Editing Apps/RotatingPolytropes/BarmodeLinearTimeDependent (section)

==Summary ''circa'' 2000==

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<tr><td align="center" bgcolor="black">[[File:Dissertation.fig3cropped.png|250px|Model A Composite Images]]</td></tr>
<tr><td align="center">Cazes' Model A</td></tr>
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This summary is drawn &#8212; largely verbatim &#8212; from a [http://www.phys.lsu.edu/astro/movie_captions/two_armed_spiral.html separate web page] that was originally intended to highlight animation sequences produced by LSU's astrophysicsl theory group.

In the mid-1980s, using three different 3D hydrodynamic simulation tools, [https://ui.adsabs.harvard.edu/abs/1986ApJ...305..281D/abstract R. H. Durisen, R. A. Gingold, J. E. Tohline &amp; A. P. Boss (1986)] &#8212; hereafter and in the references, below, DGTB86 &#8212; followed the development to nonlinear amplitudes of a two-armed, spiral-mode instability that arises naturally in rapidly rotating, self-gravitating fluids.  The particular initial model that was examined had an <math>~n = \tfrac{3}{2}</math> polytropic equation of state, an n' = 0 angular momentum distribution, and an initial ratio of rotational to gravitational potential energy T/|W| = 0.33.  The simulation tools that were used at the time were relatively crude, compared to tools that are available today.

Employing a significantly improved finite-volume simulation code and improved spatial resolution (128<sup>3</sup> grid zones), a very similar simulation has been carried out and reported by [https://ui.adsabs.harvard.edu/abs/2000ApJ...532.1051C/abstract J. E. Cazes &amp; J. E. Tohline (2000)]; see also the [https://digitalcommons.lsu.edu/gradschool_disstheses/6982/ Cazes (1999)] doctoral dissertation.  This was done, in part, to check the validity of the earlier work and, in part, to permit us to conduct a much more thorough analysis of the end-state configuration.  We will refer to this as the ''Model A'' simulation.  The initial model for this simulation was constructed using the [[AxisymmetricConfigurations/HSCF#Hachisu_Self-Consistent-Field_Technique|Hachisu Self-Consistent-Field]] (HSCF) technique; it had an <math>~n = \tfrac{3}{2}</math> polytropic equation of state, an n' = 0 angular momentum distribution, and an initial ratio of rotational to gravitational potential energy T/|W| = 0.30.  As has been cataloged in [[#Table1|Table 1, below]], a YouTube animation carrying ID = [https://youtu.be/BhRUqZe0Ly4 BhRUqZe0Ly4] shows the nonlinear development of the bar-mode &#8212; actually, two-armed, spiral-mode &#8212; instability in Model A.  The evolution is shown in the inertial reference frame and covers 46 dynamical times (20 central initial rotation periods) as defined by the properties of the initial, equilibrium axisymmetric model.  Each frame of the ''Model A'' movie displays four nested isodensity contours at 80%, 40%, 4% and 0.4% of the maximum density.  Via the trailing spiral structure, gravitational torques are able to effectively redistribute angular momentum on a dynamical time scale; a relatively small amount of material is shed into an equatorial disk.  (This disk material is not visible in the ''Model A'' movie because the disk material all has a mass density less than 0.4% of the maximum density.)  Over time, the central object (containing most of the initial object's mass) settles down into a new quasi-equilibrium configuration.  The six images displayed in the upper portion of color figure shown here, on the right, have been extracted from this animation at six different points in time (measured in terms of the central initial rotation period), as labeled.

For comparison, we also have studied the development of the same type of bar-mode (two-armed, spiral) instability in a model which will be referred to here as ''Model B''.  It had an <math>~n = \tfrac{3}{2}</math> polytropic equation of state and an initial ratio of rotational to gravitational potential energy T/|W| = 0.28, but with an angular momentum distribution specified to produce a uniform ''vortensity'' profile in the initial model.  (Vortensity is defined as the ratio of vorticity to mass density.)  A YouTube animation carrying ID = [https://youtu.be/Qh-bMto2e3k Qh-bMto2e3k] shows this evolution from an inertial frame and covers approximately 32 dynamical times as defined by the mean density of the initial model.  The instability in this model has less of a pronounced spiral character, but ultimately results in the formation of a new triaxial, quasi-equilibrium configuration with properties that are similar to the end state of the ''Model A'' evolution.

As illustrated and discussed elsewhere, we are convinced that the ellipsoidal-like configurations that have been formed through both of these model simulations are compressible analogs of Riemann S-type ellipsoids.


<div id="Table1">
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<td align="center" colspan="4"><b>Table 1: &nbsp; Accompanying YouTube Animations</b> (click on ID)</td>
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<td align="right" colspan="2" width="50%">Model:</td>
<td align="center" colspan="1">A</td>
<td align="center" colspan="1">B</td>
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<td align="right" colspan="2">Nonlinear Development of Bar-mode Instability (isodensity surfaces):</td>
<td align="center" colspan="1"><b><font color="darkgreen">[https://youtu.be/BhRUqZe0Ly4 BhRUqZe0Ly4]</font></b></td>
<td align="center" colspan="1"><b><font color="darkgreen">[https://youtu.be/Qh-bMto2e3k Qh-bMto2e3k]</font></b></td>
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<td align="right" colspan="1" rowspan="4">Quasi-Steady-State "CARE"</td>
<td align="right" colspan="1">Isodensity Surfaces:</td>
<td align="center" colspan="1"><b><font color="darkgreen">[https://youtu.be/z3RDthHnY3A z3RDthHnY3A]</font></b></td>
<td align="center" colspan="1"><b><font color="darkgreen">[https://youtu.be/Eyl81O9s62s Eyl81O9s62s]</font></b></td>
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<td align="right" colspan="1">Equatorial-Plane Fluid Flow:</td>
<td align="center" colspan="1"><b><font color="darkgreen">[https://youtu.be/fgjjzu3Nmo8 fgjjzu3Nmo8]</font></b></td>
<td align="center" colspan="1"><b><font color="darkgreen">[https://youtu.be/yggbz9z8JQ4 yggbz9z8JQ4]</font></b></td>
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<td align="right" colspan="1">Andalib's Analogous ''Prograde'' Flow-Field:</td>
<td align="center" colspan="2"><b><font color="darkgreen">[https://youtu.be/p550IDP8h-s p550IDP8h-s]</font></b></td>
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<td align="right" colspan="1">Meridional Fluid Flow:</td>
<td align="center" colspan="1"><b><font color="darkgreen">[https://youtu.be/_nXKv0k0h-Q _nXKv0k0h-Q]</font></b></td>
<td align="center" colspan="1">n/a</td>
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<td align="right" colspan="1" rowspan="4">''Cooling'' Evolution</td>
<td align="right" colspan="1">Isodensity Surfaces:</td>
<td align="center" colspan="1"><b><font color="darkgreen">[https://youtu.be/erYWHgaKYyc erYWHgaKYyc]</font></b></td>
<td align="center" colspan="1">n/a</td>
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<td align="right" colspan="1">Equatorial-Plane Fluid Flow:</td>
<td align="center" colspan="1"><b><font color="darkgreen">[https://youtu.be/nC4adQzlXrI nC4adQzlXrI]</font></b></td>
<td align="center" colspan="1">n/a</td>
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<td align="right" colspan="1">Andalib's Analogous ''Binary'' Flow-Field:</td>
<td align="center" colspan="1"><b><font color="darkgreen">[https://youtu.be/nkwRJh2tgos nkwRJh2tgos]</font></b></td>
<td align="center" colspan="1">n/a</td>
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<td align="right" colspan="1">Meridional Fluid Flow:</td>
<td align="center" colspan="1"><b><font color="darkgreen">[https://youtu.be/S0tg2jURzGE S0tg2jURzGE]</font></b></td>
<td align="center" colspan="1">n/a</td>
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