Editing Appendix/Ramblings/HybridSchemeImplications (section)

===Steady-State Velocity Field for Jacobi Ellipsoids===

In steady-state, the (Lagrangian time-derivative) operator on the left-hand-side of all three component equations maps to the following operator:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathbf{u'} \cdot \nabla</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\sum_{i=1}^3 u'_i \frac{\partial}{\partial x_i} \, ,</math>
  </td>
  <td align="right">&nbsp; &nbsp; &nbsp; &nbsp; (in Cartesian coordinates);</td>
</tr>

<tr>
  <td align="right">
<math>~\mathbf{u'} \cdot \nabla</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
u'_\varpi \frac{\partial}{\partial \varpi} 
+ \frac{u'_\varphi}{\varpi} \frac{\partial}{\partial \varphi}
+ u'_z \frac{\partial}{\partial z}
\, ,</math>
  </td>
  <td align="right">&nbsp; &nbsp; &nbsp; &nbsp; (in cylindrical coordinates);</td>
</tr>
</table>
We know, as well, that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~u'_\varpi = u'_x \cos\varphi + u'_y \sin\varphi \, ,</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~u'_\varphi = u'_y \cos\varphi - u'_x \sin\varphi \, .</math>
  </td>
</tr>
</table>
Hence, the cylindrical-coordinate-based operator may be rewritten as,

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

<tr>
  <td align="right">
<math>~\mathbf{u'} \cdot \nabla</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(  u'_x \cos\varphi + u'_y \sin\varphi ) \frac{\partial}{\partial \varpi} 
+ ( u'_y \cos\varphi - u'_x \sin\varphi )\frac{1}{\varpi} \frac{\partial}{\partial \varphi}
+ u'_z \frac{\partial}{\partial z}
\, .</math>
  </td>
</tr>
</table>

Drawing from [ [[ThreeDimensionalConfigurations/RiemannStype#Adopted_Velocity_Flow-Field|Ref04]] ] &hellip; As [https://ui.adsabs.harvard.edu/abs/2006ApJ...639..549O/abstract Ou(2006)] has pointed out, <font color="orange">the velocity field of a Riemann S-type ellipsoid as viewed from a frame rotating with angular velocity <math>~{\vec{\Omega}}_f = \boldsymbol{\hat{k}} \Omega_f</math> takes the following form:</font>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~{\mathbf{u'}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\lambda \biggl[ \boldsymbol{\hat{\imath}} \biggl(\frac{a}{b}\biggr)y - \boldsymbol{\hat{\jmath}} \biggl(\frac{b}{a}\biggr)x \biggr]  \, ,</math>
  </td>
</tr>
</table>

[https://ui.adsabs.harvard.edu/abs/2006ApJ...639..549O/abstract Ou(2006)], p. 550, &sect;2, Eq. (3)
</div>

<font color="orange">where <math>~\lambda</math> is a constant that determines the magnitude of the internal motion of the fluid, and the origin of the x-y coordinate system is at the center of the ellipsoid.  This velocity field, <math>~\mathbf{u'}</math>, is designed so that velocity vectors everywhere are always aligned with elliptical stream lines by demanding that they be tangent to the</font> equi-''effective''-potential <font color="orange"> contours, which are concentric ellipses.</font>  Hence, for Riemann S-type ellipsoids, we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~u'_x = \lambda\biggl(\frac{a}{b}\biggr)y = \lambda\biggl(\frac{a}{b}\biggr)\varpi \sin\varphi \, ;</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="center">
<math>~u'_y = -\lambda\biggl(\frac{b}{a}\biggr)x = -\lambda\biggl(\frac{b}{a}\biggr)\varpi \cos\varphi \, ;</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="left">
<math>~u'_z = 0 \, .</math>
  </td>
</tr>
</table>

So, for the velocity flow that underpins Riemann S-type ellipsoids, the cylindrical-coordinate-based operator is

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

<tr>
  <td align="right">
<math>~\mathbf{u'} \cdot \nabla</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[  \lambda\biggl(\frac{a}{b}\biggr)\varpi \sin\varphi \cos\varphi -\lambda\biggl(\frac{b}{a}\biggr)\varpi \cos\varphi \sin\varphi \biggr] \frac{\partial}{\partial \varpi} 
+ \biggl[ -\lambda\biggl(\frac{b}{a}\biggr)\varpi \cos\varphi \cos\varphi - \lambda\biggl(\frac{a}{b}\biggr)\varpi \sin\varphi \sin\varphi \biggr] \frac{1}{\varpi} \frac{\partial}{\partial \varphi}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[  \biggl(\frac{a}{b}\biggr) - \biggl(\frac{b}{a}\biggr) \biggr]\lambda \varpi \sin\varphi \cos\varphi \frac{\partial}{\partial \varpi} 
- \biggl[ \biggl(\frac{b}{a}\biggr) \cos^2\varphi  + \biggl(\frac{a}{b}\biggr) \sin^2\varphi \biggr]\lambda  \frac{\partial}{\partial \varphi}
\, .</math>
  </td>
</tr>
</table>

And, given that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathbf{\hat{e}}_\varphi \Omega_f \varpi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\Omega_f \varpi \biggl[ \boldsymbol{\hat{\jmath}} \cos\varphi - \boldsymbol{\hat{\imath}} \sin\varphi \biggr] \, ,
</math>
  </td>
</tr>
</table>
the inertial-frame velocity components are,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
v_x = \lambda\biggl(\frac{a}{b}\biggr)\varpi \sin\varphi - \Omega_f \varpi \sin\varphi 
= \biggl[ \lambda\biggl(\frac{a}{b}\biggr) - \Omega_f \biggr]\varpi\sin\varphi
\, ;</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="center">
<math>~
v_y = -\lambda\biggl(\frac{b}{a}\biggr) \varpi\cos\varphi + \Omega_f\varpi \cos\varphi 
= \biggl[\Omega_f  -\lambda\biggl(\frac{b}{a}\biggr) \biggr]\varpi\cos\varphi
\, ;</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="left">
<math>~v_z = 0 \, .</math>
  </td>
</tr>
</table>
That is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~v_\varpi = v_x \cos\varphi + v_y \sin\varphi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \lambda\biggl(\frac{a}{b}\biggr) - \Omega_f \biggr]\varpi\sin\varphi \cos\varphi
+
\biggl[\Omega_f  -\lambda\biggl(\frac{b}{a}\biggr) \biggr]\varpi\cos\varphi \sin\varphi
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \biggl(\frac{a}{b}\biggr) - \biggl(\frac{b}{a}\biggr)  \biggr]\lambda \varpi\sin\varphi \cos\varphi \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~v_\varphi = v_y \cos\varphi - v_x \sin\varphi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\Omega_f  -\lambda\biggl(\frac{b}{a}\biggr) \biggr]\varpi\cos^2\varphi 
-
\biggl[ \lambda\biggl(\frac{a}{b}\biggr) - \Omega_f \biggr]\varpi\sin^2\varphi  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\Omega_f \varpi  
-\lambda \varpi \biggl[
\biggl(\frac{b}{a}\biggr) \cos^2\varphi 
+\biggl(\frac{a}{b}\biggr) \sin^2\varphi  
\biggr] \, .
</math>
  </td>
</tr>
</table>

Note, as well, that,

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

<tr>
  <td align="right">
<math>~\frac{v_\varphi^2}{\varpi}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\varpi} \biggl\{
\Omega_f \varpi  
-\lambda \varpi \biggl[
\biggl(\frac{b}{a}\biggr) \cos^2\varphi 
+\biggl(\frac{a}{b}\biggr) \sin^2\varphi  
\biggr]
\biggr\}^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\varpi \biggl\{
\Omega_f^2  - 2\lambda \Omega_f   \biggl[
\biggl(\frac{b}{a}\biggr) \cos^2\varphi 
+\biggl(\frac{a}{b}\biggr) \sin^2\varphi  
\biggr]
+ \lambda^2  \biggl[
\biggl(\frac{b}{a}\biggr) \cos^2\varphi 
+\biggl(\frac{a}{b}\biggr) \sin^2\varphi  
\biggr]^2
\biggr\} \, .
</math>
  </td>
</tr>
</table>


Finally, then, we find that the left-hand-side of the momentum-component expressions are,
<table border="0" cellpadding="3" align="center">

<tr>
  <td align="center">
<math>\mathbf{\hat{k}}:</math>
  </td>
  <td align="right">
<math>~\frac{d v_z}{dt}  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0  \, ;
</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>\mathbf{\hat{e}}_\varpi:</math>
  </td>
  <td align="right" colspan="1">
<math>~\frac{d v_\varpi}{dt}  </math>
 </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{
\biggl[  \biggl(\frac{a}{b}\biggr) - \biggl(\frac{b}{a}\biggr) \biggr]\lambda \varpi \sin\varphi \cos\varphi \frac{\partial}{\partial \varpi} 
- \biggl[ \biggl(\frac{b}{a}\biggr) \cos^2\varphi  + \biggl(\frac{a}{b}\biggr) \sin^2\varphi \biggr]\lambda  \frac{\partial}{\partial \varphi}
\biggr\}\biggl[ \biggl(\frac{a}{b}\biggr) - \biggl(\frac{b}{a}\biggr)  \biggr]\lambda \varpi\sin\varphi \cos\varphi
</math>
  </td>
</tr>

<tr>
  <td align="center">
&nbsp;
  </td>
  <td align="right" colspan="1">
&nbsp;
 </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\lambda \biggl[ \biggl(\frac{a}{b}\biggr) - \biggl(\frac{b}{a}\biggr)  \biggr]\biggl\{
\biggl[  \biggl(\frac{a}{b}\biggr) - \biggl(\frac{b}{a}\biggr) \biggr]\lambda \varpi \sin\varphi \cos\varphi \frac{\partial}{\partial \varpi} \biggl[ \varpi\sin\varphi \cos\varphi \biggr]
- \biggl[ \biggl(\frac{b}{a}\biggr) \cos^2\varphi  + \biggl(\frac{a}{b}\biggr) \sin^2\varphi \biggr]\lambda  \frac{\partial}{\partial \varphi}\biggl[ \varpi\sin\varphi \cos\varphi \biggr]
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="center">
&nbsp;
  </td>
  <td align="right" colspan="1">
&nbsp;
 </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\lambda \biggl[ \biggl(\frac{a}{b}\biggr) - \biggl(\frac{b}{a}\biggr)  \biggr]\biggl\{
\biggl[  \biggl(\frac{a}{b}\biggr) - \biggl(\frac{b}{a}\biggr) \biggr]\lambda \varpi \sin^2\varphi \cos^2\varphi 
+ \biggl[ \biggl(\frac{b}{a}\biggr) \cos^2\varphi  + \biggl(\frac{a}{b}\biggr) \sin^2\varphi \biggr] \lambda\varpi  \biggl[ \sin^2\varphi - \cos^2\varphi \biggr]
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="center">
&nbsp;
  </td>
  <td align="right" colspan="1">
&nbsp;
 </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\lambda^2 \varpi \biggl( \frac{a}{b} - \frac{b}{a}  \biggr)
\biggl[
\biggl( \frac{a}{b}\biggr) \sin^4\varphi  
- \biggl(\frac{b}{a}\biggr) \cos^4\varphi 
\biggr] \, ;
</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>\mathbf{\hat{e}}_\varphi:</math>
  </td>
  <td align="right" colspan="1">
<math>~\frac{1}{\varpi} ~\frac{d (\varpi v_\varphi )}{dt} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\varpi}\biggl\{
\biggl[  \biggl(\frac{a}{b}\biggr) - \biggl(\frac{b}{a}\biggr) \biggr]\lambda \varpi \sin\varphi \cos\varphi \frac{\partial}{\partial \varpi} 
- \biggl[ \biggl(\frac{b}{a}\biggr) \cos^2\varphi  + \biggl(\frac{a}{b}\biggr) \sin^2\varphi \biggr]\lambda  \frac{\partial}{\partial \varphi}
\biggr\}
\biggl\{ \biggl[\Omega_f  -\lambda\biggl(\frac{b}{a}\biggr) \biggr]\varpi^2\cos^2\varphi 
-
\biggl[ \lambda\biggl(\frac{a}{b}\biggr) - \Omega_f \biggr]\varpi^2\sin^2\varphi 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right" colspan="2">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[  \biggl(\frac{a}{b}\biggr) - \biggl(\frac{b}{a}\biggr) \biggr] 2\varpi \lambda \sin\varphi \cos\varphi  
\biggl\{ \biggl[\Omega_f  -\lambda\biggl(\frac{b}{a}\biggr) \biggr]\cos^2\varphi 
-
\biggl[ \lambda\biggl(\frac{a}{b}\biggr) - \Omega_f \biggr]\sin^2\varphi 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right" colspan="2">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~+
\biggl[ \biggl(\frac{b}{a}\biggr) \cos^2\varphi  + \biggl(\frac{a}{b}\biggr) \sin^2\varphi \biggr]  
\biggl[ \frac{a}{b} - \frac{b}{a} \biggr]
2\varpi \lambda^2 \sin\varphi \cos\varphi
</math>
  </td>
</tr>

<tr>
  <td align="right" colspan="2">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(  \frac{a}{b} - \frac{b}{a} \biggr) 2\varpi \lambda \sin\varphi \cos\varphi  
\biggl\{ \biggl[\Omega_f  -\lambda\biggl(\frac{b}{a}\biggr) \biggr]\cos^2\varphi 
-
\biggl[ \lambda\biggl(\frac{a}{b}\biggr) - \Omega_f \biggr]\sin^2\varphi 
+2\lambda\biggl[ \biggl(\frac{b}{a}\biggr) \cos^2\varphi  + \biggl(\frac{a}{b}\biggr) \sin^2\varphi \biggr]  \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right" colspan="2">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(  \frac{a}{b} - \frac{b}{a} \biggr) 2\varpi \lambda \sin\varphi \cos\varphi  
\biggl\{ \biggl[\Omega_f   +\lambda\biggl(\frac{b}{a}\biggr)\biggr]\cos^2\varphi 
+
\biggl[ \Omega_f  + \lambda\biggl( \frac{a}{b}\biggr) \biggr]\sin^2\varphi 
\biggr\} \, .
</math>
  </td>
</tr>
</table>