Editing Appendix/Ramblings/AzimuthalDistortions (section)

==Empirical Construction of Eigenvector== 

===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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  <td align="center">
<b><font size="+1">Figure 2</font></b>
  </td>
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<tr><td align="center">
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.]
</td></tr>
<tr><td align="center">
[[File:ImamuraMontageTop.png|500px|Comparison with Hadley &amp; Imamura (2011)]]
</td></tr>
<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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</div>


<div align="center" id="Figure3">
<table border="1" cellpadding="3" align="center" width="60%">
<tr>
  <td align="center">
<b><font size="+1">Figure 3:</font></b> &nbsp;
Our Empirically Constructed Eigenvector
  </td>
</tr>
<tr><td align="center">
[[File:ImamuraMontage2Bottom.png|500px|Empirically Constructed Eigenfunction for Comparison with Imamura]]
</td></tr>
<tr><td align="left" colspan="2">
''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.

===Radial Eigenfunction===
It occurred to me, first, that the blue curve displayed in the left-hand panel of [http://adsabs.harvard.edu/abs/2011Ap%26SS.334....1H  HI11]'s figure 6 (our Figure 2) might be reasonably well approximated by piecing together a pair of arc-hyperbolic-tangent (ATANH) functions. In an effort to demonstrate this, I began by specifying a "midway" radial location, <math>~r_- < r_\mathrm{mid} < r_+ \, ,</math> at which the two ATANH functions meet and at which the density fluctuation is smallest.  Then I defined a function of the form,
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<tr>
  <td align="center" bgcolor="blue">&nbsp;</td>
  <td align="right">
<math>~f_\ln(\varpi)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tanh^{-1}\biggl[1 - 2 \biggl( \frac{\varpi - r_-}{r_\mathrm{mid}-r_-} \biggr) \biggr]</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; for &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>r_- < \varpi < r_\mathrm{mid} \, ;</math>
  </td>
</tr>
<tr><td colspan="6" align="center">and</td></tr>
<tr>
  <td align="center" bgcolor="green">&nbsp;</td>
  <td align="right">
<math>~f_\ln(\varpi)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tanh^{-1}\biggl[1 - 2 \biggl( \frac{\varpi - r_+}{r_\mathrm{mid}-r_+} \biggr) \biggr]</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; for &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>r_\mathrm{mid} < \varpi < r_+ \, .</math>
  </td>
</tr>
</table>
</div>

This empirically specified, two-piece <math>~f_\ln(\varpi)</math> function has been plotted in the left-hand panel of Figure 3.  (To facilitate quantitative comparison with Figure 2, the function has been normalized as explained in the boxed-in ''PRACTICAL IMPLEMENTATION'' remark that follows.)  Blue dots trace the function's behavior over the lower radial-coordinate range while green dots trace its behavior over the upper radial-coordinate range.  This plot of <math>~f_\ln(\varpi)</math> closely resembles the plot of the eigenfunction, <math>~\delta\rho/\rho (\varpi)</math> (see the left-hand panel of our Figure 2) that developed spontaneously via [http://adsabs.harvard.edu/abs/2011Ap%26SS.334....1H  HI11]'s linear stability analysis. 


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'''<font color="maroon">PRACTICAL IMPLEMENTATION:</FONT>'''  &nbsp; At the two limits, <math>~\varpi = r_-</math> and <math>~\varpi = r_+</math>, the function, <math>~f_\ln(\varpi) \rightarrow +\infty</math>; while, at the limit, <math>~\varpi = r_\mathrm{mid}</math>, the function, <math>~f_\ln(\varpi) \rightarrow -\infty</math>.  In practice, after dividing the relevant radial extent into 100 zones, we stay ''half of a radial zone'' away from these three limiting radial boundaries, so that the maximum and minimum values of <math>~f_\ln(\varpi)</math> are finite; specifically, in the example plotted here, we have set <math>~[f_\ln]_\mathrm{min} = -2.99448</math> and <math>~[f_\ln]_\mathrm{max} = 2.64665</math>.  Then we strategically employ the finite values of the function at these near-boundary limits to rescale the function such that, in the plot shown here, it lies between -3 (minimum amplitude) and 0 (maximum amplitude).
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<span id="SwitchToLog">Recognizing that the figure depicting</span> the [http://adsabs.harvard.edu/abs/2011Ap%26SS.334....1H  HI11] eigenfunction is a semi-log plot, it seems clear that the relationship between our constructed function, <math>~f_\ln(\varpi)</math>, and the eigenfunction, <math>~f_1(\varpi)</math>, is,
<div align="center">
<math>~f_1(\varpi) = e^{f_\ln(\varpi)} \, .</math>
</div>
Now, in general, the following mathematical relation holds:
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<tr>
  <td align="right">
<math>~\tanh^{-1}x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\ln\biggl( \frac{1+x}{1-x} \biggr)^{1/2} </math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; for &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>x^2 < 1 \, .</math>
  </td>
</tr>
</table>
</div>
Hence, for the innermost region of the toroidal configuration &#8212; that is, over the lower radial-coordinate range &#8212; we can set,
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<tr>
  <td align="right">
<math>~x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 - 2 \biggl( \frac{\varpi - r_-}{r_\mathrm{mid}-r_-} \biggr) </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ \frac{1+x}{1-x}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\biggl[2 - 2 \biggl( \frac{\varpi - r_-}{r_\mathrm{mid}-r_-} \biggr)\biggr] 
\biggl[2 \biggl( \frac{\varpi - r_-}{r_\mathrm{mid}-r_-} \biggr)\biggr]^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~[(r_\mathrm{mid}-r_-) - ( \varpi - r_-)]  [(\varpi - r_-)]^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\frac{r_\mathrm{mid} - \varpi}{\varpi - r_-} \, .
</math>
  </td>
</tr>
</table>
</div>
<span id="SquareRoot">Therefore we can write,</span>
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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="center" bgcolor="blue">&nbsp;</td>
  <td align="right">
<math>~f_1(\varpi) = e^{f_\ln(\varpi)}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{r_\mathrm{mid} - \varpi}{\varpi - r_-} \biggr)^{1/2}
</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; for &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>r_- < \varpi < r_\mathrm{mid} \, .</math>
  </td>
</tr>
</table>
</div>
Similarly, we find that, over the upper radial-coordinate range,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="center" bgcolor="green">&nbsp;</td>
  <td align="right">
<math>~f_1(\varpi) = e^{f_\ln(\varpi)}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{r_\mathrm{mid} - \varpi}{\varpi - r_+} \biggr)^{1/2}
</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; for &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>r_\mathrm{mid} < \varpi < r_+ \, .</math>
  </td>
</tr>
</table>
</div>

===Constant Phase Loci===

Now let's work on the phase function, <math>~\phi_1(\varpi)</math>.  The phase function displayed in the right-hand panel of our Figure 2 &#8212; that is, the phase function that developed spontaneously from the linear stability analysis performed by [http://adsabs.harvard.edu/abs/2011Ap%26SS.334....1H  HI11]  &#8212; appears to be fairly constant (''i.e.,'' the phase is independent of radius) in the innermost region of the torus and, then again, fairly constant in the outermost region of the torus with a smooth but fairly rapid phase shift of approximately <math>~\pi</math> radians between the two extremes.  This is the behavior exhibited by an arctangent (ATAN) function.  With this in mind, we have defined a new function, <math>~D(\varpi)</math>, ''in terms of'' our empirically derived radial eigenfunction, <math>~f_\ln(\varpi)</math>, as follows:

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  <td align="right">
<math>~D(\varpi)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{f_\ln(\varpi) - [f_\ln]_\mathrm{min}}{[f_\ln]_\mathrm{max} - [f_\ln]_\mathrm{min}} \, .</math>
  </td>
</tr>
</table>
</div>

It has the following behavior:  
* At the inner edge of the torus <math>~(r_-)</math>, where <math>~f_\ln(\varpi) = [f_\ln]_\mathrm{max}</math>, <math>~D(\varpi) = 1</math>;
* At <math>~r_\mathrm{mid}</math>, where <math>~f_\ln(\varpi) = [f_\ln]_\mathrm{min}</math>, <math>~D(\varpi) = 0</math>;
* At the outer edge of the torus <math>~(r_+)</math>, where again <math>~f_\ln(\varpi) = [f_\ln]_\mathrm{max}</math>, <math>~D(\varpi) = 1</math>.
This function can therefore satisfactorily serve as an argument of the ATAN function, swinging the phase by <math>~\pi/2</math> over the inner (blue) region of the torus, then swinging the phase by an additional <math>~\pi/2</math> over the outer (green) region of the torus.  If we furthermore multiply the function by a variable coefficient &#8212; call it, <math>~\aleph</math> &#8212; before feeding it to the ATAN function, we can adjust the thickness of the radial domain over which the total phase transition occurs.  What appears to work well is the following:

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<tr>
  <td align="center" bgcolor="blue">&nbsp;</td>
  <td align="right">
<math>~\phi_1(\varpi) + \frac{\pi}{2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~+~\tan^{-1}[\aleph \cdot D(\varpi)]</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; for &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>r_- < \varpi < r_\mathrm{mid} \, ;</math>
  </td>
</tr>
</table>
</div>

and

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="center" bgcolor="green">&nbsp;</td>
  <td align="right">
<math>~\phi_1(\varpi) + \frac{\pi}{2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-~\tan^{-1}[\aleph \cdot D(\varpi)] </math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; for &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>r_\mathrm{mid} < \varpi < r_+ \, .</math>
  </td>
</tr>
</table>
</div>

In the lefthand panel of Figure 4, the "constant phase loci" defined by this empirically constructed phase function have been mapped onto a polar-coordinate grid for ten different values of the leading coefficient in the range, <math>~1.0 \le \aleph \le 40.0</math>, as recorded in the bottom-right corner of the plot.  The constant phase locus created by setting <math>~\aleph = 8.0</math> has been singled out and displayed in the middle panel of Figure 4, because it closely resembles the "constant phase locus" published by HI11a (reprinted here as the righthand panel of Figure 4 to facilitate comparison).

<table border="1" cellpadding="8" align="center">
<tr><th align="center" colspan="3"><font size="+1">Figure 4</font></th></tr>
<tr>
  <td align="center" colspan="2">Constant Phase Loci Generated by Our Empirically Constructed Phase Function, <math>~\phi_1(\varpi)</math></td>
  <td align="center" rowspan="2">[http://adsabs.harvard.edu/abs/2011Ap%26SS.334....1H HI11]'s Published Constant Phase Loci</td>
</tr>
<tr><td align="center">Ten Values of <math>~\aleph</math></td><td align="center">For <math>~\aleph = 8</math></td></tr>
<tr>
<td align="center">
[[File:ImamuraPhaseMovie.gif|230px|Movie Showing Empirically Constructed Phase]]
</td>
<td align="center">
[[File:a05.png|230px|Best Match]]
</td>
<td align="center">
[[File:ImamuraOriginalPhase.png|250px|HI11a constant phase loci]]
</td>
</tr>
</table>

<!--
Then we set,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\phi(\varpi) + \phi_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{\tan^{-1}[\aleph \cdot D(\varpi)] - \frac{\pi}{2} \biggr\} + \frac{\pi}{10} \, .</math>
  </td>
</tr>
</table>
</div>

Now &#8212; as stated earlier in the [[#Figure3|caption to Figure 3]] &#8212; for the specific case being graphically illustrated here, <math>~[f_\ln]_\mathrm{min} = -2.99448</math> and <math>~[f_\ln]_\mathrm{max} = 2.64665</math>.  Hence,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\phi(\varpi) + \frac{\pi}{2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tan^{-1}\biggl\{8\cdot \biggl[ 
\frac{f_\ln(\varpi) - [f_\ln]_\mathrm{min}}{[f_\ln]_\mathrm{max} - [f_\ln]_\mathrm{min}}  
\biggr]\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tan^{-1}[a\cdot f(\varpi) + b]  \, ,
</math>
  </td>
</tr>
</table>
</div>
where, <math>~a = 1.41816</math> and <math>~b = 4.24664</math>.
-->