Editing Appendix/Ramblings/HybridSchemeImplications (section)

===Focus on Tracking Angular Momentum===

Let's begin by using <math>~\mathbf{u'}</math>, instead of <math>~{\vec{v}}_\mathrm{rot}</math>, to represent the fluid velocity vector as viewed from the rotating frame of reference.  Our foundation expression becomes,

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

<tr>
  <td align="right">
<math>~\frac{d \bold{u'}}{dt}  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{1}{\rho} \nabla P - \nabla \Phi  
- 2{\vec\Omega}_f \times \bold{u}'
- {\vec\Omega}_f \times ({\vec\Omega}_f \times \vec{x})  \, ,</math>
  </td>
</tr>
</table>
where we appreciate that we can move from the Lagrangian to an Eulerian representation by employing the operator substitution,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d}{dt}</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~\frac{\partial}{\partial t} + \mathbf{u'} \cdot \nabla </math>
  </td>
</tr>
</table>

Next, using [Ref03] as a guide, let's [[Appendix/Ramblings/HybridSchemeOld#Focus_on_Tracking_Angular_Momentum|focus on tracking angular momentum]].  We need to break the vector momentum equation, as well as the velocity vectors, into their <math>~(\bold{\hat{e}}_\varpi, \bold{\hat{e}}_\varphi, \bold{\hat{k}})</math> components.

<table border="1" cellpadding="10" align="center" width="80%"><tr><td align="left">
NOTE:  For the time being, we will write the velocity vector in terms of generic components, namely,
<div align="center">
<math>~\bold{u}' = \bold{\hat{e}}_\varpi u'_\varpi +  \bold{\hat{e}}_\varphi u'_\varphi +  \bold{\hat{k}}u'_z \, .</math>
</div>
But, eventually, we want to explicitly insert the rotating-frame velocity that underpins the equilibrium properties of Riemann S-type ellipsoids.  In Chap. 7, &sect;47, Eq. 1 (p. 130) of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], this is given in Cartesian coordinates, so we will need to convert his expressions to the equivalent cylindrical-coordinate components.
</td></tr></table>

The time-derivative on the left-hand-side of our foundation expression becomes,
<div align="center">
<table border="0" cellpadding="3">

<tr>
  <td align="right">
<math>
\frac{d\mathbf{u'}}{dt} 
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\frac{d}{dt} [ \mathbf{\hat{e}}_\varpi u'_\varpi + \mathbf{\hat{e}}_\varphi u'_\varphi + \mathbf{\hat{k}} u'_z ] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\mathbf{\hat{e}}_\varpi \frac{d u'_\varpi}{dt}  + \mathbf{\hat{e}}_\varphi \frac{d u'_\varphi}{dt}  + \mathbf{\hat{k}} \frac{d u'_z}{dt}  
+ ( u'_\varpi) \frac{d}{dt}\mathbf{\hat{e}}_\varpi  + ( u'_\varphi) \frac{d}{dt}\mathbf{\hat{e}}_\varphi    
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\mathbf{\hat{e}}_\varpi \frac{d u'_\varpi}{dt}  + \mathbf{\hat{e}}_\varphi \frac{d u'_\varphi}{dt}  + \mathbf{\hat{k}} \frac{d u'_z}{dt}  
+ \mathbf{\hat{e}}_\varphi(u'_\varpi) \frac{u'_\varphi}{\varpi}  - \mathbf{\hat{e}}_\varpi(u'_\varphi) \frac{u'_\varphi}{\varpi}   \, . 
</math>
  </td>
</tr>
</table>
</div>
We also recognize that, when expressed in cylindrical coordinates, 

<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
~{\vec{\Omega}}_f \times \vec{x}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
{\hat\mathbf{k}} \Omega_f\times (\mathbf{\hat{e}}_\varpi \varpi + \mathbf{\hat{k}}z) 
=  \mathbf{\hat{e}}_\varphi \Omega_f \varpi \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
{\vec{\Omega}}_f \times ({\vec{\Omega}}_f \times \vec{x})
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\hat{\mathbf{k}} \Omega_f \times ( \mathbf{\hat{e}}_\varphi \Omega_f \varpi )
=  - \mathbf{\hat{e}}_\varpi \Omega_f^2 \varpi \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
{\vec{\Omega}}_f \times {\mathbf{u'}} 
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
{\hat\mathbf{k}} \Omega_f\times (\mathbf{\hat{e}}_\varpi u'_\varpi + \mathbf{\hat{e}}_\varphi u'_\varphi + \mathbf{\hat{k}}u'_z) 
=  \mathbf{\hat{e}}_\varphi \Omega_f u'_\varpi - \mathbf{\hat{e}}_\varpi \Omega_f u'_\varphi  \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
{\vec{v}}_\mathrm{inertial} 
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\mathbf{u'} + \mathbf{\hat{e}}_\varphi \Omega_f \varpi \, .
</math>
  </td>
</tr>

</table>
</div>

The set of scalar momentum-component equations is obtained by "dotting" each unit vector into the vector equation.
<table border="0" cellpadding="3" align="center">

<tr>
  <td align="center">
<math>\mathbf{\hat{e}}_\varpi:</math>
  </td>
  <td align="right" colspan="1">
<math>~\frac{d u'_\varpi}{dt}  - \frac{(u'_\varphi)^2}{\varpi}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \mathbf{\hat{e}}_\varpi \cdot  \frac{\nabla P}{\rho} - \mathbf{\hat{e}}_\varpi \cdot \nabla \Phi  
+ 2 \biggl[ \Omega_f u'_\varphi \biggr]
+ \Omega_f^2 \varpi </math>
  </td>
</tr>

<tr>
  <td align="right" colspan="2">
<math>~\Rightarrow ~~~ \frac{d u'_\varpi}{dt}  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \mathbf{\hat{e}}_\varpi \cdot  \frac{\nabla P}{\rho} - \mathbf{\hat{e}}_\varpi \cdot \nabla \Phi  
+ \frac{1}{\varpi} \biggl[ (u'_\varphi)^2
+ 2 \Omega_f u'_\varphi \varpi 
+ \Omega_f^2 \varpi^2 \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right" colspan="2">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \mathbf{\hat{e}}_\varpi \cdot  \frac{\nabla P}{\rho} - \mathbf{\hat{e}}_\varpi \cdot \nabla \Phi  
+ \frac{1}{\varpi} (u'_\varphi + \Omega_f \varpi)^2 \, ;
</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>\mathbf{\hat{e}}_\varphi:</math>
  </td>
  <td align="right">
<math>~\frac{d u'_\varphi}{dt}   + \frac{u'_\varpi u'_\varphi}{\varpi}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \mathbf{\hat{e}}_\varphi \cdot \frac{\nabla P}{\rho} - \mathbf{\hat{e}}_\varphi \cdot \nabla \Phi  
- 2\biggl[ \Omega_f u'_\varpi \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right" colspan="2">
(mult. thru by &piv;) &nbsp; <math>~\Rightarrow ~~~\frac{d (\varpi u'_\varphi )}{dt} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \mathbf{\hat{e}}_\varphi \cdot \frac{\varpi \nabla P}{\rho} - \mathbf{\hat{e}}_\varphi \cdot \varpi \nabla \Phi  
- 2 \Omega_f \varpi u'_\varpi \, ;
</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>\mathbf{\hat{k}}:</math>
  </td>
  <td align="right">
<math>~\frac{d u'_z}{dt}  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \mathbf{\hat{k}} \cdot \frac{\nabla P }{\rho} - \mathbf{\hat{k}} \cdot \nabla \Phi  \, .
</math>
  </td>
</tr>
</table>

Now, recalling that <math>~\mathbf{u'} = (\mathbf{v} - \mathbf{\hat{e}}_\varphi \varpi \Omega_f)</math>, let's make the substitutions &hellip;
<table border="0" cellpadding="3" align="center">
<tr>
<td align="center">
<math>~u'_\varpi \rightarrow v_\varpi \, ,</math>&nbsp; &nbsp; &nbsp;
</td>
<td align="center">
<math>~u'_\varphi \rightarrow (v_\varphi - \varpi\Omega_f) \, ,</math>&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;
</td>
<td align="center">
<math>~u'_z \rightarrow v_z \, .</math>
</td>
</tr>
</table>
This mapping gives,
<table border="0" cellpadding="3" align="center">

<tr>
  <td align="center">
<math>\mathbf{\hat{e}}_\varphi:</math>
  </td>
  <td align="right" colspan="1">
<math>~\frac{d [\varpi v_\varphi - \varpi^2 \Omega_f]}{dt} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \mathbf{\hat{e}}_\varphi \cdot \frac{\varpi \nabla P}{\rho} - \mathbf{\hat{e}}_\varphi \cdot \varpi \nabla \Phi  
- 2 \Omega_f \varpi v_\varpi \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right" colspan="2">
<math>~\Rightarrow ~~~ \frac{d (\varpi v_\varphi )}{dt} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \mathbf{\hat{e}}_\varphi \cdot \frac{\varpi \nabla P}{\rho} - \mathbf{\hat{e}}_\varphi \cdot \varpi \nabla \Phi  
\, ;
</math>
  </td>
</tr>

<tr>
  <td align="right" colspan="2">
<math>~\Rightarrow ~~~ \frac{1}{\varpi} ~\frac{d (\varpi v_\varphi )}{dt} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \mathbf{\hat{e}}_\varphi \cdot \biggl[ \frac{\nabla P}{\rho} + \nabla \Phi  \biggr]
\, ;
</math>
  </td>
</tr>

<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>~- \mathbf{\hat{k}} \cdot \biggl[ \frac{\nabla P }{\rho} + \nabla \Phi \biggr]  \, .
</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>~
- \mathbf{\hat{e}}_\varpi \cdot \biggl[ \frac{\nabla P}{\rho} + \nabla \Phi  \biggr]
+ \frac{v_\varphi^2}{\varpi}  \, ;
</math>
  </td>
</tr>
</table>