User:Tohline/2DStructure/AxisymmetricInstabilities - Revision history

Tohline: /* Poincaré-Wavre Theorem */

2020-06-25T22:07:16Z

Poincaré-Wavre Theorem

← Older revision		Revision as of 22:07, 25 June 2020
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	</table>		</table>
	</div>		</div>

			NOTE: We should investigate how this theorem comes into play in the context of our [[User:Tohline/ThreeDimensionalConfigurations/RiemannTypeI#Riemann_Type_1_Ellipsoids\|accompanying discussion of Type 1 Riemann ellipsoids]]. These are equilibrium triaxial, uniform-density configurations in which the system's internal vorticity vector does not align with the tumble-axis of the ellipsoid and, therefore apparently <math>~\dot\varphi</math> is not independent of <math>~z</math>.

	==Høiland Criterion==		==Høiland Criterion==

Tohline: /* See Also */

2020-06-25T17:38:38Z

Tohline: /* See Also */

2020-06-25T17:33:51Z

Tohline: /* Axisymmetric Instabilities to Avoid */

2019-06-26T20:10:08Z

Axisymmetric Instabilities to Avoid

← Older revision		Revision as of 20:10, 26 June 2019
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	<tr><td align="left">		<tr><td align="left">
	<font color="indianred">		<font color="indianred">
	"For an incompressible liquid contained between concentric cylinders and rotating with angular velocity <math>~\Omega(\varpi)</math>, Rayleigh's criterion — that <math>~\varpi^4 \Omega^2</math> increase outwards — is necessary and sufficient for stability to axisymmetric disturbances. In a star rotating with angular velocity <math>~\Omega(\varpi)</math> if we continue to restrict attention to axisymmetric disturbances, this criterion must be modified by buoyancy effects; that is, some combination of the Rayleigh and Schwarzschild criteria should obtain. Such a combination has been found, for example, by [https://en.wikipedia.org/wiki/Einar_Høiland Høiland (1941)] (see also [https://archive.org/details/AllerStellarStructure Ledoux's Chapter 10, pp. 499-574 of ''Stellar Structure'')], and indicates, as one would expect, that a stable stratification of angular velocity exerts a stabilizing influence on an unstable distribution of temperature, and vice versa. The combined criterion has not been placed on the solid analytical foundation of its two component criteria, however."</font>		"For an incompressible liquid contained between concentric cylinders and rotating with angular velocity <math>~\Omega(\varpi)</math>, Rayleigh's criterion — that <math>~\varpi^4 \Omega^2</math> increase outwards — is necessary and sufficient for stability to axisymmetric disturbances. In a star rotating with angular velocity <math>~\Omega(\varpi)</math> if we continue to restrict attention to axisymmetric disturbances, this criterion must be modified by buoyancy effects; that is, some combination of the Rayleigh and Schwarzschild criteria should obtain. Such a combination has been found, for example, by [https://en.wikipedia.org/wiki/Einar_Høiland Høiland (1941)] (see also [https://archive.org/details/AllerStellarStructure Ledoux's Chapter 10, pp. 499-574 of ''Stellar Structure'' (1965)], and indicates, as one would expect, that a stable stratification of angular velocity exerts a stabilizing influence on an unstable distribution of temperature, and vice versa. The combined criterion has not been placed on the solid analytical foundation of its two component criteria, however."</font>
	</td></tr>		</td></tr>
	<tr><td align="right">		<tr><td align="right">

Tohline: /* Høiland Criterion */

2019-06-26T19:57:42Z

Høiland Criterion

← Older revision		Revision as of 19:57, 26 June 2019
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	<table border="0" cellpadding="3" align="center" width="60%">		<table border="0" cellpadding="3" align="center" width="60%">
	<tr><td align="left">		<tr><td align="left">
	<font color="~~darkgreen~~">		<font color="indianred">
	"For an incompressible liquid contained between concentric cylinders and rotating with angular velocity <math>~\Omega(\varpi)</math>, Rayleigh's criterion — that <math>~\varpi^4 \Omega^2</math> increase outwards — is necessary and sufficient for stability to axisymmetric disturbances. In a star rotating with angular velocity <math>~\Omega(\varpi)</math> if we continue to restrict attention to axisymmetric disturbances, this criterion must be modified by buoyancy effects; that is, some combination of the Rayleigh and Schwarzschild criteria should obtain. Such a combination has been found, for example, by [https://en.wikipedia.org/wiki/Einar_Høiland Høiland (1941)] (see also [https://archive.org/details/AllerStellarStructure Ledoux's Chapter 10, pp. 499-574 of ''Stellar Structure'')], and indicates, as one would expect, that a stable stratification of angular velocity exerts a stabilizing influence on an unstable distribution of temperature, and vice versa. The combined criterion has not been placed on the solid analytical foundation of its two component criteria, however."</font>		"For an incompressible liquid contained between concentric cylinders and rotating with angular velocity <math>~\Omega(\varpi)</math>, Rayleigh's criterion — that <math>~\varpi^4 \Omega^2</math> increase outwards — is necessary and sufficient for stability to axisymmetric disturbances. In a star rotating with angular velocity <math>~\Omega(\varpi)</math> if we continue to restrict attention to axisymmetric disturbances, this criterion must be modified by buoyancy effects; that is, some combination of the Rayleigh and Schwarzschild criteria should obtain. Such a combination has been found, for example, by [https://en.wikipedia.org/wiki/Einar_Høiland Høiland (1941)] (see also [https://archive.org/details/AllerStellarStructure Ledoux's Chapter 10, pp. 499-574 of ''Stellar Structure'')], and indicates, as one would expect, that a stable stratification of angular velocity exerts a stabilizing influence on an unstable distribution of temperature, and vice versa. The combined criterion has not been placed on the solid analytical foundation of its two component criteria, however."</font>
	</td></tr>		</td></tr>

Tohline: /* Høiland Criterion */

2019-06-26T19:54:41Z

Høiland Criterion

← Older revision		Revision as of 19:54, 26 June 2019
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	<tr><td align="left">		<tr><td align="left">
	<font color="darkgreen">		<font color="darkgreen">
	"For an incompressible liquid contained between concentric cylinders and rotating with angular velocity <math>~\Omega(\varpi)</math>, Rayleigh's criterion — that <math>~\varpi^4 \Omega^2</math> increase outwards — is necessary and sufficient for stability to axisymmetric disturbances. In a star rotating with angular velocity <math>~\Omega(\varpi)</math> if we continue to restrict attention to axisymmetric disturbances, this criterion must be modified by buoyancy effects; that is, some combination of the Rayleigh and Schwarzschild criteria should obtain. Such a combination has been found, for example, by [https://en.wikipedia.org/wiki/Einar_Høiland Høiland (1941)] (see also [https://archive.org/details/AllerStellarStructure Ledoux ~~1965,~~ Chapter 10)], and indicates, as one would expect, that a stable stratification of angular velocity exerts a stabilizing influence on an unstable distribution of temperature, and vice versa. The combined criterion has not been placed on the solid analytical foundation of its two component criteria, however."</font>		"For an incompressible liquid contained between concentric cylinders and rotating with angular velocity <math>~\Omega(\varpi)</math>, Rayleigh's criterion — that <math>~\varpi^4 \Omega^2</math> increase outwards — is necessary and sufficient for stability to axisymmetric disturbances. In a star rotating with angular velocity <math>~\Omega(\varpi)</math> if we continue to restrict attention to axisymmetric disturbances, this criterion must be modified by buoyancy effects; that is, some combination of the Rayleigh and Schwarzschild criteria should obtain. Such a combination has been found, for example, by [https://en.wikipedia.org/wiki/Einar_Høiland Høiland (1941)] (see also [https://archive.org/details/AllerStellarStructure Ledoux's Chapter 10, pp. 499-574 of ''Stellar Structure'')], and indicates, as one would expect, that a stable stratification of angular velocity exerts a stabilizing influence on an unstable distribution of temperature, and vice versa. The combined criterion has not been placed on the solid analytical foundation of its two component criteria, however."</font>
	</td></tr>		</td></tr>
	<tr><td align="right">		<tr><td align="right">

Tohline: /* Høiland Criterion */

2019-06-26T18:18:38Z

Høiland Criterion

← Older revision		Revision as of 18:18, 26 June 2019
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	<tr><td align="left">		<tr><td align="left">
	<font color="darkgreen">		<font color="darkgreen">
	"For an incompressible liquid contained between concentric cylinders and rotating with angular velocity <math>~\Omega(\varpi)</math>, Rayleigh's criterion — that <math>~\varpi^4 \Omega^2</math> increase outwards — is necessary and sufficient for stability to axisymmetric disturbances. In a star rotating with angular velocity <math>~\Omega(\varpi)</math> if we continue to restrict attention to axisymmetric disturbances, this criterion must be modified by buoyancy effects; that is, some combination of the Rayleigh and Schwarzschild criteria should obtain. Such a combination has been found, for example, by [https://en.wikipedia.org/wiki/Einar_Høiland Høiland (1941)] (see also Ledoux 1965), and indicates, as one would expect, that a stable stratification of angular velocity exerts a stabilizing influence on an unstable distribution of temperature, and vice versa. The combined criterion has not been placed on the solid analytical foundation of its two component criteria, however."</font>		"For an incompressible liquid contained between concentric cylinders and rotating with angular velocity <math>~\Omega(\varpi)</math>, Rayleigh's criterion — that <math>~\varpi^4 \Omega^2</math> increase outwards — is necessary and sufficient for stability to axisymmetric disturbances. In a star rotating with angular velocity <math>~\Omega(\varpi)</math> if we continue to restrict attention to axisymmetric disturbances, this criterion must be modified by buoyancy effects; that is, some combination of the Rayleigh and Schwarzschild criteria should obtain. Such a combination has been found, for example, by [https://en.wikipedia.org/wiki/Einar_Høiland Høiland (1941)] (see also [https://archive.org/details/AllerStellarStructure Ledoux 1965, Chapter 10)], and indicates, as one would expect, that a stable stratification of angular velocity exerts a stabilizing influence on an unstable distribution of temperature, and vice versa. The combined criterion has not been placed on the solid analytical foundation of its two component criteria, however."</font>
	</td></tr>		</td></tr>
	<tr><td align="right">		<tr><td align="right">

Tohline: /* Høiland Criterion */

2019-06-26T18:13:39Z

Høiland Criterion

← Older revision		Revision as of 18:13, 26 June 2019
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	<tr><td align="left">		<tr><td align="left">
	<font color="darkgreen">		<font color="darkgreen">
	"For an incompressible liquid contained between concentric cylinders and rotating with angular velocity <math>~\Omega(\varpi)</math>, Rayleigh's criterion — that <math>~\varpi^4 \Omega^2</math> increase outwards — is necessary and sufficient for stability to axisymmetric disturbances. In a star rotating with angular velocity <math>~\Omega(\varpi)</math> if we continue to restrict attention to axisymmetric disturbances, this criterion must be modified by buoyancy effects; that is, some combination of the Rayleigh and Schwarzschild criteria should obtain. Such a combination has been found, for example, by Høiland (1941; see also Ledoux 1965), and indicates, as one would expect, that a stable stratification of angular velocity exerts a stabilizing influence on an unstable distribution of temperature, and vice versa. The combined criterion has not been placed on the solid analytical foundation of its two component criteria, however."</font>		"For an incompressible liquid contained between concentric cylinders and rotating with angular velocity <math>~\Omega(\varpi)</math>, Rayleigh's criterion — that <math>~\varpi^4 \Omega^2</math> increase outwards — is necessary and sufficient for stability to axisymmetric disturbances. In a star rotating with angular velocity <math>~\Omega(\varpi)</math> if we continue to restrict attention to axisymmetric disturbances, this criterion must be modified by buoyancy effects; that is, some combination of the Rayleigh and Schwarzschild criteria should obtain. Such a combination has been found, for example, by [https://en.wikipedia.org/wiki/Einar_Høiland Høiland (1941)] (see also Ledoux 1965), and indicates, as one would expect, that a stable stratification of angular velocity exerts a stabilizing influence on an unstable distribution of temperature, and vice versa. The combined criterion has not been placed on the solid analytical foundation of its two component criteria, however."</font>
	</td></tr>		</td></tr>
	<tr><td align="right">		<tr><td align="right">

Tohline: /* Høiland Criterion */

2019-06-26T16:59:54Z

Høiland Criterion

← Older revision		Revision as of 16:59, 26 June 2019
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	<font color="darkgreen">		<font color="darkgreen">
	"For an incompressible liquid contained between concentric cylinders and rotating with angular velocity <math>~\Omega(\varpi)</math>, Rayleigh's criterion — that <math>~\varpi^4 \Omega^2</math> increase outwards — is necessary and sufficient for stability to ~~axisymmetri~~ disturbances. In a star rotating with angular velocity <math>~\Omega(\varpi)</math> if we continue to restrict attention to axisymmetric disturbances, this criterion must be modified by ~~Buoyancy~~ effects; that is, some combination of the Rayleigh and Schwarzschild criteria should obtain. Such a combination has been found, for example, by Høiland (1941; see also Ledoux 1965), and indicates, as one would expect, that a stable stratification of angular velocity exerts a stabilizing influence on an unstable distribution of temperature, and vice versa. The combined criterion has not been placed on the solid analytical foundation of its two component criteria, however."</font>		"For an incompressible liquid contained between concentric cylinders and rotating with angular velocity <math>~\Omega(\varpi)</math>, Rayleigh's criterion — that <math>~\varpi^4 \Omega^2</math> increase outwards — is necessary and sufficient for stability to axisymmetric disturbances. In a star rotating with angular velocity <math>~\Omega(\varpi)</math> if we continue to restrict attention to axisymmetric disturbances, this criterion must be modified by buoyancy effects; that is, some combination of the Rayleigh and Schwarzschild criteria should obtain. Such a combination has been found, for example, by Høiland (1941; see also Ledoux 1965), and indicates, as one would expect, that a stable stratification of angular velocity exerts a stabilizing influence on an unstable distribution of temperature, and vice versa. The combined criterion has not been placed on the solid analytical foundation of its two component criteria, however."</font>
	</td></tr>		</td></tr>
	<tr><td align="right">		<tr><td align="right">
	— Drawn from p. 475 of [https://ui.adsabs.harvard.edu/abs/1967ARA%26A...5..465L/abstract N. R. Lebovitz (1967)], ~~Annual Review of Astronomy and Astrophysics~~, 5, 465~~: ''Rotating fluid masses''~~		— Drawn from p. 475 of [https://ui.adsabs.harvard.edu/abs/1967ARA%26A...5..465L/abstract N. R. Lebovitz (1967)], ARAA, 5, 465
	</td></tr></table>		</td></tr></table>

Tohline: /* Høiland Criterion */

2019-06-26T16:56:23Z

Høiland Criterion

← Older revision		Revision as of 16:56, 26 June 2019
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	==Høiland Criterion==		==Høiland Criterion==

			<table border="0" cellpadding="3" align="center" width="60%">
			<tr><td align="left">
			<font color="darkgreen">
			"For an incompressible liquid contained between concentric cylinders and rotating with angular velocity <math>~\Omega(\varpi)</math>, Rayleigh's criterion — that <math>~\varpi^4 \Omega^2</math> increase outwards — is necessary and sufficient for stability to axisymmetri disturbances. In a star rotating with angular velocity <math>~\Omega(\varpi)</math> if we continue to restrict attention to axisymmetric disturbances, this criterion must be modified by Buoyancy effects; that is, some combination of the Rayleigh and Schwarzschild criteria should obtain. Such a combination has been found, for example, by Høiland (1941; see also Ledoux 1965), and indicates, as one would expect, that a stable stratification of angular velocity exerts a stabilizing influence on an unstable distribution of temperature, and vice versa. The combined criterion has not been placed on the solid analytical foundation of its two component criteria, however."</font>
			</td></tr>
			<tr><td align="right">
			— Drawn from p. 475 of [https://ui.adsabs.harvard.edu/abs/1967ARA%26A...5..465L/abstract N. R. Lebovitz (1967)], Annual Review of Astronomy and Astrophysics, 5, 465: ''Rotating fluid masses''
			</td></tr></table>

	As is stated on p. 166 of [ <b>[[User:Tohline/Appendix/References#T78\|<font color="red">T78</font>]] </b>], in rotating barotropic configurations, axisymmetric stability requires the simultaneous satisfaction of the following pair of conditions:		As is stated on p. 166 of [ <b>[[User:Tohline/Appendix/References#T78\|<font color="red">T78</font>]] </b>], in rotating barotropic configurations, axisymmetric stability requires the simultaneous satisfaction of the following pair of conditions:
	<div align="center">		<div align="center">

← Older revision		Revision as of 17:38, 25 June 2020
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	* [http://adsabs.harvard.edu/abs/1977ApJ...214..584M P. S. Marcus, W. H. Press, & S. A. Teukolsky (1977, ApJ, 214, 584- 597)] — ''Stablest Shapes for an Axisymmetric Body of Gravitating, Incompressible Fluid'' (includes torus with non-uniform rotation)		* [http://adsabs.harvard.edu/abs/1977ApJ...214..584M P. S. Marcus, W. H. Press, & S. A. Teukolsky (1977, ApJ, 214, 584- 597)] — ''Stablest Shapes for an Axisymmetric Body of Gravitating, Incompressible Fluid'' (includes torus with non-uniform rotation)
	<ol type="a"><li>Shortly after their equation (3.2), Marcus, Press & Teukolsky make the following statement: "… we know that an ''equilibrium'' incompressible configuration must rotate uniformly on cylinders (the famous "Poincaré-Wavre" theorem, cf. Tassoul 1977, &Sect;4.3) …"</li></ol>		<ol type="a"><li>Shortly after their equation (3.2), Marcus, Press & Teukolsky make the following statement: "… we know that an ''equilibrium'' incompressible configuration must rotate uniformly on cylinders (the famous "Poincaré-Wavre" theorem, cf. Tassoul 1977, &Sect;4.3) …"</li></ol>
			* Referring to our [[User:Tohline/ThreeDimensionalConfigurations/RiemannTypeI#Riemann_Type_1_Ellipsoids\|accompanying discussion of Type 1 Riemann ellipsoids]], it seems that uniform rotation on cylinders is not required. What's going on?


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← Older revision		Revision as of 17:33, 25 June 2020
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	=See Also=		=See Also=
	* [http://adsabs.harvard.edu/abs/1917RSPSA..93..148R Lord Rayleigh (1917, Proc. Royal Society of London. Series A, 93, 148-154)] — ''On the Dynamics of Revolving Fluids''		* [http://adsabs.harvard.edu/abs/1917RSPSA..93..148R Lord Rayleigh (1917, Proc. Royal Society of London. Series A, 93, 148-154)] — ''On the Dynamics of Revolving Fluids''
			* [http://adsabs.harvard.edu/abs/1977ApJ...214..584M P. S. Marcus, W. H. Press, & S. A. Teukolsky (1977, ApJ, 214, 584- 597)] — ''Stablest Shapes for an Axisymmetric Body of Gravitating, Incompressible Fluid'' (includes torus with non-uniform rotation)
			<ol type="a"><li>Shortly after their equation (3.2), Marcus, Press & Teukolsky make the following statement: "… we know that an ''equilibrium'' incompressible configuration must rotate uniformly on cylinders (the famous "Poincaré-Wavre" theorem, cf. Tassoul 1977, &Sect;4.3) …"</li></ol>


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