Editing ThreeDimensionalConfigurations/ChallengesPt3 (section)

===Riemann-Derived Expressions===
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''Inertial Frame'' (green with subscript "0") <br />and ''Body Frame'' (black and unsubscripted).
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  <td align="center">[[File:InertialAxes05.png|400px|Inertial and Body Frames]]</td>
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For our chosen [[ThreeDimensionalConfigurations/RiemannTypeI#Case_I|Example Type I Ellipsoid]], we have, <math>~\Omega_2 = 0.3639</math> and <math>~\Omega_3 = 0.6633</math>, in which case, <math>~\Omega_0 = 0.7566</math> and <math>~\delta = 0.5018 ~\mathrm{rad} = 28.75^\circ</math>.
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The purple (ellipsoidal) configuration is spinning with frequency, <math>~\Omega_0</math> about the <math>~z_0</math>-axis of the "inertial frame," as illustrated; that is,
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<math>~\boldsymbol\Omega</math>
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<math>~=</math>
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<math>~\boldsymbol{\hat{k}_0}\Omega_0 \, .</math>
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Also as illustrated, the "body frame," which is attached to and aligned with the principal axes of the purple ellipsoid, is tilted at an angle, <math>~\delta</math>, with respect to the inertial frame.  Hence, as viewed from the ''body'' frame, we have,
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<math>~\boldsymbol\Omega</math>
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<math>~=</math>
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<math>~\biggl[ \boldsymbol{\hat\jmath }\sin\delta + \boldsymbol{\hat{k} }\cos\delta \biggr]\Omega_0 \, .</math>
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Now, adhering to the notation used by [[Appendix/References#EFE|[<font color="red">EFE</font>] ]] &#8212; see, for example, the first paragraph of &sect;51 (p. 156) &#8212; we should write,
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<math>~\boldsymbol\Omega</math>
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<math>~=</math>
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<math>~\boldsymbol{\hat\jmath }\Omega_2 + \boldsymbol{\hat{k} }\Omega_3 
~~~~~\Rightarrow ~~~ \Omega_2 = \Omega_0\sin\delta
</math>&nbsp; &nbsp; and, &nbsp; &nbsp; 
<math>~\Omega_3 = \Omega_0\cos\delta \, .</math>
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This means that,
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<math>~\Omega_0</math>
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<math>~=</math>
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<math>~
\biggl[\Omega_2^2 + \Omega_3^2 \biggr]^{1 / 2}
</math>&nbsp; &nbsp; and, &nbsp; &nbsp; 
<math>~\delta = \tan^{-1}\biggl[ \frac{\Omega_2}{\Omega_3} \biggr] \, .</math>
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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>) &#8212; see his Eq. (154), Chapter 7, &sect;51 (p. 156).  As viewed from the rotating frame of reference, the three component expressions are,

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<math>~\dot{x} = u_1 = \boldsymbol{\hat\imath} \cdot \boldsymbol{u}</math>
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<math>~=</math>
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<math>~\biggl(\frac{a}{b}\biggr)^2 \gamma \Omega_3 y - \biggl(\frac{a}{c}\biggr)^2 \beta \Omega_2 z</math>
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<math>~=</math>
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<math>~- \biggl[ \frac{a^2}{a^2 + b^2} \biggr] \zeta_3 y + \biggl[ \frac{a^2}{a^2 + c^2} \biggr] \zeta_2 z \, ,</math>
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<math>~\dot{y} = u_2 = \boldsymbol{\hat\jmath} \cdot \boldsymbol{u}</math>
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<math>~=</math>
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<math>~- \gamma \Omega_3 x</math>
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<math>~=</math>
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<math>~+\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \zeta_3 x \, ,</math>
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<math>~\dot{z} = u_3 = \boldsymbol{\hat{k}} \cdot \boldsymbol{u}</math>
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<math>~=</math>
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<math>~+ \beta \Omega_2 x</math>
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<math>~=</math>
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<math>~- \biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2 x \, ,</math>
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<span  id="betagamma">where,</span>

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<math>~\beta</math>
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<math>~=</math>
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<math>~
- \biggl[ \frac{c^2}{a^2 + c^2} \biggr] \frac{\zeta_2}{\Omega_2}
</math>
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<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; </td>
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<math>~\gamma</math>
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<math>~=</math>
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<math>~
- \biggl[ \frac{b^2}{a^2 + b^2} \biggr] \frac{\zeta_3}{\Omega_3} \, .
</math>
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<div align="center">'''Rotating-Frame Vorticity'''</div>
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<math>~\boldsymbol{\zeta} \equiv \boldsymbol{\nabla \times}\bold{u}</math>
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<math>~=</math>
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<math>~
\boldsymbol{\hat\imath} \biggl[ \frac{\partial \dot{z} }{\partial y} - \frac{\partial \dot{y}}{\partial z} \biggr]
+ \boldsymbol{\hat\jmath} \biggl[ \frac{\partial \dot{x}}{\partial z} - \frac{\partial \dot{z}}{\partial x} \biggr]
+ \bold{\hat{k}} \biggl[ \frac{\partial \dot{y}}{\partial x} - \frac{\partial \dot{x}}{\partial y} \biggr] 
</math>
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&nbsp;
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<math>~=</math>
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<math>~
\boldsymbol{\hat\jmath} \biggl\{ 
\biggl[ \frac{a^2}{a^2 + c^2} \biggr] \zeta_2 + \biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2 
\biggr\}
+ \bold{\hat{k}} \biggl\{ 
\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \zeta_3 + \biggl[ \frac{a^2}{a^2 + b^2} \biggr] \zeta_3 
\biggr\}
</math>
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&nbsp;
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<math>~=</math>
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<math>~
\boldsymbol{\hat\jmath} ~\zeta_2
+ \bold{\hat{k}} ~\zeta_3 \, .
</math>
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For our chosen [[ThreeDimensionalConfigurations/RiemannTypeI#Case_I|Example Type I Ellipsoid]], we have, <math>~\zeta_2 = -2.2794</math> and <math>~\Omega_3 = -1.9637</math>, in which case, <math>~\zeta_\mathrm{rot} = (\zeta_2^2 + \zeta_3^2)^{1 / 2} = 2.2794</math> and <math>~\xi \equiv \tan^{-1}[\zeta_2/\zeta_3] = 4.0013 ~\mathrm{rad} = 229.26^\circ</math>.
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[[File:VorticityAxis04.png|350px|center|Vorticity Axis]]
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