Editing 3Dconfigurations/DescriptionOfRiemannTypeI (section)

====Summary Vorticity Expressions====

<ol><li>
Written in terms of the (unprimed) body-frame coordinates,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\nabla\times \boldsymbol{u}_\mathrm{EFE}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\hat\jmath} \zeta_2
+
\boldsymbol{\hat{k}}  \zeta_3 \, .
</math>
  </td>
</tr>
</table>
</li>
<li>
If we view the fluid motion from a (double-primed) frame that is tilted with respect to the (unprimed) body frame by the angle, <math>\chi</math>, such that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\tan\chi</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-\frac{\zeta_2}{\zeta_3} \, ,
</math>
  </td>
</tr>
</table>
then <math>\boldsymbol{\hat{k}''}</math> will align with the vorticity vector and the vorticity vector will have only one component, namely,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\nabla\times \boldsymbol{u''}_\mathrm{EFE}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\hat{k}''}  (\zeta_2^2 + \zeta_3^2)^{1 / 2} \, .
</math>
  </td>
</tr>
</table>

</li>
<li>
If we view the fluid motion from a (single-primed) frame that is tilted with respect to the (unprimed) body frame by an angle, <math>\theta</math>, such that the motion of Lagrangian fluid elements is everywhere parallel to the x'-y' plane &#8212; that is, such that there is no Lagrangian fluid motion in the <math>\boldsymbol{\hat{k}'}</math> direction &#8212; we find,
<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{\beta \Omega_2}{\gamma \Omega_3} = - \biggl[ \frac{c^2 (a^2 + b^2)}{b^2(a^2 + c^2)}\biggr]\frac{\zeta_2}{\zeta_3} \, ,
</math>
  </td>
</tr>
</table>
and,
<table border="0" cellpadding="5" align="center">

  <td align="right">
<math>\nabla\times \boldsymbol{u'}_\mathrm{EFE}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\hat\jmath'}\overbrace{\biggl[ \zeta_2\cos\theta 
+
\zeta_3 \sin\theta \biggr]}^{\mathrm{due~to~vertical~shear}} 
+
\boldsymbol{\hat{k}'} \underbrace{\biggl[ \zeta_3  \cos\theta
-
\zeta_2 \sin\theta \biggr]}_{\zeta_L} \, .
</math>
  </td>
</table>
</li>

<li>
[[#Vorticity_Implied_by_Lagrangian_Fluid_Motions|From below]], the contribution to the vorticity that is provided by the Lagrangian orbital-element-based description of the motion of the fluid is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\zeta_\mathrm{L}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\biggl\{ \frac{\partial \dot{y}'}{\partial x'} - \frac{\partial \dot{x}'}{\partial y'}  \biggr\}
=
\biggl[ 
\biggl(\frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr)
+
\biggl(\frac{y_\mathrm{max}}{x_\mathrm{max}}\biggr)\biggr] \dot\varphi \, .
</math>
  </td>
</tr>
</table>
<ol type="A">
  <li>
Adopting the parameter (Model001 evaluation in parentheses),
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\Lambda</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{a^2}{a^2 + b^2} \biggr] \zeta_3  \cos\theta    
-
\biggl[ \frac{a^2}{a^2 + c^2} \biggr] \zeta_2 \sin\theta 
= - 1.332892 \, ,
</math>
  </td>
</tr>
</table>
we have found that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{x_\mathrm{max}}{y_\mathrm{max}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ \Lambda \biggl[ \frac{a^2 + b^2}{b^2} \biggr] \frac{\cos\theta}{\zeta_3} \biggr\}^{1 / 2} 
= 1.025854
\, ,
</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>\dot\varphi</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ \Lambda \biggl[ \frac{b^2}{a^2 + b^2} \biggr] \frac{\zeta_3 }{\cos\theta} \biggr\}^{1 / 2}
= 1.299300
\, ,
</math>
  </td>
</tr>
</table>
which implies,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\zeta_L\biggr|_\mathrm{Model001}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
2.599446 \, .
</math>
  </td>
</tr>
</table>

  </li>
  <li>
Alternatively, we have found that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{x_\mathrm{max}}{y_\mathrm{max}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{a(c^2\cos^2\theta + b^2\sin^2\theta)^{1 / 2}}{bc}
= 1.025854
\, ,
</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>\dot\varphi</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\zeta_3}{\cos\theta}
\biggl[\frac{ abc ( c^2 \cos^2\theta + b^2 \sin^2\theta )^{1 / 2}}{c^2(a^2 + b^2)}  \biggr] 
= -1.299300
\, ,
</math>
  </td>
</tr>
</table>
which implies,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\zeta_L\biggr|_\mathrm{Model001}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-2.599446 \, .
</math>
  </td>
</tr>
</table>
  </li>
  <li>
In step #3, immediately above, we have determined that the <math>\boldsymbol{\hat{k}'}</math> component of the fluid vorticity is given by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\zeta_L</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
(\zeta_3\cos\theta - \zeta_2\sin\theta) 
= -2.599446 \, .
</math>
  </td>
</tr>
</table>

  </li>
</ol>
</li>
</ol>

<table border="1" width="80%" align="center" cellpadding="5"><tr><td align="left">
<font color="red">It appears as though we have separately derived three expressions for the quantity, <math>\zeta_L</math>.  It would be great if we could demonstrate analytically that the three expressions are, indeed, identical.</font>

Keep in mind that the definition of <math>\tan\theta</math> establishes the relation,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>b^2(a^2 + c^2)\zeta_3\sin\theta</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- c^2 (a^2 + b^2) \zeta_2 \cos\theta
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \frac{\zeta_3}{c^2(a^2 + b^2) \cos\theta}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- \frac{\zeta_2}{b^2(a^2 + c^2)  \sin\theta}
</math>
  </td>
</tr>
</table>

----

Let,
<table border="0" cellpadding="8" align="center">

<tr>
  <td align="right">
<math>\Upsilon</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
(c^2\cos^2\theta + b^2\sin^2\theta)^{1 / 2} \, .
</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>\zeta_\mathrm{L}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ 
\biggl(\frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr)
+
\biggl(\frac{y_\mathrm{max}}{x_\mathrm{max}}\biggr)\biggr] \dot\varphi 
</math>
  </td>
</tr>

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

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

\biggr\}
</math>
  </td>
</tr>

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\zeta_3}{c^2(a^2 + b^2)\cos\theta}\biggl\{
a^2 c^2\cos^2\theta + a^2b^2\sin^2\theta 
+
b^2c^2(\sin^2\theta + \cos^2\theta) 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\zeta_3}{c^2(a^2 + b^2)\cos\theta}\biggl\{
c^2(a^2 + b^2)\cos^2\theta  
\biggr\}
+
\frac{\zeta_3}{c^2(a^2 + b^2)\cos\theta}\biggl\{
b^2(a^2 + c^2)\sin^2\theta 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\zeta_3\cos\theta - \zeta_2\sin\theta \, .
</math>
  </td>
</tr>
</table>
<font color="red"><b>Q.E.D.</b></font>

----

Alternatively, pulling from the expressions that have been derived in terms of the parameter, <math>\Lambda</math>, we find,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\zeta_\mathrm{L}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ 
\biggl(\frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr)
+
\biggl(\frac{y_\mathrm{max}}{x_\mathrm{max}}\biggr)\biggr] \dot\varphi 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ \Lambda \biggl[ \frac{a^2 + b^2}{b^2} \biggr] \frac{\cos\theta}{\zeta_3} \biggr\}^{1 / 2}
\biggl\{ \Lambda \biggl[ \frac{b^2}{a^2 + b^2} \biggr] \frac{\zeta_3 }{\cos\theta} \biggr\}^{1 / 2}
+
\biggl\{ \Lambda^{-1} \biggl[ \frac{b^2}{a^2 + b^2} \biggr] \frac{\zeta_3}{\cos\theta} \biggr\}^{1 / 2}
\biggl\{ \Lambda \biggl[ \frac{b^2}{a^2 + b^2} \biggr] \frac{\zeta_3 }{\cos\theta} \biggr\}^{1 / 2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\Lambda
+
\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \frac{\zeta_3}{\cos\theta} 
</math>
  </td>
</tr>

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

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\zeta_3  \cos\theta    
-
\zeta_2\sin\theta \, .
</math>
  </td>
</tr>
</table>
<font color="red"><b>Q.E.D.</b></font>
</td></tr></table>