Jet53man: /* See Also */

2022-02-17T23:38:59Z

Jet53man: /* Hybrid Scheme */

2021-10-02T22:18:40Z

Hybrid Scheme

← Older revision		Revision as of 18:18, 2 October 2021
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	===Hybrid Scheme===		===Hybrid Scheme===

	In a separate chapter we have detailed the [[Appendix/Ramblings/~~Hybrid_Scheme_Implications~~#Hybrid_Scheme\|hybrid scheme]]. For steady-state configurations, the three components of the combined Euler + Continuity equations give,		In a separate chapter we have detailed the [[Appendix/Ramblings/HybridSchemeImplications#Hybrid_Scheme\|hybrid scheme]]. For steady-state configurations, the three components of the combined Euler + Continuity equations give,

	<table border="1" align="center" cellpadding="10" width="80%"><tr><td align="left">		<table border="1" align="center" cellpadding="10" width="80%"><tr><td align="left">

174.64.14.12: /* Ou's (2006) Detailed Force Balance */

2021-10-02T22:05:40Z

Ou's (2006) Detailed Force Balance

← Older revision		Revision as of 18:05, 2 October 2021
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	</tr>		</tr>
	</table>		</table>


	===Hybrid Scheme===		===Hybrid Scheme===

Jet53man at 17:18, 2 October 2021

2021-10-02T17:18:20Z

← Older revision		Revision as of 13:18, 2 October 2021
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	===2<sup>nd</sup>-Order TVE Expressions===		===2<sup>nd</sup>-Order TVE Expressions===

	As we have discussed in detail in an [[~~User:Tohline/~~VE/RiemannEllipsoids#Riemann_S-Type_Ellipsoids\|accompanying chapter]], the three diagonal elements of the <math>~(3 \times 3)</math> 2<sup>nd</sup>-order tensor virial equation are sufficient to determine the equilibrium values of <math>~\Pi</math>, <math>~\Omega_3</math>, and <math>~\zeta_3</math>.		As we have discussed in detail in an [[VE/RiemannEllipsoids#Riemann_S-Type_Ellipsoids\|accompanying chapter]], the three diagonal elements of the <math>~(3 \times 3)</math> 2<sup>nd</sup>-order tensor virial equation are sufficient to determine the equilibrium values of <math>~\Pi</math>, <math>~\Omega_3</math>, and <math>~\zeta_3</math>.

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	</td>		</td>
	</tr>		</tr>
	<tr><td align="center" colspan="3">[ [[~~User:Tohline/~~Appendix/References#EFE\|EFE]], <font color="#00CC00">Chapter 7, §48, Eq. (34)</font> ]</td></tr>		<tr><td align="center" colspan="3">[ [[Appendix/References#EFE\|EFE]], <font color="#00CC00">Chapter 7, §48, Eq. (34)</font> ]</td></tr>
	</table>		</table>
	and,		and,
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	</td>		</td>
	</tr>		</tr>
	<tr><td align="center" colspan="3">[ [[~~User:Tohline/~~Appendix/References#EFE\|EFE]], <font color="#00CC00">Chapter 7, §48, Eq. (33)</font> ]</td></tr>		<tr><td align="center" colspan="3">[ [[Appendix/References#EFE\|EFE]], <font color="#00CC00">Chapter 7, §48, Eq. (33)</font> ]</td></tr>
	</table>		</table>

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	===Ou's (2006) Detailed Force Balance===		===Ou's (2006) Detailed Force Balance===

	In a separate [[~~User:Tohline/~~ThreeDimensionalConfigurations/RiemannStype#Based_on_Detailed_Force_Balance\|accompanying chapter]], we have described in detail how [https://ui.adsabs.harvard.edu/abs/2006ApJ...639..549O/abstract Ou(2006)] used, essentially, the HSCF technique to solve the detailed force-balance equations. Beginning with the,		In a separate [[ThreeDimensionalConfigurations/RiemannStype#Based_on_Detailed_Force_Balance\|accompanying chapter]], we have described in detail how [https://ui.adsabs.harvard.edu/abs/2006ApJ...639..549O/abstract Ou(2006)] used, essentially, the HSCF technique to solve the detailed force-balance equations. Beginning with the,

	<div align="center">		<div align="center">
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	===Hybrid Scheme===		===Hybrid Scheme===

	In a separate chapter we have detailed the [[~~User:Tohline/~~Appendix/Ramblings/Hybrid_Scheme_Implications#Hybrid_Scheme\|hybrid scheme]]. For steady-state configurations, the three components of the combined Euler + Continuity equations give,		In a separate chapter we have detailed the [[Appendix/Ramblings/Hybrid_Scheme_Implications#Hybrid_Scheme\|hybrid scheme]]. For steady-state configurations, the three components of the combined Euler + Continuity equations give,

	<table border="1" align="center" cellpadding="10" width="80%"><tr><td align="left">		<table border="1" align="center" cellpadding="10" width="80%"><tr><td align="left">
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	==Compressible Structures==		==Compressible Structures==

	Here we draw heavily on the published work of [http://adsabs.harvard.edu/abs/1996ApJS..105..181K Korycansky & Papaloizou] (1996, ApJS, 105, 181; hereafter KP96) that we have reviewed in a [[~~User:Tohline/~~Apps/Korycansky_Papaloizou_1996#Korycansky_and_Papaloizou_.281996.29\|separate chapter]], and on the doctoral dissertation of [https://digitalcommons.lsu.edu/gradschool_disstheses/6650/ Saied W. Andalib (1998)].		Here we draw heavily on the published work of [http://adsabs.harvard.edu/abs/1996ApJS..105..181K Korycansky & Papaloizou] (1996, ApJS, 105, 181; hereafter KP96) that we have reviewed in a [[Apps/Korycansky_Papaloizou_1996#Korycansky_and_Papaloizou_.281996.29\|separate chapter]], and on the doctoral dissertation of [https://digitalcommons.lsu.edu/gradschool_disstheses/6650/ Saied W. Andalib (1998)].

	===Part I===		===Part I===
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	- {\vec{\Omega}}_f \times ({\vec{\Omega}}_f \times \vec{x}) - 2{\vec{\Omega}}_f \times \bold{u} \, ,</math> </div>		- {\vec{\Omega}}_f \times ({\vec{\Omega}}_f \times \vec{x}) - 2{\vec{\Omega}}_f \times \bold{u} \, ,</math> </div>

	we next note — as we have done in our [[~~User:Tohline/~~PGE/Euler#in_terms_of_the_vorticity:\|broader discussion of the Euler equation]] — that,		we next note — as we have done in our [[PGE/Euler#in_terms_of_the_vorticity:\|broader discussion of the Euler equation]] — that,
	<div align="center">		<div align="center">
	<math>		<math>
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	- {\vec{\Omega}}_f \times ({\vec{\Omega}}_f \times \vec{x}) - 2{\vec{\Omega}}_f \times \bold{u} \, ,</math> </div>		- {\vec{\Omega}}_f \times ({\vec{\Omega}}_f \times \vec{x}) - 2{\vec{\Omega}}_f \times \bold{u} \, ,</math> </div>

	we next note — as we have done in our [[~~User:Tohline/~~PGE/Euler#in_terms_of_the_vorticity:\|broader discussion of the Euler equation]] — that,		we next note — as we have done in our [[PGE/Euler#in_terms_of_the_vorticity:\|broader discussion of the Euler equation]] — that,
	<div align="center">		<div align="center">
	<math>		<math>
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	<table border="1" align="center" width="90%" cellpadding="10"><tr><td align="left">		<table border="1" align="center" width="90%" cellpadding="10"><tr><td align="left">
	<font color="red">'''IMPORTANT NOTE''' (by Tohline on 22 September 2020):</font>    As I have come to appreciate today — after studying the relevant sections of both [[~~User:Tohline/~~Appendix/References#EFE\|EFE]] and [[~~User:Tohline/~~Appendix/References#BT87\|BT87]] — this is an example of a heterogeneous density distribution whose gravitational potential has an analytic prescription. As is discussed in a [[~~User:Tohline/~~ThreeDimensionalConfigurations/HomogeneousEllipsoids#Inhomogeneous_Ellipsoids_Leading_to_Ferrers_Potentials\| separate chapter]], the potential that it generates is sometimes referred to as a ''Ferrers'' potential, for the exponent, n = 1.		<font color="red">'''IMPORTANT NOTE''' (by Tohline on 22 September 2020):</font>    As I have come to appreciate today — after studying the relevant sections of both [[Appendix/References#EFE\|EFE]] and [[Appendix/References#BT87\|BT87]] — this is an example of a heterogeneous density distribution whose gravitational potential has an analytic prescription. As is discussed in a [[ThreeDimensionalConfigurations/HomogeneousEllipsoids#Inhomogeneous_Ellipsoids_Leading_to_Ferrers_Potentials\| separate chapter]], the potential that it generates is sometimes referred to as a ''Ferrers'' potential, for the exponent, n = 1.

	In our [[~~User:Tohline/~~ThreeDimensionalConfigurations/HomogeneousEllipsoids#GravFor1\|accompanying discussion]] we find that,		In our [[ThreeDimensionalConfigurations/HomogeneousEllipsoids#GravFor1\|accompanying discussion]] we find that,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

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	</td>		</td>
	</tr>		</tr>
	<tr><td align="center" colspan="3">[ [[~~User:Tohline/~~Appendix/References#EFE\|EFE]], <font color="#00CC00">§21, Eq. (107)</font> ]</td></tr>		<tr><td align="center" colspan="3">[ [[Appendix/References#EFE\|EFE]], <font color="#00CC00">§21, Eq. (107)</font> ]</td></tr>
	</table>		</table>

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	</td>		</td>
	</tr>		</tr>
	<tr><td align="center" colspan="3">[ [[~~User:Tohline/~~Appendix/References#EFE\|EFE]], <font color="#00CC00">§21, Eq. (109)</font> ]</td></tr>		<tr><td align="center" colspan="3">[ [[Appendix/References#EFE\|EFE]], <font color="#00CC00">§21, Eq. (109)</font> ]</td></tr>
	</table>		</table>

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	====T5 Coordinates====		====T5 Coordinates====
	Let's evaluate the <math>~\nabla^2</math> operator by expressing it and its argument in terms of [[~~User:Tohline/~~Appendix/Ramblings/EllipticCylinderCoordinates#T5_Coordinates\|T5 Coordinates]]. Note that,		Let's evaluate the <math>~\nabla^2</math> operator by expressing it and its argument in terms of [[Appendix/Ramblings/EllipticCylinderCoordinates#T5_Coordinates\|T5 Coordinates]]. Note that,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

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	</table>		</table>

	The relevant [[~~User:Tohline/~~Appendix/Ramblings/EllipticCylinderCoordinates#Laplacian\|T5-Coordinate System Laplacian]] is,		The relevant [[Appendix/Ramblings/EllipticCylinderCoordinates#Laplacian\|T5-Coordinate System Laplacian]] is,

	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	</table>		</table>

	Note that, for the [[~~User:Tohline/~~Appendix/Ramblings/EllipticCylinderCoordinates#Invert_Coordinate_Mapping\|specific ''example'' case of]] <math>~\kappa^2 = 2</math>,		Note that, for the [[Appendix/Ramblings/EllipticCylinderCoordinates#Invert_Coordinate_Mapping\|specific ''example'' case of]] <math>~\kappa^2 = 2</math>,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

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	=See Also=		=See Also=

	* [[~~User:Tohline/~~ThreeDimensionalConfigurations/Challenges#Challenges_Constructing_Ellipsoidal-Like_Configurations\|Construction Challenges (Pt. 1)]]		* [[ThreeDimensionalConfigurations/Challenges#Challenges_Constructing_Ellipsoidal-Like_Configurations\|Construction Challenges (Pt. 1)]]
	* [[~~User:Tohline/~~ThreeDimensionalConfigurations/ChallengesPt2\|Construction Challenges (Pt. 2)]]		* [[ThreeDimensionalConfigurations/ChallengesPt2\|Construction Challenges (Pt. 2)]]
	* [[~~User:Tohline/~~ThreeDimensionalConfigurations/ChallengesPt3\|Construction Challenges (Pt. 3)]]		* [[ThreeDimensionalConfigurations/ChallengesPt3\|Construction Challenges (Pt. 3)]]
	* [[~~User:Tohline/~~ThreeDimensionalConfigurations/ChallengesPt4\|Construction Challenges (Pt. 4)]]		* [[ThreeDimensionalConfigurations/ChallengesPt4\|Construction Challenges (Pt. 4)]]
	* [[~~User:Tohline/~~ThreeDimensionalConfigurations/ChallengesPt5\|Construction Challenges (Pt. 5)]]		* [[ThreeDimensionalConfigurations/ChallengesPt5\|Construction Challenges (Pt. 5)]]
	* Related discussions of models viewed from a rotating reference frame:		* Related discussions of models viewed from a rotating reference frame:
	** [[~~User:Tohline/~~PGE/RotatingFrame#Rotating_Reference_Frame\|PGE]]		** [[PGE/RotatingFrame#Rotating_Reference_Frame\|PGE]]
	** <font color="red"><b>NOTE to Eric Hirschmann & David Neilsen... </b></font>I have moved the earlier contents of this page to a new Wiki location called [[~~User:Tohline/~~Apps/RiemannEllipsoids_Compressible\|Compressible Riemann Ellipsoids]].		** <font color="red"><b>NOTE to Eric Hirschmann & David Neilsen... </b></font>I have moved the earlier contents of this page to a new Wiki location called [[Apps/RiemannEllipsoids_Compressible\|Compressible Riemann Ellipsoids]].

	{{ SGFfooter }}		{{ SGFfooter }}

Jet53man: Created page with "FORCETOC =Challenges Constructing Ellipsoidal-Like Configurations= First, let..."

2021-07-24T21:39:27Z

Created page with "__FORCETOC__  =Challenges Constructing Ellipsoidal-Like Configurations= First, let..."

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← Older revision		Revision as of 19:38, 17 February 2022
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	=See Also=		=See Also=

			* [[ThreeDimensionalConfigurations/DescriptionOfRiemannTypeI\|Description of Riemann Type I Ellipsoids]]
			* [[ThreeDimensionalConfigurations/RiemannTypeI#Riemann_Type_1_Ellipsoids\|Riemann Type 1 Ellipsoids]] (old introduction)
	* [[ThreeDimensionalConfigurations/Challenges#Challenges_Constructing_Ellipsoidal-Like_Configurations\|Construction Challenges (Pt. 1)]]		* [[ThreeDimensionalConfigurations/Challenges#Challenges_Constructing_Ellipsoidal-Like_Configurations\|Construction Challenges (Pt. 1)]]
	* [[ThreeDimensionalConfigurations/ChallengesPt2\|Construction Challenges (Pt. 2)]]		* [[ThreeDimensionalConfigurations/ChallengesPt2\|Construction Challenges (Pt. 2)]]
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	* [[ThreeDimensionalConfigurations/ChallengesPt4\|Construction Challenges (Pt. 4)]]		* [[ThreeDimensionalConfigurations/ChallengesPt4\|Construction Challenges (Pt. 4)]]
	* [[ThreeDimensionalConfigurations/ChallengesPt5\|Construction Challenges (Pt. 5)]]		* [[ThreeDimensionalConfigurations/ChallengesPt5\|Construction Challenges (Pt. 5)]]
			* [[ThreeDimensionalConfigurations/ChallengesPt6\|Construction Challenges (Pt. 6)]]

	* Related discussions of models viewed from a rotating reference frame:		* Related discussions of models viewed from a rotating reference frame:
	** [[PGE/RotatingFrame#Rotating_Reference_Frame\|PGE]]		** [[PGE/RotatingFrame#Rotating_Reference_Frame\|PGE]]

ThreeDimensionalConfigurations/Challenges - Revision history

Jet53man: /* See Also */

Jet53man: /* Hybrid Scheme */

174.64.14.12: /* Ou's (2006) Detailed Force Balance */

Jet53man at 17:18, 2 October 2021

Jet53man: Created page with "__FORCETOC__ =Challenges Constructing Ellipsoidal-Like Configurations= First, let..."

Jet53man: Created page with "FORCETOC =Challenges Constructing Ellipsoidal-Like Configurations= First, let..."