Editing Appendix/Ramblings/HybridSchemeOld (section)

====Step 1====
If the focus is on tracking linear momentum components, then we need to break the vector momentum equation into its <math>(\mathbf\hat{i}, \mathbf\hat{j}, \mathbf\hat{k})</math> components.  This is done by, in turn, "dotting" each unit vector into the vector equation.  It is straightforward once we appreciate that the orientation of these Cartesian unit vectors does not vary in space and that, within the context of the rotating frame on which they are defined, these unit vectors do not vary in time.  Hence, the first term in the vector equation &#8212; the ''material'' time derivative &#8212; can be written as, 
<div align="center">
<math>
\frac{d(\rho\mathbf{u'})}{dt} = 
\frac{d}{dt} [ \mathbf{\hat{i}} (\rho u'_x) + \mathbf{\hat{j}} (\rho u'_y) + \mathbf{\hat{k}} (\rho u'_z) ] =
\mathbf{\hat{i}} \frac{d(\rho u'_x)}{dt}  + \mathbf{\hat{j}} \frac{d(\rho u'_y)}{dt}  + \mathbf{\hat{k}} \frac{d(\rho u'_z)}{dt}  \, , 
</math>
</div>
and the process of "dotting" each unit vector into the equation leads to the following set of scalar momentum-component equations:
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="center">
<math>\mathbf{\hat{i}}:</math>
  </td>
  <td align="right">
<math>
\frac{d(\rho u'_x)}{dt} + (\rho u'_x)\nabla\cdot \mathbf{u'} 
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
- \mathbf{\hat{i}}\cdot\nabla P - \rho \mathbf{\hat{i}}\cdot\nabla \Phi + 2\rho \mathbf{\hat{i}}\cdot[\mathbf{\hat{i}}\Omega_0 u'_y - \mathbf{\hat{j}}\Omega_0 u'_x] + \rho \mathbf{\hat{i}}\cdot[ \mathbf{\hat{i}}\Omega_0^2 x + \mathbf{\hat{j}}\Omega_0^2 y] \,
</math>
  </td>
</tr>

<tr>
  <td align="center">
&nbsp;
  </td>
  <td align="right">
<math>
\Rightarrow ~~~~~ \frac{\partial (\rho u'_x)}{\partial t} + \nabla\cdot [(\rho u'_x)\mathbf{u'}]
</math> 
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
- \mathbf{\hat{i}}\cdot\nabla P - \rho \mathbf{\hat{i}}\cdot\nabla \Phi + 2\rho \Omega_0 u'_y + \rho \Omega_0^2 x \, ;
</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>\mathbf{\hat{j}}:</math>
  </td>
  <td align="right">
<math>
\frac{d(\rho u'_y)}{dt} + (\rho u'_y)\nabla\cdot \mathbf{u'} 
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
- \mathbf{\hat{j}}\cdot\nabla P - \rho \mathbf{\hat{j}}\cdot\nabla \Phi + 2\rho \mathbf{\hat{j}}\cdot[\mathbf{\hat{i}}\Omega_0 u'_y - \mathbf{\hat{j}}\Omega_0 u'_x] + \rho \mathbf{\hat{j}}\cdot[ \mathbf{\hat{i}}\Omega_0^2 x + \mathbf{\hat{j}}\Omega_0^2 y] \,
</math>
  </td>
</tr>

<tr>
  <td align="center">
&nbsp;
  </td>
  <td align="right">
<math>
\Rightarrow ~~~~~ \frac{\partial (\rho u'_y)}{\partial t} + \nabla\cdot [(\rho u'_y) \mathbf{u'} ]
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
- \mathbf{\hat{j}}\cdot\nabla P - \rho \mathbf{\hat{j}}\cdot\nabla \Phi - 2\rho \Omega_0 u'_x + \rho \Omega_0^2 y \, ;
</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>\mathbf{\hat{k}}:</math>
  </td>
  <td align="right">
<math>
\frac{d(\rho u'_z)}{dt} + (\rho u'_z)\nabla\cdot \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 + 2\rho \mathbf{\hat{k}}\cdot[\mathbf{\hat{i}}\Omega_0 u'_y - \mathbf{\hat{j}}\Omega_0 u'_x] + \rho \mathbf{\hat{k}}\cdot[ \mathbf{\hat{i}}\Omega_0^2 x + \mathbf{\hat{j}}\Omega_0^2 y] \,
</math>
  </td>
</tr>

<tr>
  <td align="center">
&nbsp;
  </td>
  <td align="right">
<math>
\Rightarrow ~~~~~ \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>
In deriving these expressions, we also (a) have recognized from the start that, when expressed in Cartesian 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_0\times (\mathbf{\hat{i}}x + \mathbf{\hat{j}}y + \mathbf{\hat{k}}z) 
=  - \mathbf{\hat{i}}\Omega_0 y + \mathbf{\hat{j}}\Omega_0 x \, ,
</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_0 \times ( - \mathbf{\hat{i}}\Omega_0 y + \mathbf{\hat{j}}\Omega_0 x )
=  - \mathbf{\hat{i}}\Omega_0^2 x - \mathbf{\hat{j}}\Omega_0^2 y \, ,
</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_0\times (\mathbf{\hat{i}}u'_x + \mathbf{\hat{j}}u'_y + \mathbf{\hat{k}}u'_z) 
=  - \mathbf{\hat{i}}\Omega_0 u'_y + \mathbf{\hat{j}}\Omega_0 u'_x\, ,
</math>
  </td>
</tr>

</table>
</div>
and (b) have used the familiar operator mapping, 
<div align="center">
<math>\frac{d}{dt} \rightarrow \frac{\partial}{\partial t} + \mathbf{u'}\cdot \nabla  \, ,</math>
</div>
to shift from the total (Lagrangian) time derivative to the partial (Eulerian) time derivative, which is usually the more desirable representation for computational simulations.