%
% Chapter 1  The Governing Equations
%            1-12 VORTICITY AND DIVERGENCE EQUATIONS
%
% 2002/06/25 ���� ����   ����
% 2002/07/02 ���� ����   ����
% 2002/09/22 ���� ����   App. �ر����������γƹ�ΰ�̣�٤��ɲ�
%                        ���ߥʡ�������������
% 2002/10/01 ���� ����   2-12-C �ر��٤ȳ�®�٤δط��٤��ɵ�
%                      
%
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\markboth{2-12 ���٤�ȯ����������}{2-12 ���٤�ȯ����������}
%\markright{1-12 ���٤�ȯ����������}
%\Ddate{2002/10/01}
%\Dfile{hw-01-12.tex}

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\section{���٤�ȯ����������}

�ϵ�ʪ�����ϳؤθ������Ū�Ȥ������
���ͷ׻��ʤɤξ��,
��ʿ�̥٥��ȥ뱿ư������
(�⤷����2�ĤΥ����顼��ʬ��������)
��{\bfseries ����}({\itshape vorticity})��
{\bfseries ȯ��}({\itshape divergence})�Ȥ���
�̤�2�ĤΥ����顼���������˼�괹�����Ѥ�������
�����ʾ�礬¿��.

\subsection{����������}

�������Υ٥��ȥ뱲�٤�ñ���®�٥٥��ȥ�β�ž��Ȥä���ΤǤ���.
����ϳ�®�٤γ�ǰ����̲�������ΤǤ���.
Ω�Τβ�ž�ξ��ϱ��٤�ñ��˳�®�٤�2�ܤǤ���
\footnote{
 ���Ρֱ��٤���®�٤� 2 �ܡפȤ���ɽ���˴ؤ��Ƥ�
 {\bf 2-12-C}�򻲾ȤΤ���.
}.
�絤���絬�Ϥ�ή��ǤϿ�ʿ��®�٤���ٱ�ľ����®�٤�
���微����, ή��Ͻ��ʿ�̤Ǥ���ȹͤ���.
���Τ��Ȥ��絬�ϤʷϤǤ��ϳؤ���������,
���٤α�ľ��ʬ����äȤ���פʤ�ΤǤ��뤳�Ȥ��狼��.
���η�̤Ȥ���, ��ʿ�̤���®�Ȥ��α��٤α�ľ��ʬ
$\zeta = \Dvect{k} \cdot \nabla \times \Dvect{V}$ [m/s$^2$]
�����̤�ɬ�פȤʤ�.
���ʤߤ�, ���̺�ɸ�Ǳ��٤�ɽ������Ȱʲ��Τ褦�ˤʤ�
\footnote{
1��7���
\begin{eqnarray*}
 (\nabla \times \Dvect{V})_k
 =
 (1/r \cos \varphi)
 \left[
  \partial v / \partial \lambda
  - (\partial / \partial \varphi) (u \cos \varphi)
 \right]
\end{eqnarray*}
����Ƴ�ФǤ���.
}.
%
\begin{eqnarray}
 \zeta = (1/a \cos \varphi) \partial v / \partial \lambda
         - a^{-1} \partial u / \partial \varphi
	 + u \tan \varphi / a.
\end{eqnarray}
%

$\zeta$�λ����Ѳ��μ���Ƴ���ˤ�,
(2-16)��(2-17)��٥��ȥ�����ǵ��Ҥ�
\footnote{
 �ºݤ�(2-11)�η��ˤ��Ƥ�������������ɤ�
 ((2-11)���Τ��ϵ������ޤ������ʤ��Ƥ��ʤ�.)
},
����ʸ�κǸ�˷Ǻܤ���Ƥ�����ظ���(A-9)
\footnote{
 $(\Dvect{V} \cdot \nabla) \Dvect{V}
  = \nabla (\Dvect{V}^2 / 2) + \zeta \Dvect{k} \times \Dvect{V}$
}
��Ȥ��Τ������Ǥ���.
�ѷ��������ϰʲ��Τ褦�˽񤱤�.
%
\begin{eqnarray}
  \DD{\Dvect{V}}{t}
 =
  \DP{\Dvect{V}}{t}
  + \nabla (\Dvect{V}^2 / 2)
  + \zeta \Dvect{k} \times \Dvect{V}
  + w \DP{\Dvect{V}}{z}
 =
  - \alpha \nabla p
  - f \Dvect{k} \times \Dvect{V}
  + \Dvect{F} \\
 1 \hspace{12mm} 
 2 \hspace{14mm} 
 3 \hspace{12mm} 
 4 \hspace{15mm} 
 5 \hspace{15mm} 
 6 \hspace{10mm}
 \nonumber
\end{eqnarray}
%
������$f = 2 \Omega \sin \varphi$�ϥ��ꥪ�ꡦ�ѥ�᡼���Ǥ���.
�ޤ�, $\Dvect{V}$�����$\nabla$�Ͽ�ʿ�̤Ǥ�®�٤���ʬ�黻�ҤǤ���.
�����ʤ褦��, $\eta \equiv \zeta + f$�������
\footnote{
$\eta$ [m/s$^2$]��{\bfseries ���б���}({\itshape absolute vorticity})
�ȸƤФ���̤�,
����$\zeta$(���б��٤Ȥζ��̤Τ����
{\bfseries ���б���}({\itshape relative vorticity})
�ȸƤФ�뤳�Ȥ⤢��)
�ȥ��ꥪ�ꡦ�ѥ�᡼��$f$
({\bfseries ��������}({\itshape planetary vorticity})
�Ȥ�ƤФ��)
��­����碌����ΤǤ���.
},
3���ܤ�6���ܤι��­����碌��$\eta \Dvect{k} \times \Dvect{V}$
�Ȥ���.
������, ���������Τ�$\Dvect{k} \cdot \nabla \times $�Ǳ黻����.
��Ͽ�ο��ظ����Υ٥��ȥ�黻��Ȥ���
��ñ�ʼ��ˤǤ���.
�㤨��, ���ۤβ�ž��Ȥ�ȥ����ˤʤ�Τ�
\footnote{
��Ͽ�ο��ظ���(A-2)���
$\nabla \times \nabla a = 0$.
}
2���ܤι�Ͼä���.
�ʲ��Τ��
\footnote{
�ʲ�����Ͽ�ο��ظ���(A-7)�����Ѥ���.
\begin{eqnarray*}
  \nabla \times (\Dvect{A} \times \Dvect{B})
 =
  \Dvect{A} \nabla \cdot \Dvect{B}
  - \Dvect{B} \nabla \cdot \Dvect{A}
  + (\Dvect{B} \cdot \nabla) \Dvect{A}
  - (\Dvect{A} \cdot \nabla) \Dvect{B}
\end{eqnarray*}
}
�������, ��Ⱦ�ι�η׻��ϴ�ñ�Ǥ���.
%
\begin{eqnarray*}
  \Dvect{k} \cdot [\nabla \times (\eta \Dvect{k} \times \Dvect{V})]
 =
  \Dvect{k} \cdot [\eta \Dvect{k} \nabla \cdot \Dvect{V}
  - \Dvect{V} \nabla \cdot \eta \Dvect{k}
  - (\eta \Dvect{k} \cdot \nabla) \Dvect{V}
  + (\Dvect{V} \cdot \nabla) \eta \Dvect{k}]
\end{eqnarray*}
%
$ \Dvect{k} \cdot \nabla = 0$
\footnote{
 $\nabla$�α�ľ��ʬ�ϥ����Ǥ��뤳�Ȥ�����.
}
�Ǥ��뤳�Ȥ���,
���դο����2�Ĥι�Ͼä���.

���η�̤��ʲ���
{\bfseries ����������}({\itshape vorticity equation})�Ȥʤ�
\footnote{
 �����������γƹ�ΰ�̣�ˤĤ��Ƥ�{\bfseries 2-12-B}�򻲾�.
}.
%
\begin{eqnarray}
  \DP{\zeta}{t}
  + \Dvect{V} \cdot \nabla \eta
  + w \DP{\zeta}{t}
 =
  - \eta \nabla \cdot \Dvect{V}
  - \Dvect{k} \cdot \nabla w \times \DP{\Dvect{V}}{z}
  + \Dvect{k} \cdot \nabla p \times \nabla \alpha
  + \Dvect{k} \cdot \nabla   \times \Dvect{F} .
  \nonumber \\
 1 \hspace{12mm} 
 3,6 \hspace{13mm} 
 4 \hspace{15mm} 
 3,6 \hspace{18mm} 
 4 \hspace{20mm} 
 5 \hspace{10mm}
\end{eqnarray}
%
�����Ǥ��ֹ��(2-103)�Ǥ��ֹ���б����Ƥ���
\footnote{
�嵭�η׻����Ѥ����ط��������󤹤�.
%
\begin{eqnarray*}
  \Dvect{k} \cdot \nabla \times \DP{\Dvect{V}}{t}
 &=&
  \DP{\zeta}{t} , 
 \\
  \Dvect{k} \cdot \nabla \times w \DP{\Dvect{V}}{z}
 &=&
  \Dvect{k} \cdot
   \left[
    w \nabla \times \DP{\Dvect{V}}{z} + \nabla w \times \DP{\Dvect{V}}{z}
   \right]
 \\
 &=&
  w \DP{\zeta}{z}
  + \Dvect{k} \cdot \left[
		     \nabla w \times \DP{\Dvect{V}}{z}
		    \right],
 \\
  \Dvect{k} \cdot \nabla \times (\alpha \nabla p)
 &=&
  \Dvect{k} \cdot
   \left[
    \alpha \nabla \times \nabla p + \nabla \alpha \times \nabla p
   \right]
 \\
 &=&
  - \Dvect{k} \cdot
   \left[
    \nabla p \times \nabla \alpha
   \right].
\end{eqnarray*}
%
�ʤ�, 2, 3���ܤ�����ʸ�κǸ�˷Ǻܤ���Ƥ�����ظ���
�Τ褦�ʰʲ��δط��������Ѥ��Ƥ���.
%
\begin{eqnarray*}
  \nabla \times \nabla a = 0 ,
 \quad
  \nabla \times (a \Dvect{A})
  = a \nabla \times \Dvect{A} + \nabla a \times \Dvect{A} ,
 \quad
  \Dvect{A} \times \Dvect{B} = - \Dvect{B} \times \Dvect{A}.
\end{eqnarray*}
}.
���������������̤η��ˤ��뤿��,
$\partial f /\partial t = \partial f / \partial z = 0$
�����Ѥ���.
�����, (2-104)�ϰʲ��Τ褦�ˤʤ�.
%
\begin{eqnarray}
\lefteqn{
  \DP{(\zeta + f)}{t}
  + \Dvect{V} \cdot \nabla ( \zeta + f )
  + w \DP{(\zeta + f)}{t}
}
 \nonumber \\
 &=&
  - ( \zeta + f ) \nabla \cdot \Dvect{V}
  - \Dvect{k} \cdot \nabla w \times \DP{\Dvect{V}}{z}
  + \Dvect{k} \cdot \nabla p \times \nabla \alpha
  + \Dvect{k} \cdot \nabla   \times \Dvect{F} .
\end{eqnarray}
%
���μ��κ��դϤ��礦��$d(\zeta + f)/dt$�Ȥʤ�.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%   2-12-1 Divergence Equation   %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{ȯ����������}

3������ȯ����$\nabla_3 \cdot \Dvect{V}_3$�Ǥ���.
�������ʤ���, ���ʤϿ�ʿ�̤�ȯ��, $D = \nabla \cdot \Dvect{V}$,
��ʬ�����Ѥ��뤳�Ȥ�¿��.
��ͳ�����餫�Ǥ�����.
���ʤߤ�, ���̺�ɸ�Ǥ�
$\nabla \cdot \Dvect{V} = (1/a \ \cos \varphi) \partial u /\partial \lambda
+ a^{-1} \partial v / \partial \varphi - v \tan \varphi / a$
�Ǥ���
\footnote{
2������ȯ���μ���
%
\begin{eqnarray*}
  \nabla \cdot \Dvect{A}
 =
  \frac{1}{h_1 h_2}
   \left[
    \DP{}{x_1}(h_2 A_1) + \DP{}{x_2} (h_1 A_2)
   \right]
\end{eqnarray*}
%
�Ǥ���.
���̺�ɸ�ǹͤ����,
$x_1 = \lambda, \ x_2 = \varphi, \ h_1 = a \cos \varphi , \ h_2 = a, \ 
A_1 = u, \ A_2 = v$
�ʤΤ�, ��ʸ�Τ褦��2������ȯ���μ���������.
}.

$D$�λ��֤�����ʬ�μ��Ͽ�ʿ�̤Ǥα�ư��������2����ȯ����
�Ȥ뤳�Ȥ����뤳�Ȥ��Ǥ���.
�ޤ���,
%
\begin{eqnarray}
  \DP{D}{t} + \nabla \cdot [(\Dvect{V} \cdot \nabla )\Dvect{V}]
  + \nabla \cdot (f \Dvect{k} \times \Dvect{V})
  + \nabla w \cdot \DP{\Dvect{V}}{z} + w\DP{D}{z}
 \hspace{20mm}
 \nonumber \\
 =
  - \nabla \cdot (\alpha \nabla p) + \nabla \cdot \Dvect{F}
\end{eqnarray}
%
�Ȥʤ�.

������, ���ѤΤ���ˤϤ����������򤵤��Ÿ����,
�ǽ�Ū�ˤϽ����٤��㤤����������ñ�㲽����ɬ�פ�����.
�ä˿�ʿ�̤ΰ�ή�ब�ޤޤ�Ƥ��뤳�Ȥ����դ���.
(2-106)���ѷ������,
%
\begin{eqnarray}
 \DP{D}{t} + \Dvect{V} \cdot \nabla D + w \DP{D}{z}
  + \nabla w \cdot \DP{\Dvect{V}}{z} + D^2 - 2J(u,v)
  + (\Dvect{k} \times \Dvect{V}) \cdot \nabla f
  \nonumber \\
 - f \zeta = - \Ddiv (\alpha \nabla p) + \Ddiv \Dvect{F}
\end{eqnarray}
%
�Ȥʤ�
\footnote{
���μ��ѷ��˴ؤ��Ƥ�{\bfseries 2-12-A}����.
}.
������
$J(u,v) = (\partial u / \partial x) (\partial u / \partial y)
- (\partial u / \partial y) (\partial v/ \partial x)$
�Ǥ���, {\bfseries �䥳�ӥ���}({\itshape Jacobian})�ȸƤФ��.
�ʤ�餫��������̵���¤��,
2�ĤΥ����顼�ο�ʿ�̱�ư��������虜�虜
���˹⼡�α��٤�ȯ��������ʬ���������֤���������ͳ��̵��.
���θ������ˤ�������Ȥ��Ƥ�, ����ʪ���������ݤä����֤Ǥζ����
��ñ�ˤ��뤳�Ȥ��󤲤���.
������Ū�˸�����,
����ϼ��˵󤲤���Ū�Τ���˹Ԥ���.
1���ܤ�, �ϳ�Ū���Ϥ��ñ�ˤ��뤿��,
2���ܤ�, ��������α�ư�˥ե��륿���򤫤�����, ���������ꤹ�뤿��,
3���ܤ�, ����ͽ���ǥ���ñ�ˤ��뤿��,
���뤤�Ϥʤ�餫��¾����Ū�Τ���Ǥ���.

(2-107)�λϤ��5��Ͼ�����, �����̵��Ǥ���.
�ޤ�, �Ϥ��3�����Ȥ����Ȥ�, �����ξ�������뤳�Ȥ��Ǥ���.
Ʊ�ͤ�, ����������(2-105)�Ǥ�
��ľ®�٤�ޤ��ȥ�ȥ�å���
\footnote{
 (2-102)�ˤ�����$u \tan \varphi / a$�ι�ʤ�.
},
������$\zeta \Ddiv \Dvect{V}$
�Ϥ褯̵�뤵���.
�����Τ褦���͡��ʹ���ӽ��ϥ�������󥰤ˤ�ä�
Ŭ���ʲ��꤬Ϳ�����Ƥ�����ˤϹԤ����Ȥ��Ǥ���.
���Τ��Ȥ���4�Ϥ��о줹��.

����������줿���٤�ȯ�����������ϱ�ľ��ɸ�ϤȤ���$z$��
�ޤ�Ǥ���.
���Ǥ�¾�α�ľ��ɸ��ͤ���.
����ˤ�ä��������η��ˤ��Ѳ����ǤƤ���.
�ä�, �褯�Ȥ����ľ��ɸ��$p$�ǤϤ����Ĥ���ñ�㲽����ǽ�Ȥʤ�.
�㤨��, ��ʿ�̰��Ϥ�$- \nabla \phi$�Ȥ������ˤʤ�
\footnote{
1-9-1�μ�(2-58)����.
}.
��ä�, $\Dvect{k} \cdot \nabla \times \nabla \phi =0$�Ȥʤ�
\footnote{
ȯ���β�ž�ʤΤǥ����Ȥʤ�.
}.
���Τ褦�ˤ���, ȯ���μ�(2-107)�α��դκǽ�ι��
$- \nabla^2 \phi$�Ȥʤ�, ��(2-104)�Ǥ�
$\Dvect{k} \cdot \nabla p \times \nabla \alpha$���б�����褦��
���̵���Ƥ���.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%             �ʲ���Appendix             %%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%           ���ֹ�Υ������饤��         %%%%%%%% 
\setcounter{equation}{0}
\setcounter{figure}{0}
\def\theequation{A-\arabic{equation}}
\def\thefigure{A-\arabic{figure}}
\def\thetable{A-\arabic{table}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\begin{center}
% {\bf 
% --------------------------------------
% ��ʸ�����ޤ�
% --------------------------------------
% } 
%\end{center}

\subsection*{2-12-A\qquad ȯ���μ����ѷ�}
\setcounter{equation}{0}
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\def\thefigure{2-12-A-\arabic{figure}}
\def\thetable{2-12-A-\arabic{table}}


��(2-106)���鼰(2-107)�ؤμ��ѷ��ˤĤ��Ʋ��⤹��.

(2-106)��(2-107)����٤�ȼ�(2-106)��
%
\begin{eqnarray}
 \Deqlab{hw-02-12-(2-106)���ѷ���}
 \Ddiv \left[ \left( \Dvect{V} \cdot \nabla \right) \Dvect{V} \right]
 + \Ddiv \left( f \Dvect{k} \times \Dvect{V} \right)
\end{eqnarray}
%
��, (2-107)��
%
\begin{eqnarray}
 \Deqlab{hw-02-12-(2-107)���ѷ���}
 \Dvect{V} \cdot \nabla D + D^2 - 2 J(u,v)
 + \left( \Dvect{k} \times \Dvect{V} \right) \cdot \nabla f - f \zeta
\end{eqnarray}
%
�Ȥ��������Ϥ��Ǥ���.
�ޤ�, \Deqref{hw-02-12-(2-106)���ѷ���}����1���ܤ���׻�����.
%
\begin{eqnarray}
  \Ddiv \left[ \left( \Dvect{V} \cdot \nabla \right) \Dvect{V} \right]
 &=&
  \Ddiv
   \left[
    \left( u\DP{}{x} + v\DP{}{y} \right) u \Dvect{i}
    + \left( u\DP{}{x} + v\DP{}{y} \right) v \Dvect{j}
   \right]
 \nonumber \\
 &=&
   \DP{}{x} \left( u\DP{}{x} + v\DP{}{y} \right) u
   + \DP{}{y} \left( u\DP{}{x} + v\DP{}{y} \right) v
 \nonumber \\
 &=&
   \DP{}{x} \left( u\DP{u}{x} + v\DP{u}{y} \right)
   + \DP{}{y} \left( u\DP{v}{x} + v\DP{v}{y} \right)
 \nonumber \\
 &=&
  \DP[][]{u}{x}^2 + u \DP[2]{u}{x}
  + \DP{v}{x} \DP{u}{y} + v \frac{\partial^2 u}{\partial x \partial y}
 \nonumber \\
 & &
 \Deqlab{hw-02-12-(2-106)���ѷ������1��}
  + \DP{u}{y} \DP{v}{x} + u \frac{\partial^2 v}{\partial x \partial y}
  + \DP[][]{v}{y}^2 + v \DP[2]{v}{y}.
\end{eqnarray}
%
����, \Deqref{hw-02-12-(2-107)���ѷ���}����1���ܤ�׻�����.
%
\begin{eqnarray}
  \Dvect{V} \cdot \nabla D
 &=&
  \left( u \DP{}{x} + v \DP{}{y} \right)
  \left( \DP{u}{x} + \DP{v}{y} \right)
 \nonumber \\
 &=&
  u \DP{}{x} \left( \DP{u}{x} + \DP{v}{y} \right)
  v \DP{}{y} \left( \DP{u}{x} + \DP{v}{y} \right)
 \nonumber \\
 &=&
 \Deqlab{hw-02-12-(2-107)���ѷ������1��} 
  u \DP[2]{u}{x} + u \frac{\partial^2 v}{\partial x \partial y}
  + v \frac{\partial^2 u}{\partial x \partial y} + v \DP[2]{v}{y}.
\end{eqnarray}
%
\Deqref{hw-02-12-(2-107)���ѷ���}����2���ܤ�,
%
\begin{eqnarray}
  D^2
 &=&
  (\Ddiv \Dvect{V})^2
 \nonumber \\
 &=&
  \left( \DP{u}{x} + \DP{v}{y} \right)^2
 \nonumber \\
 &=&
 \Deqlab{hw-02-12-(2-107)���ѷ������2��}   
  \DP[][]{u}{x}^2 + 2 \DP{u}{x} \DP{v}{y} + \DP[][]{v}{y}^2
\end{eqnarray}
%
�Ȥʤ�.
�Ǹ��, \Deqref{hw-02-12-(2-107)���ѷ���}����3���ܤ�,
%
\begin{eqnarray}
  - 2J(u,v)
 &=&
 \Deqlab{hw-02-12-(2-107)���ѷ������3��}   
  - 2 \DP{u}{x} \DP{v}{y} + 2 \DP{u}{y} \DP{v}{x}
\end{eqnarray}
%
�Ȥʤ�.
\Deqref{hw-02-12-(2-107)���ѷ������1��},
\Deqref{hw-02-12-(2-107)���ѷ������2��},
\Deqref{hw-02-12-(2-107)���ѷ������3��}���,
\Deqref{hw-02-12-(2-106)���ѷ������1��}��
%
\begin{eqnarray}
\Deqlab{hw-02-12-(2-106)���ѷ������1��η��}
 \Ddiv \left[ \left( \Dvect{V} \cdot \nabla \right) \Dvect{V} \right]
 =
 \Dvect{V} \cdot \nabla D + D^2 - 2J(u,v)
\end{eqnarray}
%
�Ȥʤ�.


����, \Deqref{hw-02-12-(2-106)���ѷ���}����2���ܤ�׻�����.
��Ͽ�ο��ظ���(A-3)
$\nabla \cdot (a \Dvect{A})
= a \Ddiv \Dvect{A} - \Dvect{A} \cdot \nabla a$
���,
%
\begin{eqnarray}
 \Deqlab{hw-02-12-(2-106)���ѷ������2��}   
  \Ddiv \left( f \Dvect{k} \times \Dvect{V} \right)
 &=&
  f \Ddiv (\Dvect{k} \times \Dvect{V})
  + (\Dvect{k} \times \Dvect{V}) \cdot \nabla f
\end{eqnarray}
%
�Ȥʤ�.
������, \Deqref{hw-02-12-(2-106)���ѷ������2��}
�α�����2���\Deqref{hw-02-12-(2-107)���ѷ���}����4��Ȱ��פ���Τ�,
\Deqref{hw-02-12-(2-106)���ѷ������2��}�α�����1�ब
\Deqref{hw-02-12-(2-107)���ѷ���}����5��Ȱ��פ��뤳�Ȥ�Τ������ɤ�.
\Deqref{hw-02-12-(2-106)���ѷ������2��}�α�����1����ѷ������,
��Ͽ�ο��ظ���(A-6)
$\Ddiv (\Dvect{A} \times \Dvect{B})
= \Dvect{B} \cdot \nabla \times \Dvect{A}
- \Dvect{A} \cdot \nabla \times \Dvect{B}$
���,
%
\begin{eqnarray}
  f \Ddiv (\Dvect{k} \times \Dvect{V})
 &=&
  f \Dvect{V} \cdot (\nabla \times \Dvect{k})
  - f \Dvect{k} \cdot (\nabla \times \Dvect{V})
 \nonumber \\
 &=&
 \Deqlab{hw-02-12-(2-107)���ѷ������5��}    
  -f \zeta
\end{eqnarray}
%
�Ȥʤ�. $\Dvect{k}$�ϻ����Ѳ����ʤ�����,
$\nabla \times \Dvect{k} = 0$�Ȥʤä�.

\Deqref{hw-02-12-(2-106)���ѷ������1��η��},
\Deqref{hw-02-12-(2-106)���ѷ������2��},
\Deqref{hw-02-12-(2-107)���ѷ������5��}���
��(2-106)��(2-107)���ѷ��Ǥ���.




\subsection*{2-12-B\qquad �����������γƹ�ΰ�̣}
\setcounter{equation}{0}
\setcounter{figure}{0}
\def\theequation{2-12-B-\arabic{equation}}
\def\thefigure{2-12-B-\arabic{figure}}
\def\thetable{2-12-B-\arabic{table}}


�ʲ��α���������(��ʸ�Ǹ������(2-105)��)�γƹ�ΰ�̣�ˤĤ��ƹͤ���.
%
\begin{eqnarray}
 \Deqlab{hw-02-12-����������}
  \DD{(\zeta + f)}{t} =
  - (\zeta + f) \Ddiv \Dvect{V}
  - \Dvect{k} \cdot \nabla w \times \DP{\Dvect{V}}{z}
  + \Dvect{k} \cdot \nabla p \times \nabla \alpha
  + \Dvect{k} \cdot \nabla \times \Dvect{F}.
\end{eqnarray}


\subsubsection*{2-12-B-a \quad ȯ����({\itshape divergence term})}

\Deqref{hw-02-12-����������}�α�����1��
$-(\zeta + f) \Ddiv \Dvect{V}$��
{\bfseries ȯ����}({\itshape divergence term})�ȸƤФ��.
����ϲ�ž���Ƥ���������ѱ�ư�̤���¸���ʤ���, ���β�žȾ�¤�
�Ѳ�������ȳ�®�٤��Ѳ����뤳�Ȥ˻��Ƥ���.
ȯ�������Ǥ��뤳�Ȥ�, �����β�žȾ�¤����礹�뤳�Ȥ���������.
�դĤ�ŷ���ޤǸ�����褦���絬�Ϥʱ�ư�Ǥ����б���$(\zeta + f)$
�����Ǥ��뤫��, ȯ�������б��٤򸺾�����, ��«�����б��٤����ä�����
(\Dfigref{hw-02-12-ȯ����}����).
%
\begin{figure}[htbp]
 %\vspace*{10mm}
 \begin{center}
  \Depsf[][60mm]{./eps/vorticity.eps}
 \end{center}
 \begin{quote}
  \caption{
  {\footnotesize ȯ���ȼ�«�ˤ�뱲�٤�ȯ��(���б��٤����Ȳ���)}
  }
  \Dfiglab{hw-02-12-ȯ����}
 \end{quote}
\end{figure}
%

\subsubsection*{2-12-B-b \quad Ω������({\itshape tilting term})}

\Deqref{hw-02-12-����������}�α�����2��
$- \Dvect{k} \cdot \nabla w \times \partial \Dvect{V} / \partial z$
��{\bfseries Ω������}({\itshape tilting term}
or {\itshape twisting term})�ȸƤФ��
\footnote{
 {\bfseries ���й�}�Ȥ�ƤФ��.
}.
��ľ®�٤�ʿ����ǰ��ͤǤʤ�����, �Ϥ��ʿ�����˸����Ƥ������μ���
��ľ������Ω���夬��, ���٤α�ľ��ʬ���Ѳ�������̤򼨤�
\footnote{
 ������⼫���ʤ��Ȥ�����ľ®�٤����ͤʾ��ϱ���Ω���夬�뤳�Ȥ�
 ̵������, ���ι�Ͼ��Ǥ���.
}.
$- \Dvect{k} \cdot \nabla w \times \partial \Dvect{V} / \partial z$
�򥹥��顼ɽ�������
$\displaystyle - \DP{w}{x} \DP{v}{z} + \DP{w}{y} \DP{u}{z}$
�Ȥʤ�.
�ޤ�, $\displaystyle - \DP{w}{x} \DP{v}{z}$�ι��ͤ���.
�����Ǥ�$v$����ˤ����ˤĤ�Ƹ�����, $w$��$x$���������ä��Ƥ������
�ͤ���. ���ξ��,
$\displaystyle - \DP{w}{x} \DP{v}{z}$�����Ȥʤ�,
���б���$(\zeta + f)$�����ä���.
�����\Dfigref{hw-02-12-Ω���夬���}�ǹͤ��Ƥߤ�.
$\partial v / \partial z$����Ǥ��뤳�Ȥ���ޤˤ���褦��$x$������
���Ӥ뱲�δɤ��Ǥ���. �����$\partial w/ \partial x$�����Ǥ��뤳�Ȥ���
���α��ɤ��ޤ�������������˲�ž��, Ω���夬��.
���β�ž��������ͤ���, ���η�̱��٤����ä��뤳�Ȥ��狼�������.
$(\partial w / \partial y) (\partial u / \partial z)$�ι�˴ؤ��Ƥ�
Ʊ�ͤ˹ͤ��뤳�Ȥ��Ǥ���.
���ʤߤˤ��ι�ϥ᥽���ݤ�ε����ȯ�������������˽��פ������̤���
�ȸ�����.
%
\begin{figure}[htb]
 \begin{center}
  \Depsf[][]{./eps/tilte.eps}
 \end{center}
 \begin{quote}
  \caption{
  {\footnotesize $v$��$z$�����Ѳ��ˤ�ä�$x$������ʿ�Ԥʱ��ɤ�����,
  ���줬�����$w$��$x$�������Ѳ��ˤ�ä�Ω���夬�äƱ�ľ������
  ������ʬ����.}
  }
 \Dfiglab{hw-02-12-Ω���夬���}
 \end{quote}
\end{figure}
%


\subsubsection*{2-12-B-c \quad ����Υ��ɹ�({\itshape solenoidal term})}

\Deqref{hw-02-12-����������}�α�����3��
$\Dvect{k} \cdot \nabla p \times \nabla \alpha$��
{\bfseries ����Υ��ɹ�}({\itshape solenoidal term})�ȸƤФ��.
�����\Dfigref{hw-02-12-����Υ��ɹ�}�ǹͤ���ȴ�ñ�Ǥ���.
%
\begin{figure}[htb]
 \begin{center}
  \Depsf[][]{./eps/solenoid.eps}
 \end{center}
 \begin{quote}
  \caption{
  {\footnotesize (a)�Ǥ���$p$�̤���$\alpha$�̤��򺹤��Ƥ��뤿��,
  ����������. ������Ф�, (b)�Ǥ���$p$�̤���$\alpha$�̤�
  ʿ�ԤˤʤäƤ��뤿�ᱲ�������ʤ�.
  (Peixoto, J. P. and A. H. Oort, 1992:
   Physics of Climate.
   {\itshape Springer-Verlag}, 520pp. ���)}
  }
  \Dfiglab{hw-02-12-����Υ��ɹ�}
 \end{quote}
\end{figure}
%
(a)�Ǥ���$p$�̤���$\alpha$�̤��򺹤��Ƥ���.
���ϸ��ۤˤ�ä�ή�줬������������, ���κݤ�(a)�Τ褦��̩�ٹ�¤��
�ʤäƤ����, ̩�٤κ��ˤ�ä�ή��˺����Ǥ�, 
���η�̤Ȥ��Ʊ���������.
���줬����Υ��ɹ�θ��̤Ǥ���.
������Ф���(b)�Ǥ���$p$�̤���$\alpha$�̤�ʿ�ԤˤʤäƤ���.
���ξ��ˤ�ή��˺��ϤǤ���, ���������ʤ�.

���ʤߤ�, ��ľ��ɸ��$p$��Ȥä����, �����Τ褦��
$\nabla_p p = 0$�Ȥʤ�, ����Υ��ɹ�Ͼ��Ǥ���.
���������κǸ�Ǥ�Ҥ٤��Ƥ��뤳�ȤǤ���.

�����β���ˤϾ���(1978), ����(2000), Peixoto(1992)
�򻲾Ȥ���.





\subsection*{2-12-C\qquad ���٤ȳ�®�٤δط�}
\setcounter{equation}{0}
\setcounter{figure}{0}
\def\theequation{2-12-C-\arabic{equation}}
\def\thefigure{2-12-C-\arabic{figure}}
\def\thetable{2-12-C-\arabic{table}}


��ʸ�Ρֱ��٤ϳ�®�٤�2�ܤǤ���פȤ���ɽ���ˤĤ��ƹͻ�����.


\subsubsection*{2-12-C-a \quad ®�٥٥��ȥ�ȳ�®�٤δط�}

�������̤���꼴�μ���˹��Τ���ž��ư���Ƥ���Ȥ���.
��®�٥٥��ȥ��$\Dvect{\Omega}$, �ư���$\Dvect{r} = (x,y,z)$�ˤ�����
®�٤�$v$�Ȥ����, $\Dvect{v} = \Dvect{\Omega} \times \Dvect{r}$��
�ط���������Ω�Ĥ��Ȥ򼨤�.

��ñ�Τ���, ��ž����$z$���ȤȤ뤳�Ȥ�$\Dvect{\Omega} = (0,0, \Omega)$
�ȹͤ���.
���ξ��,
%
\begin{eqnarray}
 \Dvect{\Omega} \times \Dvect{r} =
  \left(
   \begin{array}{ccc}
    \Dvect{i} & \Dvect{j} & \Dvect{k} \\
    0 & 0 & \Omega \\
    x & y & z \\
   \end{array}
  \right) =
  \left(
   \begin{array}{c}
    -y \Omega \\ x \Omega \\ 0 \\
   \end{array}
  \right)
  \nonumber
\end{eqnarray}
%
�Ȥʤ�. ����, ®�٥٥��ȥ��
\Dfigref{hw-02-12-��®��}���ʬ����褦��
$\Dvect{v} = -y \Omega \Dvect{i} + x \Omega \Dvect{j}$�Ǥ��뤿��,
�ط���$\Dvect{v} = \Dvect{\Omega} \times \Dvect{r}$������Ω��.
%
\begin{figure}[htbp]
 \begin{center}
  \Depsf[][]{./eps/angular.eps}
 \end{center}
 \begin{quote}
  \caption{
   {\footnotesize z���μ�����ž����ʪ�Τ�®�٤�$x$, $y$��ʬ}
  }
  \Dfiglab{hw-02-12-��®��}
 \end{quote}
\end{figure}
%

\subsubsection*{2-12-C-b \quad ���٤ȳ�®�٤Ȥδط�}

�嵭�������ط���$\Dvect{v} = \Dvect{\Omega} \times \Dvect{r}$��Ȥä�
���٤ȳ�®�٤δط�����ͤ���.
���δط�����ξ�դ�$\nabla \times$�򤫤���, �Ĥޤ�ϲ�ž��Ȥ�.
�ʤ�, $\Dvect{r} = (x, y, z)$, $\Dvect{\Omega} =
(\Omega_x, \Omega_y, \Omega_z)$�Ȥ��Ƥ���.
%
\begin{eqnarray}
 \nabla \times \Dvect{v} &=&
  \nabla \times (\Dvect{\Omega} \times \Dvect{r})
  \nonumber \\
 &=&
  \nabla \times
  \left(
   \begin{array}{ccc}
    \Dvect{i} & \Dvect{j} & \Dvect{k} \\
    \Omega_x & \Omega_y & \Omega_z \\
    x & y & z \\
   \end{array}
  \right)
  \nonumber \\
 &=&
  \nabla \times
  \left(
   \begin{array}{ccc}
    z \Omega_y - y \Omega_z \\
    x \Omega_z - z \Omega_x \\
    y \Omega_x - x \Omega_y \\
   \end{array}
  \right)
  \nonumber \\
 &=&
  \left(
   \begin{array}{ccc}
    \Dvect{i} & \Dvect{j} & \Dvect{k} \\
    \DP{}{x} & \DP{}{y} & \DP{}{z} \\
    z \Omega_y - y \Omega_z & x \Omega_z - z \Omega_x &
     y \Omega_x - x \Omega_y \\
   \end{array}
  \right)
  \nonumber \\
 &=&
  \left(
   \begin{array}{c}
    2 \Omega_x \\ 2 \Omega_y \\ 2 \Omega_z \\
   \end{array}
  \right)
  \nonumber \\
 &=&
  2 \Dvect{\Omega}
  \nonumber
\end{eqnarray}
%
���μ����鱲��$\nabla \times \Dvect{v}$����®��$\Dvect{\Omega}$��
2�ܤǤ��뤳�Ȥ��狼��.
�ʤ�, ���β���ˤ�����(1999)�򻲾Ȥ���.






%\subsection*{Appendix \qquad }

%\subsubsection*{}

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