Editing Appendix/Ramblings/HybridSchemeImplications (section)

===Principal Governing Equations===

Quoting from [Ref01] &hellip; Among the [[PGE#Principal_Governing_Equations|principal governing equations]] we have included the ''inertial-frame'',

<div align="center">
<span id="ConservingMomentum:Lagrangian"><font color="#770000">'''Lagrangian Representation'''</font></span><br />
of the Euler Equation,

{{Math/EQ_Euler01}}

[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], Chap. 2, &sect;11, p. 20, Eq. (38)<br />
[<b>[[Appendix/References#BLRY07|<font color="red">BLRY07</font>]]</b>], p. 13, Eq. (1.55)

</div>

Shifting into a rotating frame characterized by the angular velocity vector,
<div align="center">
<math>~\vec{\Omega}_f \equiv \hat\mathbf{k} \Omega_f \, ,</math>
</div>
and applying the operations that are specified in the first few subsections of [Ref02], we recognize the following relationships &hellip;
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\vec{v}_\mathrm{inertial}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\vec{v}_\mathrm{rot} + {\vec\Omega}_f \times \vec{x} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\biggl[ \frac{d \vec{v}}{dt} \biggr]_\mathrm{inertial}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{d \vec{v}}{dt} \biggr]_\mathrm{rot} + 2{\vec\Omega}_f \times {\vec{v}}_\mathrm{rot}
+ {\vec\Omega}_f \times ({\vec\Omega}_f \times \vec{x})  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{d \vec{v}}{dt} \biggr]_\mathrm{rot} + 2{\vec\Omega}_f \times {\vec{v}}_\mathrm{rot}
- \frac{1}{2} \nabla | {\vec\Omega}_f \times \vec{x}|^2   
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{\partial \vec{v}}{\partial t} \biggr]_\mathrm{rot} 
+ ({\vec{v}}_\mathrm{rot} \cdot \nabla){\vec{v}}_\mathrm{rot}
+ 2{\vec\Omega}_f \times {\vec{v}}_\mathrm{rot}
- \frac{1}{2} \nabla | {\vec\Omega}_f \times \vec{x}|^2   
\, .</math>
  </td>
</tr>
</table>
Making this substitution on the left-hand-side of the above-specified "<font color="maroon">Lagrangian Representation</font> of the Euler Equation," we obtain what we have referred to [[PGE/RotatingFrame#Euler_Equation_.28rotating_frame.29|also in [Ref02]]] as the,

<div align="center">
<font color="#770000">'''Eulerian Representation'''</font><br />
of the Euler Equation <br />
<font color="#770000">'''as viewed from a Rotating Reference Frame'''</font>

<math>\biggl[\frac{\partial\vec{v}}{\partial t}\biggr]_\mathrm{rot} + ({\vec{v}}_\mathrm{rot}\cdot \nabla) {\vec{v}}_\mathrm{rot}= - \frac{1}{\rho} \nabla P - \nabla \biggl[\Phi - \frac{1}{2}|{\vec{\Omega}}_f \times \vec{x}|^2 \biggr] - 2{\vec{\Omega}}_f \times {\vec{v}}_\mathrm{rot} \, .</math> 
</div>

This form of the Euler equation also appears early in [Ref05], where we set up a discussion of the paper by [http://adsabs.harvard.edu/abs/1996ApJS..105..181K Korycansky &amp; Papaloizou] (1996, ApJS, 105, 181; hereafter KP96).  But, for now, let's back up a couple of steps and retain the ''total'' time derivative on the left-hand-side.  That is, let's select as the foundation expression the,
<div align="center">
<font color="#770000">'''Lagrangian Representation'''</font><br />
of the Euler Equation <br />
<font color="#770000">'''as viewed from a Rotating Reference Frame'''</font>

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl[ \frac{d \vec{v}}{dt} \biggr]_\mathrm{rot} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{1}{\rho} \nabla P - \nabla \Phi  
- 2{\vec\Omega}_f \times {\vec{v}}_\mathrm{rot}
- {\vec\Omega}_f \times ({\vec\Omega}_f \times \vec{x})  \, ,</math>
  </td>
</tr>
<tr><td align="center" colspan="3">
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], Chap. 2, &sect;12, p. 25, Eq. (62)
</td>
</tr>
</table>
</div>
which also serves as the foundation of most of our [Ref03] discussions.