Editing Appendix/Ramblings/HybridSchemeOld (section)

====Step 2====
Next, throughout this set of scalar equations, we replace each component of <math>~\rho\mathbf{u'}</math> with the corresponding component of <math>~(\rho\mathbf{u} - \rho {\vec{\Omega}}_f \times \vec{x})</math><math>= (\rho\mathbf{u} -\mathbf{\hat{e}}_\varphi \rho R\Omega_0)</math>, that is, we perform the following mappings:
<div align="center">
<table border="0" cellpadding="3">

<tr>
  <td align="right">
<math>
~\rho u'_R
</math>
  </td>
  <td align="center">
<math>~\rightarrow~</math>
  </td>
  <td align="left">
<math>
~\rho u_R \, ,
</math>
  </td>

<tr>
  <td align="right">
<math>
~\rho u'_\varphi
</math>
  </td>
  <td align="center">
<math>~\rightarrow~</math>
  </td>
  <td align="left">
<math>
~\rho (u_\varphi - R\Omega_0 ) \, , 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~\rho u'_z
</math>
  </td>
  <td align="center">
<math>~\rightarrow~</math>
  </td>
  <td align="left">
<math>
~\rho u_z \, .
</math>
  </td>
</tr>

</table>
</div>
As a result, the first and third of the three "hybrid" momentum-component equations become, respectively,
<div align="center">
<table border="0" cellpadding="3">

<tr>
  <td align="center">
<math>\mathbf{\hat{e}}_R:</math>
  </td>
  <td align="right">
<math>
\frac{\partial (\rho u_R)}{\partial t} + \nabla\cdot [(\rho u_R)\mathbf{u'}]
</math> 
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
- \mathbf{\hat{e}}_R \cdot\nabla P - \rho \mathbf{\hat{e}}_R \cdot\nabla \Phi 
+ \frac{\rho u^2_\varphi}{R}  \, ;
</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>\mathbf{\hat{k}}:</math>
  </td>
  <td align="right">
<math>
\frac{\partial (\rho u_z)  }{\partial t} + \nabla\cdot [(\rho u_z) \mathbf{u'} ]
</math> 
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
- \mathbf{\hat{k}}\cdot\nabla P - \rho \mathbf{\hat{k}}\cdot\nabla \Phi  \, .
</math>
  </td>
</tr>
</table>
</div>
The second of the three "hybrid" momentum component equations &#8212; the one governing conservation of angular momentum &#8212; becomes,
<div align="center">
<table border="0" cellpadding="3">

<tr>
  <td align="right">
<math>
\frac{\partial [\rho R (u_\varphi - R\Omega_0 ) ]}{\partial t} + \nabla\cdot \{ [\rho R (u_\varphi - R\Omega_0 ) ] \mathbf{u'} ]
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
- \mathbf{\hat{e}}_\varphi \cdot R\nabla P - \rho \mathbf{\hat{e}}_\varphi \cdot R \nabla \Phi 
- 2\rho R \Omega_0 u_R    \, 
</math>
  </td>
</tr>
</table>
</div>

<div align="center">
<table border="0" cellpadding="3">

<tr>
  <td align="right">
<math>
\Rightarrow ~~~~~ \frac{\partial (\rho R u_\varphi)  }{\partial t} + \nabla\cdot [(\rho R u_\varphi) \mathbf{u'} ]
- R^2 \Omega_0 \biggl[ \frac{\partial \rho }{\partial t} + \nabla\cdot ( \rho  \mathbf{u'} ) \biggr]
- ( \rho  \mathbf{u'} )\cdot\nabla (R^2 \Omega_0)
</math> 
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
- \mathbf{\hat{e}}_\varphi \cdot R\nabla P - \rho \mathbf{\hat{e}}_\varphi \cdot R \nabla \Phi 
- 2\rho R \Omega_0 u_R    \, .
</math>
  </td>
</tr>

</table>
</div>
Referencing the continuity equation, as before, the middle bracketed term on the left-hand side can be set to zero; and the last term on the left-hand side,
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
( \rho  \mathbf{u'} )\cdot\nabla (R^2\Omega_0)
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
2\rho R \Omega_0 u_R  \, ,
</math>
  </td>
</tr>
</table>
</div> 
matches and, hence, exactly cancels the Coriolis term on the right-hand side to give,

<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="center">
<math>\mathbf{\hat{e}}_\varphi:</math>
  </td>
  <td align="right">
<math>
\frac{\partial (\rho R u_\varphi)  }{\partial t} + \nabla\cdot [(\rho R u_\varphi) \mathbf{u'} ]
</math> 
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
- \mathbf{\hat{e}}_\varphi \cdot R\nabla P - \rho \mathbf{\hat{e}}_\varphi \cdot R \nabla \Phi   \, .
</math>
  </td>
</tr>
</table>
</div>