% 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
%

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\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{}

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\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}
