Apps/ImamuraHadleyCollaboration - Revision history

Jet53man: /* Characteristics of Unstable Eigenvectors in Self-Gravitating Tori */

2021-10-04T18:41:24Z

Characteristics of Unstable Eigenvectors in Self-Gravitating Tori

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	=Characteristics of Unstable Eigenvectors in Self-Gravitating Tori=		=Characteristics of Unstable Eigenvectors in Self-Gravitating Tori=
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			! style="height: 125px; width: 125px; background-color:white;" \|[[H_BookTiledMenu#Toroidal_.26_Toroidal-Like_2\|<b>Hadley &<br />Imamura<br />Collaboration</b>]]
			\|}
	Using numerical hydrodynamic techniques, the [[#See_Also\|Imamura & Hadley collaboration]] — see especially [http://adsabs.harvard.edu/abs/2011Ap%26SS.334....1H Paper I] and [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Paper II] — has studied the dynamical development of nonaxisymmetric instabilities in toroidal configurations that have a range of "star-to-disk" mass ratios and a wide variety of (initially axisymmetric) geometric structures. Our principal aim, here, is to compare the results of these numerical investigations with what is known, analytically, about normal modes of oscillation and nonaxisymmetric instabilities in [[Apps/PapaloizouPringleTori#Massless_Polytropic_Tori\|''massless'' Papaloizou-Pringle tori]]. On the analytic side, our focus is on the very informative stability analysis published by [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985; hereafter, Blaes85)].		Using numerical hydrodynamic techniques, the [[#See_Also\|Imamura & Hadley collaboration]] — see especially [http://adsabs.harvard.edu/abs/2011Ap%26SS.334....1H Paper I] and [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Paper II] — has studied the dynamical development of nonaxisymmetric instabilities in toroidal configurations that have a range of "star-to-disk" mass ratios and a wide variety of (initially axisymmetric) geometric structures. Our principal aim, here, is to compare the results of these numerical investigations with what is known, analytically, about normal modes of oscillation and nonaxisymmetric instabilities in [[Apps/PapaloizouPringleTori#Massless_Polytropic_Tori\|''massless'' Papaloizou-Pringle tori]]. On the analytic side, our focus is on the very informative stability analysis published by [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985; hereafter, Blaes85)].

Jet53man: /* Characteristics of Unstable Eigenvectors in Self-Gravitating Tori */

2021-10-04T17:37:15Z

Characteristics of Unstable Eigenvectors in Self-Gravitating Tori

← Older revision		Revision as of 13:37, 4 October 2021
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	As [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] broadened his analysis to include somewhat thicker PP tori — [[#Tori_with_Small_but_Finite_.CE.B2\|finite, but still small, values of the order parameter]], β — he discovered that the eigenfrequency associated with one — and ''only'' one — quantum number pair becomes complex: (''j'',''k'') = (0,0). (He found that the associated radial ''eigenfunction'' is also complex.) Therefore, for all ''m'', this particular mode is expected to be dynamically unstable. [[#Comparison_with_Results_from_the_Imamura_.26_Hadley_Collaboration\|Below]], we have compared the equatorial-plane structure of this mode, as specified analytically by the Blaes85 analysis, to the nonaxisymmetric structure associated with unstable ''P-modes'' that has developed in simulations of slim tori conducted by the Imamura & Hadley collaboration. The structures are not only qualitatively, but ''quantitatively'' very similar. (NOTE: While constructing this comparison, we have come to appreciate for the first time that the ''imaginary'' component of the complex radial eigenfunction plays a key role in defining the "constant phase locus" signature of an unstable nonaxisymmetric mode.) This bolsters our confidence in the reliability and applicability of, both, the Blaes85 linear perturbation analysis and the nonlinear simulations presented by the Imamura & Hadley collaboration.		As [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] broadened his analysis to include somewhat thicker PP tori — [[#Tori_with_Small_but_Finite_.CE.B2\|finite, but still small, values of the order parameter]], β — he discovered that the eigenfrequency associated with one — and ''only'' one — quantum number pair becomes complex: (''j'',''k'') = (0,0). (He found that the associated radial ''eigenfunction'' is also complex.) Therefore, for all ''m'', this particular mode is expected to be dynamically unstable. [[#Comparison_with_Results_from_the_Imamura_.26_Hadley_Collaboration\|Below]], we have compared the equatorial-plane structure of this mode, as specified analytically by the Blaes85 analysis, to the nonaxisymmetric structure associated with unstable ''P-modes'' that has developed in simulations of slim tori conducted by the Imamura & Hadley collaboration. The structures are not only qualitatively, but ''quantitatively'' very similar. <font color="red">(NOTE: While constructing this comparison, we have come to appreciate for the first time that the ''imaginary'' component of the complex radial eigenfunction plays a key role in defining the "constant phase locus" signature of an unstable nonaxisymmetric mode.)</font> This bolsters our confidence in the reliability and applicability of, both, the Blaes85 linear perturbation analysis and the nonlinear simulations presented by the Imamura & Hadley collaboration.

Jet53man: /* Establishing the Simpler Eigenvalue Problem */

2021-09-23T22:56:16Z

Establishing the Simpler Eigenvalue Problem

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	<tr>		<tr>
	<td align="left">		<td align="left">
	This digital image has been posted by permission of author, O.M. Blaes ~~(via email dated 19 July 2020)~~, and by permission of Oxford University Press on behalf of the Royal Astronomical Society ~~(via email dated 31 July 2020)~~.		This digital image has been posted by permission of author, O.M. Blaes, and by permission of Oxford University Press on behalf of the Royal Astronomical Society.<div align="center">[[File:PermissionsRectYellow.png\|75px\|link=Appendix/CopyrightPermissions#Blaes1985]]<br />Original publication website: https://academic.oup.com/mnras/article/216/3/553/1005987</div>
	<div align="center">Original publication website: https://academic.oup.com/mnras/article/216/3/553/1005987</div>
	</td>		</td>
	</tr>		</tr>

Jet53man: /* Setup */

2021-09-23T22:53:54Z

Setup

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	<table border="1" cellpadding="5" align="right" width="30%">		<table border="1" cellpadding="5" align="right" width="30%">
	<tr><td align="center" bgcolor="~~orange~~">		<tr><td align="center" bgcolor="lightgreen">
	Fig. 1 extracted without modification from p. 554 of [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)]<p></p>		Fig. 1 extracted without modification from p. 554 of [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)]<p></p>
	"''Oscillations of Slender Tori''"<p></p>		"''Oscillations of Slender Tori''"<p></p>
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	<td align="left">		<td align="left">
	This digital image has been posted by permission of author, O.M. Blaes, and by permission of Oxford University Press on behalf of the Royal Astronomical Society.		This digital image has been posted by permission of author, O.M. Blaes, and by permission of Oxford University Press on behalf of the Royal Astronomical Society.
	<div align="center">Original publication website: https://academic.oup.com/mnras/article/216/3/553/1005987</div>		<div align="center">[[File:PermissionsRectYellow.png\|75px\|link=Appendix/CopyrightPermissions#Blaes1985]]<br />Original publication website: https://academic.oup.com/mnras/article/216/3/553/1005987</div>
	</td>		</td>
	</tr>		</tr>
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	<div align="center" id="EigenvalueBlaes85">		<div align="center" id="EigenvalueBlaes85">
	<table border="1" cellpadding="5" width="80%">		<table border="1" cellpadding="5" width="80%">
	<tr><td align="center" bgcolor="~~orange~~">		<tr><td align="center" bgcolor="lightgreen">
	Equation (3.2) extracted without modification from p. 558 of [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B O. M. Blaes (1985)]<p></p>		Equation (3.2) extracted without modification from p. 558 of [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B O. M. Blaes (1985)]<p></p>
	"''Oscillations of Slender Tori''"<p></p>		"''Oscillations of Slender Tori''"<p></p>
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	<tr>		<tr>
	<td align="left">		<td align="left">
	This digital image has been posted by permission of author, O.M. Blaes, and by permission of Oxford University Press on behalf of the Royal Astronomical Society.<div align="center">Original publication website: https://academic.oup.com/mnras/article/216/3/553/1005987</div>		This digital image has been posted by permission of author, O.M. Blaes, and by permission of Oxford University Press on behalf of the Royal Astronomical Society.<div align="center">[[File:PermissionsRectYellow.png\|75px\|link=Appendix/CopyrightPermissions#Blaes1985]]<br />Original publication website: https://academic.oup.com/mnras/article/216/3/553/1005987</div>
	</td>		</td>
	</tr>		</tr>

Jet53man: /* Setup */

2021-09-23T22:49:10Z

Setup

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	<tr>		<tr>
	<td align="left">		<td align="left">
	This digital image has been posted by permission of author, O.M. Blaes ~~(via email dated 19 July 2020)~~, and by permission of Oxford University Press on behalf of the Royal Astronomical Society ~~(via email dated 31 July 2020)~~.		This digital image has been posted by permission of author, O.M. Blaes, and by permission of Oxford University Press on behalf of the Royal Astronomical Society.
	<div align="center">Original publication website: https://academic.oup.com/mnras/article/216/3/553/1005987</div>		<div align="center">Original publication website: https://academic.oup.com/mnras/article/216/3/553/1005987</div>
	</td>		</td>
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	<tr>		<tr>
	<td align="left">		<td align="left">
	This digital image has been posted by permission of author, O.M. Blaes ~~(via email dated 19 July 2020)~~, and by permission of Oxford University Press on behalf of the Royal Astronomical Society ~~(via email dated 31 July 2020)~~.		This digital image has been posted by permission of author, O.M. Blaes, and by permission of Oxford University Press on behalf of the Royal Astronomical Society.<div align="center">Original publication website: https://academic.oup.com/mnras/article/216/3/553/1005987</div>
	<div align="center">Original publication website: https://academic.oup.com/mnras/article/216/3/553/1005987</div>
	</td>		</td>
	</tr>		</tr>

Jet53man: /* Establishing the Simpler Eigenvalue Problem */

2021-09-23T16:39:31Z

Establishing the Simpler Eigenvalue Problem

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	<div align="center" id="EigenvalueBlaes85">		<div align="center" id="EigenvalueBlaes85">
	<table border="1" cellpadding="5" width="80%">		<table border="1" cellpadding="5" width="80%">
	<tr><td align="center" bgcolor="~~orange~~">		<tr><td align="center" bgcolor="lightgreen">
	Equation (1.6) — identical to Eq. (3.5) — extracted without modification from p. 555 of [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B O. M. Blaes (1985)]<p></p>		Equation (1.6) — identical to Eq. (3.5) — extracted without modification from p. 555 of [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B O. M. Blaes (1985)]<p></p>
	"''Oscillations of Slender Tori''"<p></p>		"''Oscillations of Slender Tori''"<p></p>

Jet53man: /* Tori with Small but Finite β */

2021-09-20T23:12:49Z

Tori with Small but Finite β

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	</div>		</div>

	In an [[Appendix/Ramblings/~~PPToriPt2~~#Stability_Analyses_of_PP_Tori_.28Part_2.29\|accompanying chapter]] that we have relegated to our [[Appendix/Ramblings#Ramblings\|Ramblings Appendix]], we demonstrate in detail that this pair of ''complex'' expressions does provide a (leading order) solution to the "thick torus" eigenvalue problem. Notice that if <math>~\beta</math> is set to zero, these two expressions reduce to the (purely real) expressions for the ''j'' = ''k'' = 0, slender torus mode.		In an [[Appendix/Ramblings/PPToriPt2A#Stability_Analyses_of_PP_Tori_.28Part_2.29\|accompanying chapter]] that we have relegated to our [[Appendix/Ramblings#Ramblings\|Ramblings Appendix]], we demonstrate in detail that this pair of ''complex'' expressions does provide a (leading order) solution to the "thick torus" eigenvalue problem. Notice that if <math>~\beta</math> is set to zero, these two expressions reduce to the (purely real) expressions for the ''j'' = ''k'' = 0, slender torus mode.

	===Our Additional Analysis===		===Our Additional Analysis===

Jet53man: /* Generic Formulation */

2021-09-20T23:06:35Z

Generic Formulation

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	</div>		</div>

	An equatorial-plane plot of <math>~\varphi_\mathrm{max}(\eta)</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]].		An equatorial-plane plot of <math>~\varphi_\mathrm{max}(\eta)</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]].

	<!-- COMMENT OUT		<!-- COMMENT OUT

Jet53man: /* Radial Modes in Homogeneous Spheres */

2021-09-20T22:51:36Z

Radial Modes in Homogeneous Spheres

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	===Radial Modes in Homogeneous Spheres===		===Radial Modes in Homogeneous Spheres===

	Before attempting to analyze natural modes of oscillation in a polytropic torus, it is useful to review what is known about radial oscillations in the geometrically simpler, uniform-density sphere. As we have reviewed in [[SSC/UniformDensity#The_Stability_of_Uniform-Density_Spheres\|a separate chapter]], [http://adsabs.harvard.edu/abs/1937MNRAS..97..582S Sterne (1937)] was the first to recognize that the set of eigenvectors that describe radial modes of oscillation in a homogeneous, self-gravitating sphere can be determined analytically. In the present context, it is advantageous for us to pull Sterne's solution from a discussion of the same problem presented by [https://ia600302.us.archive.org/12/items/ThePulsationTheoryOfVariableStars/Rosseland-ThePulsationTheoryOfVariableStars.pdf Rosseland (1964)]. As we have summarized in [[SSC/~~Perspective_Reconciliation~~#Eulerian_Reformulation\|a separate chapter]], the relevant eigenvalue problem is defined by the following one-dimensional, 2<sup>nd</sup>-order ODE:		Before attempting to analyze natural modes of oscillation in a polytropic torus, it is useful to review what is known about radial oscillations in the geometrically simpler, uniform-density sphere. As we have reviewed in [[SSC/Stability/UniformDensity#The_Stability_of_Uniform-Density_Spheres\|a separate chapter]], [http://adsabs.harvard.edu/abs/1937MNRAS..97..582S Sterne (1937)] was the first to recognize that the set of eigenvectors that describe radial modes of oscillation in a homogeneous, self-gravitating sphere can be determined analytically. In the present context, it is advantageous for us to pull Sterne's solution from a discussion of the same problem presented by [https://ia600302.us.archive.org/12/items/ThePulsationTheoryOfVariableStars/Rosseland-ThePulsationTheoryOfVariableStars.pdf Rosseland (1964)]. As we have summarized in [[SSC/PerspectiveReconciliation#Eulerian_Reformulation\|a separate chapter]], the relevant eigenvalue problem is defined by the following one-dimensional, 2<sup>nd</sup>-order ODE:
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	Drawing from [https://ia600302.us.archive.org/12/items/ThePulsationTheoryOfVariableStars/Rosseland-ThePulsationTheoryOfVariableStars.pdf Rosseland's (1964)] presentation — see his p. 29, and [[SSC/~~Perspective_Reconciliation~~#Eulerian_Reformulation\|our related discussion]] — the following table details the eigenvectors (radial eigenfunction and associated eigenfrequency) for the three lowest radial modes (m = 2, 4, 6) that satisfy this wave equation; the figure displayed in the right-most column has been extracted directly from p. 29 of Rosseland and shows the behavior of the lowest five radial modes (m = 2, 4, 6, 8, 10, as labeled) over the interval <math>~0 \le \chi_0 \le 1</math>.		Drawing from [https://ia600302.us.archive.org/12/items/ThePulsationTheoryOfVariableStars/Rosseland-ThePulsationTheoryOfVariableStars.pdf Rosseland's (1964)] presentation — see his p. 29, and [[SSC/PerspectiveReconciliation#Eulerian_Reformulation\|our related discussion]] — the following table details the eigenvectors (radial eigenfunction and associated eigenfrequency) for the three lowest radial modes (m = 2, 4, 6) that satisfy this wave equation; the figure displayed in the right-most column has been extracted directly from p. 29 of Rosseland and shows the behavior of the lowest five radial modes (m = 2, 4, 6, 8, 10, as labeled) over the interval <math>~0 \le \chi_0 \le 1</math>.

	<div align="center">		<div align="center">
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	We should point out that, except for the lowest (m = 2) mode, each of the radial eigenfunctions crosses zero at least once at some location(s) that resides between the center <math>~(\chi_0=0)</math> of and the surface <math>~(\chi_0 = 1)</math> of the sphere. More specifically, for each mode, <math>~m</math>, the number of such radial "nodes" is <math>~(m-2)/2</math>. The locations of these nodes is apparent from even a casual inspection of the figure presented in the right-most column of Table 2.		We should point out that, except for the lowest (m = 2) mode, each of the radial eigenfunctions crosses zero at least once at some location(s) that resides between the center <math>~(\chi_0=0)</math> of and the surface <math>~(\chi_0 = 1)</math> of the sphere. More specifically, for each mode, <math>~m</math>, the number of such radial "nodes" is <math>~(m-2)/2</math>. The locations of these nodes is apparent from even a casual inspection of the figure presented in the right-most column of Table 2.

	When these radial modes of oscillation are discussed in the astrophysics literature, the conditions that give rise to a dynamical instability are often emphasized. Specifically, each mode becomes unstable when <math>\sigma^2</math> becomes negative, which translates into a value of <math>~\gamma < \gamma_\mathrm{crit}(m)</math> — see our [[SSC/UniformDensity#Properties_of_Eigenfunction_Solutions\|related discussion of the properties of eigenfunction solutions]] in the context of [http://adsabs.harvard.edu/abs/1937MNRAS..97..582S Sterne's (1937)] analysis of this stability problem. Here we want to emphasize that all of natural modes of oscillation ''exist'' even when the configuration is dynamically stable.		When these radial modes of oscillation are discussed in the astrophysics literature, the conditions that give rise to a dynamical instability are often emphasized. Specifically, each mode becomes unstable when <math>\sigma^2</math> becomes negative, which translates into a value of <math>~\gamma < \gamma_\mathrm{crit}(m)</math> — see our [[SSC/Stability/UniformDensity#Properties_of_Eigenfunction_Solutions\|related discussion of the properties of eigenfunction solutions]] in the context of [http://adsabs.harvard.edu/abs/1937MNRAS..97..582S Sterne's (1937)] analysis of this stability problem. Here we want to emphasize that all of natural modes of oscillation ''exist'' even when the configuration is dynamically stable.

	===Singular Sturm-Liouville Problem===		===Singular Sturm-Liouville Problem===

Jet53man: /* Imamura & Hadley Collaboration */

2021-09-20T22:46:33Z

Imamura & Hadley Collaboration

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	==Background==		==Background==
	===Imamura & Hadley Collaboration===		===Imamura & Hadley Collaboration===
	Motivated especially by the numerical study of unstable nonaxisymmetric modes provided in [http://adsabs.harvard.edu/abs/2011Ap%26SS.334....1H Paper I] and [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Paper II] of the [[#See_Also\|Imamura & Hadley collaboration]], we have embarked on an effort to better understand what sparks the development of a wide range of complex eigenvectors in self-gravitating tori. Of particular interest are "P-", "J-", and "E-modes," such as the ones whose eigenfunctions are illustrated here in Table 1. (Click on a small image in the right-hand column to see a larger image.) Our initial attempts to construct fits to these eigenvectors empirically have been described in [[Appendix/Ramblings/~~Azimuthal_Distortions~~#Analyzing_Azimuthal_Distortions\|a separate chapter]]. We begin, here, a more quantitative analysis of these structures.		Motivated especially by the numerical study of unstable nonaxisymmetric modes provided in [http://adsabs.harvard.edu/abs/2011Ap%26SS.334....1H Paper I] and [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Paper II] of the [[#See_Also\|Imamura & Hadley collaboration]], we have embarked on an effort to better understand what sparks the development of a wide range of complex eigenvectors in self-gravitating tori. Of particular interest are "P-", "J-", and "E-modes," such as the ones whose eigenfunctions are illustrated here in Table 1. (Click on a small image in the right-hand column to see a larger image.) Our initial attempts to construct fits to these eigenvectors empirically have been described in [[Appendix/Ramblings/AzimuthalDistortions#Analyzing_Azimuthal_Distortions\|a separate chapter]]. We begin, here, a more quantitative analysis of these structures.

	<table border="1" cellpadding="8" align="center">		<table border="1" cellpadding="8" align="center">