Editing Apps/PapaloizouPringle84 (section)

===PP84===

Again following the lead of [http://adsabs.harvard.edu/abs/1984MNRAS.208..721P PP84], we let <math>~W^'</math> represent the (normalized) perturbation in the fluid entropy, specifically,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~W^' </math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{P^'}{\rho_0(\sigma + m{\dot\varphi}_0)} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~\frac{\partial}{\partial\varpi}\biggl(\frac{P^'}{\rho_0} \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\partial}{\partial\varpi} \biggl[ W^'(\sigma + m{\dot\varphi}_0 )\biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(\sigma + m{\dot\varphi}_0 )\frac{\partial W^'}{\partial\varpi}  + mW^'\frac{\partial  {\dot\varphi}_0 }{\partial\varpi} </math>
  </td>
</tr>
</table>
</div>
in which case the three linearized components of the Euler equation may be rewritten as,

<div align="center">
<table border="0" cellpadding="8" align="center">

<tr><td align="center" colspan="3"><font color="#770000">''' ''Linearized'' <math>\varpi</math> Component of Euler Equation'''</font></td></tr>
<tr>
  <td align="right">
<math>~{\dot\varpi}^'
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
i \biggl[ \frac{\partial W^'}{\partial\varpi}  + \frac{mW^'}{(\sigma + m{\dot\varphi}_0)}\frac{\partial  {\dot\varphi}_0 }{\partial\varpi}  
- \frac{2{\dot\varphi}_0 (\varpi {\dot\varphi}^' )}{(\sigma + m{\dot\varphi}_0)}   \biggr]
</math>
  </td>
</tr>

<tr><td align="center" colspan="3"><font color="#770000">''' ''Linearized'' <math>\varphi</math> Component of Euler Equation'''</font></td></tr>
<tr>
  <td align="right">
<math>~(\varpi {\dot\varphi}^')  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-  \frac{ mW^'}{\varpi}  + i~ \frac{{\dot\varpi}^'}{\varpi(\sigma + m{\dot\varphi}_0)}\biggl[ \frac{\partial (\varpi^2\dot\varphi_0)}{\partial\varpi} \biggr]  \, ;
</math>
  </td>
</tr>

<tr><td align="center" colspan="3"><font color="#770000">''' ''Linearized'' <math>~z</math> Component of Euler Equation'''</font></td></tr>
<tr>
  <td align="right">
<math>~
~{\dot{z}}^'  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
i~\frac{\partial W^'}{\partial z} \, . 
</math>
  </td>
</tr>
</table>

</div>

Using the second of these three relations to provide an expression for <math>~(\varpi {\dot\varphi}^')</math> in terms of <math>~W^'</math> and <math>~{\dot\varpi}^'</math>, and plugging this expression into the first relation allows us to solve for the radial component of the perturbed velocity in terms of <math>~W^'</math> and its radial derivative.  Specifically, we obtain,

<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
  <td align="right">
<math>~{\dot\varpi}^' 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~i \frac{\partial W^'}{\partial \varpi} + i~\frac{mW^'}{(\sigma + m{\dot\varphi}_0)} \biggl[ \frac{\kappa^2}{2\varpi {\dot\varphi}_0} - \frac{2 {\dot\varphi}_0 }{\varpi}\biggr]
- i~ \frac{2 {\dot\varphi}_0 }{(\sigma + m{\dot\varphi}_0)} 
\biggl[ -  \frac{ mW^'}{\varpi}  + i~ \frac{{\dot\varpi}^'}{\varpi(\sigma + m{\dot\varphi}_0)}\biggl( \frac{ \kappa^2 \varpi }{ 2{\dot\varphi}_0 } \biggr) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~i \frac{\partial W^'}{\partial \varpi} 
+ i~\frac{mW^'}{(\sigma + m{\dot\varphi}_0)} \biggl[ \frac{\kappa^2}{2\varpi {\dot\varphi}_0} \biggr]  
+ \biggl[  \frac{2 {\dot\varphi}_0 }{(\sigma + m{\dot\varphi}_0)} \biggr]\biggl[ \frac{{\dot\varpi}^'}{\varpi(\sigma + m{\dot\varphi}_0)}\biggl( \frac{ \kappa^2 \varpi }{ 2{\dot\varphi}_0 } \biggr) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~i \biggl[ \frac{\partial W^'}{\partial \varpi} 
+\biggl( \frac{\kappa^2}{2\varpi {\dot\varphi}_0} \biggr) \frac{ mW^'}{\bar\sigma}  \biggr]
+ \biggl[ {\dot\varpi}^'\biggl( \frac{ \kappa^2 }{  {\bar\sigma}^2 } \biggr) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ {\dot\varpi}^' ({\bar\sigma}^2 - \kappa^2 )</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~i \biggl[ {\bar\sigma}^2~\frac{\partial W^'}{\partial \varpi} 
+\biggl( \frac{\kappa^2}{2\varpi {\dot\varphi}_0}  \biggr) mW^' \bar\sigma   \biggr] \, ,
</math>
  </td>
</tr>
</table>
</div>

where, adopting notation from PP84,

<div align="center" id="epicyclic">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\kappa^2 \equiv \frac{2{\dot\varphi}_0}{\varpi} \biggl[ \frac{\partial (\varpi^2\dot\varphi_0)}{\partial\varpi} \biggr]</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~{\bar\sigma} \equiv (\sigma + m{\dot\varphi}_0) \, .</math>
  </td>
</tr>
</table>
</div>

This means, as well, that,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(\varpi {\dot\varphi}^')  ({\bar\sigma}^2 - \kappa^2 )
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-  \frac{ mW^'}{\varpi} ({\bar\sigma}^2 - \kappa^2 ) 
- \frac{ 1 }{\varpi \bar\sigma }\biggl[ \frac{\kappa^2 \varpi }{ 2{\dot\varphi}_0 }  \biggr]  
\biggl[ {\bar\sigma}^2~\frac{\partial W^'}{\partial \varpi} 
+\biggl( \frac{2 {\dot\varphi}_0}{\varpi} + \frac{\partial {\dot\varphi}_0}{\partial\varpi} \biggr) mW^' \bar\sigma   \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-  \frac{ m{\bar\sigma}^2 W^'}{\varpi}  +  \frac{ m\kappa^2W^'}{\varpi} 
- \frac{\kappa^2 {\bar\sigma} }{ 2{\dot\varphi}_0 }   \biggl[ ~\frac{\partial W^'}{\partial \varpi} 
+\biggl( \frac{2 {\dot\varphi}_0}{\varpi} + \frac{\partial {\dot\varphi}_0}{\partial\varpi} \biggr) \frac{mW^' }{\bar\sigma }  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-  \frac{ m{\bar\sigma}^2 W^'}{\varpi}  - \frac{\kappa^2 {\bar\sigma} }{ 2{\dot\varphi}_0 }   \biggl[ ~\frac{\partial W^'}{\partial \varpi} 
+\biggl(\frac{\partial {\dot\varphi}_0}{\partial\varpi} \biggr) \frac{mW^' }{\bar\sigma }  \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>


In summary, the three components of the perturbed velocity are:


<table border="1" cellpadding="8" align="center">
<tr>
  <th align="center">
Cylindrical-Coordinate Components of the Perturbed Velocity from PP84
  </th>
</tr>
<tr><td>

<table border="0" cellpadding="8" align="center">

<tr><td align="center" colspan="3"><font color="#770000">'''<math>\varpi</math> Component'''</font></td></tr>
<tr>
  <td align="right">
<math>~ {\dot\varpi}^' ({\bar\sigma}^2 - \kappa^2 )</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~i \biggl[ {\bar\sigma}^2~\frac{\partial W^'}{\partial \varpi} 
+\biggl( \frac{\kappa^2}{2\varpi {\dot\varphi}_0}  \biggr) mW^' \bar\sigma   \biggr] \, ,
</math>
  </td>
</tr>

<tr><td align="center" colspan="3"><font color="#770000">'''<math>\varphi</math> Component'''</font></td></tr>
<tr>
  <td align="right">
<math>~(\varpi {\dot\varphi}^')  ({\bar\sigma}^2 - \kappa^2 )
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-  \frac{ m{\bar\sigma}^2 W^'}{\varpi}  - \frac{\kappa^2 {\bar\sigma} }{ 2{\dot\varphi}_0 }   \biggl[ ~\frac{\partial W^'}{\partial \varpi} 
+\biggl(\frac{\partial {\dot\varphi}_0}{\partial\varpi} \biggr) \frac{mW^' }{\bar\sigma }  \biggr] \, .
</math>
  </td>
</tr>

<tr><td align="center" colspan="3"><font color="#770000">'''<math>~z</math> Component'''</font></td></tr>
<tr>
  <td align="right">
<math>~
~{\dot{z}}^'  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
i~\frac{\partial W^'}{\partial z} \, . 
</math>
  </td>
</tr>
</table>

where, the square of the epicyclic frequency,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\kappa^2 \equiv \frac{2{\dot\varphi}_0}{\varpi} \biggl[ \frac{\partial (\varpi^2\dot\varphi_0)}{\partial\varpi} \biggr]</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~{\bar\sigma} \equiv (\sigma + m{\dot\varphi}_0) </math>
  </td>
</tr>
</table>

</td></tr>
</table>


These three velocity-component expressions  match, respectively, equations (3.14), (3.15), and (3.16) of [http://adsabs.harvard.edu/abs/1984MNRAS.208..721P PP84].