Editing Appendix/Ramblings/HybridSchemeOld (section)

==Text Following Equation A17==
I recommend that the text following equation A17 read as follows:

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Equation (A17)
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[<font color="red">Do not indent</font>] When comparing equations (A16) and (A17), notice that the conserved quantity is the same &#8212; the ''z''-component of the angular momentum measure in the ''inertial'' frame.  The only difference in the two equations is the "transport" velocity (<math>~\mathbf{u}</math> for the nonrotating frame, <math>~\mathbf{u}'</math> for the rotating reference frame).

[<font color="red">Start new paragraph</font>] Equation (A17) is different from the more familiar formulation, where the angular momentum density as well as the transport velocity is measure with respect to the rotating frame, i.e., where the angular momentum density is expressed in terms of the azimuthal component of the transport velocity, <math>~u_\varphi'</math>.  But, as a consequence, the source term in the more familiar formulation is more complicated.  We can derive the more familiar formulation from equation (A17) by recognizing that,
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Equation (A18) thru Equation (A23)
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where the last step is accomplished by making use of the continuity relation, <math>~\partial\rho/\partial t + \nabla\cdot(\rho \mathbf{u}') = 0.</math>  Notice that all velocities now refer to <math>~\mathbf{u}'</math>, the velocity as measured in the rotating frame, which is the more familiar formulation.  The appearance of a Coriolis term is the result of choosing to measure angular momentum in the rotating frame rather than in the inertial frame.  This corresponds to "Case B <math>~(\eta = 3')</math>" in Call, Tohline, &amp; Lehner (2010).  In our hybrid scheme we have chosen to use equation (A17) instead of (A23) primarily because equation (A17) presents a simpler source term.