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Challenges Constructing EllipsoidalLike Configurations (Pt. 5)
This chapter extends the accompanying chapters titled, Construction Challenges (Pt. 1), (Pt. 2), (Pt. 3), and (Pt. 4).
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Tilted Plane Intersects Ellipsoid
In a 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) bodyframe coordinates that simultaneously satisfy the expressions,



and, 



where z_{0} is the location where the tipped plane intersects the zaxis of the body frame. Combining these two expressions, we see that an intersection between the tipped plane and the ellipsoidal surface will occur at (x, y)coordinate pairs that satisfy what we will henceforth refer to as the,
Intersection Expression  



as long as z_{0} lies within the range,



Before calling upon any of Riemann's model parameters, from geometric considerations alone we should be able to determine exactly what the expression is for any offcenter ellipse that results from slicing — at a tipped angle — the chosen ellipsoid.
Figure 1  


In the equatorial plane of the tipped coordinate system — that is, after mapping and , then setting — this intersection expression becomes,



The lightblue curve in the righthand panel of the following animation is a plot of this function for various values of (as indicated by the lightblue numerical value in the upperright corner of the figure's lefthand panel.
As it turns out — see our accompanying discussion — this expression can be rewritten as,



demonstrating that, as viewed from the x'y' plane, the (lightblue) intersection curve is always an offcenter ellipse. See also our COLLADAbased representation of these curves.
Trajectory of Lagrangian Fluid Elements
This subsection borrows heavily from an accompanying discussion.
Old Way of Thinking
It seems reasonable to assume that this offcenter ellipse expression will properly describe the orbital path of various Lagrangian fluid elements that make up the uniformdensity ellipsoid. Assuming that, when viewed from the rotatingandtipped coordinate frame, each fluid element's motion along this trajectory is oscillatory, it is reasonable to assume that the timedependent x'y' coordinate position of each fluid element is given by the expressions,



and, 



In this case, as viewed from the rotatingandtipped coordinate frame, the corresponding velocity components are,



and, 



This means that the (dimensional) velocity vector is,






New Thoughts
In our Old Way of Thinking, the hypothesized velocity flowfield was symmetric (in both directions) about the center of the elliptical trajectory. This hypothesized Lagrangian motion isn't (and cannot be) correct because an examination of EFE's derived Riemann (Eulerian) flowfield is not symmetric about the x'axis. Instead, the Eulerian flowfield displays a noticeable m = 1 contribution. Here we present an alternate hypothesis with two new features: (1) The flow is described by circulation about an center that is shifted along the y'axis away from the center of the ellipse; (2) The trajectory of Lagrangian fluid elements is described by motion in a cylindricalcoordinate system such that motion in the angular coordinate is uniform.
We will still insist that the trajectory of Lagrangian fluid elements is that of an ellipse described by the expression,



Now we will introduce a cylindrical coordinate system that is related to the x'y' coordinate system such that,



and, 



with . Mapping the other direction gives,



and, 



Using the (constraint) ellipse expression to eliminate y' from these last two expressions, we find,






Hence,



and,












The roots are …
Scratch notes:
where,

After has been evaluated for a given value of , the accompanying value of can be obtained, in principle, from either of the expressions:



or, 



1^{st} EXAMPLE: 

"plus"  "minus"  Expression used to obtain y'  
0.15708  0.82552  0.12068  0.50929  1.1548 × 10^{+2}  0.71575  +0.91355  0.95593 =  +0.86194  +0.66367  0.65616 

0.71575  0.21336  2.9845 = ϕ  π  +0.86194  +0.03652  0.15708 = ϕ 

We will assume that , with constant, and then determine how depends on and therefore, also, how it varies with time.
First, we note that transforming from the primedCartesian system to the cylindricalcoordinate system is accomplished via the relations,
Simpler Example
Positions of Lagrangian Fluid Elements
Let's determine the points of intersection of the following two expression:



and, 



Eliminating , then solving for , we find,















Given that,



the pair of roots are given by the expression,






As long as — where is an integer — the "x" coordinate that corresponds to each value of "y" can be obtained from the expression,



for the case of ,



Associated Velocities (1^{st} Try)
If we set , with constant, we appreciate that both of the coefficients, and will be functions of time. The timederivatives of the fluidelement positions will therefore depend on the timederivatives of these two coefficients. We find,






Hence,















And, [NOTE: The last term in this next expression was corrected on 5/21/2021. Needs to be incorporated into Excel spreadsheet.]



Associated Velocities (2^{nd} Try)
Let's simplify notation. Specifically, let's define,



Then we can write,



and, 



and, 



in which case,



The pair of roots (desired values of "y") are therefore given by the expression,









Let's examine the timederivative of y under the assumption that . First, note that,






Hence,
























See Also
 Riemann Type 1 Ellipsoids
 Construction Challenges (Pt. 1)
 Construction Challenges (Pt. 2)
 Construction Challenges (Pt. 3)
 Construction Challenges (Pt. 4)
 Construction Challenges (Pt. 5)
 Related discussions of models viewed from a rotating reference frame:
 PGE
 NOTE to Eric Hirschmann & David Neilsen... I have moved the earlier contents of this page to a new Wiki location called Compressible Riemann Ellipsoids.
© 2014  2021 by Joel E. Tohline 