% TITLE NumRu::GPhys::EP_Flux % % HISTORY 2004/08/09 塚原大輔 % 2004/11/12 塚原大輔 ( 最新改定 ) % 2005/02/13 石渡正樹 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%% Style Setting %%%%%%%% \documentclass[a4j,12pt,openbib]{jreport} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%% Package Include %%%%%%%% \usepackage{ascmac} \usepackage{tabularx} \usepackage{graphicx} \usepackage{amssymb} \usepackage{amsmath} \usepackage{Dennou6} %%%%%%%% PageStyle Setting %%%%%%%% \pagestyle{Dmyheadings} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%% Title Setting %%%%%%%% \Dtitle[NumRu::GPhys::EP\_Flux]{NumRu::GPhys::EP\_Flux \\数理ドキュメント} \Dauthor[地球流体電脳倶楽部]{地球流体電脳倶楽部} \Dfile{} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%% Set Counter (chapter, section etc. ) %%%%%%%% %\setcounter{chapter}{1} \setcounter{section}{0} \setcounter{equation}{0} \setcounter{page}{1} \setcounter{figure}{0} \setcounter{footnote}{0} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%% Counter Output Format %%%%%%%% \def\thesection{\arabic{chapter}.\arabic{section}} %\def\theequation{\arabic{chapter}.\arabic{section}.\arabic{equation}} %\def\thepage{\arabic{page}} %\def\thefigure{\arabic{section}.\arabic{figure}} %\def\thetable{\arabic{section}.\arabic{table}} %\def\thefootnote{\arabic{footnote}} %\def\thesection{\arabic{section}} %\def\theequation{\arabic{section}.\arabic{equation}} \def\thepage{\arabic{page}} \def\thefigure{\arabic{section}.\arabic{figure}} \def\thetable{\arabic{section}.\arabic{table}} \def\thefootnote{\arabic{footnote}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%% Dennou-Style Definition %%%%%%%% \Dparskip %\Dnoparskip %\Dparindent \Dnoparindent %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%% Local Definition %%%%%%%% \def\dfrac#1#2{{\displaystyle\frac{#1}{#2}}} \def\minicaption#1#2{\begin{quote} \caption{\footnotesize #1} \Dfiglab{#2} \end{quote}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%% Text Start %%%%%%%% \begin{document} %%% タイトルページ作成 \maketitle %%% 目次ページ作成 \tableofcontents \chapter{はじめに} NumRu::GPhys::EP\_Flux は Eliassen-Palm フラックス(EP フラックス) および残差循環を計算するメソッドを集めたモジュールである. 現状では, 鉛直座標として対数圧力座標を用いた 球座標系におけるプリミティブ方程式(準地衡風近似をしない) EP フラックスのためのメソッドだけが用意されている. 将来的には Plumb フラックスや Takaya-Nakamura フラックスを計算する メソッドもサポートする予定である. 本ドキュメントでは NumRu::GPhys::EP\_Flux で使用される 数式の解説と各メソッドの概説を行う. なお, NumRu::GPhys::EP\_Flux では, 微分演算のために, 別モジュール NumRu::Derivative および NumRu::GPhys::Derivative で 定義されるメソッドを使用している. 微分演算メソッドに関する詳細はそれぞれのモジュールのドキュメントを 参照されたい. \chapter{NumRu::GPhys::EP\_Flux で計算される緒量} 本章では NumRu::GPhys::EP\_Flux で定義される緒量の解説を行う. 数理モデルは Andrews {\it et al}.(1987) の第 3 章に基づく. \section{系の設定} \Dseclab{設定} 球面上の大気を考える. 大気の厚さは水平方向の広がりに比べ薄く, 鉛直方向に静水圧平衡が成り立つものとする. 緯度経度座標系を用い, 経度 $\lambda$ 軸を東向き, 緯度 $\phi$ 軸を北向きに正をとる. 鉛直座標には対数圧力座標$z^*$ \begin{eqnarray}\Deqlab{logp} z^* &=& -H \ln(p/p_s),\ \ \ \ H = \frac{R_{d} T_s}{g_0} \end{eqnarray} を用いる. ここで $H$ はスケールハイト, $R_{d}$ は乾燥空気の気体定数 (普遍気体定数を $R$, 乾燥空気の分子量を $w$ とすると $R_{d} = R/w$), $T_s$ は標準参照温度(定数), $g_0$は地表面における重力加速度(定数), $p$は圧力, $p_s$ は参照圧力である. $p_s$ として地表面圧力の代表値(定数)を用いる. \section{EP フラックス} \Dseclab{EP フラックス} 本モジュールでは惑星半径と後述の $\rho_s$ で規格化した EP フラックス (以降, 規格化した EP フラックス)を計算, 出力する. 規格化した EP フラックスは \begin{subequations}\Deqlab{normalized_F} \begin{align} \Deqlab{normalized_epflx_phi} \hat{F}_\phi &\equiv \sigma \cos \phi \left( \DP{\overline{u}}{z^*} \frac{\overline{v'\theta'}}{\DP{\overline{\theta}}{z^*}} - \overline{u'v'} \right), \\ \Deqlab{normalized_epflx_z^*} \hat{F}_{z^*} &\equiv \sigma \cos \phi \left( \left[ f - \Dinv{a\cos\phi}{\DP{\overline{u}\cos \phi}{\phi}} \right] \frac{\overline{v'\theta'}}{\DP{\overline{\theta}}{z^*}} - \overline{u'w'} \right) \end{align} \end{subequations} と定義される. ここで$\hat{F}_\phi$, $\hat{F}_{z^*}$ はそれぞれ 規格化された EP フラックスの $\phi$ 成分, $z^*$ 成分である. $\overline{\bullet}$ は東西オイラー平均量, $\bullet'$ は東西オイラー平均量からのずれを表す. $u, v, w$ はそれぞれ東西風速, 南北風速, 対数圧力速度で \begin{eqnarray*} (u, v, w) &\equiv& \left(a\cos\phi\DD{\lambda}{t}, a\DD{\phi}{t}, \DD{z^*}{t}\right) \end{eqnarray*} と定義される. $\theta$ は温位, $a$ は惑星半径(定数)である. $\sigma$ は \begin{align} \sigma \equiv \frac{\rho_0}{\rho_s} = \exp\left(\frac{-z^*}{H}\right), \end{align} である. ただし, $\rho_0$ は基本場の密度で \begin{eqnarray*} \Deqlab{basic_density} \rho_0(z^*) &\equiv& \rho_s e^{-z^*/H}, \hspace{2em} \rho_s \equiv p_s/RT_s \end{eqnarray*} である. $f$はコリオリパラメータで \begin{eqnarray} \Deqlab{colioli} f = 2 \Omega \sin \phi = \frac{4 \pi}{T_{rot}} \sin \phi \end{eqnarray} と定義される. $\Omega$ は自転角速度, $T_{rot}$は自転周期である. 本モジュールでは, 自転角速度を変更するためには $T_{rot}$ の値を与える仕様になっている. 一方, Andrews {\it et al}. (1987) で示されている EP フラックスは次のように定義される %%%%%%%%%%%%%%%%%%%%%% % \item EP フラックス %%%%%%%%%%%%%%%%%%%%%% \begin{subequations} \begin{align} \Deqlab{epflx_phi} {F_\phi} =& \rho_0 a \cos \phi \left(\DP{\overline{u}}{\overline{z^*}} \frac{\overline{v'\theta'}}{\DP{\overline{\theta}}{z^*}} - \overline{u'v'}\right)\\ \Deqlab{epflx_z^*} {F_z^*} =& \rho_0 a \cos \phi \left(\left[ f - \frac{\DP{\overline{u}\cos \phi}{\phi}}{a\cos\phi} \right] \frac{\overline{v'\theta'}}{\DP{\overline{\theta}}{z^*}} - \overline{u'w'}\right). \end{align} \end{subequations} ここで$F_\phi$, $F_{z^*}$はそれぞれ EP フラックスの$\phi$成分, $z^*$成分である. $F_y, F_z^*$ と $\hat{F_y}, \hat{F_z^*}$ は以下のように関係付けられる. \begin{align} \Deqlab{relation_F_vs_F^} (F_y, F_z^*) = a\rho_s(\hat{F_y}, \hat{F_{z^*}}) \end{align} \section{残差循環} 残差循環$(0, \overline{v}^*, \overline{w}^*)$は以下の形で定義される. \begin{subequations}\Deqlab{residual} \begin{align} \Deqlab{residual_v} \overline{v}^* &\equiv \overline{v} - \Dinv{\rho_0}\DP{}{z^*}\left(\rho_0\frac{\overline{v'\theta'}} {\DP{\overline{\theta}}{z^*}}\right)\notag\\ &= \overline{v} - \Dinv{\sigma}\DP{}{z^*}\left(\sigma\frac{\overline{v'\theta'}} {\DP{\overline{\theta}}{z^*}}\right)\\ \Deqlab{residual_w} \overline{w}^* &\equiv \overline{w} + \Dinv{a \cos\phi}\DP{}{\phi}\left(\cos\phi\frac{\overline{v'\theta'}} {\DP{\overline{\theta}}{z^*}}\right) \end{align} \end{subequations} \section{平均東西流の式} 規格化した EP フラックスを用いると, TEM 系における $u$ の式は 以下のようになる. \begin{eqnarray} \Deqlab{transformed_euler_mean_pe_momentum_x_with_F^}& & \DP{\overline{u}}{t} + \overline{v}^*\left[\Dinv{a\cos\phi}\DP{}{\phi}(\overline{u}\cos\phi) - f\right] + \overline{w}^*\DP{\overline{u}}{z^*} - \overline{X} = \Dinv{\sigma \cos\phi}\Ddiv\Dvect{\hat{F}}. \end{eqnarray} \section{子午面上の発散演算子} 子午面における発散演算子は, $\Dvect{F}$ を任意のベクトルした時に 以下の形で定義される. \begin{align} \Deqlab{div} \Ddiv{} \Dvect{F}= \Dinv{a \cos \phi} \DP{(\cos \phi F_{\phi})}{\phi} + \DP{F_{z^{*}}}{z^*} \end{align} \section{質量流線関数} 残差循環の質量流線関数 $\Psi^*$ を \begin{subequations} \begin{align} \sigma \overline{v}^* &= -g\Dinv{2\pi a \cos\phi }\DP{\Psi^*}{z^{*}}, \\ \sigma \overline{w}^* &= g\Dinv{2\pi a^2\cos\phi}\DP{\Psi^*}{\phi} \end{align} \end{subequations} と定義する. 上式を積分して $\Psi^*$ を求めるために, 本モジュールでは \Deqref{logp} を使用して 対数圧力座標 ($z^*$) 系から圧力座標($p$)系へ \begin{align} \DP{}{z^*}\Psi^* &= -\frac{p}{H}\DP{}{p}\Psi^* \end{align} と変換し, 大気上端($p=0$)において $\Psi^* = 0$ として積分し \begin{align} \Psi^*(\theta, p) = \frac{2\pi a \cos\phi}{g} \int_{0}^{p}\overline{v}^*\Dd p \end{align} と質量流線関数を導いている. %\footnote{ (2005/1/27 石渡) で良いんだっけ?\\ % (2005/1/27 塚原) 良いんです} \section{変数変換} EP\_Flux モジュールでは与えられたデータに応じて 変数変換を施す場合がある. その変換は以下のように行う. \vspace{5mm} 入力されるデータの鉛直軸が気圧軸であった場合, 以下の関係式を用いて高度軸に変換し, 計算を行う. \begin{subequations} \begin{align} z^* &= -H \log \left( \frac{p}{p_{00}} \right),\\ p &= p_{00} \exp \left( -\frac{z^*}{H} \right) \Deqlab{p-henkan} \end{align} \end{subequations} ここで$p$は圧力, $p_{00}$は地表面参考気圧(定数)である. \vspace{5mm} 入力が$\theta$や$w$ でなく, 気温$T$, 圧力「速度」$\omega \equiv Dp/Dt$ の場合はそれぞれを元に$w$, $\theta$を求める必要がある. 本モジュールでは以下の式 を用いて$w, \theta$を求める. \begin{align} w &= -\omega H / p\\ \theta &= T \left(\frac{p_{00}}{p}\right)^\kappa, \kappa = R/C_p \end{align} ここで$R$, $C_p$はそれぞれ乾燥空気の気体定数および定圧比熱である. %$\theta$ は \Deqref{p-henkan} を用いると %\begin{align} % \theta = T\exp \left( \frac{\kappa z^*}{H} \right ) %\end{align} %と書くこともできる %\footnotemark. %\footnotetext{ (2005/2/13 石渡) この式は必要なのでしょうか? % (2005/2/17 石渡) この式も使ってるんですよ. 場所を移動し %て %} %\chapter{メソッドのリファレンスマニュアル} % メモ: 将来的にはメソッドリファレンス(日本語)を入れる??? \appendix \chapter{プリミティブ方程式系と変形オイラー平均の復習} \Dchaplab{導出} 本章では変形オイラー平均方程式系とEP フラックス および残差循環の関係を確 認する. まず対数圧力座標系を用いた球面上の３次元プリミティブ方程式を提示 する. 次いでそのオイラー平均および変形オイラー平均方程式系を導出する. 最 後に変形オイラー平均方程式に基づき EP フラックスおよび残差循環を定義する. \section{球面上の対数圧力座標系におけるプリミティブ方程式} 球面上の対数圧力座標系におけるプリミティブ方程式は以下の通りである. ここでは Andrews {\it et al.} (1987) の (3.1.3) 式を参考にした. \begin{subequations}\Deqlab{pe} \begin{align} \Deqlab{pe_momentum_x} \DD{u}{t} &- \left(f + \frac{u\tan\phi}{a}\right)v + \Dinv{a\cos\phi}\DP{\Phi}{\lambda} = X,\\ \Deqlab{pe_momentum_y} \DD{v}{t} &+ \left(f + \frac{u\tan\phi}{a}\right)u + \Dinv{a}\DP{\Phi}{\phi} = Y, \end{align} \begin{align} \Deqlab{pe_momentum_z^*} \DP{\Phi}{z^*} & = \frac{R\theta e^{-\kappa z^*/H}}{H}, \end{align} \begin{align} \Deqlab{pe_continuity} \Dinv{a\cos\phi} & \left[ \DP{u}{\lambda} + \left( \DP{v\cos\phi}{\phi} \right) \right] + \Dinv{\rho_0}\DP{}{z^*}\left(\rho_0 w\right) = 0, \end{align} \begin{align} \Deqlab{pe_thermal} \DD{\theta}{t} &= Q, \end{align} \end{subequations} ここで $\Phi$ はジオポテンシャルハイト, $X, Y$ はそれぞれ外力の $\lambda$成分 と $\phi$成分, $\kappa=R_{d}/c_p$ ($c_p$ は等圧比熱)である. $Q$は非断熱加熱項で, \begin{eqnarray*} \Deqlab{adiabatic_heating_term} Q &=& \frac{J}{C_p}e^{\kappa z^*/H} \end{eqnarray*} である. $J$ は単位質量あたりの非断熱加熱率である. ここで明記した以外の変数の定義については \Dsecref{設定}, \Dsecref{EP フラックス} を を参照のこと. \section{オイラー平均方程式系} ある物理量 $A$ について, $\phi, z^*, t$ を固定して 東西方向にとった平均 \begin{eqnarray} \Deqlab{euler_mean} \overline{A}(\phi, z^*, t) \equiv \Dinv{2\pi}\int_0^{2\pi} A(\lambda, \phi, z^*, t) \Dd \lambda \end{eqnarray} をオイラー平均と呼ぶ. オイラー平均からのずれを $A'$ とすると \begin{eqnarray} \Deqlab{euler_eddy} A' = A - \overline{A} \end{eqnarray} である. 定義により, $\overline{A'}=0$, $\partial \overline{A}/\partial\lambda = 0$ となる. \Deqref{pe} 中の各量をオイラー平均とそこからのずれに分けて書くと \begin{subequations}\Deqlab{exp_pe} % \def\theequation{\arabic{section}.\arabic{parentequation}.\arabic{equation}} \begin{align} \Deqlab{exp_pe_momentum_x} & \DP{}{t}(\overline{u} + u') + \frac{\overline{u} + u'}{a\cos\phi}\DP{}{\lambda}(\overline{u} + u') + \frac{\overline{v} + v'}{a}\DP{}{\phi}(\overline{u} + u') + (\overline{w} + w')\DP{}{z^*}(\overline{u} + u') \notag\\ & \qquad - \left[f + \frac{\tan\phi}{a}(\overline{u} + u')\right](\overline{v} + v') + \Dinv{a\cos\phi}\DP{}{\lambda}(\overline{\Phi} + \Phi') = \overline{X} + X',\\ \Deqlab{exp_pe_momentum_y} & \DP{}{t}(\overline{v} + v') + \frac{\overline{u} + u'}{a\cos\phi}\DP{}{\lambda}(\overline{v} + v') + \frac{\overline{v} + v'}{a}\DP{}{\phi}(\overline{v} + v') + (\overline{w} + w')\DP{}{z^*}(\overline{v} + v')\notag\\ & \qquad + \left[f + \frac{\tan\phi}{a}(\overline{u} + u')\right](\overline{u} + u') + \Dinv{a}\DP{}{\phi}(\overline{\Phi} + \Phi') = \overline{Y} + Y', \\ \Deqlab{exp_pe_momentum_z^*} & \DP{}{z^*}(\overline{\Phi} + \Phi') = \frac{Re^{-\kappa z^*/H}}{H}(\overline{\theta} + \theta'),\\ \Deqlab{exp_pe_continuity} & \Dinv{a\cos\phi} \left[\DP{}{\lambda}(\overline{u} + u') + \DP{}{\phi}\{(\overline{v} + v')\cos\phi\}\right] + \Dinv{\rho_0}\DP{}{z^*}[\rho_0 (\overline{w} + w')] = 0,\\ \Deqlab{exp_pe_thermal} & \DP{}{t}(\overline{\theta} + \theta') + \frac{\overline{u} + u'}{a\cos\phi}\DP{}{\lambda}(\overline{\theta} + \theta') + \frac{\overline{v} + v'}{a}\DP{}{\phi}(\overline{\theta} + \theta') + (\overline{w} + w')\DP{}{z^*}(\overline{\theta} + \theta')\notag\\ & \qquad = \overline{Q} + Q' \end{align} \end{subequations} となる. 上記を変形して, 左辺に平均量と平均量同士の積の項を, 右辺にそれ以外の項をまとめると \begin{subequations}\Deqlab{exp2_pe} % \def\theequation{\arabic{section}.\arabic{parentequation}.\arabic{equation}} \begin{align} \Deqlab{exp2_pe_momentum_x} & \DP{\overline{u}}{t} + \frac{\overline{u}}{a\cos\phi}\DP{\overline{u}}{\lambda} + \frac{\overline{v}}{a}\DP{\overline{u}}{\phi} + \overline{w}\DP{\overline{u}}{z^*} - f\overline{v} - \frac{\tan\phi}{a} \overline{u} \ \overline{v} + \Dinv{a\cos\phi}\DP{\overline{\Phi}}{\lambda} - \overline{X} \notag\\ & \qquad = - \DP{u'}{t} - \frac{\overline{u}}{a\cos\phi}\DP{u'}{\lambda} - \frac{u'}{a\cos\phi}\DP{\overline{u}}{\lambda} - \frac{u'}{a\cos\phi}\DP{u'}{\lambda} \notag\\ & \qquad \qquad - \frac{\overline{v}}{a}\DP{u'}{\phi} - \frac{v'}{a}\DP{\overline{u}}{\phi} - \frac{v'}{a}\DP{u'}{\phi} - \overline{w}\DP{u'}{z^*} - w'\DP{\overline{u}}{z^*} - w'\DP{u'}{z^*} + fv'\notag\\ & \qquad \qquad + \frac{\tan\phi}{a} \overline{u} v' + \frac{\tan\phi}{a} u' \overline{v} + \frac{\tan\phi}{a} u'v' %\notag\\ % & \qquad - \Dinv{a\cos\phi}\DP{\Phi'}{\lambda} + X',\\ % \Deqlab{exp2_pe_momentum_y} & \DP{\overline{v}}{t} + \frac{\overline{u}}{a\cos\phi}\DP{\overline{v}}{\lambda} + \frac{\overline{v}}{a}\DP{\overline{v}}{\phi} + \overline{w}\DP{\overline{v}}{z^*} + f\overline{u} + \frac{\tan\phi}{a}(\overline{u})^2 + \Dinv{a}\DP{\overline{\Phi}}{\phi} - \overline{Y} \notag\\ & \qquad = - \DP{v'}{t} - \frac{\overline{u}}{a\cos\phi}\DP{v'}{\lambda} - \frac{u'}{a\cos\phi}\DP{\overline{v}}{\lambda} - \frac{u'}{a\cos\phi}\DP{v'}{\lambda}\notag\\ & \qquad \qquad - \frac{\overline{v}}{a}\DP{v'}{\phi} - \frac{v'}{a}\DP{\overline{v}}{\phi} - \frac{v'}{a}\DP{v'}{\phi} %\notag\\ % & \qquad \qquad - \overline{w}\DP{v'}{z^*} - w'\DP{\overline{v}}{z^*} - w'\DP{v'}{z^*} %\notag\\ % & \qquad - fu'\notag\\ & \qquad \qquad - 2\frac{\tan\phi}{a}\overline{u}u' - \frac{\tan\phi}{a}(u')^2 %\notag\\ % & \qquad - \Dinv{a\cos\phi}\DP{\Phi'}{\phi} + Y',\\ % \Deqlab{exp2_pe_momentum_z^*} & \DP{\overline{\Phi}}{z^*} - \frac{Re^{-\kappa z^*/H}}{H}\overline{\theta} = - \DP{\Phi'}{z^*} + \frac{Re^{-\kappa z^*/H}}{H}\theta',\\ \Deqlab{exp2_pe_continuity} & \Dinv{a\cos\phi} \left[\DP{\overline{u}}{\lambda} + \DP{}{\phi}(\overline{v}\cos\phi)\right] + \Dinv{\rho_0}\DP{}{z^*}(\rho_0 \overline{w}) \notag\\ & \qquad = - \Dinv{a\cos\phi}\left[ \DP{u'}{\lambda} + \DP{}{\phi}(v'\cos\phi) \right] - \Dinv{\rho_0}\DP{}{z^*}(\rho_0 w') ,\\ \Deqlab{exp2_pe_thermal} & \DP{\overline{\theta}}{t} + \frac{\overline{u}}{a\cos\phi}\DP{\overline{\theta}}{\lambda} + \frac{\overline{v}}{a}\DP{\overline{\theta}}{\phi} + \overline{w}\DP{\overline{\theta}}{z^*} - \overline{Q} \notag\\ & \qquad = - \DP{\theta'}{t} - \frac{\overline{u}}{a\cos\phi}\DP{\theta'}{\lambda} - \frac{u'}{a\cos\phi}\DP{\overline{\theta}}{\lambda} - \frac{u'}{a\cos\phi}\DP{\theta'}{\lambda} \notag \\ & \qquad \qquad - \frac{\overline{v}}{a}\DP{\theta'}{\phi} - \frac{v'}{a}\DP{\overline{\theta}}{\phi} - \frac{v'}{a}\DP{\theta'}{\phi} - \overline{w}\DP{\theta'}{z^*} - w'\DP{\overline{\theta}}{z^*} - w'\DP{\theta'}{z^*} + Q' \end{align} \end{subequations} と書ける. \Deqref{exp2_pe} をオイラー平均すると, \begin{subequations}\Deqlab{euler_mean_pe} % \def\theequation{\arabic{section}.\arabic{parentequation}.\arabic{equation}} \begin{align} \Deqlab{euler_mean_pe_momentum_x} & \DP{\overline{u}}{t} + \Dinv{a}\overline{v}\DP{\overline{u}}{\phi} + \overline{w}\DP{\overline{u}}{z^*} - f\overline{v} - \frac{\tan\phi}{a} \overline{u} \ \overline{v} - \overline{X} \notag\\ & \qquad = - \Dinv{a\cos\phi}\overline{u'\DP{u'}{\lambda}} - \Dinv{a}\overline{v'\DP{u'}{\phi}} - \overline{w'\DP{u'}{z^*}} + \frac{\tan\phi}{a}\overline{u'v'},\\ % \Deqlab{euler_mean_pe_momentum_y} & \DP{\overline{v}}{t} + \frac{\overline{v}}{a}\DP{\overline{v}}{\phi} + \overline{w} \DP{\overline{v}}{z^*} + f \overline{u} + \frac{\tan \phi}{a} (\overline{u})^2 + \Dinv{a}\DP{\overline{\Phi}}{\phi} - \overline{Y} \notag\\ & \qquad = - \Dinv{a \cos \phi} \overline{ u' \DP{v'}{\lambda} } - \Dinv{a} \overline{{v'}\DP{v'}{\phi}} - \overline{w'\DP{v'}{z^*}} - \frac{\tan \phi}{a} \overline{u'^2},\\ % \Deqlab{euler_mean_pe_momentum_z^*} & \DP{\overline{\Phi}}{z^*} - \frac{Re^{-\kappa z^*/H}}{H}\overline{\theta} = 0,\\ \Deqlab{euler_mean_pe_continuity} & \Dinv{a\cos\phi} \left[ \DP{}{\phi}(\overline{v}\cos\phi) \right] + \Dinv{\rho_0}\DP{}{z^*}(\rho_0 \overline{w}) = 0,\\ % \Deqlab{euler_mean_pe_thermal} & \DP{\overline{\theta}}{t} + \frac{\overline{v}}{a}\DP{\overline{\theta}}{\phi} + \overline{w}\DP{\overline{\theta}}{z^*} - \overline{Q} = - \Dinv{a\cos\phi}\overline{u'\DP{\theta'}{\lambda}} - \Dinv{a}\overline{v'\DP{\theta'}{\phi}} - \overline{w'\DP{\theta'}{z^*}} \end{align} \end{subequations} となる. ここで \Deqref{exp2_pe_continuity}, \Deqref{euler_mean_pe_continuity} から東西平均からのずれに関する連続の式 \begin{eqnarray} \Deqlab{euler_eddy_pe_continuity} \Dinv{a\cos\phi}\left[\DP{u'}{\lambda} + \DP{}{\phi}(v'\cos\phi)\right] + \Dinv{\rho_0}\DP{}{z^*}(\rho_0 w') = 0 \end{eqnarray} が得られる. \vspace{5mm} \Deqref{euler_eddy_pe_continuity} を使って \Deqref{euler_mean_pe_momentum_x} を変形する. \Deqref{euler_eddy_pe_continuity} に $u'$ をかけて オイラー平均をとると \begin{eqnarray} \Dinv{a \cos \phi} \overline{u' \DP{u'}{\lambda}} + \Dinv{a} \overline{ u' \DP{v'}{\phi} } - \frac{\tan \phi}{a} \overline{ u' v' } + \overline{ u' \DP{w'}{z^*} } + \Dinv{\rho_0} \DP{\rho_0}{z^*} \overline{ u' w' } = 0 \end{eqnarray} これを \Deqref{euler_mean_pe_momentum_x} に加えると \begin{align*} \DP{\overline{u}}{t}& + \Dinv{a}\overline{v}\DP{\overline{u}}{\phi} + \overline{w}\DP{\overline{u}}{z^*} - f\overline{v} - \frac{\tan\phi}{a}\overline{u}\overline{v} - \overline{X} \notag\\ & = - \frac{2}{a\cos\phi} \overline{u'\DP{u'}{\lambda}} - \Dinv{a}\overline{v'\DP{u'}{\phi}} - \overline{w'\DP{u'}{z^*}} %\notag\\ % & - \Dinv{a}\overline{u'\DP{v'}{\phi}} + \frac{2\tan\phi}{a}\overline{u'v'} - \overline{u'\DP{w'}{z^*}} - \Dinv{\rho_0} \DP{\rho_0}{z^*} \overline{u'w'} \end{align*} ここで \begin{align*} - \frac{2}{a\cos\phi} \overline{ u' \DP{u'}{\lambda} } & = - \Dinv{a\cos\phi}\overline{\DP{(u')^2}{\lambda}} = 0,\\ % - \Dinv{a}\overline{v'\DP{u'}{\phi}} - \Dinv{a}\overline{u'\DP{v'}{\phi}} + \frac{2\tan\phi}{a}\overline{u'v'} & = - \Dinv{a\cos^2\phi}\DP{}{\phi}(\overline{v'u'}\cos^2\phi), \\ % - \overline{w'\DP{u'}{z^*}} - \overline{u'\DP{w'}{z^*}} - \Dinv{\rho_0} \DP{\rho_0}{z^*} \overline{u'w'} & = - \Dinv{\rho_0}\DP{}{z^*}(\rho_0\overline{w'u'}) \end{align*} を用いると, \begin{align*} & \DP{\overline{u}}{t} + \Dinv{a}\overline{v}\DP{\overline{u}}{\phi} + \overline{w}\DP{\overline{u}}{z^*} - f\overline{v} - \frac{\tan\phi}{a} \overline{u} \ \overline{v} - \overline{X} \notag \\ & \qquad = - \Dinv{a\cos^2\phi}\DP{}{\phi}(\overline{v'u'}\cos^2\phi) - \Dinv{\rho_0}\DP{}{z^*}(\rho_0\overline{w'u'}) \end{align*} と書くことができる. \Deqref{euler_mean_pe_momentum_y} に関しても同様に, \Deqref{euler_eddy_pe_continuity} に $v'$ をかけて オイラー平均をとった式 \begin{eqnarray} \Dinv{a \cos \phi} \overline{ v' \DP{u'}{\lambda} } + \Dinv{a} \overline{ v' \DP{v'}{\phi} } + \frac{\tan \phi}{a} \overline{ v'^2 } + \overline{ v' \DP{w'}{z^*} } + \Dinv{\rho_0} \DP{\rho_0}{z^*} \overline{ v' w' } = 0 \end{eqnarray} を \Deqref{euler_mean_pe_momentum_y} に加えると \begin{align*} & \DP{\overline{v}}{t} + \frac{\overline{v}}{a} \DP{\overline{v}}{\phi} + \overline{w} \DP{\overline{v}}{z^*} + f \overline{u} + \frac{\tan\phi}{a} (\overline{u})^2 + \Dinv{a} \DP{\overline{\Phi}}{\phi} - \overline{Y} \notag\\ & \qquad = - \Dinv{a\cos\phi}\overline{u'\DP{v'}{\lambda}} - \Dinv{a}\overline{{v'}\DP{v'}{\phi}} - \overline{w'\DP{v'}{z^*}} - \frac{\tan\phi}{a} \overline{u'^2} \notag\\ & \qquad \qquad - \Dinv{a \cos \phi} \overline{v' \DP{u'}{\lambda}} - \Dinv{a} \overline{ v' \DP{v'}{\phi} } + \frac{\tan \phi}{a} \overline{ v'^2 } - \overline{ v' \DP{w'}{z^*} } - \Dinv{\rho_0} \DP{\rho_0}{z^*} \overline{ v' w' } \end{align*} が得られる. ここで \begin{eqnarray} - \Dinv{a\cos\phi}\overline{u'\DP{v'}{\lambda}} - \Dinv{a \cos \phi} \overline{v' \DP{u'}{\lambda}} & = & - \Dinv{a\cos\phi}\overline{\DP{(u' v')}{\lambda}} = 0, \nonumber \\ % - \Dinv{a} \overline{ v' \DP{v'}{\phi} } - \Dinv{a} \overline{ v' \DP{v'}{\phi} } + \frac{\tan \phi}{a} \overline{ v'^2 } & = & - \Dinv{a \cos \phi} \DP{}{\phi} \left( \cos \phi \overline{v'^2} \right) \nonumber \\ % - \overline{w'\DP{v'}{z^*}} - \overline{ v' \DP{w'}{z^*} } - \Dinv{\rho_0} \DP{\rho_0}{z^*} \overline{ v' w' } & = & - \Dinv{\rho_0} \DP{}{z^*} \left( \rho_0 \overline{ v' w' } \right) \end{eqnarray} を用いると \begin{align*} & \DP{\overline{v}}{t} + \frac{\overline{v}}{a} \DP{\overline{v}}{\phi} + \overline{w}\ DP{\overline{v}}{z^*} + f \overline{u} + \frac{\tan\phi}{a}(\overline{u})^2 + \Dinv{a} \DP{\overline{\Phi}}{\phi} - \overline{Y} \notag\\ & \qquad = - \Dinv{a \cos \phi} \DP{}{\phi} \left( \cos \phi \overline{v'^2} \right) - \frac{\tan\phi}{a} \overline{u'^2} - \Dinv{\rho_0} \DP{}{z^*} \left( \rho_0 \overline{ v' w' } \right) \end{align*} と書くことができる. \Deqref{euler_mean_pe_thermal} についても同様に, \Deqref{euler_eddy_pe_continuity} に $\theta'$ をかけて オイラー平均をとった式 \begin{eqnarray} \Dinv{a \cos \phi} \overline{\theta' \DP{u'}{\lambda}} + \Dinv{a} \overline{ \theta' \DP{v'}{\phi} } - \frac{\tan \phi}{a} \overline{ \theta' v' } + \overline{ \theta' \DP{w'}{z^*} } + \Dinv{\rho_0} \DP{\rho_0}{z^*} \overline{ \theta' w' } = 0 \end{eqnarray} を \Deqref{euler_mean_pe_thermal} に加えると \begin{align*} & \DP{\overline{\theta}}{t} + \frac{\overline{v}}{a}\DP{\overline{\theta}}{\phi} + \overline{w}\DP{\overline{\theta}}{z^*} - \overline{Q} \notag\\ & \qquad = - \Dinv{a\cos\phi}\overline{u'\DP{\theta'}{\lambda}} - \Dinv{a}\overline{v'\DP{\theta'}{\phi}} - \overline{w'\DP{\theta'}{z^*}} \notag\\ & \qquad \qquad - \Dinv{a \cos \phi} \overline{\theta' \DP{u'}{\lambda}} - \Dinv{a} \overline{ \theta' \DP{v'}{\phi} } + \frac{\tan \phi}{a} \overline{ \theta' v' } - \overline{ \theta' \DP{w'}{z^*} } - \Dinv{\rho_0} \DP{\rho_0}{z^*} \overline{ \theta' w' } \end{align*} が得られる. ここで \begin{eqnarray} - \Dinv{a \cos \phi}\overline{u' \DP{\theta'}{\lambda}} - \Dinv{a \cos \phi} \overline{\theta' \DP{u'}{\lambda}} & = & - \Dinv{a\cos\phi}\overline{\DP{(u' \theta')}{\lambda}} = 0, \nonumber \\ % - \Dinv{a} \overline{ v' \DP{\theta'}{\phi} } - \Dinv{a} \overline{ \theta' \DP{v'}{\phi} } + \frac{\tan \phi}{a} \overline{ \theta' v' } & = & - \Dinv{a \cos \phi} \DP{}{\phi} \left( \cos \phi \overline{v' \theta'} \right) \nonumber \\ % - \overline{w'\DP{\theta'}{z^*}} - \overline{ \theta' \DP{w'}{z^*} } - \Dinv{\rho_0} \DP{\rho_0}{z^*} \overline{ \theta' w' } & = & - \Dinv{\rho_0} \DP{}{z^*} \left( \rho_0 \overline{ w' \theta' } \right) \nonumber \end{eqnarray} を用いると \begin{align*} & \DP{\overline{\theta}}{t} + \frac{\overline{v}}{a}\DP{\overline{\theta}}{\phi} + \overline{w}\DP{\overline{\theta}}{z^*} - \overline{Q} = - \Dinv{a \cos \phi} \DP{}{\phi} \left( \cos \phi \overline{v' \theta'} \right) - \Dinv{\rho_0} \DP{}{z^*} \left( \rho_0 \overline{ w' \theta' } \right) \end{align*} となる. \vspace{5mm} 以上をまとめると, 以下の{\bfseries オイラー平均方程式}が得られる. \begin{screen} \begin{subequations} \Deqlab{new_euler_mean_pe} %\setcounter{equation}{0} %\begin{itemize} %%%%%%%%%%%%%%%% %\item 運動方程式 %%%%%%%%%%%%%%%% \begin{align} \Deqlab{new_euler_mean_pe_momentum_x} \DP{\overline{u}}{t} & + \Dinv{a}\overline{v} \DP{\overline{u}}{\phi} + \overline{w} \DP{\overline{u}}{z^*} - f\overline{v} - \frac{\tan\phi}{a} \overline{u} \ \overline{v} - \overline{X} \notag\\ & \qquad = - \Dinv{a\cos^2\phi} \DP{}{\phi} (\overline{v'u'} \cos^2 \phi) - \Dinv{\rho_0} \DP{}{z^*}(\rho_0\overline{w'u'}),\\ % \Deqlab{new_euler_mean_pe_momentum_y} \DP{\overline{v}}{t} & + \frac{\overline{v}}{a} \DP{\overline{v}}{\phi} + \overline{w} \DP{\overline{v}}{z^*} + f \overline{u} + \frac{\tan\phi}{a} (\overline{u})^2 + \Dinv{a} \DP{\overline{\Phi}}{\phi} - \overline{Y} \notag\\ & \qquad = - \Dinv{a\cos\phi}\DP{}{\phi}(\overline{v'^2} \cos\phi) - \Dinv{\rho_0}\DP{}{z^*}(\rho_0\overline{v' w'}) - \overline{u'^2}\frac{\tan\phi}{a}, \end{align} %%%%%%%%%%%%%%%% %\item 静水圧平衡の式 %%%%%%%%%%%%%%%% \begin{align} \Deqlab{new_euler_mean_pe_momentum_z^*} \DP{\overline{\Phi}}{z^*} - \frac{Re^{-\kappa z^*/H}}{H}\overline{\theta} = 0, \end{align} %%%%%%%%%%%%%%%% %\item 連続の式 %%%%%%%%%%%%%%%% \begin{align} \Deqlab{new_euler_mean_pe_continuity} \Dinv{a\cos\phi}& \DP{}{\phi}(\overline{v}\cos\phi) + \Dinv{\rho_0}\DP{}{z^*}(\rho_0 \overline{w}) = 0, \end{align} %%%%%%%%%%%%%%%% %\item 熱力学の式 %%%%%%%%%%%%%%%% \begin{align} \Deqlab{new_euler_mean_pe_thermal} \DP{\overline{\theta}}{t} + \frac{\overline{v}}{a}\DP{\overline{\theta}}{\phi} + \overline{w}\DP{\overline{\theta}}{z^*} - \overline{Q} = - \Dinv{a\cos\phi}\DP{}{\phi}(\overline{v'\theta'}\cos\phi) - \Dinv{\rho_0}\DP{}{z^*}(\rho_0\overline{w'\theta'}). \end{align} %\end{itemize} \end{subequations} \end{screen} \section{変形オイラー平均方程式系} \Deqref{new_euler_mean_pe} を EP フラックス, 残差循環を用いて書き直す. EP フラックス, 残差循環は以下のように定義する. \begin{subequations} % \begin{itemize} %%%%%%%%%%%%%%%% % \item 残差循環 %%%%%%%%%%%%%%%% \begin{align} \Deqlab{residual_v_app} \overline{v}^* & = \overline{v} - \Dinv{\rho_0} \DP{}{z^*} \left( \rho_0 \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \\ \Deqlab{residual_w_app} \overline{w}^* & = \overline{w} + \Dinv{a \cos\phi} \DP{}{\phi} \left( \cos \phi \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \end{align} \end{subequations} %%%%%%%%%%%%%%%% % \item EP フラックス %%%%%%%%%%%%%%%% \begin{eqnarray*} {F_\phi} &=& \rho_0 a \cos \phi \left(\DP{\overline{u}}{\overline{z^*}} \frac{\overline{v'\theta'}}{\DP{\overline{\theta}}{z^*}} - \overline{u'v'}\right) \\ {F_z^*} &=& \rho_0 a \cos \phi \left(\left[ f - \frac{\DP{\overline{u}\cos \phi}{\phi}}{a\cos\phi} \right] \frac{\overline{v'\theta'}}{\DP{\overline{\theta}}{z^*}} - \overline{u'w'}\right) \end{eqnarray*} % \end{itemize} \vspace{5mm} まず連続の式を書き換える. \Deqref{new_euler_mean_pe_continuity} に \Deqref{residual_v_app}, \Deqref{residual_w_app} を代入すると \begin{align*} & \Dinv{a \cos \phi} \DP{}{\phi}\left[ \left\{ \overline{v}^* + \Dinv{\rho_0} \DP{}{z^*} \left( \rho_0 \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \right\} \cos\phi \right] \\ & \qquad + \Dinv{\rho_0} \DP{}{z^*} \left[ \rho_0 \left\{ \overline{w}^* - \Dinv{a \cos\phi} \DP{}{\phi} \left( \cos \phi \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \right\} \right] = 0, \\ % & \Dinv{a \cos \phi} \DP{}{\phi} \left( \overline{v}^* \cos\phi \right) + \Dinv{\rho_0} \DP{}{z^*} \left( \rho_0 \overline{w}^* \right) \\ & \qquad + \Dinv{a \cos \phi} \DP{}{\phi} \left\{ \Dinv{\rho_0} \DP{}{z^*} \left( \rho_0 \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \cos\phi \right\} - \Dinv{\rho_0} \DP{}{z^*} \left\{ \rho_0 \Dinv{a \cos\phi} \DP{}{\phi} \left( \cos \phi \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \right\} = 0. \end{align*} この第三項と第四項だけを取り出すと \begin{align*} & \qquad \Dinv{a \cos \phi} \DP{}{\phi} \left\{ \Dinv{\rho_0} \DP{}{z^*} \left( \rho_0 \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \cos\phi \right\} - \Dinv{\rho_0} \DP{}{z^*} \left\{ \rho_0 \Dinv{a \cos\phi} \DP{}{\phi} \left( \cos \phi \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \right\} \\ & = \Dinv{a \cos \phi} \left[ \DP{}{\phi} \left\{ \Dinv{\rho_0} \DP{}{z^*} \left( \rho_0 \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \cos\phi \right\} - \Dinv{\rho_0} \DP{}{z^*} \left\{ \rho_0 \DP{}{\phi} \left( \cos \phi \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \right\} \right] \\ & = \Dinv{a \cos \phi} \left[ \Dinv{\rho_0} \DP{}{\phi} \left\{ \DP{}{z^*} \left( \rho_0 \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \cos\phi \right) \right\} - \Dinv{\rho_0} \DP{}{z^*} \left\{ \DP{}{\phi} \left(\rho_0 \cos \phi \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \right\} \right] \\ & = 0. \end{align*} したがって, 連続の式は以下のようになる. \begin{eqnarray} \Dinv{a \cos \phi} \DP{}{\phi} \left( \overline{v}^* \cos\phi \right) + \Dinv{\rho_0} \DP{}{z^*} \left( \rho_0 \overline{w}^* \right) = 0. \end{eqnarray} \vspace{5mm} 次に $u$ の式を書き換える. \Deqref{new_euler_mean_pe_momentum_x} に \Deqref{residual_v_app}, \Deqref{residual_w_app} を代入すると \begin{align} \DP{\overline{u}}{t}& + \Dinv{a} \left[ \overline{v}^* + \Dinv{\rho_0} \DP{}{z^*} \left( \rho_0 \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \right] \DP{\overline{u}}{\phi} + \left[ \overline{w}^* - \Dinv{a \cos\phi} \DP{}{\phi} \left( \cos \phi \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \right] \DP{\overline{u}}{z^*} \nonumber \\ & \qquad \qquad - f \left[ \overline{v}^* + \Dinv{\rho_0} \DP{}{z^*} \left( \rho_0 \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \right] - \frac{\tan \phi}{a} \overline{u} \left[ \overline{v}^* + \Dinv{\rho_0} \DP{}{z^*} \left( \rho_0 \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \right] - \overline{X} \nonumber \\ & \qquad = - \Dinv{a\cos^2\phi} \DP{}{\phi} (\overline{v'u'} \cos^2 \phi) - \Dinv{\rho_0} \DP{}{z^*} (\rho_0\overline{w'u'}),\nonumber \\ % \DP{\overline{u}}{t}& + \frac{\overline{v}^*}{a} \DP{\overline{u}}{\phi} + \overline{w}^* \DP{\overline{u}}{z^*} - f \overline{v}^* - \frac{\tan \phi}{a} \overline{u} \ \overline{v}^* - \overline{X} \nonumber \\ & \qquad = - \Dinv{a\cos^2\phi} \DP{}{\phi} (\overline{v'u'} \cos^2 \phi) + \Dinv{a \cos\phi} \DP{}{\phi} \left( \cos \phi \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \DP{\overline{u}}{z^*} \nonumber \\ & \qquad \qquad + f \Dinv{\rho_0} \DP{}{z^*} \left( \rho_0 \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) - \Dinv{\rho_0} \DP{}{z^*} (\rho_0\overline{w'u'}) \nonumber \\ & \qquad \qquad - \Dinv{\rho_0 a} \DP{}{z^*} \left( \rho_0 \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \DP{\overline{u}}{\phi} + \frac{\tan \phi}{a} \overline{u} \Dinv{\rho_0} \DP{}{z^*} \left( \rho_0 \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right), \nonumber \\ % \DP{\overline{u}}{t}& + \frac{\overline{v}^*}{a \cos \phi} \DP{}{\phi} \left( \overline{u} \cos \phi \right) + \overline{w}^* \DP{\overline{u}}{z^*} - f \overline{v}^* - \overline{X} \nonumber \\ & \qquad = - \Dinv{\rho_0 a^2 \cos^2 \phi} \DP{}{\phi} (\rho_0 a \overline{v'u'} \cos^2 \phi) + \Dinv{a \cos\phi} \DP{}{\phi} \left( \cos \phi \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \DP{\overline{u}}{z^*} \nonumber \\ & \qquad \qquad + \frac{1}{\rho_0 a \cos \phi} \DP{}{z^*} \left( f \rho_0 a \cos \phi \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) - \frac{1}{\rho_0 a \cos \phi} \DP{}{z^*} (\rho_0 a \cos \phi \overline{w'u'}) \nonumber \\ & \qquad \qquad - \Dinv{\rho_0 a} \DP{}{z^*} \left( \rho_0 \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \DP{\overline{u}}{\phi} + \frac{\tan \phi}{a} \overline{u} \Dinv{\rho_0} \DP{}{z^*} \left( \rho_0 \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \Deqlab{tem-u-tochuu} \end{align} \Deqref{tem-u-tochuu} の右辺を以下のように変形する. \begin{align} & - \Dinv{\rho_0 a^2 \cos^2 \phi} \DP{}{\phi} (\rho_0 a \overline{v'u'} \cos^2 \phi) + \Dinv{\rho_0 a^2 \cos^2 \phi} \rho_0 a \cos \phi \DP{\overline{u}}{z^*} \DP{}{\phi} \left( \cos \phi \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \nonumber \\ & \qquad \qquad + \frac{1}{\rho_0 a \cos \phi} \DP{}{z^*} \left( f \rho_0 a \cos \phi \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) - \frac{1}{\rho_0 a \cos \phi} \DP{}{z^*} (\rho_0 a \cos \phi \overline{w'u'}) \nonumber \\ & \qquad \qquad - \Dinv{\rho_0 a} \DP{}{z^*} \left( \rho_0 \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \DP{\overline{u}}{\phi} \right) + \Dinv{\rho_0 a} \rho_0 \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \DP{}{z^*} \left( \DP{\overline{u}}{\phi} \right) \nonumber \\ & \qquad \qquad + \frac{\tan \phi}{\rho_0 a} \DP{}{z^*} \left( \overline{u} \rho_0 \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) - \frac{\tan \phi}{\rho_0 a} \rho_0 \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \DP{}{z^*} \left( \overline{u} \right) \nonumber \\ % & = \Dinv{\rho_0 a^2 \cos^2 \phi} \left[ - \DP{}{\phi} (\rho_0 a \overline{v'u'} \cos^2 \phi) + \rho_0 a \cos \phi \DP{\overline{u}}{z^*} \DP{}{\phi} \left( \cos \phi \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \right] \nonumber \\ & \qquad + \Dinv{\rho_0 a} \rho_0 \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \DP{}{z^*} \left( \DP{\overline{u}}{\phi} \right) - \frac{\tan \phi}{\rho_0 a} \rho_0 \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \DP{\overline{u}}{z^*} \nonumber \\ & \qquad + \frac{1}{\rho_0 a \cos \phi} \DP{}{z^*} \left[ \left( f \rho_0 a \cos \phi \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) - \rho_0 a \cos \phi \overline{w'u'} \right] \nonumber \\ & \qquad - \Dinv{\rho_0 a} \DP{}{z^*} \left( \rho_0 \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \DP{\overline{u}}{\phi} \right) + \frac{\tan \phi}{\rho_0 a} \DP{}{z^*} \left( \overline{u} \rho_0 \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \nonumber \\ % & = \Dinv{\rho_0 a^2 \cos^2 \phi} \left[ - \DP{}{\phi} (\rho_0 a \overline{v'u'} \cos^2 \phi) + \rho_0 a \cos \phi \DP{\overline{u}}{z^*} \DP{}{\phi} \left( \cos \phi \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \right] \nonumber \\ & \qquad + \Dinv{\rho_0 a^2 \cos^2 \phi} \left[ \rho_0 a \cos^2 \phi \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \DP{}{z^*} \left( \DP{\overline{u}}{\phi} \right) - \rho_0 a \cos^2 \phi \tan \phi \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \DP{\overline{u}}{z^*} \right] \nonumber \\ & \qquad + \frac{1}{\rho_0 a \cos \phi} \DP{}{z^*} \left[ \left( f \rho_0 a \cos \phi \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) - \rho_0 a \cos \phi \overline{w'u'} \right] \nonumber \\ & \qquad + \Dinv{\rho_0 a \cos \phi} \left[ - \cos \phi \DP{}{z^*} \left( \rho_0 \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \DP{\overline{u}}{\phi} \right) + \cos \phi \tan \phi \DP{}{z^*} \left( \overline{u} \rho_0 \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \right] \nonumber \\ % & = \Dinv{\rho_0 a^2 \cos^2 \phi} \left[ - \DP{}{\phi} (\rho_0 a \overline{v'u'} \cos^2 \phi) + \rho_0 a \cos \phi \DP{\overline{u}}{z^*} \DP{}{\phi} \left( \cos \phi \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \right] \nonumber \\ & \qquad + \Dinv{\rho_0 a^2 \cos^2 \phi} \left[ \rho_0 a \cos^2 \phi \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \DP{}{\phi} \left( \DP{\overline{u}}{z^*} \right) + \cos \phi \DP{}{\phi} \left( \rho_0 a \cos \phi \right) \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \DP{\overline{u}}{z^*} \right] \nonumber \\ & \qquad + \frac{1}{\rho_0 a \cos \phi} \DP{}{z^*} \left[ f \rho_0 a \cos \phi \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} - \rho_0 a \cos \phi \overline{w'u'} \right] \nonumber \\ & \qquad + \Dinv{\rho_0 a \cos \phi} \DP{}{z^*} \left[ - \rho_0 \cos \phi \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \DP{\overline{u}}{\phi} + \sin \phi \overline{u} \rho_0 \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right] \Deqlab{tem-u-uhen} \end{align} \Deqref{tem-u-uhen} の第一項と第二項だけ取り出すと \begin{align*} % & \Dinv{\rho_0 a^2 \cos^2 \phi} \left[ - \DP{}{\phi} (\rho_0 a \overline{v'u'} \cos^2 \phi) + \rho_0 a \cos \phi \DP{\overline{u}}{z^*} \DP{}{\phi} \left( \cos \phi \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \right] \\ & \qquad + \Dinv{\rho_0 a^2 \cos^2 \phi} \left[ \rho_0 a \cos^2 \phi \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \DP{}{\phi} \left( \DP{\overline{u}}{z^*} \right) + \cos \phi \DP{}{\phi} \left( \rho_0 a \cos \phi \right) \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \DP{\overline{u}}{z^*} \right] \\ % & = \Dinv{\rho_0 a^2 \cos^2 \phi} \left[ - \DP{}{\phi} (\rho_0 a \overline{v'u'} \cos^2 \phi) \right] \\ & \qquad + \Dinv{\rho_0 a^2 \cos^2 \phi} \left[ \rho_0 a \cos^2 \phi \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \DP{}{\phi} \left( \DP{\overline{u}}{z^*} \right) + \DP{\overline{u}}{z^*} \DP{}{\phi} \left(\rho_0 a \cos^2 \phi \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \right] \\ % & = \Dinv{\rho_0 a^2 \cos^2 \phi} \left[ - \DP{}{\phi} (\rho_0 a \overline{v'u'} \cos^2 \phi) \right] + \Dinv{\rho_0 a^2 \cos^2 \phi} \left[ \DP{}{\phi} \left(\rho_0 a \cos^2 \phi \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \DP{\overline{u}}{z^*} \right) \right] \\ % & = \Dinv{\rho_0 a^2 \cos^2 \phi} \DP{}{\phi} \left[ - \rho_0 a \overline{v'u'} \cos^2 \phi + \rho_0 a \cos^2 \phi \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \DP{\overline{u}}{z^*} \right] \\ % & = \Dinv{\rho_0 a^2 \cos^2 \phi} \DP{}{\phi} \left[ \rho_0 a \cos^2 \phi \left\{ \DP{\overline{u}}{z^*} \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} - \overline{v'u'} \right\} \right] \\ % & = \Dinv{\rho_0 a^2 \cos^2 \phi} \DP{}{\phi} \left( \cos \phi F^{*}_{\phi} \right) \end{align*} \Deqref{tem-u-uhen} の第三項と第四項だけ取り出すと \begin{align*} & \frac{1}{\rho_0 a \cos \phi} \DP{}{z^*} \left[ f \rho_0 a \cos \phi \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} - \rho_0 a \cos \phi \overline{w'u'} \right] % \\ % & \qquad + \Dinv{\rho_0 a \cos \phi} \DP{}{z^*} \left[ - \rho_0 \cos \phi \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \DP{\overline{u}}{\phi} + \sin \phi \overline{u} \rho_0 \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right] \\ & = \frac{1}{\rho_0 a \cos \phi} \DP{}{z^*} \left[ \rho_0 a \cos \phi \left\{ f \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} - \overline{w'u'} - \frac{\overline{v'\theta'}} {a \overline{\DP{\theta}{z^*}}} \DP{\overline{u}}{\phi} + \sin \phi \overline{u} \frac{\overline{v'\theta'}} {a \cos \phi \overline{\DP{\theta}{z^*}}} \right\} \right] \\ & = \frac{1}{\rho_0 a \cos \phi} \DP{}{z^*} \left[ \rho_0 a \cos \phi \left\{ f \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} - \left( \cos \phi \DP{\overline{u}}{\phi} - \sin \phi \overline{u} \right) \frac{\overline{v'\theta'}} {a \cos \phi \overline{\DP{\theta}{z^*}}} - \overline{w'u'} \right\} \right] \\ & = \frac{1}{\rho_0 a \cos \phi} \DP{}{z^*} \left[ \rho_0 a \cos \phi \left\{ f \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} - \DP{(\overline{u} \cos \phi)}{\phi} \frac{\overline{v'\theta'}} {a \cos \phi \overline{\DP{\theta}{z^*}}} - \overline{w'u'} \right\} \right] \\ & = \frac{1}{\rho_0 a \cos \phi} \DP{}{z^*} \left[ \rho_0 a \cos \phi \left\{ \left( f - \frac{\DP{(\overline{u} \cos \phi)}{\phi}} {a \cos \phi} \right) \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} - \overline{w'u'} \right\} \right] \\ & = \frac{1}{\rho_0 a \cos \phi} \DP{F^{*}_{z}}{z^*} \end{align*} 以上より, \Deqref{tem-u-tochuu} は次のようになる. \begin{align*} & \DP{\overline{u}}{t} + \frac{\overline{v}^*}{a \cos \phi} \DP{}{\phi} \left( \overline{u} \cos \phi \right) + \overline{w}^* \DP{\overline{u}}{z^*} - f \overline{v}^* - \overline{X} = \Dinv{\rho_0 a^2 \cos^2 \phi} \DP{}{\phi} \left( \cos \phi F^{*}_{\phi} \right) + \frac{1}{\rho_0 a \cos \phi} \DP{F^{*}_{z}}{z^*}, \nonumber \\ % & \DP{\overline{u}}{t} + \frac{\overline{v}^*}{a \cos \phi} \DP{}{\phi} \left( \overline{u} \cos \phi \right) + \overline{w}^* \DP{\overline{u}}{z^*} - f \overline{v}^* - \overline{X} = \Dinv{\rho_0 a \cos \phi} \Ddiv{\Dvect{F}}. \end{align*} ここで, 子午面内の発散を以下のように表した. \begin{align} \Ddiv{\Dvect{F}} = \Dinv{a \cos \phi } \DP{(\cos \phi F_{\phi})}{\phi} + \DP{F_{z^{*}}}{z^*} \end{align} \vspace{5mm} 次に熱力学の式を書き換える. \Deqref{new_euler_mean_pe_thermal} に \Deqref{residual_v_app}, \Deqref{residual_w_app} を代入すると \begin{align*} & \DP{\overline{\theta}}{t} + \frac{1}{a} \left[ \overline{v}^* + \Dinv{\rho_0} \DP{}{z^*} \left( \rho_0 \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \right] \DP{\overline{\theta}}{\phi} + \left[ \overline{w}^* - \Dinv{a \cos\phi} \DP{}{\phi} \left( \cos \phi \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \right] \DP{\overline{\theta}}{z^*} - \overline{Q} \\ & \qquad = - \Dinv{a\cos\phi}\DP{}{\phi}(\overline{v'\theta'}\cos\phi) - \Dinv{\rho_0}\DP{}{z^*}(\rho_0\overline{w'\theta'}), \\ % & \DP{\overline{\theta}}{t} + \frac{\overline{v}^*}{a} \DP{\overline{\theta}}{\phi} + \overline{w}^* \DP{\overline{\theta}}{z^*} - \overline{Q} \\ & \qquad = - \Dinv{\rho_0 a} \DP{}{z^*} \left( \rho_0 \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \DP{\overline{\theta}}{\phi} + \Dinv{a \cos\phi} \DP{}{\phi} \left( \cos \phi \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \DP{\overline{\theta}}{z^*} \\ & \qquad \qquad - \Dinv{a\cos\phi}\DP{}{\phi}(\overline{v'\theta'}\cos\phi) - \Dinv{\rho_0}\DP{}{z^*}(\rho_0\overline{w'\theta'}) \\ \end{align*} となる. この右辺を更に変形すると \begin{align*} & - \Dinv{\rho_0} \DP{}{z^*} \left( \rho_0 \frac{\overline{v'\theta'}} {a \overline{\DP{\theta}{z^*}}} \right) \DP{\overline{\theta}}{\phi} + \Dinv{a \cos\phi} \DP{}{\phi} \left( \cos \phi \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \DP{\overline{\theta}}{z^*} \\ & \qquad - \Dinv{a\cos\phi}\DP{}{\phi}(\overline{v'\theta'}\cos\phi) - \Dinv{\rho_0}\DP{}{z^*}(\rho_0\overline{w'\theta'}) \\ % = & - \Dinv{\rho_0} \DP{}{z^*} \left( \rho_0 \frac{\overline{v'\theta'}} {a \overline{\DP{\theta}{z^*}}} \DP{\overline{\theta}}{\phi} \right) + \frac{\overline{v'\theta'}} {a \overline{\DP{\theta}{z^*}}} \DP{}{z^*}\DP{\overline{\theta}}{\phi} \\ & \qquad + \Dinv{a \cos\phi} \left[ \DP{}{\phi} \left( \cos \phi \overline{v'\theta'} \right) \frac{1}{\overline{\DP{\theta}{z^*}}} + \cos \phi \overline{v'\theta'} \DP{}{\phi} \left( \overline{\DP{\theta}{z^*}} \right)^{-1} \right] \DP{\overline{\theta}}{z^*} \\ & \qquad - \Dinv{a\cos\phi}\DP{}{\phi}(\overline{v'\theta'}\cos\phi) - \Dinv{\rho_0}\DP{}{z^*}(\rho_0\overline{w'\theta'}) \\ % = & - \Dinv{\rho_0} \DP{}{z^*} \left( \rho_0 \frac{\overline{v'\theta'}} {a \overline{\DP{\theta}{z^*}}} \DP{\overline{\theta}}{\phi} \right) + \frac{\overline{v'\theta'}} {a \overline{\DP{\theta}{z^*}}} \DP{}{z^*}\DP{\overline{\theta}}{\phi} + \Dinv{a} \overline{v'\theta'} \DP{}{\phi} \left( \overline{\DP{\theta}{z^*}} \right)^{-1} \DP{\overline{\theta}}{z^*} - \Dinv{\rho_0}\DP{}{z^*}(\rho_0\overline{w'\theta'}) \\ % = & - \Dinv{\rho_0} \DP{}{z^*} \left[ \rho_0 \frac{\overline{v'\theta'}} {a \overline{\DP{\theta}{z^*}}} \DP{\overline{\theta}}{\phi} + \rho_0\overline{w'\theta'} \right] + \frac{\overline{v'\theta'}}{a} \left[ \frac{1} {\overline{\DP{\theta}{z^*}}} \DP{}{z^*}\DP{\overline{\theta}}{\phi} + \DP{}{\phi} \left( \overline{\DP{\theta}{z^*}} \right)^{-1} \DP{\overline{\theta}}{z^*} \right] \\ % = & - \Dinv{\rho_0} \DP{}{z^*} \left[ \rho_0 \left( \frac{\overline{v'\theta'}} {a \overline{\DP{\theta}{z^*}}} \DP{\overline{\theta}}{\phi} + \overline{w'\theta'} \right) \right] + \frac{\overline{v'\theta'}}{a} \DP{}{\phi} \left( \frac{ \DP{\overline{\theta}}{z^*} } { \overline{\DP{\theta}{z^*}} } \right) \\ % = & - \Dinv{\rho_0} \DP{}{z^*} \left[ \rho_0 \left( \frac{\overline{v'\theta'}} {a \overline{\DP{\theta}{z^*}}} \DP{\overline{\theta}}{\phi} + \overline{w'\theta'} \right) \right]. \end{align*} これより, 熱力学の式は以下のようになる. \begin{align*} \DP{\overline{\theta}}{t} + \frac{\overline{v}^*}{a} \DP{\overline{\theta}}{\phi} + \overline{w}^* \DP{\overline{\theta}}{z^*} - \overline{Q} = - \Dinv{\rho_0} \DP{}{z^*} \left[ \rho_0 \left( \frac{\overline{v'\theta'}} {a \overline{\DP{\theta}{z^*}}} \DP{\overline{\theta}}{\phi} + \overline{w'\theta'} \right) \right]. \end{align*} \vspace{5mm} 最後に $v$ の式について考える. \Deqref{new_euler_mean_pe_momentum_y} に \Deqref{residual_v_app}, \Deqref{residual_w_app} を代入すると \begin{align*} & \DP{}{t} \left[ \overline{v}^* + \Dinv{\rho_0} \DP{}{z^*} \left( \rho_0 \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \right] + \frac{1}{a} \left[ \overline{v}^* + \Dinv{\rho_0} \DP{}{z^*} \left( \rho_0 \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \right] \DP{}{\phi} \left[ \overline{v}^* + \Dinv{\rho_0} \DP{}{z^*} \left( \rho_0 \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \right] \\ & \qquad \qquad + \left[ \overline{w}^* - \Dinv{a \cos\phi} \DP{}{\phi} \left( \cos \phi \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \right] \DP{}{z^*} \left[ \overline{v}^* + \Dinv{\rho_0} \DP{}{z^*} \left( \rho_0 \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \right] \\ & \qquad \qquad + f \overline{u} + \frac{\tan\phi}{a} (\overline{u})^2 + \Dinv{a} \DP{\overline{\Phi}}{\phi} - \overline{Y} \\ & \qquad = - \Dinv{a\cos\phi}\DP{}{\phi}(\overline{v'^2} \cos\phi) - \Dinv{\rho_0}\DP{}{z^*}(\rho_0\overline{v' w'}) - \overline{u'^2}\frac{\tan\phi}{a}, \\ % & f \overline{u} + \frac{\tan\phi}{a} (\overline{u})^2 + \Dinv{a} \DP{\overline{\Phi}}{\phi} \\ & \qquad = - \DP{}{t} \left[ \overline{v}^* + \Dinv{\rho_0} \DP{}{z^*} \left( \rho_0 \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \right] - \frac{1}{a} \left[ \overline{v}^* + \Dinv{\rho_0} \DP{}{z^*} \left( \rho_0 \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \right] \DP{}{\phi} \left[ \overline{v}^* + \Dinv{\rho_0} \DP{}{z^*} \left( \rho_0 \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \right] \\ & \qquad \qquad - \left[ \overline{w}^* - \Dinv{a \cos\phi} \DP{}{\phi} \left( \cos \phi \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \right] \DP{}{z^*} \left[ \overline{v}^* + \Dinv{\rho_0} \DP{}{z^*} \left( \rho_0 \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \right] \\ & \qquad \qquad - \Dinv{a\cos\phi} \DP{}{\phi}(\overline{v'^2} \cos \phi) - \Dinv{\rho_0} \DP{}{z^*}(\rho_0\overline{v' w'}) - \overline{u'^2} \frac{\tan\phi}{a} + \overline{Y} \end{align*} Andrews {\it et al.} (1987) によれば, この式の右辺の量は 左辺に比べれば小さい. 右辺の項を全てまとめて $G$ と書くと $v$ の式は次のようになる. \begin{align*} \overline{u} \left( f + \frac{\tan\phi}{a} \overline{u} \right) + \Dinv{a} \DP{\overline{\Phi}}{\phi} = G. \end{align*} \vspace{5mm} 以上をまとめると, 以下の{\bfseries 変形オイラー平均方程式}が得られる. \begin{screen} \begin{subequations}\Deqlab{transformed_euler_mean_pe} %\def\theequation{\arabic{section}.\arabic{parentequation}\alph{equation}} %\begin{itemize} %%%%%%%%%%%%%%%% %\item 運動方程式 %%%%%%%%%%%%%%%% \begin{align} \Deqlab{transformed_euler_mean_pe_momentum_x}& \DP{\overline{u}}{t} + \overline{v}^* \left[ \Dinv{a\cos\phi}\DP{}{\phi}(\overline{u}\cos\phi) - f \right] + \overline{w}^*\DP{\overline{u}}{z^*} - \overline{X} = \Dinv{\rho_0 a \cos\phi}\Ddiv\Dvect{F}, \\ \Deqlab{transformed_euler_mean_pe_momentum_y}& \overline{u} \left( f + \overline{u}\frac{\tan\phi}{a} \right) + \Dinv{a}\DP{\overline{\Phi}}{\phi} = G. \end{align} %%%%%%%%%%%%%%%% %\item 静水圧平衡の式 %%%%%%%%%%%%%%%% \begin{align} \Deqlab{transformed_euler_mean_pe_momentum_z^*} \DP{\overline{\Phi}}{z^*} - \frac{Re^{-\kappa z^*/H}}{H}\overline{\theta} = 0. \end{align} %%%%%%%%%%%%%%%% %\item 連続の式 %%%%%%%%%%%%%%%% \begin{align} \Deqlab{transformed_euler_mean_pe_continuity} \Dinv{a\cos\phi}&\left[ \DP{}{\phi}(\overline{v}^*\cos\phi)\right] + \Dinv{\rho_0}\DP{}{z^*}(\rho_0 \overline{w}^*) = 0. \end{align} %%%%%%%%%%%%%%%% %\item 熱力学の式 %%%%%%%%%%%%%%%% \begin{align} \Deqlab{transformed_euler_mean_pe_thermal} \DP{\overline{\theta}}{t} + \frac{\overline{v}^*}{a}\DP{\overline{\theta}}{\phi} + \overline{w}^*\DP{\overline{\theta}}{z^*} - \overline{Q} = - \Dinv{\rho_0}\DP{}{z^*} \left[\rho_0 \left( \overline{v'\theta'}\frac{\DP{\overline{\theta}}{\phi}} {a\DP{\overline{\theta}}{z^*}} + \overline{w'\theta'} \right) \right]. \end{align} %\end{itemize} \end{subequations} \end{screen} \clearpage \begin{thebibliography}{1} \bibitem{AHL1987} D.G. Andrews, J.R. Holton, and C.B. Leovy. Middle atmosphere dynamics, International Geophysics Series. Academic Press, 1987 \bibitem{H1975} J.R. Holton. The Dynamic Meteorology of the Stratosphere and Mesosphere, American Meteorological Society, 1975 \end{thebibliography} \end{document}