Editing Appendix/Ramblings/HybridSchemeOld (section)

=Component Forms=

Let's split the vector Euler equation into its three scalar components; various examples are identified in Table 1.

<div align="center">
<table border="1" cellpadding="5">
<tr>
  <th align="center" rowspan="2">
Example #
  </th>
  <th align="center" colspan="2">
Grid
  </th>
  <th align="center" colspan="2">
Momentum Vector
  </th>
</tr>

<tr>
  <th align="center">
Basis
  </th>
  <th align="center">
Rotating?
  </th>
  <th align="center">
Basis
  </th>
  <th align="center">
Frame
  </th>
</tr>

<tr>
  <td align="center">
1
  </td>
  <td align="center">
Cartesian
  </td>
  <td align="center">
No
  </td>
  <td align="center">
Cartesian
  </td>
  <td align="center">
Inertial
  </td>
</tr>

<tr>
  <td align="center">
2
  </td>
  <td align="center">
Cylindrical
  </td>
  <td align="center">
Yes <math>~(\Omega_0)</math>
  </td>
  <td align="center">
Cylindrical
  </td>
  <td align="center">
Rotating <math>~(\Omega_0)</math>
  </td>
</tr>

<tr>
  <td align="center">
3
  </td>
  <td align="center">
Cylindrical
  </td>
  <td align="center">
Yes <math>~(\Omega_0)</math>
  </td>
  <td align="center">
Cylindrical
  </td>
  <td align="center">
Rotating <math>~(\omega_0)</math>
  </td>
</tr>
</table>
</div>

In the following expressions, we will use <math>~\vec{v}</math> to denote the fluid velocity when it is associated with the rate of fluid transport across the coordinate grid, and we will use <math>~\vec{u}</math> to denote the fluid velocity when it is associated with the momentum density that is being advected.  In all cases, it should be understood that <math>~\vec{v} = \vec{u}</math>, as both vectors refer to the same fluid velocity.  In addition, we will use a "prime" notation to indicate when a velocity is being viewed from a rotating frame of reference; specifically, we will consider rotation about the <math>~z</math>-axis of the coordinate system, that is,
<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>~v_\phi - R\Omega_0 \, ,</math>
  </td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>~u'_\phi</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~u_\phi - R\omega_0 \, ,</math>
  </td>
</tr>
</table>
</div>
but we will not insist that the two rotation frequencies, <math>~\Omega_0</math> and <math>~\omega_0</math>, have the same value.  Hence, in general, <math>~(\vec{u})' \ne (\vec{v})'</math>.  It is worth emphasizing that, because we will only be considering frame rotation about the <math>z</math>-axis, the cylindrical <math>R</math> and <math>z</math> components of the velocity are interchangeable, that is: <math>~u'_R = v'_R = u_R = v_R</math>; and <math>~u'_z = v'_z = u_z = v_z</math>.

===Example #1===
This is certainly the most familiar component set.
<div align="center">
<table border="0" cellpadding="3">

<tr>
  <td align="right">
<math>\boldsymbol{\hat{e}}_x: ~~~\frac{\partial (\rho v_x)}{\partial t} + \nabla\cdot[(\rho v_x) \vec{v}~]</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
-~\frac{\partial P}{\partial x} - \rho \frac{\partial \Phi}{\partial x} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\boldsymbol{\hat{e}}_y: ~~~\frac{\partial (\rho v_y)}{\partial t} + \nabla\cdot[(\rho v_y) \vec{v}~]</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
-~\frac{\partial P}{\partial y} - \rho \frac{\partial \Phi}{\partial y} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\boldsymbol{\hat{e}}_z: ~~~\frac{\partial (\rho v_z)}{\partial t} + \nabla\cdot[(\rho v_z) \vec{v}~]</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
-~\frac{\partial P}{\partial z} - \rho \frac{\partial \Phi}{\partial z} \, ,
</math>
  </td>
</tr>
</table>
</div>
where, for any one of the three scalar PDEs, advection of the relevant component of the momentum density, <math>~\psi_i</math>, is handled via the operation,

<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
\nabla\cdot[\psi_{i} \vec{v} ] 
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\frac{\partial (\psi_i v_x)}{\partial x} + \frac{\partial (\psi_i v_y)}{\partial y} + \frac{\partial (\psi_i v_z)}{\partial z} \, .
</math>
  </td>
</tr>
</table>
</div>

===Example #2===
This component set has been spelled out in, for example, 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)].
<div align="center">
<table border="0" cellpadding="3">

<tr>
  <td align="right">
<math>\boldsymbol{\hat{e}}_R: ~~~~~~~\frac{\partial (\rho v_R)}{\partial t} + \nabla\cdot[(\rho v_R) \vec{v}~]</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
-~\frac{\partial P}{\partial R} - \rho \frac{\partial \Phi}{\partial R} + \frac{(\rho R v_\phi)^2}{\rho R^3} + \rho\Omega_0^2 R + \frac{2\Omega_0 (\rho R v_\phi)}{R} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
-~\frac{\partial P}{\partial R} - \rho \frac{\partial \Phi}{\partial R} + \frac{\rho}{R} (v_\phi + R\Omega_0)^2 \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\boldsymbol{\hat{e}}_\phi: ~~~\frac{\partial (\rho R v_\phi)}{\partial t} + \nabla\cdot[(\rho R v_\phi) \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} - 2\rho (\Omega_0 R )v_R \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\boldsymbol{\hat{e}}_z: ~~~~~~~~\frac{\partial (\rho v_z)}{\partial t} + \nabla\cdot[(\rho v_z) \vec{v}~]</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
-~\frac{\partial P}{\partial z} - \rho \frac{\partial \Phi}{\partial z} \, ,
</math>
  </td>
</tr>
</table>
</div>
where, as noted above,

<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
\nabla\cdot[\psi_{i} \vec{v} ] 
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\frac{\partial (\psi_i v_R)}{\partial R} + \frac{1}{R} \frac{\partial (\psi_i v_\phi)}{\partial\phi} + \frac{\partial (\psi_i v_z)}{\partial z} \, .
</math>
  </td>
</tr>
</table>
</div>

===Example #3===

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

<tr>
  <td align="right">
<math>~\boldsymbol{\hat{e}}_R:</math>
  </td>
  <td align="right">
<math>~\frac{\partial (\rho u'_R)}{\partial t} + \nabla\cdot[\rho u'_R (\vec{v})'~]</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
-~\frac{\partial P}{\partial R} - \rho \frac{\partial \Phi}{\partial R} + \frac{\rho}{R} (v'_\phi + R\Omega_0)^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
-~\frac{\partial P}{\partial R} - \rho \frac{\partial \Phi}{\partial R} + 
\frac{\rho (v'_\phi)^2}{R} + 2\rho \Omega_0 v'_\phi + \rho \Omega_0^2 R  \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\boldsymbol{\hat{e}}_\phi:</math>
  </td>
  <td align="right">
<math>~\frac{\partial \{\rho R [u'_\phi + R(\Omega_0 - \omega_0)]\} }{\partial t} + 
\nabla\cdot[ \{ \rho R [u'_\phi + R(\Omega_0 - \omega_0)] \} (\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} - 2\rho R\omega_0 v'_R \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\boldsymbol{\hat{e}}_z:</math>
  </td>
  <td align="right">
<math>~\frac{\partial (\rho u'_z)}{\partial t} + \nabla\cdot[\rho u'_z (\vec{v})'~]</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
-~\frac{\partial P}{\partial z} - \rho \frac{\partial \Phi}{\partial z} \, ,
</math>
  </td>
</tr>
</table>
</div>
where, as noted above,
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>~u'_\phi</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~u_\phi - R\omega_0 \, ,</math>
  </td>
</tr>
</table>
</div>

and, for any one of the three scalar PDEs, advection of the relevant component of the momentum density, <math>~\psi_i</math>, is handled via the operation,
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
\nabla\cdot[\psi_{i} (\vec{v})' ] 
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\frac{\partial (\psi_i v'_R)}{\partial R} + \frac{1}{R} \frac{\partial (\psi_i v'_\phi)}{\partial\phi} + \frac{\partial (\psi_i v'_z)}{\partial z} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\frac{\partial (\psi_i v_R)}{\partial R} + \frac{1}{R} \frac{\partial [\psi_i (v_\phi - R\Omega_0)]}{\partial\phi} + \frac{\partial (\psi_i v_z)}{\partial z} \, .
</math>
  </td>
</tr>
</table>
</div>