Editing Appendix/Ramblings/ToHadleyAndImamura (section)

====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.