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\begin{document}

\Dtitle[�絤����ʬ�ŷ���ή]{ 
\Huge ����ʬ�ŷ��絤�ˤ����� 2 ���������ϳإ�ǥ���꼰��
 \\
%\Large Two dimensional moist convection including condensation of major
%atmospheric component \\
%\large --- Basic numerical experiments on CO$_2$ ice cloud convection in
%Martian atmosphere ---
}
\Dauthor[���� ã��]{ 
\Huge
���� ã�� \\
YAMASHITA Tatsuya \\
�̳�ƻ������ر����������칶 \\
}
\Ddate[2012/03/27]{2012 ǯ 3 �� 27 ��}
\Dfile{}

\maketitle

\tableofcontents

\markboth{����������}
         {����������}

\chapter{����������}

�� 1 �ϤǤϱ���ή��ǥ� deepconv/arare4 ���Ѥ����Ƥ�������������ˤĤ�
����������. 

\section{�絤����ʬ�ζŷ���θ���� 2 �����వ����������}

�ܥ�ǥ�Ǥϻ����������ϤȤ��ƽవ����������(Klemp and Wilhelmson, 1978)��
�絤����ʬ�ζŷ���̤��θ�������(Odaka {\it et al.}, 2005) ���Ѥ���. 
�వ���������ϤȤ������������ϤΤ���, ���ܾ��Ƴ���������������Ԥ����
��ؤ�. 
�����������ϤȤ��絤�ΰ��������θ�����������Ϥ�ؤ�. 
�వ���������Ϲ������������Ŭ�Ѥ�̵ͭ�ˤ��, �వ���������Ϥȴ���������
�����Ϥ�ʬ�व���. 
�వ���������Ǥϻ�����ʬ�˥⡼���̻���ʬ��ˡ(Klemp and Wilhelmson, 1978)
���Ѥ��뤳�Ȥ�����, �׻��Υ����Ȥ򲼤��뤳�Ȥ������
\footnote{
�⡼���̻���ʬ��ˡ�ξܺ٤ˤĤ��Ƥ�Υ�����Υɥ�����ȤǽҤ٤�. 
}. 
����, ���������������Ͻవ���������������٤�����ͥ��Ƥ����Τ�, �⡼
���̻���ʬ��ˡ��Ŭ�Ѥ���Ȱ���˷׻����뤳�Ȥ�����ʤ����Ȥ���Ŧ����Ƥ�
��(MacDonald {\it et al.}, 2000). 
�����Ƿ׻��θ�Ψ����Ż뤷�ƽవ���������Ϥ���Ѥ��뤳�ȤȤ���. 


�ޤ��ܥ�ǥ�ǤϿ�ʿ 1 ����, ��ľ 1 ������ 2 ����ľ��ľ���ɸ�Ϥ���Ѥ�
��. 
�ܸ���Ǥϼ���ʬ�ŷ���θ������ή�����������򤹤뤳�Ȥ���Ū�Ȥ��Ƥ����, 
���̤� 3 �������⸽�ݤβ��Ϥ�Ԥ��䤹�� 2 �����ǵ�����Ԥ����ȤȤ���. 


�ʲ�, ����ʬ�ŷ���θ�����వ���������ϤˤĤ�����������. 
���ѿ��ϴ��ܾ���ʬ�Ⱦ�����ʬ���¤�ɽ���Ǥ���Ȳ��ꤹ��. 
������ʬ�ϴ��ܾ���ʬ����٤ƽ�ʬ�������Ȳ��ꤷ, ��ʪ���̤λ���ȯŸ������
���ˤ����� 2 �����������̵�뤹��. 
â���ܸ���Ǥ���ή�����ܤ����׻���Ԥʤ��Τ�, ��ή��ˤĤ��Ƥ� 2 ������
���̤�̵�뤷�ʤ�. 
��������ԤʤäƤ���వ���������ϤǤϼ�����¸�ڤӥ��ͥ륮����¸����
Ω���ʤ����Ȥ����դ��ʤ���Фʤ�ʤ�
\footnote{
����ʬ�ŷ���θ�����వ���������ϤǤΥ��ͥ륮����������Ƴ�е�
�ӥ��ͥ륮������¸�ε����ˤĤ��Ƥ�, ����(2009) ����Ͽ A �򻲾Ȥ��줿��. 
}. 
�వ���������ϤǤϿ�ʿή® $u$, ��ľή® $w$, ���� $\theta$, �������ʡ�
�ؿ� $\Pi$, ��̩�� $\rho_s$ ��ͽ��, ���� $p$, ̩�� $\rho$, ���� $T$ 
�ʤɤ�¾��ʪ���̤ˤĤ��Ƥ�ͽ���ѿ����Ѥ��ƿ���Ū�˻��Ф���. 
����, �������ʡ��ؿ��ϰʲ��Τ褦����������. 
\begin{eqnarray}
\theta &\equiv& T \left( \frac{p_0}{p} \right)^{\frac{R}{c_{pv}}},
 \Deqlab{Pottemp} \\
\Pi &\equiv& \left( \frac{p}{p_0} \right)^{\frac{R}{c_{pv}}} =
 \frac{T}{\theta}. 
 \Deqlab{Exner}
\end{eqnarray}
������ $p_0$, $R$, $c_{pv}$ �Ϥ��줾���వ��, ñ�̼��̤�����ε������,
�갵��Ǯ��ɽ��, $R \simeq 188.9$ [J K$^{-1}$ kg$^{-1}$], $c_{pv} \simeq
734.1$ [J K$^{-1}$ kg$^{-1}$] �Ǥ���. 
�ܥ�ǥ�Ǥ���ɽ�̰��Ϥ��వ�ϤˤȤ뤳�Ȥˤ�, $p_0 = 700$ [Pa] �Ȥ���. 


�ܥ�ǥ�ˤ����Ƥ��Ż߾��֤ˤ��봥�礷����ʿ���ͤʴ��ܾ��Ϳ����. 
ʪ���� $\phi$ �δ��ܾ���ʬ�� $\overline{\phi}$, ������ʬ��
$\phi^{\prime}$ ��ɽ�����Ȥˤ����, ��ͽ���ѿ��ϰʲ��Τ褦��ɽ�����. 
\begin{eqnarray}
u(x,z,t) &=& u^{\prime}(x,z,t), \Deqlab{ZA} \\
w(x,z,t) &=& w^{\prime}(x,z,t), \Deqlab{ZB} \\
\theta(x,z,t) &=& \theta^{\prime}(x,z,t) + \overline{\theta}(z),
 \Deqlab{ZC} \\ 
\Pi(x,z,t) &=& \Pi^{\prime}(x,z,t) + \overline{\Pi}(z), \Deqlab{ZD} \\
\rho_s(x,z,t) &=& {\rho_s}^{\prime}(x,z,t). \Deqlab{ZE}
\end{eqnarray}
ή®�ڤӱ�̩�٤δ��ܾ���ʬ�ϥ����Ǥ���Τ�, �ʸ�ή®�ڤӱ�̩�٤ξ�����ʬ
�ˤĤ��Ƥϴ�ñ�ΰ٤˥ץ饤�൭�� `` $^{\prime}$ '' ���ά����. 
���ܾ���Ż߾��֤ˤ���Τ�, �ſ尵ʿ�դμ�
\begin{equation}
\DP{\overline{p}}{z} = - \overline{\rho} g \Deqlab{hydrostat1}
\end{equation}
������Ω��. 
â�� $g$ �Ͻ��ϲ�®�٤Ǥ���. 
\Deqref{hydrostat1} �� $\overline{\Pi}$, $\overline{\theta}$ ��ɽ������
��
\begin{equation}
\DP{\overline{\Pi}}{z} = - \frac{g}{c_{pv} \overline{\theta}}
 \Deqlab{hydrostat2} 
\end{equation}
�Ȥʤ�. 


��ñ�ΰ��絤�� CO$_2$ �Τߤ��������ΤȤ�, ���۵��ΤȤߤʤ����ΤȤ���. 
�ޤ������β�ž���̤�̵�뤹��. 
���ΤȤ�����ʬ�ŷ���θ�����వ���������ϰʲ��Τ褦�˵��Ҥ����
\footnote{
\Deqref{AA}, \Deqref{AB} ��Ƴ�ФˤĤ��Ƥ���Ͽ A, 
\Deqref{AD} ��Ƴ�ФˤĤ��Ƥ���Ͽ B �򻲾Ȥ��줿��. 
}. 
\begin{eqnarray}
\DP{u}{t} 
&=& 
- u \DP{u}{x} - w \DP{u}{z} - c_{pv} \overline{\theta}
 \DP{\Pi^{\prime}}{x} + D_u + D_{u}^{sfc}, 
 \Deqlab{AA} \\
\DP{w}{t} 
&=& 
- u \DP{w}{x} - w \DP{w}{z} - c_{pv} \overline{\theta}
 \DP{\Pi^{\prime}}{z} + D_w 
 + g 
   \left( 
    \frac{\theta^{\prime}}{\overline{\theta}}
    - \frac{R}{p_0} 
      \frac{\overline{\theta}}{\overline{\Pi}^{c_{vv} / R}}
      \rho_s
   \right),
  \Deqlab{AB} \\
\DP{\theta^{\prime}}{t} 
&=&
- u \DP{\theta^{\prime}}{x} - w \DP{\theta^{\prime}}{z}
- w \DP{\overline{\theta}}{z} 
 \nonumber \\
& &
+ \Dinv{\overline{\Pi}}
(Q_{cond} + Q_{rad} + Q_{dis}) + D_{\theta}
+ D_{\theta}^{sfc}, 
 \Deqlab{AC} \\
\DP{\Pi^{\prime}}{t} &=&
- \frac{\overline{c}^2}{c_{pv} \overline{\rho} \overline{\theta}^2}
\left[ \DP{ (\overline{\rho} \overline{\theta} u) }{x} 
+ \DP{ (\overline{\rho} \overline{\theta} w) }{z} \right]
 \nonumber \\
& &
+ \frac{\overline{c}^2}{c_{pv} \overline{\theta}^2}
  \Dinv{\overline{\Pi}} 
  ( Q_{cond} + Q_{rad} + Q_{dis} )
- \frac{\overline{c}^2}{c_{pv} \overline{\rho} \overline{\theta}^2}
 M_{cond}, 
 \Deqlab{AD} \\
\DP{\rho_s}{t} &=& 
- \DP{(\rho_s u)}{x} - \DP{(\rho_s w)}{z}
+ M_{cond} + M_{fall} + D_{\rho_s}. \Deqlab{AE}
\end{eqnarray}
������ $\overline{c}$ �ϲ�®�δ��ܾ���ʬ��ɽ��, �ʲ��δط����ˤ������
���. 
\begin{equation}
{\overline{c}}^2 
= \frac{c_{pv}}{c_{vv}} R \overline{T}
= \frac{c_{pv}}{c_{vv}} R \overline{\theta} \overline{\Pi}. 
 \Deqlab{Sound}
\end{equation}
â�� $c_{vv}$, $c_{pv}$ �Ϥ��줾�� CO$_2$ ���Τ�������Ǯ, �갵��Ǯ��ɽ��, 
$c_{vv} \simeq 545.2$ [J K$^{-1}$ kg$^{-1}$], 
$c_{pv} \simeq 734.1$ [J K$^{-1}$ kg$^{-1}$], 
�Ǥ���. 
$D_u$, $D_w$, $D_{\theta}$, $D_{\rho_s}$ �Ϥ��줾���ʿή®, ��ľή®,
����, ��̩�٤˴ؤ�����ή�Ȼ���Ǥ���. 
$D_u^{sfc}$, $D_{\theta}^{sfc}$ �Ϥ��줾����ɽ�̤���α�ư��, Ǯ�Υե��
������͢����ɽ����Ǥ���. 
$Q_{cond}$, $Q_{rad}$, $Q_{dis}$ �Ϥ��줾�첹�٤��Ф�����Ǯ��Ǯ��, ����
��Ǯ��, �����Ǯ��Ǥ���. 
�ޤ� $M_{cond}$, $M_{fall}$ �Ϥ��줾��ñ�̻���ñ������������ζŷ���, ñ
�̻���ñ������������α�γ����̤Ǥ���. 
$D_u$, $D_w$, $D_{\theta}$, $D_{\rho_s}$ �ζ���Ū��ɽ���ˤĤ��Ƥ�
1.2 ��ǽҤ٤�. 
�ޤ� $M_{cond}$, $M_{fall}$ �ζ���Ū��ɽ���ˤĤ��Ƥ� 1.3 ��ǽҤ٤�. 

Ǯ�ϳؤμ� \Deqref{AC}, ���������� \Deqref{AD} ���������Ǯ��Ǯ��
 $Q_{cond}$, $Q_{rad}$, $Q_{dis}$ �ˤĤ��ƽҤ٤�. 
��Ǯ��Ǯ��ˤĤ��Ƥ�
\begin{equation}
Q_{cond} = \frac{L M_{cond}}{c_{pv} \overline{\rho}} \Deqlab{AF}
\end{equation}
��ɽ��. 
������ $L$ ��ñ�̼���������� CO$_2$ ����Ǯ�Ǥ���, �ܥ�ǥ�Ǥϰ���Ȥ�
�ʤ��� $L \simeq 5.86 \times 10^5$ [J kg$^{-1}$] �Ȥ���. 
$Q_{dis}$ �ζ���Ū��ɽ���ˤĤ��Ƥ� 1.2 ��ǽҤ٤�. 
$Q_{rad}$ ���ۤ˻���ȯŸ��򤯤��ȤϤ�����, �¸�����˱����ƹ��٤Ȼ��֤�
��¸����ѥ�᡼���Ȥ���Ϳ����. 

���������� \Deqref{AD} �ˤĤ������դ��٤����򤤤��Ĥ��Ҥ٤�. 
\Deqref{AD} �α����� 2 �������Ǯ��Ǯ��ȼ��Ǯ��ĥ�θ���,  
\Deqref{AD} �α����� 3 ��϶ŷ�ˤ���絤���̤θ�����ȼ�������㲼�θ���
��ɽ���Ƥ���. 
Klemp and Wilhelmson(1978) �Ǥ�, \Deqref{AD} ��
������ 2 ��ڤ��� 3 ��Ͻ�ʬ��������ΤȤ���̵�뤷�Ƥ��뤬, �ܥ�ǥ�Ǥ�
̵�뤷�ʤ�. 
\Deqref{AF} ���Ѥ����, \Deqref{AD} �ϰʲ��Τ褦�˽񤭴�������
\footnote{
\Deqref{AH} ��Ƴ�ФˤĤ��Ƥ���Ͽ B �򻲾Ȥ��줿��. 
}. 
\begin{eqnarray}
\DP{\Pi^{\prime}}{t} 
 &=&
 - \frac{\overline{c}^2}{c_{pv} \overline{\rho} \overline{\theta}^2}
 \left[ \DP{ (\overline{\rho} \overline{\theta} u) }{x} 
  + \DP{ (\overline{\rho} \overline{\theta} w) }{z} \right]
 \nonumber \\
 & &
 + \frac{\overline{c}^2}{c_{pv} \overline{\rho} \overline{\theta}}
   \left( \frac{L}{c_{pv} \overline{T}} - 1 \right) M_{cond}
 + \frac{\overline{c}^2}{c_{pv} \overline{\theta}^2 \overline{\Pi}}
  ( Q_{rad} + Q_{dis} ). 
 \Deqlab{AH}
\end{eqnarray}
�����˰�ξ�� $O(\overline{T}) \sim 150$ [K] �Ǥ���Τ�, $c_{pv} \simeq
734.1$ [J K$^{-1}$ kg$^{-1}$], $L \simeq 5.86 \times 10^5$ [J kg$^{-1}$]
��� $L / (c_{pv} \overline{T}) > 1 $ ������Ω��. 
¨�������˰�� CO$_2$ �ζŷ뤬���������, ���Ϥ����ä��뤳�Ȥˤʤ�. 

�ʲ�, �������ˤĤ��ƽҤ٤�. 
��ʿ�����ζ����Ǥϼ�����������Ŭ�Ѥ�, ��ľ�����ζ����Ǥϱ���̵������
Ŭ�Ѥ���. 
�׻��ΰ�ο�ʿ������Ĺ���� $L$ �Ȥ����, Ǥ�դ�ʪ���� $\phi$ ���Ф��Ʋ�
�����٤���ʿ�����ζ�������
\begin{equation}
\phi(x,z,t) = \phi(x+L,z,t) \Deqlab{BoundaryA}
\end{equation}
��ɽ�����. 
�ޤ��׻��ΰ�α�ľ������Ĺ���� $D$, $w$ �ʳ���ͽ���ѿ��� $\psi$ �Ȥ����,
��ľ�����ζ�������
\begin{eqnarray}
w &=& 0, \quad \quad (z = 0, D) \Deqlab{boundaryB} \\
\DP{\psi}{z} &=& 0, \quad \quad (z = 0, D) \Deqlab{boundaryC}
\end{eqnarray}
��ɽ�����. 


\section{��ή�ѥ�᥿�ꥼ�������}

���̤��絤�α�ư�򵭽Ҥ�����, ��ή��ư(�����ϱ�ư)���θ����ɬ�פ�����. 
���������ͥ�ǥ�ξ��ͭ�¸Ĥγʻ��������֤��Ʒ׻���Ԥ���, �ʻ����ֳ֤�
��⾮���ʥ�������(���֥���åɥ�������)�θ��ݤ��갷�����Ȥ�����ʤ�. 
�����ǥ��֥���åɥ����������ή��ư�˴ؤ���ʪ���̤�ʻ������ʪ���̤ȷ�
�ӤĤ���ɽ������(��ή�ѥ�᥿�ꥼ�������). 

\subsection{��ή�Ȼ�}

�ܥ�ǥ�Ǥ� Klemp and Wilhelmson(1978), ���ڡ��縶(2001)�˽��ä�
��ή�Ȼ���, ��ή�Ȼ�����, �����Ǯ���ɽ������Τ� 1.5 ���Υ����������
�ǥ���Ѥ���. 
1.5 ���Υ����������ǥ���Ѥ���� $D_u$, $D_w$, $D_\theta$,
$D_{\rho_s}$ �ϰʲ��Τ褦��ɽ�������. 
\begin{eqnarray}
D_u 
&=& - \DP{}{x} 
    \left[ - 2 K_m \DP{u}{x} + \frac{2}{3} E_{turb} \right]
    - \DP{}{z} \left[ - K_m \left( \DP{u}{x} + \DP{w}{z} \right) \right] 
    , \Deqlab{TurbA} \\
D_w 
&=& - \DP{}{x} \left[ - K_m \left( \DP{w}{x} + \DP{u}{z} \right) \right]
    - \DP{}{z} 
    \left[ - 2 K_m \DP{w}{z} + \frac{2}{3} E_{turb} \right]
    , \Deqlab{TurbB} \\
D_{\theta}
&=& - \DP{}{x} \left[ K_h \DP{\theta^{\prime}}{x} \right]
    - \DP{}{z} \left[ K_h \DP{(\theta^{\prime} + \overline{\theta})}{z}
	       \right], \Deqlab{TurbC} \\
D_{\rho_s}
&=& - \DP{}{x} \left[ K_h \DP{\rho_s}{x} \right]
    - \DP{}{z} \left[ K_h \DP{\rho_s}{z}  \right] \Deqlab{TurbD} 
\end{eqnarray}
â�� $K_m$, $K_h$ �Ϥ��줾�챿ư��, Ǯ���Ф�����ή�Ȼ������Ǥ���, 
$E_{turb}$ �ϥ��֥���åɥ����������ή��ư���ͥ륮���Ǥ���. 
�������Ϥ˴�Ť�, $E_{turb}$ �� $K_m$ ��ɽ�������
\begin{equation}
E_{turb} = \left( \frac{K_m}{C_m l} \right)^2 \Deqlab{TurbE}
\end{equation}
�Ȥʤ�. 
������ $l$ �Ϻ����Υ�Ǥ���, ��ʿ�����γʻҴֳ� $\Delta x$ �ڤӱ�ľ����
�γʻҴֳ� $\Delta z$ ���Ѥ��� $l = (\Delta x \Delta z)^{1/2}$ ��ɽ����
��. 
�ޤ� $C_m$ ��̵��������Ǥ���. 
$K_h$ �ˤĤ��Ƥ� Deardorff(1972) �˽��ä�
\begin{equation}
K_h = 3 K_m \Deqlab{TurbF}
\end{equation}
��Ϳ����. 
1.5 ���Υ����������ǥ�Ǥϰʲ��˼��� $K_m$ �λ���ȯŸ���, ����ή��
���������
\footnote{
�� \Deqref{TurbG} ��Ƴ�ФˤĤ��Ƥ�, ��Ͽ C �򻲾Ȥ��줿��. 
}. 
\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 }
       \left(
        \Dinv{\overline{\theta}} \DP{\theta}{z}
        - \frac{R \overline{\theta}}{p_0 \overline{\Pi}^{c_v/R}}
          \DP{\rho_s}{z}
       \right)
    \nonumber \\
& & + {C_m}^2 l^2 \left[ \left( \DP{u}{x} \right)^2 + \left( \DP{w}{z}
      \right)^2 \right] 
    + \frac{{C_m}^2 l^2}{2} \left( \DP{u}{z} + \DP{w}{x} \right)^2
    \nonumber \\
& & - \frac{K_m}{3} \left( \DP{u}{x} + \DP{w}{z} \right) 
    + \Dinv{2} \left( \DP[2]{{K_m}^2}{x} + \DP[2]{{K_m}^2}{z} \right)
    \nonumber \\
& & + \left( \DP{K_m}{x} \right)^2 + \left( \DP{K_m}{z} \right)^2
    - \frac{C_{\varepsilon}}{2 C_m l^2} {K_m}^2. 
 \Deqlab{TurbG}
\end{eqnarray}
������ $C_{\varepsilon}$ ��̵��������Ǥ���. 
̵����������ͤˤĤ��Ƥ� Deardorff(1975) �˽��ä� $C_m = C_{\varepsilon}
= 0.2$ �Ȥ���. 

�����Ǯ�� $Q_{dis}$ �ˤĤ��Ƥ�
\begin{equation}
Q_{dis} 
= \frac{C_{\varepsilon}}{c_{pv} l} E_{turb}^{3/2} 
= \frac{C_{\varepsilon}}{c_{pv} l} \left( \frac{K_m}{C_m l} \right)^3
 \Deqlab{TurbH}
\end{equation}
��Ϳ����. 

\subsection{��ɽ�ե�å���}

��ɽ�̤���α�ư��, Ǯ�Υե�å����򤽤줾�� $F_{u}$, $F_{\theta}$ 
�Ȥ����, ��ɽ�̤���α�ư��, Ǯ��͢���� 
$D_u^{sfc}$, $D_{\theta}^{sfc}$ �Ϥ��줾��
\begin{eqnarray}
D_u^{sfc}
 &=&
  - \Dinv{\rho} \DP{F_{u}}{z}, \\
D_{\theta}^{sfc}
 &=&
  - \Dinv{\Pi \rho c_p} \DP{F_{\theta}}{z}
\end{eqnarray}
��ɽ�����. 
�ܥ�ǥ�Ǥ�, ��ɽ�̤���α�ư��, Ǯ�Υե�å�����ɽ���Ȥ���,
Louis(1979) ���ñ����������¾(2001)�ˤ��Х륯ˡ���꼰������Ѥ��Ƥ���. 
�ʲ��Ǥ�, �����絤���Ф���Х륯ˡ��ɽ���򼨤�.

����¾(2001) �Ǥ� $D_u^{sfc}$, $D_{\theta}^{sfc}$ ��ʲ��Τ褦��ɽ��. 
\begin{eqnarray}
F_{u} 
 &=& -C_D V_{sfc} \rho u_{z_1}, \\
F_{\theta}    
 &=& - c_p C_D V_{sfc} \rho (T_{z_1} - T_{sfc}). 
\end{eqnarray}
������ $C_D$ �ϥХ륯����, $z_1$ �ϥ�ǥ�ǲ��ع���, $T_{sfc}$ ����ɽ��
����, $V_{sfc}$ ����ɽ����®�Ǥ���. 
��ɽ�̤ˤ����륨�����ʡ��ؿ�, ̩�٤���ǥ�ǲ��ع��٤Ǥ��ͤ��������Ȥ�, 
\begin{eqnarray}
D_{u}^{sfc}
 &\approx&
  \DP{}{z} (C_D V_{sfc} u_{z_1}), \\
D_{\theta}^{sfc}
 &\approx&
  \DP{}{z} 
   \left[ 
    C_D V_{sfc} \left( \theta_{z_1} - \frac{T_{sfc}}{\Pi_{z_1}} \right) 
   \right]
\end{eqnarray}
�ȶ������. 
$V_{sfc}$ ��
\begin{eqnarray}
V_{sfc} = \sqrt{u_{z_1}^2 + v_0^2}
\end{eqnarray}
��ɽ�����.
$v_0$ �������ᤤ�Ƥ��ʤ��Ƥ����ͤˤ���Ǯ���ȯ�Ȥ��ä�����Ū��
���ݤ������뤳�Ȥ��ݾڤ���٤���®�κ����ͤǤ���, �ܥ�ǥ�Ǥ� $v_0 = 0$
m/s �Ȥ��Ƥ���. 
$C_D$ �ϥХ륯����㡼�ɥ���� $Ri_{B}$ �δؿ��Ǥ���, 
\begin{eqnarray}
C_D
 &=&
 \begin{cases}
  C_{DN} 
   \left( 1 - \frac{A_1 Ri_{B}}{1 + A_3 |Ri_{B}|^{1/2}} \right), 
   (Ri_{B} < 0) \\
  C_{DN} \Dinv{1 + A_2 Ri_{B}}, (Ri_{B} > 0)
 \end{cases} \\
Ri_{B}
 &=&
  \frac{g z_1 (\theta_1^{\prime} + \overline{\theta}_1 - T_{sfc})}
  { \overline{\theta_1}^x {\rm max} (u_1^2, u_{min}^2)}
\end{eqnarray}
��ɽ�����. 
������ $u_{min}$ �ϥХ륯����㡼�ɥ�������Ф�����®�κ����ͤǤ���, 
$u_{min} = 0.1$ m/s �Ȥ��Ƥ���. 
�ޤ� $C_{DN}$ ����Ω���ؤǤΥХ륯����, $A_1$, $A_2$, $A_3$ ���������
�Ǥ���, 
\begin{eqnarray}
C_{DN}
 &=&
  \left[ \frac{\kappa}{\ln (z_1/z_0)} \right]^2, 
  \nonumber \\
A_1
 &=& 9.4, 
  \nonumber \\
A_2
 &=& 4.7, 
  \nonumber \\
A_3
 &=& 7.4 C_{DN} A_1 \left( \frac{z_1}{z_0} \right)^{1/2}
\end{eqnarray}
�Ǥ���. 
������ $\kappa=0.35$ �ϥ���ޥ����, $z_0$ ������Ĺ�Ǥ���, 
Sutton {\it et al}.(1978) �˽���, $10^{-2}$ m �Ȥ���. 

%\subsection{��ɽ�̥ե�å���}
%
%
%��ɽ�̤����Ǯ�ե�å��� $F_h$ �ˤĤ��Ƥϴ�ά�����줿�Х륯ˡ(����,
%1994)���Ѥ���
%\begin{equation}
%F_h = c_D V_{lowest} (T_{gnd} - T_{lowest}) / \Delta z \Deqlab{TurbI} 
%\end{equation}
%��ɽ������. 
%������ $c_D$, $V_{lowest}$, $T_{gnd}$, $T_{lowest}$ �Ϥ��줾��Х륯����,
%���ز�ü�Ǥο�ʿ��, ��ɽ�̲���, ���ز�ü�Ǥβ��٤�ɽ��. 
%�Х륯�����ϰ����ͤǤ���, $c_D = 1.5 \times 10^{-3}$ �Ȥ���. 
%���ز�ü�Ǥο�ʿ����
%\begin{equation}
%V_{sfc} = \sqrt{u_{lowest}^2 + u_0^2} \Deqlab{TurbJ}
%\end{equation}
%��Ϳ����. 
%������ $u_{lowest}$ ��ͽ�󤷤�����줿���ز�ü�ο�ʿ���Ǥ���, $u_0$ ����
%���Ǥ���. 
%$u_0$ �� $u_{lowest}$ �����˼夯�ʤä����Ǥ⤢�����٤��礭������ɽ��
%�ե�å������ݾڤ����褦�ˤ���٤�����Ǥ���, $u_0 = 3.0$ [m/s] �Ȥ���. 
%$T_{gnd}$ ���ۤ˲򤫤���, �ѥ�᡼���Ȥ��ư����ͤ�Ϳ����. 
%

\section{��ʪ���ѥ�᥿�ꥼ�������}


1.3 ��Ǥ�ñ�̻���ñ������������ζŷ��� $M_{cond}$ �ڤ�ñ�̻���ñ������
������α�γ����� $M_{fall}$ ��ɽ�����ˤĤ��ƽҤ٤�. 


�ϵ��絤�α���ή��ǥ�Ǥ�, ��˰�¾��֤Ͻֻ��˲�ä�����ΤȤߤʤ�, ˰
�¾��֤Ȥʤ�褦�˶ŷ��̤�Ĵ�����뤳�Ȥ�¿��
(�㤨�� Soong and Ogura, 1973 ). 
�����������絤�ξ��, ��˰�¾��֤�ͭ�¤λ���¸�ߤ����뤳�Ȥ���¬���¸���
�����ˤ�äƼ�������Ƥ���
(�㤨�� Colaprete {\it et al.}, 2003; Glandorf {\it et al.}, 2002; 
M\"{a}\"{a}tt\"{a}nen {\it et al.}, 2005). 
�������ܥ�ǥ�Ǥϱ�γ��ͭ�¤λ��֤򤫤�����Ĺ�����ΤȤ����꼰����Ԥ�
��. 


�����絤�Ǥβ��١����ϴĶ����Ǥ�, CO$_2$ �ϵ��갿���ϸ���Τ����줫�ξ�
�֤�¸�ߤ���. 
���ä��ܥ�ǥ�ǹͤ���٤���ʪ�������� CO$_2$ �絤�� CO$_2$ ɹ�����Ѵ���
�ߤǤ���. 
�ϵ��絤�ξ��, ɹ���ˤη����Υᥫ�˥�����絤���¸�ߤ�����γ�Ҥαƶ���
�������ԶѰ�˷�����, ���αƶ�������ʤ��Ѱ�˷����� 2 �����ʬ�व���
���Ȥ��Τ��Ƥ���(�㤨�п���(2000)). 
�ϵ��α���̸�ʤɤ�����ʬ���ԶѰ�˷����ˤ�ä�������ȹͤ����Ƥ���. 
�����絤��ˤϥ�����γ�Ҥ� H$_2$O ɹ�ǥ����ƥ��󥰤��줿�����Ȥ���ͷ����
����, �ԶѰ�˷�����������ȹͤ�����(Gooding, 1986). 
�������ܥ�ǥ�Ǥϳ˷����Ȥ����ԶѰ�˷����Τߤ�ͤ���. 


��γ����Ĺ�����ϳȻ���Ĺ��ʻ����Ĺ�� 2 �����ʬ�व���. 
�Ȼ���Ĺ�Ȥϵ������ʬ�Ҥ��Ȼ��ˤ�äƱ�γɽ�̤˰�ư���ŷ뤹�뤳�Ȥˤ��
�Ʊ�γ����Ĺ���뤳�Ȥ�ؤ�. 
ʻ����Ĺ�Ȥϱ�γƱ�Τ����ͤ�, ���Τ��뤳�Ȥˤ�äƱ�γ����Ĺ���뤳�Ȥ��
��. 
Tobie {\it et al.}(2003) �ˤ���, �ŷ�ˤȤʤ�����Ȥ�ʿ��Ⱦ�¤��� 0.5
 [$\mu$m] �Ǥ���(Ockert-Bell {\it et al.}, 1997), �Ļ�����Ф���ŵ��Ū��
����Ū�����������褽 0.1 �Ȥʤ�����絤�ξ��, Ⱦ�� 50 [$\mu$m] ���٤α�
γ��ʻ����Ĺ�ǽ����Τ��פ�����֤ϳȻ���Ĺ�ǽ����Τ��פ�����֤���
��ʬĹ��. 
�̼�(2006) ��ή���¸�ߤ��ʤ������絤�ǤγȻ���Ĺ�ˤĤ��Ƥο��ͷ׻���
�Ԥʤ�, �Ȼ���Ĺ�λ��֥������뤬 1 -- 20 [s] ���٤Ǥ��뤳��, ��γȾ�¤�
5 -- 25 [$\mu$m] �Ȥʤ뤳�Ȥ򸫽Ф���. 
��γȾ�¤��������ۤɳȻ���Ĺ��ʻ����Ĺ����٤�ͥ���Ȥʤ�Τ�, �����絤��
���ǤϳȻ���Ĺ��ͥ���Ȥʤ�ȹͤ�����. 
�������ܥ�ǥ�Ǥ�ʻ����Ĺ�ˤ���γ��Ĺ��̵��Ǥ���Ȳ��ꤷ, �Ȼ���Ĺ��
�ߤ�ͤ���. 

\subsection{�ŷ��̤��꼰��}

�ʲ�, ñ�̻�������������ζŷ��� $M_{cond}$ �μ�갷���ˤĤ��ƽҤ٤�. 
�ܥ�ǥ�Ǥ�ñ�̼��̤ε���˴ޤޤ��ŷ�ˤθĿ��ڤ�Ⱦ�¤϶���Ū������Ū
�˰��ͤȲ��ꤹ��. 
�ޤ���γ��Ⱦ�¤ϳƳʻ���ˤ����ƶ���Ū�˰���Ǥ���Ȳ��ꤹ��. 
���˱�γ�ϵ���ζŷ�ˤ�ˤȤ��Ʒ�������, ��γ���Ȥ����Ȥʤ�Ȳ��ꤹ��. 
���ΤȤ�, ñ�̻���ñ������������ζŷ���
$M_{cond}$ �� Tobie {\it et al.} (2003) Ʊ��, �ʲ��Τ褦��ɽ�����
\footnote{
\Deqref{CloudB} ��Ƴ�ФˤĤ��Ƥ���Ͽ D �򻲾Ȥ��줿��. 
}. 
\begin{equation}
M_{cond} = \frac{4 \pi r_d \overline{\rho} N^{*}}{R_h} (S - 1). 
 \Deqlab{CloudB}
\end{equation}
$R_h$, $S$, $r_d$ �Ϥ��줾��Ǯ͢���˴ؤ��뷸��, ˰����, ��γȾ�¤Ǥ���, 
\begin{eqnarray}
R_h &=& \frac{L^2}{k R T^2} = \frac{L^2}{k R \theta^2 \Pi^2},
 \Deqlab{CloudC} \\
S &=& \frac{p}{p_{\ast}} = \frac{{p_0}^{\frac{c_{pv}}{R}} \Pi}{p_{\ast}}
 \Deqlab{CloudE} \\
r_d &=&
 \left( 
  r_{aero}^3 + \frac{3 \rho_s}{4 \rho_I \pi \overline{\rho} N^{*}} 
 \right)^{1/3}
 \Deqlab{CloudA}
\end{eqnarray}
��ɽ�����. 
â�� $k$, $p_{\ast}$ �Ϥ��줾��Ǯ�Ȼ�����, CO$_2$ ��˰�¾�����, 
$\rho_I$ �� CO$_2$ ɹ��̩��, $r_d$ �ϱ�γȾ��, $r_{aero}$ �϶ŷ�
�ˤ�Ⱦ��, $N^{*}$ ��ñ�̼��̤ε���˴ޤޤ��ŷ�ˤο��Ǥ���. 
�ܥ�ǥ�Ǥ� $\rho_I = 1.565 \times 10^3$ [kg/m$^3$] ��Ϳ��, $r_{aero}$,
$N^{*}$ �ϼ¸��˱�����Ϳ����. 
\Dfigref{radi-dens} �˱�̩�٤δؿ��Ȥ��Ʊ�γȾ�¤�ץ��åȤ�����Τ򼨤�. 
CO$_2$ ��˰�¾������ˤĤ��Ƥ�Ⱦ�и����Ǥ��� Antoine �μ�
(Antoine, 1888)�˴�Ť�, 
\begin{equation}
p_{\ast} \simeq \exp \left( A_{ant} - \frac{B_{ant}}{T} \right)
 \Deqlab{CloudG}
\end{equation}
�Ȥ���. 
������ $A_{ant}$, $B_{ant}$ �ϼ¸��ˤ����ޤ뷸���Ǥ���, CO$_2$ �ξ��
$A_{ant} = 27.4$, $B_{ant} = 3103$ �Ǥ���(���ع��ز�, 1999). 

\subsection{��γ��̤��꼰��}

�ʲ�, ñ�̻�������������α�γ��� $M_{fall}$ �μ�갷���ˤĤ��ƽҤ٤�. 
$M_{fall}$ �� Kessler(1969) ��Ʊ�ͤ�, ��γ�ν�ü®�� $V_{term}$ �Ǥΰ�ή
�Ȥ���ɽ������. 
¨��
\begin{equation}
M_{fall} = \DP{}{z} \left( \rho_s V_{term} \right) \Deqlab{CloudI}
\end{equation}
��ɽ��. 
�������絤̩�٤Ͼ�����, �����ʱ�γ��������γ�Ҥν�ü®�٤� Stokes §��ɾ
��������ͤ���ͭ�դ��礭���ʤ뤳�Ȥ���Ŧ����Ƥ���. 
�����Ǳ�γ�ν�ü®�٤ˤĤ��Ƥ�, Conrath(1975) ����¿���β����絤����ǹ�
�ʤ��Ƥ���Τ�Ʊ�ͤ�, Cunningham ������ Stokes §��Ŭ�Ѥ���
\begin{equation}
V_{term} = C_{sc} \frac{2 r_d^2 g \rho_I}{9 \eta} \Deqlab{CloudJ}
\end{equation}
��ɽ��(Stokes, 1845; Cunningham, 1910). 
$\eta$ ��Ǵ�������Ǥ���, Sutherland �θ����˽���, 
\begin{equation}
\eta 
 = 
 \eta_{ref} 
 \left( \frac{T_{ref} + C_{CO_2}}{\overline{T} + C_{CO_2}} \right)
 \left( \frac{\overline{T}}{T_{ref}} \right)^{3/2} 
 = 
 \eta_{ref} 
 \left( 
  \frac{T_{ref} + C_{CO_2}}{\overline{\theta} \overline{\Pi} + C_{CO_2}} 
 \right)
 \left( 
  \frac{\overline{\theta} \overline{\Pi}}{T_{ref}} 
 \right)^{3/2} 
 \Deqlab{CloudM}
\end{equation}
��ɽ������(Sutherland, 1893). 
$\eta_{ref}$, $T_{ref}$, $C_{CO_2}$ �Ϥ��줾��Ǵ�������δ����, ���٤δ�
����, CO$_2$ �˴ؤ��� Sutherland ����Ǥ���, $\eta_{ref} = 1.47 \times
10^{-5}$ [Pa $\cdot$ s], $T_{ref} = 293$ [K], $C_{CO_2} = 240$ [K] ��Ϳ��
��(��Ωŷʸ��, 2004). 
$C_{sc}$ ��������γ�Ҥˤ����� Stokes §����Τ�����������뷸��
(Cunningham ��������)�Ǥ���, 
\begin{equation}
C_{sc} 
 = 1 + \frac{4}{3} K_n
 \Deqlab{CloudK}
\end{equation}
��Ϳ����(Rossow, 1978: Tobie {\it et al.}, 2003). 
â�� $K_n \equiv \lambda / r_d$ �ϱ�γ���Ф��� Knudsen ���Ǥ���
\footnote{
Cunnigham(1910) �ˤ�äƽ���Ϳ����줿����������ɽ����
\begin{equation}
C_{sc} = 1 + 1.255 K_n
 \nonumber
\end{equation}
�Ǥ���. 
���θ������٤��ɤ� Cunningham ������ɽ���������Ĥ���Ƥ��줿. 
�㤨�� Davies(1945) �Ǥϰʲ���ɽ������Ƥ��줿. 
\begin{equation}
C_{sc} 
 = 1 + K_n
    \left[ 
     1.257 + 0.400 \exp \left( - \frac{1.10}{K_n} \right)
    \right]. 
 \nonumber
\end{equation}
}. 
$\lambda$ �� CO$_2$ ��ʿ�Ѽ�ͳ�����Ǥ���, 
\begin{equation}
\lambda 
 = 
  \frac{k_B \overline{T}}{\sqrt{2} \pi \sigma^2 \overline{p}} 
 = 
  \frac{k_B \overline{\theta}}
  {\sqrt{2} \pi \sigma^2 p_0 \overline{\Pi}^{c_{vv}/R}} 
 \Deqlab{CloudL}
\end{equation}
��Ϳ����(Chapman and Cowling, 1970). 
$k_B$ �� Boltzmann ���, $\sigma$ �� CO$_2$ ʬ�Ҥ�ľ�¤Ǥ���, $k_B =
1.38 \times 10^{-23}$ [m$^2$ kg s$^{-2}$ K$^{-1}$], $\sigma =
3.3 \times 10^{-10}$ [m] �Ǥ���(Golden and Sircar, 1994). 
\Dfigref{Vterm} �˲����˴������
�ŷ�����ն�($T \sim 150$[K], $p \sim 400$[Pa]), 
������Ǯ�ؾ�ü($T \sim 135$[K], $p \sim 90$[Pa])
�Ǥα�γ�ν�ü®�٤򼨤�. 
%
%
%
\begin{figure}
 \begin{center}
%  \includegraphics[width=6.0cm,angle=270]{radius-dens.ps}
  \Depsf[][][width=6.0cm,angle=270]{ps/radius-dens.ps}
\caption{
��γȾ�¤��̩�٤δؿ��Ȥ��ƥץ��åȤ������. 
�����ϲ����˴�����ζŷ�����ն����ħŪ�ʲ��ٰ��Ͼ��
($T=150$[K], $p=400$[Pa])�����ꤷ�����, 
�����ϲ����˴�����μ�����Ǯ�ؾ�ü����ħŪ�ʲ��ٰ��Ͼ��
($T=135$[K], $p=90$[Pa])�����ꤷ����ΤǤ���. 
�ŷ�˿�̩�٤� $5.0 \times 10^{8}$[kg$^{-1}$] �Ȥ��Ƥ���. 
\Dfiglab{radi-dens}
}
 \end{center}
  \end{figure}
%
%
%
\begin{figure}
 \begin{center}
%  \includegraphics[width=6.0cm,angle=270]{Vterm.ps}
  \Depsf[][][width=6.0cm,angle=270]{ps/Vterm.ps}
\caption{
��γ�ν�ü®�٤��γȾ�¤δؿ��Ȥ��ƥץ��åȤ������. 
�����ϲ����˴�����ζŷ�����ն����ħŪ�ʲ��ٰ��Ͼ��
($T=150$[K], $p=400$[Pa])�����ꤷ�����, 
�����ϲ����˴�����μ�����Ǯ�ؾ�ü����ħŪ�ʲ��ٰ��Ͼ��
($T=135$[K], $p=90$[Pa])�����ꤷ����ΤǤ���. 
\Dfiglab{Vterm}
}
 \end{center}
  \end{figure}
%
%
%

\clearpage

\chapter*{����ʸ��}

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Soong, S-T., Ogura, Y., 1973: 
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Stokes, G. G., 1845: 
On the effect of internal friction on the motion of pendulum, 
Transactions of the Cambridge Philosophical Society,
$\bf{9}$, part2, 8 -- 14

Sutherland, W., 1893: 
The viscosity of gases and molecular force. 
Philosophical Magazine, S. 5, $\bf{36}$, 507 -- 531

Sutton, J. L., C. B. Leovy, and J. E. Tillman, 1978: 
Diurnal variations
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���� ����, ���� ���, ���� ����, �� �Ͳ�, 2001: 
2 �����������Ϥ��Ѥ��������絤������ή�ο��ͷ׻�, 
�ʤ���, $\bf{20}$, �ʤ���ޥ����ǥ��� 2001, \\
http://www.nagare.or.jp/mm/2001/odaka/

���ع��ز���, 1999: 
���ع�������, ����, 1339 pp

�̼� ����, 2006: 
�絤����ʬ�ζŷ���θ���������絤�����ϳإ�ǥ�γ�ȯ.  
�̳�ƻ�����ر��ϵ������ʳ��칶������ʸ, 99 pp \\
http://www.gfd-dennou.org/arch/prepri/2006/hokudai/kitamo/paper/pub/main.pdf

��Ωŷʸ����, 2004: 
����ǯɽ, ����, 1015 pp

���� �µ�, �縶 �ƻ�, 2001: 
CReSS �桼�������������� 2 ��. 253 pp \\
http://cf.tokyo.rist.or.jp/CReSS.top.html

���� ��, 2000: ���ȱ��ε��ݳ�. ī�ҽ�Ź, 196 pp

���� ã��, 2009:
�絤����ʬ�ζŷ���θ���� 2 ����������ή
--- �����絤��� CO$_2$ ɹ����ȼ����ή�˴ؤ������Ū���ͼ¸� --- .
�̳�ƻ�����ر����������칶������ʸ, 179 pp

\clearpage

\setcounter{equation}{0}
\setcounter{chapter}{1}
\renewcommand{\thechapter}{\Alph{chapter}}
\renewcommand{\theequation}{\thechapter.\arabic{equation}}

\chapter*{��ϿA \quad ��ư��������Ƴ��}

%\addcontentsline{toc}{chapter}
%{��ϿA \quad ��ư��������Ƴ��}

\markboth{��ư��������Ƴ��}
         {��ư��������Ƴ��}

����Ͽ�Ǥϱ�ư������ \Deqref{AA}, \Deqref{AB} ��Ƴ�Ф�Ԥʤ�. 
����̩�� $\rho_s$ ������̩�� $\rho_v$ ����٤ƽ�ʬ���������, 
����Ū�ʰ�����ή�Τ���������
\begin{eqnarray}
\frac{D u}{D t}
 &=& - \Dinv{\rho} \DP{p}{x} + D_u, \Deqlab{EOM1} \\
\frac{D w}{D t}
 &=& - \Dinv{\rho} \DP{p}{z} - g + D_w \Deqlab{EOM2}
\end{eqnarray}
��ɽ�����. 
â�� $\rho \equiv \rho_v + \rho_s$ ����̩�٤Ǥ���. 
���Ϸ����Ϥ� $p$ ������� $\theta$, $\Pi$ ��ɽ����, 
\begin{eqnarray}
- \Dinv{\rho} \DP{p}{x_i}
 &=&
  - \frac{\rho_v}{\rho} \Dinv{\rho_v} \DP{p}{x_i}
  \nonumber \\
 &=&
  - \frac{\rho_v}{\rho} \frac{R \theta \Pi}{p} 
    \DP{}{x_i} \left[ p_0 \Pi^{c_{pv} / R} \right]
  \nonumber \\
 &=&
  - \frac{\rho_v}{\rho} \frac{R \theta \Pi}{p} 
    \frac{p_0 c_{pv}}{R} \Pi^{(c_{pv} / R) - 1} \DP{\Pi}{x_i} 
  \nonumber \\
 &=&
  - \frac{\rho_v}{\rho} \frac{R \theta \Pi}{p} 
    p_0 \Pi^{c_{pv} / R} \frac{c_{pv}}{R \Pi} \DP{\Pi}{x_i} 
  \nonumber \\
 &=&
  - \frac{\rho_v}{\rho} \frac{R \theta \Pi}{p} 
    p \frac{c_{pv}}{R \Pi} \DP{\Pi}{x_i}
  \nonumber \\
 &=&
  - \frac{\rho_v}{\rho} c_{pv} \theta \DP{\Pi}{x_i}
\Deqlab{EOM3}
\end{eqnarray}
�Ȥʤ�. 
â�����ѷ�����������۵��Τξ��������� 
$p = \rho_v R T = \rho_v R \theta \Pi$ ���Ѥ���. 
\Deqref{EOM3}��\Deqref{EOM1}, \Deqref{EOM2}�����������, 
\begin{eqnarray}
\frac{D u}{D t}
 &=& - \frac{\rho_v}{\rho} c_{pv} \theta \DP{\Pi}{x}
     + D_u, \Deqlab{EOM4} \\
\frac{D w}{D t}
 &=& - \frac{\rho_v}{\rho} c_{pv} \theta \DP{\Pi}{z}
      - g + D_w \Deqlab{EOM5}
\end{eqnarray}
�Ȥʤ�. 

���� \Deqref{EOM4} ������������. 
���ܾ�ˤϱ���¸�ߤ��ʤ������ꤷ�Ƥ���Τ�, 
 $\overline{\rho} = \overline{\rho_v}$, 
 $\rho^{\prime} = \rho_v^{\prime} + \rho_s^{\prime}$, 
 $\rho = \overline{\rho_v} + \rho_v^{\prime} + \rho_s^{\prime}$
�Ȥʤ�. 
\Deqref{EOM4} �γ��ѿ�����ܾ���ʬ�Ⱦ�����ʬ���¤�ɽ��, ������ʬ�� 2 ��
�ʾ���̤Ͻ�ʬ��������ΤȤ���̵�뤹���, 
\begin{eqnarray}
\DP{u}{t} + u \DP{u}{x} + w \DP{u}{z}
 &=&
  - \frac{\overline{\rho_v} + \rho_v^{\prime}}
         {\overline{\rho} + \rho^{\prime}}
  c_{pv} (\overline{\theta} + \theta^{\prime})
  \DP{\Pi^{\prime}}{x} + D_{u}
   \nonumber \\
 &=&
  - \frac{\overline{\rho_v} + \rho_v^{\prime}}
         {\overline{\rho}}
  \Dinv{1 + (\rho^{\prime} / \overline{\rho})}
  c_{pv} (\overline{\theta} + \theta^{\prime})
  \DP{\Pi^{\prime}}{x} + D_{u}
   \nonumber \\
 &\simeq&
  - \frac{\overline{\rho_v} + \rho_v^{\prime}}
         {\overline{\rho}}
  \left( 1 - \frac{\rho^{\prime}}{\overline{\rho}} \right)
  c_{pv} (\overline{\theta} + \theta^{\prime})
  \DP{\Pi^{\prime}}{x} + D_{u}
   \nonumber \\
 &=&
  - \frac{\overline{\rho_v} + \rho_v^{\prime}}
         {\overline{\rho_v}}
  \left( 1 - \frac{\rho^{\prime}}{\overline{\rho}} \right)
  c_{pv} (\overline{\theta} + \theta^{\prime})
  \DP{\Pi^{\prime}}{x} + D_{u}
   \nonumber \\
 &=&
  - \left( 1 + \frac{\rho_v^{\prime}}{\overline{\rho_v}} \right)
  \left( 1 - \frac{\rho^{\prime}}{\overline{\rho}} \right)
  c_{pv} (\overline{\theta} + \theta^{\prime})
  \DP{\Pi^{\prime}}{x} + D_{u}
   \nonumber \\
 &\simeq&
  -  c_{pv} \overline{\theta} \DP{\Pi^{\prime}}{x} + D_{u}, 
 \nonumber \\
\DP{u}{t} 
 &=&
 - u \DP{u}{x} - w \DP{u}{z}
 - c_{pv} \overline{\theta} \DP{\Pi^{\prime}}{x} + D_{u} 
\Deqlab{EOM6}
\end{eqnarray}
�Ȥʤ�, \Deqref{AA} ��������. 

���� \Deqref{EOM5} ������������. 
���ܾ�ˤ������ſ尵ʿ�դ�����Ω�ĤȤ�, \Deqref{EOM5} ���
\begin{eqnarray}
0 = - c_{pv} \overline{\theta} \DP{\overline{\Pi}}{z} - g 
\Deqlab{EOM7}
\end{eqnarray}
�Ȥʤ�. 
\Deqref{EOM5} ����������, \Deqref{EOM7} ���Ѥ��ƽ񤭴������, 
\begin{eqnarray}
\DP{w}{t} + u \DP{w}{x} + w \DP{w}{z}
 &=&
  - \frac{\overline{\rho_v} + \rho_v^{\prime}}
         {\overline{\rho} + \rho^{\prime}}
  c_{pv} (\overline{\theta} + \theta^{\prime})
  \left( \DP{\overline{\Pi}}{z} + \DP{\Pi^{\prime}}{z} \right) 
  - g + D_{w}
   \nonumber \\
 &\simeq&
  - \left( 1 + \frac{\rho_v^{\prime}}{\overline{\rho_v}} \right)
  \left( 1 - \frac{\rho^{\prime}}{\overline{\rho}} \right)
  c_{pv} (\overline{\theta} + \theta^{\prime})
  \left( \DP{\overline{\Pi}}{z} + \DP{\Pi^{\prime}}{z} \right) 
  \nonumber \\
 & &
  - g + D_{w}
   \nonumber \\
 &\simeq&
  -  c_{pv} \overline{\theta} \DP{\overline{\Pi}}{z} 
  -  c_{pv} \overline{\theta} \DP{\Pi^{\prime}}{z} 
   \nonumber \\
 & &
  -  \frac{\rho_v^{\prime}}{\overline{\rho_v}} 
     c_{pv} \overline{\theta} \DP{\overline{\Pi}}{z} 
  +  \frac{\rho^{\prime}}{\overline{\rho}} 
     c_{pv} \overline{\theta} \DP{\overline{\Pi}}{z} 
  -  c_{pv} \theta^{\prime} \DP{\overline{\Pi}}{z} 
  - g + D_{w}
   \nonumber \\
 &=&
  -  c_{pv} \overline{\theta} \DP{\Pi^{\prime}}{z} 
  -  \frac{\rho_v^{\prime}}{\overline{\rho}} 
     c_{pv} \overline{\theta} \DP{\overline{\Pi}}{z} 
  +  \frac{\rho^{\prime}}{\overline{\rho}} 
     c_{pv} \overline{\theta} \DP{\overline{\Pi}}{z} 
  -  c_{pv} \theta^{\prime} \DP{\overline{\Pi}}{z} 
  + D_{w}
   \nonumber \\
 &=&
  -  c_{pv} \overline{\theta} \DP{\Pi^{\prime}}{z} 
  +  \frac{\rho_s^{\prime}}{\overline{\rho}} 
     c_{pv} \overline{\theta} \DP{\overline{\Pi}}{z} 
  -  c_{pv} \theta^{\prime} \DP{\overline{\Pi}}{z} 
  + D_{w}
   \nonumber \\
 &=&
  -  c_{pv} \overline{\theta} \DP{\Pi^{\prime}}{z} 
  -  \frac{\rho_s^{\prime}}{\overline{\rho}}  g
  +  \frac{ \theta^{\prime}}{\overline{\theta}} g
  + D_{w}
   \nonumber \\
 &=&
  -  c_{pv} \overline{\theta} \DP{\Pi^{\prime}}{z} 
  + g 
   \left(
     \frac{\theta^{\prime}}{\overline{\theta}}
     - \frac{R \overline{\theta}}{p_0 \overline{\Pi}^{c_{vv}/R}}
       \rho_s^{\prime}
   \right)
  + D_{w}, \nonumber
\end{eqnarray}
\begin{eqnarray}
\DP{w}{t} 
 &=&
  - u \DP{w}{x} - w \DP{w}{z}
  \nonumber \\
 & &
  -  c_{pv} \overline{\theta} \DP{\Pi^{\prime}}{z} 
  + g 
   \left(
     \frac{\theta^{\prime}}{\overline{\theta}}
     - \frac{R \overline{\theta}}{p_0 \overline{\Pi}^{c_{vv}/R}}
       \rho_s^{\prime}
   \right)
  + D_{w}
\Deqlab{EOM8}
\end{eqnarray}
�Ȥʤ�, \Deqref{AB} ��������. 

\clearpage
\setcounter{chapter}{2}
\setcounter{equation}{0}

\chapter*{��ϿB \quad ������������Ƴ��}

\addcontentsline{toc}{chapter}
{��ϿB \quad ������������Ƴ��}

\markboth{������������Ƴ��}
         {������������Ƴ��}

����Ͽ�Ǥϰ��������� \Deqref{AD}, \Deqref{AH} ��Ƴ�Ф�Ԥʤ�. 
������������ $\theta$, $\Pi$ ��ɽ�������, 
\begin{eqnarray}
\rho_v
 = \frac{p}{R T}
 = \frac{ p_0 \Pi^{c_{pv}/R}}{R \Pi \theta}
 = \frac{p_0}{R} \frac{\Pi^{c_{vv} / R}}{\theta}
 \Deqlab{Peq1}
\end{eqnarray}
�Ȥʤ�. 
\Deqref{Peq1} �� Lagrange ��ʬ��Ȥ��, 
\begin{eqnarray}
\frac{D \rho_v}{D t}
 &=&
  \frac{D}{D t} 
   \left( \frac{p_0}{R} \frac{\Pi^{c_{vv} / R}}{\theta} \right)
   \nonumber \\
 &=&
  \frac{p_0 c_{vv}}{R^2}
   \frac{ \Pi^{c_{vv}/R - 1}}{\theta} \frac{D \Pi}{D t}
  - \frac{p_0}{R}
    \frac{\Pi^{c_{vv}/R}}{\theta^2} \frac{D \theta}{D t}
   \nonumber \\
 &=&
  \frac{c_{vv}}{R \Pi}
  \left( \frac{p_0}{R} \frac{\Pi^{c_{vv} / R}}{\theta} \right)
  \frac{D \Pi}{D t}
  - 
  \Dinv{\theta}
  \left( \frac{p_0}{R} \frac{\Pi^{c_{vv} / R}}{\theta} \right)
  \frac{D \theta}{D t}
   \nonumber \\
 &=& 
  \frac{c_{vv} \rho_v}{R \Pi} \frac{D \Pi}{D t}
   - \frac{\rho_v}{\theta} \frac{D \theta}{D t}
 \Deqlab{Peq2}
\end{eqnarray}
������Ϣ³�μ�
\begin{equation}
\frac{D \rho_v}{D t}
 = - \rho_v \nabla \cdot \Dvect{u} - M_{cond}
 \Deqlab{Peq3}
\end{equation}
�ڤ�Ǯ�ϳؤμ�
\begin{equation}
\frac{D \theta}{D t}
 = 
  \Dinv{\Pi}
   \left( Q_{cond} + Q_{dis} + Q_{rad} \right)
%   \left( \frac{L M_{cond}}{c_{pv} \rho} + Q_{dis} + Q_{rad} \right)
 \Deqlab{Peq4}
\end{equation}
�� \Deqref{Peq2} �����������, 
\begin{eqnarray}
\frac{D \Pi}{D t}
 &=&
  \frac{R \Pi}{c_{vv}}
  \left( 
   \Dinv{\rho_v} \frac{D \rho_v}{D t} 
   + \Dinv{\theta} \frac{D \theta}{D t}
  \right)
  \nonumber \\
 &=&
  \frac{R \Pi}{c_{vv}}
  \left[
   - \nabla \cdot \Dvect{u}
   - \Dinv{\rho_v} M_{cond}
   + \Dinv{\theta \Pi}
    \left( Q_{cond} + Q_{dis} + Q_{rad} \right)
  \right]
  \nonumber \\
 &=&
  \frac{c^2}{c_{pv} \theta}
  \left[
   - \nabla \cdot \Dvect{u}
   - \Dinv{\rho_v} M_{cond}
   + \Dinv{\theta \Pi}
    \left( Q_{cond} + Q_{dis} + Q_{rad} \right)
  \right]
  \nonumber \\
 &=&
   - \frac{c^2}{c_{pv} \theta} \nabla \cdot \Dvect{u}
   - \frac{c^2}{c_{pv} \rho_v \theta} M_{cond}
   + \frac{c^2 }{c_{pv} \theta^2 \Pi} 
     \left( Q_{cond} + Q_{dis} + Q_{rad} \right)
 \Deqlab{Peq5}
\end{eqnarray}
�Ȥʤ�. 
â��Ǯ�ϳؤμ��ˤ�����, ��ή�Ȼ����¾�ι����٤ƽ�ʬ�������Ȥ���̵�뤷
��. 
�ޤ� $c$ �ϲ�®�Ǥ���, 
\begin{equation}
c^2 
 = \frac{c_{pv}}{c_{vv}} R \theta \Pi 
 \Deqlab{Peq6}
\end{equation}
�Ǥ���. 
$M_{cond}$, $Q_{*}$ ��������ʬ�Ǥ��뤳�Ȥ����դ���, \Deqref{Peq5} ������
����Ԥʤ���, 
\begin{eqnarray}
\DP{\Pi^{\prime}}{t}
 &=&
   - w^{\prime} \DP{\overline{\Pi}}{z}
   - \frac{\overline{c^2}}{c_{pv} \overline{\theta}} 
     \nabla \cdot \Dvect{u}^{\prime}
   - \frac{\overline{c^2}}{c_{pv} \overline{\rho} \overline{\theta}} 
     M_{cond}
    \nonumber \\
 & &
   + \frac{\overline{c^2} }{c_{pv} \overline{\theta}^2 \overline{\Pi}} 
     \left( Q_{cond + }Q_{dis} + Q_{rad} \right)
 \Deqlab{Peq7}
\end{eqnarray}
�Ȥʤ�. 
������
\begin{eqnarray}
\overline{\rho} \overline{\theta}
 &=& 
  \frac{p_0}{R} \overline{\Pi}^{c_{vv}/R}
   \Deqlab{Peq8} \\
\overline{c^2}
 &=& 
  \frac{c_{pv}}{c_{vv}} R \overline{\theta} \overline{\Pi}
   \Deqlab{Peq9}
\end{eqnarray}
�Ȥʤ뤳�Ȥ����ܤ����, 
\begin{eqnarray}
\DP{}{z} \left( \overline{\rho} \overline{\theta} \right)
 &=&
  \DP{}{z}
   \left( \frac{p_0}{R} \overline{\Pi}^{c_{vv}/R} \right)
    \nonumber \\
 &=&
  \frac{c_{vv}}{R} \frac{p_0}{R} \overline{\Pi}^{c_{vv}/R - 1}
   \DP{\overline{\Pi}}{z}
    \nonumber \\
 &=&
  \frac{c_{vv}}{R} 
  \frac{\overline{\rho} \overline{\theta}}{\overline{\Pi}}
   \DP{\overline{\Pi}}{z}
    \nonumber \\
 &=&
  \frac{c_{pv} \overline{\rho} \overline{\theta}^2}{\overline{c^2}}
   \DP{\overline{\Pi}}{z}
   \Deqlab{Peq10}
\end{eqnarray}
�Ȥʤ�. 
\Deqref{Peq10} ���
\begin{eqnarray}
- w^{\prime} \DP{\overline{\Pi}}{z}
 &=& 
    - \frac{\overline{c^2}}{c_{pv} \overline{\rho} \overline{\theta}^2}
    w^{\prime}
     \DP{}{z} \left( \overline{\rho} \overline{\theta} \right)
  \nonumber \\
 &=&
    - \frac{\overline{c^2}}{c_{pv} \overline{\rho} \overline{\theta}^2}
    \Dvect{u}^{\prime} \cdot \nabla
     \left( \overline{\rho} \overline{\theta} \right)
  \Deqlab{Peq11}
\end{eqnarray}
�Ȥʤ�. 
\Deqref{Peq11} �� \Deqref{Peq7} �����������, 
\begin{eqnarray}
\DP{\Pi^{\prime}}{t}
 &=&
    - \frac{\overline{c^2}}{c_{pv} \overline{\rho} \overline{\theta}^2}
    \Dvect{u}^{\prime} \cdot \nabla
     \left( \overline{\rho} \overline{\theta} \right)
   - \frac{\overline{c^2}}{c_{pv} \overline{\theta}} 
     \nabla \cdot \Dvect{u}^{\prime}
    \nonumber \\
 & &
   + \frac{\overline{c^2} }{c_{pv}^2 \overline{\theta}^2 \overline{\Pi}} 
     \left( Q_{cond} + Q_{dis} + Q_{rad} \right)
   - \frac{\overline{c^2}}{c_{pv} \overline{\rho} \overline{\theta}} 
     M_{cond}
    \nonumber \\
 &=&
    - \frac{\overline{c^2}}{c_{pv} \overline{\rho} \overline{\theta}^2}
    \left[ 
     \Dvect{u}^{\prime} \cdot \nabla 
      \left( \overline{\rho} \overline{\theta} \right)
     + \overline{\rho} \overline{\theta} 
      \nabla \cdot \Dvect{u}^{\prime}
    \right]
    \nonumber \\
 & &
   + \frac{\overline{c^2} }{c_{pv}^2 \overline{\theta}^2 \overline{\Pi}} 
     \left( Q_{cond} + Q_{dis} + Q_{rad} \right)
   - \frac{\overline{c^2}}{c_{pv} \overline{\rho} \overline{\theta}} 
     M_{cond}
    \nonumber \\
 &=&
   - \frac{\overline{c^2}}{c_{pv} \overline{\rho} \overline{\theta}^2}
     \nabla \cdot
     \left( \overline{\rho} \overline{\theta} \Dvect{u}^{\prime} \right)
   + \frac{\overline{c^2} }{c_{pv} \overline{\theta}^2 \overline{\Pi}} 
     \left( Q_{cond} + Q_{dis} + Q_{rad} \right)
   - \frac{\overline{c^2}}{c_{pv} \overline{\rho} \overline{\theta}} 
     M_{cond}
 \nonumber \\
 \Deqlab{Peq12}
\end{eqnarray}
�Ȥʤ�, \Deqref{AD} ��������. 
���� \Deqref{Peq12} ��
\begin{equation}
Q_{cond} = \frac{L M_{cond}}{c_{pv} \overline{\rho}} 
 \Deqlab{Peq13}
\end{equation}
���������� $Q_{cond}$ ��õ���, 
\begin{eqnarray}
\DP{\Pi^{\prime}}{t}
 &=&
   - \frac{\overline{c^2}}{c_{pv} \overline{\rho} \overline{\theta}^2}
     \nabla \cdot
     \left( \overline{\rho} \overline{\theta} \Dvect{u}^{\prime} \right)
   + \frac{\overline{c^2} }{c_{pv} \overline{\theta}^2 \overline{\Pi}} 
     \left( 
      \frac{L M_{cond}}{c_{pv} \overline{\rho}}  + Q_{dis} + Q_{rad} 
     \right)
 \nonumber \\
 & &
   - \frac{\overline{c^2}}{c_{pv} \overline{\rho} \overline{\theta}} 
     M_{cond}
 \nonumber \\
 &=&
   - \frac{\overline{c^2}}{c_{pv} \overline{\rho} \overline{\theta}^2}
     \nabla \cdot
     \left( \overline{\rho} \overline{\theta} \Dvect{u}^{\prime} \right)
   + \frac{\overline{c^2} }{c_{pv} \overline{\theta}^2 \overline{\Pi}} 
     \cdot \frac{L M_{cond}}{c_{pv} \overline{\rho}}
   - \frac{\overline{c^2}}{c_{pv} \overline{\rho} \overline{\theta}} 
     M_{cond}
 \nonumber \\
 & &
   + \frac{\overline{c^2} }{c_{pv} \overline{\theta}^2 \overline{\Pi}}  
     \left( Q_{dis} + Q_{rad} \right)
 \nonumber \\
 &=&
   - \frac{\overline{c^2}}{c_{pv} \overline{\rho} \overline{\theta}^2}
     \nabla \cdot
     \left( \overline{\rho} \overline{\theta} \Dvect{u}^{\prime} \right)
   + \frac{\overline{c^2} }{c_{pv} \overline{\rho} \overline{\theta}} 
     \left( 
      \frac{L}{c_{pv} \overline{\theta} \overline{\Pi}} - 1
     \right)  M_{cond}
 \nonumber \\
 & &
   + \frac{\overline{c^2} }{c_{pv} \overline{\theta}^2 \overline{\Pi}}  
     \left( Q_{dis} + Q_{rad} \right)
 \nonumber \\
 &=&
   - \frac{\overline{c^2}}{c_{pv} \overline{\rho} \overline{\theta}^2}
     \nabla \cdot
     \left( \overline{\rho} \overline{\theta} \Dvect{u}^{\prime} \right)
   + \frac{\overline{c^2} }{c_{pv} \overline{\rho} \overline{\theta}} 
     \left( \frac{L}{c_{pv} \overline{T}} - 1 \right)  M_{cond}
 \nonumber \\
 & &
   + \frac{\overline{c^2} }{c_{pv} \overline{\theta}^2 \overline{\Pi} } 
     \left( Q_{dis} + Q_{rad} \right)
 \Deqlab{Peq14}
\end{eqnarray}
�Ȥʤ�, \Deqref{AH} ��������. 

\clearpage
\setcounter{chapter}{3}
\setcounter{equation}{0}

\chapter*{��ϿC \quad ����ʬ�ŷ�Ϥˤ�������ή�ѥ�᥿�ꥼ�������}

\addcontentsline{toc}{chapter}
{��ϿC \quad ����ʬ�ŷ�Ϥˤ�������ή�ѥ�᥿�ꥼ�������}

\markboth{��ή�ѥ�᥿�ꥼ�������}
         {��ή�ѥ�᥿�ꥼ�������}

\section*{��ή�ѥ�᥿�ꥼ�������}

Klemp and Wilhelmson (1978) ����� CReSS ���Ѥ����Ƥ��� 1.5 ���Υ�����
���㡼���Ѥ���. ���ΤȤ���ή��ư���ͥ륮���λ���ȯŸ��������,
%
\begin{eqnarray}
 \DLag{E_{turb}}{t} &=& B + S + D_{E}
  - \left(\frac{C_{\varepsilon}}{l}\right)
    E_{turb}^{\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_v^{\prime}} ,
 \\
 S &=&
  - \overline{(u_{i}^{\prime} u_{j}^{\prime})}
  \DP{u_{i}}{x_{j}} ,
 \\
 D_{E} &=& \DP{}{x_{j}} \left(K_{m} \DP{E_{turb}}{x_{j}} \right)
\end{eqnarray}
�Ǥ���.
����ʬ�ŷ�Ϥˤ����Ƥ�
\begin{eqnarray}
\theta_v^{\prime}
 &=&
  \theta^{\prime}
  - \frac{\overline{\theta}}{\overline{\rho}} \rho_s^{\prime}
  \nonumber \\
 &=&
  \theta^{\prime}
  - \frac{R \overline{\theta}^2}{p_0 \overline{\Pi}^{c_v/R}}
    \rho_s^{\prime}
  \Deqlab{BuoyTurb101}
\end{eqnarray}
�Ǥ���Τ�,
\begin{eqnarray}
B
 &=&
  \frac{g_{j}}{\overline{\theta}}
  \overline{u_{j}^{\prime} \theta_v^{\prime}}
  \nonumber \\
 &=&
  \frac{g_{j}}{\overline{\theta}}
  \overline{
   u_{j}^{\prime}
   \left(
    \theta^{\prime}
    - \frac{R \overline{\theta}^2}{p_0 \overline{\Pi}^{c_v/R}}
      \rho_s^{\prime}
   \right)}
  \nonumber \\
 &=&
  \frac{g_{j}}{\overline{\theta}}
   \overline{ u_{j}^{\prime} \theta^{\prime}}
  - g_{j} \frac{R \overline{\theta}}{p_0 \overline{\Pi}^{c_v/R}}
   \overline{ u_{j}^{\prime} \rho_s }
  \nonumber \\
  \Deqlab{BuoyTurb2}
\end{eqnarray}
��ɽ�����.
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_{turb},
  \Deqlab{�쥤�Υ륺����1}
\\
  \overline{u_{j}^{\prime} \phi^{\prime} }
  &=& - K_{h} \DP{\phi}{x_{j}}.
  \Deqlab{�쥤�Υ륺����2}
\end{eqnarray}
%
������ $\phi^{\prime}$ ��Ǥ�դΥ����顼��ξ�����ʬ,
$K_{m}$ �ϱ�ư�̤��Ф��뱲Ǵ�������Ǥ���,
$E_{turb}$ �ϥ��֥���åɥ����������ή��ư���ͥ륮��,
$K_{h}$ �ϱ��Ȼ������Ǥ���.
$K_{m}$, $K_{h}$ �� $E_{turb}$ ���Ѥ��ưʲ��Τ褦��Ϳ������.
%
 \begin{eqnarray}
  K_{m} &=& C_{m} E_{turb}^{\frac{1}{2}} l,
   \Deqlab{B:E}\\
  K_{h} &=& 3 K_{m}.
 \end{eqnarray}
%
�ѥ�᡼�� $C_{\varepsilon}, C_{m}$ �ϤȤ�� 0.2 �Ǥ���.

\subsection*{2 �����ξ���ɽ��}

2 �����ξ���\Deqref{B:dEdt}���γƹ��񤭲���.
���Ϥˤ����ή���ͥ륮���������,
%
\begin{eqnarray}
 B
  &=&
  \frac{g_{j}}{\overline{\theta}}
   \overline{ u_{j}^{\prime} \theta^{\prime}}
  - g_{j} \frac{R \overline{\theta}}{p_0 \overline{\Pi}^{c_v/R}}
   \overline{ u_{j}^{\prime} \rho_s }
  \nonumber \\
  &=&
  \frac{g}{\overline{\theta}}
  \overline{w^{\prime}  \theta^{\prime}}
  - g \frac{R \overline{\theta}}{p_0 \overline{\Pi}^{c_v/R}}
   \overline{ w^{\prime} \rho_s }
  \nonumber \\
  &=&
  - \frac{g}{\overline{\theta}}
  \left( K_{h} \DP{\theta}{z} \right)
  + g \frac{R \overline{\theta}}{p_0 \overline{\Pi}^{c_v/R}}
  \left( K_{h} \DP{\rho_s}{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_{turb}
    \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_{turb}
   \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_{turb}
   \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_{turb}
   \right\}
  \DP{w}{x_{j}}
  \nonumber \\
  &=&
   \left\{
      2 K_{m} \left(\DP{u}{x} \right)
          - \frac{2}{3} E_{turb}
   \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_{turb}
   \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_{turb} \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_{turb}}{x_{j}} \right),
  \nonumber \\
  &=& \DP{}{x} \left(K_{m} \DP{E_{turb}}{x} \right)
      + \DP{}{z} \left(K_{m} \DP{E_{turb}}{z} \right)
\Deqlab{De}
\end{eqnarray}
%
�Ǥ���.
�ʾ�� \Deqref{B}, \Deqref{S}, \Deqref{De} ���� \Deqref{B:dEdt} ��
���������뤳�Ȥǰʲ��μ�������.
%
 \begin{eqnarray}
  \DLag{E_{turb}}{t}
  &=&
   - \frac{g}{\overline{\theta}}
      \left( K_{h} \DP{\theta}{z} \right)
   + g \frac{R \overline{\theta}}{p_0 \overline{\Pi}^{c_v/R}}
   \left( K_{h} \DP{\rho_s}{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_{turb} \left( \DP{u}{x} + \DP{w}{z} \right)
\nonumber \\
  &&+ \DP{}{x} \left(K_{m} \DP{E_{turb}}{x} \right)
      + \DP{}{z} \left(K_{m} \DP{E_{turb}}{z} \right)
   - \left(\frac{C_{\varepsilon}}{l}\right)
          E_{turb}^{\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�
����. ������ 6, 7 �����ή���ͥ륮���Ȼ����񤭲�����,
%
 \begin{eqnarray}
\DP{}{x} \left(K_{m} \DP{E_{turb}}{x}\right)
       &+& \DP{}{z} \left(K_{m} \DP{E_{turb}}{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}}  \DLag{K_{m}}{t}
  &=&
   - \frac{g}{\overline{\theta}}
      \left( K_{h} \DP{\theta}{z} \right)
   + g \frac{R \overline{\theta}}{p_0 \overline{\Pi}^{c_v/R}}
   \left( K_{h} \DP{\rho_s}{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 } \frac{K_{h}}{K_{m}}
      \left(
       \Dinv{\overline{\theta}} \DP{\theta}{z}
       - \frac{R \overline{\theta}}{p_0 \overline{\Pi}^{c_v/R}}
         \DP{\rho_s}{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 }
      \left(
       \Dinv{\overline{\theta}} \DP{\theta}{z}
       - \frac{R \overline{\theta}}{p_0 \overline{\Pi}^{c_v/R}}
         \DP{\rho_s}{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}
%
��������.



\clearpage
\setcounter{chapter}{4}
\setcounter{equation}{0}

\chapter*{��ϿD \quad \Deqref{CloudB} ��Ƴ��}

\addcontentsline{toc}{chapter}
{��ϿD \quad \Deqref{CloudB} ��Ƴ��}

\markboth{\Deqref{CloudB} ��Ƴ��}
         {\Deqref{CloudB} ��Ƴ��}

����$m$���ı�γ���ޤ����絤�γȻ��ˤ�ä���Ĺ���������ͤ���. 
�Ȼ��ˤ���γ����Ĺ��
\begin{eqnarray}
    \DD{m}{t} = 4 \pi r_d^2 D \DP{\rho}{r}
\end{eqnarray}
��ɽ�����. $r$�ϱ�γ���濴����ε�Υ, 
$D$���絤��ʬ�ҳȻ������Ǥ���. 
�絤�ξ��֤���狼�������Ǥ���Ȳ��ꤷ�Ƥ��μ���$r$�ˤĤ��Ʋ򤯤�
\begin{eqnarray}
    \DD{m}{t} = - 4 \pi r_d D (\rho_{r_d} - \rho_{\infty})
\end{eqnarray}
�Ȥʤ�. ������, �������Ȥ���$r=r_d$��$\rho = \rho_{r_d}$, 
$r\rightarrow \infty$��$\rho \rightarrow \rho_{\infty}$���Ѥ���. 
���������۵��Τξ���������$p = \rho R T$�����������
\begin{eqnarray}
    \DD{m}{t} = - 4 \pi r_d D 
               \left(\frac{p_*(T_{r_d})}{RT_{r_d}}
                - \frac{p_{\infty}}{RT_{\infty}}\right)
\end{eqnarray}
�Ȥʤ�. ���μ���ξ�դ�$p*(T_{\infty})$�ǳ���
\begin{eqnarray}
    \frac{1}{p_*(T_{\infty})} \DD{m}{t}
  &=& - \frac{4 \pi r_d D}{R} \left(\frac{p_*(T_{r_d})}{p_*(T_{r_d})T_{r_d}}
          - \frac{p_{\infty}}{p_*(T_{r_d})T_{\infty}}\right)
  \nonumber \\
  &=& - \frac{4 \pi r_d D}{RT_{\infty}} \left(\frac{p_*(T_{r_d})}{p_*(T_{\infty})}
          - \frac{p_{\infty}}{p_*(T_{\infty})}\right)
    \Deqlab{��γ���̤λ����Ѳ�}
\end{eqnarray}
�Ȥʤ�. ������, �Ǹ���ѷ��ˤ�$(T_{r_d}-T_{\infty})/T_{\infty} \ll 1$���Ѥ���.

%�絤�γȻ��ϳȻ�������
%\begin{eqnarray}
%    \DP{\rho}{t} = D \Dlapla \rho
%\end{eqnarray}
%�ˤ�ä�ɽ�����. ��������狼�������ʾ��֤��ꤹ���
%\begin{eqnarray}
%    D \frac{1}{r^2} \DP{}{r} \left(r^2  \DP{\rho}{r} \right) = 0
%\end{eqnarray}
%�Ȥʤ�. 

�ŷ뤬����������Ǯ�����������. 
������Ǯ��Ǯ��Ƴ�ˤ�ä�͢�������Ȳ��ꤹ���
\begin{eqnarray}
    L \DD{m}{t} = - 4 \pi r^2 k \DD{T}{r}
\end{eqnarray}
������Ω��. ������$k$���絤��Ǯ�Ȼ������Ǥ���. 
�絤̩�٤γȻ���������Ʊ�ͤˤ��μ���򤯤�
\begin{eqnarray}
    L \DD{m}{t} = 4 \pi r k (T_{r_d} - T_{\infty})
    \Deqlab{��Ǯ�λ����Ѳ�}
\end{eqnarray}
�Ȥʤ�. ������, �������Ȥ���$r=r_d$��$T = T_{r_d}$, 
$r\rightarrow \infty$��$T \rightarrow T_{\infty}$���Ѥ���. 
�����ǥ��饦������-����ڥ�����μ�
\begin{eqnarray}
    \DD{p_*}{T} = \frac{L p_*}{R T^2}
    \nonumber
\end{eqnarray}
����ʬ�����
\begin{eqnarray}
    \ln \frac{p_*(T_{r_d})}{p_*(T_{\infty})}
    &=& \frac{L}{R}\frac{T_{r_d} - T_{\infty}}{T_{r_d}T_{\infty}}
    \nonumber \\
    &\sim& \frac{L}{R}\frac{T_{r_d} - T_{\infty}}{T_{\infty}^2}
\end{eqnarray}
�Ȥʤ�. ��ä�
\begin{eqnarray}
    \frac{p_*(T_{r_d})}{p_*(T_{\infty})}
    &=& \exp\left(\frac{L}{R}\frac{T_{r_d} - T_{\infty}}{T_{r_d}T_{\infty}}\right)
    \nonumber \\
    &\sim& 1 + \frac{L}{R}\frac{T_{r_d} - T_{\infty}}{T_{\infty}^2}
    \nonumber \\
    &\sim& 1 + \frac{L^2}{4 \pi r_d k R T_{\infty}^2} \DD{m}{t}
    \Deqlab{˰�¾�������μ�}
\end{eqnarray}
�Ȥʤ�. �Ǹ���ѷ��ˤϼ�\Deqref{��Ǯ�λ����Ѳ�}���Ѥ���. 

��\Deqref{��γ���̤λ����Ѳ�}, \Deqref{˰�¾�������μ�}���
\begin{eqnarray}
    \DD{m}{t}
  &=& - \frac{4 \pi r D p_*(T_{\infty})}{RT_{\infty}} 
          \left(1 + \frac{L^2}{4 \pi r_d k R T_{\infty}^2} \DD{m}{t}
          - \frac{p_{\infty}}{p_*(T_{\infty})}\right)
\end{eqnarray}
�Ȥʤ�. ���μ������������
\begin{eqnarray}
    \left( \frac{RT_{\infty}}{D p_*(T_{\infty})}
  + \frac{L^2}{k R T_{\infty}^2} \right) \DD{m}{t}
  = 4 \pi r_d \left(\frac{p_{\infty}}{p_*(T_{\infty})} - 1 \right)
\end{eqnarray}
�Ȥʤ�. ������
\begin{eqnarray}
    R_m &=& \frac{RT_{\infty}}{D p_*(T_{\infty})}
    \nonumber \\
    R_h &=& \frac{L^2}{k R T_{\infty}^2}
    \nonumber \\
    S &=& \frac{p_{\infty}}{p_*(T_{\infty})} 
    \nonumber 
\end{eqnarray}
�Ȥ�����
\begin{eqnarray}
  \DD{m}{t} = \frac{4 \pi r_d}{R_m + R_h} (S - 1)
\end{eqnarray}
�Ȥʤ�. $R_m$, $R_h$�Ϥ��줾�����͢��, Ǯ͢���˴ط����뷸���Ǥ���. 
$S$��˰����Ǥ���. 

ñ������������α�γ�θĿ���$N$�Ȥ�, ��γ���礭��������Ʊ���Ǥ���Ȳ���
�����ñ������������ζŷ���$M_{cond}$��
\begin{eqnarray}
    M_{cond} = N \DD{m}{t} = \frac{4 \pi r_d N}{R_m + R_h} (S - 1)
\end{eqnarray}
��Ϳ������. 
�����˴�����Ǥβ��١����Ͼ������ꤹ���, $R_m \ll R_h$ �Ǥ���. 
���ä�
\begin{eqnarray}
    M_{cond} \sim \frac{4 \pi r_d N}{R_h} (S - 1)
\end{eqnarray}
�Ȥʤ�. 

\clearpage
\setcounter{chapter}{5}
\setcounter{equation}{0}

\chapter*{��ϿE \quad ���ؤ�˰�������, ���ؤ������Ǥ�����ܾ첹��ʬ�ۤ�
�Ĥ���}

\addcontentsline{toc}{chapter}
{��ϿE \quad ���ؤ�˰�������, ���ؤ������ʴ��ܾ첹��ʬ��}

\markboth{���ؤ�˰�������, ���ؤ������ʴ��ܾ첹��ʬ��}
         {���ؤ�˰�������, ���ؤ������ʴ��ܾ첹��ʬ��}

Colaprete {\it et al}.(2003) �ο��ͼ¸��ǤϽ����˰���椬����Ǥ��벹��
ʬ�ۤ�Ϳ���Ƥ���. 
�����Ǥϲ��ؤ�˰�������, ���ؤ������Ǥ�����ܾ첹��ʬ�� $\overline{T}$
��Ϳ�������ͤ���. 
���ؤǤβ��� $T_{iso}$, ���ؤǤ�˰������� $S_0$, ��ɽ�̰��� $p_s$ �ϴ�
�ΤǤ���Ȥ���. 
�褺˰���������ؤǤβ���ʬ�� $T_{S_0}(z)$ �����. 
���ܾ�ˤ����Ƥ��ſ尵ʿ�դ�����Ω�ĤΤ�, 
\begin{eqnarray}
\DD{p}{z} = - \rho g
 \Deqlab{Tprof1} 
\end{eqnarray}
�Ȥʤ�. 
˰�¾������� $p_{*}$ �Ȥ����, 
\begin{eqnarray}
p = S_0 p_{*}
 \Deqlab{Tprof2} 
\end{eqnarray}
������Ω��. 
�� \Deqref{Tprof2} �� \Deqref{Tprof1} �����������, 
\begin{eqnarray}
\DD{p}{z}
 = S_0 \DD{p_{*}}{z}
 = S_0 \DD{p_{*}}{T} \DD{T}{z}
 = - \rho g
 \Deqlab{Tprof3} 
\end{eqnarray}
�Ȥʤ�. 
�����ǥ��饦������������ڥ�����μ�
\begin{eqnarray}
\DD{p_{*}}{T} = \frac{L p_{*}}{R T^2}
 \Deqlab{Tprof4} 
\end{eqnarray}
�ڤ����۵��Τξ��������� $p = \rho R T$ ���Ѥ����, 
\begin{eqnarray}
S_0 \frac{L p_{*}}{R T^2} \DD{T}{z}
 &=& - \frac{S_0 p_{*}}{R T} g, 
  \nonumber \\
\DD{T}{z}
 &=& - \frac{g}{L} T
 \Deqlab{Tprof5} 
\end{eqnarray}
�Ȥʤ�. 
$z=0$ �Ǥβ��٤� $T_0$ �Ȥ�, �� \Deqref{Tprof5} �� $z=0$ ���� $z$ �ޤ���
ʬ�����, 
\begin{eqnarray}
T_{S_0}(z) = T_0 \exp \left( - \frac{g}{L} z \right)
 \Deqlab{Tprof6} 
\end{eqnarray}
�Ȥʤ�. 
Antoine �μ�
\begin{eqnarray}
p_{*} = \exp \left(A_{ant} - \frac{B_{ant}}{T} \right)
 \Deqlab{Tprof7} 
\end{eqnarray}
�ˤ����� $z=0$ �ξ���ͤ����, 
\begin{eqnarray}
\frac{p_s}{S_0} 
 &=& \exp \left(A_{ant} - \frac{B_{ant}}{T_0} \right), 
  \nonumber \\
T_0
 &=& \frac{B_{ant}}{A_{ant} - \ln (p_s / S_0)}
 \Deqlab{Tprof8} 
\end{eqnarray}
�Ȥʤ�. 
�� \Deqref{Tprof8} �� \Deqref{Tprof7} �����������, 
\begin{eqnarray}
T_{S_0} (z)
 =
  \frac{B_{ant}}{A_{ant} - \ln (p_s / S_0)}
  \exp \left( - \frac{g}{L} z \right)
 \Deqlab{Tprof9}
\end{eqnarray}
�Ȥʤ�. 
���˲��ٰ����ʬ�� $T=T_{iso}$ ��˰��������ʬ�� $T=T_{S_0}$ �������
�٤� $z_{T}$ �Ȥ����, �� \Deqref{Tprof9} ���
\begin{eqnarray}
T_{iso}
 &=&
  \frac{B_{ant}}{A_{ant} - \ln (p_s / S_0)}
  \exp \left( - \frac{g}{L} z_{T} \right), 
  \nonumber \\
z_T
 &=&
  - \frac{L}{g} 
  \ln \left[ \frac{A - \ln (p_s/S_0)}{B} T_{iso} \right]
 \Deqlab{Tprof10}
\end{eqnarray}
�Ȥʤ�. 
�ʾ���, 
\begin{equation}
\overline{T}(z) 
 = 
 \begin{cases}
  \frac{B_{ant}}{A_{ant} - \ln (p_s / S_0)}
  \exp \left( - \frac{g}{L} z \right), 
  \hspace{5mm} (z \leq z_T) \\
  T_{iso}
  \hspace{43mm} (z > z_T)
 \end{cases}
 \Deqlab{Tprof11}
\end{equation}
�Ȥʤ�. 
â������ʬ�ۤ� \Deqref{Tprof11} ��Ϳ������, $z=z_{T}$ �˲��ٸ��ۤ���
Ϣ³��¸�ߤ�, �ɽ�Ū���礭�ʲ��٤γȻ��������Ƥ��ޤ����Ȥˤʤ�. 
�����Ǽºݤο��ͷ׻��Ǥ�, �ʲ��Τ褦�˽Ťߴؿ� $W(z)$ ��Ƴ����, ���ٰ���
��ʬ�ۤ�˰�������β���ʬ�ۤ�ʤ�餫�ˤĤʤ�����Τ�Ϳ���Ƥ���. 
\begin{eqnarray}
\overline{T}(z)
 &=&
  [1 - W(z)]
  \frac{B_{ant}}{A_{ant} - \ln (p_s / S_0)}
  \exp \left( - \frac{g}{L} z \right)
  + W(z) T_{iso}, 
 \Deqlab{Tprof12} \\
W(z)
 &=&
  \Dinv{2} 
  \left[ \tanh \left( \frac{z - z_T}{\Delta h} \right) + 1 \right]. 
 \Deqlab{Tprof13}
\end{eqnarray}
â�� $\Delta h$ �ϲ���ʬ�ۤ����ܤ���ħŪ��Ĺ����������Ǥ���. 

\end{document}