Editing Appendix/Ramblings/PPToriPt1A (section)

====General Formulation====
From my initial focused reading of the analysis presented by [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)], I conclude that, in the infinitely slender torus case, unstable modes in PP tori exhibit eigenvectors of the form,
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<math>~\frac{\delta W}{W_0} \equiv \biggl[ \frac{W(\eta,\theta)}{C} - 1 \biggr]e^{im\Omega_p t}e^{-y_2 (\Omega_0 t)} </math>
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<math>~=</math>
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<math>~\biggl\{ f_m(\eta,\theta)e^{-i[m\phi_m + k\theta]} \biggr\}  \, ,</math>
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<!-- [<font color="red"><b>NOTE:</b></font>   Initially, I wrote "+ k" rather than "- k" in the exponent of the exponential term on the RHS of this expression; but experience shows that "- k" is required to achieve proper behavior of the "constant phase locus" plot, as displayed below.]  -->
where we have written the perturbation amplitude in a manner that conforms with the notation that we have used in  [[Appendix/Ramblings/AzimuthalDistortions#Figure1|Figure 1 of a related, but more general discussion]].  As is summarized in &sect;1.3 of Blaes (1985), for "thick" (but, actually, still quite thin) tori, "exactly one exponentially growing mode exists for each value of the azimuthal wavenumber <math>~m</math>," and its complex amplitude takes the following form (see his equation 1.10):

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<math>~f_m(\eta,\theta)</math>
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<math>~=</math>
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<math>
~\beta^2 m^2 \biggl[ 2\eta^2 \cos^2\theta - \frac{3\eta^2}{4(n+1)} - \frac{(4n+1)}{4(n+1)^2} 
\pm 4i\biggl(\frac{3}{2n+2}\biggr)^{1/2} \eta\cos\theta\biggr] 
+ \mathcal{O}(\beta^3) \, .
</math>
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Aside from an arbitrary leading scale factor, we should therefore find that the amplitude (modulus) of the enthalpy perturbation is,
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<math>~\biggl|\frac{\delta W}{W_0} \biggr|</math>
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<math>~=</math>
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<math>~\sqrt{[\mathrm{Re}(f_m)]^2+ [\mathrm{Im}(f_m)]^2} \, ;</math>
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and the associated phase function is,
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<math>~m\phi_m - k\theta</math>
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<math>~=</math>
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<math>~\tan^{-1} \biggl\{ \frac{\mathrm{Re}(f_m)}{\mathrm{Im}(f_m)} \biggr\} \, .</math>
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[<font color="red"><b>NOTE:</b></font>  Initially, I expected the argument inside the arctan function to be the ratio, <math>~\mathrm{Im}(f_m)/\mathrm{Re}(f_m)</math>; but experience shows that the reciprocal of this ratio is required to achieve proper behavior of the "constant phase locus" plot, as displayed below.]
-->

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<math>~m\phi_\mathrm{max}(\varpi)</math>
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<math>~=</math>
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<math>~ 
-\biggl\{ \tan^{-1}\biggl[ \frac{\mathrm{Re}(f_m)}{\mathrm{Im}(f_m)} \biggr] +\frac{\pi}{2} +k\theta \biggr\}
\, .</math>
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Now, keeping in mind that, for the time being, we are only interested in examining the shape of the unstable eigenvector in the ''equatorial plane'' of the torus, we can set,
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<math>~\cos\theta ~~ \rightarrow ~~ \pm 1 \, .</math>
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Hence, we have,
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<math>~\frac{1}{\beta^4 m^4}\biggl|\frac{\delta W}{W_0} \biggr|^2</math>
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<math>~=</math>
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<math>~\biggl[2\eta^2  - \frac{3\eta^2}{4(n+1)} - \frac{(4n+1)}{4(n+1)^2}\biggr]^2
+ 16\biggl[\biggl(\frac{3}{2n+2}\biggr)^{1/2} \eta \biggr]^2 </math>
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&nbsp;
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<math>~=</math>
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<math>~\frac{1}{[2^2(n+1)^2]^2}\biggl[2^3(n+1)^2\eta^2  - 3(n+1)\eta^2 - (4n+1) \biggr]^2
+ \frac{2^3 \cdot 3\eta^2}{(n+1)}   </math>
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&nbsp;
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<math>~=</math>
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<math>~\frac{1}{[2^2(n+1)^2]^2}\biggl[(4n+1) - (n+1)[2^3(n+1)-3]\eta^2  \biggr]^2
+ \frac{2^3 \cdot 3\eta^2}{(n+1)}   </math>
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<math>~\Rightarrow ~~~~ \biggl[\frac{2(n+1)}{\beta m} \biggr]^4 \biggl|\frac{\delta W}{W_0} \biggr|^2</math>
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<math>~=</math>
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<math>~\biggl[(4n+1) - (n+1)[2^3(n+1)-3]\eta^2  \biggr]^2
+ 2^7 \cdot 3(n+1)^3\eta^2 \, .  </math>
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Also, keeping in mind that, because of the <math>~\cos\theta</math> factor, the sign on the imaginary term flips its sign when switching from the "inner" region to the "outer" region of the torus,

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<math>~m\phi_\mathrm{max}</math>
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<math>~=</math>
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<math>~- \biggl\{ \tan^{-1}\biggl[ 
\frac{(4n+1) - (n+1)[2^3(n+1)-3]\eta^2  }{2^{7/2}\cdot 3^{1/2}(n+1)^{3/2}\eta} 
\biggr]+\frac{\pi}{2}\biggr\}</math>
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&nbsp; &nbsp; &nbsp; &nbsp; over &nbsp; &nbsp; &nbsp; &nbsp; 
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inner <math>~(\theta=0)</math> region of the torus;
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<tr><td colspan="6" align="center">while</td></tr>
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  <td align="center" bgcolor="green">&nbsp;</td>
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<math>~m\phi_\mathrm{max}</math>
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<math>~=</math>
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<math>~- \biggl\{ \tan^{-1}\biggl[-  
\frac{(4n+1) - (n+1)[2^3(n+1)-3]\eta^2  }{2^{7/2}\cdot 3^{1/2}(n+1)^{3/2}\eta}\biggr\}  +\frac{3\pi}{2}
\biggr\}</math>
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&nbsp; &nbsp; &nbsp; &nbsp; over &nbsp; &nbsp; &nbsp; &nbsp; 
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outer <math>~(\theta=\pi)</math> region of torus.
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