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% Chapter 3   Scale Analysis
%             3-3 BAROCLINIC EQUATION
%
% 2002/11/04  ���� ����   ����
% 2002/11/05  ���� ����   ����
% 2002/12/26  ���� ����   ���߸�ν���
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\section{����������}

\begin{itembox}[l]{����}
 �����絤�Υ�������󥰤򤪤��ʤ�.
 ������ʹߤϱ�ľ��ɸ�Ȥ���$Z \equiv - \ln (p/p_0)$���Ѥ���.
 $p$-�Ϥ�$Z$-�Ϥ�̩�ܤʴط��ˤ��뤳�Ȥ���, ��2�Ϥ�Ƴ����$p$-�Ϥ�
 �������Ϥ����Ѥ���$Z$-�Ϥ��������Ϥ�Ƴ�Ф�, �����Υ�������󥰤򤪤�
 �ʤ�. �ʤ�, ��ư�������ϱ��٤�ȯ���μ����ѷ�����.
 ��������󥰤η�̤����Ѥ���������̤Ϥ��θ����Ǥ����ʤ�.
\end{itembox}

������Ǥϵ��ΤǤ����ϵ��Ǥη����������˥���������Ϥ��Ѥ���.
�ʹߤϰʲ��α�ľ��ɸ���Ѥ���
\footnote{
 ���κ�ɸ�Ϥ�{\bf ����$p$��ɸ��}�ȸƤФ�, ���Ϲ�ή�ϤǻȤ���.
 ���ʤߤ�, $Z = - H \ln (p/p_0)$�Ȥ�����⤢��.
}.
%
\begin{eqnarray}
 Z \equiv - \ln (p/p_0).
\end{eqnarray}
%
$p_0$��ɸ�೤�̤ΰ��ϤǤ���. �������������ſ尵�μ��ˤ�ä�
��ľ��ɸ$Z$�ϼºݤι���$z$, �ڤӥ����ݥƥ󥷥��$\phi = gz$��
�ʲ��Τ褦�ʴط������
\footnote{
 ��ñ�˸������Ƥߤ�.
 $\partial \phi / \partial Z
 = - g \{\partial z / \partial (\ln p/p_0)\}$�Ǥ���, $z(p)$��
 ������̣�Ǥ�ñĴ�ؿ��ʤΤ�,
 %
 \begin{eqnarray}
  g \DP{z}{\left( - \ln p / p_0 \right)}
   = g \bigg/ \DP{}{z} \left( - \ln \frac{p}{p_0} \right)
   = g \bigg/ \DP{p}{z} \DP{}{p} \left( - \ln \frac{p}{p_0} \right)
   = g \bigg/ \left( - \frac{1}{p} \right) \DP{p}{z}
   \nonumber
 \end{eqnarray}
 %
 �Ȥʤ�. $p(z)$��ޤ�������̣�Ǥ�ñĴ�ؿ��ʤΤ�,
 %
 \begin{eqnarray}
  g \bigg/ \left( - \frac{1}{p} \right) \DP{p}{z}
   = - p g\DP{z}{p}
   \nonumber
 \end{eqnarray}
 %
 �Ȥʤ�.
}.
%
\begin{eqnarray}
 RT = p \alpha = - p g \DP{z}{p} = \DP{\phi}{Z}.
\end{eqnarray}
%
$Z$�������ΰ�Ĥ�, $Z$���ºݤι⤵�򥹥�����ϥ���$H=RT/g$�ǳ�ä���Τ�
�ۤ����������ȤǤ���
\footnote{
 �Ĥޤ�, ñ��$Z$�Ϥ��礦�ɥ�������ϥ���$H$ʬ�ι⤵�Ȥ������Ȥˤʤ�.
 (4-32)�򥹥�����ϥ��Ȥǽ񤭴�����Ȱʲ��Τ褦�ˤʤ�.
 %
 \begin{eqnarray}
  H = \DP{z}{Z}.
   \nonumber
 \end{eqnarray}
}.
����Ϥ��褽8km�Ǥ���.

���ηϤ���2�Ϥΰ��Ϻ�ɸ�Ϥ�̩�ܤʴط��ˤ���. ��ľ��ʬ�ϰʲ��δط��ˤ���.
%
\begin{eqnarray}
 \DP{}{p} = - \frac{1}{p} \DP{}{Z}.
\end{eqnarray}
%
¾���ѿ��ˤ������ʬ��Ʊ�ͤǤ���.
$Z$®��$\dot{Z}$��$\omega = \partial p / \partial t$�Ȱʲ��δط��ˤ���.
%
\begin{eqnarray}
 \dot{Z} = -\dot{p}/p = - \omega / p.
\end{eqnarray}
%

$Z$-�ϤǤα�ư�������ϰʲ��Τ褦�ˤʤ�
\footnote{
 $p$-�ϤǤα�ư������(���ϤϽ���).
 $$
  \DD{\Dvect{V}}{t} + \nabla \phi + f \Dvect{k} \times \Dvect{V} = 0
  \eqno \textrm{(2-58)}
 $$
 ��(4-33)��(4-34)�δط������Ѥ��뤳�Ȥ�(4-35)��������.
}.
%
\begin{eqnarray}
 \DP{\Dvect{V}}{t} + \Dvect{V} \cdot \nabla \Dvect{V}
  + \dot{Z} \DP{\Dvect{V}}{Z} + \nabla \phi
  + f \Dvect{k} \times \Dvect{V} = 0.
\end{eqnarray}
%
(4-32)��Ȥ���Ǯ�ϳ���1ˡ§�ϰʲ��Τ褦�˵��ҤǤ���
\footnote{
 (4-36)��$p$-�ϤǤ�Ǯ�ϳؤμ�.
 $$
  c_p \DD{T}{t} - \alpha \omega = Q.
  \eqno \textrm{(2-57b)}
 $$
 ��(4-32)��(4-34)���Ѥ����ѷ�������ΤǤ���($\omega = dp/dt$).
 (2-57b)��(4-32)��(4-34)�����������,
 %
 \begin{eqnarray}
  \frac{c_p}{R} \DD{}{t} \DP[][]{\phi}{Z} + \dot{Z} p \alpha &=& Q
   \nonumber \\
  \frac{1}{\kappa} \DD{}{t} \DP[][]{\phi}{Z} + \dot{Z} \DP[][]{\phi}{Z} &=& Q
   \nonumber \\
  \DD{}{t} \DP[][]{\phi}{Z} + \kappa \dot{Z} \DP[][]{\phi}{Z} &=& \kappa Q
   \nonumber 
 \end{eqnarray}
 %
 �Ȥʤ�. �Ǹ��$d/dt$�ι��Ÿ�������(4-36)�ˤʤ�.
}.
%
\begin{eqnarray}
 \DP{}{t} \DP{\phi}{Z} + \Dvect{V} \cdot \nabla \DP{\phi}{Z}
  + \dot{Z} \DP{}{Z} \left( \DP{\phi}{Z} + \kappa \phi \right)
  = \kappa Q.
\end{eqnarray}
%
$\kappa = R c_p^{-1}$�Ǥ���, $Q$��ñ�̻��֤�����˲ä���줿Ǯ�̤Ǥ���.
Ϣ³�μ���
%
\begin{eqnarray}
 \Ddiv \Dvect{V} + \DP{\dot{Z}}{Z} - \dot{Z} = 0
\end{eqnarray}
�Ȥʤ�
\footnote{
 �����$p$-�Ϥˤ�����Ϣ³�μ�.
 $$
  \Ddiv \Dvect{V} + \DP{\omega}{p} = 0.
  \eqno \textrm{(2-60)}
 $$
 ��(4-33), (4-34)���ѷ�������ΤǤ���.
 ��1����Ѳ����ʤ��Τ���2����ѷ����Ƥߤ��,
 %
 \begin{eqnarray}
  \DP{\omega}{p} = - \DP{}{p} \left( p \dot{Z} \right)
   = - p \DP{\dot{Z}}{p} - \dot{Z}
   = - p \left( - \frac{1}{p} \right) \DP{\dot{Z}}{Z} - \dot{Z}
   = \DP{\dot{Z}}{Z} - \dot{Z}
   \nonumber
 \end{eqnarray}
 %
 �Ȥʤ�. ���, (4-37)��������.
}.

�����ǤΥ���������ϤǤ�, ̵�����ѿ���$R_O, \varepsilon, L/a$�ˤʤ�.
$a$���ϵ��Ⱦ�¤Ǥ���. $\varepsilon$��{\bfseries ��ž�ե롼�ɿ�}
({\itshape rotational Froude number})�Ǥ���. ����ʪ���̤��絤����Ū������
��ȿ���㤹���̤Ǥ���\footnote{���Τ��Ȥ˴ؤ��ƤϤ�����θ�Ⱦ�ˤƾҲ𤵤�
��.}, ������Ѥ���$F$�Ȼ����褦�ʤ�ΤǤ���.
�������ä���ͳ����, ������Ѥ�����������Ϥ����Ǥ��͡���ʪ���̤ˤ�
���Ѥ����.
��ʬ�ϰʲ��Τ褦�˸��Ѥ����
\footnote{
 $Z$�Υ��������(4-31)�Ǥ���$Z \equiv - \ln (p/p_0)$����������.
}.
%
\begin{eqnarray}
 \DP{}{t} \sim \frac{V}{L},
  \quad
  \nabla \sim \frac{1}{L},
  \quad
  \DP{}{Z} \sim 1.
\end{eqnarray}
%
������$\nabla f \sim \beta = 2 \Omega \cos \varphi /a$���㳰�Ǥ���.
$\beta$�˴ؤ��뤳�μ��ϼ����(4-56)��������.
$Z$��̵�����̤Ǥ���, $\partial / \partial Z \sim 1$�ϱ�ľ��������Ǥ���
8km��������
\footnote{
 �����$z$�Υ��������Ǥ���, �����褽��������ϥ���$H$�Ǥ���.
 (4-32)����������ط���$H = \partial z/ \partial Z$���θ�����,
 ���礦��$\partial / \partial Z \sim 1$�Ǥ��뤳�Ȥ��狼��.
}
���б�����. �������ή���ξ���Ȥ��Ƥ�Ŭ�ڤǤ���.
®�٤������Ʊ�ͤ˲�ž��ȯ����ʬ�����--��(4-13), (4-14)����--.
�����ݥƥ󥷥��ϰʲ��Τ褦��ʬ�����.
%
\begin{eqnarray}
 \phi = \overline{\phi} (Z) + \phi' (x, y ,Z, t).
\end{eqnarray}
%
$\overline{\phi}$��ɸ���絤�ǤΥ����ݥƥ󥷥��Ǥ���.
��°�ѿ��Υ�������ϰʲ��Τ褦�ˤʤ�.
%
\begin{eqnarray}
 \Dvect{V}_{\Psi} \sim V, \quad \Dvect{V}_{\chi} \sim R_1 V,
  \\
 \dot{Z} \sim R_1 V/L, \hspace{10mm}
  \\
 \phi' \sim f V/L. \hspace{12mm}
\end{eqnarray}
%
$R_1$�Ϥޤ����ꤷ�Ƥ��ʤ���, ��������󥰼��Τ������Ʊ�ͤǤ���.
$\dot{Z}$�Υ��������,
Ϣ³�μ�(4-37)�ˤ�������1�ब��2, 3��ȥХ�󥹤���Ȳ��ꤷ, 
$\Dvect{V}_{\chi}$�Υ���������Ѥ��뤳�Ȥ�����줿.
$\phi'$�Υ�������󥰤��Ϲ�ή���������(4-8)����ľ�ܤ�äƤ���.

���٤���٤�ȯ���Ͼ�������ͽ�ۤ����Τ�, ��ư�������ϱ��٤μ���ȯ���μ�
���֤�������Τ������Ǥ���.
���ηϤˤ����뱲�٤μ��ϰʲ��Τ褦�ˤʤ�. ���줾��ι�β��ˤ���Τ�
��������Ǥ���.
%
\begin{eqnarray}
 \DP{\zeta}{t} \
  + \ \Dvect{V}_{\Psi} \cdot \nabla \zeta \
  + \quad \Dvect{V}_{\chi} \cdot \nabla \zeta \quad
  + \quad \Dvect{V}_{\Psi} \cdot \nabla f \quad
  + \qquad \Dvect{V}_{\chi} \cdot \nabla f \qquad
  + \quad \dot{Z} \DP{\zeta}{Z} 
  \nonumber \\
 \frac{V^2}{L^2} \hspace{11mm}
  \frac{V^2}{L^2} \hspace{13mm}
  \frac{R_1 V^2}{L^2} \hspace{6mm}
  \frac{(\beta a/f) L/a}{R_O} \frac{V^2}{L^2} \hspace{5mm}
  \frac{(\beta a/f) L/a}{R_O} \frac{R_1 V^2}{L^2} \hspace{7mm}
  \frac{R_1 V^2}{L^2}
  \nonumber \\
 \nonumber \\
 + \quad fD \quad + \quad \zeta D \quad
  + \Dvect{k} \cdot \nabla \dot{Z} \times \DP{\Dvect{V}_{\Psi}}{Z}
  + \Dvect{k} \cdot \nabla \dot{Z} \times \DP{\Dvect{V}_{\chi}}{Z}
  = 0.
  \nonumber \\
 \frac{R_1}{R_O} \frac{V^2}{L^2} \hspace{7mm}
  \frac{R_1 V^2}{L^2} \hspace{13mm}
  \frac{R_1 V^2}{L^2} \hspace{18mm}
  \frac{{R_1}^2 V^2}{L^2} \hspace{13mm} 
\end{eqnarray}
%
�����ǤϿ����˥�������$\zeta \sim V/L, D \sim R_1 V/L$���Ѥ���.
�ޤ�$R_O = V/fL$���Ѥ������ƤΥ������뤬$V^2/L^2$���̤Υե�������
�Ȥ���褦������.
(��4, 5��Υ�������ˤ������򤪤��ʤä�).
ȯ���������Ȥ��γơ��ι�Υ�������ϰʲ��Τ褦�ˤʤ�.
%
\begin{eqnarray}
 \DP{D}{t} + \Ddiv (\Dvect{V}_{\Psi} \cdot \nabla \Dvect{V}_{\Psi})
  + \Ddiv
   \left(
    \Dvect{V}_{\Psi} \cdot \nabla \Dvect{V}_{\chi}
    + \Dvect{V}_{\chi} \cdot \nabla \Dvect{V}_{\Psi}
   \right)
   \hspace{50mm}
   \nonumber \\
 + \Ddiv (\Dvect{V}_{\chi} \cdot \nabla \Dvect{V}_{\chi})
  + \nabla \dot{Z} \cdot \DP{\Dvect{V}_{\Psi}}{Z}
  + \dot{Z} \DP{D}{Z}
  \nonumber \\
 \frac{R_1 V^2}{L^2} \hspace{14mm}
  \frac{V^2}{L^2} \hspace{26mm}
  \frac{R_1 V^2}{L^2} \hspace{18mm}
  \frac{{R_1}^2 V^2}{L^2} \hspace{14mm}
  \frac{R_1 V^2}{L^2} \hspace{6mm}
  \frac{{R_1}^2 V^2}{L^2} 
  \nonumber \\
 \nonumber \\
 + \nabla \dot{Z} \cdot \DP{\Dvect{V}_{\chi}}{Z}
  + \Dlapla \phi' \quad
  - \quad f \zeta \quad
  - \quad \Dvect{k} \times \nabla f \cdot \Dvect{V}_{\Psi} \quad
  - \quad \Dvect{k} \times \nabla f \cdot \Dvect{V}_{\chi} \quad
  = 0. \hspace{10mm}
  \nonumber \\
 \frac{{R_1}^2 V^2}{L^2} \hspace{5mm}
  \frac{1}{R_O} \frac{V^2}{L^2} \hspace{7mm}
  \frac{1}{R_O} \frac{V^2}{L^2} \hspace{5mm}
  \frac{(\beta a/f) L/a}{R_O} \frac{V^2}{L^2} \hspace{7mm}
  \frac{(\beta a/f) L/a R_1}{R_O} \frac{V^2}{L^2} \hspace{18mm}
\end{eqnarray}
%
Ϣ³�μ�(4-37)�򥹥�����󥰤����, �ʲ��Τ褦�����Ƥι�Υ��������
Ʊ�����������Ǥ���. �����
$\dot{Z}$�Υ��������(4-41)��Ϣ³�μ���3�ब���ƥХ�󥹤���褦��
���Ʒ�᤿��ΤʤΤ�����, �����Ǥ���.
%
\begin{eqnarray}
 D \ + \ \DP{\dot{Z}}{Z} \ - \ \dot{Z} \ = \ 0.
  \\
 \frac{R_1 V}{L} \hspace{4mm}
  \frac{R_1 V}{L} \hspace{3mm}
  \frac{R_1 V}{L} \hspace{7mm}
  \nonumber
\end{eqnarray}
%
Ǯ�ϳ���1ˡ§(4-36)�ϰʲ��Τ褦�˥�������󥰤����
\footnote{
 ��Ҥ���뤬, $\Gamma(Z)$�Υ��������$\overline{\phi}$�Ǥ���.
 ��ä�$\dot{Z} \Gamma(Z)$�Υ��������$\overline{\phi} R_1 V/L$�Ȥʤ�.
 ��ž�ե롼�ɿ��������������$F = f^2 L^2 / \overline{\phi}$�ʤΤ�,
 $\varepsilon = f^2 L^2 / \overline{\phi}$�Ȥʤ�.
 ���β�ž�ե롼�ɿ��ȥ����ӡ���$R_O = V/fL$�Ǥ��Υ�������򵭽Ҥ���.
 %
 \begin{eqnarray}
  \overline{\phi} R_1 \frac{V}{L}
   = \frac{f^2 L^2}{\varepsilon} R_1 \frac{V}{L}
   = R_1 f V^2 \frac{fL}{V} \frac{1}{\varepsilon}
   = \frac{R_1 f V^2}{R_O \varepsilon}
   \nonumber.
 \end{eqnarray}
}.
%
\begin{eqnarray}
 \DP{}{t} \DP{\phi'}{Z}
  + \Dvect{V}_{\Psi} \cdot \nabla \DP{\phi'}{Z}
  + \Dvect{V}_{\chi} \cdot \nabla \DP{\phi'}{Z} \
  + \ \dot{Z} \Gamma (Z) \ 
 + \ \dot{Z} \DP{}{Z}
  \left(
   \DP{\phi'}{Z} + \kappa \phi'
  \right)
  = \kappa Q.
  \nonumber \\
 fV^2 \hspace{13mm} 
  fV^2 \hspace{14mm} 
  R_1 fV^2 \hspace{11mm} 
  \frac{R_1 f V^2}{R_O \varepsilon} \hspace{17mm} 
  R_1 fV^2 \hspace{25mm}
\end{eqnarray}
%
�ʤ�,
%
\begin{eqnarray}
 \Gamma (Z) = \DP{}{Z}
  \left(
   \DP{\overline{\phi}}{Z} + \kappa \overline{\phi}
  \right)
  =
  \left(
   \DP{\overline{T}}{Z} + \kappa \overline{T}
  \right)
  \simeq
  \frac{H^2 g}{\overline{T}}
  \left(
   \frac{g}{c_p} + \frac{1}{H}\DP{\overline{T}}{Z}
  \right)
\end{eqnarray}
%
�Ǥ���.
(4-47)�ˤ������ѷ��ˤ��ſ尵�μ�(4-32)���Ѥ�, $H \simeq R \overline{T} / g$
�Ȳ��ꤷ��. (����$\overline{T}$��$Z$�δؿ��Ǥ���).
$\Gamma$��{\bfseries ��Ū������}({\itshape static stability})�ȸƤФ��.
�����$\Gamma$�����粹�ٸ�Ψ��ɸ���絤�β��ٸ�Ψ�κ������㤷�Ƥ��뤫��
�Ǥ���
\footnote{
 $g/c_p$�ϴ��粹�ٸ�Ψ��ɽ��. ɸ���絤�β��ٸ�Ψ��
 $- \partial \overline{T} / \partial z$����, (4-32)�򥹥�����ϥ��Ȥ��ѷ�
 �����ط���$H = \partial z/ \partial Z$���
 $- (1/H)(\partial \overline{T} / \partial Z)$�Ȥʤ�.
 ����2�Ĥ�{\bfseries ��}��
 $g/c_p + (1/H)(\partial \overline{T} / \partial Z)$�Ȥʤ�.
}.
�ѿ�$\varepsilon$ (��ž�ե롼�ɿ�)�ϰʲ��Τ褦����������
\footnote{
 $\dot{��}\dot{ž}$�ǤϤʤ�{\bf �ե롼�ɿ�}
 ({\it Froude number\/})��¸�ߤ���.
 �㤨��$F = U/\sqrt{gH}$�ʤɤǤ���. 
}.
%
\begin{eqnarray}
 \varepsilon \equiv f^2 L^2 \Gamma^{-1}.
\end{eqnarray}
%
�����̤�{\bfseries ����㡼�ɥ����}({\itshape Richardson's number})
�ǽ�ľ���Ȱʲ��Τ褦�ˤʤ�.
%
\begin{eqnarray}
 \varepsilon = \frac{1}{{R_O}^2 R_i}.
\end{eqnarray}
%
����㡼�ɥ�����Ȥϰʲ��Τ褦������������Ǥ���.
%
\begin{eqnarray}
 R_i = \Gamma / V^2.
\end{eqnarray}
%
����ϱ�ľ��®������2����Ф�����Ū�����٤���Ǥ���.
�ʹߤβ��ϤǤ�, (4-46)�ˤ������Ǯ�ι����Ȥ�.
�ʤ��ʤ�, �ʹߤβ��ϤǤϲ�ǮΨ\footnote{$Q$�Τ��ȤǤ���.}�ϥ���������Ϥ�
�ƶ���ڤܤ��ۤ��礭���Ϥʤ�����Ǥ���.
�ʤ�, ���������, ����������Ϥ˱ƶ���ڤܤ����ɤ�����¾�ι�Ȥ���Ӥ�
�����ʤä�Ĵ�٤뤳�Ȥˤʤ�.

�ޤ�(4-50)�������㡼�ɥ�������Ѥ��, ������Ѥ���(4-48)����
$\varepsilon$��׻�����Τ������Ǥ���.
�ʹߤ��ͤ��Ѥ��ƥ���㡼�ɥ������׻�����.
$g \sim 10$ m sec$^{-1}$, $\overline{T} \sim 300^{\circ}$,
$g/c_p + H^{-1} \partial T / \partial Z \sim 3.5^{\circ} \times 10^{-3}$
m$^{-1}$, $H \sim 10^4$ m , $V \sim 10 $m sec$^{-1}$
\footnote{
 (4-47)���,
 %
 \begin{eqnarray}
  R_i &=& \frac{H^2 g}{\overline{T} V^2}
   \left(
    \frac{g}{c_p} + \frac{1}{H} \DP{\overline{T}}{Z}
   \right)
   \nonumber \\
  &=& \frac{(10^4)^2 \times 10}{300 \times 10^2} \times 3.5 \times 10^{-3}
   \nonumber \\
  &\simeq& 100
   \nonumber
 \end{eqnarray}
}.
%
\begin{eqnarray}
 R_i \sim 100.
\end{eqnarray}




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% } 
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%%%%%%%%   Set Counter (chapter, section etc. ) %%%%%%%%
%\setcounter{chapter}{2}
%\setcounter{section}{0}
%\setcounter{footnote}{0}
\setcounter{equation}{0}
%\setcounter{page}{1}
\setcounter{figure}{0}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%        Counter Output Format           %%%%%%%%
\def\theequation{A-\arabic{equation}}
%\def\thesection{\arabic{chapter}-\arabic{section}}
%\def\thesubsection{\arabic{chapter}-\arabic{section}-\arabic{subsection}}
\def\thefigure{A-\arabic{figure}}
\def\thetable{A-\arabic{table}}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%      Appendix        %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\subsection*{Appendix \qquad }

%\subsubsection*{}


%%%%%%%%              Text End                  %%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
