Editing Apps/ImamuraHadleyCollaboration (section)

=====Visual Comparison=====
With the above information in hand, we can now directly compare the eigenfunction of the unstable, ''m'' = 2 mode discovered and defined analytically by [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] with the eigenfunction of unstable, ''m'' = 2 modes found by the [[#See_Also|Imamura &amp; Hadley collaboration]] while using numerical hydrodynamic techniques to simulate the evolution of similarly slim tori.  Panel A of Figure 3 displays a pair of plots, extracted from [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Hadley et al. (2014; Paper II)], that present information regarding the ''m'' = 2 mode, nonaxisymmetric structure that developed in the equatorial plane of their model P4.  The model P4 disk/torus is geometrically slim <math>~(\beta \approx 0.18)</math> and its mass, although not zero, is only 1% of the mass of the central star <math>~(M_*/M_\mathrm{disk} = 10^2)</math>.   Panel B of Figure 3 displays a similar pair of plots from a separate, [[Appendix/Ramblings/HadleyAndImamuraSupplementaryDatabase#Supplementary_Dataset_Generated_by_Hadley_.26_Imamura_Collaboration|unpublished Hadley et al. model evolution]]; in this model, the disk/torus mass is only 0.1% that of the central star <math>~(M_*/M_\mathrm{disk} = 10^3)</math>.  In both panels, the blue curves show the geometric structure of non-axisymmetric, equatorial-plane density fluctuations, <math>~|\rho^'/\rho_0|_2</math>, and the red curves show the structure of the "enthalpy + gravity" perturbation, <math>~\mathcal{W}</math>:  The left-hand plot displays log<sub>10</sub> of (the modulus of) the amplitude versus radius; while the right-hand plot displays the unstable mode's constant phase locus.  In each of these plots, the radial coordinate is, <math>~\chi \equiv \varpi/\varpi_0</math>.  


Panel C of Figure 3 presents similar plots that we have generated to show the ''m'' = 2 mode, non-axisymmetric, equatorial-plane density and enthalpy fluctuations predicted by the [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] analysis.  More specifically, on the left, we have plotted (blue curve) <math>~\log_{10}|g_{0,0,2}|</math> versus <math>~\chi</math>, for <math>~\beta = 0.12</math> and a normalization coefficient, <math>~C = 4</math>; and (red curve) <math>~\log_{10}\mathcal{W}</math> versus <math>~\chi</math>, where we have used a normalization coefficient, <math>~C = 2</math>.  (See the third column of Table 5 for example data values.)  On the right, we present an equatorial-plane plot of (see columns five and six of Table 5) the constant phase locus, <math>~\tfrac{1}{2}\alpha_{0,0,2}</math> versus <math>~\chi</math>, and &#8212; reflecting the ''m'' = 2 mode structure &#8212; its twin phase locus, shifted in entirety by <math>~\pi</math> radians. [Note that, in the right-hand plot we have flipped the sign of <math>~\varphi_\mathrm{max}</math> in order to match the sign convention exhibited by the constant phase locus plots presented by the [[#See_Also|Imamura &amp; Hadley collaboration]].]

 
<div align="center" id="Figure3">
<table border="1" align="center" cellpadding="5">
<tr><th><font size="+1">Figure 3:  Comparison</font></th></tr>
<tr><th><font size="+1">Panel A: &nbsp;&nbsp; Model P4 from [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Hadley et al. (2014; Paper II)];</font><p></p>
<math>~M_*/M_\mathrm{disk} = 100</math></th></tr>
<tr><td align="right">
[[File:Montage01Apart1.png|500px|center|Imamura &amp; Hadley collaboration]]
</td></tr>
<tr><th><font size="+1">Panel B: &nbsp;&nbsp; [[Appendix/Ramblings/HadleyAndImamuraSupplementaryDatabase#Supplementary_Dataset_Generated_by_Hadley_.26_Imamura_Collaboration|Unpublished model]] from Imamura &amp; Hadley collaboration;</font><p></p>
<math>~M_*/M_\mathrm{disk} = 1000</math></th></tr>
<tr><td align="right">
[[File:Montage01Apart3.png|500px|center|Imamura &amp; Hadley collaboration]]
</td></tr>
<tr><th><font size="+1">Panel C: &nbsp;&nbsp; [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] Analytic Solution with</font> <math>~(\beta, m, C) = (0.12, 2, 0.25)</math>;<p></p>
<math>~M_*/M_\mathrm{disk} = \infty</math></th></tr>
<tr><td align="right">
[[File:BlaesAnalyticCombined3.png|500px|center|Analytic Solution from Blaes(1985)]]
</td></tr>
</table>
</div>