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: 参考文献 : プリミティブ方程式系と変形オイラー平均の復習 : オイラー平均方程式系   目次

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.
\begin{subequations}\begin{align}
 \overline{v}^* 
 & =
 \overline{v} 
 - \Dinv{...
...\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \right)
 \end{align}\end{subequations}


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



まず連続の式を書き換える. ([*]) に ([*]), ([*]) を代入すると

  $\displaystyle \Dinv{a \cos \phi}
 \DP{}{\phi}\left[
 \left\{ 
 \overline{v}^* 
...
...v'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \right)
 \right\}
 \cos\phi \right]$    
  $\displaystyle \qquad
 + \Dinv{\rho_0}
 \DP{}{z^*} 
 \left[ \rho_0 
 \left\{
 \o...
...ne{v'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \right)
 \right\}
 \right]
 = 0,$    
  $\displaystyle \Dinv{a \cos \phi}
 \DP{}{\phi}
 \left(
 \overline{v}^* \cos\phi
 \right)
 + \Dinv{\rho_0}
 \DP{}{z^*} 
 \left( \rho_0 \overline{w}^* \right)$    
  $\displaystyle \qquad
 + \Dinv{a \cos \phi}
 \DP{}{\phi}
 \left\{ 
 \Dinv{\rho_0...
...c{\overline{v'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \right)
 \right\}
 = 0.$    

この第三項と第四項だけを取り出すと

  $\displaystyle \qquad 
 \Dinv{a \cos \phi}
 \DP{}{\phi}
 \left\{ 
 \Dinv{\rho_0}...
...
 \frac{\overline{v'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \right)
 \right\}$    
  $\displaystyle = 
 \Dinv{a \cos \phi}
 \left[
 \DP{}{\phi}
 \left\{ 
 \Dinv{\rho...
...overline{v'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \right)
 \right\}
 \right]$    
  $\displaystyle = 
 \Dinv{a \cos \phi}
 \left[
 \Dinv{\rho_0}
 \DP{}{\phi}
 \left...
...overline{v'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \right)
 \right\}
 \right]$    
  $\displaystyle = 0.$    

したがって, 連続の式は以下のようになる.
$\displaystyle \Dinv{a \cos \phi}
\DP{}{\phi}
\left(
\overline{v}^* \cos\phi
\right)
+ \Dinv{\rho_0}
\DP{}{z^*}
\left( \rho_0 \overline{w}^* \right) = 0.$     (A.14)



次に $ u$ の式を書き換える. ([*]) に ([*]), ([*]) を代入すると

$\displaystyle \DP{\overline{u}}{t}$ $\displaystyle + \Dinv{a}
 \left[ 
 \overline{v}^* 
 + \Dinv{\rho_0} \DP{}{z^*}
...
...eta'}}
 {\overline{\DP{\theta}{z^*}}}
 \right)
 \right]
 \DP{\overline{u}}{z^*}$    
  $\displaystyle \qquad \qquad
 - f 
 \left[ 
 \overline{v}^* 
 + \Dinv{\rho_0} \D...
...ne{v'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \right)
 \right]
 - \overline{X}$    
  $\displaystyle \qquad
 = - \Dinv{a\cos^2\phi} \DP{}{\phi} (\overline{v'u'} \cos^2 \phi)
 - \Dinv{\rho_0} \DP{}{z^*} (\rho_0\overline{w'u'}),$    
$\displaystyle \DP{\overline{u}}{t}$ $\displaystyle + \frac{\overline{v}^*}{a} \DP{\overline{u}}{\phi}
 + \overline{w...
...ine{v}^* 
 - \frac{\tan \phi}{a} \overline{u} \ \overline{v}^* 
 - \overline{X}$    
  $\displaystyle \qquad
 = - \Dinv{a\cos^2\phi} \DP{}{\phi} (\overline{v'u'} \cos^...
...line{v'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \right) \DP{\overline{u}}{z^*}$    
  $\displaystyle \qquad \qquad
 + f \Dinv{\rho_0} \DP{}{z^*}
 \left( \rho_0 
 \fra...
...\DP{\theta}{z^*}}}
 \right)
 - \Dinv{\rho_0} \DP{}{z^*} (\rho_0\overline{w'u'})$    
  $\displaystyle \qquad \qquad
 - \Dinv{\rho_0 a} \DP{}{z^*}
 \left( \rho_0 
 \fra...
...( \rho_0 
 \frac{\overline{v'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \right),$    
$\displaystyle \DP{\overline{u}}{t}$ $\displaystyle + \frac{\overline{v}^*}{a \cos \phi} \DP{}{\phi} 
 \left( \overli...
...)
 + \overline{w}^* \DP{\overline{u}}{z^*}
 - f \overline{v}^* 
 - \overline{X}$    
  $\displaystyle \qquad
 = - \Dinv{\rho_0 a^2 \cos^2 \phi} 
 \DP{}{\phi} (\rho_0 a...
...line{v'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \right) \DP{\overline{u}}{z^*}$    
  $\displaystyle \qquad \qquad
 + \frac{1}{\rho_0 a \cos \phi} 
 \DP{}{z^*}
 \left...
... \frac{1}{\rho_0 a \cos \phi} 
 \DP{}{z^*} (\rho_0 a \cos \phi \overline{w'u'})$    
  $\displaystyle \qquad \qquad
 - \Dinv{\rho_0 a} \DP{}{z^*}
 \left( \rho_0 
 \fra...
...t( \rho_0 
 \frac{\overline{v'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \right)$ (A.15)

([*]) の右辺を以下のように変形する.

  $\displaystyle - \Dinv{\rho_0 a^2 \cos^2 \phi} 
 \DP{}{\phi} (\rho_0 a \overline...
...\cos \phi 
 \frac{\overline{v'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \right)$    
  $\displaystyle \qquad \qquad
 + \frac{1}{\rho_0 a \cos \phi} 
 \DP{}{z^*}
 \left...
... \frac{1}{\rho_0 a \cos \phi} 
 \DP{}{z^*} (\rho_0 a \cos \phi \overline{w'u'})$    
  $\displaystyle \qquad \qquad
 - \Dinv{\rho_0 a} \DP{}{z^*}
 \left( \rho_0 
 \fra...
...rline{\DP{\theta}{z^*}}}
 \DP{}{z^*}
 \left( 
 \DP{\overline{u}}{\phi}
 \right)$    
  $\displaystyle \qquad \qquad
 + \frac{\tan \phi}{\rho_0 a}
 \DP{}{z^*}
 \left( \...
...eta'}}
 {\overline{\DP{\theta}{z^*}}}
 \DP{}{z^*}
 \left( \overline{u}
 \right)$    
  $\displaystyle =
 \Dinv{\rho_0 a^2 \cos^2 \phi} 
 \left[
 - \DP{}{\phi} (\rho_0 ...
...
 \frac{\overline{v'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \right) 
 \right]$    
  $\displaystyle \qquad
 + \Dinv{\rho_0 a} 
 \rho_0 
 \frac{\overline{v'\theta'}}
...
...ac{\overline{v'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \DP{\overline{u}}{z^*}$    
  $\displaystyle \qquad
 + \frac{1}{\rho_0 a \cos \phi} 
 \DP{}{z^*}
 \left[
 \lef...
...line{\DP{\theta}{z^*}}}
 \right)
 - \rho_0 a \cos \phi \overline{w'u'}
 \right]$    
  $\displaystyle \qquad
 - \Dinv{\rho_0 a} \DP{}{z^*}
 \left( \rho_0 
 \frac{\over...
...u} \rho_0 
 \frac{\overline{v'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \right)$    
  $\displaystyle =
 \Dinv{\rho_0 a^2 \cos^2 \phi} 
 \left[
 - \DP{}{\phi} (\rho_0 ...
...
 \frac{\overline{v'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \right) 
 \right]$    
  $\displaystyle \qquad
 + \Dinv{\rho_0 a^2 \cos^2 \phi} 
 \left[
 \rho_0 a \cos^2...
...ine{v'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \DP{\overline{u}}{z^*}
 \right]$    
  $\displaystyle \qquad
 + \frac{1}{\rho_0 a \cos \phi} 
 \DP{}{z^*}
 \left[
 \lef...
...line{\DP{\theta}{z^*}}}
 \right)
 - \rho_0 a \cos \phi \overline{w'u'}
 \right]$    
  $\displaystyle \qquad
 + \Dinv{\rho_0 a \cos \phi} 
 \left[
 - \cos \phi
 \DP{}{...
... 
 \frac{\overline{v'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \right)
 \right]$    
  $\displaystyle =
 \Dinv{\rho_0 a^2 \cos^2 \phi} 
 \left[
 - \DP{}{\phi} (\rho_0 ...
...
 \frac{\overline{v'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \right) 
 \right]$    
  $\displaystyle \qquad
 + \Dinv{\rho_0 a^2 \cos^2 \phi} 
 \left[
 \rho_0 a \cos^2...
...ine{v'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \DP{\overline{u}}{z^*}
 \right]$    
  $\displaystyle \qquad
 + \frac{1}{\rho_0 a \cos \phi} 
 \DP{}{z^*}
 \left[
 f \r...
...}
 {\overline{\DP{\theta}{z^*}}}
 - \rho_0 a \cos \phi \overline{w'u'}
 \right]$    
  $\displaystyle \qquad
 + \Dinv{\rho_0 a \cos \phi} 
 \DP{}{z^*}
 \left[
 - \rho_...
...u} \rho_0 
 \frac{\overline{v'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \right]$ (A.16)

([*]) の第一項と第二項だけ取り出すと

  $\displaystyle \Dinv{\rho_0 a^2 \cos^2 \phi} 
 \left[
 - \DP{}{\phi} (\rho_0 a \...
...
 \frac{\overline{v'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \right) 
 \right]$    
  $\displaystyle \qquad
 + \Dinv{\rho_0 a^2 \cos^2 \phi} 
 \left[
 \rho_0 a \cos^2...
...ine{v'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \DP{\overline{u}}{z^*}
 \right]$    
  $\displaystyle =
 \Dinv{\rho_0 a^2 \cos^2 \phi} 
 \left[
 - \DP{}{\phi} (\rho_0 a \overline{v'u'} \cos^2 \phi)
 \right]$    
  $\displaystyle \qquad
 + \Dinv{\rho_0 a^2 \cos^2 \phi} 
 \left[
 \rho_0 a \cos^2...
...
 \frac{\overline{v'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \right) 
 \right]$    
  $\displaystyle =
 \Dinv{\rho_0 a^2 \cos^2 \phi} 
 \left[
 - \DP{}{\phi} (\rho_0 ...
...ta'}}
 {\overline{\DP{\theta}{z^*}}}
 \DP{\overline{u}}{z^*}
 \right) 
 \right]$    
  $\displaystyle = \Dinv{\rho_0 a^2 \cos^2 \phi} 
 \DP{}{\phi}
 \left[
 - \rho_0 a...
...ine{v'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \DP{\overline{u}}{z^*}
 \right]$    
  $\displaystyle = \Dinv{\rho_0 a^2 \cos^2 \phi} 
 \DP{}{\phi}
 \left[
 \rho_0 a \...
...'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 - \overline{v'u'}
 \right\}
 \right]$    
  $\displaystyle = \Dinv{\rho_0 a^2 \cos^2 \phi} 
 \DP{}{\phi}
 \left(
 \cos \phi F^{*}_{\phi}
 \right)$    

([*]) の第三項と第四項だけ取り出すと

  $\displaystyle \frac{1}{\rho_0 a \cos \phi} 
 \DP{}{z^*}
 \left[
 f \rho_0 a \co...
...u} \rho_0 
 \frac{\overline{v'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \right]$    
  $\displaystyle = 
 \frac{1}{\rho_0 a \cos \phi} 
 \DP{}{z^*}
 \left[
 \rho_0 a \...
...rline{v'\theta'}}
 {a \cos \phi \overline{\DP{\theta}{z^*}}}
 \right\}
 \right]$    
  $\displaystyle = 
 \frac{1}{\rho_0 a \cos \phi} 
 \DP{}{z^*}
 \left[
 \rho_0 a \...
...{a \cos \phi \overline{\DP{\theta}{z^*}}}
 - \overline{w'u'}
 \right\}
 \right]$    
  $\displaystyle = 
 \frac{1}{\rho_0 a \cos \phi} 
 \DP{}{z^*}
 \left[
 \rho_0 a \...
...{a \cos \phi \overline{\DP{\theta}{z^*}}}
 - \overline{w'u'}
 \right\}
 \right]$    
  $\displaystyle = 
 \frac{1}{\rho_0 a \cos \phi} 
 \DP{}{z^*}
 \left[
 \rho_0 a \...
...'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 - \overline{w'u'}
 \right\}
 \right]$    
  $\displaystyle = \frac{1}{\rho_0 a \cos \phi} 
 \DP{F^{*}_{z}}{z^*}$    

以上より, ([*]) は次のようになる.

  $\displaystyle \DP{\overline{u}}{t}
 + \frac{\overline{v}^*}{a \cos \phi} \DP{}{...
...hi}
 \right)
 + \frac{1}{\rho_0 a \cos \phi} 
 \DP{F^{*}_{z}}{z^*}, 
 \nonumber$    
  $\displaystyle \DP{\overline{u}}{t}
 + \frac{\overline{v}^*}{a \cos \phi} \DP{}{...
...\overline{v}^* 
 - \overline{X} 
 = \Dinv{\rho_0 a \cos \phi} \Ddiv{\Dvect{F}}.$    

ここで, 子午面内の発散を以下のように表した.

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



次に熱力学の式を書き換える. ([*]) に ([*]), ([*]) を代入すると

  $\displaystyle \DP{\overline{\theta}}{t}
 + \frac{1}{a} 
 \left[ 
 \overline{v}^...
...P{\theta}{z^*}}}
 \right)
 \right]
 \DP{\overline{\theta}}{z^*}
 - \overline{Q}$    
  $\displaystyle \qquad 
 =
 - \Dinv{a\cos\phi}\DP{}{\phi}(\overline{v'\theta'}\cos\phi)
 - \Dinv{\rho_0}\DP{}{z^*}(\rho_0\overline{w'\theta'}),$    
  $\displaystyle \DP{\overline{\theta}}{t}
 + \frac{\overline{v}^*}{a} \DP{\overline{\theta}}{\phi}
 + \overline{w}^* \DP{\overline{\theta}}{z^*}
 - \overline{Q}$    
  $\displaystyle \qquad 
 = - \Dinv{\rho_0 a} \DP{}{z^*}
 \left( \rho_0 
 \frac{\o...
...v'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \right) \DP{\overline{\theta}}{z^*}$    
  $\displaystyle \qquad \qquad
 - \Dinv{a\cos\phi}\DP{}{\phi}(\overline{v'\theta'}\cos\phi)
 - \Dinv{\rho_0}\DP{}{z^*}(\rho_0\overline{w'\theta'})$    

となる. この右辺を更に変形すると

  $\displaystyle - \Dinv{\rho_0} \DP{}{z^*}
 \left( \rho_0 
 \frac{\overline{v'\th...
...v'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \right) \DP{\overline{\theta}}{z^*}$    
  $\displaystyle \qquad
 - \Dinv{a\cos\phi}\DP{}{\phi}(\overline{v'\theta'}\cos\phi)
 - \Dinv{\rho_0}\DP{}{z^*}(\rho_0\overline{w'\theta'})$    
$\displaystyle =$ $\displaystyle - \Dinv{\rho_0} \DP{}{z^*}
 \left( \rho_0 
 \frac{\overline{v'\th...
...eta'}}
 {a \overline{\DP{\theta}{z^*}}}
 \DP{}{z^*}\DP{\overline{\theta}}{\phi}$    
  $\displaystyle \qquad 
 + \Dinv{a \cos\phi} 
 \left[ 
 \DP{}{\phi} \left( \cos \...
...( \overline{\DP{\theta}{z^*}} \right)^{-1}
 \right] \DP{\overline{\theta}}{z^*}$    
  $\displaystyle \qquad
 - \Dinv{a\cos\phi}\DP{}{\phi}(\overline{v'\theta'}\cos\phi)
 - \Dinv{\rho_0}\DP{}{z^*}(\rho_0\overline{w'\theta'})$    
$\displaystyle =$ $\displaystyle - \Dinv{\rho_0} \DP{}{z^*}
 \left( \rho_0 
 \frac{\overline{v'\th...
...{\overline{\theta}}{z^*}
 - \Dinv{\rho_0}\DP{}{z^*}(\rho_0\overline{w'\theta'})$    
$\displaystyle =$ $\displaystyle - \Dinv{\rho_0} \DP{}{z^*}
 \left[ \rho_0 
 \frac{\overline{v'\th...
... \overline{\DP{\theta}{z^*}} \right)^{-1}
 \DP{\overline{\theta}}{z^*}
 \right]$    
$\displaystyle =$ $\displaystyle - \Dinv{\rho_0} \DP{}{z^*}
 \left[ \rho_0 
 \left(
 \frac{\overli...
... \frac{ \DP{\overline{\theta}}{z^*} }
 { \overline{\DP{\theta}{z^*}} }
 \right)$    
$\displaystyle =$ $\displaystyle - \Dinv{\rho_0} \DP{}{z^*}
 \left[ \rho_0 
 \left(
 \frac{\overli...
...^*}}}
 \DP{\overline{\theta}}{\phi}
 + \overline{w'\theta'}
 \right) 
 \right].$    

これより, 熱力学の式は以下のようになる.

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



最後に $ v$ の式について考える. ([*]) に ([*]), ([*]) を代入すると

  $\displaystyle \DP{}{t}
 \left[ 
 \overline{v}^* 
 + \Dinv{\rho_0} \DP{}{z^*}
 \...
... 
 \frac{\overline{v'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \right)
 \right]$    
  $\displaystyle \qquad \qquad
 + \left[
 \overline{w}^* 
 - \Dinv{a \cos\phi} 
 \...
... 
 \frac{\overline{v'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \right)
 \right]$    
  $\displaystyle \qquad \qquad
 + f \overline{u}
 + \frac{\tan\phi}{a} (\overline{u})^2
 + \Dinv{a} \DP{\overline{\Phi}}{\phi}
 - \overline{Y}$    
  $\displaystyle \qquad
 = - \Dinv{a\cos\phi}\DP{}{\phi}(\overline{v'^2} \cos\phi)...
...\rho_0}\DP{}{z^*}(\rho_0\overline{v' w'})
 - \overline{u'^2}\frac{\tan\phi}{a},$    
  $\displaystyle f \overline{u}
 + \frac{\tan\phi}{a} (\overline{u})^2
 + \Dinv{a} \DP{\overline{\Phi}}{\phi}$    
  $\displaystyle \qquad
 = - \DP{}{t}
 \left[ 
 \overline{v}^* 
 + \Dinv{\rho_0} \...
... 
 \frac{\overline{v'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \right)
 \right]$    
  $\displaystyle \qquad \qquad
 - \left[
 \overline{w}^* 
 - \Dinv{a \cos\phi} 
 \...
... 
 \frac{\overline{v'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \right)
 \right]$    
  $\displaystyle \qquad \qquad
 - \Dinv{a\cos\phi} \DP{}{\phi}(\overline{v'^2} \co...
...}(\rho_0\overline{v' w'})
 - \overline{u'^2} \frac{\tan\phi}{a}
 + \overline{Y}$    

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.

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



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.
\begin{subequations}\begin{align}&
 \DP{\overline{u}}{t}
 + \overline{v}^*
 \lef...
...}{z^*}} + \overline{w'\theta'}
 \right)
 \right].
 \end{align}\end{subequations}


next up previous contents
: 参考文献 : プリミティブ方程式系と変形オイラー平均の復習 : オイラー平均方程式系   目次
Tsukahara Daisuke 平成17年2月19日