Jet53man: /* Stability Analyses of PP Tori */

2021-10-05T00:36:19Z

Stability Analyses of PP Tori

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	=Stability Analyses of PP Tori=		=Stability Analyses of PP Tori=

	<font color="red"><b>[Comment by J. E. Tohline on 24 May 2016]</b></font>   This chapter contains a set of technical notes and accompanying discussion that I put together several months ago as I was trying to gain a foundational understanding of the results of a large study of instabilities in self-gravitating tori published by the Imamura & Hadley collaboration. I have come to appreciate that some of the logic and interpretation of published results that are presented, below, has serious flaws. Therefore, anyone reading this should be quite cautious in deciding what subsections provide useful insight. I have written a separate chapter titled, "[[~~User:Tohline/~~Apps/ImamuraHadleyCollaboration#Characteristics_of_Unstable_Eigenvectors_in_Self-Gravitating_Tori\|Characteristics of Unstable Eigenvectors in Self-Gravitating Tori]]," that contains a much more trustworthy analysis of this very interesting problem.		<font color="red"><b>[Comment by J. E. Tohline on 24 May 2016]</b></font>   This chapter contains a set of technical notes and accompanying discussion that I put together several months ago as I was trying to gain a foundational understanding of the results of a large study of instabilities in self-gravitating tori published by the Imamura & Hadley collaboration. I have come to appreciate that some of the logic and interpretation of published results that are presented, below, has serious flaws. Therefore, anyone reading this should be quite cautious in deciding what subsections provide useful insight. I have written a separate chapter titled, "[[Apps/ImamuraHadleyCollaboration#Characteristics_of_Unstable_Eigenvectors_in_Self-Gravitating_Tori\|Characteristics of Unstable Eigenvectors in Self-Gravitating Tori]]," that contains a much more trustworthy analysis of this very interesting problem.

	{{ SGFworkInProgress }}		{{ SGFworkInProgress }}

	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_\mathrm{max}(\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/~~Apps/WoodwardTohlineHachisu94#Online_Movies\|here]]; these videos supplement the published work of [http://adsabs.harvard.edu/abs/1994ApJ...420..247W Woodward, Tohline & Hachisu (1994)].		As has been summarized in [[Appendix/Ramblings/ToHadleyAndImamura#Summary_for_Hadley_.26_Imamura\|an accompanying chapter]] — also see our related [[Appendix/Ramblings/AzimuthalDistortions#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_\mathrm{max}(\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 [[Apps/WoodwardTohlineHachisu94#Online_Movies\|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 three publications:		Here, we focus specifically on instabilities that arise in so-called (non-self-gravitating) [[Apps/PapaloizouPringleTori#Massless_Polytropic_Tori\|Papaloizou-Pringle tori]] and will draw heavily from three publications:
	* [http://adsabs.harvard.edu/abs/1987MNRAS.225..267P Papaloizou & Pringle (1987), MNRAS, 225, 267] (''aka'' PPIII) — ''The dynamical stability of differentially rotating discs.   III.''		* [http://adsabs.harvard.edu/abs/1987MNRAS.225..267P Papaloizou & Pringle (1987), MNRAS, 225, 267] (''aka'' PPIII) — ''The dynamical stability of differentially rotating discs.   III.''
	* [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985), MNRAS, 216, 553] (''aka'' Blaes85) — ''Oscillations of slender tori.''		* [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985), MNRAS, 216, 553] (''aka'' Blaes85) — ''Oscillations of slender tori.''
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	===Equilibrium Configuration===		===Equilibrium Configuration===
	In our [[~~User:Tohline/~~Apps/PapaloizouPringleTori#Solution\|separate discussion of PP84]], we showed that the equilibrium structure of a PP-torus is defined by the enthalpy distribution,		In our [[Apps/PapaloizouPringleTori#Solution\|separate discussion of PP84]], we showed that the equilibrium structure of a PP-torus is defined by the enthalpy distribution,
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	Normalizing this expression by the enthalpy at the "center" — ''i.e.,'' at the pressure maximum — of the torus which, as we have [[~~User:Tohline/~~Apps/PapaloizouPringleTori#Pressure_Maximum\|already shown]], is		Normalizing this expression by the enthalpy at the "center" — ''i.e.,'' at the pressure maximum — of the torus which, as we have [[Apps/PapaloizouPringleTori#Pressure_Maximum\|already shown]], is
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	</div>		</div>

	Now, in our review of [[~~User:Tohline/~~Apps/PapaloizouPringleTori#Model_as_Described_by_Kojima\|Kojima's (1986)]] work, we showed that the square of the Mach number at the "center" of the torus is,		Now, in our review of [[Apps/PapaloizouPringleTori#Model_as_Described_by_Kojima\|Kojima's (1986)]] work, we showed that the square of the Mach number at the "center" of the torus is,

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	This matches equation (2.2) of Blaes85, if we acknowledge that Blaes uses <math>~f</math> — instead of the parameter notation, <math>~\Theta_H</math>, found in [[~~User:Tohline/~~SSC/Structure/Polytropes#Governing_Relations\|our discussion of equilibrium polytropic configurations]] — to denote the normalized enthalpy; that is,		This matches equation (2.2) of Blaes85, if we acknowledge that Blaes uses <math>~f</math> — instead of the parameter notation, <math>~\Theta_H</math>, found in [[SSC/Structure/Polytropes#Governing_Relations\|our discussion of equilibrium polytropic configurations]] — to denote the normalized enthalpy; that is,

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	Following the same set of steps that were followed in [[~~User:Tohline/~~Appendix/Ramblings/PPTori#Outer_.5Binferior_sign.5D_Solution\|determining the "outer" solution]], here we find: <math>~Q</math> remains the same; <math>~R</math> has the same magnitude, but changes sign; and, hence, <math>~D</math> remains the same. We therefore have,		Following the same set of steps that were followed in [[Appendix/Ramblings/PPTori#Outer_.5Binferior_sign.5D_Solution\|determining the "outer" solution]], here we find: <math>~Q</math> remains the same; <math>~R</math> has the same magnitude, but changes sign; and, hence, <math>~D</math> remains the same. We therefore have,
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	===Analytically Prescribed Eigenvector===		===Analytically Prescribed Eigenvector===
	====Our Notation====		====Our Notation====
	As is explicitly defined in [[~~User:Tohline/~~Appendix/Ramblings/~~Azimuthal_Distortions~~#Figure1\|Figure 1 of our accompanying detailed notes]], we have chosen to represent the spatial structure of an eigenfunction in the equatorial-plane of toroidal-like configurations via the expression,		As is explicitly defined in [[Appendix/Ramblings/AzimuthalDistortions#Figure1\|Figure 1 of our accompanying detailed notes]], we have chosen to represent the spatial structure of an eigenfunction in the equatorial-plane of toroidal-like configurations via the expression,
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	An equatorial-plane plot of <math>~\phi_\mathrm{max}(\varpi)</math> should produce the "constant phase locus" referenced, for example, in recent papers from the [[~~User:Tohline/~~Appendix/Ramblings/~~To_Hadley_and_Imamura~~#Summary_for_Hadley_.26_Imamura\|Imamura & Hadley collaboration]].		An equatorial-plane plot of <math>~\phi_\mathrm{max}(\varpi)</math> should produce the "constant phase locus" referenced, for example, in recent papers from the [[Appendix/Ramblings/ToHadleyAndImamura#Summary_for_Hadley_.26_Imamura\|Imamura & Hadley collaboration]].

	<!-- SECOND ATTEMPT		<!-- SECOND ATTEMPT
	This is the form that has been adopted broadly by the numerical simulation community, as graphical displays of <math>~f_m(\varpi)</math> and <math>~\phi_m(\varpi)</math> have been used to study the structure of unstable eigenmodes — see, for example, our discussion of [[~~User:Tohline/~~Appendix/Ramblings/~~To_Hadley_and_Imamura~~#Summary_for_Hadley_.26_Imamura\|simulation results published by the Imamura & Hadley collaboration]]. Multiplying this expression through by its complex conjugate gives the square of the modulus of the function.		This is the form that has been adopted broadly by the numerical simulation community, as graphical displays of <math>~f_m(\varpi)</math> and <math>~\phi_m(\varpi)</math> have been used to study the structure of unstable eigenmodes — see, for example, our discussion of [[Appendix/Ramblings/ToHadleyAndImamura#Summary_for_Hadley_.26_Imamura\|simulation results published by the Imamura & Hadley collaboration]]. Multiplying this expression through by its complex conjugate gives the square of the modulus of the function.
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	<!-- [<font color="red"><b>NOTE:</b></font> Initially, I wrote "+ k" rather than "- k" in the exponent of the exponential term on the RHS of this expression; but experience shows that "- k" is required to achieve proper behavior of the "constant phase locus" plot, as displayed below.] -->		<!-- [<font color="red"><b>NOTE:</b></font> Initially, I wrote "+ k" rather than "- k" in the exponent of the exponential term on the RHS of this expression; but experience shows that "- k" is required to achieve proper behavior of the "constant phase locus" plot, as displayed below.] -->
	where we have written the perturbation amplitude in a manner that conforms with the notation that we have used in [[~~User:Tohline/~~Appendix/Ramblings/~~Azimuthal_Distortions~~#Figure1\|Figure 1 of a related, but more general discussion]]. As is summarized in §1.3 of Blaes (1985), for "thick" (but, actually, still quite thin) tori, "exactly one exponentially growing mode exists for each value of the azimuthal wavenumber <math>~m</math>," and its complex amplitude takes the following form (see his equation 1.10):		where we have written the perturbation amplitude in a manner that conforms with the notation that we have used in [[Appendix/Ramblings/AzimuthalDistortions#Figure1\|Figure 1 of a related, but more general discussion]]. As is summarized in §1.3 of Blaes (1985), for "thick" (but, actually, still quite thin) tori, "exactly one exponentially growing mode exists for each value of the azimuthal wavenumber <math>~m</math>," and its complex amplitude takes the following form (see his equation 1.10):

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	=====Step 3=====		=====Step 3=====
	From our [[~~User:Tohline/~~Apps/PapaloizouPringle84#isolatingBlaes85\|accompanying discussion of the Blaes85 derivation]], we expect the following equality to hold (see his equations 4.1 and 4.2):		From our [[Apps/PapaloizouPringle84#isolatingBlaes85\|accompanying discussion of the Blaes85 derivation]], we expect the following equality to hold (see his equations 4.1 and 4.2):
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Jet53man: /* Stability Analyses of PP Tori */

Jet53man: Created page with "__FORCETOC__ =Stability Analyses of PP Tori= [Comment by J. E. Tohline on 24 May 2016] This chapt..."

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