Editing Apps/ImamuraHadleyCollaboration (section)

===Papaloizou &amp; Pringle (1985)===
Performing their linear stability analysis in cylindrical coordinates, [http://adsabs.harvard.edu/abs/1985MNRAS.213..799P Papaloizou &amp; Pringle (1985)] considered temporal and spatial variations in, for example, the pressure of the form,
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<math>~p(\varpi,\varphi,z,t) = p_0(\varpi,z) + p^'(\varpi,z)e^{i(m\varphi + \sigma t)} \, .</math>
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Rather than focusing on <math>~p^'</math>, however, they chose to build the governing PDE around an enthalpy-like perturbation defined as,
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<math>~W(\varpi,z) =  \frac{p^'}{\rho_0\bar\sigma} \, ,</math>
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where, <math>~\bar\sigma \equiv (\sigma + m\Omega)</math>, and <math>~\Omega(\varpi)</math> is the fluid's circular orbital frequency in the initially axisymmetric, equilibrium torus.  The governing PDE appears as equation (2.19) in [http://adsabs.harvard.edu/abs/1985MNRAS.213..799P Papaloizou &amp; Pringle (1985)]; for convenience and clarity, that key equation has been extracted from their paper and displayed in the following framed image.

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Equation (2.19) extracted without modification from p. 803 of [http://adsabs.harvard.edu/abs/1985MNRAS.213..799P Papaloizou &amp; Pringle (1985)]<p></p>
"''The dynamical stability of differentially rotating discs. II''"<p></p>
Monthly Notices of the Royal Astronomical Society, vol. 213, pp. 799-820 &copy; Royal Astronomical Society
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[[File:PP85Eq2.19.png|500px|center|Papaloizou and Pringle (1985, MNRAS, 213, 799)]]
<!-- [[Image:AAAwaiting01.png|400px|center|Whitworth (1981) Eq. 5]] -->
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Note that, in this equation, <math>~\kappa(\varpi)</math> is the (radially dependent) epicyclic frequency in the torus, and <math>~h^'</math> is the radial derivative of the configuration's specific angular momentum.  As has been realized by a number of groups &#8212; and as we have demonstrated in [[Apps/PapaloizouPringle84#Analyses_of_Configurations_with_Uniform_Specific_Angular_Momentum|our accompanying detailed discussion]] &#8212; this governing PDE simplifies considerably when considering only PP tori that have uniform specific angular momentum because both <math>~\kappa^2</math> and <math>~h^'</math> are zero.  Hence, also, <math>~D = \bar\sigma^2</math>.  For such systems, the governing PDE is,

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<math>~\frac{ {\bar\sigma}^2  \rho_0^2 W}{\Gamma p_0  }
</math>
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<math>~=</math>
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<math>~
- \frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl[  \rho_0 \varpi \cdot \frac{\partial W}{\partial \varpi} \biggr]
+  \frac{\rho_0 m^2  W }{\varpi^2} 
- \frac{\partial}{\partial z} \biggl(\rho_0 \frac{\partial W}{\partial z} \biggr) 
\, .
</math>
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Fundamentally, this is the governing PDE that [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] used in his examination of the stability of PP tori.  But he chose to shift to dimensionless variables and to employ a different meridional-plane coordinate system in his analysis.  In an [[Apps/PapaloizouPringle84#Equivalent_Dimensionless_Expression|accompanying discussion]], we show step-by-step how this expression morphs into the governing PDE that serves as the focus of the Blaes85 analysis.  In what follows, we provide a brief summary of this mathematical transformation.


<span id="DensityPerturbation1">For later use,</span> let's show how the eigenfunction for the density perturbation, <math>~\rho^'</math>, can be obtained from <math>~W</math>.  We have,
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<math>~\rho^'</math>
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<math>~=</math>
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<math>~\frac{\rho_0 p^'}{\gamma_g p_0} = \frac{\rho_0^2 \bar\sigma W}{\gamma_g p_0} \, .</math>
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