Editing Apps/PapaloizouPringle84 (section)

====Equivalent Dimensionless Expression====

Now, as should be clear from [[SR#Barotropic_Structure|our introductory description of barotropic structures]], because the initial, unperturbed [[Apps/PapaloizouPringleTori#Solution|Papaloizou-Pringle torus]] is a polytropic configuration, the functions <math>~P_0(\varpi,z)</math> and <math>~\rho_0(\varpi,z)</math> can both be expressed in terms of the [[Appendix/Ramblings/PPToriPt1A#Equilibrium_Configuration|dimensionless enthalpy distribution]],

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<math>~\Theta_H(\varpi,z) </math>
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<math>~=</math>
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<math>~1 - \frac{1}{\beta^2}\biggl[ \chi^{-2} 
- 2 ( \chi^2 + \zeta^2 )^{-1/2} + 1 \biggr] \, ,</math>
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where,

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<math>~\chi \equiv \frac{\varpi}{\varpi_0} \, ,</math>
&nbsp; &nbsp; &nbsp; &nbsp; 
<math>~\zeta \equiv \frac{z}{\varpi_0} \, ,</math>
&nbsp; &nbsp; &nbsp; &nbsp;
<math>~\beta^2 \equiv \frac{2n}{\mathfrak{M}_0^2} \, ,</math>
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<math>~</math>
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<math>~</math>
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and <math>~\mathfrak{M}_0</math> is the Mach number of the circular, azimuthal flow at the pressure and density maximum.  Specifically (see also equation 1.1 of [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes85]),

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<math>~P_0 = P_\mathrm{max} \Theta_H^{n+1} </math> 
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; 
<math>~\rho_0 = \rho_\mathrm{max} \Theta_H^{n} \, .</math> 
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Making these state-variable substitutions in the PDE that we have just presented for comparison with Kojima's work, we have,

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<math>~
0 
</math>
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<math>~=</math>
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<math>~
\frac{\rho_\mathrm{max}}{\varpi} \frac{\partial}{\partial\varpi} \biggl[  \Theta_H^n \varpi \cdot \frac{\partial W^'}{\partial \varpi} \biggr]
+ \rho_\mathrm{max}\frac{\partial}{\partial z} \biggl(\Theta_H^n \frac{\partial W^'}{\partial z} \biggr) 
+ \rho_\mathrm{max}\biggl\{ \frac{ n{\bar\sigma}^2  \Theta_H^{(n-1)} \rho_\mathrm{max} }{(n+1) P_\mathrm{max}  } 
-  \frac{\Theta_H^n m^2 }{\varpi^2} \biggr\} W^' \, .
</math>
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In an effort to make this entire expression dimensionless, let's define a dimensionless enthalpy perturbation via the relation,
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<math>~\delta W</math>
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<math>~\equiv \biggl[ \frac{\Omega_0 \rho_\mathrm{max}}{P_\mathrm{max}} \biggr]W^' \, ,</math>
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<math>~</math>
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and multiply our expression through by <math>~(\varpi_0^2 \Omega_0/P_\mathrm{max})</math>.  This gives,

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<math>~
0 
</math>
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<math>~=</math>
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<math>~
\frac{1}{\chi} \frac{\partial}{\partial\chi} \biggl[  \Theta_H^n \chi \cdot \frac{\partial (\delta W) }{\partial \chi} \biggr]
+ \frac{\partial}{\partial \zeta} \biggl[ \Theta_H^n \frac{\partial (\delta W) }{\partial \zeta} \biggr] 
+ \biggl[ 
\Theta_H^{n-1} \biggl( \frac{{\bar\sigma}}{\Omega_0} \biggr)^2
\frac{ n\Omega_0^2 \rho_\mathrm{max} \varpi_0^2}{(n+1)P_\mathrm{max}  }-  \frac{\Theta_H^n m^2 }{\chi^2} \biggr]\delta W 
</math>
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&nbsp;
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<math>~=</math>
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<math>~
\Theta_H^n \cdot \frac{\partial^2 (\delta W) }{\partial \chi^2} 
+\Theta_H^n \cdot  \frac{\partial^2 (\delta W) }{\partial \zeta^2} 
+ \biggl[ \Theta_H^{n-1} \biggl( \frac{{\bar\sigma}}{\Omega_0} \biggr)^2 \frac{2n}{\beta^2} -  \frac{\Theta_H^n m^2 }{\chi^2} \biggr]\delta W 
</math>
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&nbsp;
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&nbsp;
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<math>~
+\frac{1}{\chi} \frac{\partial (\delta W) }{\partial \chi} \biggl[\Theta_H^n + n\chi \Theta_H^{n-1} \frac{\partial \Theta_H}{\partial\chi} \biggr]
+ n \Theta_H^{n-1} ~\frac{\partial (\delta W) }{\partial \zeta} \frac{\partial \Theta_H}{\partial \zeta} 
</math>
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&nbsp;
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<math>~=</math>
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<math>~\Theta_H^{n-1} \biggl\{
\Theta_H \cdot \frac{\partial^2 (\delta W) }{\partial \chi^2} 
+\Theta_H \cdot  \frac{\partial^2 (\delta W) }{\partial \zeta^2} 
+ \biggl[\frac{\Theta_H}{ \chi } + n \frac{\partial \Theta_H}{\partial\chi} \biggr]\frac{\partial (\delta W) }{\partial \chi} 
</math>
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&nbsp;
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&nbsp;
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<math>~
+ \biggl[ n \frac{\partial \Theta_H}{\partial \zeta} \biggr] \frac{\partial (\delta W) }{\partial \zeta} 
+ \biggl[ \frac{2n }{\beta^2}\biggl( \frac{{\bar\sigma}}{\Omega_0} \biggr)^2  -  \frac{m^2 \Theta_H}{\chi^2} \biggr]\delta W 
\biggr\} \, .
</math>
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[It appears as though I'm on the right track because this expression is very similar to equation (3.2) of Blaes85!]