Editing ThreeDimensionalConfigurations/ChallengesPt2 (section)

===Riemann-Derived Velocity Components===

====Inertial-Frame Expressions====
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,

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

<tr>
  <td align="right">
<math>~\dot{x} = u_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{a}{b}\biggr)^2 \gamma \Omega_3 y - \biggl(\frac{a}{c}\biggr)^2 \beta \Omega_2 z</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\dot{y} = u_2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \gamma \Omega_3 x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~+\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \zeta_3 x \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\dot{z} = u_3</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~+ \beta \Omega_2 x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2 x \, ,</math>
  </td>
</tr>
</table>
where,

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

<tr>
  <td align="right">
<math>~\beta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \biggl[ \frac{c^2}{a^2 + c^2} \biggr] \frac{\zeta_2}{\Omega_2}
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~\gamma</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \biggl[ \frac{b^2}{a^2 + b^2} \biggr] \frac{\zeta_3}{\Omega_3} \, .
</math>
  </td>
</tr>
</table>

<table border="1" cellpadding="8" width="90%" align="center"><tr><td align="left">
<div align="center">'''Rotating-Frame Vorticity'''</div>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\boldsymbol{\zeta} \equiv \boldsymbol{\nabla \times}\bold{u}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\boldsymbol{\hat\jmath} ~\zeta_2
+ \bold{\hat{k}} ~\zeta_3 \, .
</math>
  </td>
</tr>
</table>
</td></tr></table>

In the inertial frame, the velocity components are,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\bold{u}^{(0)} = \bold{u} + \boldsymbol{\Omega \times}\bold{x}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
( \boldsymbol{\hat\imath} \dot{x}
+ \boldsymbol{\hat\jmath} \dot{y}
+ \bold{\hat{k}} \dot{z} )
+
(\boldsymbol{\hat\jmath}\Omega_2 + \boldsymbol{\hat{k}}\Omega_3) \boldsymbol\times (\boldsymbol{\hat\imath} x + \boldsymbol{\hat\jmath}y + \boldsymbol{\hat{k}}z)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
( \boldsymbol{\hat\imath} \dot{x}
+ \boldsymbol{\hat\jmath} \dot{y}
+ \bold{\hat{k}} \dot{z} )
+
\Omega_2(\boldsymbol{\hat\imath}z -\boldsymbol{\hat{k}} x)
+ \Omega_3 (\boldsymbol{\hat\jmath}x - \boldsymbol{\hat\imath}y)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\boldsymbol{\hat\imath}(\dot{x} + \Omega_2 z - \Omega_3y)
+
\boldsymbol{\hat\jmath}(\dot{y} +  \Omega_3x)
+
\boldsymbol{\hat{k}}(\dot{z} - \Omega_2 x)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\boldsymbol{\hat\imath}\biggl[ \biggl(\frac{a}{b}\biggr)^2 \gamma \Omega_3 y - \biggl(\frac{a}{c}\biggr)^2 \beta \Omega_2 z + \Omega_2 z - \Omega_3y\biggr]
+
\boldsymbol{\hat\jmath}\biggr[ - \gamma \Omega_3 x +  \Omega_3x \biggr]
+
\boldsymbol{\hat{k}}\biggl[ + \beta \Omega_2 x - \Omega_2 x \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\boldsymbol{\hat\imath}\biggl\{ \biggl[ \biggl(\frac{a}{b}\biggr)^2 \gamma - 1 \biggr]\Omega_3y + \biggl[ 1 -  \biggl(\frac{a}{c}\biggr)^2 \beta\biggr] \Omega_2 z\biggr\}
+
\boldsymbol{\hat\jmath} ( 1- \gamma ) \Omega_3 x
+
\boldsymbol{\hat{k}} ( \beta -1 ) \Omega_2 x \, .
</math>
  </td>
</tr>
</table>

<table border="1" cellpadding="8" width="90%" align="center"><tr><td align="left">
<div align="center">'''Inertial-Frame Vorticity'''</div>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\boldsymbol{\zeta}^{(0)} = \boldsymbol{\nabla \times}\bold{u}^{(0)}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\boldsymbol{\hat\imath} \biggl[ 0\biggr]
+ \boldsymbol{\hat\jmath} \biggl\{ \biggl[ 1 -  \biggl(\frac{a}{c}\biggr)^2 \beta\biggr] \Omega_2 + (1-\beta)\Omega_2 \biggr\}
+ \bold{\hat{k}} \biggl\{ (1-\gamma)\Omega_3 +  \biggl[1 -  \biggl(\frac{a}{b}\biggr)^2 \gamma \biggr]\Omega_3 \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\boldsymbol{\hat\jmath} \biggl[ 1 -  \biggl(\frac{a}{c}\biggr)^2 \beta  + (1-\beta) \biggr]\Omega_2
+ \bold{\hat{k}} \biggl[ (1-\gamma) +  1 -  \biggl(\frac{a}{b}\biggr)^2 \gamma  \biggr] \Omega_3 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\boldsymbol{\hat\jmath} \biggl[ 2 -  \biggl( 1 + \frac{a^2}{c^2} \biggr)\beta   \biggr]\Omega_2
+ \bold{\hat{k}} \biggl[ 2 - \biggl(1 +  \frac{a^2}{b^2} \biggr) \gamma  \biggr] \Omega_3 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\boldsymbol{\hat\jmath} \biggl[ 2 + \frac{\zeta_2}{\Omega_2}   \biggr]\Omega_2
+ \bold{\hat{k}} \biggl[ 2 + \frac{\zeta_3}{\Omega_3}  \biggr] \Omega_3 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2\boldsymbol{\Omega} + \boldsymbol{\zeta} \, .
</math>
  </td>
</tr>
</table>
</td></tr></table>

====Coefficient Expression in Tipped Plane ====
In order for our expressions for the body-frame velocity components to align with Riemann's velocity components, we see, first, that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\tan\theta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\dot{z}}{\dot{y}}  
=
- \frac{\zeta_2}{\zeta_3} \biggl[ \frac{a^2 + b^2}{a^2 + c^2} \biggr]  \frac{c^2}{b^2} \, .
</math>
  </td>
</tr>
</table>
As a result, we find that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{b^2 c^2}{a^2}\biggl[\frac{x_\mathrm{max}}{y_\mathrm{max}}  \biggr]^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
c^2 \cos^2\theta + b^2\sin^2\theta 
=
\frac{c^2 + b^2\tan^2\theta}{1 + \tan^2\theta}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ c^2 + \frac{b^2 \zeta_2^2}{\zeta_3^2} \biggl( \frac{a^2 + b^2}{a^2 + c^2} \biggr)^2 \frac{c^4}{b^4}  \biggr]
\biggl[1 + \frac{\zeta_2^2}{\zeta_3^2} \biggl( \frac{a^2 + b^2}{a^2 + c^2} \biggr)^2  \frac{c^4}{b^4}  \biggr]^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \biggl[\frac{x_\mathrm{max}}{y_\mathrm{max}}  \biggr]^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~a^2
\biggl[ b^2 \zeta_3^2 (a^2 + c^2)^2 + c^2 \zeta_2^2 ( a^2 + b^2)^2  \biggr]
\biggl[b^4\zeta_3^2 (a^2 + c^2)^2 + c^4 \zeta_2^2 ( a^2 + b^2 )^2  \biggr]^{-1} 
\, .
</math>
  </td>
</tr>
</table>

Finally, setting the (square of the) two expressions for the <math>~\dot{y}</math> velocity component equal to one another gives,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl[ \frac{b^2}{a^2 + b^2} \biggr]^2 \zeta_3^2 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~{\dot\varphi}^2 \biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr]^2 \cos^2\theta</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~{\dot\varphi}^2 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\zeta_3^2\biggl[ \frac{b^2}{a^2 + b^2} \biggr]^2 
\biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr]^2 \biggl[1 + \tan^2\theta \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
a^2\zeta_3^2\biggl[ \frac{ b^2}{a^2 + b^2} \biggr]^2 
\biggl[ b^2 \zeta_3^2 (a^2 + c^2)^2 + c^2 \zeta_2^2 ( a^2 + b^2)^2  \biggr]
\biggl[b^4\zeta_3^2 (a^2 + c^2)^2 + c^4 \zeta_2^2 ( a^2 + b^2 )^2  \biggr]^{-1} 
\biggl\{ 1 + \biggl[ - \frac{\zeta_2}{\zeta_3} \biggl( \frac{a^2 + b^2}{a^2 + c^2} \biggr)  \frac{c^2}{b^2} \biggr]^2 \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{ a^2 b^4 \zeta_3^2}{(a^2 + b^2)^2} \biggr] 
\biggl[ b^2 \zeta_3^2 (a^2 + c^2)^2 + c^2 \zeta_2^2 ( a^2 + b^2)^2  \biggr]
\biggl[b^4\zeta_3^2 (a^2 + c^2)^2 + c^4 \zeta_2^2 ( a^2 + b^2 )^2  \biggr]^{-1} 
\biggl\{ \frac{b^4 \zeta_3^2 (a^2 + c^2)^2 + c^4 \zeta_2^2 (a^2 + b^2)^2}{b^4 \zeta_3^2 (a^2 + c^2)^2} \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{ a^2 }{(a^2 + b^2)^2(a^2 + c^2)^2}  
\biggl[ b^2 \zeta_3^2 (a^2 + c^2)^2 + c^2 \zeta_2^2 ( a^2 + b^2)^2  \biggr]
</math>
  </td>
</tr>
</table>