Editing Appendix/Ramblings/HybridSchemeOld (section)

===Angular Momentum Conservation===
When the three vector components of the Euler equation (of motion) are projected onto a nonrotating cylindrical coordinate grid, the azimuthal component of the Euler equation may be written as,
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
\frac{d(\rho \varpi v_\phi)}{dt} + (\rho \varpi v_\phi) \nabla\cdot \vec{v}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
-\frac{\partial P}{\partial\phi} - \rho \frac{\partial\Phi}{\partial\phi} \, .
</math>
  </td>
</tr>
</table>
</div>
For this equation, the source term is identified as,
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
~S
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
-\frac{\partial P}{\partial\phi} - \rho \frac{\partial\Phi}{\partial\phi} \, ,
</math>
  </td>
</tr>
</table>
</div>
and <math>~\psi = (\rho\varpi v_\phi)</math> is the ''inertial-frame'' angular momentum density, as measured with respect to the <math>~z</math>-coordinate axis.  This corresponds the scalar equation and representation referred to as "Case B (<math>~\eta=3</math>)" in [http://adsabs.harvard.edu/abs/2010CQGra..27q5002C CTL (2010)].

<div align="center">
<table border="1" cellpadding="5" width="100%">
<tr>
  <td align="center" colspan="2">
From Tables 6.1 &amp; 6.2 of [http://adsabs.harvard.edu/abs/2010CQGra..27q5002C Call, Tohline, &amp; Lehner (2010)]
<br>
'''Case B''' <math>~(\eta = 3)</math>
<br>
with the following replacements: 
&nbsp;&nbsp; <math>~(\rho h)_\mathrm{CTL} \rightarrow \rho</math>&nbsp;&nbsp; ;
&nbsp;&nbsp; <math>~(R)_\mathrm{CTL} \rightarrow \varpi</math>  ;
&nbsp;&nbsp; <math>~(R u^\phi)_\mathrm{CTL} \rightarrow \varpi\dot\phi = v_\phi</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>~\psi_{(3)}</math>
  </td>
  <td align="center">
<math>~S_{(3)}</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>~\rho \varpi v_\phi</math>
  </td>
  <td align="center">
<math>~ - \frac{\partial P}{\partial\phi} - \rho \frac{\partial \Phi}{\partial\phi}</math>
  </td>
</tr>
</table>
</div>


As foreshadowed above &#8212; see the [[#Recognizing_Statements_of_Conservation|subsection titled, ''Recognizing Statements of Conservation'']] &#8212; the angular momentum of a Lagrangian fluid element will be conserved if the "source" term, <math>~S = 0</math>.  This situation will arise if, at the fluid element's location, the azimuthal pressure variation, <math>~\partial P/\partial\phi</math>, and the azimuthal variation in the gravitational potential, <math>~\partial \Phi/\partial\phi</math>, are both zero, or if the two balance one another (''i.e.,''<math>~\partial P/\partial\phi=-\rho\partial\Phi/\partial\phi</math>).

Based on the above discussion, we can equally well view the flow from a frame of reference that is rotating with a constant angular velocity, <math>~\Omega_0</math>, and write, 
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
\frac{d(\rho \varpi v_\phi)}{dt} + (\rho \varpi v_\phi) \nabla\cdot \vec{u}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
-\frac{\partial P}{\partial\phi} - \rho \frac{\partial\Phi}{\partial\phi} \, ,
</math>
  </td>
</tr>
</table>
</div>
where, as before,
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
\vec{u} 
</math>
  </td>
  <td align="center">
<math>~\equiv~</math>
  </td>
  <td align="left">
<math>
\vec{v} - \boldsymbol{\hat{e}}_\phi \varpi\Omega_0 \, .
</math>
  </td>
</tr>
</table>
</div>
Also, following the earlier discussion, if one wants to follow the time-variation of the fluid's inertial-frame angular momentum at a fixed location in inertial space, then the appropriate Eulerian representation of this azimuthal component of the equation of motion is,
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
\frac{\partial (\rho \varpi v_\phi)}{\partial t} + \nabla\cdot [(\rho \varpi v_\phi) \vec{v}]
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
S \, .
</math>
  </td>
</tr>
</table>
</div>
If, however, one wants to follow the time-variation of the fluid's inertial-frame angular momentum at a fixed location on a rotating coordinate grid, then the appropriate Eulerian representation of this azimuthal component of the equation of motion is obtained by replacing the "transport" velocity, <math>~\vec{v}</math> with <math>~\vec{u}</math>; specifically, 
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
\frac{\partial (\rho \varpi v_\phi)}{\partial t} + \nabla\cdot [(\rho \varpi v_\phi) \vec{u}]
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
S \, .
</math>
  </td>
</tr>
</table>
</div>