変形オイラー平均方程式系

() 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.

$${F_\phi} = \rho_0 a \cos \phi \left(\DP{\overline{u}}{\overline{z^*}} \frac{\overline{v'\theta'}}{\DP{\overline{\theta}}{z^*}} - \overline{u'v'}\right)$$

$${F_z^*} = \rho_0 a \cos \phi \left(\left[ f - \frac{\DP{\overline{u}\cos \p... ...rac{\overline{v'\theta'}}{\DP{\overline{\theta}}{z^*}} - \overline{u'w'}\right)$$

まず連続の式を書き換える. () に (), () を代入すると

$$\Dinv{a \cos \phi} \DP{}{\phi}\left[ \left\{ \overline{v}^* ... ...v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \right\} \cos\phi \right]$$

$$\qquad + \Dinv{\rho_0} \DP{}{z^*} \left[ \rho_0 \left\{ \o... ...ne{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \right\} \right] = 0,$$

$$\Dinv{a \cos \phi} \DP{}{\phi} \left( \overline{v}^* \cos\phi \right) + \Dinv{\rho_0} \DP{}{z^*} \left( \rho_0 \overline{w}^* \right)$$

$$\qquad + \Dinv{a \cos \phi} \DP{}{\phi} \left\{ \Dinv{\rho_0... ...c{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \right\} = 0.$$

この第三項と第四項だけを取り出すと

$$\qquad \Dinv{a \cos \phi} \DP{}{\phi} \left\{ \Dinv{\rho_0}... ... \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \right\}$$

$$= \Dinv{a \cos \phi} \left[ \DP{}{\phi} \left\{ \Dinv{\rho... ...overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \right\} \right]$$

$$= \Dinv{a \cos \phi} \left[ \Dinv{\rho_0} \DP{}{\phi} \left... ...overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \right\} \right]$$

$$= 0.$$

したがって, 連続の式は以下のようになる.

$$\Dinv{a \cos \phi} \DP{}{\phi} \left( \overline{v}^* \cos\phi \right) + \Dinv{\rho_0} \DP{}{z^*} \left( \rho_0 \overline{w}^* \right) = 0.$$ (A.14)

次に $u$ の式を書き換える. () に (), () を代入すると

$$\DP{\overline{u}}{t} + \Dinv{a} \left[ \overline{v}^* + \Dinv{\rho_0} \DP{}{z^*} ... ...eta'}} {\overline{\DP{\theta}{z^*}}} \right) \right] \DP{\overline{u}}{z^*}$$

$$\qquad \qquad - f \left[ \overline{v}^* + \Dinv{\rho_0} \D... ...ne{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \right] - \overline{X}$$

$$\qquad = - \Dinv{a\cos^2\phi} \DP{}{\phi} (\overline{v'u'} \cos^2 \phi) - \Dinv{\rho_0} \DP{}{z^*} (\rho_0\overline{w'u'}),$$

$$\DP{\overline{u}}{t} + \frac{\overline{v}^*}{a} \DP{\overline{u}}{\phi} + \overline{w... ...ine{v}^* - \frac{\tan \phi}{a} \overline{u} \ \overline{v}^* - \overline{X}$$

$$\qquad = - \Dinv{a\cos^2\phi} \DP{}{\phi} (\overline{v'u'} \cos^... ...line{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \DP{\overline{u}}{z^*}$$

$$\qquad \qquad + f \Dinv{\rho_0} \DP{}{z^*} \left( \rho_0 \fra... ...\DP{\theta}{z^*}}} \right) - \Dinv{\rho_0} \DP{}{z^*} (\rho_0\overline{w'u'})$$

$$\qquad \qquad - \Dinv{\rho_0 a} \DP{}{z^*} \left( \rho_0 \fra... ...( \rho_0 \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right),$$

$$\DP{\overline{u}}{t} + \frac{\overline{v}^*}{a \cos \phi} \DP{}{\phi} \left( \overli... ...) + \overline{w}^* \DP{\overline{u}}{z^*} - f \overline{v}^* - \overline{X}$$

$$\qquad = - \Dinv{\rho_0 a^2 \cos^2 \phi} \DP{}{\phi} (\rho_0 a... ...line{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \DP{\overline{u}}{z^*}$$

$$\qquad \qquad + \frac{1}{\rho_0 a \cos \phi} \DP{}{z^*} \left... ... \frac{1}{\rho_0 a \cos \phi} \DP{}{z^*} (\rho_0 a \cos \phi \overline{w'u'})$$

$$\qquad \qquad - \Dinv{\rho_0 a} \DP{}{z^*} \left( \rho_0 \fra... ...t( \rho_0 \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right)$$ (A.15)

() の右辺を以下のように変形する.

$$- \Dinv{\rho_0 a^2 \cos^2 \phi} \DP{}{\phi} (\rho_0 a \overline... ...\cos \phi \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right)$$

$$\qquad \qquad + \frac{1}{\rho_0 a \cos \phi} \DP{}{z^*} \left... ... \frac{1}{\rho_0 a \cos \phi} \DP{}{z^*} (\rho_0 a \cos \phi \overline{w'u'})$$

$$\qquad \qquad - \Dinv{\rho_0 a} \DP{}{z^*} \left( \rho_0 \fra... ...rline{\DP{\theta}{z^*}}} \DP{}{z^*} \left( \DP{\overline{u}}{\phi} \right)$$

$$\qquad \qquad + \frac{\tan \phi}{\rho_0 a} \DP{}{z^*} \left( \... ...eta'}} {\overline{\DP{\theta}{z^*}}} \DP{}{z^*} \left( \overline{u} \right)$$

$$= \Dinv{\rho_0 a^2 \cos^2 \phi} \left[ - \DP{}{\phi} (\rho_0 ... ... \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \right]$$

$$\qquad + \Dinv{\rho_0 a} \rho_0 \frac{\overline{v'\theta'}} ... ...ac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \DP{\overline{u}}{z^*}$$

$$\qquad + \frac{1}{\rho_0 a \cos \phi} \DP{}{z^*} \left[ \lef... ...line{\DP{\theta}{z^*}}} \right) - \rho_0 a \cos \phi \overline{w'u'} \right]$$

$$\qquad - \Dinv{\rho_0 a} \DP{}{z^*} \left( \rho_0 \frac{\over... ...u} \rho_0 \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right)$$

$$= \Dinv{\rho_0 a^2 \cos^2 \phi} \left[ - \DP{}{\phi} (\rho_0 ... ... \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \right]$$

$$\qquad + \Dinv{\rho_0 a^2 \cos^2 \phi} \left[ \rho_0 a \cos^2... ...ine{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \DP{\overline{u}}{z^*} \right]$$

$$\qquad + \frac{1}{\rho_0 a \cos \phi} \DP{}{z^*} \left[ \lef... ...line{\DP{\theta}{z^*}}} \right) - \rho_0 a \cos \phi \overline{w'u'} \right]$$

$$\qquad + \Dinv{\rho_0 a \cos \phi} \left[ - \cos \phi \DP{}{... ... \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \right]$$

$$= \Dinv{\rho_0 a^2 \cos^2 \phi} \left[ - \DP{}{\phi} (\rho_0 ... ... \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \right]$$

$$\qquad + \Dinv{\rho_0 a^2 \cos^2 \phi} \left[ \rho_0 a \cos^2... ...ine{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \DP{\overline{u}}{z^*} \right]$$

$$\qquad + \frac{1}{\rho_0 a \cos \phi} \DP{}{z^*} \left[ f \r... ...} {\overline{\DP{\theta}{z^*}}} - \rho_0 a \cos \phi \overline{w'u'} \right]$$

$$\qquad + \Dinv{\rho_0 a \cos \phi} \DP{}{z^*} \left[ - \rho_... ...u} \rho_0 \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right]$$ (A.16)

() の第一項と第二項だけ取り出すと

$$\Dinv{\rho_0 a^2 \cos^2 \phi} \left[ - \DP{}{\phi} (\rho_0 a \... ... \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \right]$$

$$\qquad + \Dinv{\rho_0 a^2 \cos^2 \phi} \left[ \rho_0 a \cos^2... ...ine{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \DP{\overline{u}}{z^*} \right]$$

$$= \Dinv{\rho_0 a^2 \cos^2 \phi} \left[ - \DP{}{\phi} (\rho_0 a \overline{v'u'} \cos^2 \phi) \right]$$

$$\qquad + \Dinv{\rho_0 a^2 \cos^2 \phi} \left[ \rho_0 a \cos^2... ... \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \right]$$

$$= \Dinv{\rho_0 a^2 \cos^2 \phi} \left[ - \DP{}{\phi} (\rho_0 ... ...ta'}} {\overline{\DP{\theta}{z^*}}} \DP{\overline{u}}{z^*} \right) \right]$$

$$= \Dinv{\rho_0 a^2 \cos^2 \phi} \DP{}{\phi} \left[ - \rho_0 a... ...ine{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \DP{\overline{u}}{z^*} \right]$$

$$= \Dinv{\rho_0 a^2 \cos^2 \phi} \DP{}{\phi} \left[ \rho_0 a \... ...'\theta'}} {\overline{\DP{\theta}{z^*}}} - \overline{v'u'} \right\} \right]$$

$$= \Dinv{\rho_0 a^2 \cos^2 \phi} \DP{}{\phi} \left( \cos \phi F^{*}_{\phi} \right)$$

() の第三項と第四項だけ取り出すと

$$\frac{1}{\rho_0 a \cos \phi} \DP{}{z^*} \left[ f \rho_0 a \co... ...u} \rho_0 \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right]$$

$$= \frac{1}{\rho_0 a \cos \phi} \DP{}{z^*} \left[ \rho_0 a \... ...rline{v'\theta'}} {a \cos \phi \overline{\DP{\theta}{z^*}}} \right\} \right]$$

$$= \frac{1}{\rho_0 a \cos \phi} \DP{}{z^*} \left[ \rho_0 a \... ...{a \cos \phi \overline{\DP{\theta}{z^*}}} - \overline{w'u'} \right\} \right]$$

$$= \frac{1}{\rho_0 a \cos \phi} \DP{}{z^*} \left[ \rho_0 a \... ...{a \cos \phi \overline{\DP{\theta}{z^*}}} - \overline{w'u'} \right\} \right]$$

$$= \frac{1}{\rho_0 a \cos \phi} \DP{}{z^*} \left[ \rho_0 a \... ...'\theta'}} {\overline{\DP{\theta}{z^*}}} - \overline{w'u'} \right\} \right]$$

$$= \frac{1}{\rho_0 a \cos \phi} \DP{F^{*}_{z}}{z^*}$$

以上より, () は次のようになる.

$$\DP{\overline{u}}{t} + \frac{\overline{v}^*}{a \cos \phi} \DP{}{... ...hi} \right) + \frac{1}{\rho_0 a \cos \phi} \DP{F^{*}_{z}}{z^*}, \nonumber$$

$$\DP{\overline{u}}{t} + \frac{\overline{v}^*}{a \cos \phi} \DP{}{... ...\overline{v}^* - \overline{X} = \Dinv{\rho_0 a \cos \phi} \Ddiv{\Dvect{F}}.$$

ここで, 子午面内の発散を以下のように表した.

$$\Ddiv{\Dvect{F}} = \Dinv{a \cos \phi } \DP{(\cos \phi F_{\phi})}{\phi} + \DP{F_{z^{*}}}{z^*}$$ (A.17)

次に熱力学の式を書き換える. () に (), () を代入すると

$$\DP{\overline{\theta}}{t} + \frac{1}{a} \left[ \overline{v}^... ...P{\theta}{z^*}}} \right) \right] \DP{\overline{\theta}}{z^*} - \overline{Q}$$

$$\qquad = - \Dinv{a\cos\phi}\DP{}{\phi}(\overline{v'\theta'}\cos\phi) - \Dinv{\rho_0}\DP{}{z^*}(\rho_0\overline{w'\theta'}),$$

$$\DP{\overline{\theta}}{t} + \frac{\overline{v}^*}{a} \DP{\overline{\theta}}{\phi} + \overline{w}^* \DP{\overline{\theta}}{z^*} - \overline{Q}$$

$$\qquad = - \Dinv{\rho_0 a} \DP{}{z^*} \left( \rho_0 \frac{\o... ...v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \DP{\overline{\theta}}{z^*}$$

$$\qquad \qquad - \Dinv{a\cos\phi}\DP{}{\phi}(\overline{v'\theta'}\cos\phi) - \Dinv{\rho_0}\DP{}{z^*}(\rho_0\overline{w'\theta'})$$

となる. この右辺を更に変形すると

$$- \Dinv{\rho_0} \DP{}{z^*} \left( \rho_0 \frac{\overline{v'\th... ...v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \DP{\overline{\theta}}{z^*}$$

$$\qquad - \Dinv{a\cos\phi}\DP{}{\phi}(\overline{v'\theta'}\cos\phi) - \Dinv{\rho_0}\DP{}{z^*}(\rho_0\overline{w'\theta'})$$

$$= - \Dinv{\rho_0} \DP{}{z^*} \left( \rho_0 \frac{\overline{v'\th... ...eta'}} {a \overline{\DP{\theta}{z^*}}} \DP{}{z^*}\DP{\overline{\theta}}{\phi}$$

$$\qquad + \Dinv{a \cos\phi} \left[ \DP{}{\phi} \left( \cos \... ...( \overline{\DP{\theta}{z^*}} \right)^{-1} \right] \DP{\overline{\theta}}{z^*}$$

$$\qquad - \Dinv{a\cos\phi}\DP{}{\phi}(\overline{v'\theta'}\cos\phi) - \Dinv{\rho_0}\DP{}{z^*}(\rho_0\overline{w'\theta'})$$

$$= - \Dinv{\rho_0} \DP{}{z^*} \left( \rho_0 \frac{\overline{v'\th... ...{\overline{\theta}}{z^*} - \Dinv{\rho_0}\DP{}{z^*}(\rho_0\overline{w'\theta'})$$

$$= - \Dinv{\rho_0} \DP{}{z^*} \left[ \rho_0 \frac{\overline{v'\th... ... \overline{\DP{\theta}{z^*}} \right)^{-1} \DP{\overline{\theta}}{z^*} \right]$$

$$= - \Dinv{\rho_0} \DP{}{z^*} \left[ \rho_0 \left( \frac{\overli... ... \frac{ \DP{\overline{\theta}}{z^*} } { \overline{\DP{\theta}{z^*}} } \right)$$

$$= - \Dinv{\rho_0} \DP{}{z^*} \left[ \rho_0 \left( \frac{\overli... ...^*}}} \DP{\overline{\theta}}{\phi} + \overline{w'\theta'} \right) \right].$$

これより, 熱力学の式は以下のようになる.

$$\DP{\overline{\theta}}{t} + \frac{\overline{v}^*}{a} \DP{\overli... ...^*}}} \DP{\overline{\theta}}{\phi} + \overline{w'\theta'} \right) \right].$$

最後に $v$ の式について考える. () に (), () を代入すると

$$\DP{}{t} \left[ \overline{v}^* + \Dinv{\rho_0} \DP{}{z^*} \... ... \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \right]$$

$$\qquad \qquad + \left[ \overline{w}^* - \Dinv{a \cos\phi} \... ... \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \right]$$

$$\qquad \qquad + f \overline{u} + \frac{\tan\phi}{a} (\overline{u})^2 + \Dinv{a} \DP{\overline{\Phi}}{\phi} - \overline{Y}$$

$$\qquad = - \Dinv{a\cos\phi}\DP{}{\phi}(\overline{v'^2} \cos\phi)... ...\rho_0}\DP{}{z^*}(\rho_0\overline{v' w'}) - \overline{u'^2}\frac{\tan\phi}{a},$$

$$f \overline{u} + \frac{\tan\phi}{a} (\overline{u})^2 + \Dinv{a} \DP{\overline{\Phi}}{\phi}$$

$$\qquad = - \DP{}{t} \left[ \overline{v}^* + \Dinv{\rho_0} \... ... \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \right]$$

$$\qquad \qquad - \left[ \overline{w}^* - \Dinv{a \cos\phi} \... ... \frac{\overline{v'\theta'}} {\overline{\DP{\theta}{z^*}}} \right) \right]$$

$$\qquad \qquad - \Dinv{a\cos\phi} \DP{}{\phi}(\overline{v'^2} \co... ...}(\rho_0\overline{v' w'}) - \overline{u'^2} \frac{\tan\phi}{a} + \overline{Y}$$

Andrews 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.

$$\overline{u} \left( f + \frac{\tan\phi}{a} \overline{u} \right) + \Dinv{a} \DP{\overline{\Phi}}{\phi} = G.$$

以上をまとめると, 以下の変形オイラー平均方程式が得られる.