Editing Apps/ImamuraHadleyCollaboration (section)

=====Plots of a Few Example Eigenvectors=====

[http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] makes the following brief comments regarding the role of the indices, ''j'' and ''k'', in defining the geometric distortions associated with each normal mode.  "<font color="green">The significance of ''k'' is obvious &#8212; it is simply the number of wavelengths on a given isobaric surface</font> [''i.e.'', varying <math>~\theta</math> while holding <math>~\eta</math> constant] <font color="green">so that there are 2''k'' nodal lines radiating from the torus centre at</font> <math>~\eta = 0</math>."  Alternatively, moving radially through the torus at a fixed <math>~\theta</math>, "<font color="green">In addition to the</font> [radial] <font color="green">node at <math>~\eta = 0</math> which occurs for <math>~k \ne 0</math>, there are nodal surfaces at ''j'' other distinct values of <math>~\eta</math> inside the torus</font>."


The plots presented below in Figures 1 and 2 are intended to further illustrate the radial structure of various normal modes, and to begin to draw a connection between the analytically describable attributes of ''slender'' PP tori and the broad range of self-gravitating tori whose stability has been investigated numerically by the [[#See_Also|Imamura &amp; Hadley collaboration]].  For nine separate "slender torus" normal modes, panels A, B, &amp; C of Figure 1 display the radial variation of:  (Middle) <math>~\delta W_{j,k}^{(0)}</math> versus <math>~\eta</math>; (Left) the associated Jacobi polynomial, <math>~J_{j}^{n-1,|k|}</math> versus <math>~x</math>, where <math>~x = (2\eta^2-1)</math>; and (Right) <math>\log_{10}|\delta W_{j,k}^{(0)}|</math> versus <math>~\eta</math>.  In panel A, all three displayed modes have ''j'' = 0; in panel B, all modes have ''j'' = 1; and in panel C, all modes have ''j'' = 2.  In all three of these panels (A, B, C), blue curves are associated with ''k'' = 0; red curves are associated with ''k'' = 1; and green curves are associated with ''k'' = 2.  


The left-most segment of each panel (A, B, C) displays the specific Jacobi polynomial that seeds the oscillatory behavior of that panel's associated radial eigenvector; they have been shown here in order to emphasize overlap with solutions to the singular Sturm-Liouville problem as illustrated in [[#Table3|Table 3, above]].  In reality, the right-most segment of each panel (A, B, C) presents the same information as is presented in the middle segment, but by plotting the log of the absolute value of the radial eigenfunction we are displaying that information in a manner that aligns with the means of presentation used by the [[#See_Also|Imamura &amp; Hadley collaboration]].  Because the log of this function goes to minus infinity whenever the eigenfunction crosses zero, it is particularly easy to identify the number and location of radial nodes in the right-most segment of each panel.  Referencing again the comment by [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)], there is a node at <math>~\eta = 0</math>, except when ''k = 0'' (the blue curves in each panel of our Figure 1); in addition, there are nodal surfaces at ''j'' other distinct values of <math>~\eta</math> inside the torus &#8212; that is, between <math>~\eta = 0</math> and  <math>~\eta = 1</math> or between <math>~\eta = 0</math> and  <math>~\eta = -1</math>.


<div align="center">
<table border="1" cellpadding="5" align="center">
<tr><th><font size="+1">Figure 1:  Blaes85 Eigenfunctions for Slender Tori with <math>~(n,q) = (\tfrac{3}{2},2)</math></font></th></tr>
<tr><th><font size="+1">Panel A: &nbsp;&nbsp; j = 0 and (k = 0, 1, 2)</font></th></tr>
<tr><td align="center">
[[File:N1.5j0_Combinedsmall.png|750px|center|j0 Eigenfunction from Blaes85]]
</td></tr>
<tr><th><font size="+1">Panel B: &nbsp;&nbsp; j = 1 and (k = 0, 1, 2)</font></th></tr>
<tr><td align="center">
[[File:N1.5j1_Combinedsmall.png|750px|center|j1 Eigenfunction from Blaes85]]
</td></tr>
<tr><th><font size="+1">Panel C: &nbsp;&nbsp; j = 2 and (k = 0, 1, 2)</font></th></tr>
<tr><td align="center">
[[File:N1.5j2_Combinedsmall.png|750px|center|j2 Eigenfunction from Blaes85]]
</td></tr>
</table>
</div>


<span id="CorotationMode">
<font color="maroon"><b>COROTATION MODE:</b></font> As [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] has pointed out, for any azimuthal number, ''m'', the simplest mode occurs for ''j'' = ''k'' = 0. </span> In this case, the [[#Analytic_Solution|analytic expression]] for the ''slender torus'' eigenfrequency is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\sigma^{(0)}_{0,0,m}  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-m \Omega_0 \, ,</math>
  </td>
</tr>
</table>
</div>
and, the associated analytic expression for the eigenfunction is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\delta W_{0,0,m}^{(0)}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~C ~\exp[i(m\varphi + \sigma^{(0)}_{0,0,m} t)]  ~J_0^{n-1,0}(2\eta^2-1) </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>C~\exp[im(\varphi - \Omega_0 t)] \, ,</math>
  </td>
</tr>
</table>
</div>

where, <math>~C</math> is an arbitrary, overall scale factor.  A plot showing the radial structure of this "simplest mode" &#8212; assuming <math>~C=1</math> &#8212; is provided by the blue curve in the middle segment and in the right-hand segment of panel A.  The perturbation, <math>~\delta W_{0,0,m}^{(0)}</math>, has a constant amplitude throughout the configuration.  Note, however, that as the left-most segment of panel D in Figure 2 shows, the fractional ''density'' perturbation is not uniform throughout the configuration; this is primarily because an extra factor of <math>~(1-\eta^2)^{-1}</math> appears in the expression for <math>~\rho^'/\rho_0</math> &#8212; [[#DensityPerturbation2|see above]].

While this <font color="maroon"><b>COROTATION</b></font> mode exhibits a rather boring structure relative to other modes, it plays a key role in the Blaes (1985) publication.  As his analysis is expanded to include the examination of oscillations in tori with finite &#8212; but still small &#8212; equilibrium values of <math>~\beta</math>, he finds that, for all ''m'', both the eigenfunction and the eigenfrequency of the ''j'' = ''k'' = 0 mode exhibit nonzero imaginary components.  (More on this, [[#Tori_with_Small_but_Finite_.CE.B2|immediately below]].)


<div align="center">
<table border="1" cellpadding="5" align="center" width="750px">
<tr><th colspan="3"><font size="+1">Figure 2:  Mass Density Eigenfunctions for Slender Tori with <math>~(n,q) = (\tfrac{3}{2},2)</math></font></th></tr>
<tr><th colspan="3"><font size="+1">Panel D: &nbsp;&nbsp; j = 0 (left), j = 1 (middle), and j = 2 (right) and, in each case, (k = 0, 1, 2)</font></th></tr>
<tr><td align="right" colspan="3">
[[File:LogDensityCombinedSmall01.png|700px|right|j0 Eigenfunction from Blaes85]]
</td></tr>
<tr><th colspan="3"><font size="+1">Panel E: &nbsp;&nbsp; Models J1 (left), E3 (middle) and E2 (right) from Figs. 2 &amp; 4 of [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Hadley, et al. (2014)]</font><p></p>
(blue curves show log10 of mass-density fluctuation amplitudes)</th></tr>
<tr>
  <td align="right" width="250px">[[File:ImamuraPaper2Fig2ModelJ1.png|right|250px|Model J1 Radial Eigenfunction]]</td>
  <td align="right" width="250px">[[File:ImamuraPaper2Fig4ModelE3Revised.png|right|220px|Model E3 Radial Eigenfunction]]</td>
  <td align="right" width="250px">[[File:ImamuraPaper2Fig4Eigenfunction.png|right|200px|Model E2 Radial Eigenfunction]]</td>
</tr>
</table>
</div>


<font color="maroon"><b>MODEL E3 from [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Hadley, et al. (2014)]:</b></font> In the middle segment of Figure 2's panel E, we have re-printed the semi-log plot of the magnitude of the radial eigenfunction that developed in model E3 from [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Hadley, et al. (2014)].  Both curves in the plot exhibit two radial nodes:  One associated with the co-rotation radius, which we have purposely aligned with the node at <math>~\eta = 0</math> that appears in the middle segment of panel D; and one that lies between the center of the torus and the surface.  This association strongly suggests that the unstable mode found in model E3 displays an underlying ''linear'' radial eigenfunction akin to the "''j'' = 1" Jacobi polynomial. Also, because the model E3 eigenfunction exhibits a node at co-rotation, we conclude that <math>~k \ne 0</math>; and, while the spacing between the two E3 nodes suggests a value of the index ''k'' greater than unity, identifying the precise value of ''k'' may require an examination of the node count above/below the equatorial plane along one or more isobaric surfaces.


<font color="maroon"><b>MODEL E2 from [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Hadley, et al. (2014)]:</b></font>  In the right-most segment of Figure 2's panel E, we have re-printed the semi-log plot of the magnitude of the radial eigenfunction that developed in model E2 from [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Hadley, et al. (2014)].  In addition to the radial node aligned with co-rotation, both curves in the plot exhibit (at least) four radial nodes.  This strongly suggests that this unstable mode displays an underlying radial eigenfunction that is a polynomial of degree, ''j'' = 4 (or higher).


<font color="maroon"><b>MODEL J1 from [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Hadley, et al. (2014)]:</b></font>  In the left-most segment of Figure 2's panel E, we have re-printed the semi-log plot of the magnitude of the radial eigenfunction that developed in model J1 from [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Hadley, et al. (2014)].  In all respects, the blue curve (associated with the nonaxisymmetric structure of the density fluctuation) resembles the red curve displayed in the left-most segment of panel D.  Because this <font color="maroon"><b>MODEL J1</b></font> eigenfunction exhibits a single radial node that is closely aligned with the center of the torus strongly suggests that the unstable mode corresponds to an oscillation mode having <math>~j = 0</math>, but <math>~k \ne 0</math>.  Given its similarity to the red curve in left-most panel of panel D, we suspect that <math>~k = 1</math>.