Jet53man: /* See Also */

2022-02-17T23:39:55Z

Jet53man: /* See Also */

2022-02-06T02:19:26Z

Jet53man at 22:23, 2 October 2021

2021-10-02T22:23:59Z

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← Older revision		Revision as of 19:39, 17 February 2022
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	=See Also=		=See Also=

	* [[ThreeDimensionalConfigurations/RiemannTypeI#Riemann_Type_1_Ellipsoids\|Riemann Type 1 Ellipsoids]]		* [[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/ChallengesPt5\|Construction Challenges (Pt. 5)]]		* [[ThreeDimensionalConfigurations/ChallengesPt5\|Construction Challenges (Pt. 5)]]
	* [[ThreeDimensionalConfigurations/ChallengesPt6\|Construction Challenges (Pt. 6)]]		* [[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]]

← Older revision		Revision as of 22:19, 5 February 2022
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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]]

← Older revision		Revision as of 18:23, 2 October 2021
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	=Challenges Constructing Ellipsoidal-Like Configurations (Pt. 2)=		=Challenges Constructing Ellipsoidal-Like Configurations (Pt. 2)=

	This chapter extends an [[~~User:Tohline/~~ThreeDimensionalConfigurations/Challenges\|accompanying chapter titled, ''Construction Challenges (Pt. 1)'']]. The focus here is on an SCF technique that will incorporate specification of a Lagrangian flow-flied.		This chapter extends an [[ThreeDimensionalConfigurations/Challenges\|accompanying chapter titled, ''Construction Challenges (Pt. 1)'']]. The focus here is on an SCF technique that will incorporate specification of a Lagrangian flow-flied.
	==Motivation==		==Motivation==

	===Where Are We Headed?===		===Where Are We Headed?===
	In a [[~~User:Tohline/~~ThreeDimensionalConfigurations/RiemannTypeI#Try_Again\|separate discussion]], we have shown that, as viewed from a frame that "tumbles" with the (purple) body of a Type 1 Riemann ellipsoid, each Lagrangian fluid element moves along an elliptical path in a plane that is tipped by an angle <math>~\theta</math> about the x-axis of the body. (See the yellow-dotted orbits in Figure panels 1a and 1b below). As viewed from the (primed) coordinates associated with this tipped plane, by definition, z' = 0 and dz'/dt = 0, and the planar orbit is defined by the expression for an,		In a [[ThreeDimensionalConfigurations/RiemannTypeI#Try_Again\|separate discussion]], we have shown that, as viewed from a frame that "tumbles" with the (purple) body of a Type 1 Riemann ellipsoid, each Lagrangian fluid element moves along an elliptical path in a plane that is tipped by an angle <math>~\theta</math> about the x-axis of the body. (See the yellow-dotted orbits in Figure panels 1a and 1b below). As viewed from the (primed) coordinates associated with this tipped plane, by definition, z' = 0 and dz'/dt = 0, and the planar orbit is defined by the expression for an,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
	<tr>		<tr>
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	<span id="VorticitySetup">Also,</span> along the lines of our [[~~User:Tohline/~~ThreeDimensionalConfigurations/Challenges#Riemann_S-type_Ellipsoids\|accompanying discussion of Riemann S-Type Ellipoids]], it is useful to develop the expression for the fluid vorticity as viewed from the rotating- and tipped-reference frame.		<span id="VorticitySetup">Also,</span> along the lines of our [[ThreeDimensionalConfigurations/Challenges#Riemann_S-type_Ellipsoids\|accompanying discussion of Riemann S-Type Ellipoids]], it is useful to develop the expression for the fluid vorticity as viewed from the rotating- and tipped-reference frame.
	<table border="1" cellpadding="8" width="90%" align="center"><tr><td align="left">		<table border="1" cellpadding="8" width="90%" align="center"><tr><td align="left">
	<div align="center">'''Vorticity'''</div>		<div align="center">'''Vorticity'''</div>
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	====Body Frame====		====Body Frame====

	In a [[~~User:Tohline/~~ThreeDimensionalConfigurations/RiemannTypeI#TippedPlane\|an early subsection of the accompanying discussion]], we have pointed out that the intersection of each Lagrangian fluid element's tipped orbital plane with the surface of the (purple) ellipsoidal surface is given by the (unprimed) body-frame coordinates that simultaneously satisfy the expressions,		In a [[ThreeDimensionalConfigurations/RiemannTypeI#TippedPlane\|an early subsection of the accompanying discussion]], we have pointed out that the intersection of each Lagrangian fluid element's tipped orbital plane with the surface of the (purple) ellipsoidal surface is given by the (unprimed) body-frame coordinates that simultaneously satisfy the expressions,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

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	====Tipped Orbital Plane====		====Tipped Orbital Plane====

	A table provided in [[~~User:Tohline/~~ThreeDimensionalConfigurations/RiemannTypeI#Try_Again\|our accompanying discussion]] shows how to transform from the body-frame coordinates (unprimed) to the (primed) frame that aligns with the Lagrangian fluid element's orbit. Specifically,		A table provided in [[ThreeDimensionalConfigurations/RiemannTypeI#Try_Again\|our accompanying discussion]] shows how to transform from the body-frame coordinates (unprimed) to the (primed) frame that aligns with the Lagrangian fluid element's orbit. Specifically,
	<table border="1" width="50%" cellpadding="8" align="center">		<table border="1" width="50%" cellpadding="8" align="center">
	<tr>		<tr>
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	<span id="Result1">which, as indicated,</span> allows us to identify the appropriate expression for the y-offset, <math>~y_0</math>.		<span id="Result1">which, as indicated,</span> allows us to identify the appropriate expression for the y-offset, <math>~y_0</math>.
	<table border="1" align="center" cellpadding="10" width="60%" bordercolor="orange">		<table border="1" align="center" cellpadding="10" width="60%" bordercolor="orange">
	<tr><td align="center" bgcolor="lightblue">'''RESULT 1'''<br />(compare with [[~~User:Tohline/~~ThreeDimensionalConfigurations/ChallengesPt4#Result2\|Result 2]])</td></tr>		<tr><td align="center" bgcolor="lightblue">'''RESULT 1'''<br />(compare with [[ThreeDimensionalConfigurations/ChallengesPt4#Result2\|Result 2]])</td></tr>
	<tr><td align="left">		<tr><td align="left">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	====Body Frame Velocities====		====Body Frame Velocities====

	From the already-referenced table provided in [[~~User:Tohline/~~ThreeDimensionalConfigurations/RiemannTypeI#Try_Again\|our accompanying discussion]], we can transform this pair of expressions for the velocity components in the "tipped orbit" frame — remember that the third component, <math>~\dot{z}' = 0</math> — into the (three-component) velocities of the body frame using the expressions,		From the already-referenced table provided in [[ThreeDimensionalConfigurations/RiemannTypeI#Try_Again\|our accompanying discussion]], we can transform this pair of expressions for the velocity components in the "tipped orbit" frame — remember that the third component, <math>~\dot{z}' = 0</math> — into the (three-component) velocities of the body frame using the expressions,
	<table border="1" width="75%" cellpadding="8" align="center">		<table border="1" width="75%" cellpadding="8" align="center">
	<tr>		<tr>
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	====Inertial-Frame Expressions====		====Inertial-Frame Expressions====
	As we have summarized in an [[~~User:Tohline/~~ThreeDimensionalConfigurations/RiemannTypeI#EFEvelocities\|accompanying discussion]] of Riemann Type 1 ellipsoids, [[~~User:Tohline/~~Appendix/References#EFE\|[<font color="red">EFE</font>] ]] provides an expression for the velocity vector of each fluid element, given its instantaneous ''body''-coordinate position (x, y, z) = (x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>) — see his Eq. (154), Chapter 7, §51 (p. 156). As viewed from the rotating frame of reference, the three component expressions are,		As we have summarized in an [[ThreeDimensionalConfigurations/RiemannTypeI#EFEvelocities\|accompanying discussion]] of Riemann Type 1 ellipsoids, [[Appendix/References#EFE\|[<font color="red">EFE</font>] ]] provides an expression for the velocity vector of each fluid element, given its instantaneous ''body''-coordinate position (x, y, z) = (x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>) — see his Eq. (154), Chapter 7, §51 (p. 156). As viewed from the rotating frame of reference, the three component expressions are,

	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	===Example Equilibrium Model===		===Example Equilibrium Model===

	These key parameters have been drawn from [[~~User:Tohline/~~Appendix/References#EFE\|[<font color="red">EFE</font>] ]]Chapter 7, Table XIII (p. 170):		These key parameters have been drawn from [[Appendix/References#EFE\|[<font color="red">EFE</font>] ]]Chapter 7, Table XIII (p. 170):
	<table width="80%" align="center" cellpadding="8" border="0">		<table width="80%" align="center" cellpadding="8" border="0">
	<tr><td align="left"><math>~a = a_1 = 1</math></td></tr>		<tr><td align="left"><math>~a = a_1 = 1</math></td></tr>
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	</table>		</table>

	As a consequence — see [[~~User:Tohline/~~ThreeDimensionalConfigurations/RiemannTypeI#Try_Again\|an accompanying discussion]] for details — the values of other parameters are …		As a consequence — see [[ThreeDimensionalConfigurations/RiemannTypeI#Try_Again\|an accompanying discussion]] for details — the values of other parameters are …

	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	===COLLADA-Based Representation===		===COLLADA-Based Representation===

	As has been described in an [[~~User:Tohline/~~ThreeDimensionalConfigurations/RiemannTypeI#Figure3\|accompanying discussion of Riemann Type 1 ellipsoids]], we have used COLLADA to construct an animated and interactive 3D scene that displays in purple the surface of an example Type I ellipsoid; panels a and b of Figure 1 show what this ellipsoid looks like when viewed from two different perspectives. (As a reminder — see the [[#explanation\| explanation accompanying Figure 2 of that accompanying discussion]] — the ellipsoid is tilted about the x-coordinate axis at an angle of 61.25° to the equilibrium spin axis, which is shown in green.) Yellow markers also have been placed in this 3D scene at each of the coordinate locations specified in the [[#ExampleTrajectories\|table that accompanies that discussion]]. From the perspective presented in Figure 3b, we can immediately identify three separate, nearly circular trajectories; the largest one corresponds to our choice of z<sub>0</sub> = -0.25, the smallest corresponds to our choice of z<sub>0</sub> = -0.60, and the one of intermediate size correspond to our choice of z<sub>0</sub> = -0.4310. When viewed from the perspective presented in Figure 1a, we see that these three trajectories define three separate planes; each plane is tipped at an angle of θ = -19.02° to the ''untilted'' equatorial, x-y plane of the purple ellipsoid.		As has been described in an [[ThreeDimensionalConfigurations/RiemannTypeI#Figure3\|accompanying discussion of Riemann Type 1 ellipsoids]], we have used COLLADA to construct an animated and interactive 3D scene that displays in purple the surface of an example Type I ellipsoid; panels a and b of Figure 1 show what this ellipsoid looks like when viewed from two different perspectives. (As a reminder — see the [[#explanation\| explanation accompanying Figure 2 of that accompanying discussion]] — the ellipsoid is tilted about the x-coordinate axis at an angle of 61.25° to the equilibrium spin axis, which is shown in green.) Yellow markers also have been placed in this 3D scene at each of the coordinate locations specified in the [[#ExampleTrajectories\|table that accompanies that discussion]]. From the perspective presented in Figure 3b, we can immediately identify three separate, nearly circular trajectories; the largest one corresponds to our choice of z<sub>0</sub> = -0.25, the smallest corresponds to our choice of z<sub>0</sub> = -0.60, and the one of intermediate size correspond to our choice of z<sub>0</sub> = -0.4310. When viewed from the perspective presented in Figure 1a, we see that these three trajectories define three separate planes; each plane is tipped at an angle of θ = -19.02° to the ''untilted'' equatorial, x-y plane of the purple ellipsoid.

	<div align="center">		<div align="center">
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	=See Also=		=See Also=

	* [[~~User:Tohline/~~ThreeDimensionalConfigurations/RiemannTypeI#Riemann_Type_1_Ellipsoids\|Riemann Type 1 Ellipsoids]]		* [[ThreeDimensionalConfigurations/RiemannTypeI#Riemann_Type_1_Ellipsoids\|Riemann Type 1 Ellipsoids]]
	* [[~~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]].

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ThreeDimensionalConfigurations/ChallengesPt2 - Revision history

Jet53man: /* See Also */

Jet53man: /* See Also */

Jet53man at 22:23, 2 October 2021

Jet53man: Created page with "__FORCETOC__ =Challenges Constructing Ellipsoidal-Like Configurations (Pt. 2)= T..."

Jet53man: Created page with "FORCETOC =Challenges Constructing Ellipsoidal-Like Configurations (Pt. 2)= T..."