Editing Appendix/Ramblings/ToHadleyAndImamura (section)

===Blaes (1985)===
====His Derived Eigenfunction====
Via an analytic perturbation analysis, [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] shows that, in slender PP tori with uniform specific angular momentum, the spatial structure of unstable modes can be expressed as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{W(\eta,\theta)}{W_0} -1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{ f_m(\eta,\theta)e^{-i[m\phi + k\theta]} \biggr\}  \, ,</math>
  </td>
</tr>
</table>
</div>

where (see his equation 1.10),

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

<tr>
  <td align="right">
<math>~f_m(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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>
  </td>
</tr>
</table>
</div>

In this expression, <math>~n</math> is the polytropic index, and <math>~\beta</math> is a dimensionless parameter that specifies the relative thickness of the torus; specifically, in terms of the ratio of the inner-to-outer edge of the torus, <math>~r_-/r_+</math>,
<div align="center">
<math>~\beta \equiv \frac{1- r_-/r_+}{1+ r_-/r_+} \, .</math>
</div>

Notice that this eigenfunction derived by [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] describes a spatial perturbation that is not confined to the equatorial plane of the torus.  The parameter pair, <math>~(\eta,\theta)</math>, defines a two-dimensional, polar coordinate system in a plane lying perpendicular to the equatorial plane that contains the torus's (nearly circular) cross-section:  <math>~\eta</math> is a dimensionless radial coordinate measuring distance from the center of the cross-section &#8212; <math>~\eta = 0</math> at the center of the cross-section and <math>~\eta = 1</math> at the surface of the torus; and <math>~\theta</math> is the polar-coordinate oriented such that <math>~\theta = 0</math> points toward the "inner" edge of the torus, <math>~\theta = \pi/2</math> points vertically, straight "up," and <math>~\theta = \pi</math> points toward the "outer" edge of the torus.  

Paralleling the [[#Generic_Formulation|generic formulation that has just been discussed]], this is a complex function having, to lowest order, the following real and imaginary parts:

<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>~(\beta m)^2\biggl[
2\eta^2 \cos^2\theta - \frac{3\eta^2}{4(n+1)} - \frac{(4n+1)}{4(n+1)^2} \biggr]
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{(\beta m)^2}{4(n+1)^2}\biggl[2^7\cdot 3(n+1)^3
\eta^2\cos^2\theta \biggr]^{1/2}\, .
</math>
  </td>
</tr>
</table>
</div>

We should therefore find that the amplitude (modulus) of the perturbation is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl|\frac{W}{W_0} -1\biggr|</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\sqrt{\mathcal{A}_\mathrm{Blaes}^2+ \mathcal{B}_\mathrm{Blaes}^2} \, ;</math>
  </td>
</tr>
</table>
</div>
and the associated "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{\mathcal{A}_\mathrm{Blaes}}{\mathcal{B}_\mathrm{Blaes}} \biggr]  - k\theta 
\, .</math>
  </td>
</tr>
</table>
</div>
<!-- OLD, INCORRECT DESCRIPTION THAT IGNORES LEADING "1" IN DEFINITION OF REAL COMPONENT...
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~f_m(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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>
  </td>
</tr>
</table>
</div>

In this expression, <math>~n</math> is the polytropic index, and <math>~\beta</math> is a dimensionless parameter that specifies the relative thickness of the torus; specifically, in terms of the ratio of the inner-to-outer edge of the torus, <math>~r_-/r_+</math>,
<div align="center">
<math>~\beta \equiv \frac{1- r_-/r_+}{1+ r_-/r_+} \, .</math>
</div>

Notice that this eigenfunction derived by [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] describes a spatial perturbation that is not confined to the equatorial plane of the torus.  The parameter pair, <math>~(\eta,\theta)</math>, defines a two-dimensional, polar coordinate system in a plane lying perpendicular to the equatorial plane that contains the torus's (nearly circular) cross-section:  <math>~\eta</math> is a dimensionless radial coordinate measuring distance from the center of the cross-section &#8212; <math>~\eta = 0</math> at the center of the cross-section and <math>~\eta = 1</math> at the surface of the torus; and <math>~\theta</math> is the polar-coordinate oriented such that <math>~\theta = 0</math> points toward the "inner" edge of the torus, <math>~\theta = \pi/2</math> points vertically, straight "up," and <math>~\theta = \pi</math> points toward the "outer" edge of the torus.  

Paralleling the [[#Generic_Formulation|generic formulation that has just been discussed]], this is a complex function having, to lowest order, the following real and imaginary parts:

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

<tr>
  <td align="right">
<math>~\mathcal{A}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~2\eta^2 \cos^2\theta - \frac{3\eta^2}{4(n+1)} - \frac{(4n+1)}{4(n+1)^2} 
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
<math>~\mathcal{B}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
4\biggl(\frac{3}{2n+2}\biggr)^{1/2} \eta\cos\theta 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{4(n+1)^2}\biggl[2^7\cdot 3(n+1)^3
\eta^2\cos^2\theta \biggr]^{1/2}\, .
</math>
  </td>
</tr>
</table>
</div>

We should therefore find that the amplitude (modulus) of the enthalpy perturbation is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{(m\beta)^2}\biggl|\frac{\delta W}{W_0} \biggr|</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\sqrt{\mathcal{A}^2+ \mathcal{B}^2} \, ;</math>
  </td>
</tr>
</table>
</div>
and the associated "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{\mathcal{A}}{\mathcal{B}} \biggr]  - k\theta 
\, .</math>
  </td>
</tr>
</table>
</div>

END OLD INCORRECT DESCRIPTION  -->

====Nice Features====
After summarizing, above, our efforts to develop (by empirical techniques) mathematical expressions that qualitatively match the shape of unstable eigenfunctions in toroidal configurations, we put together a subsection titled, "[[#Additional_Comments|Additional Comments]]," to highlight ways in which the empirical fits were successful and ways in which they fell short of expectations.  Here, following our presentation of Blaes's (1985) analytically derived eigenfunction for slim PP tori, we highlight elements of his (physically justified) eigenfunction that explain the origin of many features that were highlighted, above.

* The "amplitude" plot and a plot of the "constant phase locus" ''should'', indeed, appear to be interdependent because they both depend on the functional forms of <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>; in the end, however, the two plots are formally independent of one another.
* The square of the modulus &#8212; that is, the "amplitude" plot &#8212; should always be the sum of two independent functions, both of which are intrinsically positive.
* If either <math>~\mathcal{A}</math> or <math>~\mathcal{B}</math> is defined by a function that crosses zero (perhaps multiple times) with a naturally continuous derivative, that function can quite naturally give rise to a "log-amplitude" plot that shoots toward minus infinity and exhibits a discontinuous derivative after the function is squared to become a piece of the "modulus" expression.
* It is now easy to understand why the earlier "empirically derived" constant phase loci were phrased in terms of the arctangent function.
* Each "constant phase locus" plot should naturally be composed of two (smoothly joined) pieces:  One defined over the inner portion of the torus &#8212; <math>~\eta</math> goes from zero to 1 while <math>~\theta = 0</math>; and one defined over the outer portion of the torus &#8212; <math>~\eta</math> goes from zero to 1 while <math>~\theta = \pi</math>.
* Because (at least for slim PP-tori) the ratio, <math>~\mathcal{A}/\mathcal{B}</math> contains an overall factor that is an odd power of <math>~\cos\theta</math>, the argument of the arctangent function will automatically flip signs as we move from the "inner" region of the torus to the "outer" region of the torus.
* We now appreciate that a plot of "constant phase locus" is smooth across the mid-point (across the cross-sectional center) of the torus because the definition of <math>~\phi_\mathrm{max}</math> naturally contains a phase shift of <math>~k\theta</math>.

====Movie====

Figure 3 displays, for various values of the polytropic index, <math>~0.0 \le n\le 5.0</math>, the radial structure of the Blaes85 <math>~m=2</math> eigenfunction for a slim PP-torus having <math>~\beta = 0.18</math>.  More specifically, for each frame of the animation, the figure displays as a function of the dimensionless radial coordinate, <math>~x</math>, (left panel) the behavior of
<div align="center">
'''Left:'''&nbsp; &nbsp; <math>~\sqrt{\mathcal{A}_\mathrm{Blaes}^2 + \mathcal{B}_\mathrm{Blaes}^2}</math>
</div>
on a semi-log plot; and (right panel) the "constant phase locus,"
<div align="center">
'''Right:'''&nbsp; &nbsp; <math>~\frac{1}{2} \tan^{-1} \biggl[ \frac{\mathcal{A}_\mathrm{Blaes}}{\mathcal{B}_\mathrm{Blaes}} \biggr] - \theta \, .</math>
</div>

<div align="center" id="Fig3">
<table border="1" align="center" cellpadding="2" width="610px">
<tr>
  <td align="center" colspan="2">
<font size="+1"><b>Figure 3:</b></font> &nbsp; The Blaes85 Analytic Eigenfunction
  </td>
</tr>

<tr><td align="center" bgcolor="gray" colspan="2">
[[File:Blaes85Analytic.gif|600px|Amp and Phase from Blaes (1985)]]
</td></tr>
<tr>
  <td align="left" colspan="2">
CAPTION: &nbsp; Radial dependence of the (left) amplitude and (right) "constant phase locus" of the Blaes85 analytic eigenfunction, for various values of the polytropic Index, <math>~n</math>, as indicated in the bottom-right corner of the left panel.
  </td>
</tr>
</table>
</div>