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% Chapter 1  The Governing Equations
%            1-5 FIRST LAW OF THERMODYNAMICS
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% 2002/05/29 ���� ����    ����
% 2002/07/04 ���� ����    ����, ��dI/dt = c dT/dt�� �˴ؤ����ɵ�
%                         (2-10)����Ƴ�Ф���­
% 2002/07/27 ���� ����    �ޥ��䡼�δط����θ�������
% 2002/08/03 ���� ����    ���̤˴ؤ��ƽ���
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\section{Ǯ�ϳ����ˡ§}

Ǯ�ϳ����ˡ§�ϥ��ͥ륮����¸§�ΰ�ĤǤ���.
���ͥ륮����¸§�Ȥ�,
�֤���ϤΥ��ͥ륮�����Ѳ����̤Ϥ��ηϤζ�����������ꤹ�륨�ͥ륮����
�̤��������פȤ������Ȥ��̣����.
����Ū��, ����ˡ§�ϰ��֥��ͥ륮��, ��ư���ͥ륮��, Ǯ���ͥ륮��,
���ͥ��ͥ륮��, �������ͥ륮��, �ŵ����ͥ륮��, ���إ��ͥ륮���ʤ�
�Ȥ��ä����Ƥμ���Υ��ͥ륮���˴ؤ�������Ω��.
������, �����ǵ���Ǯ�ϳؤ�ͤ�����ˤ�
����ñ�㲽�����ط������Ѥ���.
��ǾҲ𤹤뤬, �˥塼�ȥ�α�ư��������
�ϳإ��ͥ륮���ΥХ�󥹤�ɽ�����Ƥ���.
�������ä�,
��ư���ͥ륮���Ƚ��Ͼ�ˤ����֥��ͥ륮�����¤�
���Ϥ��໤�Ϥˤ��Ż���������.
���Τ��Ȥ��餳��������ι��Ǯ�ϳ����ˡ§�����������.
���Ȥ�, ���Τΰ��Ϥ��໤�Ϥˤ��Ż����ĤäƤ��뤬,
����ϵ��Τ���ĥ���ѷ��Ȥ��ư���.

����, ����, �������ŵ����ͥ륮���������Ǥ���ʤ��,
Ǯ�ϳ����ˡ§�ϰʲ��δ�ñ�ʷ��Ǥ���魯���Ȥ��Ǥ���.
%
\begin{eqnarray*}
 Q=\DD{I}{t}+W.
\end{eqnarray*}
%
������$Q$ [J kg$^{-1}$ s$^{-1}$]��ñ�̻��֤�ñ�����Ѥ�
�ä���줿Ǯ���ͥ륮���򤢤�路,
$W$ [J kg$^{-1}$ s$^{-1}$]��ñ�̻��֤�ñ�����Ѥ�
���Τ���ĥ(�ޤ��ϼ���)���뤳�ȤǤ��μ��Ϥ˵ڤܤ��Ż��򤢤�路,
$dI/dt$ [J kg$^{-1}$ s$^{-1}$]�Ϥ���ʪ�Τ�ñ������ʬ���������ͥ륮����
�����Ѳ��򤢤�魯.
���۵��Τ�ͤ������,
ñ�����Ѥ�������������ͥ륮���λ����Ѳ���$c_v dT/dt$ [J kg$^{-1}$ s$^{-1}$]
��Ϳ������
\footnote{
$dI/dt = c_v dT/dt$��Ƴ�Ф˴ؤ��Ƥ�{\bf 2-5-B}����.
}.
������$c_v$ [J kg$^{-1}$ K$^{-1}$] ��������Ǯ�Ǥ���
\footnote{
������Ǯ�Ȥ�
�����Ѱ���ξ���, 1kg��ʪ����1K�����夲��Τ�ɬ�פ�(Ǯ)���ͥ륮����
�Τ��ȤǤ���.
}.
����, ��Ǵ�����Τβĵ���ĥ�ˤ��ñ�̼���, ñ�̻��֤ǤλŻ���
$p \ d\alpha/dt$ [J kg$^{-1}$ s$^{-1}$]
��Ϳ������
\footnote{
$p$ [N/m$^2$], $\alpha$ [m$^3$/kg]��� $p \ d\alpha/dt$ ��ñ�̤�
[N m$^3$/m$^2$ kg t]��[N m/kg t]��[J/kg t] �Ȥʤ�.
}.
������ɽ����Ǯ�ϳ�����������������Ȱʲ��μ���������.
$$
 Q=c_v \DD{T}{t}+p\DD{\alpha}{t}.
 \eqno (2\textrm{-}9a)
$$

(2-8)����Ȥ��Ȱʲ��Τ褦�ʷ��Ǥ�񤯤��Ȥ��Ǥ���
\footnote{
{\bf 2-5-A}����.
}.
$$
 Q=c_p\DD{T}{t}-\alpha \DD{p}{t}=c_p T\DD{(\ln \theta)}{t}.
 \eqno (2\textrm{-}9b)
$$
\setcounter{equation}{9}
������
\begin{eqnarray*}
 \theta \equiv T(p_0/p)^{\kappa}, \quad \kappa \equiv R/c_p, 
 \quad c_p - c_v = R, \quad p_0 = 1000 {\rm mb}
\end{eqnarray*}
%
�Ǥ���. $c_p$ [J kg$^{-1}$ K$^{-1}$]���갵��Ǯ�Ǥ���,
$p_0$ [N/m$^2$]���Ͼ�Ǥε����Ǥ���
\footnote{
$\theta$ [K]��{\bf ����}({\itshape potential temperature})
�ȸƤФ���̤���, ��ʸ�ǤϿ�����Ƥ��ʤ�.
���̤�1��9��ؤ��Ȥ��䤹����ľ��ɸ�ϡ٤ˤ��о줹��.
}.

��ξ����Ѳ���ͤ������(�㤨�о���������Τ��Ѳ�������)
�ˤϼ�(2-9b)�ϰʲ��Τ褦�ʷ���Ȥ�
\footnote{
���μ���Ƴ�Ф˴ؤ��Ƥ�{\bf 2-5-C}����.
}.
%
\begin{eqnarray}
 Q-L\DD{q}{t}-c_l l\DD{T}{t}=c_p\DD{T}{t}-\alpha\DD{p}{t}.
\end{eqnarray}
%
������$q$ [kg kg$^{-1}$]�Ͽ�����˴ؤ����漾
\footnote{
�漾�ȤϿ������̩��$\rho_W$ [kg/m$^3$]�μ��������̩��
$\rho$ [kg/m$^3$]���Ф�����$q$ [kg kg$^{-1}$]�Τ��ȤǤ���.
 $$
  q \equiv \frac{\rho_W}{\rho}.
 $$
�ʤ�, ���̤ä���Τ˺�����$r$ [kg kg$^{-1}$]�Ȥ�����Τ����뤬,
����Ͽ������̩��$\rho_W$ �δ��������̩��$\rho_d$ [kg/m$^{3}$]
���Ф�����Ǥ���.
 $$
  r \equiv \frac{\rho_W}{\rho_d} . 
 $$
$\rho = \rho_W + \rho_d$�Ǥ��뤫��, �漾�Ⱥ�����δ֤˰ʲ��Τ褦��
�ط�������Ω�Ĥ��Ȥ�狼�������.
 $$
  q \equiv \frac{\rho_W}{\rho} = \frac{r}{1+r}.
 $$
�漾�Ⱥ�����ϼ�������ΰ��̡���ĥ���Ǯ����Ѥˤϱƶ����ʤ�����,
���˿�����ζŷ롦��ȯ�ˤΤ߱ƶ������.
}
�Ǥ���.
$l$ [kg kg$^{-1}$]��ñ�̼��̤��絤��ο�(����)�μ��̤Ǥ���.
$L$ [J/kg]����Ǯ�Ǥ���.
$c_l$ [J kg$^{-1}$ K$^{-1}$]�Ͽ�(����)����Ǯ�Ǥ���.
�����Ǥ�$Q$����Ǯ�ʳ���Ǯ���ͥ륮����ɽ���Ƥ���.



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%%%%%%%%           ���ֹ�Υ������饤��         %%%%%%%% 
\setcounter{equation}{0}
\def\theequation{A-\arabic{equation}}
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%\begin{center}
% {\bf 
% --------------------------------------
% ��ʸ�����ޤ�
% --------------------------------------
% } 
%\end{center}


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%%%%      Appendix 1      %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\theequation{2-5-A-\arabic{equation}}

\subsection*{2-5-A \qquad (2-9b)��Ƴ��}

\subsubsection*{2-5-A-a \quad (2-9b)�ο���μ���Ƴ��}

��(2-9a)�ȼ�(2-8)���Ѥ���(2-9b)�ο���μ���Ƴ�Ф���.

(2-8)��ξ�դ�$t$����ʬ����.
%
\begin{eqnarray}
 \DD{(p \alpha)}{t} = \DD{(RT)}{t} \nonumber .
\end{eqnarray}
%
�������$R$�ʳ��ϻ����Ѳ�����Τ�,
%
\begin{eqnarray}
 p \DD{\alpha}{t} + \alpha \DD{p}{t} = R \DD{T}{t} \nonumber \\
 p \DD{\alpha}{t} = - \alpha \DD{p}{t} + R \DD{T}{t} \nonumber  .
\end{eqnarray}
%
���μ���(2-9a)�����������,
%
\begin{eqnarray}
 Q=c_v\DD{T}{t} - \alpha \DD{p}{t} + R \DD{T}{t} \nonumber  .
\end{eqnarray}
%
�Ȥʤ�, �������갵��Ǯ$c_p$ [J kg$^{-1}$ K$^{-1}$]
($c_p = c_v + R$)��Ƴ�������
(2-9b)�ο���μ��ޤǤ���ޤ�.
%
\begin{eqnarray}
  Q=c_p\DD{T}{t}-\alpha \DD{p}{t} \nonumber.
\end{eqnarray}

\subsubsection*{2-5-A-b \quad (2-9b)�α��դ�Ƴ��}

(2-9b)�α��դ�Ƴ�Ф򤪤��ʤ�.
������, ����(�����ۤɵ�᤿��ʬ���ص�Ū�ˤ����Ƥ�)�����
���դ�Ƴ�Ф��񤷤��ä�����,
�����Ǥϱ��դ������դ����ΤߤȤ���.

(2-9b)�α��դǤ���$c_p T \dfrac{d(\ln \theta)}{dt}$���ѷ����Ƥ���.
�ʤ�, ��ʸ��Ǥ��������Ƥ���
%
\begin{eqnarray*}
 \theta = T(p_0/p)^{\kappa}, \quad \kappa = R/c_p, 
 \quad c_p - c_v = R, \\
 p \alpha = RT  \hspace*{3cm}
\end{eqnarray*}
%
�����Ѥ���.
%
\begin{eqnarray}
 \ln \theta
  &=& \ln T \left( \frac{p_0}{p} \right)^{\kappa} \nonumber \\
  &=& \ln T + \kappa \ln \left( \frac{p_0}{p} \right) \nonumber \\
  &=& \ln T + \kappa \ln p_0 - \kappa \ln p , \nonumber \\ \nonumber \\
 \DD{\ln \theta}{t}
  &=& \DD{\ln T}{t} - \DD{\kappa \ln p}{t} \nonumber \\
  &=& \DD{T}{t}\DD{\ln T}{T} - \kappa \DD{p}{t}\DD{\ln p}{p} \nonumber \\
  &=& \frac{1}{T}\DD{T}{t} - \kappa \frac{1}{p}\DD{p}{t}, \nonumber \\
 c_p T \DD{\ln \theta}{t}
  &=& c_p T \left\{ \frac{1}{T}\DD{T}{t} - \frac{R}{c_p} \frac{1}{p}\DD{p}{t} \right\} \nonumber \\
  &=& c_p \DD{T}{t} - \frac{TR}{p}\DD{p}{t} \nonumber \\
  &=& c_p \DD{T}{t} - \alpha \DD{p}{t} \nonumber .
\end{eqnarray}
%
���Τ褦��, (2-9b)��Ƴ�Ф�Ԥ����Ȥ��Ǥ���.



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%%%%      Appendix 2      %%%%
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\def\theequation{2-5-B-\arabic{equation}}

\subsection*{2-5-B \qquad $dI/dt = c_v dT/dt$��Ƴ��}

$dI/dt = c_v dT/dt$�δط�����Ƴ�Ф���.
�ʤ�, ��ʸ�Ǥ��������ͥ륮����$I$��ɽ������Ƥ��뤬,
����Ū�ˤ�$U$��ɽ����뤳�Ȥ�¿���Τ�, �����Ǥ�
�������ͥ륮����$U$��ɽ������.

�������ͥ륮��$U$ [J]�ϲ���$T$������$V$�˰�¸�����
�ͤ�����Τ�����ʬ$dU$�ϰʲ��Τ褦�˵��ҤǤ���.
%
\begin{eqnarray}
 \Deqlab{hw-01-05-U������ʬ}
 dU = \DP[][V]{U}{T} dT + \DP[][T]{U}{V} dV.
\end{eqnarray}
%
Ǯ�ϳشط������,
%
\begin{eqnarray}
 \Deqlab{hw-01-05-Ǯ�ϳشط���}
 dU = TdS - pdV,
\end{eqnarray}
%
������Ǯ��������,
%
\begin{eqnarray}
 \Deqlab{hw-01-05-������Ǯ}
 c_v = \left( \DD{Q}{T} \right)_{V} = \DP[][V]{U}{T}
\end{eqnarray}
%
������Ω��
\footnote{
������Ǯ�μ��ο���ι�ȱ��ι�Ȥ��������Τ�, 
Ǯ�ϳ����ˡ§$dQ = dU + pdV$���狼�������.
}.
%
\Deqref{hw-01-05-Ǯ�ϳشط���}���겹��������ʬ�����,
%
\begin{eqnarray}
 \Deqlab{hw-01-05-�겹�Ǥ�������ʬ}
 \DP[][T]{U}{V} = T \DP[][T]{S}{V} - p
\end{eqnarray}
%
�Ȥʤ�.
����, ��ͳ���ͥ륮���δط���
$dF = -S dT -p dV$����ʲ��δط���������Ω��
\footnote{
����ʬ$dF$�ϰʲ��Τ褦��ɽ�����Ȥ��Ǥ���.
%
\begin{eqnarray*}
 dF = \DP[][V]{F}{T} dT + \DP[][T]{F}{V} dV.
\end{eqnarray*}
%
����Ⱦ嵭�δط����Ȥ��б����,
%
\begin{eqnarray*}
 \DP[][V]{F}{T} = - S &,& \qquad \DP[][T]{F}{V} = -p \\
 \frac{\partial^2 F}{\partial T \partial V} = - \DP[][T]{S}{V},
 &,& \qquad
 \frac{\partial^2 F}{\partial V \partial T} = - \DP[][V]{p}{T} 
\end{eqnarray*}
%
�Ȥʤ뤿��,  \Deqref{hw-01-05-����ȥ��ԡ��Ȱ��Ϥδط���}������Ω��. 
}.
%
\begin{eqnarray}
 \Deqlab{hw-01-05-����ȥ��ԡ��Ȱ��Ϥδط���}
 \DP[][T]{S}{V} = \DP[][V]{p}{T}.
\end{eqnarray}
%
\Deqref{hw-01-05-�겹�Ǥ�������ʬ}��
\Deqref{hw-01-05-����ȥ��ԡ��Ȱ��Ϥδط���}�����������
%
\begin{eqnarray}
 \DP[][T]{U}{V} &=& T \DP[][V]{p}{T} - p  \nonumber \\
                &=& T \left( \frac{n R}{V} \right) - p   \nonumber \\
 \Deqlab{hw-01-05-U��V�˰�¸���ʤ�}
                &=& 0
\end{eqnarray}
%
�Ȥʤ�.
2���ܤϾ��������������Ѥ��Ƥ���.
���Τ��Ȥ���$U$��$V$�ˤϰ�¸����, $T$�ˤΤ߰�¸���Ƥ��뤷�Ƥ��뤳�Ȥ�
�狼��.

\Deqref{hw-01-05-U������ʬ}��\Deqref{hw-01-05-������Ǯ}��
\Deqref{hw-01-05-U��V�˰�¸���ʤ�}�����������
%
\begin{eqnarray}
 \Deqlab{hw-01-05-�������ͥ륮���Ȳ��٤δط���}
 dU = c_v dT
\end{eqnarray}

{\bf 2-5-B-a \quad ��­}

��ͳ���ͥ륮���δط�������
\Deqref{hw-01-05-����ȥ��ԡ��Ȱ��Ϥδط���}
��Ƴ���줿��, Ʊ�ͤˤ��Ƥ���¾�Υ��ͥ륮���δط�������
�����褦�ʴط�����Ƴ�����Ȥ��Ǥ���.
�ʲ��Ϥ����б��Ǥ���.
Ƴ�Фϼ�ͳ���ͥ륮���λ���Ʊ��.
%
\begin{eqnarray*}
 dU = TdS - pdV  \quad &��& \quad \DP[][S]{T}{V} = - \DP[][V]{p}{S}, \\
 dH = TdS + Vdp  \quad &��& \quad \DP[][S]{T}{p} =   \DP[][p]{V}{S}, \\
 dG =-SdT + Vdp  \quad &��& \quad \DP[][T]{S}{p} = - \DP[][p]{V}{T}.
\end{eqnarray*}
%


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%%%%      Appendix 3      %%%%
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\def\theequation{2-5-C-\arabic{equation}}

\subsection*{2-5-C \qquad ��ξ����Ѳ���ͤ���Ǯ�ϳشط���}

(2-10)��Ƴ�Ф��Ƥߤ�.
ñ�̼��̤��絤��(�����ο�(����)��ޤ�)��
��������μ��̤�$m_d$ [kg kg$^{-1}$], ������μ��̤�$m_v$ [kg kg$^{-1}$],
��(����)�μ��̤�$m_l$ [kg kg$^{-1}$]�Ȥ�,
����������갵��Ǯ��$c_d$, ��������갵��Ǯ��$c_v$,
��(����)���갵��Ǯ��$c_l$�Ȥ�,
��Ǯ��ȯ�����θ��, �ʲ��Τ褦�ˤʤ�.
%
\begin{eqnarray*}
  Q + L \DD{m_l}{t}
 =
  \left(
   c_l m_l + c_v m_v + c_d m_d
  \right)
  \DD{T}{t}
 - \alpha \DD{p}{t}.
\end{eqnarray*}
%
�夬���ä�����̤ȿ����������������̤ϰ��פ���, �Ĥޤ�
$d m_l / dt = - d m_v / dt$�Ǥ���.
�ޤ�, $1 = m_l + m_v + m_d$����$m_d \gg m_v$, $m_d \gg m_l$
�Ǥ��뤳�Ȥ���$c_v m_v + c_d m_d \simeq c_d$�ȶ������.
����餫��,
%
\begin{eqnarray*}
  Q - L \DD{m_v}{t}
 =
  \left(
   c_l m_l + c_d
  \right)
  \DD{T}{t}
 - \alpha \DD{p}{t}
\end{eqnarray*}
%
�Ȥʤ�.
��ʸ�Ǥ�, $m_v = q$, $m_l =l$, $c_d = c_p$�ȤʤäƤ���Τ�,
%
\begin{eqnarray*}
 Q-L\DD{q}{t}-c_l l\DD{T}{t}=c_p\DD{T}{t}-\alpha\DD{p}{t}
\end{eqnarray*}
%
�Ȥʤ�.






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