Editing Appendix/Ramblings/AzimuthalDistortions (section)

==Adopted Notation==

Beginning with equation (2) of [http://adsabs.harvard.edu/abs/1990ApJ...361..394T TH90] but ignoring variations in the vertical coordinate direction, the mass density is given by the expression,
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<math>~\rho</math>
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<math>~=</math>
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<math>~\rho_0 \biggl[ 1 + f(\varpi)e^{-i(\omega t - m\phi)} \biggr] \, ,</math>
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</table>
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where it is understood that <math>~\rho_0</math>, which defines the structure of the initial axisymmetric equilibrium configuration, is generally a function of the cylindrical radial coordinate, <math>~\varpi</math>.

Using the subscript, <math>~m</math>, to identify the time-invariant coefficients and functions that characterize the intrinsic eigenvector of each azimuthal eigen-mode, and acknowledging that the associated eigenfrequency will in general be imaginary, that is,
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<math>~\omega_m</math>
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<math>~=</math>
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<math>~\omega_R + i\omega_I \, ,</math>
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we expect each unstable mode to display the following behavior:
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<math>~\biggl[ \frac{\rho}{\rho_0} - 1 \biggr]</math>
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<math>~=</math>
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<math>~f_m(\varpi)e^{-i[\omega_R t + i \omega_I t - m\phi_m(\varpi)]}  </math>
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&nbsp;
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<math>~=</math>
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<math>~\biggl\{ f_m(\varpi)e^{-im\phi_m(\varpi)}\biggr\} e^{-i\omega_R t } \cdot  e^{\omega_I t}  </math>
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&nbsp;
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<math>~=</math>
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<math>~\biggl\{ f_m(\varpi)e^{-i[\omega_R t + m\phi_m(\varpi)]} \biggr\} e^{\omega_I t}  \, .</math>
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</table>
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Adopting [http://adsabs.harvard.edu/abs/1986PThPh..75..251K Kojima's (1986)] notation, that is, defining,
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<math>~y_1 \equiv \frac{\omega_R}{\Omega_0} - m</math>
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&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp;
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<math>~y_2 \equiv \frac{\omega_I}{\Omega_0} \, ,</math>
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the eigenvector's behavior can furthermore be described by the expression,
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<math>~\biggl[ \frac{\rho}{\rho_0} - 1 \biggr]</math>
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<math>~=</math>
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<math>~\biggl\{ f_m(\varpi)e^{-i[(y_1+m) (\Omega_0 t) + m\phi_m(\varpi)]} \biggr\} e^{y_2 (\Omega_0 t)}  </math>
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&nbsp;
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<math>~=</math>
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<math>~\biggl\{ f_m(\varpi)e^{-im[(y_1/m+1) (\Omega_0 t) + \phi_m(\varpi)]} \biggr\} e^{y_2 (\Omega_0 t)}  \, .</math>
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Note that, as viewed from a frame of reference that is rotating with the mode pattern frequency,
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<math>\Omega_p \equiv \frac{\omega_R}{m} = \Omega_0\biggl(\frac{y_1}{m}+1\biggr) \, ,</math>
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we should find an eigenvector of the form,
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<math>~\biggl[ \frac{\delta\rho}{\rho_0}\biggr]_\mathrm{rot} \equiv \biggl[ \frac{\rho}{\rho_0} - 1 \biggr]e^{im\Omega_p t}</math>
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<math>~=</math>
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<math>~\biggl\{ f_m(\varpi)e^{-im[\phi_m(\varpi)]} \biggr\} e^{y_2 (\Omega_0 t)}  \, ,</math>
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</table>
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whose relative amplitude &#8212; with a radial structure as specified inside the curly braces &#8212; is undergoing a uniform exponential growth but is otherwise unchanging.  

Drawing from figure 2 of [http://adsabs.harvard.edu/abs/1994ApJ...420..247W WTH94], our Figure 1, immediately below, illustrates how the behavior of each factor in this expression can reveal itself during a numerical simulation that follows the time-evolutionary development of an unstable, nonaxisymmetric eigenmode.  The initial model for this depicted evolution (model O3 from Table 1 of [http://adsabs.harvard.edu/abs/1994ApJ...420..247W WTH94]) is a zero-mass &#8212; that is, it is a [[Apps/PapaloizouPringleTori#Massless_Polytropic_Tori|Papaloizou-Pringle like torus]] &#8212; with [[SR#Barotropic_Structure|polytropic index]],<math>~n = 3</math>, and a rotation-law profile defined by [[AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential|uniform specific angular momentum]].  
* The top-left panel shows how, at any radial location, the phase angle, <math>~\phi_1/(2\pi)</math>, for the <math>~m=1</math> eigenmode, varies with time, <math>~t/t_\mathrm{rot}</math>, where, <math>~t_\mathrm{rot} \equiv 2\pi/\Omega_0</math> is the rotation period at the density maximum;
* Using a semi-log plot, the top-right panel shows the exponential growth of the amplitude of three separate modes:  The dominant unstable mode, displaying the largest amplitude, is <math>~m = 1</math>.
* Using a semi-log plot (log amplitude versus fractional radius, <math>~\varpi/r_+</math>), the bottom-left panel displays the shape of the eigenfunction, <math>~f_1(\varpi)</math>, for the unstable, <math>~m=1</math> mode;
* The bottom-right panel displays the radial dependence of the equatorial-plane phase angle, <math>~\phi_1(\varpi)</math>, for the unstable, <math>~m=1</math> mode; this is what [http://adsabs.harvard.edu/abs/2011Ap%26SS.334....1H HI11] refer to as the "constant phase locus."


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<b><font size="+1">Figure 1</font></b>
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'''Four panels extracted<sup>&dagger;</sup> from Figure 2, p. 252 of<br />[http://adsabs.harvard.edu/abs/1994ApJ...420..247W J. W. Woodward, J. E. Tohline &amp; I. Hachisu (1994)]'''<br />
''The Stability of Thick, Self-gravitating Disks in Protostellar Systems''<br />
ApJ, vol. 420, pp. 247-267 &copy; [http://aas.org/ American Astronomical Society]<br />[https://doi.org/10.1086/173556 https://doi.org/10.1086/173556]<br />
[[File:PermissionsRectYellow.png|75px|link=Appendix/CopyrightPermissions#WTH94]]
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[[File:Diagram01.png|550px|Rearranged Figure 2 from Woodward, Tohline, and Hachisu (1994)]]
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<tr><td align="left"><sup>&dagger;</sup>As displayed here, the layout of figure panels (a, b, c, d) has been modified from the original publication layout; otherwise, each panel is unmodified.</td></tr>
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