ThreeDimensionalConfigurations/DescriptionOfRiemannTypeI

Description of Riemann Type I Ellipsoids

Type I Riemann Ellipsoids

Example Equilibrium Model

This particular set of seven key parameters has been drawn from [EFE] Chapter 7, Table XIII (p. 170). The tabular layout presented here, also appears in a related discussion labeled, Challenges Pt. 2.

$a=a_1=1$

$b=a_2=1.25$

$c=a_3=0.4703$

${\mathrm{\Omega }}_2=0.3639$

${\mathrm{\Omega }}_3=0.6633$

${\zeta }_2=-2.2794$

${\zeta }_3=-1.9637$

As a consequence — see an accompanying discussion (alternatively, ChallengesPt6) for details — the values of other parameters are …

					Example Values
$\tan \theta$	$=$	$-\frac{{\zeta }_2}{{\zeta }_3}\left[\frac{a^2+b^2}{a^2+c^2}\right]\frac{c^2}{b^2}=-0.344793$			$\theta =$	$-19.0238^{\circ }$
$\mathrm{\Lambda }$	$\equiv$	$\left[\frac{a^2}{a^2+b^2}\right]{\zeta }_3\cos \theta -\left[\frac{a^2}{a^2+c^2}\right]{\zeta }_2\sin \theta$			$\mathrm{\Lambda }=$	$-1.332892$
$\frac{y_0}{z_0}$	$=$	$\left[\frac{a^2}{a^2+c^2}\right]\frac{{\zeta }_2}{\mathrm{\Lambda }}=-\frac{b^2\sin \theta }{(c^2{\cos }^2\theta +b^2{\sin }^2\theta )}$			$\frac{y_0}{z_0}=$	$+1.400377$
$\frac{x_{\mathrm{m}\mathrm{a}\mathrm{x}}}{y_{\mathrm{m}\mathrm{a}\mathrm{x}}}$	$=$	$\left\{\mathrm{\Lambda }\right[\frac{a^2+b^2}{b^2}]\frac{\cos \theta }{{\zeta }_3}{\}}^{1/2}$			$\frac{x_{\mathrm{m}\mathrm{a}\mathrm{x}}}{y_{\mathrm{m}\mathrm{a}\mathrm{x}}}=$	$+1.025854$
	$=$	$\frac{(c^2{\cos }^2\theta +b^2{\sin }^2\theta {)}^{1/2}}{bc}$
$\dot{\phi }$	$=$	$\left\{\mathrm{\Lambda }\right[\frac{b^2}{a^2+b^2}]\frac{{\zeta }_3}{\cos \theta }{\}}^{1/2}$			$\dot{\phi }=$	$+1.299300$

EFE Rotating Cartesian Frame

Concentric triaxial ellipsoids are defined by the expression,

$P$

$=$

$(\frac{x}{a}{)}^2+(\frac{y}{b}{)}^2+(\frac{z}{c}{)}^2,$

where $0\le P\le 1$ is a constant. As viewed from the rotating reference frame, the velocity flow-field everywhere inside $(0\le P<1)$, and on the surface $(P=1)$ of the Type I Riemann ellipsoid is given by the expression — see, for example, an accompanying discussion of the Riemann flow-field,

${\mathbf{𝐮}}_{\mathrm{E}\mathrm{F}\mathrm{E}}$

$=$

$\hat{\mathit{ı}}\{-[\frac{a^2}{a^2+b^2}]{\zeta }_{3}y+[\frac{a^2}{a^2+c^2}\left]{\zeta }_{2}z\right\}+\hat{\mathit{ȷ}}\{+[\frac{b^2}{a^2+b^2}\left]{\zeta }_{3}x\right\}+\hat{\mathbf{k}}\{-[\frac{c^2}{a^2+c^2}\left]{\zeta }_{2}x\right\}.$

In an accompanying discussion, we have shown that,

${\mathbf{𝐮}}_{\mathrm{E}\mathrm{F}\mathrm{E}}\cdot \mathrm{\nabla }P$

$=$

$0,$

which means that, at every location inside and on the surface of the configuration, the velocity vector is orthogonal to a vector that is normal to the ellipsoidal surface at that location.

Tilted Coordinate System

Figure 1:   Tilted Reference Frame
$\hat{\mathit{ı}}$ $\to$ ${\hat{\mathit{ı}}}^{\prime },$ $\hat{\mathit{ȷ}}$ $\to$ ${\hat{\mathit{ȷ}}}^{\prime }\cos \theta -{\hat{\mathit{k}}}^{\prime }\sin \theta ,$ $\hat{\mathit{k}}$ $\to$ ${\hat{\mathit{ȷ}}}^{\prime }\sin \theta +{\hat{\mathit{k}}}^{\prime }\cos \theta .$		$x$ $\to$ $x^{\prime },$ $y$ $\to$ $y^{\prime }\cos \theta -z^{\prime }\sin \theta ,$ $z-z_0$ $\to$ $y^{\prime }\sin \theta +z^{\prime }\cos \theta .$

As we have detailed in our accompanying discussion, as viewed from this "tipped" frame, the concentric ellipsoidal surfaces of a Type I Riemann ellipsoid are defined by the expression,

$P^{\prime }$

$=$

$[\frac{y^{\prime }\cos \theta -z^{\prime }\sin \theta }{b}{]}^2+[\frac{z_0+z^{\prime }\cos \theta +y^{\prime }\sin \theta }{c}{]}^2+(\frac{x^{\prime }}{a}{)}^2.$

and the velocity flow-field is given by the expression,

${{\mathit{u}}^{\prime }}_{\mathrm{E}\mathrm{F}\mathrm{E}}$	$=$	${\hat{\mathit{ı}}}^{\prime }\{-[\frac{a^2}{a^2+b^2}]{\zeta }_3(y^{\prime }\cos \theta -z^{\prime }\sin \theta )+[\frac{a^2}{a^2+c^2}]{\zeta }_2(z_0+y^{\prime }\sin \theta +z^{\prime }\cos \theta )\}$
		$+[{\hat{\mathit{ȷ}}}^{\prime }\cos \theta -{\hat{\mathbf{k}}}^{\prime }\sin \theta ]\left\{\right[\frac{b^2}{a^2+b^2}\left]{\zeta }_3{x}^{\prime }\right\}+[{\hat{\mathit{ȷ}}}^{\prime }\sin \theta +{\hat{\mathbf{k}}}^{\prime }\cos \theta ]\{-[\frac{c^2}{a^2+c^2}\left]{\zeta }_2{x}^{\prime }\right\}.$

We also have explicitly demonstrated that, for any arbitrarily chosen value of the tilt angle, $\theta$,

${{\mathbf{𝐮}}^{\prime }}_{\mathrm{E}\mathrm{F}\mathrm{E}}\cdot \mathrm{\nabla }{P}^{\prime }$

$=$

$0.$

Preferred Tilt

As we discuss elsewhere, if we specifically choose,

$\tan \theta$

$=$

$-\frac{\beta {\mathrm{\Omega }}_2}{\gamma {\mathrm{\Omega }}_3}=-\left[\frac{c^2(a^2+b^2)}{b^2(a^2+c^2)}\right]\frac{{\zeta }_2}{{\zeta }_3}.$

the component of the flow-field in the ${\hat{\mathbf{k}}}^{\prime }$ direction vanishes; that is, in this specific case, as viewed from the tilted reference frame, all of the fluid motion is in confined to the x'-y' plane. Notice that this plane is not parallel to any of the three principal planes of the Type I Riemann ellipsoid.

See Also

• Description of Riemann Type I Ellipsoids
• Riemann Type 1 Ellipsoids (old introduction)
• Construction Challenges (Pt. 1)
• Construction Challenges (Pt. 2)
• Construction Challenges (Pt. 3)
• Construction Challenges (Pt. 4)
• Construction Challenges (Pt. 5)
• Construction Challenges (Pt. 6)

• 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.