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\Dtitle{��×͊w���f���̕������n}
\Dauthor{���� ��}
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\begin{document}
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\Dtitle{��×͊w���f���̕������n}
\author{�_�ˑ�w���w���n���f���Ȋw�� 
\\ �n������јf����C�Ȋw������
\\ �w��4�N�@�@�@���ց@��}
\date{2009/02/28}
\maketitle
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\begin{abstract}
~~�{�_���ł͋C�ی����m�[�g~(1999)~�̐ē��a�Y~��,~�w����~��×͊w���f���̕��ށx
�Ɋ�Â��Ĕ�×͊w�������n�̒莮�����s����.

�܂��͔�×͊w�n���f�����g���K�v����,~�×͊w�������n�Ɣ�×͊w�������n�ł̈����^���̃X�P�[����,~
�p����ߎ��̈Ⴂ�̔�r��������,~�×͊w�n���f���Ɣ�×͊w�n���f���ł̍~���\���̌��ʂ��r���邱�Ƃ�
�l�@����.~
��×͊w�������n�̒莮���ɂ��Ă�,~�܂�,~��×͊w�������n�͘A���̎��̋ߎ��̎d���ɂ�蕪��
�����̂ł��̕��ނɂ��Ă���q�ׂ�.~
�����ĕ��ނ��ꂽ�������n���Ƃ̊�b�������n�̒莮�����s����.~
��×͊w�������n�͉��������̉^��,~���Ȃ킿�Η����d�v�ɂȂ�悤�ȉ^���������̂ɓK���Ă���.~
���̕������n�ōs����ߎ��Ƃ��Ă�,~�A���̎��ŕ\������C�̖��x�̋ߎ���,~��̕ϐ�����{��ƕ΍��ɕ���,~
��{��̒l�����������ɂ݈̂ˑ�����Ƃ�����,~�萔�Ƃ��Ĉ����Ƃ����悤�ȋߎ�,~��{��܂��ł̕������n��
���`�ߎ��Ȃǂ�����.~�����,~���ꂼ��̋ߎ��ɉ�����,~���ނ��ꂽ�������n���ǂ̂悤�Ȍ��ۂɈ����ׂ����ɂ��Ă�
�ȒP�ɒ莮���Ƌ��ɍs����.
\end{abstract}
\tableofcontents
\chapter{�͂��߂�}
\markright{�͂��߂�}
~~�ϗ��_�͋����㏸���ɂ�蔭�B��,~�������W�����J�������炷.~
������,~�V�C�\��ɂ����ďW�����J��\�����邱�Ƃ͑�Ϗd�v�ł���.~�V�C�\��̂��߂�
���l�V�~�����[�V�������f���ł���C�ے���NHM~�ł�,~���ۂɔ�×͊w�������n���p�����Ă���.~
���̂���,~���������̉^�����d�v�ł���ϗ��_�𐔒l�V�~�����[�V��������ɂ͔�×͊w�������n���K���邾�낤.~ 
NHM~�̑��ɂ�,~NCAR~(�č���C�Ȋw�Z���^�[)~��PSU~(�y���V���o�j�A�B����w)~�ɂ�郁�\�X�P�[��
���f����MM5,~CSU~(�R�����h�B����w)~��RAMS~�Ȃǂ̐��l�V�~�����[�V�������f���ɂ�
��×͊w�������n���p�����Ă���.~
������,~��×͊w�������n��p�������l�V�~�����[�V�������f��~(��×͊w�n���f��)~
��,~�Η��Ȃǂ̏��X�P�[���̌��ۂ𗝉����邽�߂̌�������Ƃ��ĊJ������,~�C�ے��̓V�C�\�񃂃f���Ȃǂɂ�
�×͊w�������n�Ɋ�Â����̂��p�����Ă���.~������,~��L�̂悤�Ȑ��l�V�~�����[�V�������f����
����悤�Ɍ��݂ł͔�×͊w���f������Ƃ��ėp������悤�ɂȂ���.~

�{�����ł͔�×͊w�������n�Ɋ�Â����l�V�~�����[�V�������f���̕K�v���̍l�@,~
��×͊w�������n�̕��ށE�莮��������.~
�{���̍\���͈ȉ��̂Ƃ���ł���.~��2~�͂Ŕ�×͊w�n���f���̕K�v���ɂ��čl�@����.~
��3~�͂Ŕ�×͊w�������n�̕��ނɂ��ĘA���̎���p���Ď�����.~
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�����đ�5~�͂��܂Ƃ߂ł���.
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\chapter{��×͊w�n���f���̕K�v��}
\markright{��×͊w�n���f���̕K�v��}
\section{�×͊w�n�Ɣ�×͊w�n}
��ɂ��q�ׂ��悤��,~����,~�C�ے��ŗp�����Ă����V�C�\�񃂃f���Ȃǂ͐×͊w�������n�Ɋ�Â����̂ł�����.~
�܂���,~�Ȃ��×͊w�n����ɗp�����Ă����̂����l����.~
���̂��߂�,~�×͊w�n�Ɣ�×͊w�n�ň�����^���̈Ⴂ�ɂ��ĊȒP�ɏq�ׂ�.
�×͊w�n�ł͑�C�̉��������̉����x���ɂ߂ď��������̂ƌ��Ȃ�,~
�����̈��͌��z�Əd�͂��������Ƃ���×͊w�ߎ�,
%
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\begin{equation}
\frac{1}{\rho} \frac{\partial p}{\partial z} + g = 0
\end{equation}
%
%
�����Ă���.~������$\rho$~�͑�C�̖��x,~$p$~�͋C��,~$g$~�͏d�͉����x,~$z$~�͉��������̒����X�P�[���ł���.~
�×͊w�n�ň����Ă���̂�,~(�^���̐����X�P�[��)~$\gg$~(�^���̉����X�P�[��)~�Ƃ�����ł���.
���Ă͂܂���̂Ƃ��Ă�,~���ϋK��~(����~km)~�̋C�ی��ۂ������ł���,~
��Ƃ��Ăӂ��V�C�}�Ɍ����鉷�ђ�C����ړ������C��,~�C���̒J������Ȃǂł���.

���ɔ�×͊w�n�ł�,~�������̉����x���ƂĂ��傫�����ߐ×͊w�ߎ����g���Ȃ�����l����.
~���̂���3~�����̉^�����������������ƂƂȂ�×͊w�n�����v�Z�����G�ƂȂ�.~
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���̉^���������ł���.

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�×͊w�������n���p�����Ă����̂��낤.~

�v�Z���ȒP���ł���Ƃ����_�݂̂ɒ��ڂ����,~�×͊w�������n��p����΂����悤�Ɏv����.~
����ɂ�������炸,~��×͊w�������n��p����͉̂��̂Ȃ̂��낤��.~����ɂ��Č������邽��,~���߂ł�
�×͊w�E��×͊w�������n�Ɋ�Â����l�V�~�����[�V�������f���ɂ��\���l�̐��x���r���čl�@����.
%
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\section{�×͊w�n�E��×͊w�n���f���̗\���l�̔�r}
Kato~(1997)~�ł͐×͊w�E��×͊w�������n�Ɋ�Â����l���f���ł̃V�~
�����[�V�������ʂ̔�r���s����.~�}2.2.1~�͋C�ی������ŊJ�����ꂽ��×͊w���f��~(Ikawa and Saito,~1991)~�Ƃ��̃��f��
�̉����̉^����������×͊w���t�̎��ɒu���������×͊w���f��~(Kato and Saito,~1995)~��p���č~���Ɋւ���\������,~
���̗\���l���r�������̂ł���.~
%
%
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���̃��f���ł͎O�̍~���X�L�[��~(PES, PMS, MAS)~��p���Ă���.~�~���X�L�[����,~���l�V�~�����[�V�������f���ɂ�����
�~�����ۂ����邱�Ƃ��ǂ̂悤�ɕ\���̂������߂Ă�����̂ł���̂�,~�~���X�L�[�����قȂ�ꍇ�ł̗\���l�̔�r���s���Ă���.~
%\footnote{�~���X�L�[���Ƃ͐��l�V�~�����[�V�������f�����`�������ł̍~���ߒ��̕\���̎d���ł���,~���ꂼ��ȉ��̂悤�ɂȂ��Ă���
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%�X�̐ω_�𒼐ڗ\�񂷂邽�߂ɕ���\����~km~�ȉ��̔�×͊w���f���ł悭�p������.~
%���̉׏d�͈ȉ��̎��ɂ���悤�ɉ����̉^��������~($\alpha = 1$)~��×͊w�̎�~($\alpha = 0$)~��ʂ��ĉE�ӑ�O���ɕ��̕���
%�Ƃ��ē���.~
%\begin{equation}
%\alpha \biggl( \rho \frac{dw}{dt} - \nu \nabla^2 w \biggr) = -\frac{\partial p}{\partial z} + B - g \rho (q_c + q_r)\nonumber
%\end{equation}
%������,~$\nu$~�͓��S���W��,~$B$~�͕���,~$q_c,q_r$~�͂��ꂼ��_���ƉJ���̍�����ł���.
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%����������܂ł̊Ԃ̏����̌��ʂ�������Ă���.
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�X�L�[���ɂ���đ����̈Ⴂ�͂��邪,~
�×͊w�n���f���͔�×͊w�n���f���ɔ�ׂ�,~���~���ʁE�~���ʐς��ߑ�]�����Ă��邱�Ƃ��킩��.~


�×͊w�n�͉^���������ɐ×͊w�ߎ���p���邱�ƂŌv�Z���ȗ����ł���Ƃ������_������.~
����ɑ΂�,~��×͊w�n�ł͐×͊w�ߎ����g��Ȃ�����$3$~�����̉^�����������K�v�ł���,~�������n�����G�ɂȂ�.
~������,~�㏸������z����ϗ��_�𐔒l�V�~�����[�V��������ɂ͔�×͊w�������n���K�v�ł���.~
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���g�����Ƃ������I�Ȃ��̂ƂȂ���.
\begin{figure}[h]
\begin{center}
\includegraphics[width=4cm,angle=90,trim= 230 0 120 0,clip]{fig-2-1.eps}
\caption{\footnotesize{�ϕ�����~(9����)~�̊Ԃɔ�×͊w���f�����\�z����,~1~mm~�ȏ�̍~�����������̈�ʐ�$P_{a>1}$~�y�ё��~����$P_{total}$,
~��×͊w���f���ɑ΂���×͊w���f���̗\���l�̔�.~H~��N~�͂��ꂼ��×͊w~(hydrostatic)~���f���Ɣ�×͊w~(Non-hydrostatic)~���f��
���Ӗ�����.~PES,~PMS,~MAS~�͈قȂ�O�̍~���X�L�[���ł���.}}
\label{fig:Pedlosky-1987}
\end{center}
\end{figure}
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\chapter{��×͊w�������n�̕���}
\markright{��×͊w�������n�̕���}
~~�����ł͔�×͊w�������n�̕��ނɂ��ďq�ׂ�.~

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%
%
\[\frac{\partial \rho}{\partial t} + \frac{\partial \rho u}{\partial x}+ \frac{\partial \rho v}{\partial y} + \frac{\partial \rho w}{\partial z} =0\]
%
%
�ɂ�����$\partial \rho / \partial t$~�����邩�ǂ����ŕ\�����.~
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�e���������n�Ɣ�e���������n�͖��x�̋ߎ��̒��x�ɂ���Ă���ɕ��ނł���.~���ނ��ꂽ�e�������n�̒莮����
���͂ōs�����̂Ƃ�,~�����ł͂��ꂼ��̕������n�̖��̂Ɩ��x�̋ߎ��̒��x�ɂ��ċL��.

���S���k�������n:~�ߎ����s��Ȃ������ȘA���̎�
%
\[\frac{\partial \rho}{\partial t} + \frac{\partial \rho u}{\partial x}+ \frac{\partial \rho v}{\partial y} + \frac{\partial \rho w}{\partial z} =0\]
%
��p����.

�����k�������n:~��C�̈��k���͔F�߂邪,~���x��
\[\rho = \overline{\rho}(z) + \rho ' (x,y,z,t) \]
�̂悤�ɐ�����l�Ȋ�{��̒l$\overline{\rho}(z)$~�Ƃ�������̕΍�$\rho '(x,y,z,t)$~�ɕ���,~
���x����{��̂܂��Ő��`�ߎ�����
%
%
\begin{equation}
\frac{1}{\overline{\rho}} \frac{d \rho}{d t} + \biggl( \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} \biggr) =0\nonumber
\end{equation}
%
%
��p����.~������$d/dt$~�̓��O�����W�������ł���.

��e���������n:~�����k���f���Ɠ��l�ɖ��x�𐅕���l�Ȋ�{��Ƃ�������̕΍��ɕ���,~��C�̈��k���𖳎������A���̎�
\[
\frac{\partial \overline{\rho} u}{\partial x} + \frac{\partial \overline{\rho} v}{\partial y} + \frac{\partial \overline{\rho} w}{\partial z} =0
\]
��p���Ă���.~�Ȃ�,~�����ł�,~��{��̒l�ɔ�ׂĕ΍����͂邩�ɏ������Ƃ���
$\nabla \cdot (\rho' \Dvect{u})$~�𖳎����Ă���.

�񈳏k�������n:~��C�̈��k���𖳎���,~��e���������n�Ɠ��l��,~���x����{��ƕ΍��ɕ����čl���邪,~
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\[
\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} =0
\]
�ł���.~������,~��{��̒l�ɔ�ׂĕ΍����͂邩�ɏ������Ƃ���$\nabla \cdot (\rho' \Dvect{u})$~�𖳎����Ă���.
%�Ȃ�,~��Ƃ��đ��͂Ŕ�×͊w���f���Ƃ��ċ�����MM5,~RAMS,~NHM~�𕪗ނ��Ă���.
%
%$\star$~MM5:~���S���k�n
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%$\star$~RAMS:~�����k
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%$\star$~NHM~:~���S���k�n

�Ō�Ƀ��f���̕��ނɂ��Ă܂Ƃ߂��ȒP�Ȏ��`�}���ڂ��Ă���.
%
%
\begin{figure}[h]
\begin{center}
\includegraphics[width=10cm,angle=90,trim= 0 0 0 0,clip]{bunruihyo.eps}
\caption{\footnotesize{��×͊w�������n�̕��ނɂ��Ă̎��`�}.}}
\label{fig:Pedlosky-1987}
\end{center}
\end{figure}
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\chapter{��×͊w�������n�̒莮��}
\markright{��×͊w�������n�̒莮��}
\section{���̗͊w�ɂ������b������}
3~�͂ł��q�ׂ��悤��,~��×͊w�������n�͖��x�̋ߎ��̎d���ɂ���Ă������̕������n�ɕ��ނ����.~
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���̕������n�͑�C�̏�Ԃ�\�����߂̂��̂ł��邩�痬�̗͊w�ɂ������b�������n�ƌ���,~������,~���x��
�ߎ����s���Ă��Ȃ����Ƃ��犮�S���k�������n�ł�����.~
�����Ō�����C�̏�ԂƂ͋C����C���Ȃǂ̕����ʂł���.~�܂�,~���s�������W�n�Ŋ�����C�ɂ��čl���邽��,~
��C�̏�Ԃ�,~�C��$T$,~�C��$p$,~����3~����~$(u,v,w)$~�ŕ\�������.~�����ɖ��x$\rho$~��
�������ȉ���6~�̊�b�����������݂���.~
�Ȃ�,~��������̒莮���ł͊ȒP�̂��ߒn����]�̉e���𖳎���,~������C���������̂Ƃ��Ă���.

�E�A���̎�:
\begin{equation}
\Deqlab{renzoku}
\frac{\partial \rho}{\partial t} + \frac{\partial \rho u}{\partial x}+ \frac{\partial \rho v}{\partial y} + \frac{\partial \rho w}{\partial z} =0
\end{equation}
������$\rho$~�͑�C�̖��x,~$(u,v,w)$~�͕�����3~�����ł���.

�E�i�r�G�E�X�g�[�N�X�̉^��������:
\begin{align}
\Deqlab{�i�r�Gx}
\frac{du}{dt} + \frac{1}{\rho} \frac{\partial p}{\partial x} &= \nu \nabla^2 u, \\
\Deqlab{�i�r�Gy}
\frac{dv}{dt} + \frac{1}{\rho} \frac{\partial p}{\partial y} &= \nu \nabla^2 v, \\ 
\Deqlab{�i�r�Gz}
\frac{dw}{dt} + \frac{1}{\rho} \frac{\partial p}{\partial z} +g &= \nu \nabla^2 w. 
\end{align}
%
������$p$~�͋C��,~$g$~�͏d�͉����x,~$\nu$~�͓��S�����ł���.~
�e�������̍��ӑ�񍀂͈��͌X�x��,~�E�ӂ͗����S������\��.~
�܂�,~�����̉^���������̍��ӑ�O���͒P�ʑ̐ς�����̏d�͂ł���.

�E�M�͊w�̎�
\footnote{(4.1.5)~�̓��o�͕t�^A.1~���Q��.}:
\begin{equation}
\Deqlab{�M��}
\frac{d \theta}{dt} = \frac{\mathcal{Q}}{C_p \pi} + \frac{\kappa}{\pi}\nabla^2 T.
\end{equation}
������$T$~�͉��x,~$\theta$~�͉���,~$\pi$~�̓G�N�X�i�[�֐�~(�������������C��),~
$\mathcal{Q}$~�͔�f�M���M,~$C_p$~�͊�����C�̒舳��M,~$R$~�͊�����C�ɑ΂���C�̒萔,~$\kappa$~�͔M�g�U���ł���.~
����̓G���g���s�[�̒�`����M�͊w���@���Ɖ���$\theta$~�ɂ���ĕ\�������̂ł���.

�E��ԕ�����:
\begin{equation}
\Deqlab{���1}
p = \rho RT.
\end{equation}
%

�܂�,~$\theta$~��$\pi$~�͂��ꂼ��ȉ��Œ�`�����.
%
%
\begin{align}
\Deqlab{pi}
\pi &\equiv \biggl(\frac{p}{p_0}\biggr)^{R/C_p}, \\
\Deqlab{theta}
\theta &\equiv \frac{T}{\pi}.
\end{align}
%
������$p_0$~�͕W���C���ł���.

�����܂ł�6~�̕ϐ��ɑ΂���6~�̕����������݂��邽�ߗ��̗͊w�̊�b�������n��
������,~�]���ϐ��Ƃ���$p$�̑����$\pi$��p����ꍇ�̉^���������������Ă���.

$\Deqref{pi}$,~$\Deqref{theta}$~����ԕ�����~$\Deqref{���1}$~��,
\begin{equation}
\Deqlab{���2}
\rho = \frac{p_0}{R \theta} \biggl(\frac{p}{p_0}\biggr)^{C_v / C_p}
\end{equation}
%
%
�ƂȂ�
%
%
\footnote{$\Deqref{���2}$~�̓��o������.~(4.1.7),~(4.1.8)~���
\[\frac{1}{T}= \frac{1}{\pi \theta} = \frac{1}{\theta}\biggl(\frac{p}{p_0}\biggr)^{-R/C_p}\tag{a1}\]
�ł��邩��,~$\Deqref{���1}$~�ɑ������,
\begin{equation}
\rho = \frac{p}{R \theta}\biggl(\frac{p}{p_0}\biggr)^{-R/C_p} = \frac{1}{R \theta} p^{\frac{C_p-R}{C_p}}p_0^{\frac{R}{C_p}}
     = \frac{1}{R \theta} p^{\frac{C_p-R}{C_p}}p_0^{-\frac{C_p-R}{C_p}}
     = \frac{p_0}{R \theta} \biggl(\frac{p}{p_0}\biggr)^{C_v /C_p} \nonumber \tag{4.1.10}
\end{equation}
�𓾂�.}
%
%
.~������$C_v$~�͊�����C�̒�ϔ�M�ł���,~$C_v = C_p - R$~�ł���.~����~(4.1.7),~(4.1.8),~$\Deqref{���2}$~���,
\begin{equation}
\Deqlab{dpdpi}
\frac{1}{\rho}dp = C_p \theta d \pi
\end{equation}
�ƂȂ�
%
%
\footnote{$\Deqref{dpdpi}$~�̓��o�͕t�^A.2~���Q��.}
%
%
%
%
.~�����,~(4.1.10)~��~(4.1.2)~�`~(4.1.4)~�ɗp����,
\begin{align}
\Deqlab{�i�r�Gpix}
\frac{du}{dt} + C_p \theta \frac{\partial \pi}{\partial x} &= \nu \nabla^2 u, \\
\Deqlab{�i�r�Gpiy}
\frac{dv}{dt} + C_p \theta \frac{\partial \pi}{\partial y} &= \nu \nabla^2 v ,\\ 
\Deqlab{�i�r�Gpiz}
\frac{dw}{dt} + C_p \theta \frac{\partial \pi}{\partial z} +g &= \nu \nabla^2 w
\end{align}
%
%
�𓾂�.~����ɂ��$p$~�̑����$\pi$~��p�����^��������������ꂽ.
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\section{��e���������n}
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\setcounter{equation}{0}
���̐߂ł͔�×͊w�������n�̓���,~��C�̈��k���𖳎������������n�ł����e���������n�̒莮�����s��.
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\subsection{��e���������n}
��e���������n�͑�C�̈��k���𖳎������������n��,~��̕ϐ�$\phi(x,y,z,t)$~�͈ȉ��̂悤��
������l�Ȋ�{��$\overline{\phi}(z)$~�Ƃ�������̕΍�$\phi(x,y,z,t)$~�ɕ����Ď�舵����.
%
%
\begin{equation}
\Deqlab{hensu}
p=\overline{p}+p',~\rho = \overline{\rho} + \rho ',~\theta = \overline{\theta} + \theta '.
\end{equation}
%
%

�A���̎���
%
%
\begin{equation}
\Deqlab{renzoku2}
\frac{\partial \overline{\rho} u}{\partial x}+ \frac{\partial \overline{\rho} v}{\partial y} + \frac{\partial \overline{\rho} w}{\partial z} =0
\end{equation}
%
%
�ł���.~�����ő�C�̈��k���̖����̂��ߖ��x�̎��ԕω����𖳎���,~�܂�,~��{��̒l�ɔ�ׂĕ΍����͂邩�ɏ������Ƃ���
$\nabla\cdot(\rho'\Dvect{u})$~���������Ă���.~

~�^����������
%%
\begin{align}
\Deqlab{�i�r�Gx2}
\frac{du}{dt} + \frac{1}{\overline{\rho}} \frac{\partial p'}{\partial x} &= \nu \nabla^2 u, \\
\Deqlab{�i�r�Gy2}
\frac{dv}{dt} + \frac{1}{\overline{\rho}} \frac{\partial p'}{\partial y} &= \nu \nabla^2 v, \\ 
\Deqlab{�i�r�Gz2}
\frac{dw}{dt} + \frac{1}{\overline{\rho}} \biggl(\frac{\partial p'}{\partial z} +\rho 'g\biggr) &= \nu \nabla^2 w 
\end{align}
%
�ƂȂ�
\footnote{$\Deqref{�i�r�Gx2}$~�`~$\Deqref{�i�r�Gz2}$~�̓��o�͕t�^A.3~���Q��.}
.~������,~��{��̒l�ɑ΂��Ă͐×͊w�̎�
\begin{equation}
\Deqlab{seiriki}
\frac{1}{\overline{\rho}} \frac{\partial \overline{p}}{\partial z} + g = 0
\end{equation}
%
���g���Ă���.

�M�͊w�̎���~(4.1.5)~�Ɋ�{��ƕ΍��ɕ��������ʂ����������̂�,
%
%
\begin{equation}
\Deqlab{netsuriki}
\frac{\partial \theta'}{\partial t} = \frac{\mathcal{Q}}{C_p \pi} + \frac{\kappa}{\pi}\nabla^2 T
\end{equation}
�ł���,~��ԕ�������~$\Deqref{���1}$~��p����.

�����̉^���������͏�ԕ��������疧�x�΍��𓱂����Ƃł���ɕό`�ł���̂ňȉ�
�Œ莮��������.

(4.1.6)�`(4.1.8)~���,
\begin{equation}
\Deqlab{mitsudohensa}
\rho ' \simeq -\overline{\rho} \frac{\theta '}{\overline{\theta}} + \frac{p'}{\overline{C_s^2}}
\end{equation}
%
�𓾂�\footnote{$\Deqref{mitsudohensa}$~�̓��o�͕t�^A.4~���Q��.}.
~������,~$\Deqref{mitsudohensa}$~�̉E�ӂ͊�{��̒l�ɂ����`�ߎ���\��.~�܂�$C_s^2$~�͉����̎���ł���,~
%
\begin{equation}
\Deqlab{onsoku}
C_s^2 = \frac{C_p}{C_v}RT
\end{equation}
%
�ŕ\����,~�C���▧�x�Ɠ��l��,~��{��̒l$\overline{C_s^2}(z)$~�ƕ΍�$C_s'(x,y,z,t)$~�ɕ������Ă���.~
���̂悤�ɖ��x�̕΍������ʂ̕΍��ƋC���̕΍��ɂ���ĕ\���ł���.~
(4.2.3)��~$\Deqref{�i�r�Gz2}$~�ɑ������Ɖ����̉^����������,
%
%
\begin{equation}
\Deqlab{�i�r�Gz3}
\frac{dw}{dt} + \frac{1}{\overline{\rho}} \biggl(\frac{\partial p'}{\partial z} +\frac{p'}{\overline{C_s^2}}g\biggr) = \frac{\theta '}{\overline{\theta}}g + \nu \nabla^2 w
\end{equation}
%
%
�ƂȂ�.~���ӑ�O��,~�E�ӑ�ꍀ�͖��x�̕΍��ɂ�镂�͂�\��.

�����܂łŕ������n�͕��Ă��邪,~�^�����������ȉ��̌`�ɕό`���邱�Ƃ��ł���.

�܂��̓t���b�N�X�`��
\footnote{�t���b�N�X�`���Ƃ�,~�����ʂ̎��ԕω�����ԂɌŒ肵�������̐ςւ̗����E���o�ɂ��\�����邱�Ƃł���.}
�̉^���������𓱏o����.

�A���̎�~$\Deqref{renzoku2}$~��p�����,
\begin{align}
\Deqlab{renzoku3}
\overline{\rho} \frac{d \phi}{dt} &= \overline{\rho} \biggl(\frac{\partial \phi}{\partial t} + u\frac{\partial \phi}{\partial x} + v \frac{\partial \phi}{\partial y} + w\frac{\partial \phi}{\partial z} \biggr) \nonumber \\
                                  &= \frac{\partial \overline{\rho} \phi}{\partial t} + \frac{\partial \overline{\rho} u \phi}{\partial x} + \frac{\partial \overline{\rho} v \phi}{\partial y} + \frac{\partial \overline{\rho} w \phi}{\partial z} 
\end{align}
�𓾂�
\footnote{$\Deqref{renzoku3}$~�𓱏o����.~�܂�,$\displaystyle \frac{d}{dt} = \frac{\partial}{\partial t} + (\Dvect{u} \cdot \nabla)$~
�ł��邩��
\[
\rho \frac{d \phi}{dt} = \rho \biggl(\frac{\partial \phi}{\partial t} + u\frac{\partial \phi}{\partial x} + v \frac{\partial \phi}{\partial y} + w\frac{\partial \phi}{\partial z} \biggr) \tag{a2}
\]
�ł���.~����
\[
\frac{\partial \rho u \phi}{\partial x} = \phi \frac{\partial \rho u}{\partial x} + \rho u \frac{\partial \phi}{\partial x},~\frac{\partial \rho v \phi}{\partial y} = \phi \frac{\partial \rho v}{\partial y} + \rho v \frac{\partial \phi}{\partial y},~\frac{\partial \rho w \phi}{\partial z} = \phi \frac{\partial \rho w}{\partial z} + \rho w \frac{\partial \phi}{\partial z}
\]
�ł���.~������ӁX����,~$\rho = \overline{\rho} + \rho'$~�Ƃ��Đ��`�ߎ��ɂ��,~$\rho'$~���܂ލ��𖳎���,
~�����$\Deqref{renzoku2}$~��p�����,
\begin{align}
\frac{\partial \overline{\rho} u \phi}{\partial x} + \frac{\partial \overline{\rho} v \phi}{\partial y} + \frac{\partial \overline{\rho} w \phi}{\partial z} &= \phi \biggl(\frac{\partial \overline{\rho} u}{\partial x} + \frac{\partial \overline{\rho} v}{\partial y} + \frac{\partial \overline{\rho} w}{\partial z} \biggr) +  \overline{\rho} \biggl( u \frac{\partial \phi}{\partial x} + v \frac{\partial \phi}{\partial y} + w \frac{\partial \phi}{\partial z} \biggr) \nonumber \\
&=�@\overline{\rho} \biggl( u \frac{\partial \phi}{\partial x} + v \frac{\partial \phi}{\partial y} + w \frac{\partial \phi}{\partial z} \biggr) \nonumber
\end{align}
�𓾂�.~�����~(a2)~�ɑ������~$\Deqref{renzoku3}$~�ƂȂ�.
}
.~������^��������~$\Deqref{�i�r�Gx2}$,~$\Deqref{�i�r�Gy2}$,~$\Deqref{�i�r�Gz3}$~�ɗp����ƃt���b�N�X�`���̉^���������������,~
%
%
\begin{align}
\Deqlab{�i�r�Gfx1}
\frac{\partial \overline{\rho} u}{\partial t} + \biggl( \frac{\partial \overline{\rho} u^2}{\partial x} +\frac{\partial \overline{\rho} uv}{\partial y} + \frac{\partial \overline{\rho} uw}{\partial z} \biggr) + \frac{\partial p'}{\partial x} &= \overline{\rho} \nu \nabla u ,\\
\Deqlab{�i�r�Gfy1}
\frac{\partial \overline{\rho} v}{\partial t} + \biggl( \frac{\partial \overline{\rho} vu}{\partial x} +\frac{\partial \overline{\rho} v^2}{\partial y} + \frac{\partial \overline{\rho} vw}{\partial z} \biggr) + \frac{\partial p'}{\partial y} &= \overline{\rho} \nu \nabla v, \\
\Deqlab{�i�r�Gfz1}
\frac{\partial \overline{\rho} w}{\partial t} + \biggl( \frac{\partial \overline{\rho} wu}{\partial x} +\frac{\partial \overline{\rho} wv}{\partial y} + \frac{\partial \overline{\rho} w^2}{\partial z} \biggr) + \frac{\partial p'}{\partial z} + \frac{p'}{\overline{C_s^2}}g&= \overline{\rho} \frac{\theta'}{\overline{\theta}}g + \overline{\rho} \nu \nabla w 
\end{align}
%
%
�ƂȂ�.~�����Ŋe�������̍��ӑ�񍀂̓t���b�N�X�`���̈ڗ����ł���.~
�t���b�N�X�`���ւ̏��������ɂ�,~���ʋ��E��ʂ��Ă̎��ʈڗ����v�Z���₷��
�Ȃ�Ƃ������_������.

���ɋC���Ƃ���$p$~�̑����$\pi$~��p����������n�ɂ�����^���������𓱏o����.

�܂�,~��{��ɑ΂���×͊w�̎���,
%
%
\begin{equation}
\Deqlab{seiriki2}
C_p \overline{\theta} \frac{\partial \overline{\pi}}{\partial z} + g = 0
\end{equation}
%
%
�Ə���������
\footnote{$\Deqref{seiriki2}$~�𓱏o����.~$\Deqref{dpdpi}$~���Čf�����,
\[\frac{1}{\rho}dp = C_p \theta d \pi \tag{4.1.10}\]
�ł��邩��,~���̗��ӂ�$z$~�ŕΔ�������
\[
\frac{1}{\rho}\frac{\partial p}{\partial z} = C_p \theta \frac{\partial \pi}{\partial z}
\]
�������{��̒l�ɓK�p����̂Ő��`�ߎ����s��,
\[
\frac{1}{\overline{\rho}}\frac{\partial \overline{p}}{\partial z} = C_p \overline{\theta} \frac{\partial \overline{\pi}}{\partial z}
\]
�Ƃ���.~�����~(4.2.6)~�ɑ������
\[C_p \overline{\theta} \frac{\partial \overline{\pi}}{\partial z} + g = 0 \tag{4.2.15}\]�𓾂�.}
%
%
.~$\Deqref{seiriki2}$~��p����
%
%
\begin{equation}
\Deqlab{seiriki3}
C_p \theta \frac{\partial{\pi}}{\partial z} + g = C_p (\overline{\theta} +\theta ') \frac{\partial (\overline{\pi} + \pi ')}{\partial z} + g 
\simeq C_p \overline{\theta} \frac{\partial \pi '}{\partial z} - \frac{\theta '}{\overline{\theta}}g
\end{equation}
�𓾂�
\footnote{$\Deqref{seiriki3}$~�̓��o�͕t�^A.5~���Q��.}
%
%
.~$\Deqref{seiriki3}$~�ɂ����~$\Deqref{�i�r�Gx2}$~�`~$\Deqref{�i�r�Gz2}$~��,
%
%
\begin{align}
\Deqlab{�i�r�Gpix2}
\frac{du}{dt} + C_p \overline{\theta} \frac{\partial \pi '}{\partial x} &= \nu \nabla^2 u, \\
\Deqlab{�i�r�Gpiy2}
\frac{dv}{dt} + C_p \overline{\theta} \frac{\partial \pi '}{\partial y} &= \nu \nabla^2 v, \\ 
\Deqlab{�i�r�Gpiz2}
\frac{dw}{dt} + C_p \overline{\theta} \frac{\partial \pi '}{\partial z} &= \frac{\theta '}{\overline{\theta}}g + \nu \nabla^2 w 
\end{align}
%
%
�Ƌߎ������.~$\Deqref{�i�r�Gz3}$~��~$\Deqref{�i�r�Gpiz2}$~���r����Ƃ킩��悤��,~$\Deqref{�i�r�Gpiz2}$~�ɂ�
���������̉^���������ɕ��͂Ɋւ���C���΍��̍����o�Ă��Ȃ����Ƃ�,~�\��ϐ���$\pi$~��p����
�����̈�̗��_�ł���.

�ʏ�, ���̂̈��k���̉e���������ł��Ȃ��傫���ɂȂ�̂̓}�b�n���ɂ���0.3 �ȏ�ƌ�����.~
����͂��悻����400~$\mathrm{km}$~���炢�ł���.~����䂦�ɒʏ�̋C�ی��ۂł�
��C�̈��k���͖����ł���قǏ������ƌ���,~���̂��ߔ�e���ߎ��͗ǂ����x�Ő��藧���̂ł���.~
������,~������l�Ȋ�{��܂��ł̐��`�ߎ����s������,~�Ⴆ�΋C���΍����傫���Ȃ�悤�ȏ�,~���C���ƒ�C�����ǂ�����܂��
���܂��悤�Ȑ��������ɍL���̈��,~�������̊Ԋu������,~���Ȃ킿�����������傫���̈�Ȃǂɂ͕s�����ł��낤.

%��e���ߎ��͒ʏ�̋C�ی��ۂɑ΂��ĊT���ėǂ����x�łȂ肽��.~
%�������������n����{��̖��x�Ő��`�����邽�ߖ��x�ω����傫���Ȃ�悤�ȏ�ł͌덷���傫���Ȃ��Ă��܂�.
%%�܂�,~�A���̎���~(4.1.1')~�̂悤�ɋߎ����邱�Ƃ͗̈���̎��ʕۑ���O��ɂ��邽��,~���̂܂܂ł͕��ϋC���̕ω�
%%��\���ł��Ȃ�.~�܂�,~�R���܂ޏꍇ�ɂ͋C�������������G�ɂȂ�,~�}�b�v�t�@�N�^�[�Ȃǂ��l������̂�
%%�ȒP�ł͂Ȃ�.~���̂��ߍL���̈��ΏۂƂ����C����̗\����܂ޒ����Ԃ̗\������Ȃǂɂ͕s�����ł���.
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\subsection{�񈳏k�������n}
�񈳏k�������n��,~��C�̈��k���͖�����,~��̕ϐ������ԁE�ꏊ�ɂ��Ȃ���l�Ȓl�Ƃ��Ă̊�{��$\phi_0$~��
��������̕΍�$\phi'(x,y,z,t)$~�ɕ����čl��,~���Ȃ킿�ȉ��̂悤�ɂȂ�.
\[
\rho = \rho_0 + \rho' ,~ p = p_0 + p' ,~ \theta = \theta_0 + \theta'
\]
%
�A���̎���,~�ϐ��Ɋ�{��̒l��p���邽�ߔ񈳏k�̎�
\begin{equation}
\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0
\end{equation} 
�ƂȂ�.~���̏ꍇ�̕������n�̓u�V�l�X�N�n�ƌĂ΂��.

�^����������,
%
%
\begin{align}
\frac{du}{dt} + \frac{1}{\rho_0} \frac{\partial p'}{\partial x} &= \nu \nabla^2 u \\
\frac{dv}{dt} + \frac{1}{\rho_0} \frac{\partial p'}{\partial y} &= \nu \nabla^2 v \\
\frac{dw}{dt} + \frac{1}{\rho_0} \frac{\partial p'}{\partial z} &= \frac{\theta '}{\theta_0}g + \nu \nabla^2 w
\end{align}
�ƂȂ�.~�����͔�e�����f���̉^�����������ȗ����������̂ł���.~
���Ȃ킿,~$\Deqref{�i�r�Gx2}$,~$\Deqref{�i�r�Gy2}$,~$\Deqref{�i�r�Gz3}$~�ɂ���
��̕ϐ�~($\rho,p,\theta$)~��
��{��̒l~($\rho_0,p_0,\theta_0$)~�ƕ΍�~($\rho',p',\theta'$)~�ɕ���,~
�����~$\Deqref{�i�r�Gz3}$~�̍��ӑ�񍀂̕��͂Ɋւ���C���΍��������������̂ɂȂ��Ă���.

�M�͊w�̎��Ə�ԕ������͂��ꂼ��,~$\Deqref{netsuriki}$~��~$\Deqref{���1}$~�ł���.

�񈳏k�n���f����,~��b�����⍂�x�ɂ�閧�x�ω����傫���Ȃ�Ȃ��󂢑Η���Ώۂɂ����V�~�����[�V�����ɗp�����邱�Ƃ�����.~
������,~��{��̒l��萔�Ƃ�����,~�����̉^�����������畂�͂Ɋւ���C���΍������������肷��悤�ɋߎ����傫������,~
�Η����S�w�̂悤�ɍL���͈͂ł̍Č������ɂ͎g���Ȃ�.
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\section{�e���������n}
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��������͑�C�̈��k�����l������e���������n�̒莮�����s��.
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\subsection{�����k�������n}
�����k�������n��,~�A���̎��ɑ�C�̈��k����F��,~���U����C����\�񂷂邪,~
���x�͊�{��̒l��p����.~�܂���̕ϐ��͔�e���������n�Ɠ��l�Ɋ�{��ƕ΍��ɕ����Ĉ���.

�����k���f���͋C����$p$~��p����ꍇ��$\pi$~��p����ꍇ�ƂŘA���̎����قȂ邽��
���ꂼ��ɂ��Ă̕������n�𓱂�.

�E�C����$p$~��p����ꍇ

�A���̎��͈ȉ��̂��̂�p����.
%
%
\begin{equation}
\Deqlab{zyunasshukup}
\frac{\partial \rho '}{\partial t} + \frac{\partial \overline{\rho} u}{\partial x} + \frac{\partial \overline{\rho} v}{\partial y} + \frac{\partial \overline{\rho} w}{\partial z} =0.
\end{equation}
�����ł�$\displaystyle \DP{\overline{\rho}}{t} + \nabla\cdot(\rho'\Dvect{u})$~�𖳎������Ă���.~�܂�,~
���ӑ�ꍀ�ɑ�C�̈��k�����\����Ă���.
%

�^���������͔�e���������n�ɂ�����t���b�N�X�`���̉^�����������g��,~$\Deqref{�i�r�Gfx1}$~�`$\Deqref{�i�r�Gfz1}$~�ł���.

�M�͊w�̎��Ə�ԕ������͂��ꂼ��~$\Deqref{netsuriki}$~��$\Deqref{���1}$~�ł���.

$p$~��p����ꍇ�̋C���̗\���������
%
%
\begin{equation}
\Deqlab{dpdt1}
\frac{\partial p'}{\partial t} = \overline{C_s^2}\biggl\{\frac{\overline{\rho}}{\overline{\theta}} \frac{\partial \theta'}{\partial t} - \biggl( \frac{\partial \overline{\rho} u}{\partial x} + \frac{\partial \overline{\rho} v}{\partial y} + \frac{\partial \overline{\rho} w}{\partial z}\biggr)\biggr\}
\end{equation}
%
%
�ƂȂ�\footnote{\Deqref{dpdt1}~�̓��o�͕t�^A.6~���Q��.}.~
�E�Ӓ����ʓ��̑�ꍀ����C�̔M�c���Ɋւ����f�M���ƔM�g�U�ł���,~��񍀈ȉ�����C�̎��ʃt���b�N�X�̔��U��\��,~�A���̎����爳�k���������Ă��邱�Ƃ��킩��.~

�������n�͘A���̎��E�^���������E�M�͊w�̎��E��ԕ������ŕ���.~�������C���̗\������������߂邱�Ƃ�,~
���x�E�����E�C���E���x��6~�̖��m�ϐ��ɑ΂���,~6~�̎��ԂɊւ���1~�K�����������ŕ������n�������悤�ɂȂ�.~

�C����$p$~��p����ꍇ�̏����k�������n��,~�t���b�N�X�`���̉^����������p���Ă���.~
������,~���̉^����������,~��e���������n�ɂ�����^���������ł��邽��,~��C�̈��k���𖳎������A���̎���p���ē�������̂ł���.~
���̂��ߑ�C�̈��k����F�߂鏀���k�������n�ł͌����ɂ͉^�����������t���b�N�X�`���ɂ͂ł���,~�덷�������Ă��܂�.

�E�C����$\pi$~��p����ꍇ

�A���̎���
%
%
\begin{equation}
\Deqlab{zyunasshukupi}
\frac{1}{\overline{\rho}} \frac{d \rho}{dt} + \biggl( \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} \biggr) =0
\end{equation}
%
%
�ƂȂ�\footnote{$\Deqref{zyunasshukupi}$~�̓��o�͕t�^A.7~���Q��.}.

�^���������͔�e�����f���Ɠ��l�ɐ��`������~$\Deqref{�i�r�Gpix2}$~�`~$\Deqref{�i�r�Gpiz2}$~��p����.

�M�͊w�̎��Ə�ԕ������͂��ꂼ��~$\Deqref{netsuriki}$~��$\Deqref{���1}$~�ł���.

����$\pi$~�ɂ��C���̗\�񎮂𓱏o����.~��ԕ�����~$\Deqref{���2}$~�̕ό`,
%
%
\begin{equation}
\Deqlab{���3}
\pi = \biggl( \frac{\rho R \theta}{p_0}\biggr)^{R/C_v}
\end{equation}
%
%
����
\footnote{$\Deqref{���3}$~��$\Deqref{���2}$~�̗��ӂ�$R/C_v$~�悵,~$\pi$~�ɂ��Đ�������Γ�����.}
$\pi$~�̑ΐ���������`�ߎ�������,
%
%
\begin{equation}
\Deqlab{dpidt1}
\frac{1}{\pi} \frac{d \pi}{dt} = \frac{R}{C_v} \biggl( \frac{1}{\overline{\rho}} \frac{d \rho}{dt} + \frac{1}{\overline{\theta}} \frac{d \theta}{dt} \biggr)
\end{equation}
%
%
��
%
%
\footnote{\Deqref{dpidt1}~�̓��o�͕t�^A.8~���Q��.}
%
%
,~�����k�̘A���̎�~$\Deqref{zyunasshukupi}$,~�����̎���̎�~$\Deqref{onsoku}$~��p����
%
%
$\pi$~�̗\�������,
%
%
\begin{equation}
\Deqlab{dpidt2}
\frac{d \pi}{dt} = -\frac{R \pi}{C_v} \biggl( \frac{\partial u}{\partial x} +\frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} \biggr) + \frac{C_s^2}{C_p \overline{\theta}^2} \frac{d \theta}{dt} 
\end{equation}
�𓾂�
%
%
\footnote{$\Deqref{dpidt2}$~��~$\Deqref{onsoku}$~��~$\Deqref{zyunasshukupi}$~��~$\Deqref{dpidt1}$~�ɑ����,~$\theta'$�𖳎�����Γ�����.}
%
%
.~�E�ӑ�ꍀ�������̔��U��\��,~��C�̈��k���Ɋ֌W���Ă���,~
�E�ӑ�񍀂��M�c���Ɋ֌W����C���ω��ł���,~��f�M���M�ƔM�g�U�ŕ\�����.~
�C����$p$~��p����ꍇ�Ɠ��l��,~$\pi$~�ɂ��C���̗\�񎮂����߂邱�Ƃ�,~
���x�E�����E�C���E���x��6~�̖��m�ϐ��ɑ΂���,~6~�̎��ԂɊւ���1~�K�����������ŕ������n�������悤�ɂȂ�.

�����k�������n�ł͔�e���������n�ł͖���������C�̈��k�����l�������A���̎���p���Ă���.~���̂���
6~�̖��m�ϐ��ɑ΂���,~6~�̎��ԂɊւ���1~�K������������������.~��e���������n�̊�b��������
��ԕ������̂悤�Ȏ��Ԕ����̍����܂܂Ȃ��f�f����,~�^���������̂悤�Ȏ��Ԕ����̍����܂ޗ\�񎮂̍��݂���
���̂ł��邱�Ƃ��l�����,~�����k�������n�͂��̊�b�������n���܂Ƃ߂₷���Ƃ������_������.~
���������x�ɂ��Ă�,~��C�̈��k���𖳎������A���̎��ɂ���ē���ꂽ��e���������n�̃t���b�N�X�`���̉^��������
��p���邽��,~��C�̈��k�����l�������A���̎���p���鏀���k�������n�ł͌����ɂ͉^�����������t���b�N�X�`���ɂ�
�ł���,~��e���������n�Ƃقړ����x�̐��x�ɂȂ��Ă��܂�.
�܂�,~�����k�������n���K����̈�ɂ��Ă�,~
��e���������n�Ɠ��l�ɐ�����l�Ȋ�{��ɂ����`�ߎ����s�����Ƃ���,~�����̈�Ɍ����邾�낤.~
������,~��C�̈��k���͍l�����Ă��邽��,~�\���J�̂悤�ɏu�ԍő啗�����ƂĂ��傫���Ȃ�,~��C�̈��k��
�̍l�����K�v�Ȍ��ۂɎg����_�Ŕ�e���������n���D��Ă��邾�낤.
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\subsection{���S���k�������n}
���S���k�������n�͑�C�̈��k����F��,~�ߎ����s��Ȃ������ȘA���̎��Ə�ԕ���������C����\�񂷂���̂ł���.~ 
�����ŏ�̕ϐ��͊�{��̒l$\overline{\phi}$~�ƕ΍�$\phi'$~�ɕ����Ĉ���.~
�Ȃ�,~��{��̒l�̈����ɂ��Đ�ɂ܂Ƃ߂Ă���.~
�C���̊�{��̒l$\overline{p}(z)$~�ɂ��Ă͐×͊w�̎�~$\Deqref{seiriki}$~��p����.~
�܂�,~�������n�ɑ΂��Ċ�{��̂܂��ł̐��`�ߎ��͍s��Ȃ�.

�A���̎���,~$\Deqref{renzoku}$�ł���.

�A���̎�~$\Deqref{renzoku}$~��p�����,
\begin{align}
\Deqlab{renzoku4}
\rho \frac{d \phi}{dt} &= \rho \biggl( \frac{\partial \phi}{\partial t} + u\frac{\partial \phi}{\partial x} + v\frac{\partial \phi}{\partial y} + w\frac{\partial \phi}{\partial z} \biggr) \nonumber \\
                       &= \frac{\partial \rho \phi}{\partial t}  + \frac{\partial \rho u \phi}{\partial x} + \frac{\partial \rho v \phi}{\partial y} + \frac{\partial \rho w \phi}{\partial z} - \phi \biggl( \frac{\partial \rho}{\partial t} + \frac{\partial \rho u}{\partial x}+ \frac{\partial \rho v}{\partial y} + \frac{\partial \rho w}{\partial z} \biggr)\nonumber \\
                       &= \frac{\partial \rho \phi}{\partial t}  + \frac{\partial \rho u \phi}{\partial x} + \frac{\partial \rho v \phi}{\partial y} + \frac{\partial \rho w \phi}{\partial z}
\end{align}
�ƂȂ�,~�C����$p$~��p����t���b�N�X�`���̃��f���ł͉^���������͈ȉ��̂悤�ɂȂ�.
\begin{align}
\Deqlab{�i�r�Gfx2}
\frac{\partial \rho u}{\partial t} + \biggl( \frac{\partial \rho u^2}{\partial x} +\frac{\partial \rho uv}{\partial y} + \frac{\partial \rho uw}{\partial z} \biggr) + \frac{\partial p'}{\partial x} &= \nu \rho \nabla^2 u, \\
\Deqlab{�i�r�Gfy2}
\frac{\partial \rho v}{\partial t} + \biggl( \frac{\partial \rho vu}{\partial x} +\frac{\partial \rho v^2}{\partial y} + \frac{\partial \rho vw}{\partial z} \biggr) + \frac{\partial p'}{\partial y} &= \nu \rho \nabla^2 v, \\
\Deqlab{�i�r�Gfz2}
\frac{\partial \rho w}{\partial t} + \biggl( \frac{\partial \rho wu}{\partial x} +\frac{\partial \rho wv}{\partial y} + \frac{\partial \rho w^2}{\partial z} \biggr) + \frac{\partial p'}{\partial z} &= -\rho ' g + \nu \rho \nabla^2 w.
\end{align}
%
%
�t���b�N�X�`���ł͑��ʋ��E��ʂ��Ă̎��ʈڗ����v�Z���₷�����Ƃ����_�ł���.~
�܂�,~���ӑ�ꍀ�͎��ʃt���b�N�X�̎��ԕω�,~��񍀂͈ڗ���,~��O���͈��͌X�x�͂ł���,~
�E�ӂ�$\nabla^2$~���|���鍀�͊g�U���ł���.~
�����̉^���������̉E�ӑ�ꍀ�͖��x�̕΍��ɂ�镂�͂ł���,~��ԕ�������p���Ė��x�΍����璼�ڋ��߂���.~

�M�͊w�̎��Ə�ԕ������͂��ꂼ��~$\Deqref{netsuriki}$~��\Deqref{���1}�ł���.

�����ŋC���̕΍��͏�ԕ���������,
%
%
\begin{equation}
\Deqlab{mitsudohensa2}
p' = {C_s^2}\biggl(\rho ' + \rho \frac{\theta '}{\theta}\biggr)
\end{equation}
%
%
�ƂȂ�.~$\Deqref{mitsudohensa2}$~��4.2~�߂ŏ�ԕ��������瓾�����x�̕΍��̎�~$\Deqref{mitsudohensa}$~��
���`�ߎ����s���Ă��Ȃ����̂�,~�C���΍�
�ɂ��Đ����������̂ł���.~$\Deqref{mitsudohensa2}$~�ƘA���̎�~$\Deqref{renzoku}$~��p���ċC���̗\�������
\begin{equation}
\Deqlab{dpdt2}
\frac{\partial p'}{\partial t} = - C_s^2 \biggl( \frac{\partial \rho u}{\partial x} + \frac{\partial \rho v}{\partial y} + \frac{\partial \rho w}{\partial z} \biggr) + C_s^2 \frac{\rho}{\theta} \frac{\partial \theta'}{\partial t}
\end{equation}
�𓾂�
\footnote{$\Deqref{dpdt2}$~��$\Deqref{mitsudohensa2}$~��$t$~�Ɋւ��ĕΔ�����,~�A���̎�~$\Deqref{renzoku}$~
��������Γ�����.}
.~�E�ӑ�ꍀ�͕����ɂ���C�̔��U�ɋN������C���ω��ł���.~����͘A���̎��ƌ���ׂ�Ƃ킩��悤��
$\partial \rho / \partial t$~�Ɠ��l�ł���,~��C�̈��k����\�����Ƃ��킩��.~
�E�ӑ�񍀂͔M�I�ȋC���ω���\���Ă���,~��C�̔M�c����\���Ă���.~���̔M�I�ȕω��Ƃ�,~
�M�͊w�̎�����킩��悤�ɔ�f�M���M�ƔM�g�U�ɂ����̂ł���.~
�㎮�͋C���̎��ԕω����C�̎��ʃt���b�N�X$\rho \Dvect{u}$~�̔��U�Ɖ��x�̎��ԕω��ɒu�������Ă���.~
����Č��݂̕���, �C��, ���x�̒l��^����΋C���̒P�ʎ��ԓ�����̕ω��ʂ������,~�C���̗\��l���킩��.~

�������n�͘A���̎�,~�^��������,~�M�̎�,~��ԕ������ŕ��Ă���.~������,~�C���̗\�񎮂𓱏o���邱�Ƃ�,~
���x�E�����E�C���E���x��6~�̖��m�ϐ��ɑ΂���,~6~�̎��ԂɊւ���1~�K�����������ŕ������n�������悤�ɂȂ�.

�����܂ł͋C����$p$~��p����������n������ꂽ��,~�C����$\pi$~��p���ĕ\�����Ƃ��ł���̂ňȉ��ɓ���.~

�A���̎���4.3~�߂ŋ��߂������k�̘A���̎�~$\Deqref{zyunasshukupi}$~����`�ߎ�����O�̎��ł���,~����͖��x�Ɋ�{��̒l��
�p���Ȃ����̂�,~�ȉ��̂悤�ɂȂ�.~
%
%
\begin{equation}
\Deqlab{asshuku}
\frac{1}{\rho} \frac{d \rho}{dt} + \biggl( \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} \biggr) =0 
\end{equation}
%
%

�^����������4.1~�߂Ŏ�����,~$\Deqref{�i�r�Gpix}$~�`~$\Deqref{�i�r�Gpiz}$~�ł���.

�M�̎�,~��ԕ������͋C����$p$~��p����ꍇ�Ɠ��l��,~���ꂼ��~$\Deqref{netsuriki}$~��\Deqref{���1}�ł���.

����,~$\pi$~��ΐ���������
\begin{equation}
\Deqlab{dpidt3}
\frac{1}{\pi} \frac{d \pi}{dt} = \frac{R}{C_v} \biggl( \frac{1}{\rho} \frac{d \rho}{dt} + \frac{1}{\theta} \frac{d \theta}{dt} \biggr)
\end{equation}
��
\footnote{$\Deqref{dpidt3}$~��~$\Deqref{dpidt1}$~�����߂���@�Ɠ��l�ɂ��ē�����.}
,~�A���̎�~$\Deqref{asshuku}$~�Ɖ����̎�~$\Deqref{onsoku}$~��p���ċC��������
\begin{equation}
\Deqlab{dpidt4}
\frac{d \pi}{dt} = -\frac{R \pi}{C_v} \biggl(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z}\biggr) + \frac{C_s^2}{C_p \theta^2}\frac{d\theta}{dt}
\end{equation}
�𓾂�.~�C����$p$~��p����ꍇ�Ɠ��l��,~$\pi$~�ɂ��C���̗\�񎮂����߂邱�Ƃ�,~
���x�E�����E�C���E���x��6~�̖��m�ϐ��ɑ΂���,~6~�̎��ԂɊւ���1~�K�����������ŕ������n�������悤�ɂȂ�.

���S���k���f���͕������n�ɋߎ����܂܂Ȃ��_�ŏ����k�n���e���n�������x�̓_�ŏ���,~������̈�
�Ɋւ��鐧�����������Ȃ�ƌ����邾�낤.~
������,~�ߎ����s���Ă��Ȃ������v�Z�ʂ͑���,~�����I�Ɋ��S���k�������n�Ɋ�Â����f����
���p�ł��邩�͌v�Z�@�̐��\�ɑ傫���ˑ����Ă��܂����낤.
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\chapter{�܂Ƃ�}
\markright{�܂Ƃ�}
~~�{�_���ł͔�×͊w�������n���f���ɂ���,~���̕K�v���ɂ��čl�@��,~
�A���̎��ɂ��������n�̕��ނƒ莮�����s���Ă���.

�×͊w�n�Ɣ�×͊w�n�ł̉^���X�P�[���̈Ⴂ��,~���l���f���ɂ��\���l�̔�r
�Ȃǂ���,~�ϗ��_�̂悤�ȏ㏸������z����悤�Ȍ��ۂ𐔒l�V�~�����[�V��������ɂ�
��×͊w�������n���K���邱�Ƃ��킩����.

�������n�̕��ނɂ��Ă�,~�A���̎��ɂ������C�̈��k���̗L������������n��
��ʂ���.~����Ɋ�b���������ŗp�����閧�x�̊�{��̒l��,~������l�Ȃ���,~��������
���ԁE��Ԃɂ��ĂƂ��Ɉ�l�Ȃ��̂Ƃ��Ĉ����ꍇ�Ƃŕ��ނ���邱�Ƃ�������.

�莮���ɂ��Ă�,~�A���̎��ɂ�镪�ނɊ�Â��Ċe�������n�̒莮�����s����.~�����ł�
���ꂼ��̕������n�ɂ��Ă܂Ƃ߂�.~

�قƂ�ǋߎ����s��Ȃ����S���k�������n�͐��x�̍������l���f�������邪,~
�����ϐ��������镪,~�v�Z�ʂ������Ȃ��Ă��܂�.~���̂��Ƃ�,~���͂ŏq�ׂ��悤��3~������
�^����������p���邽�߂ɐ×͊w�n�����������n�����G�Ȕ�×͊w�n�ɂ�����傫�Ȗ��_�ł���,~
��قǂ̐��\�̌v�Z�@��p���Ȃ����,~���S���k�n���f���̗��p�͌����I�łȂ����낤.~
�����k�������n�ł�,~��C�̈��k����F�߂���,~������l�Ȋ�{��܂��̐��`�ߎ����s����.~
�����,~�^���������̋ߎ��ɗp�����A���̎��͑�C�̈��k���𖳎��������̂ł�����.~
��{��܂��̐��`�ߎ��ɂ��,~���C��,~��C�����ǂ�����܂܂��悤�ȋC���΍����傫���Ȃ�L��ł�
�g�p�ɂ͌����Ȃ���,~��C�̈��k��������$\partial \rho' / \partial t$~�͕������n�Ɋ܂ނ���,~
�\���J�̂悤�ȕ������ƂĂ��傫�����ۂ��������Ƃ͂ł��邾�낤.~
���ɔ�e���������n�͑�C�̈��k����F�߂�,~������l�Ȋ�{��܂��̐��`�ߎ����s���Ă���.~
�����k�������n�Ƃ̈Ⴂ������Ƒ�C�̈��k���𖳎�����Ƃ����_�ȊO�͕ς��Ȃ��������n��,~
������̈�͏����k�n�Ɠ��l�ɋ����͂Ȃ邪,~�������ɒ[�ɑ傫���Ȃ�Ȃ��ʏ�̋C�ی��ۂɂ͗ǂ��ߎ��Ő��藧���낤.
�Ō�ɔ񈳏k�������n�ɂ��Ăł��邪,~����͑�C�̈��k���𖳎���,~�ϐ��̊�{��̒l��萔�Ƃ��Ĉ����Ă���.~
���̂��ߍ��x�ɂ�閧�x�ω���������,~�󂢑Η��𐔒l�V�~�����[�V��������ɂ͓K������̂ł���.
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\chapter*{�ӎ�}
\addcontentsline{toc}{chapter}{�ӎ�}
\markright{�ӎ�}
�яˉ��,~��R�����y�����ɂ͌����e�[�}�ƕ��j�𒸂��܂���.
�����F�K����,~���M������~���X�ؗm������,~�X����傳��,~����^�炳��,~�[���N�j����ɂ�
�M�d�Ȏ��Ԃ������Ē���,~��w��,~�����𒸂��܂���.

�n������јf����C�Ȋw�������̊F�l�ɂ͐������̏���,~�䋦�͂𒸂��܂���.
�����ŉ��߂Đ[�����Ӓv���܂�.
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\chapter*{�t�^A~�莮���ɂ����鎮�̓��o}
\addcontentsline{toc}{chapter}{�t�^A~�莮���ɂ����鎮�̓��o}
\markright{�t�^A~�莮���ɂ����鎮�̓��o}
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\section*{�t�^A.1~�M�͊w�̎�~(4.1.5)~�̓��o}
%�܂��K�v�Ȏ����Čf����.
%\[p = \rho RT \tag{4.1.6}\]
%\[\pi = \biggl(\frac{p}{p_0}\biggr)^{R/C_p} \tag{4.1.7}\]
%\[\theta = \frac{T}{\pi} \tag{4.1.8}\]
�M�͊w���@����
%
\[
U = -W + Q
\]
%
�ł���.~������$U$~�͓����G�l���M�[,~$W$~�͌n���O�E�ɂ���d��,~$Q$~�͊O�E����n�ɓ������M�ʂł���.~
(A.1.1)~�ɂ�����,~�P�ʎ��ʂ�����̓����G�l���M�[��$e$~�Ƃ���Ə㎮��
%
%
\begin{align}
\frac{de}{dt} &= -p\frac{dV}{dt} + \mathcal{Q} \nonumber \\
              &= -p\frac{d}{dt}\biggl(\frac{1}{\rho}\biggr) + \mathcal{Q}
\end{align}
%
%
�ƂȂ�.~������$p$~�͌n����O�E�ɂ͂��炭����,~$V$~�͑̐�,~$\rho$~�͖��x,~$\mathcal{Q}$~��
�����M���ɂ��P�ʎ��ʓ���̉��M���ł���.~����ɔM�g�U���l������,
\begin{equation}
\frac{de}{dt} = -p\frac{d}{dt}\biggl(\frac{1}{\rho}\biggr) +\kappa \nabla^2 T + \mathcal{Q}
\end{equation}
%
�𓾂�.~������$\kappa$~�͔M�g�U��,~$T$~�͐�Ή��x�ł���.~���ɃG���g���s�[$S$~�̒�`
\[
S \equiv \int \frac{dQ}{T}
\]
���,~$\displaystyle d S \equiv \frac{d Q}{T}$~��p����,
%
%
\begin{equation}
T \frac{dS}{dt} = \frac{de}{dt} + p\frac{dV}{dt} = \frac{de}{dt} + p\frac{d}{dt}\biggl(\frac{1}{\rho}\biggr)
\end{equation}
%
%
�ƂȂ�.~�����~(A.1.2),~(A.1.3)~���
%
%
\begin{equation}
T \frac{dS}{dt} = \kappa \nabla^2 T + \mathcal{Q}
\end{equation}
%
%
�𓾂�.

����,~������C�����肵�Ă���̂�
%
%
\[\rho = \rho (p,T)\]
�܂���
\[S = S (p,T)\]
%
%
�Ƃ�����.~������$S$~��$p$~��$T$~�̓�̓Ɨ��ϐ��݂̂Ɉˑ�����Ƃ����̂�
\begin{equation}
dS = \biggl(\frac{\partial S}{\partial p}\biggr)_T dp + \biggl(\frac{\partial S}{\partial T}\biggr)_p dT
\end{equation}
�ƂȂ�.
%~�����,~(A.1.4),~(A.1.5)~���
%\begin{align}
%T\biggl(\frac{\partial S}{\partial p}\biggr)_T \frac{dp}{dt} + T\biggl(\frac{\partial S}{\partial T}\biggr)_p \frac{dT}{dt} &= \kappa \nabla^2 T + \mathcal{Q} \nonumber \\
%T\biggl(\frac{\partial S}{\partial p}\biggr)_T \frac{dp}{dt} + C_p \frac{dT}{dt} &= 
%\end{align}
%�ƂȂ�.
%\footnote{�����Œ舳��M�̒�`$\displaystyle C_p = T\biggl(\frac{\partial S}{\partial T}\biggr)_p$~��p����.}.~
������,~�M�u�X�̎��R�G�l���M�[$G(p,T)$~�ɂ���,
\[G=U-TS+pV\]
���,
%
%
\begin{align}
dG &= dU -TdS -SdT + pdV + Vdp \nonumber \\
   &= (dU + pdV -TdS) - SdT + Vdp \nonumber \\
   &= (dU + dW - dQ) - SdT + Vdp \nonumber \\
   &= -SdT + \frac{1}{\rho}dp
\end{align}
%
%
�ł���.~(A.1.6)~�̉�����2~�s�ډE�ӑ�ꍀ�͔M�͊w���@���ɂ��[���Ƃ���.~
�܂�$G$~��$p$~��$T$~�̓�̓Ɨ��ϐ��݂̂Ɉˑ���,
%
%
\begin{equation}
dG = \biggl(\frac{\partial G}{\partial T}\biggr)_p dT +\biggl(\frac{\partial G}{\partial p}\biggr)_T dp
\end{equation}
%
%
�ł��邩��,~(A.1.7)~��(A.1.8)~��ӁX��r����
\begin{align}
\biggl(\frac{\partial G}{\partial T}\biggr)_p &= -S, \\
\biggl(\frac{\partial G}{\partial p}\biggr)_T &= \frac{1}{\rho}
\end{align}
�𓾂�.~(A.1.9),~(A.1.10)~�ɂ���
\[
\biggl( \frac{\partial}{\partial p}\biggl(\frac{\partial G}{\partial T}\biggr)_p\biggr)_T = \biggl( \frac{\partial}{\partial T}\biggl(\frac{\partial G}{\partial p}\biggr)_T\biggr)_p
\]
��p�����,
\begin{equation}
-\biggl(\frac{\partial S}{\partial p}\biggr)_T = \frac{\partial}{\partial T} \biggl(\frac{1}{\rho}\biggr)_p = -\frac{1}{\rho^2} \biggl(\frac{\partial \rho}{\partial T}\biggr)_p, \nonumber
\end{equation}
%
%
�ł���̂�,
%
%
\begin{equation}
\biggl(\frac{\partial S}{\partial p}\biggr)_T = \frac{1}{\rho^2} \biggl(\frac{\partial \rho}{\partial T}\biggr)_p
\end{equation}
%
%
�ƂȂ�.
%~������~(A.1.11)~��~(A.1.6)~�ɑ������
%
%
%\begin{equation}
%\frac{T}{\rho^2} \biggl(\frac{\partial \rho}{\partial T}\biggr)_p \frac{dp}{dt} + C_p \frac{dT}{dt} = \kappa \nabla^2 T + \mathcal{Q}, \nonumber
%\end{equation}
%
%
%
%
%
%\begin{equation}
%-\frac{T}{\rho} \alpha \frac{dp}{dt} + C_p \frac{dT}{dt} = \kappa \nabla^2 T + \mathcal{Q}
%\end{equation}
%�𓾂�.~������$\alpha$~�͔M�c���W���ł���,~
%
%
%\[
%\alpha = -\frac{1}{\rho} \biggl(\frac{\partial \rho}{\partial T}\biggr)_p
%\]
%
%
%�ŕ\�����.
�܂�,~�舳��M$C_p$~�̒�`���,
\begin{equation}
C_p = T\biggl(\frac{\partial S}{\partial T}\biggr)_p
\end{equation}
�ł��邩��,~(A.1.5)~��~(A.1.10),~(A.1.11)~��p����,
%
%
\begin{equation}
dS = \frac{1}{\rho^2} \biggl(\frac{\partial \rho}{\partial T}\biggr)_p dp + \frac{C_p}{T}dT
\end{equation}
�ƂȂ�.~����Ɋ�����C�Ɋւ����ԕ������͗��z�C�̂̏ꍇ�Ɠ��l��
\[\rho = \frac{p}{RT}\]
�Ƃł���̂�,~(A.1.12)~�͂���ɕό`�ł�,
\begin{align}
dS &= \biggl(\frac{RT}{p}\biggr)^2 \biggl(-\frac{p}{RT^2}\biggl) dp + \frac{C_p}{T}dT \nonumber \\
   &= -R\frac{dp}{p} + C_p \frac{dT}{T}
\end{align}
�ƂȂ�.~(A.1.13)~�𗼕Ӑϕ���,~�ϕ��萔��$S_0$~�Ƃ����,
\begin{align}
S &= C_p \ln T - R \ln p + S_0 \nonumber \\
  &= C_p\biggl(\ln T -\frac{R}{C_p}\ln p\biggr) + S_0 \nonumber \\
  &= C_p \ln T \biggl(\frac{1}{p} \biggr)^{R/C_p} + S_0 ,
\end{align}
�ł���,~�㎮�𐮗�����
%
%
\begin{align}
\frac{S-S_0}{C_p} &= \ln T \biggl(\frac{1}{p} \biggr)^{R/C_p} \nonumber \\
e^{\frac{S-S_0}{C_p}} &= T \biggl(\frac{1}{p} \biggr)^{R/C_p}
\end{align}
�ƂȂ�.~������,~�W���C��$p_0 = const$~��p����,~(A.1.15)~�̗��ӂ�$p_0^{R/C_p}$~���������,
%
%
\begin{equation}
p_0^{R/C_p} e^{\frac{S-S_0}{C_p}} = T \biggl(\frac{p_0}{p}\biggr)^{R/C_p}
\end{equation}
�ƂȂ�.~$\Deqref{pi}$,~$\Deqref{theta}$~���(A.1.16)~��
%
%
\begin{equation}
p_0^{R/C_p} e^{\frac{S-S_0}{C_p}} = \theta
\end{equation}
�ł���,~���ӂ�$t$~�Ŕ�������
\begin{equation}
\frac{d \theta}{dt}= \frac{1}{C_p} p_0^{R/C_p} e^{\frac{S-S_0}{C_p}} \frac{dS}{dt}
\end{equation}
�ƂȂ�.~(A.1.18)~��~(A.1.17)~���������,
\begin{equation}
\frac{d \theta}{dt}= \frac{\theta}{C_p}\frac{dS}{dt}
\end{equation}
�𓾂�.~(A.1.4)~��~(A.1.19)~�ɑ�������
\begin{equation}
\frac{d \theta}{dt} = \frac{\theta}{C_p T} (\kappa \nabla^2 T + \mathcal{Q})
\end{equation}
�ł���.~�Ō��~$\Deqref{pi}$~���$\displaystyle \frac{\theta}{T} = \frac{1}{\pi}$~�ł��邩��,
\begin{equation}
\frac{d \theta}{dt} = \frac{\mathcal{Q}}{C_p \pi} + \frac{\kappa}{\pi}\nabla^2 T \nonumber \tag{4.1.5}
\end{equation}
�����߂�ꂽ.
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\newpage
\section*{�t�^A.2~\Deqref{dpdpi}~�̓��o}
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$\Deqref{dpdpi}$~�̓��o������.~
%�܂��K�v�Ȏ����Čf����.
%\[p = \rho RT \tag{4.1.6}\]
%\[\pi = \biggl(\frac{p}{p_0}\biggr)^{R/C_p} \tag{4.1.7}\]
%\[\theta = \frac{T}{\pi} \tag{4.1.8}\]
~�G�N�X�i�[�֐�~$\Deqref{pi}$~���
\[
p^{R/C_p} = p_0^{R/C_p} \pi
\]
�ł��邩��,~����𗼕Ӕ�������,
\begin{equation}
\frac{R}{C_p}p^{\frac{R}{C_p}-1} dp = p_0^{R/C_p} d \pi \nonumber 
\end{equation}
�ł���,~�����ό`����,
\begin{equation}
dp = \frac{C_p}{R} p \biggl(\frac{p}{p_0}\biggr)^{-R/C_p}d \pi = \frac{C_p}{R} p \frac{1}{\pi} d\pi
\end{equation}
�ƂȂ�.~
�܂�,~��ԕ�����~$\Deqref{���1}$~��~����~$\Deqref{theta}$~���,
\begin{equation}
\frac{1}{\rho}=\frac{R\theta\pi}{p}
\end{equation}
�ł��邩��~(A.2.1),~(A.2.2)~���,~
\[
\frac{1}{\rho}dp = \frac{R\theta\pi}{p}\cdot\biggl(\frac{C_p}{R} p \frac{1}{\pi} d\pi\biggr) =C_p \theta d \pi \tag{4.1.10}
\]
�𓾂�.
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\section*{�t�^A.3~\Deqref{�i�r�Gx2}~�`~\Deqref{�i�r�Gz2}~�̓��o}
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$\Deqref{�i�r�Gx2}$~�`~$\Deqref{�i�r�Gz2}$~�𓱏o����.

�܂�$\Deqref{�i�r�Gx2}$~���狁�߂�.~
$\Deqref{�i�r�Gx}$~��$\Deqref{hensu}$~���������,
%
%
\begin{align}
(\overline{\rho} + \rho ' )\frac{d u}{dt} + \frac{\partial}{\partial x}(\overline{p} + p') &= \nu \nabla^2 u, \nonumber \\
\overline{\rho}\biggl( 1 + \frac{\rho '}{\overline{\rho}}\biggr) \frac{d u}{dt} + \frac{\partial p'}{\partial x}&= \nu \nabla^2 u \nonumber
\end{align}
�ƂȂ�.~������$\displaystyle \mathrm{O} \biggl(\frac{\rho '}{\overline{\rho}}\biggr) \ll \mathrm{O}(1)$~��p�����
$\Deqref{�i�r�Gx2}$~��������.~�܂�,~���l�ɂ���$\Deqref{�i�r�Gy2}$~��
$\Deqref{�i�r�Gy}$~��$\Deqref{hensu}$~��������Ƌ��߂���.
%
%

�Ō��$\Deqref{�i�r�Gz2}$~�����߂�.~$\Deqref{�i�r�Gz}$~��$\Deqref{hensu}$~���������,
\begin{align}
(\overline{\rho} + \rho ' )\frac{d z}{dt} + \frac{\partial}{\partial z}(\overline{p} + p') + (\overline{\rho} + \rho ')g &= \nu \nabla^2 z \nonumber \\
\overline{\rho}\biggl( 1 + \frac{\rho '}{\overline{\rho}}\biggr) \frac{d z}{dt} + \frac{\partial \overline{p}(z)}{\partial z} + \frac{\partial p'}{\partial z} + \overline{\rho}g + \rho 'g &= \nu \nabla^2 z \nonumber
\end{align}
%
�����Ŋ�{��ɑ΂���×͊w�̎�~$\Deqref{seiriki}$~��,~$\displaystyle \mathrm{O} \biggl(\frac{\rho '}{\overline{\rho}}\biggr) \ll \mathrm{O}(1)$~
��p���Đ��������,
\[
\frac{d z}{dt} + \frac{1}{\overline{\rho}} \biggl( \frac{\partial p'}{\partial z}+ \rho 'g \biggr) = \nu \nabla^2 z \tag{4.2.5}
\]
�ƂȂ�,~$\Deqref{�i�r�Gz2}$~�����߂�ꂽ.
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\section*{�t�^A.4~\Deqref{mitsudohensa}~�̓��o}
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%
%
%(4.2.3)~�̓��o������.
%�܂��K�v�Ȏ����Čf����.
%\[p = \rho RT \tag{4.1.6}\]
%\[\pi = \biggl(\frac{p}{p_0}\biggr)^{R/C_p} \tag{4.1.7}\]
%\[\theta = \frac{T}{\pi} \tag{4.1.8}\]
%\[C_s^2 = \frac{C_p}{C_v}RT \tag{4.2.4}\]
$\Deqref{���1}$~�`~$\Deqref{theta}$~���,
\begin{align}
\rho ' &= \biggl(\frac{p}{R \theta \pi}\biggr)' \nonumber \\
       &= \biggl\{\frac{p_0}{R \theta} \biggl( \frac{p}{p_0} \biggr)^{C_v /C_p}\biggr\}' \nonumber \\
       &= \frac{p_0}{R} \biggl( \frac{p}{p_0} \biggr)^{C_v /C_p} \biggl(-\frac{\theta '}{\theta ^2} \biggr) + \frac{p_0}{R \theta} \frac{C_v}{C_p} \biggl( \frac{p}{p_0} \biggr)^{C_v /C_p -1} \frac{p'}{p_0} \nonumber \\
       &= -\frac{p_0}{R \theta} \frac{p}{p_0}\biggl( \frac{p}{p_0} \biggr)^{C_v/C_p -1} \biggl(\frac{\theta '}{\theta} \biggr) + \frac{C_v}{R C_p} \frac{1}{\theta} \biggl(\frac{p}{p_0}\biggr)^{-R/C_p} p'  \nonumber \\
       &= -\frac{p}{R \theta}\biggl( \frac{p}{p_0} \biggr)^{-R/C_p} \frac{\theta '}{\theta} + \frac{C_v}{R C_p} \frac{\pi}{T} \frac{1}{\pi} p' \nonumber \\
       &= -\frac{p}{R \theta \pi} \frac{\theta '}{\theta} + \frac{C_v}{C_p RT} p' \nonumber \\
       &= -\rho \frac{\theta '}{\theta} + \frac{p '}{C_s^2} \nonumber \\
       &\simeq -\overline{\rho} \frac{\theta '}{\overline{\theta}} + \frac{p '}{\overline{C_s^2}} \nonumber \tag{4.2.3}
\end{align}
�ƂȂ�.~�㎮�̍Ō�̍s�ւ̕ό`��$\rho$,~$\theta$,~$C_s^2$~�ɑ΂����{��܂��̐��`�ߎ���\��,~
$\rho'$,~$\theta'$,~$C_s^{'2} $~�𖳎����Ă���.
%$( \frac{\overline{p} + p'}{p_0})^{R/C_p}$
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\section*{�t�^A.5~\Deqref{seiriki3}~�̓��o}
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$\Deqref{seiriki3}$~�𓱏o����.
%�܂��K�v�Ȏ����Čf����.
%\begin{equation}
%p=\overline{p}+p',\rho = \overline{\rho} + \rho ',\theta = \overline{\theta} + \theta ' , \tag{4.2.1}
%\end{equation}
%\begin{equation}
%C_p \overline{\theta} \frac{\partial \overline{\pi}}{\partial z} + g = 0 . \tag{4.2.10}
%\end{equation}
(4.2.1)~��~$\Deqref{seiriki2}$~�ɑ������,
\begin{align}
C_p (\overline{\theta} + \theta') \frac{\partial{\pi}}{\partial z} + g &= C_p (\overline{\theta} +\theta ') \frac{\partial (\overline{\pi} + \pi ')}{\partial z} + g \nonumber \\
&= C_p (\overline{\theta} +\theta ') \frac{\partial \pi '}{\partial z} + C_p \theta ' \frac{\partial \overline{\pi}}{\partial z} + \biggl( C_p \overline{\theta} \frac{\partial \overline{\pi}}{\partial z} + g \biggr) \nonumber 
\end{align}
�ƂȂ�.~������~$\Deqref{seiriki2}$~����㎮�E�ӓ�s�ڂ̑�O���̓[���ƂȂ�,~�����~$\Deqref{seiriki2}$~��
\[C_p \frac{\partial \overline{\pi}}{\partial z} = - \frac{g}{\overline{\theta}}\]
�ƕό`���đ�������
\begin{align}
C_p \theta \frac{\partial{\pi}}{\partial z} + g &=C_p \theta \frac{\partial \pi '}{\partial z} - \frac{\theta '}{\overline{\theta}}g \nonumber \\
                                                &\simeq C_p \overline{\theta} \frac{\partial \pi '}{\partial z} - \frac{\theta '}{\overline{\theta}}g \tag{4.2.16}
\end{align}
�𓾂�.~�����ŏ㎮�̓�s�ڂւ̕ό`�͐��`�ߎ���\��.
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\section*{�t�^A.6~\Deqref{dpdt1}~�̓��o}
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$\Deqref{dpdt1}$~�𓱏o����.
%�K�v�Ȏ����Čf����.
%
%
%\begin{equation}
%\rho ' \simeq -\overline{\rho} \frac{\theta '}{\overline{\theta}} + \frac{p'}{\overline{C_s^2}},\tag{4.2.3}
%\end{equation}
%\begin{equation}
%\frac{1}{\overline{\rho}} \frac{\partial \rho '}{\partial t} + \biggl( \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} \biggr) =0. \tag{4.3.1}
%\end{equation}
%
%
$\Deqref{mitsudohensa}$~���,~
%
%
\begin{equation}
p' \simeq \overline{C_s^2} \biggl( \rho ' + \frac{\theta '}{\overline{\theta}} \overline{\rho} \biggr)
\end{equation}
%
%
%%%�ƂȂ�.~�E�ӂ�$\overline{\rho}$,~$\overline{\theta}$,~$\overline{C_s^2}$~��
%%%$\rho$,~$\theta$,~$C_s^2$~�ɑ΂����{��܂��̐��`�ߎ��ɂ����̂ł���.~
����ė��ӂ�$t$~�Ŕ�����,~$\Deqref{zyunasshukup}$~��p�����
%
%
\begin{align}
\frac{\partial p '}{\partial t} &\simeq \overline{C_s^2} \biggl( \frac{\overline{\theta}}{\overline{\rho}} \frac{\partial \theta '}{\partial t} + \frac{\partial \rho '}{\partial t} \biggr) \nonumber \\
                                &= \overline{C_s^2}\biggl\{\frac{\overline{\rho}}{\overline{\theta}} \frac{\partial \theta '}{\partial t} - \biggl( \frac{\partial \overline{\rho} u}{\partial x} + \frac{\partial \overline{\rho} v}{\partial y} + \frac{\partial \overline{\rho} w}{\partial z}\biggr)\biggr\} \tag{4.3.2}
\end{align}
������ꂽ.~������$\displaystyle \frac{\partial \overline{C_s^2}}{\partial t}$,~
$\displaystyle \frac{\partial\overline{\rho}}{\partial t}$,~$\displaystyle \frac{\partial \overline{\theta
}}{\partial t}$�𖳎����Ă���.
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\section*{�t�^A.7~\Deqref{zyunasshukupi}~�̓��o}
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(4.3.3)~�𓱏o����.
%�܂��͕K�v�Ȏ����Čf����.
%\begin{equation}
%\frac{\partial \rho}{\partial t} + \frac{\partial \rho u}{\partial x}+ \frac{\partial \rho v}{\partial y} + \frac{\partial \rho w}{\partial z} =0 \tag{4.1.1}
%\end{equation}
$\Deqref{renzoku}$~��p����$d\rho / dt$~�����߂��,
\begin{align}
\frac{d \rho}{dt} &= \frac{\partial \rho}{\partial t} + u\frac{\partial \rho}{\partial x} + v\frac{\partial \rho}{\partial y} + w\frac{\partial \rho}{\partial z} \nonumber \\
                  &= \frac{\partial \rho}{\partial t} + \biggl(\frac{\partial \rho u}{\partial x} - \rho \frac{\partial u}{\partial x}\biggr) + \biggl(\frac{\partial \rho v}{\partial y} - \rho \frac{\partial v}{\partial y}\biggr) + \biggl(\frac{\partial \rho w}{\partial z} - \rho \frac{\partial w}{\partial z}\biggr) \nonumber \\
                  &= \biggl( \frac{\partial \rho}{\partial t} + \frac{\partial \rho u}{\partial x} + \frac{\partial \rho v}{\partial y} + \frac{\partial \rho w}{\partial z} \biggr) - \rho \biggl( \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} \biggr) \nonumber \\
                  &= - (\overline{\rho} + \rho') \biggl( \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} \biggr)
\end{align}
�ƂȂ�.~������$\overline{\rho} \gg \rho'$~�ɂ��$\rho'\nabla\cdot\Dvect{u}$~�𖳎�����,
\begin{equation}
\frac{1}{\overline{\rho}} \frac{d \rho}{dt} + \biggl( \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} \biggr) =0 \tag{4.3.3}
\end{equation}
�𓾂�.
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\newpage
\section*{�t�^A.8~\Deqref{dpidt1}~�̓��o}
\setcounter{equation}{0}
\renewcommand{\theequation}{A.8.\arabic{equation}}
$\Deqref{dpidt2}$~�𓱏o����.
%~�K�v�Ȏ����Čf����.
%\[\pi = \biggl( \frac{\rho R \theta}{p_0}\biggr)^{R/C_v}.\tag{4.3.4}\]
$\Deqref{���3}$~�̗��ӂ̑ΐ����Ƃ��,
\begin{align}
\ln \pi &= \frac{R}{C_v} ( \ln \rho R \theta - \ln p_0 ) \nonumber \\
        &= \frac{R}{C_v} \{\ln \rho + \ln \theta  + \ln (R/p_0) \} 
\end{align}
$\ln (R/p_0)$~�͒萔�ł��邩��,~���ӂ������$t$~�Ŕ��������,
\begin{equation}
\frac{1}{\pi} \frac{d \pi}{dt} = \frac{R}{C_v} \biggl( \frac{1}{\rho} \frac{d \rho}{dt} + \frac{1}{\theta} \frac{d \theta}{dt} \biggr)
\end{equation}
�ł���.~�������`�ߎ�����
\begin{equation}
\frac{1}{\overline{\pi}} \frac{d \pi}{dt} = \frac{R}{C_v} \biggl( \frac{1}{\overline{\rho}} \frac{\partial \rho '}{\partial t} + \frac{1}{\overline{\theta}} \frac{d \theta}{dt} \biggr) \tag{4.3.5}
\end{equation}
�𓾂�.
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\chapter*{�Q�l����}
\addcontentsline{toc}{chapter}{�Q�l����}
\markright{�Q�l����}
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