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As has been summarized in [[User:Tohline/Appendix/Ramblings/To_Hadley_and_Imamura#Summary_for_Hadley_.26_Imamura|an accompanying chapter]] — also see our related [[User:Tohline/Appendix/Ramblings/Azimuthal_Distortions#Analyzing_Azimuthal_Distortions|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 [[#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 & 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 & 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''. | |||
==PP III== | ==PP III== | ||
Revision as of 18:42, 18 February 2016
Stability Analyses of PP Tori
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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, 267 — The dynamical stability of differentially rotating discs. III. — hereafter, PPIII — and (2) Blaes (1985), MNRAS, 216, 553 — Oscillations of slender tori.
PP III
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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 |
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See Also
- Imamura & Hadley collaboration:
- Paper I: K. Hadley & J. N. Imamura (2011, Astrophysics and Space Science, 334, 1-26), "Nonaxisymmetric instabilities in self-gravitating disks. I. Toroids" — In this paper, Hadley & Imamura perform linear stability analyses on fully self-gravitating toroids; that is, there is no central point-like stellar object and, hence, <math>~M_*/M_d = 0.0</math>.
- Paper II: K. Z. Hadley, P. Fernandez, J. N. Imamura, E. Keever, R. Tumblin, & W. Dumas (2014, Astrophysics and Space Science, 353, 191-222), "Nonaxisymmetric instabilities in self-gravitating disks. II. Linear and quasi-linear analyses" — In this paper, the Imamura & Hadley collaboration performs "an extensive study of nonaxisymmetric global instabilities in thick, self-gravitating star-disk systems creating a large catalog of star/disk systems … for star masses of <math>~0.0 \le M_*/M_d \le 10^3</math> and inner to outer edge aspect ratios of <math>~0.1 < r_-/r_+ < 0.75</math>."
- Paper III: K. Z. Hadley, W. Dumas, J. N. Imamura, E. Keever, & R. Tumblin (2015, Astrophysics and Space Science, 359, article id. 10, 23 pp.), "Nonaxisymmetric instabilities in self-gravitating disks. III. Angular momentum transport" — In this paper, the Imamura & Hadley collaboration carries out nonlinear simulations of nonaxisymmetric instabilities found in self-gravitating star/disk systems and compares these results with the linear and quasi-linear modeling results presented in Papers I and II.
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© 2014 - 2021 by Joel E. Tohline |