Difference between revisions of "User:Tohline/Appendix/Ramblings/PPTori"

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In what follows, we will draw heavily from two publications:  (1) [http://adsabs.harvard.edu/abs/1987MNRAS.225..267P Papaloizou & Pringle (1987), MNRAS, 225, 267] — ''The dynamical stability of differentially rotating discs.   III.'' — hereafter, PPIII — and (2)  [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985), MNRAS, 216, 553] — ''Oscillations of slender tori''.
As has been summarized in [[User:Tohline/Appendix/Ramblings/To_Hadley_and_Imamura#Summary_for_Hadley_.26_Imamura|an accompanying chapter]] &#8212; also see our related [[User:Tohline/Appendix/Ramblings/Azimuthal_Distortions#Analyzing_Azimuthal_Distortions|detailed notes]] &#8212; we have been trying to understand why unstable nonaxisymmetric eigenvectors have the shapes that they do in rotating toroidal configurations.  For any azimuthal mode, <math>~m</math>, we are referring both to the radial dependence of the distortion amplitude, <math>~f_m(\varpi)</math>, and the radial dependence of the phase function, <math>~\phi_m(\varpi)</math> &#8212; the latter is what the [[#See_Also|Imamura and Hadley collaboration]] refer to as a "constant phase locus."  Some old videos showing the development over time of various self-gravitating "constant phase loci" can be found [[User:Tohline/Appendix/Ramblings/Azimuthal_Distortions#Analyzing_Azimuthal_Distortions|here]]; these videos supplement the published work of [http://adsabs.harvard.edu/abs/1994ApJ...420..247W Woodward, Tohline &amp; Hachisu (1994)].
 
Here, we focus specifically on instabilities that arise in so-called (non-self-gravitating) [[User:Tohline/Apps/PapaloizouPringleTori#Massless_Polytropic_Tori|Papaloizou-Pringle tori]] and will draw heavily from two publications:  (1) [http://adsabs.harvard.edu/abs/1987MNRAS.225..267P Papaloizou &amp; Pringle (1987), MNRAS, 225, 267] &#8212; ''The dynamical stability of differentially rotating discs. &nbsp; III.'' &#8212; hereafter, PPIII &#8212; and (2)  [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985), MNRAS, 216, 553] &#8212; ''Oscillations of slender tori''.


==PP III==
==PP III==

Revision as of 18:42, 18 February 2016

Stability Analyses of PP Tori

Whitworth's (1981) Isothermal Free-Energy Surface
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As has been summarized in an accompanying chapter — also see our related detailed notes — we have been trying to understand why unstable nonaxisymmetric eigenvectors have the shapes that they do in rotating toroidal configurations. For any azimuthal mode, <math>~m</math>, we are referring both to the radial dependence of the distortion amplitude, <math>~f_m(\varpi)</math>, and the radial dependence of the phase function, <math>~\phi_m(\varpi)</math> — the latter is what the Imamura and Hadley collaboration refer to as a "constant phase locus." Some old videos showing the development over time of various self-gravitating "constant phase loci" can be found here; these videos supplement the published work of Woodward, Tohline & Hachisu (1994).

Here, we focus specifically on instabilities that arise in so-called (non-self-gravitating) Papaloizou-Pringle tori and will draw heavily from two publications: (1) Papaloizou & Pringle (1987), MNRAS, 225, 267The dynamical stability of differentially rotating discs.   III. — hereafter, PPIII — and (2) Blaes (1985), MNRAS, 216, 553Oscillations of slender tori.

PP III

Figure 2 extracted without modification from p. 274 of J. C. B. Papaloizou & J. E. Pringle (1987)

"The Dynamical Stability of Differentially Rotating Discs.   III"

MNRAS, vol. 225, pp. 267-283 © The Royal Astronomical Society

Figure 2 from PP III

See Also


Whitworth's (1981) Isothermal Free-Energy Surface

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