
z\right)^{1/2}$ �Ȥ���.  $B$ �� $S$ �Ϥ��줾�����Ϥ�ή����ѷ�®�٤ˤ�
����ή���ͥ륮��������, $D_{E}$ ����ή���ͥ륮���Ȼ���, �� 4 �����ή
���ͥ륮���ξû���Ǥ���,
%
\begin{eqnarray}
 B &=& 
  \frac{g_{j}}{\overline{\theta}} 
  \overline{u^{\prime}_{j} \theta^{\prime}} ,
 \\
 S &=& 
  - \overline{(u_{i}^{\prime} u_{j}^{\prime})}
  \DP{u_{i}}{x_{j}} ,
 \\
 D_{E} &=& \DP{}{x_{j}} \left(K_{m} \DP{E}{x_{j}} \right)
\end{eqnarray}
%
�Ǥ���. 1.5 ���Υ��������㡼�Ǥ�, �쥤�Υ륺���Ϥ�ʲ��Τ褦���������.
%
\begin{eqnarray}
  \overline{(u_{i}^{\prime} u_{j}^{\prime})}
  &=& - K_{m} \left(\DP{u_{i}}{x_{j}}
	  	        + \DP{u_{j}}{x_{i}}\right)
	  + \frac{2}{3} \delta_{ij} E,
  \Deqlab{�쥤�Υ륺����1}
\\
  \overline{u_{j}^{\prime} \theta } &=& - K_{h}\DP{\theta}{x_{j}}.
  \Deqlab{�쥤�Υ륺����2}
\end{eqnarray}
%
������ $K_{m}$ �ϱ�ư�̤��Ф��뱲Ǵ�������Ǥ���, $E$ �ϥ��֥���åɥ�
���������ή��ư���ͥ륮��, $K_{h}$ �ϱ��Ȼ������Ǥ���.
$K_{m}$, $K_{h}$ �� $E$ ���Ѥ��ưʲ��Τ褦��Ϳ������.
%
 \begin{eqnarray}
  K_{m} &=& C_{m} E^{\frac{1}{2}} l,
   \Deqlab{B:E}\\
  K_{h} &=& 3 K_{m}.
 \end{eqnarray}
%
�ѥ�᡼�� $C_{\varepsilon}, C_{m}$ �ϤȤ�� 0.2 �Ǥ���. 
a
\subsection{��ή��ư���ͥ륮����������Ƴ��}

Klemp and Wilhelmson (1978) �Ǥ�\Deqref{B:dEdt}�ˤĤ���, ��Deardroff
  (1975), Mellor and Yamada (1974), Schemm and Lipps (1976) ���Ѥ����
�Ƥ��뼰������Τ�ΤǤ���פȤ������Ҥ���, ����Ƴ�Фξܺ٤ˤĤ��Ƥϲ�
�⤵��Ƥ��ʤ�. ����椨�絤��۴ĥ�ǥ�Ǥ褯�Ѥ����Ƥ��� Mellor
and Yamada (1974, 1982) �Υѥ�᥿�ꥼ�������Ȥ��б��������ƤǤ�
��. �����ǰʲ��Ǥ� Mellor and Yamada (1973, 1974) ���꼰���μ��˱��
�Ƽ�\Deqref{B:dEdt}, \Deqref{�쥤�Υ륺����1}, \Deqref{�쥤�Υ륺����
2} ��Ƴ�Ф�Ԥ�.

�ͤ��Ƥ��륵�֥���åɥ���������ˤ�����, ̩�٤ϰ���, ưǴ��������Ȼ�
�����ʤɤ�ʪ������ϰ���Ȥ���. ��ȯ���Ȥʤ���������, Mellor and
Yamada (1973) �μ� (7) ����� (8)
\begin{eqnarray}
  \DP{\overline{u^{\prime}_{i}u^{\prime}_{j}}}{t}
      &+& \DP{}{x_{k}}\left[
          u_{k}\overline{u^{\prime}_{i}u^{\prime}_{j}}
       + \overline{u^{\prime}_{k}u^{\prime}_{i}u^{\prime}_{j}}
       -  \nu \DP{}{x_{k}}\overline{u^{\prime}_{i}u^{\prime}_{j}}\right] 
       \nonumber \\ 
      &+& \DP{}{x_{j}}\overline{pu^{\prime}_{i}} 
        + \DP{}{x_{i}}\overline{pu^{\prime}_{j}}
        + f_{k}(\varepsilon _{jkl}\overline{u_{l}^{\prime}u^{\prime}_{i}}
        + \varepsilon _{ikl}\overline{u_{l}^{\prime}u^{\prime}_{j}}) 
	\nonumber \\
      &=& -\overline{u^{\prime}_{k}u^{\prime}_{i}}\DP{u_{j}}{x_{k}}
          -\overline{u^{\prime}_{k}u^{\prime}_{j}}\DP{u_{i}}{x_{k}}
          -\beta(g_{j}\overline{u^{\prime}_{i}\theta }
          + g_{i}\overline{u^{\prime}_{j}\theta }) \nonumber \\
      &+& \overline{p\left(\DP{u^{\prime}_{i}}{x_{j}} 
       + \DP{u^{\prime}_{j}}{x_{i}}\right)}
       -  2\nu \overline{\DP{u^{\prime}_{i}}{x_{k}}\DP{u^{\prime}_{j}}{x_{k}}},
           \Deqlab{MY1974:eq(7)}
    \end{eqnarray}
    \begin{eqnarray}
      \DP{\overline{u^{\prime}_{j}\theta^{\prime}}}{t}
      &+& 
      \DP{}{x_{k}}\left[
        u_{k}\overline{\theta^{\prime} u^{\prime}_{j}} 
      + \overline{u^{\prime}_{k}u^{\prime}_{j}\theta^{\prime} }
      - \alpha \overline{u^{\prime}_{j}\DP{\theta^{\prime} }{x_{k}}}
      - \nu \overline{\theta^{\prime} \DP{u^{\prime}_{j}}{x_{k}}}\right] 
      + \DP{}{x_{j}}\overline{p\theta^{\prime} } 
      + \varepsilon _{jkl}f_{k}\overline{u^{\prime}_{l}\theta^{\prime} } 
      \nonumber \\
      &=& -\overline{u^{\prime}_{j}u^{\prime}_{k}}\DP{\theta }{x_{k}} 
          -\overline{\theta^{\prime} u^{\prime}_{k}}\DP{u_{j}}{x_{k}} 
          -\beta g_{j}\overline{(\theta^{\prime})^{2}} 
          + \overline{p\DP{\theta^{\prime} }{x_{k}}}
          -(\alpha + \nu )
            \overline{\DP{u^{\prime}_{j}}{x_{k}}\DP{\theta^{\prime} }{x_{k}}}.
         \Deqlab{MY1974:eq(8)}
\end{eqnarray}
�����, \Deqref{MY1974:eq(7)}�ˤ����� $i=j$ �Ȥ�����
\begin{eqnarray}
  \DP{q^{2}}{t} + u_{k}\DP{q^{2}}{x_{k}} 
    + \DP{}{x_{k}}\left(\overline{u^{\prime}_{k}u^{\prime}_{j}u^{\prime}_{j}}
    - \nu \DP{q^{2}}{x_{k}}\right)
    &=&
    - \overline{u^{\prime}_{j}u^{\prime}_{k}}\DP{u_{j}}{x_{k}} 
    - \DP{}{x_{k}}\overline{pu^{\prime}_{j}} \nonumber \\ 
   && + g_{j}\beta\overline{u^{\prime}_{j}\theta^{\prime}}
    - 2\nu \overline{\left(\DP{u^{\prime}_{j}}{x_{k}}\right)}
    \Deqlab{q��ͽ��}
\end{eqnarray}
������ 
\begin{eqnarray*}
  q &\equiv&  \sqrt{\overline{(u^{\prime}_{i})^{2}}}  \\
    & = &    \sqrt{2E} 
\end{eqnarray*}
��, $\nu, \alpha, \beta$ �Ϥ��줾��ưǴ������, �Ȼ����������Ǯ��
ĥΨ, $g_{j}$ �Ͻ��ϲ�®�٥٥��ȥ���� $j$ ��ʬ�Ǥ���.

\Deqref{MY1974:eq(7)}�����\Deqref{MY1974:eq(8)}�˸���밵�Ϥ˴ؤ�����ع�
����� 3 ��������̤ˤĤ��Ƥϰʲ��β���򤪤�. 

\begin{enumerate}
  \item $\overline{p\left(\DP{u^{\prime}_{i}}{x_{j}} + \DP{u^{\prime}_{i}}{x_{j}}\right)}$
        (���Ϥˤ�뱿ư���ͥ륮���κ�ʬ��)

        \[
          = - \frac{q}{3l_{1}}(\overline{u^{\prime}_{i}u^{\prime}_{j}} - 
                              \frac{\delta _{ij}}{3}q^{2})
            + Cq^{2}\left(\DP{u_{i}}{x_{j}} 
            + \DP{u_{j}}{x_{i}}\right),
        \]
        �Ȥ���. ������ $l_{1}$ ����ή����ħŪ�ʥ�������,
        $C$ ��̵����������Ǥ���.

  \item $\overline{p\DP{\theta^{\prime} }{x_{k}}}$ (���Ϥˤ��Ǯ���ͥ륮����ʬ��)

        1. ��Ƴ�Ф�Ʊ�ͤιͻ��ˤ�ä�,
        \[
          = -\frac{q}{3l_{2}}\overline{u^{\prime}_{i}\theta^{\prime} }
        \]
        �Ȥ���. �����Ǥ����Υ�������� $l_{2}$ �Ȥ���.

  \item $2\nu \overline{\DP{u^{\prime}_{i}}{x_{k}}\DP{u^{\prime}_{j}}{x_{k}}}$
        (Ǵ���ˤ�뻶��)

        Ǵ���˴�Ϳ����褦�ʾ���������θ��ݤ�����Ū�Ȥߤ� $q$ �Τ�
        ��ɽ������.
        \[
          = \frac{2}{3}\frac{q^{3}}{\Lambda _{1}}\delta _{ij}.
        \]
        ������ $\Lambda _{1}$ ��Ǵ���εڤ���ħŪ��������Ǥ���.

   \item $(\alpha + \nu )
          \overline{\DP{u^{\prime}_{j}}{x_{k}}\DP{\theta^{\prime} }{x_{k}}}$

        \[
          = 0
        \]
	�Ȥ���. 

  \item $\overline{u^{\prime}_{k}u^{\prime}_{i}u^{\prime}_{j}}, 
        \overline{u^{\prime}_{k}u^{\prime}_{j}\theta^{\prime} }, 
        \overline{u^{\prime}_{k}(\theta^{\prime})^{2}}$ 

        ®����ư�ˤ�� $\overline{u^{\prime}_{k}u^{\prime}_{i}u^{\prime}_{j}}, 
        \overline{u^{\prime}_{k}u^{\prime}_{j}\theta^{\prime} }, 
        \overline{u^{\prime}_{k}(\theta^{\prime})^{2}}$ �ȹͤ����Τ褦�ˤ���.
        \begin{eqnarray*}
          \overline{u^{\prime}_{k}u^{\prime}_{i}u^{\prime}_{j}} 
          &=& -q\lambda _{1}\left(
            \DP{\overline{u^{\prime}_{i}u^{\prime}_{j}}}{x_{k}} + 
            \DP{\overline{u^{\prime}_{i}u^{\prime}_{k}}}{x_{j}} + 
            \DP{\overline{u^{\prime}_{j}u^{\prime}_{j}}}{x_{i}}\right), \\
          \overline{u^{\prime}_{k}u^{\prime}_{j}\theta^{\prime} } 
          &=& -q\lambda _{2}\left( 
            \DP{\overline{u^{\prime}_{k}\theta^{\prime} }}{x_{j}} + 
            \DP{\overline{u^{\prime}_{j}\theta^{\prime} }}{x_{k}}\right), \\
          \overline{u^{\prime}_{k}(\theta^{\prime})^{2}} &=& -q\lambda _{3}
                   \DP{\overline{(\theta^{\prime})^{2}}}{x_{k}}. 
        \end{eqnarray*}
        ������ $\lambda _{i}(i=1,2,3)$ �Ϥ��줾�����ħŪ��������Ǥ���.

   \item $\overline{pu^{\prime}_{i}}, \overline{p\theta^{\prime} }$
        (������ư�ˤ��Ȼ�)

        \[
            \overline{pu^{\prime}_{i}} =  \overline{p\theta^{\prime} } = 0
        \]
        �Ȥ���. ���ζ���� 
        Deardroff (1975), Schemm and Lipps (1976) �Ǥ�Ԥ��Ƥ���.

   \item $f_{k}(\varepsilon _{jkl}\overline{u_{l}^{\prime}u^{\prime}_{i}}
         + \varepsilon _{ikl}\overline{u_{l}^{\prime}u^{\prime}_{j}}), \;
         f_{k}\varepsilon _{jkl}\overline{u_{l}^{\prime}\theta^{\prime} }$ 
	 (���ꥪ���)

        \[
           f_{k}\varepsilon _{jkl}\overline{u_{l}^{\prime}u^{\prime}_{i}} =
           f_{k}\varepsilon _{ikl}\overline{u_{l}^{\prime}u^{\prime}_{j}} = 0,
        \]
        \[
           f_{k}\varepsilon _{jkl}\overline{u_{l}^{\prime}\theta^{\prime}} = 0
        \]
	�Ȥ���. ���ζ���� 
        Deardroff (1975), Schemm and Lipps (1976) �Ǥ�Ԥ��Ƥ���.

   \item $\alpha \overline{u^{\prime}_{j}\DP{\theta^{\prime} }{x_{k}}}, \;
          \nu \overline{\theta^{\prime} \DP{u^{\prime}_{j}}{x_{k}}}$

        \begin{equation}
          = 0
	\end{equation}
	�Ȥ���. 

\end{enumerate}
�ʾ�ζ����\Deqref{MY1974:eq(7)}, \Deqref{MY1974:eq(8)}, \Deqref{q��ͽ��}
���Ф��ƹԤ���, �ʲ��μ�������.
\begin{eqnarray}
    \DD{\overline{u^{\prime}_{i}u^{\prime}_{j}}}{t}
          &-& \DP{}{x_{k}}\left[
              q\lambda _{1}\left(
                \DP{\overline{u^{\prime}_{i}u^{\prime}_{j}}}{x_{k}} + 
                \DP{\overline{u^{\prime}_{i}u^{\prime}_{k}}}{x_{j}} + 
                \DP{\overline{u^{\prime}_{j}u^{\prime}_{i}}}{x_{i}}\right)
		-\nu\DP{}{x_{k}}\overline{u^{\prime}_{i}u^{\prime}_{j}}\right] 
              \nonumber \\
          &=& - \overline{u^{\prime}_{k}u^{\prime}_{i}}\DP{u_{j}}{x_{k}}
              - \overline{u^{\prime}_{k}u^{\prime}_{j}}\DP{u_{i}}{x_{k}}
	      - \beta(g_{j}\overline{u^{\prime}_{i}\theta^{\prime} }
              + g_{i}\overline{u^{\prime}_{j}\theta^{\prime} })
              \nonumber \\
          & & - \frac{q}{3l_{1}}(\overline{u^{\prime}_{i}u^{\prime}_{j}}
              - \frac{\delta _{ij}}{3}q^{2})
              + Cq^{2}\left(\DP{u_{i}}{x_{j}} 
              + \DP{u_{j}}{x_{i}}\right)
              - \frac{2}{3}\frac{q^{3}}{\Lambda _{1}}\delta _{ij},
	      \Deqlab{MY1974:Level4(1)}
              \\
    \DD{\overline{u^{\prime}_{j}\theta^{\prime} }}{t}
          &-& \DP{}{x_{k}}\left[q\lambda _{2}\left(
              \DP{\overline{u^{\prime}_{k}\theta^{\prime} }}{x_{j}}
              + \DP{\overline{u^{\prime}_{j}\theta^{\prime} }}{x_{k}}\right)\right]
              \nonumber \\
          &=& - \overline{u^{\prime}_{j}u^{\prime}_{k}}\DP{\theta }{x_{k}} 
              - \overline{\theta^{\prime} u^{\prime}_{k}}\DP{u^{\prime}_{j}}{x_{k}} 
              - \beta g_{j}\overline{(\theta^{\prime})^{2}} 
              - \frac{q}{3l_{2}}\overline{u^{\prime}_{j}\theta^{\prime} }, 
	      \Deqlab{MY1974:Level4(2)}
             \\
  \DD{q^{2}}{t} 
    &+& \DP{}{x_{k}}\left[
         q\lambda _{1}\left(
               2\DP{q^{2}}{x_{k}} + 
                \DP{\overline{u^{\prime}_{j}u^{\prime}_{k}}}{x_{j}}\right)
        - \nu \DP{q^{2}}{x_{k}}\right]
   \nonumber \\
   &=&
    - 2\overline{u^{\prime}_{j}u^{\prime}_{k}}\DP{u_{j}}{x_{k}} 
    + 2g_{j}\beta\overline{u^{\prime}_{j}\theta^{\prime}}
    - 2\frac{q^{3}}{\Lambda _{1}}
	      \Deqlab{MY1974:Level4(3)}
\end{eqnarray}
������
\[
  \DD{}{t} \equiv \DP{}{t} + u_{k}\DP{}{x_{k}}
\]
�Ǥ���. ������ Mellor and Yamada (1974) �� Level 4 ��ǥ�μ����б�
���뼰�Ǥ���.

��\Deqref{MY1974:Level4(1)}, \Deqref{MY1974:Level4(2)},
\Deqref{MY1974:Level4(3)}���Ф�, ����˰ʲ��ζ����ä���.
\begin{itemize}
  \item ��\Deqref{MY1974:Level4(1)}��, ���դ��� 4 ����� 5 �������θ����.
        �����\Deqref{MY1974:Level1(1)}�Ǥ� $C=1/3$ �Ȥ���.

  \item ��\Deqref{MY1974:Level4(2)}��, ���դ��� 1 ����� 4 �������θ����.
        ����� $\overline{u^{\prime}_{j}u^{\prime}_{k}}\sim q^{2}\delta_{jk}/3$ �Ȥ���.

  \item ��\Deqref{MY1974:Level4(3)}��, ���դ� 3 ����ֹ��̵�뤹��.
\end{itemize}
�����ζ����Ԥ���, ��\Deqref{MY1974:Level4(1)},
\Deqref{MY1974:Level4(2)}, \Deqref{MY1974:Level4(3)}��
\begin{eqnarray}
  \overline{u^{\prime}_{i}u^{\prime}_{j}} 
    &=& \frac{\delta _{ij}}{3}q^{2}
       - ql_{1}\left(\DP{u_{i}}{x_{j}} + \DP{u_{j}}{x_{i}}\right)
    \Deqlab{MY1974:Level1(1)}
    \\
   \overline{u^{\prime}_{j}\theta^{\prime}}
    &=& - ql_{2}\DP{\theta }{x_{j}} 
    \Deqlab{MY1974:Level1(2)}
             \\
  \DD{q^{2}}{t} 
   &=&
    - 2\overline{u^{\prime}_{j}u^{\prime}_{k}}\DP{u_{j}}{x_{k}} 
    + 2g_{j}\beta\overline{u^{\prime}_{j}\theta^{\prime}}
    + \DP{}{x_{k}}\left[\nu \DP{q^{2}}{x_{k}}\right]
    - 2\frac{q^{3}}{\Lambda _{1}}
    \Deqlab{MY1974:Level3(1)}
\end{eqnarray}
�Ȥʤ�. \Deqref{MY1974:Level1(1)}�� Mellor and Yamada (1974) �� Level
1 ��ǥ�� $\overline{u^{\prime}_{i}u^{\prime}_{j}}$ �μ��Ǥ���.
\Deqref{MY1974:Level1(2)}�� Mellor and Yamada (1974) �� Level 1 ��ǥ�
�� $\overline{u^{\prime}_{j}\theta^{\prime}}$ �μ���
$\overline{(\theta^{\prime})^{2}}$ �ι��̵�뤷����Τ��б�����.
\Deqref{MY1974:Level3(1)} �� Mellor and Yamada (1974) �� Level 3 ���
��� $q^{2}$ �μ��ˤ�����, 3 ����ع��̵�뤷Ǵ���Ȼ����Ĥ�����Τ�
�б�����.

$ql_{1}=K_{m}, ql_{2}=K_{h}$ �Ȥ�, 
$q$ �� $E$ ��ɽ��ưǴ����������ή�Ȼ��������֤��������
\begin{eqnarray}
  \overline{u^{\prime}_{i}u^{\prime}_{j}} 
    &=& \frac{2}{3}\delta _{ij}E
       - K_{m}\left(\DP{u_{i}}{x_{j}} + \DP{u_{j}}{x_{i}}\right)
    \\
   \overline{u^{\prime}_{j}\theta^{\prime}}
    &=& - K_{h}\DP{\theta }{x_{k}} 
             \\
  \DD{E}{t} 
   &=&
    - \overline{u^{\prime}_{j}u^{\prime}_{k}}\DP{u_{j}}{x_{k}} 
    + g_{j}\beta\overline{u^{\prime}_{j}\theta^{\prime}}
    + \DP{}{x_{k}}\left[K_{m} \DP{q^{2}}{x_{k}}\right]
    - \frac{2^{3/2}}{\Lambda _{1}}E^{3/2}
  \Deqlab{��ή���ͥ륮���μ�}
\end{eqnarray}
�Ȥʤ�. ���۵��Τξ�� $\beta = 1/\theta$ �Ǥ��뤳�Ȥ����դ����, ��
\Deqref{��ή���ͥ륮���μ�}�ϻ����η��������\Deqref{B:dEdt}�˰���
����.

�ʾ���, Klemp and Wilhelmson (1978) ����ή�ѥ�᥿�ꥼ��������,
Mellor and Yamada (1974) �� Level 3 ��ǥ�� Level 1 ��ǥ�Ȥ��Ȥߤ�
�碌����Τ����򤹤뤳�Ȥ��Ǥ���. Klemp and Wilhelmson (1978) ��Ʊ�ͤ�
��ή��ư���ͥ륮���Τ�ͽ��¾������̤Ͽ���Ū�˵����ǥ�Ȥ��� 
Mellor and Yamada (1974) �� Level 2.5 ��ǥ뤬����. ������ Level 2.5 
��ǥ�� Level 3 ��ǥ�� Level 2 ��ǥ�Ȥ��ȹ礻�Ǥ��뤳�Ȥ����դ�ɬ
�פǤ���.

\subsection{2 �����ξ���ɽ��}

2 �����ξ���\Deqref{B:dEdt}���γƹ��񤭲���. ���Ϥˤ����ή���ͥ�
�����������,
%
\begin{eqnarray}
 B &=& 
  \frac{g_{j}}{\overline{\theta}} 
  \overline{u^{\prime}_{j} \theta^{\prime}} 
  \nonumber \\
  &=& 
  - \frac{g}{\overline{\theta}} 
  \overline{w^{\prime}  \theta^{\prime}} 
  \nonumber \\
  &=& 
  - \frac{g}{\overline{\theta}} 
  \left( K_{h} \DP{\theta}{z} \right)
\Deqlab{B}
\end{eqnarray}
%
�Ǥ���. ����ή����ѷ�®�٤ˤ����ή���ͥ륮�������� $S$ ��, 
%
\begin{eqnarray}
 S 
  &=& 
  - \overline{(u_{i}^{\prime} u_{j}^{\prime})}
  \DP{u_{i}}{x_{j}} 
  \nonumber \\
  &=& 
  - \left\{
     - K_{m} \left(\DP{u_{i}}{x_{j}} + \DP{u_{j}}{x_{i}}\right)
	  + \frac{2}{3} \delta_{ij} E
    \right\}
  \DP{u_{i}}{x_{j}}
  \nonumber \\
  &=& 
   \left\{ 
      K_{m} \left(\DP{u_{i}}{x_{j}} + \DP{u_{j}}{x_{i}}\right)
	  - \frac{2}{3} \delta_{ij} E
   \right\} 
  \DP{u_{i}}{x_{j}} 
  \nonumber \\
  &=& 
   \left\{ 
    K_{m} \left(\DP{u}{x_{j}} + \DP{u_{j}}{x}\right)
    - \frac{2}{3} \delta_{1j} E
   \right\} 
  \DP{u}{x_{j}} 
  \nonumber \\
  &&+  
   \left\{ 
      K_{m} \left(\DP{w}{x_{j}} + \DP{u_{j}}{z}\right)
	  - \frac{2}{3} \delta_{3j} E
   \right\} 
  \DP{w}{x_{j}} 
  \nonumber \\
  &=& 
   \left\{ 
      2 K_{m} \left(\DP{u}{x} \right)
	  - \frac{2}{3} E
   \right\} 
  \DP{u}{x}
  +  
   K_{m} \left( \DP{w}{x} + \DP{u}{z} \right)
  \DP{u}{z}
  \nonumber \\
  &&+  
   K_{m} \left(\DP{w}{x} + \DP{u}{z}\right)
  \DP{w}{x}
  +  
   \left\{ 
     2 K_{m} \left(\DP{w}{z}  \right)
	  - \frac{2}{3} E
   \right\} 
  \DP{w}{z} 
  \nonumber \\
 &=&  
   2 K_{m} \left\{ 
       \left( \DP{u}{x} \right)^{2}
     + \left( \DP{w}{z} \right)^{2}
    \right\}
  +  K_{m}
     \left(\DP{u}{z} + \DP{w}{x}\right)^{2}
     \nonumber \\
 && - \frac{2}{3} E \left( \DP{u}{x} + \DP{w}{z} \right) 
\Deqlab{S}
\end{eqnarray}
%
�Ǥ���. ��ή���ͥ륮���Ȼ��� $D_{E}$ ��, 
%
\begin{eqnarray}
 D_{E} &=& \DP{}{x_{j}} \left(K_{m} \DP{E}{x_{j}} \right), 
  \nonumber \\
  &=& \DP{}{x} \left(K_{m} \DP{E}{x} \right)  
      + \DP{}{x} \left(K_{m} \DP{E}{x} \right)  
\Deqlab{De}
\end{eqnarray}
%
�Ǥ���. �ʾ�� \Deqref{B}, \Deqref{S}, \Deqref{De} ���� \Deqref{B:dEdt} ��
���������뤳�Ȥǰʲ��μ�������. 
%
 \begin{eqnarray}
  \DD{E}{t}
  &=& 
   - \frac{g}{\overline{\theta}} 
      \left( K_{h} \DP{\theta}{z} \right)
\nonumber \\
  &&+ 2 K_{m} \left\{ 
       \left( \DP{u}{x} \right)^{2}
     + \left( \DP{w}{z} \right)^{2}
    \right\}
  +  K_{m}
     \left(\DP{u}{z} + \DP{w}{x}\right)^{2}
  - \frac{2}{3} E \left( \DP{u}{x} + \DP{w}{z} \right) 
\nonumber \\
  &&+ \DP{}{x} \left(K_{m} \DP{E}{x} \right)  
      + \DP{}{x} \left(K_{m} \DP{E}{x} \right)  
   - \left(\frac{C_{\varepsilon}}{l}\right)
	  E^{\frac{3}{2}}.
\Deqlab{B:TurbE}	  
\end{eqnarray}
%

\subsection{��ή�Ȼ��������Ѥ���ɽ��}

\Deqref{B:TurbE} ���� \Deqref{B:E} �����Ѥ��� $K_{m}$ �˴ؤ��뼰���ѷ�
����. ���դ���ή���ͥ륮���Ȼ����񤭲�����,
%
 \begin{eqnarray}
\DP{}{x} \left(K_{m} \DP{E}{x}\right)
       &+& \DP{}{z} \left(K_{m} \DP{E}{z}\right)
\nonumber \\
   &=& \frac{1}{C_{m}^{2} l^{2}}
       \Biggl\{\DP{}{x}
               \left(K_{m} \DP{K_{m}^{2}}{x}\right)
             + \DP{}{z}
	        \left(K_{m} \DP{K_{m}^{2}}{z}\right)
       \Biggr\}
  \nonumber \\
   &=& \frac{1}{C_{m}^{2} l^{2}}
       \Biggl\{K_{m} \DP[2]{K_{m}^{2}}{x}
               + \DP{K_{m}}{x}
                 \DP{K_{m}^{2}}{x}
               + K_{m} \DP[2]{K_{m}^{2}}{z}
               + \DP{K_{m}}{z}
                 \DP{K_{m}^{2}}{z}
       \Biggr\}
  \nonumber \\
   &=& \frac{K_{m}}{C_{m}^{2} l^{2}}
       \left(\DP[2]{K_{m}^{2}}{x}
	       + \DP[2]{K_{m}^{2}}{z}
       \right)
   + \frac{2 K_{m}}{C_{m}^{2} l^{2}}
         \Biggl\{\left(\DP{K_{m}}{x}\right)^{2}
	 + \left(\DP{K_{m}}{z}\right)^{2}
	 \Biggr\}
\nonumber \\
 \end{eqnarray}
%
�Ȥʤ�Τ�, \Deqref{B:TurbE} �����ѷ������, 
%
 \begin{eqnarray}
\frac{2 K_{m}}{C_{m}^{2} l^{2}}  \DD{K_{m}}{t}
  &=& 
   - \frac{g}{\overline{\theta}} 
      \left( K_{h} \DP{\theta}{z} \right)
  + 2 K_{m} \left\{ 
       \left( \DP{u}{x} \right)^{2}
     + \left( \DP{w}{z} \right)^{2}
    \right\}
\nonumber \\
  &&
  +  K_{m}
     \left(\DP{u}{z} + \DP{w}{z}\right)^{2}
  - \frac{2}{3} \frac{K_{m}^{2}}{C_{m}^{2} l^{2}} \left( \DP{u}{x} + \DP{w}{z} \right) 
\nonumber \\
   && 
   + \frac{K_{m}}{C_{m}^{2} l^{2}}
       \left(\DP[2]{K_{m}^{2}}{x}
	       + \DP[2]{K_{m}^{2}}{z}
       \right)
   + \frac{2 K_{m}}{C_{m}^{2} l^{2}}
         \Biggl\{\left(\DP{K_{m}}{x}\right)^{2}
	 + \left(\DP{K_{m}}{z}\right)^{2}
	 \Biggr\}
\nonumber \\
  && 
   - \frac{C_{\varepsilon}}{C_{m}^{3} {l}^{4}} 
	  K_{m}^{3}.
\end{eqnarray}
���������������,
\begin{eqnarray}
 \DP{K_{m}}{t}
  &=& 
   - \left( 
      u \DP{K_{m}}{x} + w \DP{K_{m}}{z}
     \right)
   - \frac{g C_{m}^{2} l^{2}}{ 2 \overline{\theta}} \frac{K_{h}}{K_{m}}
      \left(\DP{\theta}{z} \right)
\nonumber \\
  && 
  + \left( C_{m}^{2} l^{2} \right) \left\{ 
       \left( \DP{u}{x} \right)^{2}
     + \left( \DP{w}{z} \right)^{2}
    \right\}
\nonumber \\
  &&
  +  \frac{ C_{m}^{2} l^{2} }{2}
     \left( \DP{u}{z} + \DP{w}{z}\right)^{2}
  - \frac{K_{m}}{3}
     \left( \DP{u}{x} + \DP{w}{z} \right) 
\nonumber \\
   && 
   + \Dinv{2}
       \left(\DP[2]{K_{m}^{2}}{x}
	       + \DP[2]{K_{m}^{2}}{z}
       \right)
   + \left(\DP{K_{m}}{x}\right)^{2}
   + \left(\DP{K_{m}}{z}\right)^{2}
\nonumber \\
  && 
   - \frac{C_{\varepsilon}}{2 C_{m} l^{2}} K_{m}^{2}
\nonumber 
\end{eqnarray}
% 
�Ȥʤ�, ������ $C_{m} = C_{\varepsilon} = 0.2$ �� $K_{h} = 3 K_{m}$ �Ȥ���
�ط����Ѥ����, 
%
\begin{eqnarray}
 \DP{K_{m}}{t}
  &=& 
   - \left( 
      u \DP{K_{m}}{x} + w \DP{K_{m}}{z}
     \right)
   - \frac{3 g C_{m}^{2} l^{2}}{ 2 \overline{\theta}} 
      \left(\DP{\theta}{z} \right)
\nonumber \\
  && 
  + \left( C_{m}^{2} l^{2} \right) \left\{ 
       \left( \DP{u}{x} \right)^{2}
     + \left( \DP{w}{z} \right)^{2}
    \right\}
\nonumber \\
  &&
  +  \frac{ C_{m}^{2} l^{2} }{2}
     \left( \DP{u}{z} + \DP{w}{x}\right)^{2}
  - \frac{K_{m}}{3}
     \left( \DP{u}{x} + \DP{w}{z} \right) 
\nonumber \\
   && 
   + \Dinv{2}
       \left(\DP[2]{K_{m}^{2}}{x}
	       + \DP[2]{K_{m}^{2}}{z}
       \right)
   + \left(\DP{K_{m}}{x}\right)^{2}
   + \left(\DP{K_{m}}{z}\right)^{2}
\nonumber \\
  && 
   - \Dinv{2 l^{2}} K_{m}^{2}
\Deqlab{close_1.5}
\end{eqnarray}
%
�Ȥʤ�. 

