Editing ThreeDimensionalConfigurations/ChallengesPt2 (section)

===Where Are We Headed?===
In a [[ThreeDimensionalConfigurations/RiemannTypeI#Try_Again|separate discussion]], we have shown that, as viewed from a frame that "tumbles" with the (purple) body of a Type 1 Riemann ellipsoid, each Lagrangian fluid element moves along an elliptical path in a plane that is tipped by an angle <math>~\theta</math> about the x-axis of the body.  (See the yellow-dotted orbits in Figure panels 1a and 1b below).  As viewed from the (primed) coordinates associated with this tipped plane, by definition, z' = 0 and dz'/dt = 0, and the planar orbit is defined by the expression for an,
<table border="0" cellpadding="5" align="center">
<tr>
<td align="center" colspan="3"><font color="maroon">'''Off-Center Ellipse'''</font></td>
</tr>
<tr>
  <td align="right">
<math>~1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{x'}{x_\mathrm{max}} \biggr)^2 + \biggl(\frac{y' - y_0}{y_\mathrm{max}} \biggr)^2 \, .</math>
  </td>
</tr>
</table>
As a function of time, the x'-y' coordinates and associated velocity components of each Lagrangian fluid element are given by the expressions,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~x_\mathrm{max}\cos(\dot\varphi t)</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;
  <td align="right">
<math>~y' - y_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~y_\mathrm{max}\sin(\dot\varphi t) \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\dot{x}'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- x_\mathrm{max}~ \dot\varphi \cdot \sin(\dot\varphi t) = (y_0 - y') \biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr] \dot\varphi </math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;
  <td align="right">
<math>~\dot{y}' </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~y_\mathrm{max}~\dot\varphi \cdot \cos(\dot\varphi t) = x' \biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}}\biggr] \dot\varphi \, .</math>
  </td>
</tr>
</table>

Notice that this is a divergence-free flow-field:

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

<tr>
  <td align="right">
<math>~\nabla \cdot \vec{v'}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\partial \dot{x}'}{\partial x'} + \frac{\partial \dot{y}'}{\partial y'} + \frac{\partial \dot{z}'}{\partial z'}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\partial }{\partial x'} \biggl[ (y' - y_0) \dot\varphi \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr)  \biggr] 
+ 
\frac{\partial }{\partial y'} \biggl[ x'  \dot\varphi \biggl( \frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr)  \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0\, .
</math>
  </td>
</tr>
</table>

</td></tr></table>


<span id="VorticitySetup">Also,</span> along the lines of our [[ThreeDimensionalConfigurations/Challenges#Riemann_S-type_Ellipsoids|accompanying discussion of Riemann S-Type Ellipoids]], it is useful to develop the expression for the fluid vorticity as viewed from the rotating- and tipped-reference frame.
<table border="1" cellpadding="8" width="90%" align="center"><tr><td align="left">
<div align="center">'''Vorticity'''</div>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\boldsymbol{\zeta'} \equiv \boldsymbol{\nabla \times}\bold{v'}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\boldsymbol{\hat\imath'} \biggl[ \frac{\partial \cancelto{}{\dot{z}'} }{\partial y'} - \frac{\partial \dot{y}'}{\partial z'} \biggr]
+ \boldsymbol{\hat\jmath'} \biggl[ \frac{\partial \dot{x}'}{\partial z'} - \frac{\partial \cancelto{}{\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\imath'} (x' \dot\varphi )\frac{\partial }{\partial z'} \biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max} }\biggr]
+ \boldsymbol{\hat\jmath'} \biggl\{
\dot\varphi (y_0 - y')\frac{\partial }{\partial z'}\biggl[  \frac{x_\mathrm{max}}{y_\mathrm{max} } \biggr]
+ \dot\varphi \biggl[  \frac{x_\mathrm{max}}{y_\mathrm{max} } \biggr] \frac{\partial y_0}{\partial z'}
\biggr\}
+ \bold{\hat{k}'} \biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max} } - \frac{x_\mathrm{max} }{y_\mathrm{max} } \biggr] \dot\varphi \, .
</math>
  </td>
</tr>
</table>
Further evaluation is completed, [[#Vorticity_Determination|below]], after we determine how <math>~y_0</math> and <math>~[x_\mathrm{max}/y_\mathrm{max}]^{\pm 1}</math> depend on <math>~z_0</math>; and after appreciating that, in order to introduce the functional dependence on <math>~z' \ne 0</math> in every relevant expression, we need to make the replacement, <math>~z_0 \rightarrow (z_0 + z'\cos\theta)</math>.  <font color="red"><==&nbsp; &nbsp; Figure this out!</font>
</td></tr></table>


In the subsections of this chapter that follow, we provide analytic expressions for these various quantities &#8212; <math>~x_\mathrm{max}, y_\mathrm{max}, y_0, \dot\varphi</math> &#8212; in terms of the properties of any chosen Type 1 Riemann ellipsoid.