Editing Appendix/Ramblings/AzimuthalDistortions (section)

===Summary===
While studying the series of three papers that was published recently by the [[#See_Also|Imamura &amp; Hadley collaboration]], I was particularly drawn to the pair of plots presented in Figure 6 &#8212; and, again, in the top portion of Figure 13 &#8212; of [http://adsabs.harvard.edu/abs/2011Ap%26SS.334....1H  HI11].  This pair of plots has been reprinted here, without modification, as our Figure 2.  As in the bottom two panels of our Figure 1, the curves delineated by the blue dots in this pair of [http://adsabs.harvard.edu/abs/2011Ap%26SS.334....1H  HI11] plots display (on the left) the shape of the eigenfunction, <math>~f_1(\varpi)</math>, and (on the right) the "constant phase locus," <math>~\phi_1(\varpi)</math>, for an unstable, <math>~m=1</math> mode.  In this case, the initial model for the depicted evolution is the equilibrium model from Table 2 of [http://adsabs.harvard.edu/abs/2011Ap%26SS.334....1H  HI11] that has <math>~T/|W| = 0.253</math>; it is a fully self-gravitating torus with [[SR#Barotropic_Structure|polytropic index]], <math>~n = 3/2</math>, and a rotation-law profile defined by a [[AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential|"Keplerian" angular velocity profile]].

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<b><font size="+1">Figure 2</font></b>
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Panel pair extracted<sup>&dagger;</sup> without modification from the top-most segment of Figure 13, p. 12 of [http://adsabs.harvard.edu/abs/2011Ap%26SS.334....1H K. Hadley &amp; J. N. Imamura (2011)]<p></p>
"''Nonaxisymmetric Instabilities of Self-Gravitating Disks. &nbsp; ''I'' Toroids''"<p></p>
''Astrophysics and Space Science'', 334, 1 - 26 &copy; [http://www.springer.com/gp/about-springer/company-information/locations/springer-science-business-media-llc Springer Science+Business Media B.V.]
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[[File:ImamuraMontageTop.png|500px|Comparison with Hadley &amp; Imamura (2011)]]
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<tr><td align="left"><sup>&dagger;</sup>This pair of plots also appears, by itself, as Figure 6 on p. 12 of [http://adsabs.harvard.edu/abs/2011Ap%26SS.334....1H K. Hadley &amp; J. N. Imamura (2011)].</td></tr>
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<b><font size="+1">Figure 3:</font></b> &nbsp;
Our Empirically Constructed Eigenvector
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[[File:ImamuraMontage2Bottom.png|500px|Empirically Constructed Eigenfunction for Comparison with Imamura]]
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''Left panel'':  A plot of our empirically constructed radial amplitude function, <math>~f_\ln(\varpi)</math>; the function has been normalized as explained in the boxed-in ''PRACTICAL IMPLEMENTATION'' remark, below.   ''Right panel'':  A plot of our empirically constructed phase function, <math>~\phi_1(\varpi)</math> with <math>~\aleph = 8.0</math>; after extraction from the animation sequence presented in Figure 4, here each point along this "constant phase locus" has been shifted by an additional phase of <math>~\pi/10</math> in order to better highlight its resemblance to the [http://adsabs.harvard.edu/abs/2011Ap%26SS.334....1H HI11] "constant phase locus" plot shown in the righthand panel of our Figure 2.  In both panels, blue dots trace the function's behavior over the inner region of the torus <math>~(r_- < \varpi < r_\mathrm{mid})</math> and green dots trace the function's behavior over the outer region of the torus <math>~(r_\mathrm{mid} < \varpi < r_+)</math>.
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As is described in the subsections that follow, we have devised two related and relatively simple analytic expressions whose behaviors, as a function of <math>~\varpi</math>, qualitatively resemble the two blue, [http://adsabs.harvard.edu/abs/2011Ap%26SS.334....1H  HI11] curves.  Our two empirically constructed functions have been plotted in Figure 3, immediately below Figure 2, to aid visual comparison with the eigenfunctions that were generated by [http://adsabs.harvard.edu/abs/2011Ap%26SS.334....1H  HI11] via a proper stability analysis.  Next we describe the thought process that led to the construction of the amplitude and phase eigenfunctions presented in Figure 3.