Editing Apps/PapaloizouPringle84 (section)

==Linearized Principal Governing Equations in Cylindrical Coordinates==
We begin by drawing [[Cylindrical3D/Linearization#Summary|from an accompanying derivation]] the relevant set of linearized principal governing equations, written in cylindrical coordinates but, following the lead of [http://adsabs.harvard.edu/abs/1984MNRAS.208..721P Papaloizou &amp; Pringle] (1984, MNRAS, 208, 721-750; hereafter, PP84), express each perturbation in the form,
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<math>~q^'~~\rightarrow~~ q^' (\varpi,z) f_\sigma</math>
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&nbsp; &nbsp; &nbsp; &nbsp; where, &nbsp; &nbsp; &nbsp; &nbsp; 
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<math>~f_\sigma \equiv e^{i(m\varphi + \sigma t)} \, ,</math>
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and, set <math>~\Phi^' = 0</math> &#8212; hence, the Poisson equation becomes irrelevant &#8212; because the torus is assumed not to be self-gravitating and the background (point source) potential, <math>~\Phi_0</math>, is assumed to be unchanging.

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Set of Linearized Principal Governing Equations in Cylindrical Coordinates
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<tr><td align="center" colspan="3"><font color="#770000">'''Continuity Equation'''</font></td></tr>
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<math>~\frac{\partial (\rho^' f_\sigma) }{\partial t} + ( {\dot\varphi}_0 )\frac{\partial (\rho^' f_\sigma)}{\partial \varphi} 
</math>
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<math>~=</math>
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<math>~
- \frac{f_\sigma}{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \rho_0 \varpi {\dot\varpi}^' \biggr] 
- \frac{1}{\varpi} \frac{\partial }{\partial \varphi} \biggl[ \rho_0 \varpi {\dot\varphi}^' f_\sigma\biggr]
- f_\sigma\frac{\partial}{\partial z} \biggl[ \rho_0 {\dot{z}}^' \biggr] 
</math>
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<tr><td align="center" colspan="3"><font color="#770000">'''<math>\varpi</math> Component of Euler Equation'''</font></td></tr>
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<math>~
\frac{\partial ({\dot\varpi}^'f_\sigma) }{\partial t} + 
( {\dot\varphi}_0 ) \frac{\partial ( {\dot\varpi}^'f_\sigma)}{\partial\varphi} 
-  2\varpi ( {\dot\varphi}_0 {\dot\varphi}^' f_\sigma)
</math>
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<math>~=</math>
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<math>~
-  f_\sigma\frac{\partial}{\partial\varpi}\biggl( \frac{P^'}{\rho_0} \biggr)  
</math>
  </td>
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<tr><td align="center" colspan="3"><font color="#770000">'''<math>\varphi</math> Component of Euler Equation'''</font></td></tr>
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<math>~\frac{\partial (\varpi {\dot\varphi}^' f_\sigma)}{\partial t} + 
( \dot\varphi_0)\frac{\partial (\varpi{\dot\varphi}^' f_\sigma)}{\partial\varphi} +
\frac{{\dot\varpi}^' f_\sigma}{\varpi}\biggl[ \frac{\partial (\varpi^2\dot\varphi_0)}{\partial\varpi} \biggr] 
</math>
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<math>~=</math>
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<math>~-  \frac{ 1}{\varpi} \biggl[ \frac{\partial }{\partial \varphi} \biggl(\frac{P^'f_\sigma}{\rho_0}\biggr) \biggr] 
</math>
  </td>
</tr>

<tr><td align="center" colspan="3"><font color="#770000">'''<math>~z</math> Component of Euler Equation'''</font></td></tr>
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<math>~
\frac{\partial ({\dot{z}}^' f_\sigma)}{\partial t} 
+ (\dot\varphi_0) \frac{\partial ({\dot{z}}^' f_\sigma)}{\partial\varphi}  
</math>
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<math>~=</math>
  </td>
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<math>~
-  f_\sigma \frac{\partial}{\partial z}\biggl( \frac{P^'}{\rho_0} \biggr) 
</math>
  </td>
</tr>

<tr><td align="center" colspan="3"><font color="#770000">'''Adiabatic Form of the 1<sup>st</sup> Law of Thermodynamics'''</font></td></tr>
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<math>~\frac{P^' f_\sigma}{P_0}</math>
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<math>~=</math> 
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<math>~ \frac{\gamma (\rho^' f_\sigma)}{\rho_0} </math>
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Next, taking derivatives of <math>~f_\sigma</math>, where indicated, then dividing every equation through by <math>~f_\sigma</math> gives,

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<tr><td align="center" colspan="3"><font color="#770000">''' ''Linearized'' Adiabatic Form of the 1<sup>st</sup> Law of Thermodynamics'''</font></td></tr>
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<math>~\frac{P^' }{P_0}</math>
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<math>~=</math> 
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<math>~ \frac{\gamma \rho^' }{\rho_0} \, ;</math>
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<tr><td align="center" colspan="3"><font color="#770000">''' ''Linearized'' <math>\varpi</math> Component of Euler Equation'''</font></td></tr>
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<math>~{\dot\varpi}^'[i(\sigma + m{\dot\varphi}_0)] 
-  2 {\dot\varphi}_0 (\varpi {\dot\varphi}^' )
</math>
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<math>~=</math>
  </td>
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<math>~
-  \frac{\partial}{\partial\varpi}\biggl( \frac{P^'}{\rho_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>
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<math>~(\varpi {\dot\varphi}^')[i(\sigma + m{\dot\varphi}_0)]  
+ \frac{{\dot\varpi}^'}{\varpi}\biggl[ \frac{\partial (\varpi^2\dot\varphi_0)}{\partial\varpi} \biggr] 
</math>
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<math>~=</math>
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<math>~-  \frac{ im}{\varpi} \biggl(\frac{P^'}{\rho_0}\biggr)   \, ;
</math>
  </td>
</tr>

<tr><td align="center" colspan="3"><font color="#770000">''' ''Linearized'' <math>~z</math> Component of Euler Equation'''</font></td></tr>
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<math>~
~{\dot{z}}^'[i(\sigma + m{\dot\varphi}_0)]  
</math>
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<math>~=</math>
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<math>~
-  \frac{\partial}{\partial z}\biggl( \frac{P^'}{\rho_0} \biggr)  \, ;
</math>
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</tr>

<tr><td align="center" colspan="3"><font color="#770000">''' ''Linearized'' Continuity Equation'''</font></td></tr>
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<math>~\rho^'[i(\sigma + m{\dot\varphi}_0)]  + i m\rho_0  {\dot\varphi}^' 
</math>
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<math>~=</math>
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<math>~
- \frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \rho_0 \varpi {\dot\varpi}^' \biggr] 
- \frac{\partial}{\partial z} \biggl[ \rho_0 {\dot{z}}^' \biggr] \, .
</math>
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These five equations match, respectively, equations (3.8) - (3.12) of [http://adsabs.harvard.edu/abs/1984MNRAS.208..721P PP84].