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% An Introduction to Atmospheric Gravity Waves (2002) 
% 1st edition
%
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\begin{document}
%\appendix
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%\setcounter{equation}{0}
%\def\thechapter{\Alph{chapter}}
%\def\thesection{\Alph{cahpter}.\arabic{section}}
%\def\thesubsection{\Alph{chapter}.\arabic{section}.\arabic{subsection}}
%\def\thefigure{\Alph{chapter}.\arabic{figure}}
%\def\theequation{\Alph{chapter}.\arabic{equation}}

%\appendix
%%
%\def\thechapter{\Alph{chapter}}
%\def\thesection{\Alph{chapter}.\arabic{section}}
%\def\theequation{\Alph{chapter}.\arabic{equation}}
%\def\thepage{\arabic{page}}
%\def\thefigure{\Alph{chapter}.\arabic{figure}}
%\def\thetable{\Alph{chapter}.\arabic{table}}


\chapter{Goya \& Miyahara (1999)の方程式の導出}

Goya \& Miyahara (1999)で扱う方程式系を導出する.
鉛直2次元の系を考えると
運動方程式は %重力場中の Euler 方程式
\begin{align}
	\frac{Du}{Dt}&=-\frac{1}{\rho}\pdv{p}{x} +D_u
	\label{x-motion_orig.eq},
	\\
	\frac{Dw}{Dt}&=-\frac{1}{\rho}\pdv{p}{z}-\rho g +D_w
	\label{z-motion_orig.eq}
\end{align}
である. 連続の式は
\begin{equation}
	\frac{D\rho}{Dt}
	+\rho\left(\pdv{u}{x}+\pdv{w}{z}\right)
	=0
	\label{continuity_orig.eq}
\end{equation}
である. 熱力学の式は
\begin{equation}
	c_p\frac{DT}{Dt}=\frac{1}{\rho}\frac{Dp}{Dt}+D_T
	\label{thermo_orig.eq}
\end{equation}
である.
また理想気体の状態方程式は
\begin{equation}
	p=\rho R T
	\label{state_orig.eq}
\end{equation}
である.
記号は以下のとおりに定義している:
\begin{align*}
	u &:\text{水平速度},\\
	w &:\text{鉛直速度},\\
	p &:\text{圧力},\\
	\rho &:\text{密度},\\
	T &:\text{温度},\\
	D_{u,w,T} &:\text{散逸の効果}.
\end{align*}

%力学変数[$ u,w,T $]と$ \rho $の積を$ \tilde{} $つきの量で表す:
%例えば
%\begin{equation}
%	\tilde{u}\equiv u(x,z,t)\cdot\rho(x,z,t)
%\end{equation}
%である.
%さらに初期の基本場からの変化を表す場合は$ \delta $つきの量で表す.
%すなわち
%\begin{align}
%	&\delta\tilde{u}=\tilde{u}(x,z,t)-\tilde{u}_B(z),
%	\;\;\;\;\;
%	\tilde{u}_B\equiv\rho_B(z)u_B(z),
%	\label{tilde_u.eq}
%	\\
%	&\delta\tilde{w}(x,z,t)=\tilde{w}(x,z,t),
%	\label{tilde_w.eq}
%	\\
%	&\delta\tilde{T}=\tilde{T}(x,z,t)-\tilde{T}_B(z),
%	\;\;\;\;\;
%	\tilde{T}_B\equiv\rho_B(z)T_B(z).
%		\label{tilde_T.eq}
%\end{align}
\eqref{tilde_u.eq}, \eqref{tilde_w.eq}
を用いて\eqref{continuity_orig.eq},
\eqref{state_orig.eq}を書き換えると
\begin{equation}
	\pdv{\rho}{t}+\pdv{\tilde{u}}{x}+\pdv{\tilde{w}}{z}
	=0,
	\tag{\ref{1d.eq}}
\end{equation}
\begin{equation}
	p=R\tilde{T}
	\tag{\ref{1e.eq}}
\end{equation}
となる.


\eqref{1a.eq}, \eqref{1b.eq}を導出する. \eqref{x-motion_orig.eq}の
物質微分を展開し, \eqref{tilde_u.eq}, 
\eqref{tilde_w.eq}を用いて整理すると
\begin{align}
	&\rho\left(\pdv{u}{t}+u\pdv{u}{x}+w\pdv{u}{z}\right)
	=-\pdv{p}{x}+\rho D_x
	\notag \\
	&\pdv{\tilde{u}}{t}
	=-\pdv{\tilde{u}u}{x}-\pdv{\tilde{u}w}{z}
	+u\left(
	\pdv{\rho}{t}
	+\pdv{\tilde{u}}{x}+\pdv{\tilde{w}}{z}
	\right)
	-\pdv{p}{x}+\rho D_u
	\label{x-motion.eq2}
\end{align}
ここで
\begin{equation*}
	\pdv{\tilde{u}}{t}
	=\pdv{}{t}\left(\delta\tilde{u}+\tilde{u}_B\right)
	=\pdv{\delta\tilde{u}}{t}
\end{equation*}
であることと, \eqref{1d.eq}を用いると\eqref{x-motion.eq2}は
\begin{equation}\label{x-motion.eq3}
	\pdv{\delta\tilde{u}}{t}
	=-\pdv{\tilde{u}u}{x}-\pdv{\tilde{u}w}{z}
	-\pdv{p}{x}+\rho D_u
\end{equation}

と書ける. \eqref{z-motion_orig.eq}も同様に変形すると
\begin{equation}\label{z-motion.eq3}
\pdv{\delta\tilde{w}}{t}
=-\pdv{\tilde{w}u}{x}-\pdv{\tilde{w}w}{z}
-\pdv{p}{z}-\rho g+\rho D_w
\end{equation}
である.

\eqref{1c.eq}を導出する.
\eqref{1e.eq}の物質微分をとると
\begin{equation*}
	\frac{Dp}{Dt}=R\frac{D\tilde{T}}{Dt}.
\end{equation*}
これを用いて\eqref{thermo_orig.eq}から$ Dp/Dt $を
消去すると,
\begin{align}
	\rho c_p\frac{DT}{Dt}
	=R\frac{D\tilde{T}}{Dt}+\rho D_T
	\notag\\
	c_p\left(
		\frac{D\tilde{T}}{Dt}-T\frac{D\rho}{Dt}
	\right)
	=R\frac{D\tilde{T}}{Dt}+\rho D_T
\end{align}
である.
これに\eqref{continuity_orig.eq}を用い,
物質微分を微分を展開して整理すると,
\begin{align}
	c_v\frac{D\tilde{T}}{Dt}
	&=-c_p\tilde{T}\left(\pdv{u}{x}+\pdv{w}{z}\right)
	+\rho D_T
	\notag \\
	\pdv{\delta\tilde{T}}{t}
	&=-\pdv{\tilde{u}T}{x}-\pdv{\tilde{w}T}{z}
	-\frac{R\tilde{T}}{c_v}
	\left(\pdv{u}{x}+\pdv{w}{z}\right)
	+\frac{\rho}{c_v}D_T
\end{align}
となる.

\begin{align}
	\pdv{\delta\tilde{u}}{t}
	=&-\pdv{\tilde{u}u}{x}-\pdv{\tilde{u}w}{z}
	-\pdv{p}{x}
	-K_H\pdv[4]{\delta\tilde{u}}{x}
	+K_V\pdv[2]{\delta\tilde{u}}{z}
	-R_f\delta\tilde{u}
	\tag{\ref{1a.eq}}
	\\
	\pdv{\delta\tilde{w}}{t}
	=&-\pdv{\tilde{w}u}{x}-\pdv{\tilde{w}w}{z}
	-\pdv{p}{z}-\rho g
	-K_H\pdv[4]{\delta\tilde{w}}{x}
	+K_V\pdv[2]{\delta\tilde{w}}{z}
	-R_f\delta\tilde{w}
	\tag{\ref{1b.eq}}
	\\
	\pdv{\delta\tilde{T}}{t}
	=&-\pdv{\tilde{u}T}{x}-\pdv{\tilde{w}T}{z}
	-\frac{R\tilde{T}}{c_v}
	\left(\pdv{u}{x}+\pdv{w}{z}\right)
	-K_H\pdv[4]{\delta\tilde{T}}{x}
	+K_V\pdv[2]{\delta\tilde{T}}{z}
	-\alpha\delta\tilde{T}
	\tag{\ref{1c.eq}}
	\\
	&\pdv{\rho}{t}+\pdv{\tilde{u}}{x}+\pdv{\tilde{w}}{z}
	=0,
	\tag{\ref{1d.eq}}
	\\
	&p=R\tilde{T}
	\tag{\ref{1e.eq}}
\end{align}


\chapter{差分化の方法と精度について}
\setcounter{equation}{0}
\section{一階微分}
水平方向・鉛直方向ともに差分化には中央差分近似を用いる.
格子間隔を$ \Delta x $とすると, 一階微分は
\begin{equation}
	 \pdv{f}{x}
	=\frac
		{f(x+\Delta x)-f(x-\Delta x)}
		{2\Delta x}
	\label{1diff}
\end{equation}
と表される. 
$ f(x+\Delta x) $と$ f(x-\Delta x) $を
それぞれ二次の項まで Taylor 展開し,
\eqref{1diff}に代入すると
\begin{align}
	 \pdv{f}{x}
	=&\frac{1}{2\Delta x}
	\left[
		 \left\{
			f(x)
			+\pdv{f}{x}\Delta x
			+\frac{1}{2!}\pdv[2]{f}{x}\Delta x^2
			+\mathcal{O}(\Delta x^3)
		 \right\}
	\right.
	\notag
	\\
	&\;\;\;\;\;\;\;\; 
	\left.
		-\left\{
			f(x)
			-\pdv{f}{x}\Delta x
			+\frac{1}{2!}\pdv[2]{f}{x}\Delta x^2
			-\mathcal{O}(\Delta x^3)
		\right\}
	\right]
	\notag
	\\
	=&\pdv{f}{x}+\mathcal{O}(\Delta x^2)
	\label{diff1_check}
\end{align}
となるので, この近似は$ \Delta x $について二次精度の近似である.

\section{二階微分}

二階微分の中央差分近似を導出する.
二階微分は一階微分の差分として表すことができるので,
\begin{align}
	 \pdv[2]{f}{x}
	&=\frac{1}{\Delta x}
	\left[
		\pdv{}{x}f\left(x+\frac{\Delta x}{2}\right)
		-\pdv{}{x}f\left(x-\frac{\Delta x}{2}\right)
	\right]
	\notag
	\\
	&=\frac{1}{\Delta x}
	\left[
		\frac{f(x+\Delta x)-f(x)}{\Delta x}
		-\frac{f(x)-f(x-\Delta x)}{\Delta x}
	\right]
	\notag
	\\
	&=\frac
		{f(x+\Delta x)-2f(x)+f(x-\Delta x)}
		{\Delta x^2}
	\label{2diff}
\end{align}
である.
$ f(x+\Delta x),\ f(x-\Delta x) $を3次の項まで
 Taylor 展開すると
\begin{equation}
	 f(x\pm \Delta x)
	=f(x)
	\pm \pdv{f}{x}\Delta x
	+ \frac{1}{2!}\pdv[2]{f}{x}\Delta x^2
	\pm \frac{1}{3!}\pdv[3]{f}{x}\Delta x^3
	+\mathcal{O}(\Delta x^4)
\end{equation}
となるので, \eqref{2diff}に代入して整理すると
\begin{equation}
	 \pdv[2]{f}{x}
	=\frac{1}{\Delta x^2}
	\left[
		\frac{1}{2!}\pdv[2]{f}{x}2\Delta x^2
		+\mathcal{O}(\Delta x^4)
	\right]
	=\pdv[2]{f}{x}+\mathcal{O}(\Delta x^2)
\end{equation}
となり, \eqref{2diff}は
$ \Delta x $について二次精度の近似であることがわかる.

\section{四階微分}
$ f(x+\Delta x/2),\ f(x-\Delta x/2) $での
三階微分の差分を考える.
$ f(x+\Delta x/2),\ f(x-\Delta x/2) $の三階微分は
\eqref{2diff}を用いてそれぞれ
\begin{align}
	 \pdv[3]{}{x}f(x+\Delta x/2)
	&=\frac{1}{\Delta x}
		\left[
			\pdv[2]{f(x+\Delta x)}{x}
			-\pdv[2]{f(x)}{x}
		\right]
	\notag
	\\
	&=\frac
		{f(x+2\Delta x)
		-3f(x+\Delta x)
		+3f(x)
		-f(x-\Delta x)
		}
		{\Delta x^3},
	\label{3diff_p}
	\\
	 \pdv[3]{}{x}f(x-\Delta x/2)
	 &=\frac{1}{\Delta x}
	 \left[
	 \pdv[2]{f(x)}{x}
	 -\pdv[2]{f(x-\Delta x)}{x}
	 \right]
	 \notag
	 \\
	 &=\frac
	 {f(x+\Delta x)
	 	-3f(x)
	 	+3f(x-\Delta x)
	 	-f(x-2\Delta x)
	 }
	 {\Delta x^3}
	 \label{3diff_m}
\end{align}
である.
\eqref{3diff_p}, \eqref{3diff_m}を用いると
$ f(x) $の四階微分は
\begin{align}
	 \pdv[4]{f}{x}
	&=\frac{1}{\Delta x}
	\left[
		\pdv[3]{}{x}f(x+\Delta x/2)
		-\pdv[3]{}{x}f(x-\Delta x/2)
	\right]
	\notag
	\\
	&=\frac{
		f(x+2\Delta x)
		-4f(x+\Delta x)
		+6f(x)
		-4f(x-\Delta x)
		+f(x-2\Delta x)
		}
		{\Delta x^4}
	\label{4diff}
\end{align}
と書ける.
また$ f(x\pm 2\Delta x),\ f(x\pm\Delta x) $を
5次の項まで展開すると,
\begin{align}
	 f(x\pm 2\Delta x)
	=&f(x)
	\pm \pdv{f}{x} (2 \Delta x)
	+\frac{1}{2!}\pdv[2]{f}{x} (2\Delta x)^2
	\pm \frac{1}{3!}\pdv[3]{f}{x} (2\Delta x)^3
	\notag
	\\
	&+\frac{1}{4!}\pdv[4]{f}{x} (2\Delta x)^4
	\pm \frac{1}{5!}\pdv[5]{f}{x} (2\Delta x)^5
	+\mathcal{O} (\Delta x^6),
	\\
	 f(x\pm \Delta x)
	 =&f(x)
	 \pm \pdv{f}{x}\Delta x
	 +\frac{1}{2!}\pdv[2]{f}{x}\Delta x^2
	 \pm \frac{1}{3!}\pdv[3]{f}{x}\Delta x^3
	 \notag
	 \\
	 &+\frac{1}{4!}\pdv[4]{f}{x}\Delta x^4
	 \pm \frac{1}{5!}\pdv[5]{f}{x}\Delta x^5
	 +\mathcal{O}(\Delta x^6).
\end{align}
これを\eqref{4diff}に代入して整理すると
\begin{align}
	 \pdv[4]{f}{x}
	=&\frac{2}{\Delta x^4}
	\left[
		\left\{
			\frac{1}{2!}\pdv[2]{f}{x} (2\Delta x)^2
			+\frac{1}{4!}\pdv[4]{f}{x} (2\Delta x)^4
		\right\}
		\right.
		\notag
		\\
		&\;\;\;\;\;\;\;
		\left.
		-4\left\{
			\frac{1}{2!}\pdv[2]{f}{x}\Delta x^2
			+\frac{1}{4!}\pdv[4]{f}{x}\Delta x^4
		\right\}
		+\mathcal{O}(\Delta x^6)
	\right]
	\notag
	\\
	=&\pdv[4]{f}{x}+\mathcal{O}(\Delta x^2)
\end{align}
となるので, この近似は$ \Delta x $について二次精度を持つ.

%\chapter{Goya \& Miyahara (1999)の方程式の差分化}
%%\setcounter{equation}{0}
%
%\section{運動量方程式}
%
%\begin{align}
%	\qty(\pdv{\delta\tilde{u}}{t})_{i+\frac{1}{2},j}
%	=
%	&-\frac{\tilde{u}_{i+1,j}u_{i+1,j}
%			-\tilde{u}_{i,j}u_{i,j}}
%			{\Delta x}
%	\notag\\
%	&-\frac{\tilde{u}_{i+\frac{1}{2},j+\frac{1}{2}}
%			w_{i+\frac{1}{2},j+\frac{1}{2}}
%			-\tilde{u}_{i+\frac{1}{2},j-\frac{1}{2}}
%			w_{i+\frac{1}{2},j-\frac{1}{2}}}
%			{\Delta z}
%	\notag\\
%	&-\frac{p_{i+1,j}-p_{i,j}}{\Delta x}
%	\notag\\
%	&-K_H\frac{
%			\delta\tilde{u}_{i+\frac{5}{2},j}
%			-4\delta\tilde{u}_{i+\frac{3}{2},j}
%			+6\delta\tilde{u}_{i+\frac{1}{2},j}
%			-4\delta\tilde{u}_{i-\frac{1}{2},j}
%			\delta\tilde{u}_{i-\frac{3}{2},j}	
%		}
%		{(\Delta x)^4}
%	\notag\\
%	&+K_V\frac{
%			\delta\tilde{u}_{i+\frac{1}{2},j+1}
%			-2\delta\tilde{u}_{i+\frac{1}{2},j}
%			+\delta\tilde{u}_{i+\frac{1}{2},j-1}
%		}
%	{(\Delta z)^2}
%	\notag\\
%	&-Rf_j\delta\tilde{u}_{i+\frac{1}{2},j}
%\end{align}
%
%\begin{align}
%	\qty(\pdv{\delta\tilde{w}}{t})_{i,j+\frac{1}{2}}
%	=
%	&-\frac{\tilde{w}_{i+\frac{1}{2},j+\frac{1}{2}}
%			u_{i+\frac{1}{2},j+\frac{1}{2}}
%			-\tilde{w}_{i-\frac{1}{2},j+\frac{1}{2}}
%			u_{i-\frac{1}{2},j+\frac{1}{2}}}
%		{\Delta x}
%	\notag\\
%	&-\frac{\tilde{w}_{i+1,j}w_{i+1,j}
%			-\tilde{w}_{i,j}w_{i,j}}
%		{\Delta z}
%	\notag\\
%	&-\frac{\delta p_{i,j+1}-\delta p_{i,j}}{\Delta z}
%	 -g\frac{\delta\rho_{i,j+1}+\delta\rho_{i,j}}{2}
%	\notag\\
%	&-K_H\frac{
%			\delta\tilde{w}_{i+2,j+\frac{1}{2}}
%			-4\delta\tilde{w}_{i+1,j+\frac{1}{2}}
%			+6\delta\tilde{u}_{i,j+\frac{1}{2}}
%			-4\delta\tilde{u}_{i-1,j+\frac{1}{2}}
%			\delta\tilde{u}_{i-2,j+\frac{1}{2}}	
%		}
%		{(\Delta x)^4}
%	\notag\\
%	&+K_V\frac{
%			\delta\tilde{w}_{i,j++\frac{3}{2}}
%			-2\delta\tilde{u}_{i,j+\frac{1}{2}}
%			+\delta\tilde{u}_{i,j-\frac{1}{2}}
%		}
%		{(\Delta z)^2}
%	\notag\\
%	&-Rf_j\delta\tilde{w}_{i,j+\frac{1}{2}}
%\end{align}
%ここで基本場について静水圧平衡が成り立つことから
%\begin{equation*}
%	\frac{(p_B)_{i,j+1}-(p_B)_{i,j}}{\Delta z}
%	+g\frac{(\rho_B)_{i,j+1}+(\rho_B)_{i,j}}{2}
%	=0
%\end{equation*}
%であることを用いた.
%
%\section{連続の式}
%
%\begin{equation}\label{ContEq_d}
%\qty(\pdv{\rho}{t})_{i,j}
%=-\frac{\tilde{u}_{i+\frac{1}{2},j}
%	-\tilde{u}_{i-\frac{1}{2},j}}
%{\Delta x}
%-\frac{\tilde{w}_{i,j+\frac{1}{2}}
%	-\tilde{u}_{i,j-\frac{1}{2}}}
%{\Delta z}
%\end{equation}
%
%\section{熱力学の式}
%\begin{align}
%	\qty(\pdv{\delta\tilde{T}}{t})_{i,j}
%	=
%	&-\frac{
%			\tilde{u}_{i+\frac{1}{2},j}T_{i+\frac{1}{2},j}
%			-\tilde{u}_{i-\frac{1}{2},j}T_{i-\frac{1}{2},j}
%		}{\Delta x}
%	\notag \\
%	&-\frac{
%			\tilde{w}_{i,j+\frac{1}{2}}T_{i,j+\frac{1}{2}}
%			-\tilde{w}_{i,j-\frac{1}{2}}T_{i,j-\frac{1}{2}}
%		}{\Delta z}
%	\notag \\
%	&+\frac{R\tilde{T}_{I,j}}{c_v}
%	\qty[
%		\frac{u_{i+\frac{1}{2},j}-u_{i-\frac{1}{2},j}}{\Delta x}
%		+\frac{w_{i,j+\frac{1}{2}}-w_{i,j-\frac{1}{2}}}{\Delta z}
%	]
%	\notag \\
%	&-K_H\frac{
%			\delta\tilde{T}_{i+2,j}
%			-4\delta\tilde{T}_{i+1,j}
%			+6\delta\tilde{T}_{i,j}
%			-4\delta\tilde{T}_{i-1,j}
%			\delta\tilde{T}_{i-2,j}}
%			{(\Delta x)^4}
%	\notag \\
%	&+K_V\frac{
%			\delta\tilde{T}_{i,j+1}
%			-2\delta\tilde{T}_{i,j}
%			+\delta\tilde{T}_{i,j-1}}
%			{(\Delta z)^2}
%	\notag \\
%	&-\alpha_{i,j}\delta\tilde{T}_{i,j}
%\end{align}


\end{document}

