Editing Appendix/Ramblings/HybridSchemeOld (section)

====Step 3====
'''<font color="red">Cartesian Grid:</font>''' Assuming the numerical simulation will be conducted on a Cartesian coordinate mesh, the divergence (advection) term on the left-hand-side should be evaluated by breaking the transport velocity into its three Cartesian components,
<div align="center">
<math>~\mathbf{u'} = (u'_x, u'_y, u'_z) \, ,</math>
</div>
and, on the right-hand-side, the projection of the spatial operators should be written in the familiar form,
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<math>\mathbf{\hat{i}}\cdot\nabla \rightarrow \frac{\partial}{\partial x} \, ,</math>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<math>\mathbf{\hat{j}}\cdot\nabla \rightarrow \frac{\partial}{\partial y} \, ,</math>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<math>\mathbf{\hat{k}}\cdot\nabla \rightarrow \frac{\partial}{\partial z} \, .</math>
</div>
In summary, then, the relevant set of momentum conservation equations is,

<div align="center">
<table border="1" cellpadding="5" width="80%">
<tr>
  <td align="center" bgcolor="lightgreen">
'''Cartesian Components of the Inertial-Frame Momentum'''<br>
<font size="-1">advected across a</font><br>
'''Rotating, Cartesian Coordinate Mesh'''
  </td>
</tr>
<tr><td align="center">
<table border="0" cellpadding="3">

<tr>
  <td align="right">
<math>
\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>
- \frac{\partial}{\partial x}  P - \rho \frac{\partial}{\partial x} \Phi + \rho \Omega_0 u_y 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\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>
- \frac{\partial}{\partial y}  P - \rho \frac{\partial}{\partial y} \Phi - \rho \Omega_0 u_x
</math>
  </td>
</tr>

<tr>
  <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>
- \frac{\partial}{\partial z} P - \rho \frac{\partial}{\partial z}\Phi  
</math>
  </td>
</tr>

</table>
</td></tr>
<tr>
  <td align="left">
This is the set of equations that has served as the foundation of the ''Cartesian'' simulations reported in Byerly, Adelstein-Lelbach, Tohline, &amp; Marcello (2014).
  </td>
</tr>
</table>
</div>


'''<font color="red">Cylindrical Grid:</font>''' If, instead, the numerical simulation is to be conducted on a cylindrical coordinate mesh, the spatial operators on both sides of the component momentum equations should be broken down into their cylindrical-coordinate components.  In concert with this, the divergence (advection) term on the left-hand-side should be evaluated by breaking the transport velocity into its three cylindrical components,
<div align="center">
<math>~\mathbf{u'} = (u'_R, u'_\varphi, u'_z) \, .</math>
</div>
Furthermore, recognizing that, when written in cylindrical coordinates, the gradient operator is,
<div align="center">
<math>
\nabla = \mathbf{\hat{e}}_R \frac{\partial}{\partial R} + 
\mathbf{\hat{e}}_\varphi \frac{1}{R} \frac{\partial}{\partial \varphi} +\mathbf{\hat{e}}_z \frac{\partial}{\partial z} \, ,
</math>
</div>
and that the unit vectors in cylindrical coordinates can be related to their Cartesian counterparts via the mappings,
<div align="center">
<math>\mathbf{\hat{e}}_R = \mathbf{\hat{i}} \cos\varphi + \mathbf{\hat{j}} \sin\varphi \, ,</math>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<math>\mathbf{\hat{e}}_\varphi = \mathbf{\hat{j}} \cos\varphi - \mathbf{\hat{i}} \sin\varphi \, ,</math>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<math>\mathbf{\hat{e}}_z = \mathbf{\hat{k}}  \, ,</math><br>
</div>
the relevant projections of the gradient operator on the right-hand-sides of the governing equations should take the form,
<div align="center">
<table border="0" cellpadding="3">

<tr>
  <td align="right">
<math>
\mathbf{\hat{i}}\cdot\nabla
</math>
  </td>
  <td align="center">
<math>~\rightarrow~</math>
  </td>
  <td align="left">
<math>
\biggl[ \cos\varphi \frac{\partial}{\partial R} - 
\frac{\sin\varphi}{R} \frac{\partial}{\partial \varphi}\biggr] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\mathbf{\hat{j}}\cdot\nabla
</math>
  </td>
  <td align="center">
<math>~\rightarrow~</math>
  </td>
  <td align="left">
<math>
\biggl[ \sin\varphi \frac{\partial}{\partial R} + 
\frac{\cos\varphi}{R} \frac{\partial}{\partial \varphi}\biggr] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\mathbf{\hat{k}}\cdot\nabla
</math>
  </td>
  <td align="center">
<math>~\rightarrow~</math>
  </td>
  <td align="left">
<math>
\frac{\partial}{\partial z} \, .
</math>
  </td>
</tr>
</table>
</div>

In summary, then, the relevant set of momentum conservation equations is,

<div align="center">
<table border="1" cellpadding="5">
<tr>
  <td align="center" bgcolor="lightgreen">
'''Cartesian Components of the Inertial-Frame Momentum'''<br>
<font size="-1">advected across a</font><br>
'''Rotating, Cylindrical Coordinate Mesh'''
  </td>
</tr>
<tr><td align="center">
<table border="0" cellpadding="3">

<tr>
  <td align="right">
<math>
\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>
- \biggl[ \cos\varphi \frac{\partial}{\partial R} - 
\frac{\sin\varphi}{R} \frac{\partial}{\partial \varphi}\biggr] P - 
\rho \biggl[ \cos\varphi \frac{\partial}{\partial R} - 
\frac{\sin\varphi}{R} \frac{\partial}{\partial \varphi}\biggr] \Phi + \rho \Omega_0 u_y 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\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>
- \biggl[ \sin\varphi \frac{\partial}{\partial R} + 
\frac{\cos\varphi}{R} \frac{\partial}{\partial \varphi}\biggr]  P - \rho \biggl[ \sin\varphi \frac{\partial}{\partial R} + 
\frac{\cos\varphi}{R} \frac{\partial}{\partial \varphi}\biggr] \Phi - \rho \Omega_0 u_x  
</math>
  </td>
</tr>

<tr>
  <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>
- \frac{\partial}{\partial z} P - \rho \frac{\partial}{\partial z}\Phi  
</math>
  </td>
</tr>

</table>
</td></tr>
</table>
</div>