Editing Appendix/Ramblings/HybridSchemeOld (section)

===An Element of the Hybrid Scheme===

This last equation displays one subtle, but valuable, element of the hybrid scheme developed by [http://adsabs.harvard.edu/abs/2010CQGra..27q5002C Call, Tohline, &amp; Lehner (2010)].  The velocity component, <math>~v_\phi</math>, that appears in the formulation of the relevant conserved quantity &#8212; the inertial-frame angular momentum density &#8212; is drawn from the velocity vector, <math>~\vec{v}</math>, which is different from the transport velocity vector, <math>~\vec{u}</math>, that defines the Eulerian frame from which the dynamical evolution of the system is being viewed.  This equation is usually written, instead, in a form such that the angular momentum density is expressed in terms of the azimuthal component of the transport velocity; see, for example, equation (7) in [http://adsabs.harvard.edu/abs/1978ApJ...224..497N Norman &amp; Wilson (1978)] and equation (12) in  [http://adsabs.harvard.edu/abs/1997ApJ...490..311N New &amp; Tohline (1997)].  In this more familiar formulation, the momentum density and the transport velocity both directly refer to the same frame of reference.  But, as a consequence, the source term is more complicated.  

The more familiar formulation &#8212; including its modified source term &#8212; can be derived from our "hybrid" formulation by recognizing that,
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<math>
~v_\phi
</math>
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<math>~=~</math>
  </td>
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<math>
~u_\phi + \varpi\Omega_0 \, .
</math>
  </td>
</tr>
</table>
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So we can write,

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<tr>
  <td align="right">
<math>
\frac{\partial [\rho \varpi (u_\phi + \varpi\Omega_0 ) ]}{\partial t} + \nabla\cdot \{[\rho \varpi ( u_\phi + \varpi\Omega_0)] \vec{u} \}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
~S_{\phi i} \, ,
</math>
  </td>
</tr>
</table>
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where, as shorthand, we have used,
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<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
~S_{\phi i}
</math>
  </td>
  <td align="center">
<math>~\equiv~</math>
  </td>
  <td align="left">
<math>
- \frac{\partial P}{\partial\phi} - \rho \frac{\partial \Phi}{\partial\phi} \, .
</math>
  </td>
</tr>
</table>
</div>
This implies,
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<table border="0" cellpadding="3">

<tr>
  <td align="right">
<math>
\frac{\partial (\rho \varpi u_\phi )}{\partial t} + \nabla\cdot [ (\rho \varpi u_\phi) \vec{u} ]
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
S_{\phi i} - \frac{\partial [\rho \varpi (\varpi\Omega_0 ) ]}{\partial t} - \nabla\cdot \{[\rho \varpi (\varpi\Omega_0)] \vec{u} \} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
S_{\phi i} - \varpi^2\Omega_0 \biggl\{ \frac{\partial \rho}{\partial t} + \nabla\cdot (\rho \vec{u} ) \biggr\} - \rho \vec{u}\cdot \nabla(\varpi^2 \Omega_0)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
S_{\phi i} - 2\rho \varpi u_\varpi \Omega_0 \, .
</math>
  </td>
</tr>
</table>
</div>
As we see, all terms involving the velocity now explicitly refer to <math>~\vec{u}</math> and, hence, to the velocity as measured in the rotating reference frame.  But the source now includes a Coriolis term.  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)].

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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>
as before: 
&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>
<br>additional replacements:
&nbsp;&nbsp; <math>~(\bar\omega u^{t'})_\mathrm{CTL} \rightarrow \Omega_0</math>  ;
&nbsp;&nbsp; <math>~u^R \rightarrow v_\varpi = u_\varpi</math>
  </td>
</tr>

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

<tr>
  <td align="center">
<math>~\rho \varpi (v_\phi - \varpi\Omega_0) = \rho \varpi u_\phi </math>
  </td>
  <td align="center">
<math>~ - \frac{\partial P}{\partial\phi} - \rho \frac{\partial \Phi}{\partial\phi} - 2\rho\varpi u_\varpi \Omega_0</math>
  </td>
</tr>
</table>
</div>