# LaTeX2HTML 2K.1beta (1.48)
# Associate images original text with physical files.


$key = q/w;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img10.png"
 ALT="$ w$">|; 

$key = q/displaystyle=;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5.png"
 ALT="$\displaystyle =$">|; 

$key = q/{subequations}align{hat{F}_phi<col_mark>equivsigmacosphileft(DP{overline{u}}{z^*theta}}{z^*}}-overline{u'w'}right)align{{subequations};MSF=1.6;LFS=12;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="555" HEIGHT="128" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img17.png"
 ALT="\begin{subequations}\begin{align}
 \hat{F}_\phi &amp;\equiv \sigma
 \cos \phi \left(...
...verline{\theta}}{z^*}} - \overline{u'w'}
 \right)
 \end{align}\end{subequations}">|; 

$key = q/phi,z^*,t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img76.png"
 ALT="$ \phi, z^*, t$">|; 

$key = q/{subequations}align{<col_mark>DP{overline{u}}{t}+frac{overline{u}}{acosphi}DP{ov{theta}}{z^*}-w'DP{theta'}{z^*}+Q'align{{subequations};MSF=1.6;LFS=12;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="556" HEIGHT="805" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img83.png"
 ALT="\begin{subequations}\begin{align}
 &amp; \DP{\overline{u}}{t}
 + \frac{\overline{u}}...
...erline{\theta}}{z^*}
 - w'\DP{\theta'}{z^*}
 + Q'
 \end{align}\end{subequations}">|; 

$key = q/T;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img58.png"
 ALT="$ T$">|; 

$key = q/R_{d};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img8.png"
 ALT="$ R_{d}$">|; 

$key = q/displaystyleDP{overline{theta}}{t}+frac{overline{v}}{a}DP{overline{theta}}{phi}+overline{w}DP{overline{theta}}{z^*}-overline{Q}notag;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="200" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img113.png"
 ALT="$\displaystyle \DP{\overline{\theta}}{t}
 + \frac{\overline{v}}{a}\DP{\overline{\theta}}{\phi}
 + \overline{w}\DP{\overline{\theta}}{z^*}
 - \overline{Q} 
 \notag$">|; 

$key = q/displaystyletheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img63.png"
 ALT="$\displaystyle \theta$">|; 

$key = q/displaystyle-frac{2}{acosphi}overline{u'DP{u'}{lambda}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img91.png"
 ALT="$\displaystyle - \frac{2}{a\cos\phi} \overline{ u' \DP{u'}{\lambda} }$">|; 

$key = q/displaystyle-Dinv{rho_0a^2cos^2phi}DP{}{phi}(rho_0aoverline{v'u'}cos^2phi)+Dinv{osphifrac{overline{v'theta'}}{overline{DP{theta}{z^*}}}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="594" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img149.png"
 ALT="$\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)$">|; 

$key = q/displaystyleDP{overline{theta}}{t}+frac{1}{a}left[overline{v}^*+Dinv{rho_0}DP{}{theta}{z^*}}}right)right]DP{overline{theta}}{z^*}-overline{Q};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="647" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img175.png"
 ALT="$\displaystyle \DP{\overline{\theta}}{t}
 + \frac{1}{a} 
 \left[ 
 \overline{v}^...
...P{\theta}{z^*}}}
 \right)
 \right]
 \DP{\overline{\theta}}{z^*}
 - \overline{Q}$">|; 

$key = q/displaystyle=frac{1}{rho_0acosphi}DP{}{z^*}left[rho_0acosphileft{ffrac{overline{rline{v'theta'}}{acosphioverline{DP{theta}{z^*}}}right}right];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="595" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img168.png"
 ALT="$\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]$">|; 

$key = q/displaystylew;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img61.png"
 ALT="$\displaystyle w$">|; 

$key = q/displaystyleDP{}{t}left[overline{v}^*+Dinv{rho_0}DP{}{z^*}left(rho_0frac{overlinrac{overline{v'theta'}}{overline{DP{theta}{z^*}}}right)right];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="692" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img190.png"
 ALT="$\displaystyle \DP{}{t}
 \left[ 
 \overline{v}^* 
 + \Dinv{\rho_0} \DP{}{z^*}
 \...
... 
 \frac{\overline{v'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \right)
 \right]$">|; 

$key = q/kappa=R_{d}slashc_p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img69.png"
 ALT="$ \kappa=R_{d}/c_p$">|; 

$key = q/displaystyleqquad+Dinv{rho_0}DP{}{z^*}left[rho_0left{overline{w}^*-Dinv{acosphi}ine{v'theta'}}{overline{DP{theta}{z^*}}}right)right}right]=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="458" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img130.png"
 ALT="$\displaystyle \qquad
 + \Dinv{\rho_0}
 \DP{}{z^*} 
 \left[ \rho_0 
 \left\{
 \o...
...ne{v'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \right)
 \right\}
 \right]
 = 0,$">|; 

$key = q/displaystyleDinv{acosphi}overline{u'DP{u'}{lambda}}+Dinv{a}overline{u'DP{v'}{philine{u'DP{w'}{z^*}}+Dinv{rho_0}DP{rho_0}{z^*}overline{u'w'}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="503" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img87.png"
 ALT="$\displaystyle \Dinv{a \cos \phi} \overline{u' \DP{u'}{\lambda}}
+ \Dinv{a} \ove...
...line{ u' \DP{w'}{z^*} }
+ \Dinv{\rho_0} \DP{\rho_0}{z^*} \overline{ u' w' }
= 0$">|; 

$key = q/displaystylerho_se^{-z^*slashH},hspace{2em}rho_sequivp_sslashRT_s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="220" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img32.png"
 ALT="$\displaystyle \rho_s e^{-z^*/H}, \hspace{2em} \rho_s \equiv p_s/RT_s$">|; 

$key = q/displaystyleDP{}{z^*}Psi^*;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img51.png"
 ALT="$\displaystyle \DP{}{z^*}\Psi^*$">|; 

$key = q/H;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img7.png"
 ALT="$ H$">|; 

$key = q/displaystyleqquad+Dinv{rho_0a}rho_0frac{overline{v'theta'}}{overline{DP{theta}{zline{v'theta'}}{overline{DP{theta}{z^*}}}DP{overline{u}}{z^*};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="377" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img153.png"
 ALT="$\displaystyle \qquad
 + \Dinv{\rho_0 a} 
 \rho_0 
 \frac{\overline{v'\theta'}}
...
...ac{\overline{v'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \DP{\overline{u}}{z^*}$">|; 

$key = q/displaystyleqquad+frac{1}{rho_0acosphi}DP{}{z^*}left[left(frho_0acosphifrac{overrline{DP{theta}{z^*}}}right)-rho_0acosphioverline{w'u'}right];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="457" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img154.png"
 ALT="$\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]$">|; 

$key = q/F_phi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img38.png"
 ALT="$ F_\phi$">|; 

$key = q/Psi^*;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img49.png"
 ALT="$ \Psi^*$">|; 

$key = q/displaystylerho_0(z^*);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img31.png"
 ALT="$\displaystyle \rho_0(z^*)$">|; 

$key = q/p=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img53.png"
 ALT="$ p=0$">|; 

$key = q/displaystylesigmaequivfrac{rho_0}{rho_s}=expleft(frac{-z^*}{H}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="192" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img29.png"
 ALT="$\displaystyle \sigma \equiv \frac{\rho_0}{\rho_s} = \exp\left(\frac{-z^*}{H}\right),$">|; 

$key = q/displaystyleqquadqquad-Dinv{acosphi}overline{v'DP{u'}{lambda}}-Dinv{a}overline{verline{v'DP{w'}{z^*}}-Dinv{rho_0}DP{rho_0}{z^*}overline{v'w'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="546" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img103.png"
 ALT="$\displaystyle \qquad \qquad
 - \Dinv{a \cos \phi} \overline{v' \DP{u'}{\lambda}...
...verline{ v' \DP{w'}{z^*} }
 - \Dinv{\rho_0} \DP{\rho_0}{z^*} \overline{ v' w' }$">|; 

$key = q/{subequations}align{sigmaoverline{v}^*<col_mark>=-gDinv{2piacosphi}DP{Psi^*}{z^{=gDinv{2pia^2cosphi}DP{Psi^*}{phi}align{{subequations};MSF=1.6;LFS=12;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="555" HEIGHT="98" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img50.png"
 ALT="\begin{subequations}\begin{align}
 \sigma \overline{v}^* &amp;= -g\Dinv{2\pi a \cos\...
...w}^* &amp;= g\Dinv{2\pi a^2\cos\phi}\DP{\Psi^*}{\phi}
 \end{align}\end{subequations}">|; 

$key = q/displaystyleqquadqquad-Dinv{rho_0a}DP{}{z^*}left(rho_0frac{overline{v'theta'}}{oho_0frac{overline{v'theta'}}{overline{DP{theta}{z^*}}}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="493" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img144.png"
 ALT="$\displaystyle \qquad \qquad
 - \Dinv{\rho_0 a} \DP{}{z^*}
 \left( \rho_0 
 \fra...
...( \rho_0 
 \frac{\overline{v'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \right),$">|; 

$key = q/C_p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img65.png"
 ALT="$ C_p$">|; 

$key = q/displaystyle-Dinv{rho_0}DP{}{z^*}left(rho_0overline{w'theta'}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img121.png"
 ALT="$\displaystyle - \Dinv{\rho_0} \DP{}{z^*} \left( \rho_0 \overline{ w' \theta' } \right)$">|; 

$key = q/displaystylez^*;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4.png"
 ALT="$\displaystyle z^*$">|; 

$key = q/omegaequivDpslashDt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img59.png"
 ALT="$ \omega \equiv Dp/Dt$">|; 

$key = q/displaystyleDinv{acosphi}DP{}{phi}left(overline{v}^*cosphiright)+Dinv{rho_0}DP{}{z^*}left(rho_0overline{w}^*right)=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="338" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img137.png"
 ALT="$\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.$">|; 

$key = q/v;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img189.png"
 ALT="$ v$">|; 

$key = q/displaystyle-Dinv{a}overline{v'DP{theta'}{phi}}-Dinv{a}overline{theta'DP{v'}{phi}}+frac{tanphi}{a}overline{theta'v'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="253" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img118.png"
 ALT="$\displaystyle - \Dinv{a} \overline{ v' \DP{\theta'}{\phi} }
- \Dinv{a} \overline{ \theta' \DP{v'}{\phi} }
+ \frac{\tan \phi}{a} \overline{ \theta' v' }$">|; 

$key = q/R_{d}=Rslashw;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img11.png"
 ALT="$ R_{d} = R/w$">|; 

$key = q/F_{z^*};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img39.png"
 ALT="$ F_{z^*}$">|; 

$key = q/displaystyleqquad=-Dinv{acosphi}overline{u'DP{theta'}{lambda}}-Dinv{a}overline{v'DP{theta'}{phi}}-overline{w'DP{theta'}{z^*}}notag;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="332" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img114.png"
 ALT="$\displaystyle \qquad = 
 - \Dinv{a\cos\phi}\overline{u'\DP{\theta'}{\lambda}}
 ...
...inv{a}\overline{v'\DP{\theta'}{\phi}}
 - \overline{w'\DP{\theta'}{z^*}}
 \notag$">|; 

$key = q/displaystyle-Dinv{a}overline{v'DP{u'}{phi}}-Dinv{a}overline{u'DP{v'}{phi}}+frac{2tanphi}{a}overline{u'v'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="270" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img93.png"
 ALT="$\displaystyle - \Dinv{a}\overline{v'\DP{u'}{\phi}}
 - \Dinv{a}\overline{u'\DP{v'}{\phi}}
 + \frac{2\tan\phi}{a}\overline{u'v'}$">|; 

$key = q/displaystyleDP{overline{u}}{t}+frac{overline{v}^*}{acosphi}DP{}{phi}left(overlin_{phi}right)+frac{1}{rho_0acosphi}DP{F^{*}_{z}}{z^*},nonumber;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="736" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img172.png"
 ALT="$\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$">|; 

$key = q/p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img14.png"
 ALT="$ p$">|; 

$key = q/displaystyleqquadqquad-Dinv{acosphi}DP{}{phi}(overline{v'theta'}cosphi)-Dinv{rho_0}DP{}{z^*}(rho_0overline{w'theta'});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="403" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img179.png"
 ALT="$\displaystyle \qquad \qquad
 - \Dinv{a\cos\phi}\DP{}{\phi}(\overline{v'\theta'}\cos\phi)
 - \Dinv{\rho_0}\DP{}{z^*}(\rho_0\overline{w'\theta'})$">|; 

$key = q/displaystyleqquadqquad+frac{1}{rho_0acosphi}DP{}{z^*}left(frho_0acosphifrac{overt)-frac{1}{rho_0acosphi}DP{}{z^*}(rho_0acosphioverline{w'u'});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="589" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img147.png"
 ALT="$\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'})$">|; 

$key = q/displaystyleqquad+Dinv{rho_0a^2cos^2phi}left[rho_0acos^2phifrac{overline{v'theta'theta'}}{overline{DP{theta}{z^*}}}DP{overline{u}}{z^*}right];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="572" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img156.png"
 ALT="$\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]$">|; 

$key = q/displaystyle-Dinv{acosphi}DP{}{phi}left(cosphioverline{v'theta'}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="190" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img119.png"
 ALT="$\displaystyle - \Dinv{a \cos \phi}
\DP{}{\phi} \left( \cos \phi \overline{v' \theta'} \right)$">|; 

$key = q/displaystyleqquad=-Dinv{acos^2phi}DP{}{phi}(overline{v'u'}cos^2phi)-Dinv{rho_0}DP{}{z^*}(rho_0overline{w'u'}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="408" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img140.png"
 ALT="$\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'}),$">|; 

$key = q/displaystyle=frac{1}{rho_0acosphi}DP{F^{*}_{z}}{z^*};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img171.png"
 ALT="$\displaystyle = \frac{1}{\rho_0 a \cos \phi} 
 \DP{F^{*}_{z}}{z^*}$">|; 

$key = q/G;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img198.png"
 ALT="$ G$">|; 

$key = q/displaystyleqquad=-Dinv{rho_0a^2cos^2phi}DP{}{phi}(rho_0aoverline{v'u'}cos^2phi)'theta'}}{overline{DP{theta}{z^*}}}right)DP{overline{u}}{z^*};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="559" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img146.png"
 ALT="$\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^*}$">|; 

$key = q/displaystyleqquad+Dinv{rho_0a^2cos^2phi}left[rho_0acos^2phifrac{overline{v'thetarac{overline{v'theta'}}{overline{DP{theta}{z^*}}}right)right];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="587" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img162.png"
 ALT="$\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]$">|; 

$key = q/displaystyleqquad=-Dinv{acosphi}overline{u'DP{v'}{lambda}}-Dinv{a}overline{{v'}D}}-overline{w'DP{v'}{z^*}}-frac{tanphi}{a}overline{u'^2}notag;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="423" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img102.png"
 ALT="$\displaystyle \qquad
 = - \Dinv{a\cos\phi}\overline{u'\DP{v'}{\lambda}}
 - \Din...
...hi}}
 - \overline{w'\DP{v'}{z^*}}
 - \frac{\tan\phi}{a} \overline{u'^2}
 \notag$">|; 

$key = q/Psi^*=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img54.png"
 ALT="$ \Psi^* = 0$">|; 

$key = q/A;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img75.png"
 ALT="$ A$">|; 

$key = q/lambda;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1.png"
 ALT="$ \lambda$">|; 

$key = q/displaystyle=Dinv{rho_0a^2cos^2phi}left[-DP{}{phi}(rho_0aoverline{v'u'}cos^2phi)rac{overline{v'theta'}}{overline{DP{theta}{z^*}}}right)right];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="557" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img152.png"
 ALT="$\displaystyle =
 \Dinv{\rho_0 a^2 \cos^2 \phi} 
 \left[
 - \DP{}{\phi} (\rho_0 ...
...
 \frac{\overline{v'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \right) 
 \right]$">|; 

$key = q/displaystyleqquadqquad+fDinv{rho_0}DP{}{z^*}left(rho_0frac{overline{v'theta'}}{otheta}{z^*}}}right)-Dinv{rho_0}DP{}{z^*}(rho_0overline{w'u'});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="384" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img143.png"
 ALT="$\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'})$">|; 

$key = q/displaystyle-Dinv{rho_0}DP{}{z^*}left(rho_0frac{overline{v'theta'}}{aoverline{DP}{aoverline{DP{theta}{z^*}}}DP{}{z^*}DP{overline{theta}}{phi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="312" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img182.png"
 ALT="$\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}$">|; 

$key = q/A';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img78.png"
 ALT="$ A'$">|; 

$key = q/sigma;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img28.png"
 ALT="$ \sigma$">|; 

$key = q/{subequations}align{overline{v}^*<col_mark>equivoverline{v}-Dinv{rho_0}DP{}{z^*}}}{DP{overline{theta}}{z^*}}right)align{{subequations};MSF=1.6;LFS=12;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="555" HEIGHT="191" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img44.png"
 ALT="\begin{subequations}\begin{align}
 \overline{v}^* &amp;\equiv \overline{v} 
 - \Dinv...
...v'\theta'}}
 {\DP{\overline{\theta}}{z^*}}\right)
 \end{align}\end{subequations}">|; 

$key = q/X,Y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img68.png"
 ALT="$ X, Y$">|; 

$key = q/u;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img45.png"
 ALT="$ u$">|; 

$key = q/displaystyle{F_phi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img125.png"
 ALT="$\displaystyle {F_\phi}$">|; 

$key = q/displaystylef=2Omegasinphi=frac{4pi}{T_{rot}}sinphi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="203" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img34.png"
 ALT="$\displaystyle f = 2 \Omega \sin \phi = \frac{4 \pi}{T_{rot}} \sin \phi$">|; 

$key = q/p_{00};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img57.png"
 ALT="$ p_{00}$">|; 

$key = q/R;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img9.png"
 ALT="$ R$">|; 

$key = q/{subequations}align{<col_mark>DP{overline{u}}{t}+overline{v}^*left[Dinv{acosphi}}}+overline{w'theta'}right)right].align{{subequations};MSF=1.6;LFS=12;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="555" HEIGHT="316" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img200.png"
 ALT="\begin{subequations}\begin{align}&amp;
 \DP{\overline{u}}{t}
 + \overline{v}^*
 \lef...
...}{z^*}} + \overline{w'\theta'}
 \right)
 \right].
 \end{align}\end{subequations}">|; 

$key = q/displaystylerho_0acosphileft(DP{overline{u}}{overline{z^*}}frac{overline{v'theta'}}{DP{overline{theta}}{z^*}}-overline{u'v'}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="225" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img126.png"
 ALT="$\displaystyle \rho_0 a
\cos \phi \left(\DP{\overline{u}}{\overline{z^*}}
\frac{\overline{v'\theta'}}{\DP{\overline{\theta}}{z^*}} -
\overline{u'v'}\right)$">|; 

$key = q/displaystyle+Dinv{a}overline{v}DP{overline{u}}{phi}+overline{w}DP{overline{u}}{zine{v}-frac{tanphi}{a}overline{u}overline{v}-overline{X}notag;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="312" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img89.png"
 ALT="$\displaystyle + \Dinv{a}\overline{v}\DP{\overline{u}}{\phi}
 + \overline{w}\DP{...
...erline{v} 
 - \frac{\tan\phi}{a}\overline{u}\overline{v}
 - \overline{X} \notag$">|; 

$key = q/displaystyle=Dinv{rho_0a^2cos^2phi}DP{}{phi}left(cosphiF^{*}_{phi}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="222" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img166.png"
 ALT="$\displaystyle = \Dinv{\rho_0 a^2 \cos^2 \phi} 
 \DP{}{\phi}
 \left(
 \cos \phi F^{*}_{\phi}
 \right)$">|; 

$key = q/displaystyleDinv{acosphi}DP{}{phi}left(overline{v}^*cosphiright)+Dinv{rho_0}DP{}{z^*}left(rho_0overline{w}^*right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="298" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img131.png"
 ALT="$\displaystyle \Dinv{a \cos \phi}
 \DP{}{\phi}
 \left(
 \overline{v}^* \cos\phi
 \right)
 + \Dinv{\rho_0}
 \DP{}{z^*} 
 \left( \rho_0 \overline{w}^* \right)$">|; 

$key = q/hat{F}_phi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img18.png"
 ALT="$ \hat{F}_\phi$">|; 

$key = q/displaystyle-Dinv{rho_0}DP{}{z^*}left[rho_0left(frac{overline{v'theta'}}{aoverlirac{DP{overline{theta}}{z^*}}{overline{DP{theta}{z^*}}}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="416" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img186.png"
 ALT="$\displaystyle - \Dinv{\rho_0} \DP{}{z^*}
 \left[ \rho_0 
 \left(
 \frac{\overli...
... \frac{ \DP{\overline{\theta}}{z^*} }
 { \overline{\DP{\theta}{z^*}} }
 \right)$">|; 

$key = q/displaystyleqquad=-Dinv{rho_0a}DP{}{z^*}left(rho_0frac{overline{v'theta'}}{overlta'}}{overline{DP{theta}{z^*}}}right)DP{overline{theta}}{z^*};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="498" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img178.png"
 ALT="$\displaystyle \qquad 
 = - \Dinv{\rho_0 a} \DP{}{z^*}
 \left( \rho_0 
 \frac{\o...
...v'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \right) \DP{\overline{\theta}}{z^*}$">|; 

$key = q/w,theta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img60.png"
 ALT="$ w, \theta$">|; 

$key = q/displaystyle=-frac{2}{acosphi}overline{u'DP{u'}{lambda}}-Dinv{a}overline{v'DP{u'erline{u'DP{w'}{z^*}}-Dinv{rho_0}DP{rho_0}{z^*}overline{u'w'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="671" HEIGHT="69" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img90.png"
 ALT="$\displaystyle = - \frac{2}{a\cos\phi} \overline{u'\DP{u'}{\lambda}}
 - \Dinv{a}...
...
 - \overline{u'\DP{w'}{z^*}}
 - \Dinv{\rho_0} \DP{\rho_0}{z^*} \overline{u'w'}$">|; 

$key = q/{subequations}align{<col_mark>DP{}{t}(overline{u}+u')+frac{overline{u}+u'}{acospark><col_mark>qquad=overline{Q}+Q'align{{subequations};MSF=1.6;LFS=12;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="607" HEIGHT="501" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img82.png"
 ALT="\begin{subequations}\begin{align}
 &amp; \DP{}{t}(\overline{u} + u') 
 + \frac{\over...
... \theta')\notag\  
 &amp; \qquad = \overline{Q} + Q'
 \end{align}\end{subequations}">|; 

$key = q/displaystyleDP{overline{u}}{t};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img88.png"
 ALT="$\displaystyle \DP{\overline{u}}{t}$">|; 

$key = q/displaystylefrac{J}{C_p}e^{kappaz^*slashH};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img73.png"
 ALT="$\displaystyle \frac{J}{C_p}e^{\kappa z^*/H}$">|; 

$key = q/displaystyle=-Dinv{acos^2phi}DP{}{phi}(overline{v'u'}cos^2phi),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="229" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img94.png"
 ALT="$\displaystyle = - \Dinv{a\cos^2\phi}\DP{}{\phi}(\overline{v'u'}\cos^2\phi),$">|; 

$key = q/displaystyle(F_y,F_z^*)=arho_s(hat{F_y},hat{F_{z^*}});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="191" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img42.png"
 ALT="$\displaystyle (F_y, F_z^*) = a\rho_s(\hat{F_y}, \hat{F_{z^*}})$">|; 

$key = q/displaystyleDinv{acosphi}overline{v'DP{u'}{lambda}}+Dinv{a}overline{v'DP{v'}{philine{v'DP{w'}{z^*}}+Dinv{rho_0}DP{rho_0}{z^*}overline{v'w'}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="490" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img100.png"
 ALT="$\displaystyle \Dinv{a \cos \phi} \overline{ v' \DP{u'}{\lambda} }
+ \Dinv{a} \o...
...line{ v' \DP{w'}{z^*} }
+ \Dinv{\rho_0} \DP{\rho_0}{z^*} \overline{ v' w' }
= 0$">|; 

$key = q/displaystyleoverline{u}left(f+frac{tanphi}{a}overline{u}right)+Dinv{a}DP{overline{Phi}}{phi}=G.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="247" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img199.png"
 ALT="$\displaystyle \overline{u} 
 \left( f + \frac{\tan\phi}{a} \overline{u} \right) 
 + \Dinv{a} \DP{\overline{\Phi}}{\phi}
 = G.$">|; 

$key = q/hat{F_y},hat{F_z^*};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img41.png"
 ALT="$ \hat{F_y}, \hat{F_z^*}$">|; 

$key = q/displaystyle-Dinv{a}overline{v'DP{v'}{phi}}-Dinv{a}overline{v'DP{v'}{phi}}+frac{tanphi}{a}overline{v'^2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="247" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img106.png"
 ALT="$\displaystyle - \Dinv{a} \overline{ v' \DP{v'}{\phi} }
- \Dinv{a} \overline{ v' \DP{v'}{\phi} }
+ \frac{\tan \phi}{a} \overline{ v'^2 }$">|; 

$key = q/displaystyle-Dinv{acosphi}DP{}{phi}left(cosphioverline{v'^2}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="188" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img107.png"
 ALT="$\displaystyle - \Dinv{a \cos \phi}
\DP{}{\phi} \left( \cos \phi \overline{v'^2} \right)$">|; 

$key = q/displaystyleDinv{rho_0a^2cos^2phi}left[-DP{}{phi}(rho_0aoverline{v'u'}cos^2phi)+rac{overline{v'theta'}}{overline{DP{theta}{z^*}}}right)right];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="537" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img160.png"
 ALT="$\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]$">|; 

$key = q/rho_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img30.png"
 ALT="$ \rho_0$">|; 

$key = q/displaystyleDP{overline{u}}{t}+Dinv{a}overline{v}DP{overline{u}}{phi}+overline{wine{v}-frac{tanphi}{a}overline{u}overline{v}-overline{X}notag;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="353" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img97.png"
 ALT="$\displaystyle \DP{\overline{u}}{t}
 + \Dinv{a}\overline{v}\DP{\overline{u}}{\ph...
...e{v} 
 - \frac{\tan\phi}{a} \overline{u}  \overline{v}
 - \overline{X}
 \notag$">|; 

$key = q/displaystylePsi^*(theta,p)=frac{2piacosphi}{g}int_{0}^{p}overline{v}^*Ddp;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="243" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img55.png"
 ALT="$\displaystyle \Psi^*(\theta, p) = \frac{2\pi a \cos\phi}{g} \int_{0}^{p}\overline{v}^*\Dd p$">|; 

$key = q/displaystyleqquad+Dinv{rho_0acosphi}left[-cosphiDP{}{z^*}left(rho_0frac{overlinerac{overline{v'theta'}}{overline{DP{theta}{z^*}}}right)right];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="580" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img157.png"
 ALT="$\displaystyle \qquad
 + \Dinv{\rho_0 a \cos \phi} 
 \left[
 - \cos \phi
 \DP{}{...
... 
 \frac{\overline{v'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \right)
 \right]$">|; 

$key = q/displaystyleqquad=-Dinv{acosphi}DP{}{phi}(overline{v'theta'}cosphi)-Dinv{rho_0}DP{}{z^*}(rho_0overline{w'theta'}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="390" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img176.png"
 ALT="$\displaystyle \qquad 
 =
 - \Dinv{a\cos\phi}\DP{}{\phi}(\overline{v'\theta'}\cos\phi)
 - \Dinv{\rho_0}\DP{}{z^*}(\rho_0\overline{w'\theta'}),$">|; 

$key = q/displaystyleqquadqquad+frac{tanphi}{rho_0a}DP{}{z^*}left(overline{u}rho_0frac{ova'}}{overline{DP{theta}{z^*}}}DP{}{z^*}left(overline{u}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="443" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img151.png"
 ALT="$\displaystyle \qquad \qquad
 + \frac{\tan \phi}{\rho_0 a}
 \DP{}{z^*}
 \left( \...
...eta'}}
 {\overline{\DP{\theta}{z^*}}}
 \DP{}{z^*}
 \left( \overline{u}
 \right)$">|; 

$key = q/partialoverline{A}slashpartiallambda=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img81.png"
 ALT="$ \partial \overline{A}/\partial\lambda = 0$">|; 

$key = q/displaystyle-Dinv{acosphi}overline{DP{(u'theta')}{lambda}}=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="174" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img117.png"
 ALT="$\displaystyle - \Dinv{a\cos\phi}\overline{\DP{(u' \theta')}{\lambda}}
= 0,$">|; 

$key = q/{subequations}align{DP{overline{u}}{t}<col_mark>+Dinv{a}overline{v}DP{overline{uP{}{z^*}(rho_0overline{w'theta'}).align{{subequations};MSF=1.6;LFS=12;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="555" HEIGHT="393" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img123.png"
 ALT="\begin{subequations}\begin{align}
 \DP{\overline{u}}{t}
 &amp; + \Dinv{a}\overline{v...
...nv{\rho_0}\DP{}{z^*}(\rho_0\overline{w'\theta'}).
 \end{align}\end{subequations}">|; 

$key = q/displaystylerho_0acosphileft(left[f-frac{DP{overline{u}cosphi}{phi}}{acosphi}rigine{v'theta'}}{DP{overline{theta}}{z^*}}-overline{u'w'}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="312" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img128.png"
 ALT="$\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)$">|; 

$key = q/displaystyle+Dinv{a}left[overline{v}^*+Dinv{rho_0}DP{}{z^*}left(rho_0frac{overli'}}{overline{DP{theta}{z^*}}}right)right]DP{overline{u}}{z^*};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="574" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img138.png"
 ALT="$\displaystyle + \Dinv{a}
 \left[ 
 \overline{v}^* 
 + \Dinv{\rho_0} \DP{}{z^*}
...
...eta'}}
 {\overline{\DP{\theta}{z^*}}}
 \right)
 \right]
 \DP{\overline{u}}{z^*}$">|; 

$key = q/Q;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img71.png"
 ALT="$ Q$">|; 

$key = q/hat{F}_{z^*};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img19.png"
 ALT="$ \hat{F}_{z^*}$">|; 

$key = q/displaystyle+frac{overline{v}^*}{a}DP{overline{u}}{phi}+overline{w}^*DP{overlineline{v}^*-frac{tanphi}{a}overline{u}overline{v}^*-overline{X};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="339" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img141.png"
 ALT="$\displaystyle + \frac{\overline{v}^*}{a} \DP{\overline{u}}{\phi}
 + \overline{w...
...ine{v}^* 
 - \frac{\tan \phi}{a} \overline{u}  \overline{v}^* 
 - \overline{X}$">|; 

$key = q/displaystyle=Dinv{rho_0a^2cos^2phi}DP{}{phi}left[rho_0acos^2phileft{DP{overline{theta'}}{overline{DP{theta}{z^*}}}-overline{v'u'}right}right];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="393" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img165.png"
 ALT="$\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]$">|; 

$key = q/displaystyleqquad=-Dinv{acosphi}DP{}{phi}left(cosphioverline{v'^2}right)-frac{taline{u'^2}-Dinv{rho_0}DP{}{z^*}left(rho_0overline{v'w'}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="483" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img110.png"
 ALT="$\displaystyle \qquad 
 = - \Dinv{a \cos \phi} 
 \DP{}{\phi} \left( \cos \phi \o...
...line{u'^2}
 - \Dinv{\rho_0} \DP{}{z^*} \left( \rho_0 \overline{ v' w' } \right)$">|; 

$key = q/displaystyle(u,v,w);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img23.png"
 ALT="$\displaystyle (u, v, w)$">|; 

$key = q/{subequations}align{{F_phi}=<col_mark>rho_0acosphileft(DP{overline{u}}{overline{heta}}{z^*}}-overline{u'w'}right).align{{subequations};MSF=1.6;LFS=12;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="555" HEIGHT="128" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img37.png"
 ALT="\begin{subequations}\begin{align}
 {F_\phi} =&amp; \rho_0 a
 \cos \phi \left(\DP{\ov...
...overline{\theta}}{z^*}} - \overline{u'w'}\right).
 \end{align}\end{subequations}">|; 

$key = q/displaystyleqquad=-Dinv{acos^2phi}DP{}{phi}(overline{v'u'}cos^2phi)-Dinv{rho_0}DP{}{z^*}(rho_0overline{w'u'});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="403" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img98.png"
 ALT="$\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'})$">|; 

$key = q/displaystyleqquadqquad-fleft[overline{v}^*+Dinv{rho_0}DP{}{z^*}left(rho_0frac{ov{v'theta'}}{overline{DP{theta}{z^*}}}right)right]-overline{X};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="627" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img139.png"
 ALT="$\displaystyle \qquad \qquad
 - f 
 \left[ 
 \overline{v}^* 
 + \Dinv{\rho_0} \D...
...ne{v'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \right)
 \right]
 - \overline{X}$">|; 

$key = q/displaystyleqquadqquad+left[overline{w}^*-Dinv{acosphi}DP{}{phi}left(cosphifrac{rac{overline{v'theta'}}{overline{DP{theta}{z^*}}}right)right];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="582" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img191.png"
 ALT="$\displaystyle \qquad \qquad
 + \left[
 \overline{w}^* 
 - \Dinv{a \cos\phi} 
 \...
... 
 \frac{\overline{v'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \right)
 \right]$">|; 

$key = q/displaystylefrac{1}{rho_0acosphi}DP{}{z^*}left[frho_0acosphifrac{overline{v'thetrho_0frac{overline{v'theta'}}{overline{DP{theta}{z^*}}}right];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="775" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img167.png"
 ALT="$\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]$">|; 

$key = q/displaystyleqquadDinv{acosphi}DP{}{phi}left{Dinv{rho_0}DP{}{z^*}left(rho_0frac{orac{overline{v'theta'}}{overline{DP{theta}{z^*}}}right)right};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="648" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img133.png"
 ALT="$\displaystyle \qquad 
 \Dinv{a \cos \phi}
 \DP{}{\phi}
 \left\{ 
 \Dinv{\rho_0}...
...
 \frac{\overline{v'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \right)
 \right\}$">|; 

$key = q/displaystyleqquadqquad-Dinv{rho_0a}DP{}{z^*}left(rho_0frac{overline{v'theta'}}{orho_0frac{overline{v'theta'}}{overline{DP{theta}{z^*}}}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="485" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img148.png"
 ALT="$\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)$">|; 

$key = q/displaystyle-Hln(pslashp_s),H=frac{R_{d}T_s}{g_0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="224" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6.png"
 ALT="$\displaystyle -H \ln(p/p_s),    H = \frac{R_{d} T_s}{g_0}$">|; 

$key = q/T_{rot};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img36.png"
 ALT="$ T_{rot}$">|; 

$key = q/displaystyle=-omegaHslashp;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img62.png"
 ALT="$\displaystyle = -\omega H / p$">|; 

$key = q/displaystyleQ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img72.png"
 ALT="$\displaystyle Q$">|; 

$key = q/displaystyleDdiv{Dvect{F}}=Dinv{acosphi}DP{(cosphiF_{phi})}{phi}+DP{F_{z^{*}}}{z^*};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="279" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img174.png"
 ALT="$\displaystyle \Ddiv{\Dvect{F}} 
 = \Dinv{a \cos \phi } \DP{(\cos \phi F_{\phi})}{\phi} + \DP{F_{z^{*}}}{z^*}$">|; 

$key = q/displaystyle=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img136.png"
 ALT="$\displaystyle = 0.$">|; 

$key = q/displaystyleqquad+Dinv{acosphi}DP{}{phi}left{Dinv{rho_0}DP{}{z^*}left(rho_0frac{{overline{v'theta'}}{overline{DP{theta}{z^*}}}right)right}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="704" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img132.png"
 ALT="$\displaystyle \qquad
 + \Dinv{a \cos \phi}
 \DP{}{\phi}
 \left\{ 
 \Dinv{\rho_0...
...c{\overline{v'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \right)
 \right\}
 = 0.$">|; 

$key = q/displaystyle-Dinv{acosphi}overline{u'DP{v'}{lambda}}-Dinv{acosphi}overline{v'DP{u'}{lambda}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="245" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img104.png"
 ALT="$\displaystyle - \Dinv{a\cos\phi}\overline{u'\DP{v'}{\lambda}}
- \Dinv{a \cos \phi} \overline{v' \DP{u'}{\lambda}}$">|; 

$key = q/displaystyle-Dinv{acosphi}overline{DP{(u'v')}{lambda}}=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="175" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img105.png"
 ALT="$\displaystyle - \Dinv{a\cos\phi}\overline{\DP{(u' v')}{\lambda}}
= 0,$">|; 

$key = q/phi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2.png"
 ALT="$ \phi$">|; 

$key = q/displaystyleqquadqquad-Dinv{acosphi}overline{theta'DP{u'}{lambda}}-Dinv{a}overliheta'DP{w'}{z^*}}-Dinv{rho_0}DP{rho_0}{z^*}overline{theta'w'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="551" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img115.png"
 ALT="$\displaystyle \qquad \qquad
 - \Dinv{a \cos \phi} \overline{\theta' \DP{u'}{\la...
...theta' \DP{w'}{z^*} }
 - \Dinv{\rho_0} \DP{\rho_0}{z^*} \overline{ \theta' w' }$">|; 

$key = q/displaystyle=frac{1}{rho_0acosphi}DP{}{z^*}left[rho_0acosphileft{left(f-frac{DP{theta'}}{overline{DP{theta}{z^*}}}-overline{w'u'}right}right];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="481" HEIGHT="78" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img170.png"
 ALT="$\displaystyle = 
 \frac{1}{\rho_0 a \cos \phi} 
 \DP{}{z^*}
 \left[
 \rho_0 a \...
...'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 - \overline{w'u'}
 \right\}
 \right]$">|; 

$key = q/displaystyleqquadqquad-Dinv{rho_0a}DP{}{z^*}left(rho_0frac{overline{v'theta'}}{oline{DP{theta}{z^*}}}DP{}{z^*}left(DP{overline{u}}{phi}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="460" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img150.png"
 ALT="$\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)$">|; 

$key = q/u';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img86.png"
 ALT="$ u'$">|; 

$key = q/displaystyleDP{overline{v}}{t}+frac{overline{v}}{a}DP{overline{v}}{phi}+overlineoverline{u})^2+Dinv{a}DP{overline{Phi}}{phi}-overline{Y}notag;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="412" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img101.png"
 ALT="$\displaystyle \DP{\overline{v}}{t}
 + \frac{\overline{v}}{a} \DP{\overline{v}}{...
...(\overline{u})^2
 + \Dinv{a} \DP{\overline{\Phi}}{\phi}
 - \overline{Y}
 \notag$">|; 

$key = q/displaystyle-overline{w'DP{v'}{z^*}}-overline{v'DP{w'}{z^*}}-Dinv{rho_0}DP{rho_0}{z^*}overline{v'w'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="251" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img108.png"
 ALT="$\displaystyle - \overline{w'\DP{v'}{z^*}}
- \overline{ v' \DP{w'}{z^*} }
- \Dinv{\rho_0} \DP{\rho_0}{z^*} \overline{ v' w' }$">|; 

$key = q/displaystyleDinv{acosphi}overline{theta'DP{u'}{lambda}}+Dinv{a}overline{theta'DPta'DP{w'}{z^*}}+Dinv{rho_0}DP{rho_0}{z^*}overline{theta'w'}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="495" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img112.png"
 ALT="$\displaystyle \Dinv{a \cos \phi} \overline{\theta' \DP{u'}{\lambda}}
+ \Dinv{a}...
...ta' \DP{w'}{z^*} }
+ \Dinv{\rho_0} \DP{\rho_0}{z^*} \overline{ \theta' w' }
= 0$">|; 

$key = q/displaystyle-overline{w'DP{u'}{z^*}}-overline{u'DP{w'}{z^*}}-Dinv{rho_0}DP{rho_0}{z^*}overline{u'w'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="253" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img95.png"
 ALT="$\displaystyle - \overline{w'\DP{u'}{z^*}}
 - \overline{u'\DP{w'}{z^*}}
 - \Dinv{\rho_0} \DP{\rho_0}{z^*} \overline{u'w'}$">|; 

$key = q/displaystyleDP{overline{u}}{t}+overline{v}^*left[Dinv{acosphi}DP{}{phi}(overlineline{u}}{z^*}-overline{X}=Dinv{sigmacosphi}DdivDvect{hat{F}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="522" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img46.png"
 ALT="$\displaystyle \DP{\overline{u}}{t}
+ \overline{v}^*\left[\Dinv{a\cos\phi}\DP{}{...
...\overline{u}}{z^*}
- \overline{X} =
\Dinv{\sigma \cos\phi}\Ddiv\Dvect{\hat{F}}.$">|; 

$key = q/theta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img26.png"
 ALT="$ \theta$">|; 

$key = q/displaystyleqquad+frac{1}{rho_0acosphi}DP{}{z^*}left[frho_0acosphifrac{overline{}}{overline{DP{theta}{z^*}}}-rho_0acosphioverline{w'u'}right];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="427" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img158.png"
 ALT="$\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]$">|; 

$key = q/displaystyleDP{overline{theta}}{t}+frac{overline{v}}{a}DP{overline{theta}}{phi}+right)-Dinv{rho_0}DP{}{z^*}left(rho_0overline{w'theta'}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="558" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img122.png"
 ALT="$\displaystyle \DP{\overline{\theta}}{t}
 + \frac{\overline{v}}{a}\DP{\overline{...
...ight)
 - \Dinv{\rho_0} \DP{}{z^*} \left( \rho_0 \overline{ w' \theta' } \right)$">|; 

$key = q/displaystyle-overline{w'DP{theta'}{z^*}}-overline{theta'DP{w'}{z^*}}-Dinv{rho_0}DP{rho_0}{z^*}overline{theta'w'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="251" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img120.png"
 ALT="$\displaystyle - \overline{w'\DP{\theta'}{z^*}}
- \overline{ \theta' \DP{w'}{z^*} }
- \Dinv{\rho_0} \DP{\rho_0}{z^*} \overline{ \theta' w' }$">|; 

$key = q/displaystyleqquad-Dinv{acosphi}DP{}{phi}(overline{v'theta'}cosphi)-Dinv{rho_0}DP{}{z^*}(rho_0overline{w'theta'});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="365" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img181.png"
 ALT="$\displaystyle \qquad
 - \Dinv{a\cos\phi}\DP{}{\phi}(\overline{v'\theta'}\cos\phi)
 - \Dinv{\rho_0}\DP{}{z^*}(\rho_0\overline{w'\theta'})$">|; 

$key = q/displaystyle+frac{overline{v}^*}{acosphi}DP{}{phi}left(overline{u}cosphiright)+overline{w}^*DP{overline{u}}{z^*}-foverline{v}^*-overline{X};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="342" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img145.png"
 ALT="$\displaystyle + \frac{\overline{v}^*}{a \cos \phi} \DP{}{\phi} 
 \left( \overli...
...)
 + \overline{w}^* \DP{\overline{u}}{z^*}
 - f \overline{v}^* 
 - \overline{X}$">|; 

$key = q/displaystyleDdiv{}Dvect{F}=Dinv{acosphi}DP{(cosphiF_{phi})}{phi}+DP{F_{z^{*}}}{z^*};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="279" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img48.png"
 ALT="$\displaystyle \Ddiv{} \Dvect{F}= \Dinv{a \cos \phi} \DP{(\cos \phi F_{\phi})}{\phi} 
 + \DP{F_{z^{*}}}{z^*}$">|; 

$key = q/displaystyleequiv;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img24.png"
 ALT="$\displaystyle \equiv$">|; 

$key = q/displaystyleA'=A-overline{A};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img79.png"
 ALT="$\displaystyle A' = A - \overline{A}$">|; 

$key = q/{subequations}align{DD{u}{t}<col_mark>-left(f+frac{utanphi}{a}right)v+Dinv{acospgn{align{DD{theta}{t}<col_mark>=Q,align{{subequations};MSF=1.6;LFS=12;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="555" HEIGHT="339" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img66.png"
 ALT="\begin{subequations}\begin{align}
 \DD{u}{t} &amp;- \left(f + \frac{u\tan\phi}{a}\ri...
...nd{align}
 
 \begin{align}
 \DD{\theta}{t} &amp;= Q, 
 \end{align}\end{subequations}">|; 

$key = q/displaystyle=Dinv{rho_0a^2cos^2phi}DP{}{phi}left[-rho_0aoverline{v'u'}cos^2phi+r'theta'}}{overline{DP{theta}{z^*}}}DP{overline{u}}{z^*}right];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="451" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img164.png"
 ALT="$\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]$">|; 

$key = q/displaystyleDP{overline{u}}{t}+frac{overline{v}^*}{acosphi}DP{}{phi}left(overlin-foverline{v}^*-overline{X}=Dinv{rho_0acosphi}Ddiv{Dvect{F}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="518" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img173.png"
 ALT="$\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}}.$">|; 

$key = q/z^*;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3.png"
 ALT="$ z^*$">|; 

$key = q/displaystyle=Dinv{acosphi}left[DP{}{phi}left{Dinv{rho_0}DP{}{z^*}left(rho_0frac{erline{v'theta'}}{overline{DP{theta}{z^*}}}right)right}right];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="597" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img134.png"
 ALT="$\displaystyle = 
 \Dinv{a \cos \phi}
 \left[
 \DP{}{\phi}
 \left\{ 
 \Dinv{\rho...
...overline{v'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \right)
 \right\}
 \right]$">|; 

$key = q/J;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img74.png"
 ALT="$ J$">|; 

$key = q/displaystyle-Dinv{acosphi}overline{u'DP{theta'}{lambda}}-Dinv{acosphi}overline{theta'DP{u'}{lambda}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="244" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img116.png"
 ALT="$\displaystyle - \Dinv{a \cos \phi}\overline{u' \DP{\theta'}{\lambda}}
- \Dinv{a \cos \phi} \overline{\theta' \DP{u'}{\lambda}}$">|; 

$key = q/g_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img13.png"
 ALT="$ g_0$">|; 

$key = q/displaystyle=Dinv{rho_0a^2cos^2phi}left[-DP{}{phi}(rho_0aoverline{v'u'}cos^2phi)'}}{overline{DP{theta}{z^*}}}DP{overline{u}}{z^*}right)right];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="630" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img163.png"
 ALT="$\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]$">|; 

$key = q/F_y,F_z^*;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img40.png"
 ALT="$ F_y, F_z^*$">|; 

$key = q/displaystyle=-Dinv{rho_0}DP{}{z^*}(rho_0overline{w'u'});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img96.png"
 ALT="$\displaystyle = - \Dinv{\rho_0}\DP{}{z^*}(\rho_0\overline{w'u'})$">|; 

$key = q/displaystyleDP{overline{theta}}{t}+frac{overline{v}^*}{a}DP{overline{theta}}{phi^*}}}DP{overline{theta}}{phi}+overline{w'theta'}right)right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="515" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img188.png"
 ALT="$\displaystyle \DP{\overline{\theta}}{t}
 + \frac{\overline{v}^*}{a} \DP{\overli...
...^*}}}
 \DP{\overline{\theta}}{\phi}
 + \overline{w'\theta'}
 \right) 
 \right].$">|; 

$key = q/theta';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img111.png"
 ALT="$ \theta'$">|; 

$key = q/displaystyleDinv{acosphi}DP{}{phi}left[left{overline{v}^*+Dinv{rho_0}DP{}{z^*}le{v'theta'}}{overline{DP{theta}{z^*}}}right)right}cosphiright];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="362" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img129.png"
 ALT="$\displaystyle \Dinv{a \cos \phi}
 \DP{}{\phi}\left[
 \left\{ 
 \overline{v}^* 
...
...v'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \right)
 \right\}
 \cos\phi \right]$">|; 

$key = q/displaystyle-Dinv{rho_0}DP{}{z^*}left(rho_0overline{v'w'}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="144" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img109.png"
 ALT="$\displaystyle - \Dinv{\rho_0} \DP{}{z^*} \left( \rho_0 \overline{ v' w' } \right)$">|; 

$key = q/a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img27.png"
 ALT="$ a$">|; 

$key = q/displaystyle=frac{1}{rho_0acosphi}DP{}{z^*}left[rho_0acosphileft{ffrac{overline{}{acosphioverline{DP{theta}{z^*}}}-overline{w'u'}right}right];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="626" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img169.png"
 ALT="$\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]$">|; 

$key = q/displaystyleqquadqquad+foverline{u}+frac{tanphi}{a}(overline{u})^2+Dinv{a}DP{overline{Phi}}{phi}-overline{Y};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="321" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img192.png"
 ALT="$\displaystyle \qquad \qquad
 + f \overline{u}
 + \frac{\tan\phi}{a} (\overline{u})^2
 + \Dinv{a} \DP{\overline{\Phi}}{\phi}
 - \overline{Y}$">|; 

$key = q/displaystyleqquad=-Dinv{acosphi}DP{}{phi}(overline{v'^2}cosphi)-Dinv{rho_0}DP{}{z^*}(rho_0overline{v'w'})-overline{u'^2}frac{tanphi}{a},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="474" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img193.png"
 ALT="$\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},$">|; 

$key = q/displaystyle=-Dinv{acosphi}overline{DP{(u')^2}{lambda}}=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="188" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img92.png"
 ALT="$\displaystyle = - \Dinv{a\cos\phi}\overline{\DP{(u')^2}{\lambda}}
 = 0,$">|; 

$key = q/displaystyle-Dinv{rho_0}DP{}{z^*}left(rho_0frac{overline{v'theta'}}{aoverline{DPta'}}{overline{DP{theta}{z^*}}}right)DP{overline{theta}}{z^*};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="439" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img180.png"
 ALT="$\displaystyle - \Dinv{\rho_0} \DP{}{z^*}
 \left( \rho_0 
 \frac{\overline{v'\th...
...v'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \right) \DP{\overline{\theta}}{z^*}$">|; 

$key = q/displaystyleleft(acosphiDD{lambda}{t},aDD{phi}{t},DD{z^*}{t}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="194" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img25.png"
 ALT="$\displaystyle \left(a\cos\phi\DD{\lambda}{t}, a\DD{\phi}{t}, \DD{z^*}{t}\right)$">|; 

$key = q/overline{bullet};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img20.png"
 ALT="$ \overline{\bullet}$">|; 

$key = q/displaystyle=Tleft(frac{p_{00}}{p}right)^kappa,kappa=RslashC_p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="198" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img64.png"
 ALT="$\displaystyle = T \left(\frac{p_{00}}{p}\right)^\kappa, \kappa = R/C_p$">|; 

$key = q/displaystyleDinv{acosphi}left[DP{u'}{lambda}+DP{}{phi}(v'cosphi)right]+Dinv{rho_0}DP{}{z^*}(rho_0w')=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="397" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img85.png"
 ALT="$\displaystyle \Dinv{a\cos\phi}\left[\DP{u'}{\lambda}
+ \DP{}{\phi}(v'\cos\phi)\right]
+ \Dinv{\rho_0}\DP{}{z^*}(\rho_0 w')
= 0$">|; 

$key = q/displaystyle=Dinv{acosphi}left[Dinv{rho_0}DP{}{phi}left{DP{}{z^*}left(rho_0frac{erline{v'theta'}}{overline{DP{theta}{z^*}}}right)right}right];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="601" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img135.png"
 ALT="$\displaystyle = 
 \Dinv{a \cos \phi}
 \left[
 \Dinv{\rho_0}
 \DP{}{\phi}
 \left...
...overline{v'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \right)
 \right\}
 \right]$">|; 

$key = q/displaystyleoverline{A}(phi,z^*,t)equivDinv{2pi}int_0^{2pi}A(lambda,phi,z^*,t)Ddlambda;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="299" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img77.png"
 ALT="$\displaystyle \overline{A}(\phi, z^*, t) \equiv \Dinv{2\pi}\int_0^{2\pi} A(\lambda, \phi, z^*, t) \Dd \lambda$">|; 

$key = q/Phi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img67.png"
 ALT="$ \Phi$">|; 

$key = q/bullet';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img21.png"
 ALT="$ \bullet'$">|; 

$key = q/{subequations}align{<col_mark>DP{overline{u}}{t}+Dinv{a}overline{v}DP{overline{u{phi}}-overline{w'DP{theta'}{z^*}}align{{subequations};MSF=1.6;LFS=12;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="555" HEIGHT="390" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img84.png"
 ALT="\begin{subequations}\begin{align}
 &amp; \DP{\overline{u}}{t}
 + \Dinv{a}\overline{v...
...\theta'}{\phi}}
 - \overline{w'\DP{\theta'}{z^*}}
 \end{align}\end{subequations}">|; 

$key = q/displaystyleqquad=-DP{}{t}left[overline{v}^*+Dinv{rho_0}DP{}{z^*}left(rho_0frac{rac{overline{v'theta'}}{overline{DP{theta}{z^*}}}right)right];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="764" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img195.png"
 ALT="$\displaystyle \qquad
 = - \DP{}{t}
 \left[ 
 \overline{v}^* 
 + \Dinv{\rho_0} \...
... 
 \frac{\overline{v'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \right)
 \right]$">|; 

$key = q/displaystyle=-frac{p}{H}DP{}{p}Psi^*;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img52.png"
 ALT="$\displaystyle = -\frac{p}{H}\DP{}{p}\Psi^*$">|; 

$key = q/displaystyle-Dinv{rho_0}DP{}{z^*}left[rho_0left(frac{overline{v'theta'}}{aoverli^*}}}DP{overline{theta}}{phi}+overline{w'theta'}right)right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="279" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img187.png"
 ALT="$\displaystyle - \Dinv{\rho_0} \DP{}{z^*}
 \left[ \rho_0 
 \left(
 \frac{\overli...
...^*}}}
 \DP{\overline{\theta}}{\phi}
 + \overline{w'\theta'}
 \right) 
 \right].$">|; 

$key = q/p_s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img15.png"
 ALT="$ p_s$">|; 

$key = q/displaystyleqquadqquad-left[overline{w}^*-Dinv{acosphi}DP{}{phi}left(cosphifrac{rac{overline{v'theta'}}{overline{DP{theta}{z^*}}}right)right];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="582" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img196.png"
 ALT="$\displaystyle \qquad \qquad
 - \left[
 \overline{w}^* 
 - \Dinv{a \cos\phi} 
 \...
... 
 \frac{\overline{v'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \right)
 \right]$">|; 

$key = q/c_p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img70.png"
 ALT="$ c_p$">|; 

$key = q/displaystyleDP{overline{theta}}{t}+frac{overline{v}^*}{a}DP{overline{theta}}{phi}+overline{w}^*DP{overline{theta}}{z^*}-overline{Q};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="215" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img177.png"
 ALT="$\displaystyle \DP{\overline{\theta}}{t}
 + \frac{\overline{v}^*}{a} \DP{\overline{\theta}}{\phi}
 + \overline{w}^* \DP{\overline{\theta}}{z^*}
 - \overline{Q}$">|; 

$key = q/displaystylefoverline{u}+frac{tanphi}{a}(overline{u})^2+Dinv{a}DP{overline{Phi}}{phi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="193" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img194.png"
 ALT="$\displaystyle f \overline{u}
 + \frac{\tan\phi}{a} (\overline{u})^2
 + \Dinv{a} \DP{\overline{\Phi}}{\phi}$">|; 

$key = q/Omega;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img35.png"
 ALT="$ \Omega$">|; 

$key = q/f;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img33.png"
 ALT="$ f$">|; 

$key = q/displaystyleqquad=-Dinv{acos^2phi}DP{}{phi}(overline{v'u'}cos^2phi)+Dinv{acosphi'theta'}}{overline{DP{theta}{z^*}}}right)DP{overline{u}}{z^*};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="507" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img142.png"
 ALT="$\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^*}$">|; 

$key = q/overline{A'}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img80.png"
 ALT="$ \overline{A'}=0$">|; 

$key = q/displaystyle-Dinv{rho_0}DP{}{z^*}left(rho_0frac{overline{v'theta'}}{aoverline{DPne{theta}}{z^*}-Dinv{rho_0}DP{}{z^*}(rho_0overline{w'theta'});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="658" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img184.png"
 ALT="$\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'})$">|; 

$key = q/v';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img99.png"
 ALT="$ v'$">|; 

$key = q/displaystyleqquad+Dinv{acosphi}left[DP{}{phi}left(cosphioverline{v'theta'}right)line{DP{theta}{z^*}}right)^{-1}right]DP{overline{theta}}{z^*};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="520" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img183.png"
 ALT="$\displaystyle \qquad 
 + \Dinv{a \cos\phi} 
 \left[ 
 \DP{}{\phi} \left( \cos \...
...( \overline{\DP{\theta}{z^*}} \right)^{-1}
 \right] \DP{\overline{\theta}}{z^*}$">|; 

$key = q/displaystyle-Dinv{rho_0}DP{}{z^*}left[rho_0frac{overline{v'theta'}}{aoverline{DPline{DP{theta}{z^*}}right)^{-1}DP{overline{theta}}{z^*}right];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="585" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img185.png"
 ALT="$\displaystyle - \Dinv{\rho_0} \DP{}{z^*}
 \left[ \rho_0 
 \frac{\overline{v'\th...
... \overline{\DP{\theta}{z^*}} \right)^{-1}
 \DP{\overline{\theta}}{z^*}
 \right]$">|; 

$key = q/displaystyle=Dinv{rho_0a^2cos^2phi}left[-DP{}{phi}(rho_0aoverline{v'u'}cos^2phi)right];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="299" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img161.png"
 ALT="$\displaystyle =
 \Dinv{\rho_0 a^2 \cos^2 \phi} 
 \left[
 - \DP{}{\phi} (\rho_0 a \overline{v'u'} \cos^2 \phi)
 \right]$">|; 

$key = q/displaystyle{F_z^*};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img127.png"
 ALT="$\displaystyle {F_z^*}$">|; 

$key = q/u,v,w;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img22.png"
 ALT="$ u, v, w$">|; 

$key = q/displaystyleqquad-Dinv{rho_0a}DP{}{z^*}left(rho_0frac{overline{v'theta'}}{overlirho_0frac{overline{v'theta'}}{overline{DP{theta}{z^*}}}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="423" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img155.png"
 ALT="$\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)$">|; 

$key = q/Dvect{F};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img47.png"
 ALT="$ \Dvect{F}$">|; 

$key = q/{subequations}align{overline{v}^*<col_mark>=overline{v}-Dinv{rho_0}DP{}{z^*}left}}{overline{DP{theta}{z^*}}}right)align{{subequations};MSF=1.6;LFS=12;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="555" HEIGHT="129" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img124.png"
 ALT="\begin{subequations}\begin{align}
 \overline{v}^* 
 &amp; =
 \overline{v} 
 - \Dinv{...
...\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \right)
 \end{align}\end{subequations}">|; 

$key = q/T_s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img12.png"
 ALT="$ T_s$">|; 

$key = q/(0,overline{v}^*,overline{w}^*);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img43.png"
 ALT="$ (0, \overline{v}^*, \overline{w}^*)$">|; 

$key = q/rho_s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img16.png"
 ALT="$ \rho_s$">|; 

$key = q/displaystyleqquadqquad-Dinv{acosphi}DP{}{phi}(overline{v'^2}cosphi)-Dinv{rho_0}Dho_0overline{v'w'})-overline{u'^2}frac{tanphi}{a}+overline{Y};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="525" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img197.png"
 ALT="$\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}$">|; 

$key = q/{subequations}align{z^*<col_mark>=-Hlogleft(frac{p}{p_{00}}right),<row_mark>p<co=p_{00}expleft(-frac{z^*}{H}right)align{{subequations};MSF=1.6;LFS=12;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="555" HEIGHT="105" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img56.png"
 ALT="\begin{subequations}\begin{align}
 z^* &amp;= -H \log \left( \frac{p}{p_{00}} \right...
... 
 p &amp;= p_{00} \exp \left( -\frac{z^*}{H} \right)
 \end{align}\end{subequations}">|; 

$key = q/displaystyleqquad+Dinv{rho_0acosphi}DP{}{z^*}left[-rho_0cosphifrac{overline{v'thrho_0frac{overline{v'theta'}}{overline{DP{theta}{z^*}}}right];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="441" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img159.png"
 ALT="$\displaystyle \qquad
 + \Dinv{\rho_0 a \cos \phi} 
 \DP{}{z^*}
 \left[
 - \rho_...
...u} \rho_0 
 \frac{\overline{v'\theta'}}
 {\overline{\DP{\theta}{z^*}}}
 \right]$">|; 

1;

