Editing Appendix/Ramblings/HybridSchemeOld (section)

==Setting the Stage==

===Recognizing Statements of Conservation===
When dealing with the compressible fluid equations, we will often encounter hyperbolic PDEs of the following form:
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
\frac{d\psi}{dt} + \psi \nabla\cdot \vec{v}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
S \, ,
</math>
  </td>
</tr>
</table>
</div>
where we are using <math>~\vec{v}</math> to represent the velocity field of the fluid as viewed from an ''inertial frame of reference'', and the total (as opposed to partial) time derivative indicates the time-rate of change of <math>~\psi</math> is being measured in a so-called ''Lagrangian'' fashion, that is, at the location of some fluid element and ''moving along with'' that fluid element.  

When we encounter a situation in which the "source" term, <math>~S</math>, on the right-hand side is zero, we will be able to identify the scalar variable, <math>~\psi</math>, as the volume density of some conserved quantity.  For example, the continuity equation &#8212; which is a mathematical statement of mass conservation &#8212; has the form,
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
{{ Math/EQ_Continuity01}}
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; or, equivalently, &nbsp; &nbsp; &nbsp;
  </td>
  <td align="right">
<math>
\frac{d\ln\rho}{dt} 
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
~- \nabla\cdot \vec{v} \, ,
</math>
  </td>
</tr>
</table>
</div>
where, <math>~\rho</math> is the mass per unit volume or, simply, the mass density of the fluid element.  Clearly, when the mass of a Lagrangian fluid element is conserved, the fluid element's mass density changes only in accordance with the divergence of the local velocity field. 

Similarly, if we are following the evolution of a fluid that expands and contracts adiabatically, we should expect to encounter an equation of the form, 
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
\frac{ds}{dt} + s\nabla\cdot \vec{v}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
0 \, ,
</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; or, equivalently, &nbsp; &nbsp; &nbsp;
  </td>
  <td align="right">
<math>
\frac{d\ln s}{dt} 
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
~- \nabla\cdot \vec{v} \, ,
</math>
  </td>
</tr>
</table>
</div>
where, <math>~s</math> is the entropy density of a Lagrangian fluid element.  Or, if an axisymmetric distribution of fluid is moving in an axisymmetric potential, we should expect the azimuthal component of the fluid's angular momentum to be conserved, in which case we should expect to encounter a dynamical equation of the form,
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
\frac{d(\rho \varpi v_\phi)}{dt} + (\rho \varpi v_\phi) \nabla\cdot \vec{v}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
0 \, ,
</math>
  </td>
</tr>
</table>
</div>
where, <math>~\varpi</math> is the Lagrangian fluid element's (cylindrical radial) distance measured from the symmetry axis of the underlying potential and <math>~v_\phi = \varpi\dot\phi</math> is the azimuthal component of the inertial velocity field, <math>~\vec{v}</math>, at the location of the fluid element.

===Alternative Reference Frames===

Now, we might want to examine the time-dependent behavior of a fluid system while viewing the flow from a reference frame that is more or less moving along with the fluid.  This new frame of reference need not be an inertial frame; for example, when studying a rotating fluid, we may want to view the system's evolution from a rotating frame of reference.  This will be accomplished mathematically by adjusting the dynamical equations so that the velocity that appears in the divergence term accounts for the new "frame" velocity field; specifically, we want to replace <math>~\vec{v}</math> with,
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
\vec{u} 
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\vec{v} - \vec{v}_\mathrm{frame} \, .
</math>
  </td>
</tr>
</table>
</div>
(Here, we will consider only time-independent functional expressions for the frame velocity, <math>~\vec{v}_\mathrm{frame}</math>.)  Of course, switching to the rotating frame must be done in such a way that the resulting, new PDE describes exactly the same physical behavior of the system as was described by the original equation; that is, the new equation must be derivable from the original one.  

If <math>~\vec{v}_\mathrm{frame}</math> is a divergence-free velocity field, then the transformation is trivial.  For example, if we choose a frame of reference that is rotating uniformly with angular velocity, <math>~\Omega_0</math>, then,
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
\vec{v}_\mathrm{frame} 
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\hat{e}}_\phi (\varpi \Omega_0) \, ,
</math>
  </td>
</tr>
</table>
</div>
and, utilizing cylindrical coordinates,
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
\nabla\cdot\vec{v}_\mathrm{frame} 
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\frac{\partial(0)}{\partial \varpi} + \frac{1}{\varpi}\frac{\partial(\varpi \Omega_0)}{\partial \phi}
+ \frac{\partial(0)}{\partial z} = 0 \, .
</math>
  </td>
</tr>
</table>
</div>
Hence, 
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
\frac{d\psi}{dt} + \psi \nabla\cdot \vec{u}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\frac{d\psi}{dt} + \psi \nabla\cdot [\vec{v} - \vec{v}_\mathrm{frame}]
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\frac{d\psi}{dt} + \psi \nabla\cdot \vec{v} \, ,
</math>
  </td>
</tr>
</table>
</div>
so the new generic hyperbolic PDE becomes,
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
\frac{d\psi}{dt} + \psi \nabla\cdot \vec{u}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
S \, ,
</math>
  </td>
</tr>
</table>
</div>
and we can be confident that this new PDE represents the physics of the system just as well as the original PDE.

===Eulerian Representation===
We can shift any of the PDEs from a Lagrangian to an Eulerian representation &#8212; and thereby use them to follow the time-rate of change of physical variables at a point in space that is fixed with respect to the chosen frame of reference &#8212; by using the following transformation to replace each total time derivative with a partial time derivative:
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
\frac{d\psi}{dt}
</math>
  </td>
  <td align="center">
<math>~~~\rightarrow~~~</math>
  </td>
  <td align="left">
<math>
\frac{\partial \psi}{\partial t} + \vec{u} \cdot \nabla\psi \, .
</math>
  </td>
</tr>
</table>
</div>
Hence, the "new" generic hyperbolic PDE derived above can be rewritten as,
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
\frac{\partial\psi}{\partial t} + \vec{u} \cdot \nabla\psi + \psi \nabla\cdot \vec{u}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
S \, ,
</math>
  </td>
</tr>
</table>
</div>
or, more succinctly,
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
\frac{\partial\psi}{\partial t} + \nabla\cdot (\psi \vec{u} )
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
S \, .
</math>
  </td>
</tr>
</table>
</div>
This equation also is broadly recognized as a conservation statement because, when <math>~S = 0</math>, the variable <math>~\psi</math> will represent the volume density of a conserved quantity. 

We should emphasize that the inertial-frame version of this Eulerian conservation equation can be retrieved straightforwardly by setting <math>~\Omega_0 = 0</math>, which is equivalent to setting <math>~\vec{u} = \vec{v}</math>.  It is,
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
\frac{\partial\psi}{\partial t} + \nabla\cdot (\psi \vec{v})
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
S \, .
</math>
  </td>
</tr>
</table>
</div>
The physics of the flow that is being described by this PDE is identical to the physics that is described by the preceding PDE.  But an important distinction must be made regarding how the two equations are ''interpreted.''  The "inertial frame" version of the equation (explicitly containing <math>~\vec{v}</math>) provides the time-rate of change of <math>~\psi</math> at a fixed point in ''inertial'' space, while the "new" version (explicitly containing <math>~\vec{u}</math>) provides the time-rate of change of <math>~\psi</math> at a fixed point in our "new" ''rotating'' coordinate frame.

===Angular Momentum Conservation===
When the three vector components of the Euler equation (of motion) are projected onto a nonrotating cylindrical coordinate grid, the azimuthal component of the Euler equation may be written as,
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
\frac{d(\rho \varpi v_\phi)}{dt} + (\rho \varpi v_\phi) \nabla\cdot \vec{v}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
-\frac{\partial P}{\partial\phi} - \rho \frac{\partial\Phi}{\partial\phi} \, .
</math>
  </td>
</tr>
</table>
</div>
For this equation, the source term is identified as,
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
~S
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
-\frac{\partial P}{\partial\phi} - \rho \frac{\partial\Phi}{\partial\phi} \, ,
</math>
  </td>
</tr>
</table>
</div>
and <math>~\psi = (\rho\varpi v_\phi)</math> is the ''inertial-frame'' angular momentum density, as measured with respect to the <math>~z</math>-coordinate axis.  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)].

<div align="center">
<table border="1" cellpadding="5" width="100%">
<tr>
  <td align="center" colspan="2">
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>
with the following replacements: 
&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>
  </td>
</tr>

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

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


As foreshadowed above &#8212; see the [[#Recognizing_Statements_of_Conservation|subsection titled, ''Recognizing Statements of Conservation'']] &#8212; the angular momentum of a Lagrangian fluid element will be conserved if the "source" term, <math>~S = 0</math>.  This situation will arise if, at the fluid element's location, the azimuthal pressure variation, <math>~\partial P/\partial\phi</math>, and the azimuthal variation in the gravitational potential, <math>~\partial \Phi/\partial\phi</math>, are both zero, or if the two balance one another (''i.e.,''<math>~\partial P/\partial\phi=-\rho\partial\Phi/\partial\phi</math>).

Based on the above discussion, we can equally well view the flow from a frame of reference that is rotating with a constant angular velocity, <math>~\Omega_0</math>, and write, 
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
\frac{d(\rho \varpi v_\phi)}{dt} + (\rho \varpi v_\phi) \nabla\cdot \vec{u}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
-\frac{\partial P}{\partial\phi} - \rho \frac{\partial\Phi}{\partial\phi} \, ,
</math>
  </td>
</tr>
</table>
</div>
where, as before,
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
\vec{u} 
</math>
  </td>
  <td align="center">
<math>~\equiv~</math>
  </td>
  <td align="left">
<math>
\vec{v} - \boldsymbol{\hat{e}}_\phi \varpi\Omega_0 \, .
</math>
  </td>
</tr>
</table>
</div>
Also, following the earlier discussion, if one wants to follow the time-variation of the fluid's inertial-frame angular momentum at a fixed location in inertial space, then the appropriate Eulerian representation of this azimuthal component of the equation of motion is,
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
\frac{\partial (\rho \varpi v_\phi)}{\partial t} + \nabla\cdot [(\rho \varpi v_\phi) \vec{v}]
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
S \, .
</math>
  </td>
</tr>
</table>
</div>
If, however, one wants to follow the time-variation of the fluid's inertial-frame angular momentum at a fixed location on a rotating coordinate grid, then the appropriate Eulerian representation of this azimuthal component of the equation of motion is obtained by replacing the "transport" velocity, <math>~\vec{v}</math> with <math>~\vec{u}</math>; specifically, 
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
\frac{\partial (\rho \varpi v_\phi)}{\partial t} + \nabla\cdot [(\rho \varpi v_\phi) \vec{u}]
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
S \, .
</math>
  </td>
</tr>
</table>
</div>

===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,
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
~v_\phi
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
~u_\phi + \varpi\Omega_0 \, .
</math>
  </td>
</tr>
</table>
</div>

So we can write,

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

<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>
</div>
where, as shorthand, we have used,
<div align="center">
<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,
<div align="center">
<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)].

<div align="center">
<table border="1" cellpadding="5" width="100%">
<tr>
  <td align="center" colspan="2">
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>

<tr>
  <td align="center">
<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>

===Even Broader Generalization===
As [http://adsabs.harvard.edu/abs/2010CQGra..27q5002C Call, Tohline, &amp; Lehner (2010)] point out, we are free to measure &#8212; and follow the evolution of &#8212; the angular momentum density with respect to any of a variety of different rotating frames of reference.  Specifically, we are not constrained to choose between the inertial (nonrotating) frame &#8212; in which the measured angular momentum density is <math>~(\rho\varpi v_\phi)</math> &#8212; and the "grid" frame &#8212; in which the measured angular momentum density is <math>~\rho\varpi u_\phi = \rho\varpi(v_\phi - \varpi\Omega_0)</math>.  Quite generally, we can choose to measure the angular momentum with respect to a separate "primed" frame that is rotating with angular velocity <math>~\omega_0</math> and in which the measured azimuthal component of the fluid velocity is,
<div align="center">
<math>
~v'_\phi ~= ~v_\phi - \varpi \omega_0 \, .
</math>
</div>
With this definition in hand, we also recognize that,
<div align="center">
<math>
~u_\phi = v_\phi - \varpi\Omega_0 = v'_\phi + \varpi (\omega_0 - \Omega_0) \, .
</math>
</div>
These two substitutions allow us to rewrite the angular momentum evolution equation in the forms that [http://adsabs.harvard.edu/abs/2010CQGra..27q5002C Call, Tohline, &amp; Lehner (2010)] label as Case C (<math>~\eta=3</math>) and Case C (<math>~\eta=3'</math>).

<div align="center">
<table border="1" cellpadding="5" width="100%">
<tr>
  <td align="center" colspan="2">
From Tables 6.1 &amp; 6.2 of [http://adsabs.harvard.edu/abs/2010CQGra..27q5002C Call, Tohline, &amp; Lehner (2010)]
<br>
'''Case C''' <math>~(\eta = 3)</math>
<br>
with the following replacements: 
&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>
  </td>
</tr>

<tr>
  <td align="center">
<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 v_\phi</math>
  </td>
  <td align="center">
<math>~ - \frac{\partial P}{\partial\phi} - \rho \frac{\partial \Phi}{\partial\phi}</math>
  </td>
</tr>
</table>
</div>
&nbsp;<br>
&nbsp;

<div align="center">
<table border="1" cellpadding="5" width="100%">
<tr>
  <td align="center" colspan="2">
From Tables 6.1 &amp; 6.2 of [http://adsabs.harvard.edu/abs/2010CQGra..27q5002C Call, Tohline, &amp; Lehner (2010)]
<br>
'''Case C''' <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>

<tr>
  <td align="center">
<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 - \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>
In the latter case, <math>~v'_\phi</math> becomes <math>~u_\phi</math>, as it should, when the choice is made to measure the angular momentum density in the "grid" frame, that is, when the choice is made to set <math>~\omega_0 = \Omega_0</math>.