Editing Appendix/Ramblings/ToHadleyAndImamura (section)

===Comparisons===
====For n = 3 Configurations====
Following &sect;1.3 of Blaes85, let's specifically consider the case of slim PP-tori that have a polytropic index, <math>~n=3</math>.  In this case, we have,

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

<tr>
  <td align="right">
<math>~\mathcal{A}_\mathrm{Blaes}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{ (\beta m)^2}{2^6}\biggl[a_3(\eta,\theta)\biggr]
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathcal{B}_\mathrm{Blaes}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{(\beta m)^2}{2^6}\biggl[b_3(\eta,\theta)\biggr] \, ,
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~a_3(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl\{
~[8(n+1)\cos^2\theta - 3]\eta^2 (n+1) - (4n+1) \biggr\}_{n=3}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
~2^2[2^5\cos^2\theta - 3]\eta^2  - 13 
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~b_3(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl\{ [2^7\cdot 3(n+1)^3
\eta^2\cos^2\theta ]^{1/2} \biggr\}_{n=3}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~[2^{13}\cdot 3\eta^2\cos^2\theta ]^{1/2}\, .
</math>
  </td>
</tr>
</table>
</div>

<!-- OMIT
Hence, the "constant phase locus" should be identified by the function,

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

<tr>
  <td align="right">
<math>~m\phi_\mathrm{max}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\tan^{-1}\biggl\{ \frac{[2^3/(\beta m)]^2 + a(\eta,\theta) }{  b(\eta,\theta) } \biggr\}  - k\theta
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\tan^{-1}\biggl\{ \frac{[2^3/(\beta m)]^2 - 13 
+ 2^2[2^5\cos^2\theta - 3]\eta^2 }{  2^{6}(2\cdot 3)^{1/2}\eta \cos\theta  } \biggr\}  - k\theta \, ,
</math>
  </td>
</tr>
</table>
</div>

END OMIT -->

Now, rather than examining the structural behavior of the amplitude and phase of the function, <math>~[W/W_0 -1]</math>, as we have done, above, Blaes evaluated the amplitude (only) of the function, 
<div align="center">
<math>~\frac{W}{W_0} = 1 + \mathcal{A}_\mathrm{Blaes}(\eta,\theta) + i \mathcal{B}_\mathrm{Blaes}(\eta,\theta) \, .</math>
</div>

For the specific case of <math>~n=3</math>, the square of the amplitude of ''this'' function is,

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

<tr>
  <td align="right">
<math>~\biggl|\frac{W}{W_0} \biggr|_{n=3}^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{ 1 + \frac{ (\beta m)^2}{2^6}\biggl[a_3(\eta,\theta)\biggr] \biggr\}^2 +  
\biggl\{ \frac{(\beta m)^2}{2^6}\biggl[b_3(\eta,\theta)\biggr] \biggr\}^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 + \frac{ (\beta m)^2a_3}{2^5}  
+ \frac{ (\beta m)^4 }{2^{12}} (a_3^2 + b_3^2) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 + \frac{ (\beta m)^2}{2^5}  \biggl\{ 2^2[2^5\cos^2\theta - 3]\eta^2  - 13\biggr\}
+ \mathcal{O}[(\beta m)^4] \, .
</math>
  </td>
</tr>
</table>
</div>

Therefore, to lowest order in the "slimness" parameter, <math>~\beta</math>,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl|\frac{W}{W_0} \biggr|_{n=3}</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~1 + \frac{ (\beta m)^2}{2^6}  \biggl\{ 2^2[2^5\cos^2\theta - 3]\eta^2  - 13\biggr\}
\, .
</math>
  </td>
</tr>
</table>
</div>

This (almost exactly) matches the amplitude expression derived by [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] and evaluated for the specific case of <math>~\beta = (101)^{-1/2}</math> &#8212; the relevant equation (1.12) from Blaes85 is digitally reproduced in the table that follows.  

<div align="center">
<table border="1" cellpadding="5" width="80%">
<tr><td align="center">
Equation (1.12) extracted without modification from p. 556 of [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)]<p></p>
"''Oscillations of Slender Tori''"<p></p>
Monthly Notices of the Royal Astronomical Society, vol. 216, pp. 553-563 &copy; Royal Astronomical Society
</td></tr>
<tr>
  <td align="center">
[[File:Blaes85Eq1.12.png|400px|center|Blaes (1985, MNRAS, 216, 553)]]
<!-- [[Image:AAAwaiting01.png|400px|center|Whitworth (1981) Eq. 5]] -->
  </td>
</tr>
</table>
</div>

Our expression differs from the one published in Blaes85 in one detail:  In Blaes85, the pre-factor of the second term inside the curly braces is, evidently, <math>~(\beta m)^2/2^5</math>, which is a factor of two larger than the corresponding pre-factor found in our expression.  In our attempt to re-derive the Blaes85 expression (1.12), this extra factor of two disappears when we take the square-root of both sides to obtain the modulus, rather than the square of the modulus.  Hopefully further reflection will resolve this discrepancy between our approximate expression for <math>~|W/W_0|_{n=3}</math> and the analogous one presented in Blaes85.  

====For n = 0 Configurations====

As we have [[User:Tohline/Appendix/Ramblings/PPTori#Goldreich.2C_Goodman_and_Narayan_.281986.29|discussed elsewhere]], [http://adsabs.harvard.edu/abs/1986MNRAS.221..339G Goldreich, Goodman &amp; Narayan (1986, MNRAS, 221, 339)] &#8212; hereafter, GGN86 &#8212; also used analytic techniques to analyze the properties of unstable, nonaxisymmetric eigenmodes in Papaloizou-Pringle tori.  They restricted their discussion to slim, ''incompressible'' tori, so in order to assess the overlap between the GGN86 and Blaes85 works, we will set <math>~n=0</math> in the general expressions presented in Blaes85.

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

<tr>
  <td align="right">
<math>~\mathcal{A}_\mathrm{Blaes}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{ (\beta m)^2}{2^2}\biggl[a_0(\eta,\theta)\biggr]
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathcal{B}_\mathrm{Blaes}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{(\beta m)^2}{2^2}\biggl[b_0(\eta,\theta)\biggr] \, ,
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~a_0(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl\{
~[8(n+1)\cos^2\theta - 3]\eta^2 (n+1) - (4n+1) \biggr\}_{n=0}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
~[8\cos^2\theta - 3]\eta^2  - 1 
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~b_0(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl\{ [2^7\cdot 3(n+1)^3 \eta^2\cos^2\theta ]^{1/2} \biggr\}_{n=0}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2^3 (2\cdot 3)^{1/2} \eta \cos\theta  \, .
</math>
  </td>
</tr>
</table>
</div>

From the Blaes85 analysis, then, we conclude that the unstable eigenfunction for slim, ''incompressible'' PP-tori is,

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

<tr>
  <td align="right">
<math>~\biggl[ \frac{W}{W_0}\biggr]_{n=0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 + \frac{ (\beta m)^2}{2^2}\biggl[(8\cos^2\theta - 3)\eta^2  - 1\biggr]
+ i \frac{ (\beta m)^2}{2^2}\biggl[ 2^3 (2\cdot 3)^{1/2} \eta \cos\theta \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[1 - \tfrac{1}{4}(\beta m)^2\biggr] + (\beta m)^2 \biggl[\biggl(2\cos^2\theta - \frac{3}{4}\biggr)\eta^2 \biggr]
+ 4i (\beta m)^2\biggl[ \biggl(\frac{3}{2}\biggr)^{1/2} \eta \cos\theta \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

This should be compared with equation (5.16) of GGN86, which is "the lowest order [complex] expression for the [perturbed] velocity potential," namely,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\psi </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1+\tfrac{1}{4} k^2(5x^2 - 3z^2) \mp 4i\biggl(\frac{3}{2}\biggr)^{1/2} k x \beta_\mathrm{GGN} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1+m^2 \beta^2_\mathrm{Blaes}\biggl[2\eta^2\cos^2\theta - \frac{3\eta^2}{4}\biggr]  
\mp 4i \biggl(\frac{3}{2}\biggr)^{1/2} m^2\beta^2_\mathrm{Blaes} \eta\cos\theta \, ,</math>
  </td>
</tr>
</table>
</div>

where the last expression results from [[User:Tohline/Appendix/Ramblings/PPTori#Goldreich.2C_Goodman_and_Narayan_.281986.29|our mapping of the GGN86 terminology to the Blaes85 terminology]].  The two derived expressions match in every detail, except one:  The constant term, <math>~\tfrac{1}{4}(\beta m)^2</math>, that is subtracted from unity on the right-hand side of the Blaes85 function is missing from the function derived by GGN86.  Hopefully, additional study of this problem will rectify this apparent difference between the two published analyses.

<!-- OMIT
In addition, the "constant phase locus" should be identified by the function,

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

<tr>
  <td align="right">
<math>~m\phi_\mathrm{max}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\tan^{-1}\biggl\{ \frac{[2/(\beta m)]^2 + a(\eta,\theta) }{  b(\eta,\theta) } \biggr\}  - k\theta
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\tan^{-1}\biggl\{ \frac{[2/(\beta m)]^2 - 1 
+ [8\cos^2\theta - 3]\eta^2  }{  2^3 (2\cdot 3)^{1/2} \eta \cos\theta } \biggr\}  - k\theta \, ,
</math>
  </td>
</tr>
</table>
</div>

and the square of the amplitude is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl|\frac{\delta W}{W_0} \biggr|^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{ 1 + \frac{ (\beta m)^2}{2^2}\biggl[a(\eta,\theta)\biggr] \biggr\}^2 +  
\biggl\{ \frac{(\beta m)^2}{2^2}\biggl[b(\eta,\theta)\biggr] \biggr\}^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 + \frac{ 2(\beta m)^2a}{2^2}  
+ \frac{ (\beta m)^4 }{2^4} (a^2 + b^2) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 + \frac{ (\beta m)^2}{2}  \biggl\{ [8\cos^2\theta - 3]\eta^2  - 1 \biggr\}
+ \mathcal{O}[(\beta m)^4] \, .
</math>
  </td>
</tr>
</table>
</div>
END OMIT -->


====For n = 3/2 Configurations====
The Imamura &amp; Hadley collaboration examined instabilities that develop in tori having a variety of polytropic indexes, but their focus was often  on <math>~n=3/2</math> configurations.  It is useful, therefore, to evaluate the Blaes85 eigenfunction for this set of models.  In this case,

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

<tr>
  <td align="right">
<math>~\mathcal{A}_\mathrm{Blaes}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{ (\beta m)^2}{5^2}\biggl[a_{3/2}(\eta,\theta)\biggr]
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathcal{B}_\mathrm{Blaes}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{(\beta m)^2}{5^2}\biggl[b_{3/2}(\eta,\theta)\biggr] \, ,
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~a_{3/2}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl\{
~[8(n+1)\cos^2\theta - 3]\eta^2 (n+1) - (4n+1) \biggr\}_{n=3/2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
~\frac{5}{2}[20\cos^2\theta - 3]\eta^2  - 7 
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~b_{3/2}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl\{ [2^7\cdot 3(n+1)^3 \eta^2\cos^2\theta ]^{1/2} \biggr\}_{n=3/2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~[2^{4}\cdot 3\cdot 5^3\eta^2\cos^2\theta ]^{1/2}\, .
</math>
  </td>
</tr>
</table>
</div>

Hence, the "constant phase locus" should be identified by the function,

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

<tr>
  <td align="right">
<math>~m\phi_\mathrm{max}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\tan^{-1}\biggl\{ \frac{a_{3/2}(\eta,\theta) }{  b_{3/2}(\eta,\theta) } \biggr\}  - k\theta
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\tan^{-1}\biggl\{ \frac{- 7 
+ \frac{5}{2}[20\cos^2\theta - 3]\eta^2 }{  20\cdot (3\cdot 5)^{1/2}\eta \cos\theta  } \biggr\}  - k\theta 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\tan^{-1}\biggl\{ \frac{- 14 
+ 5[20\cos^2\theta - 3]\eta^2 }{  40\cdot (3\cdot 5)^{1/2}\eta \cos\theta  } \biggr\}  - k\theta \, ,
</math>
  </td>
</tr>
</table>
</div>

and the square of the (unity-adjusted) amplitude is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl|\frac{W}{W_0} - 1 \biggr|^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{ (\beta m)^4 }{5^4} \biggl[ a_{3/2}^2 + b_{3/2}^2 \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{ (\beta m)^4 }{5^4} \biggl\{ \biggl[\frac{5}{2} (20\cos^2\theta - 3)\eta^2  - 7\biggr]^2
+ \biggl[ 2^{4}\cdot 3\cdot 5^3\eta^2\cos^2\theta \biggr]    \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{ (\beta m)^4 }{5^4} \biggl\{ \biggl[\frac{5}{2} (20\cos^2\theta - 3)\eta^2\biggr]^2
-14 \biggl[\frac{5}{2} (20\cos^2\theta - 3)\eta^2 \biggr]
+ \biggl[- 7\biggr]^2
+ 2^{4}\cdot 3\cdot 5^3\eta^2\cos^2\theta \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{ (\beta m)^4 }{5^4} \biggl\{ \frac{5^2}{2^2} (20\cos^2\theta - 3)^2\eta^4
-35 (20\cos^2\theta - 3)\eta^2 + 49
+ 6000\eta^2\cos^2\theta \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{ (\beta m)^4 }{5^4} \biggl\{49
 + [105 + 5300\cos^2\theta  ]\eta^2
+  \frac{5^2}{2^2} (20\cos^2\theta - 3)^2\eta^4\biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>


Let's compare the amplitude and phase diagrams that result from the Blaes85 analytic model with results from the model "P4" evolution reported in [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Hadley et al. (2014)], that is, from Paper II in the Imamura &amp; Hadley collaboration.  Setting <math>~n = 3/2</math> and, because this comparison is restricted to the equatorial plane, setting <math>~\theta = 0</math> (inner region of torus) or <math>~\theta = \pi</math> (outer region of torus), we have from Blaes85,

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

<tr>
  <td align="right">
<math>~\mathcal{A}(\eta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\biggl(\frac{17}{10}\biggr)\eta^2 - \frac{7}{25} 
\, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathcal{B}(\eta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\pm
\biggl(\frac{2^4\cdot 3}{5}\biggr)^{1/2} \eta \, .
</math>
  </td>
</tr>
</table>
</div>

Figure 3 presents a plot (left panel) of <math>~\tfrac{1}{4}\mathcal{A}</math> versus <math>~x</math> (salmon-colored markers) and <math>~\tfrac{1}{4}\mathcal{B}</math> versus <math>~x</math> (green markers) for a PP-torus with <math>~\beta = 0.176</math>.  This torus has the same aspect ratio as the model named "P4" in [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Paper II]:  It has an inner edge at <math>~x_- = 0.85</math>, outer edge at <math>~x_+ = 1.21</math>, and cross-sectional "center" at <math>~x_0 = 1</math>.  (For equilibrium model characteristics, also see [[User:Tohline/Appendix/Ramblings/Azimuthal_Distortions#Setup|Table 4 in our accompanying detailed discussion]]).  Via a semi-log plot, the right panel of Figure 3 displays the behavior of <math>~(\tfrac{1}{4}\mathcal{A})^2</math> and <math>~(\tfrac{1}{4}\mathcal{B})^2</math> as a function of <math>~x</math>.  

<div align="center">
<table border="1" cellpadding="0" align="center">
<tr>
  <td align="center">
<font size="+1"><b>Figure 3:</b></font>&nbsp; Real (A) and Imaginary (B) Components of the Blaes85 Analytic Eigenfunction
  </td>
</tr>
<tr><td align="center">
[[File:RealImaginaryMontage.png|600px|Analytic Eigenfunction]]
</td></tr>
</table>
</div>

Note that, although the function, <math>~\mathcal{B}(\eta)</math>, is linear in <math>~\eta</math>, the green curve in the left panel of Figure 3 is slightly curved.  This is because the horizontal axis (in both panels) is the coordinate, <math>~x</math>, rather than <math>~\eta</math>.  The conversion from <math>~\eta</math> to <math>~x</math> is provided by the root of a cubic equation, as [[User:Tohline/Appendix/Ramblings/PPTori#Blaes_.281985.29|discussed separately]].  


The panel on the right in Figure 3 explains in a qualitative sense how sharp features &#8212; in particular, steep valleys &#8212; can arise in the "amplitude" plots of simulations that study the nonlinear growth of unstable, nonaxisymmetric eigenmodes in tori.  A sharp feature can arise when either the real or the imaginary component of the eigenmode crosses zero (and thereby changes sign).  In the analytic eigenfunction expression derived by Blaes (1985), the radial dependence of the imaginary component is defined by a linear function, and it displays a ''single'' sharp feature; the radial dependence of the real component is defined by a quadratic function, and it displays a ''pair'' of sharp features.  It seems clear that, depending on the ''relative'' overall amplitude of the real and imaginary parts, the combined amplitude could display a single sharp feature, a pair of sharp features, or a milder curve with no particularly sharp features.  It is this third option that results from the specific case presented to us by the Blaes85 eigenfunction.  As is shown by the blue curve in the middle-left-hand panel of Figure 4, the Blaes85 modulus &#8212; <math>~\tfrac{1}{4}\sqrt{\mathcal{A}^2 + \mathcal{B}^2}</math> &#8212; presents a curve with no particularly sharp features.

<div align="center">
<table border="1" cellpadding="0" align="center">
<tr>
  <td align="center" colspan="2">
<font size="+1"><b>Figure 4:</b></font>&nbsp; Comparison
  </td>
</tr>
<tr>
  <td align="center" rowspan="1">
[[File:Figure4PanelTitleA.png|30px|Panel A Title]]
  </td>
  <td align="center" rowspan="3">
[[File:Montage01.png|400px|Comparison]]
  </td>
</tr>
<tr>
  <td align="center" rowspan="1">
[[File:Figure4PanelTitleB.png|30px|Panel A Title]]
  </td>
</tr>
<tr>
  <td align="center" rowspan="1">
[[File:Figure4PanelTitleC.png|30px|Panel A Title]]
  </td>
</tr>
</table>
</div>

Figure 4 displays (left) the radial dependence of the "amplitude" and (right) the radial dependence of the "constant phase locus" for the unstable modes of three models with similar &#8212; although not identical &#8212; initial equilibrium structures:  (Top row) The model with a star-to-disk mass ratio of 100 labeled "P4" in Paper II of the Imamura &amp; Hadley collaboration; (Middle row) The ''massless'' model analyzed analytically by Blaes (1985) and described herein; and (Bottom row) An unpublished model from the Imamura &amp; Hadley collaboration (private communication) with a similar aspect ratio but a star-to-disk mass ratio of 1000.



====Various Thoughts====

<ol>
<li>In the Blaes85 model, the "constant phase locus" swings through a total angle of,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~m\Delta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\tan^{-1}\biggl[ \frac{\mathcal{A}}{\mathcal{B}} \biggr]_{\eta=1} + \pi \, .</math>
  </td>
</tr>
</table>
</div>

For example, in the model displayed, above, <math>~[\mathcal{A}/\mathcal{B}]_{\eta=1} = 0.4583</math>.  Hence, <math>~2\Delta = 4.001</math> radians.</li>

<li>
We need to resolve the apparent discrepancy between the value of the leading constant that appears in the GGN86 eigenfunction versus the one that is found in the Blaes85 eigenfunction (when evaluated for <math>~n=0</math>).  Graphically, the Blaes85 amplitude function appears to make more sense, but a physically based explanation needs to be identified.
</li>

<li>
The amplitude and phase functions obtained from the Blaes85 work appear to match &#8212; qualitatively, if not quantitatively &#8212; the amplitude and phase functions published by the Imamura &amp; Hadley collaboration if we subtract the leading "unity" constant from the Blaes85 expression for <math>~W/W_0</math>.  On the other hand, when Blaes refers to the modulus of the amplitude, he includes this leading "unity" constant in evaluating the "real" component of his expression.  I do not yet fully understand why both ways of viewing the eigenfunction's amplitude &#8212; that is, both with and without including the unity term &#8212; can be physically relevant.
</li>

<li>
As a small extension of the Blaes85 analysis, we should determine what the eigenfunction is for the ''density'' perturbation, rather than for the "enthalpy" perturbation, and see how well it matches the ''blue'' amplitude and phase curves published by Hadley et al. 
</li>
</ol>