Editing Appendix/Ramblings/HybridSchemeOld (section)

====Step 1====
If the focus is on tracking angular momentum, then we need to break the vector momentum equation into its <math>(\mathbf\hat{e}_R, \mathbf\hat{e}_\varphi, \mathbf\hat{k})</math> components.  As before, this is done by "dotting" each unit vector into the vector equation. This is less straightforward than in the Cartesian case because the orientation of both the <math>\mathbf\hat{e}_R</math> and <math>\mathbf\hat{e}_\varphi</math> unit vectors vary in space.  As a result, the first term in the vector equation &#8212; the ''material'' time derivative &#8212; generates a couple of extra terms, viz., 
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<math>
\frac{d(\rho\mathbf{u'})}{dt} 
</math>
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  <td align="center">
<math>~=~</math>
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<math>
\frac{d}{dt} [ \mathbf{\hat{e}}_R (\rho u'_R) + \mathbf{\hat{e}}_\varphi (\rho u'_\varphi) + \mathbf{\hat{k}} (\rho u'_z) ] 
</math>
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&nbsp;
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<math>~=~</math>
  </td>
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<math>
\mathbf{\hat{e}}_R \frac{d(\rho u'_R)}{dt}  + \mathbf{\hat{e}}_\varphi \frac{d(\rho u'_\varphi)}{dt}  + \mathbf{\hat{k}} \frac{d(\rho u'_z)}{dt}  
+ (\rho u'_R) \frac{d}{dt}\mathbf{\hat{e}}_R  + (\rho u'_\varphi) \frac{d}{dt}\mathbf{\hat{e}}_\varphi   \, , 
</math>
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&nbsp;
  </td>
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<math>~=~</math>
  </td>
  <td align="left">
<math>
\mathbf{\hat{e}}_R \frac{d(\rho u'_R)}{dt}  + \mathbf{\hat{e}}_\varphi \frac{d(\rho u'_\varphi)}{dt}  + \mathbf{\hat{k}} \frac{d(\rho u'_z)}{dt}  
+ \mathbf{\hat{e}}_\varphi(\rho u'_R) \frac{u'_\varphi}{R}  - \mathbf{\hat{e}}_R(\rho u'_\varphi) \frac{u'_\varphi}{R}   \, . 
</math>
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We also recognize that, when expressed in cylindrical coordinates, 

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<math>
~{\vec{\Omega}}_f \times \vec{x}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
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<math>
{\hat\mathbf{k}} \Omega_0\times (\mathbf{\hat{e}}_R R + \mathbf{\hat{k}}z) 
=  \mathbf{\hat{e}}_\varphi \Omega_0 R \, ,
</math>
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<math>
{\vec{\Omega}}_f \times ({\vec{\Omega}}_f \times \vec{x})
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
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<math>
\hat{\mathbf{k}} \Omega_0 \times ( \mathbf{\hat{e}}_\varphi \Omega_0 R )
=  - \mathbf{\hat{e}}_R \Omega_0^2 R \, ,
</math>
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<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{e}}_R u'_R + \mathbf{\hat{e}}_\varphi u'_\varphi + \mathbf{\hat{k}}u'_z) 
=  \mathbf{\hat{e}}_\varphi \Omega_0 u'_R - \mathbf{\hat{e}}_R \Omega_0 u'_\varphi  \, .
</math>
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</table>
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Hence, the process of "dotting" each unit vector into the equation leads to the following set of scalar momentum-component equations:
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  <td align="center">
<math>\mathbf{\hat{e}}_R:</math>
  </td>
  <td align="right">
<math>
\frac{d(\rho u'_R)}{dt} + (\rho u'_R)\nabla\cdot \mathbf{u'} - (\rho u'_\varphi) \frac{u'_\varphi}{R}  
</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 
- 2\rho \mathbf{\hat{e}}_R \cdot [ \mathbf{\hat{e}}_\varphi \Omega_0 u'_R - \mathbf{\hat{e}}_R \Omega_0 u'_\varphi  ] 
+ \rho \mathbf{\hat{e}}_R\cdot (\mathbf{\hat{e}}_R \Omega_0^2 R) \,
</math>
  </td>
</tr>

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  <td align="center">
&nbsp;
  </td>
  <td align="right">
<math>
\Rightarrow ~~~~~ \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}{R} \biggl[ (u'_\varphi)^2 + 2R\Omega_0 u'_\varphi + (\Omega_0 R)^2 \biggr] \, 
</math>
  </td>
</tr>

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  <td align="center">
&nbsp;
  </td>
  <td align="right">
&nbsp;
  </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}{R} (u'_\varphi + R\Omega_0)^2 \, ;
</math>
  </td>
</tr>

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  <td align="center">
<math>\mathbf{\hat{e}}_\varphi:</math>
  </td>
  <td align="right">
<math>
\frac{d(\rho u'_\varphi)}{dt} + (\rho u'_\varphi)\nabla\cdot \mathbf{u'} + (\rho u'_R) \frac{u'_\varphi}{R}  
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
- \mathbf{\hat{e}}_\varphi \cdot\nabla P - \rho \mathbf{\hat{e}}_\varphi \cdot\nabla \Phi 
- 2\rho \mathbf{\hat{e}}_\varphi \cdot [ \mathbf{\hat{e}}_\varphi \Omega_0 u'_R - \mathbf{\hat{e}}_R \Omega_0 u'_\varphi  ] 
+ \rho \mathbf{\hat{e}}_\varphi\cdot (\mathbf{\hat{e}}_R \Omega_0^2 R) \,
</math>
  </td>
</tr>

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(mult. thru by R)
  </td>
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<math>
\Rightarrow ~~~~~ \frac{d(\rho R u'_\varphi)}{dt} + (\rho R u'_\varphi)\nabla\cdot \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>

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&nbsp;
  </td>
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<math>
\Rightarrow ~~~~~ \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 
- 2\rho R \Omega_0 u'_R    \, ;
</math>
  </td>
</tr>

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  <td align="center">
<math>\mathbf{\hat{k}}:</math>
  </td>
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<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{e}}_\varphi \Omega_0 u'_R - \mathbf{\hat{e}}_R \Omega_0 u'_\varphi  ] 
+ \rho \mathbf{\hat{k}}\cdot (\mathbf{\hat{e}}_R \Omega_0^2 R) \,
</math>
  </td>
</tr>

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  <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>
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<math>
- \mathbf{\hat{k}}\cdot\nabla P - \rho \mathbf{\hat{k}}\cdot\nabla \Phi \, .
</math>
  </td>
</tr>

</table>
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'''<font color="lightblue">ASIDE:</font>''' <span id="NW78">If we pause our discussion here and map this set of component equations</span> onto a (rotating) cylindrical coordinate mesh &#8212; that is, if on the right-hand-sides we implement the straightforward operator projections,
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<math>\mathbf{\hat{e}}_R \cdot\nabla \rightarrow \frac{\partial}{\partial R} \, ,</math>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<math>\mathbf{\hat{e}}_\varphi \cdot\nabla \rightarrow \frac{1}{R} \frac{\partial}{\partial \varphi} \, ,</math>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<math>\mathbf{\hat{k}}\cdot\nabla \rightarrow \frac{\partial}{\partial z} \, ,</math>
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we obtain a formulation that is familiar to the astrophysics community.  For example, as the following table of equations illustrates, it is the component set that has been spelled out in equations (5) - (7) of [http://adsabs.harvard.edu/abs/1978ApJ...224..497N Norman &amp; Wilson (1978)] and equations (11), (12), &amp; (3) of  [http://adsabs.harvard.edu/abs/1997ApJ...490..311N New &amp; Tohline (1997)].


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'''Cylindrical Components of the Rotating-Frame Momentum'''<br>
<font size="-1">advected across a</font><br>
'''Rotating, Cylindrical Coordinate Mesh'''
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  <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>
- \frac{\partial}{\partial R} P - \rho \frac{\partial}{\partial R}\Phi 
+ \frac{\rho}{R} \biggl[ (u'_\varphi)^2 + 2R\Omega_0 u'_\varphi + (\Omega_0 R)^2 \biggr] \, 
</math>
  </td>
</tr>

<tr>
  <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>
- \frac{\partial}{\partial\varphi} P - \rho \frac{\partial}{\partial\varphi}\Phi 
- 2\rho R \Omega_0 u'_R   
</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>
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  <td align="left">
[[File:NormanWilson78C.png|650px|center|Norman &amp; Wilson (1978)]]
Note:  For complete correspondence, set <math>(\gamma-1)E \rightarrow P</math> in all three component equations.
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[[File:NewTohline97B.png|650px|center|New &amp; Tohline (1997)]]
Note:  When comparing this set of equations to the set presented by Norman &amp; Wilson (1978), the definitions of the variables, <math>~\mathcal{S}</math> and <math>~\mathcal{T}</math>, must be swapped.
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[<font color="red">Comment by J. E. Tohline (April 7, 2014)</font>]  This is the set of equations that my research group has been using to simulate a wide variety of astrophysical fluid flows over the past twenty years. This is no longer our method of choice, however.   A numerical algorithm based on the hybrid scheme, as summarized below, is far preferable to an algorithm that is based on this more familiar, traditional set of equations for several reasons:
* In the hybrid scheme, the Coriolis term disappears from the source term, so it is much easier to design and implement a computational algorithm that conserves angular momentum conservation.
* Although the hybrid scheme advects ''inertial-frame'' angular momentum, it retains all of the advantages associated with using a ''rotating'' frame of reference; for example, numerical diffusion is less severe and, in general, the Courant-limited time-step is larger than would be the case if you were forced to transport fluid in the inertial frame of reference.
* The hybrid scheme facilitates transport (and conservation) of angular momentum across a (rotating) ''Cartesian'' mesh.  This facilitates the use of adaptive-mesh refinement (AMR) techniques and simplifies load-balancing on distributed memory, high-performance computers.

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