Apps/PapaloizouPringle84 - Revision history

Jet53man: /* Nonaxisymmetric Instability in Papaloizou-Pringle Tori */

2021-09-20T22:06:22Z

Nonaxisymmetric Instability in Papaloizou-Pringle Tori

← Older revision		Revision as of 18:06, 20 September 2021
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	=Nonaxisymmetric Instability in Papaloizou-Pringle Tori=		=Nonaxisymmetric Instability in Papaloizou-Pringle Tori=
	Equilibrium, axisymmetric '''structures''' are obtained by searching for time-independent, steady-state solutions to the [[AxisymmetricConfigurations/PGE#Axisymmetric_Configurations_.28Part_I.29\|identified set of simplified governing equations]].		Equilibrium, axisymmetric '''structures''' are obtained by searching for time-independent, steady-state solutions to the [[AxisymmetricConfigurations/PGE#Axisymmetric_Configurations_.28Part_I.29\|identified set of simplified governing equations]].
	~~  <br />~~
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	==Linearized Principal Governing Equations in Cylindrical Coordinates==		==Linearized Principal Governing Equations in Cylindrical Coordinates==

Jet53man: /* Equivalent Dimensionless Expression */

2021-09-20T21:57:27Z

Equivalent Dimensionless Expression

← Older revision		Revision as of 17:57, 20 September 2021
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	====Equivalent Dimensionless Expression====		====Equivalent Dimensionless Expression====

	Now, as should be clear from [[SR#Barotropic_Structure\|our introductory description of barotropic structures]], because the initial, unperturbed [[Apps/PapaloizouPringleTori#Solution\|Papaloizou-Pringle torus]] is a polytropic configuration, the functions <math>~P_0(\varpi,z)</math> and <math>~\rho_0(\varpi,z)</math> can both be expressed in terms of the [[Appendix/Ramblings/~~PPTori~~#Equilibrium_Configuration\|dimensionless enthalpy distribution]],		Now, as should be clear from [[SR#Barotropic_Structure\|our introductory description of barotropic structures]], because the initial, unperturbed [[Apps/PapaloizouPringleTori#Solution\|Papaloizou-Pringle torus]] is a polytropic configuration, the functions <math>~P_0(\varpi,z)</math> and <math>~\rho_0(\varpi,z)</math> can both be expressed in terms of the [[Appendix/Ramblings/PPToriPt1A#Equilibrium_Configuration\|dimensionless enthalpy distribution]],

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Jet53man: /* Checking Self-Consistency */

2021-09-20T21:52:27Z

Checking Self-Consistency

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	</div>		</div>

	Adopting [http://adsabs.harvard.edu/abs/1986PThPh..75..251K Kojima's (1986)] <math>~y_1</math> and <math>~y_2</math> notation, which we have discussed in a [[Appendix/Ramblings/~~Azimuthal_Distortions~~#Adopted_Notation\|separate but closely related chapter]], we therefore have,		Adopting [http://adsabs.harvard.edu/abs/1986PThPh..75..251K Kojima's (1986)] <math>~y_1</math> and <math>~y_2</math> notation, which we have discussed in a [[Appendix/Ramblings/AzimuthalDistortions#Adopted_Notation\|separate but closely related chapter]], we therefore have,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Jet53man at 21:50, 20 September 2021

2021-09-20T21:50:46Z

← Older revision		Revision as of 17:50, 20 September 2021
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	==Linearized Principal Governing Equations in Cylindrical Coordinates==		==Linearized Principal Governing Equations in Cylindrical Coordinates==
	We begin by drawing [[~~User:Tohline/Cylindrical_3D~~/Linearization#Summary\|from an accompanying derivation]] the relevant set of linearized principal governing equations, written in cylindrical coordinates but, following the lead of [http://adsabs.harvard.edu/abs/1984MNRAS.208..721P Papaloizou & Pringle] (1984, MNRAS, 208, 721-750; hereafter, PP84), express each perturbation in the form,		We begin by drawing [[Cylindrical3D/Linearization#Summary\|from an accompanying derivation]] the relevant set of linearized principal governing equations, written in cylindrical coordinates but, following the lead of [http://adsabs.harvard.edu/abs/1984MNRAS.208..721P Papaloizou & Pringle] (1984, MNRAS, 208, 721-750; hereafter, PP84), express each perturbation in the form,
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	</div>		</div>

	In line with [[~~User:Tohline/~~AxisymmetricConfigurations/SolutionStrategies#SRPtable\|our own discussion of ''simple rotation profiles'']] (but note that, in [[~~User:Tohline/~~AxisymmetricConfigurations/SolutionStrategies#SRPtable\|that chapter]], the variable we use for the power-law exponent is different from theirs), GGN86 adopt a generalized power-law rotation profile of the form (see their equation 2.1),		In line with [[AxisymmetricConfigurations/SolutionStrategies#SRPtable\|our own discussion of ''simple rotation profiles'']] (but note that, in [[AxisymmetricConfigurations/SolutionStrategies#SRPtable\|that chapter]], the variable we use for the power-law exponent is different from theirs), GGN86 adopt a generalized power-law rotation profile of the form (see their equation 2.1),
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	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	</div>		</div>

	Adopting [http://adsabs.harvard.edu/abs/1986PThPh..75..251K Kojima's (1986)] <math>~y_1</math> and <math>~y_2</math> notation, which we have discussed in a [[~~User:Tohline/~~Appendix/Ramblings/Azimuthal_Distortions#Adopted_Notation\|separate but closely related chapter]], we therefore have,		Adopting [http://adsabs.harvard.edu/abs/1986PThPh..75..251K Kojima's (1986)] <math>~y_1</math> and <math>~y_2</math> notation, which we have discussed in a [[Appendix/Ramblings/Azimuthal_Distortions#Adopted_Notation\|separate but closely related chapter]], we therefore have,
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	<math>~\kappa^2 = \frac{1}{\varpi^3} \frac{d j_0^2}{d \varpi} \, ,</math>		<math>~\kappa^2 = \frac{1}{\varpi^3} \frac{d j_0^2}{d \varpi} \, ,</math>
	</div>		</div>
	where, <math>~j_0(\varpi)\equiv \varpi^2{\dot\varphi}_0(\varpi)</math> is a function that specifies how the fluid's specific angular momentum varies with radius in the initial, unperturbed, equilibrium configuration. (See our related discussion of [[~~User:Tohline/~~AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential\|''Simple Rotation Profiles'']].) From this relation, we recognize as well that,		where, <math>~j_0(\varpi)\equiv \varpi^2{\dot\varphi}_0(\varpi)</math> is a function that specifies how the fluid's specific angular momentum varies with radius in the initial, unperturbed, equilibrium configuration. (See our related discussion of [[AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential\|''Simple Rotation Profiles'']].) From this relation, we recognize as well that,
	<div align="center">		<div align="center">
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	====Equivalent Dimensionless Expression====		====Equivalent Dimensionless Expression====

	Now, as should be clear from [[~~User:Tohline/~~SR#Barotropic_Structure\|our introductory description of barotropic structures]], because the initial, unperturbed [[~~User:Tohline/~~Apps/PapaloizouPringleTori#Solution\|Papaloizou-Pringle torus]] is a polytropic configuration, the functions <math>~P_0(\varpi,z)</math> and <math>~\rho_0(\varpi,z)</math> can both be expressed in terms of the [[~~User:Tohline/~~Appendix/Ramblings/PPTori#Equilibrium_Configuration\|dimensionless enthalpy distribution]],		Now, as should be clear from [[SR#Barotropic_Structure\|our introductory description of barotropic structures]], because the initial, unperturbed [[Apps/PapaloizouPringleTori#Solution\|Papaloizou-Pringle torus]] is a polytropic configuration, the functions <math>~P_0(\varpi,z)</math> and <math>~\rho_0(\varpi,z)</math> can both be expressed in terms of the [[Appendix/Ramblings/PPTori#Equilibrium_Configuration\|dimensionless enthalpy distribution]],

	<div align="center">		<div align="center">
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	</div>		</div>

	In a [[~~User:Tohline/~~Apps/Blaes85SlimLimit\|separate chapter]], we dissect the Blaes85 assertion that Jacobi Polynomials provide an analytic eigenvector solution to this specific "slim torus" eigenvalue problem.		In a [[Apps/Blaes85SlimLimit\|separate chapter]], we dissect the Blaes85 assertion that Jacobi Polynomials provide an analytic eigenvector solution to this specific "slim torus" eigenvalue problem.

	====GGN86====		====GGN86====

Jet53man: /* Formulation of Eigenvalue Problem */

2021-09-20T21:05:55Z

Formulation of Eigenvalue Problem

← Older revision		Revision as of 17:05, 20 September 2021
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	==Formulation of Eigenvalue Problem==		==Formulation of Eigenvalue Problem==
			{\| class="AxisymmetricConfigurations" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
			\|-
			! style="height: 125px; width: 125px; background-color:lightgreen;" \|[[H_BookTiledMenu#Toroidal_.26_Toroidal-Like_2\|<b>Defining the<br />Eigenvalue Problem</b>]]
			\|}
	In an effort to better understand the full three-dimensional structure of the eigenvector that characterizes nonaxisymmetric instabilities in Papaloizou-Pringle tori, we have studied in considerable depth five published analyses: The discovery paper, [http://adsabs.harvard.edu/abs/1984MNRAS.208..721P PP84], and papers by four separate groups that were published within a couple of years of the discovery paper — [http://adsabs.harvard.edu/abs/1985MNRAS.213..799P Papaloizou & Pringle (1985)], [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)], [http://adsabs.harvard.edu/abs/1986PThPh..75..251K Kojima (1986)], and [http://adsabs.harvard.edu/abs/1986MNRAS.221..339G Goldreich, Goodman & Narayan (1986)]. In order to quantitatively compare and contrast the results of these separate analyses, it has been necessary to understand the terminology, technical approaches, and, in particular, the variable names adopted by each group. In order to accomplish this, here we show how the two-dimensional, 2<sup>nd</sup>-order PDE that fundamentally defines the PP-torus eigenvalue problem is derived in each case; in each of the four subsequently published papers, the derivation has its roots in the original PP84 derivation, as it should.		In an effort to better understand the full three-dimensional structure of the eigenvector that characterizes nonaxisymmetric instabilities in Papaloizou-Pringle tori, we have studied in considerable depth five published analyses: The discovery paper, [http://adsabs.harvard.edu/abs/1984MNRAS.208..721P PP84], and papers by four separate groups that were published within a couple of years of the discovery paper — [http://adsabs.harvard.edu/abs/1985MNRAS.213..799P Papaloizou & Pringle (1985)], [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)], [http://adsabs.harvard.edu/abs/1986PThPh..75..251K Kojima (1986)], and [http://adsabs.harvard.edu/abs/1986MNRAS.221..339G Goldreich, Goodman & Narayan (1986)]. In order to quantitatively compare and contrast the results of these separate analyses, it has been necessary to understand the terminology, technical approaches, and, in particular, the variable names adopted by each group. In order to accomplish this, here we show how the two-dimensional, 2<sup>nd</sup>-order PDE that fundamentally defines the PP-torus eigenvalue problem is derived in each case; in each of the four subsequently published papers, the derivation has its roots in the original PP84 derivation, as it should.

Jet53man: /* Nonaxisymmetric Instability in Papaloizou-Pringle Tori */

2021-09-20T21:05:07Z

Nonaxisymmetric Instability in Papaloizou-Pringle Tori

← Older revision		Revision as of 17:05, 20 September 2021
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	<!-- __NOTOC__ will force TOC off -->		<!-- __NOTOC__ will force TOC off -->
	=Nonaxisymmetric Instability in Papaloizou-Pringle Tori=		=Nonaxisymmetric Instability in Papaloizou-Pringle Tori=
	~~{\| class="AxisymmetricConfigurations" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"~~
	\|-
	~~! style="height: 125px; width: 125px; background-color:lightgreen;" \|[[H_BookTiledMenu#Toroidal_.26_Toroidal-Like_2\|<b>Defining the<br />Eigenvalue Problem</b>]]~~
	\|}
	Equilibrium, axisymmetric '''structures''' are obtained by searching for time-independent, steady-state solutions to the [[AxisymmetricConfigurations/PGE#Axisymmetric_Configurations_.28Part_I.29\|identified set of simplified governing equations]].		Equilibrium, axisymmetric '''structures''' are obtained by searching for time-independent, steady-state solutions to the [[AxisymmetricConfigurations/PGE#Axisymmetric_Configurations_.28Part_I.29\|identified set of simplified governing equations]].
	<br />		<br />

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2021-09-20T20:56:01Z

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Apps/PapaloizouPringle84 - Revision history

Jet53man: /* Nonaxisymmetric Instability in Papaloizou-Pringle Tori */

Jet53man: /* Equivalent Dimensionless Expression */

Jet53man: /* Checking Self-Consistency */

Jet53man at 21:50, 20 September 2021

Jet53man: /* Formulation of Eigenvalue Problem */

Jet53man: /* Nonaxisymmetric Instability in Papaloizou-Pringle Tori */

Jet53man: Created page with "__FORCETOC__ =Nonaxisymmetric Instability in Papaloizou-Pringle Tori= {| class="AxisymmetricConfigurations" style="float:left; margin-ri..."

Jet53man: Created page with "FORCETOC =Nonaxisymmetric Instability in Papaloizou-Pringle Tori= {| class="AxisymmetricConfigurations" style="float:left; margin-ri..."