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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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\appendix
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\begin{subequations}\begin{align}
          \DD{u}{t}       &-  \left(f + \frac{u\tan\phi}{a}\right)v
                                         + \Dinv{a\cos\phi}\DP{\Phi}{\lambda} = X,\\
          \DD{v}{t}       &+  \left(f + \frac{u\tan\phi}{a}\right)u 
                                                    + \Dinv{a}\DP{\Phi}{\phi} = Y,
 \end{align}
 
 \begin{align}
    \DP{\Phi}{z^*} & = \frac{R\theta e^{-\kappa z^*/H}}{H},
 \end{align}
 
 \begin{align}
      \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}
          \DD{\theta}{t}  &=  Q, 
 \end{align}\end{subequations}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3345}%
$ \Phi$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3347}%
$ X, Y$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3353}%
$ \kappa=R_{d}/c_p$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3355}%
$ c_p$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3357}%
$ Q$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3360}%
$\displaystyle Q$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3364}%
$\displaystyle \frac{J}{C_p}e^{\kappa z^*/H}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3366}%
$ J$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3371}%
$ A$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3373}%
$ \phi, z^*, t$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3376}%
$\displaystyle \overline{A}(\phi, z^*, t) \equiv \Dinv{2\pi}\int_0^{2\pi} A(\lambda, \phi, z^*, t) \Dd \lambda$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3378}%
$ A'$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3381}%
$\displaystyle A' = A - \overline{A}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3383}%
$ \overline{A'}=0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3385}%
$ \partial \overline{A}/\partial\lambda = 0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\setcounter{equation}{3}
\lthtmldisplayA{subequations3387}%
\setcounter{equation}{2}
\begin{subequations}\begin{align}
    & \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') \\ 
    &  \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',\\
   &  \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',
    \\
   &  \DP{}{z^*}(\overline{\Phi} + \Phi') 
        =  \frac{Re^{-\kappa z^*/H}}{H}(\overline{\theta} + \theta'),\\
   & \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,\\
    & \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}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\setcounter{equation}{4}
\lthtmldisplayA{subequations3389}%
\setcounter{equation}{3}
\begin{subequations}\begin{align}
   &  \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} 
  \\ 
   & \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'
 - \Dinv{a\cos\phi}\DP{\Phi'}{\lambda}
           + X',\\
   &  \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}
 - \overline{w}\DP{v'}{z^*}
           - w'\DP{\overline{v}}{z^*}
           - w'\DP{v'}{z^*}
 - fu'\notag\\ 
   & \qquad \qquad
           - 2\frac{\tan\phi}{a}\overline{u}u'
           - \frac{\tan\phi}{a}(u')^2
 - \Dinv{a\cos\phi}\DP{\Phi'}{\phi}
           + Y',\\
   &  \DP{\overline{\Phi}}{z^*} - \frac{Re^{-\kappa z^*/H}}{H}\overline{\theta}
        = - \DP{\Phi'}{z^*} + \frac{Re^{-\kappa z^*/H}}{H}\theta',\\
   &  \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') ,\\
   &  \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}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\setcounter{equation}{5}
\lthtmldisplayA{subequations3391}%
\setcounter{equation}{4}
\begin{subequations}\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} 
       \\ 
   & \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'},\\
   &   \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},\\
   &  \DP{\overline{\Phi}}{z^*} 
      - \frac{Re^{-\kappa z^*/H}}{H}\overline{\theta} = 0,\\
   &  \Dinv{a\cos\phi}
      \left[
        \DP{}{\phi}(\overline{v}\cos\phi)
      \right]
        + \Dinv{\rho_0}\DP{}{z^*}(\rho_0 \overline{w}) 
        = 0,\\
   &  \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}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3394}%
$\displaystyle \Dinv{a\cos\phi}\left[\DP{u'}{\lambda}
+ \DP{}{\phi}(v'\cos\phi)\right]
+ \Dinv{\rho_0}\DP{}{z^*}(\rho_0 w')
= 0$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3396}%
$ u'$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3399}%
$\displaystyle \Dinv{a \cos \phi} \overline{u'  \DP{u'}{\lambda}}
+ \Dinv{a} \overline{ u'  \DP{v'}{\phi} }
- \frac{\tan \phi}{a} \overline{ u'  v' }
+ \overline{ u'  \DP{w'}{z^*} }
+ \Dinv{\rho_0} \DP{\rho_0}{z^*} \overline{ u' w' }
= 0$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3400}%
$\displaystyle \DP{\overline{u}}{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3401}%
$\displaystyle + \Dinv{a}\overline{v}\DP{\overline{u}}{\phi}
         + \overline{w}\DP{\overline{u}}{z^*}
         - f\overline{v} 
         - \frac{\tan\phi}{a}\overline{u}\overline{v}
         - \overline{X} \notag$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3402}%
$\displaystyle =     - \frac{2}{a\cos\phi} \overline{u'\DP{u'}{\lambda}}
         - \Dinv{a}\overline{v'\DP{u'}{\phi}}
         - \overline{w'\DP{u'}{z^*}}
 - \Dinv{a}\overline{u'\DP{v'}{\phi}}
         + \frac{2\tan\phi}{a}\overline{u'v'}
         - \overline{u'\DP{w'}{z^*}}
         - \Dinv{\rho_0} \DP{\rho_0}{z^*} \overline{u'w'}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3403}%
$\displaystyle - \frac{2}{a\cos\phi} \overline{ u' \DP{u'}{\lambda} }$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3404}%
$\displaystyle = - \Dinv{a\cos\phi}\overline{\DP{(u')^2}{\lambda}}
    = 0,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3405}%
$\displaystyle - \Dinv{a}\overline{v'\DP{u'}{\phi}}
      - \Dinv{a}\overline{u'\DP{v'}{\phi}}
      + \frac{2\tan\phi}{a}\overline{u'v'}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3406}%
$\displaystyle = - \Dinv{a\cos^2\phi}\DP{}{\phi}(\overline{v'u'}\cos^2\phi),$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3407}%
$\displaystyle - \overline{w'\DP{u'}{z^*}}
      - \overline{u'\DP{w'}{z^*}}
      - \Dinv{\rho_0} \DP{\rho_0}{z^*} \overline{u'w'}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3408}%
$\displaystyle = - \Dinv{\rho_0}\DP{}{z^*}(\rho_0\overline{w'u'})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3409}%
$\displaystyle \DP{\overline{u}}{t}
    + \Dinv{a}\overline{v}\DP{\overline{u}}{\phi}
    + \overline{w}\DP{\overline{u}}{z^*}
    - f\overline{v} 
    - \frac{\tan\phi}{a} \overline{u} \  \overline{v}
    - \overline{X}
       \notag$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3410}%
$\displaystyle \qquad
   =  - \Dinv{a\cos^2\phi}\DP{}{\phi}(\overline{v'u'}\cos^2\phi)
    - \Dinv{\rho_0}\DP{}{z^*}(\rho_0\overline{w'u'})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3412}%
$ v'$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3415}%
$\displaystyle \Dinv{a \cos \phi} \overline{ v' \DP{u'}{\lambda} }
+ \Dinv{a} \overline{ v' \DP{v'}{\phi} }
+ \frac{\tan \phi}{a} \overline{ v'^2 }
+ \overline{ v'  \DP{w'}{z^*} }
+ \Dinv{\rho_0} \DP{\rho_0}{z^*} \overline{ v' w' }
=   0$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3416}%
$\displaystyle \DP{\overline{v}}{t}
      + \frac{\overline{v}}{a} \DP{\overline{v}}{\phi}
      + \overline{w} \DP{\overline{v}}{z^*}
      + f \overline{u}
      + \frac{\tan\phi}{a} (\overline{u})^2
      + \Dinv{a} \DP{\overline{\Phi}}{\phi}
      - \overline{Y}
      \notag$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3417}%
$\displaystyle \qquad
    = - \Dinv{a\cos\phi}\overline{u'\DP{v'}{\lambda}}
      - \Dinv{a}\overline{{v'}\DP{v'}{\phi}}
      - \overline{w'\DP{v'}{z^*}}
      - \frac{\tan\phi}{a} \overline{u'^2}
      \notag$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3418}%
$\displaystyle \qquad \qquad
      -  \Dinv{a \cos \phi} \overline{v'  \DP{u'}{\lambda}}
      - \Dinv{a} \overline{ v'  \DP{v'}{\phi} }
      + \frac{\tan \phi}{a} \overline{ v'^2 }
      - \overline{ v'  \DP{w'}{z^*} }
      - \Dinv{\rho_0} \DP{\rho_0}{z^*} \overline{ v' w' }$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3421}%
$\displaystyle - \Dinv{a\cos\phi}\overline{u'\DP{v'}{\lambda}}
- \Dinv{a \cos \phi} \overline{v'  \DP{u'}{\lambda}}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3425}%
$\displaystyle - \Dinv{a\cos\phi}\overline{\DP{(u' v')}{\lambda}}
= 0,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3427}%
$\displaystyle - \Dinv{a} \overline{ v' \DP{v'}{\phi} }
- \Dinv{a} \overline{ v' \DP{v'}{\phi} }
+ \frac{\tan \phi}{a} \overline{ v'^2 }$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3431}%
$\displaystyle - \Dinv{a \cos \phi}
\DP{}{\phi} \left( \cos \phi \overline{v'^2} \right)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3433}%
$\displaystyle - \overline{w'\DP{v'}{z^*}}
- \overline{ v'  \DP{w'}{z^*} }
- \Dinv{\rho_0} \DP{\rho_0}{z^*} \overline{ v' w' }$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3437}%
$\displaystyle - \Dinv{\rho_0} \DP{}{z^*} \left( \rho_0 \overline{ v' w' } \right)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3439}%
$\displaystyle \qquad 
    =  - \Dinv{a \cos \phi} 
         \DP{}{\phi} \left( \cos \phi \overline{v'^2} \right)
       - \frac{\tan\phi}{a} \overline{u'^2}
       - \Dinv{\rho_0} \DP{}{z^*} \left( \rho_0 \overline{ v' w' } \right)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3441}%
$ \theta'$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3444}%
$\displaystyle \Dinv{a \cos \phi} \overline{\theta'  \DP{u'}{\lambda}}
+ \Dinv{a} \overline{ \theta'  \DP{v'}{\phi} }
- \frac{\tan \phi}{a} \overline{ \theta' v' }
+ \overline{ \theta'  \DP{w'}{z^*} }
+ \Dinv{\rho_0} \DP{\rho_0}{z^*} \overline{ \theta' w' }
= 0$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3445}%
$\displaystyle \DP{\overline{\theta}}{t}
         + \frac{\overline{v}}{a}\DP{\overline{\theta}}{\phi}
         + \overline{w}\DP{\overline{\theta}}{z^*}
         - \overline{Q} 
      \notag$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3446}%
$\displaystyle \qquad = 
           - \Dinv{a\cos\phi}\overline{u'\DP{\theta'}{\lambda}}
           - \Dinv{a}\overline{v'\DP{\theta'}{\phi}}
           - \overline{w'\DP{\theta'}{z^*}}
      \notag$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3447}%
$\displaystyle \qquad \qquad
    -  \Dinv{a \cos \phi} \overline{\theta'  \DP{u'}{\lambda}}
    - \Dinv{a} \overline{ \theta'  \DP{v'}{\phi} }
    + \frac{\tan \phi}{a} \overline{ \theta' v' }
    - \overline{ \theta'  \DP{w'}{z^*} }
    - \Dinv{\rho_0} \DP{\rho_0}{z^*} \overline{ \theta' w' }$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3450}%
$\displaystyle - \Dinv{a \cos \phi}\overline{u' \DP{\theta'}{\lambda}}
- \Dinv{a \cos \phi} \overline{\theta'  \DP{u'}{\lambda}}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3454}%
$\displaystyle - \Dinv{a\cos\phi}\overline{\DP{(u' \theta')}{\lambda}}
= 0,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3456}%
$\displaystyle - \Dinv{a} \overline{ v' \DP{\theta'}{\phi} }
- \Dinv{a} \overline{ \theta' \DP{v'}{\phi} }
+ \frac{\tan \phi}{a} \overline{ \theta' v' }$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3460}%
$\displaystyle - \Dinv{a \cos \phi}
\DP{}{\phi} \left( \cos \phi \overline{v' \theta'} \right)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3462}%
$\displaystyle - \overline{w'\DP{\theta'}{z^*}}
- \overline{ \theta'  \DP{w'}{z^*} }
- \Dinv{\rho_0} \DP{\rho_0}{z^*} \overline{  \theta' w'  }$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3466}%
$\displaystyle - \Dinv{\rho_0} \DP{}{z^*} \left( \rho_0 \overline{ w' \theta' } \right)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3467}%
$\displaystyle \DP{\overline{\theta}}{t}
         + \frac{\overline{v}}{a}\DP{\overline{\theta}}{\phi}
         + \overline{w}\DP{\overline{\theta}}{z^*}
         - \overline{Q} 
  =     - \Dinv{a \cos \phi} 
         \DP{}{\phi} \left( \cos \phi \overline{v' \theta'} \right)
       - \Dinv{\rho_0} \DP{}{z^*} \left( \rho_0 \overline{ w' \theta' } \right)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\setcounter{equation}{11}
\lthtmldisplayA{subequations3469}%
\setcounter{equation}{10}
\begin{subequations}\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} 
              \\  
  & \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{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}
 \begin{align}
         \DP{\overline{\Phi}}{z^*} - \frac{Re^{-\kappa z^*/H}}{H}\overline{\theta} = 0,
     \end{align}
 \begin{align}
         \Dinv{a\cos\phi}&
        \DP{}{\phi}(\overline{v}\cos\phi)
        + \Dinv{\rho_0}\DP{}{z^*}(\rho_0 \overline{w}) 
        = 0,
 \end{align}
 \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}(\overline{v'\theta'}\cos\phi)
 		- \Dinv{\rho_0}\DP{}{z^*}(\rho_0\overline{w'\theta'}).
 \end{align}\end{subequations}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\setcounter{equation}{12}
\lthtmldisplayA{subequations3474}%
\setcounter{equation}{11}
\begin{subequations}\begin{align}
 	\overline{v}^* 
 	& =
      \overline{v} 
 	- \Dinv{\rho_0} \DP{}{z^*}
 	\left( \rho_0 
 	 \frac{\overline{v'\theta'}}
 	 {\overline{\DP{\theta}{z^*}}}
 	\right)
 	\\
                   \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}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3477}%
$\displaystyle {F_\phi}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3481}%
$\displaystyle \rho_0 a
\cos \phi \left(\DP{\overline{u}}{\overline{z^*}}
\frac{\overline{v'\theta'}}{\DP{\overline{\theta}}{z^*}} -
\overline{u'v'}\right)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3483}%
$\displaystyle {F_z^*}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3487}%
$\displaystyle \rho_0 a
\cos \phi \left(\left[ f - \frac{\DP{\overline{u}\cos \phi}{\phi}}{a\cos\phi} \right]
\frac{\overline{v'\theta'}}{\DP{\overline{\theta}}{z^*}} -
\overline{u'w'}\right)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3488}%
$\displaystyle \Dinv{a \cos \phi}
      \DP{}{\phi}\left[
      \left\{ 
              \overline{v}^* 
            + \Dinv{\rho_0} \DP{}{z^*}
              \left( \rho_0 
                     \frac{\overline{v'\theta'}}
                          {\overline{\DP{\theta}{z^*}}}
              \right)
      \right\}
      \cos\phi \right]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3489}%
$\displaystyle \qquad
    + \Dinv{\rho_0}
      \DP{}{z^*} 
      \left[ \rho_0 
          \left\{
               \overline{w}^* 
             - \Dinv{a \cos\phi} 
               \DP{}{\phi}
               \left( \cos \phi 
                          \frac{\overline{v'\theta'}}
                               {\overline{\DP{\theta}{z^*}}}
               \right)
        \right\}
      \right]
        = 0,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3490}%
$\displaystyle \Dinv{a \cos \phi}
      \DP{}{\phi}
      \left(
              \overline{v}^* \cos\phi
      \right)
    + \Dinv{\rho_0}
      \DP{}{z^*} 
      \left( \rho_0  \overline{w}^*  \right)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3491}%
$\displaystyle \qquad
    + \Dinv{a \cos \phi}
      \DP{}{\phi}
      \left\{ 
             \Dinv{\rho_0} \DP{}{z^*}
              \left( \rho_0 
                     \frac{\overline{v'\theta'}}
                          {\overline{\DP{\theta}{z^*}}}
              \right) \cos\phi
      \right\}
    - \Dinv{\rho_0}
      \DP{}{z^*} 
      \left\{
              \rho_0  \Dinv{a \cos\phi} 
               \DP{}{\phi}
               \left( \cos \phi 
                          \frac{\overline{v'\theta'}}
                               {\overline{\DP{\theta}{z^*}}}
               \right)
        \right\}
        = 0.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3492}%
$\displaystyle \qquad 
      \Dinv{a \cos \phi}
      \DP{}{\phi}
      \left\{ 
             \Dinv{\rho_0} \DP{}{z^*}
              \left( \rho_0 
                     \frac{\overline{v'\theta'}}
                          {\overline{\DP{\theta}{z^*}}}
              \right) \cos\phi
      \right\}
    - \Dinv{\rho_0}
      \DP{}{z^*} 
      \left\{
              \rho_0  \Dinv{a \cos\phi} 
               \DP{}{\phi}
               \left( \cos \phi 
                          \frac{\overline{v'\theta'}}
                               {\overline{\DP{\theta}{z^*}}}
               \right)
        \right\}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3493}%
$\displaystyle = 
      \Dinv{a \cos \phi}
      \left[
         \DP{}{\phi}
         \left\{ 
             \Dinv{\rho_0} \DP{}{z^*}
              \left( \rho_0 
                     \frac{\overline{v'\theta'}}
                          {\overline{\DP{\theta}{z^*}}}
               \right) \cos\phi
          \right\}
        - \Dinv{\rho_0}
          \DP{}{z^*} 
          \left\{
              \rho_0 
               \DP{}{\phi}
               \left( \cos \phi 
                          \frac{\overline{v'\theta'}}
                               {\overline{\DP{\theta}{z^*}}}
               \right)
          \right\}
      \right]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3494}%
$\displaystyle = 
      \Dinv{a \cos \phi}
      \left[
         \Dinv{\rho_0}
         \DP{}{\phi}
         \left\{ 
              \DP{}{z^*}
              \left( \rho_0 
                     \frac{\overline{v'\theta'}}
                          {\overline{\DP{\theta}{z^*}}}
                     \cos\phi
               \right) 
          \right\}
        - \Dinv{\rho_0}
          \DP{}{z^*} 
          \left\{
               \DP{}{\phi}
               \left(\rho_0  \cos \phi 
                          \frac{\overline{v'\theta'}}
                               {\overline{\DP{\theta}{z^*}}}
               \right)
          \right\}
      \right]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3495}%
$\displaystyle = 0.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3498}%
$\displaystyle \Dinv{a \cos \phi}
\DP{}{\phi}
\left(
\overline{v}^* \cos\phi
\right)
+ \Dinv{\rho_0}
\DP{}{z^*}
\left( \rho_0  \overline{w}^*  \right) = 0.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3502}%
$\displaystyle + \Dinv{a}
      \left[ 
              \overline{v}^* 
            + \Dinv{\rho_0} \DP{}{z^*}
              \left( \rho_0 
                     \frac{\overline{v'\theta'}}
                          {\overline{\DP{\theta}{z^*}}}
              \right)
      \right]
      \DP{\overline{u}}{\phi}
    + \left[
               \overline{w}^* 
             - \Dinv{a \cos\phi} 
               \DP{}{\phi}
               \left( \cos \phi 
                          \frac{\overline{v'\theta'}}
                               {\overline{\DP{\theta}{z^*}}}
               \right)
      \right]
      \DP{\overline{u}}{z^*}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3503}%
$\displaystyle \qquad \qquad
    - f 
           \left[ 
              \overline{v}^* 
            + \Dinv{\rho_0} \DP{}{z^*}
              \left( \rho_0 
                     \frac{\overline{v'\theta'}}
                          {\overline{\DP{\theta}{z^*}}}
              \right)
      \right]
    - \frac{\tan \phi}{a} \overline{u} 
      \left[ 
              \overline{v}^* 
            + \Dinv{\rho_0} \DP{}{z^*}
              \left( \rho_0 
                     \frac{\overline{v'\theta'}}
                          {\overline{\DP{\theta}{z^*}}}
              \right)
      \right]
    - \overline{X}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3504}%
$\displaystyle \qquad
  = - \Dinv{a\cos^2\phi} \DP{}{\phi} (\overline{v'u'} \cos^2 \phi)
    - \Dinv{\rho_0} \DP{}{z^*} (\rho_0\overline{w'u'}),$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3506}%
$\displaystyle + \frac{\overline{v}^*}{a} \DP{\overline{u}}{\phi}
    + \overline{w}^* \DP{\overline{u}}{z^*}
    - f \overline{v}^* 
    - \frac{\tan \phi}{a} \overline{u} \   \overline{v}^* 
    - \overline{X}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3507}%
$\displaystyle \qquad
  = - \Dinv{a\cos^2\phi} \DP{}{\phi} (\overline{v'u'} \cos^2 \phi)
    + \Dinv{a \cos\phi} 
               \DP{}{\phi}
               \left( \cos \phi 
                          \frac{\overline{v'\theta'}}
                               {\overline{\DP{\theta}{z^*}}}
               \right) \DP{\overline{u}}{z^*}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3508}%
$\displaystyle \qquad \qquad
    + f \Dinv{\rho_0} \DP{}{z^*}
      \left( \rho_0 
                     \frac{\overline{v'\theta'}}
                          {\overline{\DP{\theta}{z^*}}}
      \right)
    - \Dinv{\rho_0} \DP{}{z^*} (\rho_0\overline{w'u'})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3509}%
$\displaystyle \qquad \qquad
    - \Dinv{\rho_0 a} \DP{}{z^*}
      \left( \rho_0 
                     \frac{\overline{v'\theta'}}
                          {\overline{\DP{\theta}{z^*}}}
      \right)
      \DP{\overline{u}}{\phi}
    + \frac{\tan \phi}{a} \overline{u}  \Dinv{\rho_0} \DP{}{z^*}
      \left( \rho_0 
                     \frac{\overline{v'\theta'}}
                          {\overline{\DP{\theta}{z^*}}}
      \right),$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3511}%
$\displaystyle + \frac{\overline{v}^*}{a \cos \phi} \DP{}{\phi} 
      \left(  \overline{u} \cos \phi \right)
    + \overline{w}^* \DP{\overline{u}}{z^*}
    - f \overline{v}^* 
    - \overline{X}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3512}%
$\displaystyle \qquad
  = - \Dinv{\rho_0 a^2 \cos^2 \phi} 
      \DP{}{\phi} (\rho_0 a \overline{v'u'} \cos^2 \phi)
    + \Dinv{a \cos\phi} 
               \DP{}{\phi}
               \left( \cos \phi 
                          \frac{\overline{v'\theta'}}
                               {\overline{\DP{\theta}{z^*}}}
               \right) \DP{\overline{u}}{z^*}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3513}%
$\displaystyle \qquad \qquad
    + \frac{1}{\rho_0 a \cos \phi} 
      \DP{}{z^*}
      \left( f \rho_0  a \cos \phi 
             \frac{\overline{v'\theta'}}
                  {\overline{\DP{\theta}{z^*}}}
      \right)
    - \frac{1}{\rho_0 a \cos \phi} 
      \DP{}{z^*} (\rho_0 a \cos \phi  \overline{w'u'})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3514}%
$\displaystyle \qquad \qquad
    - \Dinv{\rho_0 a} \DP{}{z^*}
      \left( \rho_0 
                     \frac{\overline{v'\theta'}}
                          {\overline{\DP{\theta}{z^*}}}
      \right)
      \DP{\overline{u}}{\phi}
    + \frac{\tan \phi}{a} \overline{u}  \Dinv{\rho_0} \DP{}{z^*}
      \left( \rho_0 
                     \frac{\overline{v'\theta'}}
                          {\overline{\DP{\theta}{z^*}}}
      \right)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3515}%
$\displaystyle - \Dinv{\rho_0 a^2 \cos^2 \phi} 
      \DP{}{\phi} (\rho_0 a \overline{v'u'} \cos^2 \phi)
    + \Dinv{\rho_0 a^2 \cos^2 \phi} 
      \rho_0 a \cos \phi
      \DP{\overline{u}}{z^*}
      \DP{}{\phi}
      \left( \cos \phi 
             \frac{\overline{v'\theta'}}
                  {\overline{\DP{\theta}{z^*}}}
      \right)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3517}%
$\displaystyle \qquad \qquad
    - \Dinv{\rho_0 a} \DP{}{z^*}
      \left( \rho_0 
                     \frac{\overline{v'\theta'}}
                          {\overline{\DP{\theta}{z^*}}}
             \DP{\overline{u}}{\phi}     
      \right)
    + \Dinv{\rho_0 a} 
      \rho_0 
      \frac{\overline{v'\theta'}}
           {\overline{\DP{\theta}{z^*}}}
      \DP{}{z^*}
      \left( 
             \DP{\overline{u}}{\phi}
      \right)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3518}%
$\displaystyle \qquad \qquad
    + \frac{\tan \phi}{\rho_0 a}
      \DP{}{z^*}
      \left( \overline{u} \rho_0 
             \frac{\overline{v'\theta'}}
                  {\overline{\DP{\theta}{z^*}}}
      \right)
    - \frac{\tan \phi}{\rho_0 a}
      \rho_0 
      \frac{\overline{v'\theta'}}
           {\overline{\DP{\theta}{z^*}}}
      \DP{}{z^*}
      \left( \overline{u}
      \right)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3519}%
$\displaystyle =
      \Dinv{\rho_0 a^2 \cos^2 \phi} 
      \left[
       -  \DP{}{\phi} (\rho_0 a \overline{v'u'} \cos^2 \phi)
       +  \rho_0 a \cos \phi
      \DP{\overline{u}}{z^*}
      \DP{}{\phi}
      \left( \cos \phi 
                          \frac{\overline{v'\theta'}}
                               {\overline{\DP{\theta}{z^*}}}
           \right) 
      \right]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3520}%
$\displaystyle \qquad
    + \Dinv{\rho_0 a} 
      \rho_0 
      \frac{\overline{v'\theta'}}
           {\overline{\DP{\theta}{z^*}}}
      \DP{}{z^*}
      \left( 
             \DP{\overline{u}}{\phi}
      \right) 
    - \frac{\tan \phi}{\rho_0 a}
      \rho_0 
      \frac{\overline{v'\theta'}}
           {\overline{\DP{\theta}{z^*}}}
      \DP{\overline{u}}{z^*}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3521}%
$\displaystyle \qquad
    + \frac{1}{\rho_0 a \cos \phi} 
      \DP{}{z^*}
      \left[
          \left( f \rho_0  a \cos \phi 
             \frac{\overline{v'\theta'}}
                  {\overline{\DP{\theta}{z^*}}}
          \right)
        - \rho_0 a \cos \phi  \overline{w'u'}
      \right]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3522}%
$\displaystyle \qquad
    - \Dinv{\rho_0 a} \DP{}{z^*}
      \left( \rho_0 
                     \frac{\overline{v'\theta'}}
                          {\overline{\DP{\theta}{z^*}}}
             \DP{\overline{u}}{\phi}     
      \right)
    + \frac{\tan \phi}{\rho_0 a}
      \DP{}{z^*}
      \left( \overline{u} \rho_0 
             \frac{\overline{v'\theta'}}
                  {\overline{\DP{\theta}{z^*}}}
      \right)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3524}%
$\displaystyle \qquad
    + \Dinv{\rho_0 a^2 \cos^2 \phi} 
      \left[
         \rho_0 a \cos^2 \phi
         \frac{\overline{v'\theta'}}
              {\overline{\DP{\theta}{z^*}}}
         \DP{}{z^*}
         \left( 
              \DP{\overline{u}}{\phi}
         \right) 
       - \rho_0 a \cos^2 \phi \tan \phi
         \frac{\overline{v'\theta'}}
           {\overline{\DP{\theta}{z^*}}}
         \DP{\overline{u}}{z^*}
      \right]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3526}%
$\displaystyle \qquad
    + \Dinv{\rho_0 a \cos \phi} 
      \left[
        - \cos \phi
          \DP{}{z^*}
          \left( \rho_0 
                 \frac{\overline{v'\theta'}}
                      {\overline{\DP{\theta}{z^*}}}
                 \DP{\overline{u}}{\phi}     
           \right)
        + \cos \phi \tan \phi
          \DP{}{z^*}
          \left( \overline{u} \rho_0 
             \frac{\overline{v'\theta'}}
                  {\overline{\DP{\theta}{z^*}}}
          \right)
      \right]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3529}%
$\displaystyle \qquad
    + \frac{1}{\rho_0 a \cos \phi} 
      \DP{}{z^*}
      \left[
          f \rho_0  a \cos \phi 
             \frac{\overline{v'\theta'}}
                  {\overline{\DP{\theta}{z^*}}}
        - \rho_0 a \cos \phi  \overline{w'u'}
      \right]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3530}%
$\displaystyle \qquad
    + \Dinv{\rho_0 a \cos \phi} 
      \DP{}{z^*}
      \left[
        - \rho_0 \cos \phi
          \frac{\overline{v'\theta'}}
               {\overline{\DP{\theta}{z^*}}}
          \DP{\overline{u}}{\phi}     
        + \sin \phi \overline{u} \rho_0 
          \frac{\overline{v'\theta'}}
               {\overline{\DP{\theta}{z^*}}}
      \right]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3531}%
$\displaystyle \Dinv{\rho_0 a^2 \cos^2 \phi} 
     \left[
       -  \DP{}{\phi} (\rho_0 a \overline{v'u'} \cos^2 \phi)
       +  \rho_0 a \cos \phi
      \DP{\overline{u}}{z^*}
      \DP{}{\phi}
      \left( \cos \phi 
                          \frac{\overline{v'\theta'}}
                               {\overline{\DP{\theta}{z^*}}}
           \right) 
      \right]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3533}%
$\displaystyle =
      \Dinv{\rho_0 a^2 \cos^2 \phi} 
      \left[
       -  \DP{}{\phi} (\rho_0 a \overline{v'u'} \cos^2 \phi)
      \right]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3534}%
$\displaystyle \qquad
    + \Dinv{\rho_0 a^2 \cos^2 \phi} 
      \left[
         \rho_0 a \cos^2 \phi
         \frac{\overline{v'\theta'}}
              {\overline{\DP{\theta}{z^*}}}
         \DP{}{\phi}
         \left( 
              \DP{\overline{u}}{z^*}
         \right) 
       +  \DP{\overline{u}}{z^*}
          \DP{}{\phi}
          \left(\rho_0 a \cos^2 \phi
                          \frac{\overline{v'\theta'}}
                               {\overline{\DP{\theta}{z^*}}}
           \right) 
      \right]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3535}%
$\displaystyle =
      \Dinv{\rho_0 a^2 \cos^2 \phi} 
      \left[
       -  \DP{}{\phi} (\rho_0 a \overline{v'u'} \cos^2 \phi)
      \right] 
    + \Dinv{\rho_0 a^2 \cos^2 \phi} 
      \left[
          \DP{}{\phi}
          \left(\rho_0 a \cos^2 \phi
                          \frac{\overline{v'\theta'}}
                               {\overline{\DP{\theta}{z^*}}}
                \DP{\overline{u}}{z^*}
           \right) 
      \right]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3536}%
$\displaystyle = \Dinv{\rho_0 a^2 \cos^2 \phi} 
     \DP{}{\phi}
     \left[
          -  \rho_0 a \overline{v'u'} \cos^2 \phi
          +  \rho_0 a \cos^2 \phi
                          \frac{\overline{v'\theta'}}
                               {\overline{\DP{\theta}{z^*}}}
                \DP{\overline{u}}{z^*}
      \right]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3537}%
$\displaystyle = \Dinv{\rho_0 a^2 \cos^2 \phi} 
     \DP{}{\phi}
     \left[
            \rho_0 a \cos^2 \phi
             \left\{
                  \DP{\overline{u}}{z^*}
                  \frac{\overline{v'\theta'}}
                       {\overline{\DP{\theta}{z^*}}}
               -  \overline{v'u'}
             \right\}
      \right]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3538}%
$\displaystyle = \Dinv{\rho_0 a^2 \cos^2 \phi} 
     \DP{}{\phi}
     \left(
            \cos \phi F^{*}_{\phi}
     \right)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3539}%
$\displaystyle \frac{1}{\rho_0 a \cos \phi} 
      \DP{}{z^*}
      \left[
          f \rho_0  a \cos \phi 
             \frac{\overline{v'\theta'}}
                  {\overline{\DP{\theta}{z^*}}}
        - \rho_0 a \cos \phi  \overline{w'u'}
      \right]
 + \Dinv{\rho_0 a \cos \phi} 
      \DP{}{z^*}
      \left[
        - \rho_0 \cos \phi
          \frac{\overline{v'\theta'}}
               {\overline{\DP{\theta}{z^*}}}
          \DP{\overline{u}}{\phi}     
        + \sin \phi \overline{u} \rho_0 
          \frac{\overline{v'\theta'}}
               {\overline{\DP{\theta}{z^*}}}
      \right]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3540}%
$\displaystyle = 
     \frac{1}{\rho_0 a \cos \phi} 
      \DP{}{z^*}
      \left[
         \rho_0  a \cos \phi 
         \left\{
              f \frac{\overline{v'\theta'}}
                  {\overline{\DP{\theta}{z^*}}}
           -  \overline{w'u'}
           -  \frac{\overline{v'\theta'}}
               {a \overline{\DP{\theta}{z^*}}}
              \DP{\overline{u}}{\phi}     
           + \sin \phi \overline{u} 
             \frac{\overline{v'\theta'}}
               {a \cos \phi \overline{\DP{\theta}{z^*}}}
         \right\}
      \right]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3541}%
$\displaystyle = 
     \frac{1}{\rho_0 a \cos \phi} 
      \DP{}{z^*}
      \left[
         \rho_0  a \cos \phi 
         \left\{
              f \frac{\overline{v'\theta'}}
                  {\overline{\DP{\theta}{z^*}}}
           - \left( 
                 \cos \phi
                 \DP{\overline{u}}{\phi}     
               - \sin \phi \overline{u} 
             \right) 
             \frac{\overline{v'\theta'}}
                  {a  \cos \phi \overline{\DP{\theta}{z^*}}}
           -  \overline{w'u'}
         \right\}
      \right]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3543}%
$\displaystyle = 
     \frac{1}{\rho_0 a \cos \phi} 
      \DP{}{z^*}
      \left[
         \rho_0  a \cos \phi 
         \left\{
              \left(     f 
                      -  \frac{\DP{(\overline{u} \cos \phi)}{\phi}}
                              {a  \cos \phi}
              \right)
              \frac{\overline{v'\theta'}}
                   {\overline{\DP{\theta}{z^*}}}
           -  \overline{w'u'}
         \right\}
      \right]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3544}%
$\displaystyle =  \frac{1}{\rho_0 a \cos \phi} 
      \DP{F^{*}_{z}}{z^*}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3545}%
$\displaystyle \DP{\overline{u}}{t}
    + \frac{\overline{v}^*}{a \cos \phi} \DP{}{\phi} 
      \left(  \overline{u} \cos \phi \right)
    + \overline{w}^* \DP{\overline{u}}{z^*}
    - f \overline{v}^* 
    - \overline{X} 
  =  \Dinv{\rho_0 a^2 \cos^2 \phi} 
     \DP{}{\phi}
     \left(
            \cos \phi F^{*}_{\phi}
     \right)
    + \frac{1}{\rho_0 a \cos \phi} 
      \DP{F^{*}_{z}}{z^*}, 
   \nonumber$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3546}%
$\displaystyle \DP{\overline{u}}{t}
    + \frac{\overline{v}^*}{a \cos \phi} \DP{}{\phi} 
      \left(  \overline{u} \cos \phi \right)
    + \overline{w}^* \DP{\overline{u}}{z^*}
    - f \overline{v}^* 
    - \overline{X} 
  =  \Dinv{\rho_0 a \cos \phi} \Ddiv{\Dvect{F}}.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3547}%
$\displaystyle \Ddiv{\Dvect{F}} 
     = \Dinv{a \cos \phi } \DP{(\cos \phi F_{\phi})}{\phi} + \DP{F_{z^{*}}}{z^*}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3548}%
$\displaystyle \DP{\overline{\theta}}{t}
    + \frac{1}{a} 
      \left[ 
              \overline{v}^* 
            + \Dinv{\rho_0} \DP{}{z^*}
              \left( \rho_0 
                     \frac{\overline{v'\theta'}}
                          {\overline{\DP{\theta}{z^*}}}
              \right)
      \right]
      \DP{\overline{\theta}}{\phi}
    + \left[
               \overline{w}^* 
             - \Dinv{a \cos\phi} 
               \DP{}{\phi}
               \left( \cos \phi 
                          \frac{\overline{v'\theta'}}
                               {\overline{\DP{\theta}{z^*}}}
               \right)
      \right]
      \DP{\overline{\theta}}{z^*}
    - \overline{Q}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3549}%
$\displaystyle \qquad  
  =
    - \Dinv{a\cos\phi}\DP{}{\phi}(\overline{v'\theta'}\cos\phi)
    - \Dinv{\rho_0}\DP{}{z^*}(\rho_0\overline{w'\theta'}),$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3550}%
$\displaystyle \DP{\overline{\theta}}{t}
    + \frac{\overline{v}^*}{a}  \DP{\overline{\theta}}{\phi}
    + \overline{w}^* \DP{\overline{\theta}}{z^*}
    - \overline{Q}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3551}%
$\displaystyle \qquad  
  = - \Dinv{\rho_0 a} \DP{}{z^*}
      \left( \rho_0 
                     \frac{\overline{v'\theta'}}
                          {\overline{\DP{\theta}{z^*}}}
              \right) \DP{\overline{\theta}}{\phi}
    + \Dinv{a \cos\phi} 
      \DP{}{\phi}
      \left( \cos \phi 
                          \frac{\overline{v'\theta'}}
                               {\overline{\DP{\theta}{z^*}}}
      \right) \DP{\overline{\theta}}{z^*}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3552}%
$\displaystyle \qquad \qquad
    - \Dinv{a\cos\phi}\DP{}{\phi}(\overline{v'\theta'}\cos\phi)
    - \Dinv{\rho_0}\DP{}{z^*}(\rho_0\overline{w'\theta'})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3553}%
$\displaystyle - \Dinv{\rho_0} \DP{}{z^*}
      \left( \rho_0 
                     \frac{\overline{v'\theta'}}
                          {a \overline{\DP{\theta}{z^*}}}
              \right) \DP{\overline{\theta}}{\phi}
    + \Dinv{a \cos\phi} 
               \DP{}{\phi}
               \left( \cos \phi 
                          \frac{\overline{v'\theta'}}
                               {\overline{\DP{\theta}{z^*}}}
               \right) \DP{\overline{\theta}}{z^*}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3554}%
$\displaystyle \qquad
    - \Dinv{a\cos\phi}\DP{}{\phi}(\overline{v'\theta'}\cos\phi)
    - \Dinv{\rho_0}\DP{}{z^*}(\rho_0\overline{w'\theta'})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3556}%
$\displaystyle - \Dinv{\rho_0} \DP{}{z^*}
      \left(  \rho_0 
              \frac{\overline{v'\theta'}}
                   {a \overline{\DP{\theta}{z^*}}}
              \DP{\overline{\theta}}{\phi}
      \right) 
   +  \frac{\overline{v'\theta'}}
           {a \overline{\DP{\theta}{z^*}}}
      \DP{}{z^*}\DP{\overline{\theta}}{\phi}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3557}%
$\displaystyle \qquad 
    + \Dinv{a \cos\phi} 
      \left[  
                 \DP{}{\phi} \left( \cos \phi  \overline{v'\theta'} \right)  
                 \frac{1}{\overline{\DP{\theta}{z^*}}}
             +  \cos \phi \overline{v'\theta'}
                \DP{}{\phi} \left( \overline{\DP{\theta}{z^*}} \right)^{-1}
      \right] \DP{\overline{\theta}}{z^*}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3560}%
$\displaystyle - \Dinv{\rho_0} \DP{}{z^*}
      \left(  \rho_0 
              \frac{\overline{v'\theta'}}
                   {a \overline{\DP{\theta}{z^*}}}
              \DP{\overline{\theta}}{\phi}
      \right) 
   +  \frac{\overline{v'\theta'}}
           {a \overline{\DP{\theta}{z^*}}}
      \DP{}{z^*}\DP{\overline{\theta}}{\phi}
   +  \Dinv{a}  
      \overline{v'\theta'}
      \DP{}{\phi} \left( \overline{\DP{\theta}{z^*}} \right)^{-1}
      \DP{\overline{\theta}}{z^*}
    - \Dinv{\rho_0}\DP{}{z^*}(\rho_0\overline{w'\theta'})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3562}%
$\displaystyle - \Dinv{\rho_0} \DP{}{z^*}
     \left[ \rho_0 
              \frac{\overline{v'\theta'}}
                   {a \overline{\DP{\theta}{z^*}}}
              \DP{\overline{\theta}}{\phi}
             + \rho_0\overline{w'\theta'}
     \right]
   + \frac{\overline{v'\theta'}}{a}
     \left[
              \frac{1}
                   {\overline{\DP{\theta}{z^*}}}
              \DP{}{z^*}\DP{\overline{\theta}}{\phi}
         +    \DP{}{\phi} \left( \overline{\DP{\theta}{z^*}} \right)^{-1}
              \DP{\overline{\theta}}{z^*}
      \right]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3564}%
$\displaystyle - \Dinv{\rho_0} \DP{}{z^*}
     \left[ \rho_0 
            \left(
                 \frac{\overline{v'\theta'}}
                      {a \overline{\DP{\theta}{z^*}}}
                 \DP{\overline{\theta}}{\phi}
               + \overline{w'\theta'}
            \right)  
     \right]
   + \frac{\overline{v'\theta'}}{a}
     \DP{}{\phi}
     \left(
          \frac{ \DP{\overline{\theta}}{z^*} }
                   { \overline{\DP{\theta}{z^*}} }
     \right)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3566}%
$\displaystyle - \Dinv{\rho_0} \DP{}{z^*}
     \left[ \rho_0 
            \left(
                 \frac{\overline{v'\theta'}}
                      {a \overline{\DP{\theta}{z^*}}}
                 \DP{\overline{\theta}}{\phi}
               + \overline{w'\theta'}
            \right)  
     \right].$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3567}%
$\displaystyle \DP{\overline{\theta}}{t}
    + \frac{\overline{v}^*}{a}  \DP{\overline{\theta}}{\phi}
    + \overline{w}^* \DP{\overline{\theta}}{z^*}
    - \overline{Q}
  = 
   - \Dinv{\rho_0} \DP{}{z^*}
     \left[ \rho_0 
            \left(
                 \frac{\overline{v'\theta'}}
                      {a \overline{\DP{\theta}{z^*}}}
                 \DP{\overline{\theta}}{\phi}
               + \overline{w'\theta'}
            \right)  
     \right].$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3569}%
$ v$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3570}%
$\displaystyle \DP{}{t}
        \left[ 
              \overline{v}^* 
            + \Dinv{\rho_0} \DP{}{z^*}
              \left( \rho_0 
                     \frac{\overline{v'\theta'}}
                          {\overline{\DP{\theta}{z^*}}}
              \right)
        \right]
      + \frac{1}{a} 
        \left[ 
              \overline{v}^* 
            + \Dinv{\rho_0} \DP{}{z^*}
              \left( \rho_0 
                     \frac{\overline{v'\theta'}}
                          {\overline{\DP{\theta}{z^*}}}
              \right)
        \right]
        \DP{}{\phi}
        \left[ 
              \overline{v}^* 
            + \Dinv{\rho_0} \DP{}{z^*}
              \left( \rho_0 
                     \frac{\overline{v'\theta'}}
                          {\overline{\DP{\theta}{z^*}}}
              \right)
        \right]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3571}%
$\displaystyle \qquad    \qquad
      + \left[
               \overline{w}^* 
             - \Dinv{a \cos\phi} 
               \DP{}{\phi}
               \left( \cos \phi 
                          \frac{\overline{v'\theta'}}
                               {\overline{\DP{\theta}{z^*}}}
               \right)
        \right]
        \DP{}{z^*}
        \left[ 
              \overline{v}^* 
            + \Dinv{\rho_0} \DP{}{z^*}
              \left( \rho_0 
                     \frac{\overline{v'\theta'}}
                          {\overline{\DP{\theta}{z^*}}}
              \right)
        \right]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3572}%
$\displaystyle \qquad       \qquad
      + f \overline{u}
      + \frac{\tan\phi}{a} (\overline{u})^2
      + \Dinv{a} \DP{\overline{\Phi}}{\phi}
      - \overline{Y}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3573}%
$\displaystyle \qquad
    = - \Dinv{a\cos\phi}\DP{}{\phi}(\overline{v'^2} \cos\phi)
      - \Dinv{\rho_0}\DP{}{z^*}(\rho_0\overline{v' w'})
      - \overline{u'^2}\frac{\tan\phi}{a},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3574}%
$\displaystyle f \overline{u}
      + \frac{\tan\phi}{a} (\overline{u})^2
      + \Dinv{a} \DP{\overline{\Phi}}{\phi}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3575}%
$\displaystyle \qquad
    = -  \DP{}{t}
        \left[ 
              \overline{v}^* 
            + \Dinv{\rho_0} \DP{}{z^*}
              \left( \rho_0 
                     \frac{\overline{v'\theta'}}
                          {\overline{\DP{\theta}{z^*}}}
              \right)
        \right]
      - \frac{1}{a} 
        \left[ 
              \overline{v}^* 
            + \Dinv{\rho_0} \DP{}{z^*}
              \left( \rho_0 
                     \frac{\overline{v'\theta'}}
                          {\overline{\DP{\theta}{z^*}}}
              \right)
        \right]
        \DP{}{\phi}
        \left[ 
              \overline{v}^* 
            + \Dinv{\rho_0} \DP{}{z^*}
              \left( \rho_0 
                     \frac{\overline{v'\theta'}}
                          {\overline{\DP{\theta}{z^*}}}
              \right)
        \right]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3576}%
$\displaystyle \qquad    \qquad
      - \left[
               \overline{w}^* 
             - \Dinv{a \cos\phi} 
               \DP{}{\phi}
               \left( \cos \phi 
                          \frac{\overline{v'\theta'}}
                               {\overline{\DP{\theta}{z^*}}}
               \right)
        \right]
        \DP{}{z^*}
        \left[ 
              \overline{v}^* 
            + \Dinv{\rho_0} \DP{}{z^*}
              \left( \rho_0 
                     \frac{\overline{v'\theta'}}
                          {\overline{\DP{\theta}{z^*}}}
              \right)
        \right]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3577}%
$\displaystyle \qquad       \qquad
    - \Dinv{a\cos\phi} \DP{}{\phi}(\overline{v'^2} \cos \phi)
    - \Dinv{\rho_0} \DP{}{z^*}(\rho_0\overline{v' w'})
    - \overline{u'^2} \frac{\tan\phi}{a}
    + \overline{Y}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3579}%
$ G$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3582}%
$\displaystyle \overline{u} 
      \left( f + \frac{\tan\phi}{a} \overline{u} \right) 
    + \Dinv{a} \DP{\overline{\Phi}}{\phi}
  =   G.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\setcounter{equation}{17}
\lthtmldisplayA{subequations3584}%
\setcounter{equation}{16}
\begin{subequations}\begin{align}&
       \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},    \\&
       \overline{u}
       \left( f + \overline{u}\frac{\tan\phi}{a} \right) 
     + \Dinv{a}\DP{\overline{\Phi}}{\phi}
  = G.
 \end{align}
 \begin{align}
      \DP{\overline{\Phi}}{z^*} 
    - \frac{Re^{-\kappa z^*/H}}{H}\overline{\theta}
  =   0.
 \end{align}
 \begin{align}
         \Dinv{a\cos\phi}&\left[
        \DP{}{\phi}(\overline{v}^*\cos\phi)\right]
        + \Dinv{\rho_0}\DP{}{z^*}(\rho_0 \overline{w}^*) 
        = 0.
 \end{align}
 \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(
                         \overline{v'\theta'}\frac{\DP{\overline{\theta}}{\phi}}
                                  {a\DP{\overline{\theta}}{z^*}} + \overline{w'\theta'}
                       \right)
                 \right].
 \end{align}\end{subequations}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}


\end{document}