Editing Appendix/Ramblings/ToHadleyAndImamura (section)

====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  -->