Editing Appendix/Ramblings/HybridSchemeImplications (section)

====Hybrid Scheme====

In our ''hybrid scheme,'' we will continue to use <math>~\mathcal{L}_u</math> &#8212; that is, an advection operator that incorporates the rotating-frame velocity, <math>~\bold{u}</math> &#8212; but we will switch all other velocity references to the inertial-frame velocity, <math>~\bold{v}</math>, and its components.  This will be done via the [[#Rotating_Frame|above-declared mapping]], 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\bold{u}</math>
  </td>
  <td align="center">
<math>~~\rightarrow~~</math>
  </td>
  <td align="left">
<math>~[\bold{v} - \bold{\hat{e}}_\varphi \varpi \Omega_f] \, ,</math>
  </td>
</tr>
</table>
that is, <math>~u_\varpi ~\rightarrow~ v_\varpi</math>, <math>~u_z ~\rightarrow~ v_z</math>, and <math>~u_\varphi ~\rightarrow~ (v_\varphi - \varpi \Omega_f)</math>.
The Euler equation becomes,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right" colspan="1">
<math>~\frac{\partial [\bold{v} - \bold{\hat{e}}_\varphi \varpi \Omega_f]}{\partial t} + 
\bold{\hat{e}}_\varpi \biggl[ \mathcal{L}_u v_\varpi \biggr]
+ \bold{\hat{e}}_\varphi \biggl[ \mathcal{L}_u (v_\varphi  - \varpi \Omega_f)\biggr]
+ \bold{\hat{e}}_z \biggl[ \mathcal{L}_u v_z \biggr] 
</math>  
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \bold{a} 
+ \bold{\hat{e}}_\varpi  \biggl[ \frac{v_\varphi^2}{\varpi} \biggr]
- \bold{\hat{e}}_\varphi \biggl[ \varpi \Omega_f  + v_\varphi \biggr] \frac{v_\varpi}{\varpi} 
\, ,
</math>
  </td>
</tr>
</table>
where we recognize that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{L}_u v_i</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(\bold{u}\cdot \bold\nabla) v_i \, .</math>
  </td>
</tr>
</table>
Now, given that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial}{\partial t} \biggl[ \bold{\hat{e}}_\varphi \varpi \Omega_f \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, ,</math>
  </td>
</tr>
</table>
and,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{L}_u (\varpi \Omega_f)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[
u_\varpi \frac{\partial }{\partial \varpi} + \frac{u_\varphi }{\varpi}\frac{\partial }{\partial \varphi} + u_z \frac{\partial }{\partial z} 
\biggr] (\varpi \Omega_f) = u_\varpi \Omega_f 
= v_\varpi \Omega_f \, ,
</math>
  </td>
</tr>
</table>
the Euler equation becomes,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right" colspan="1">
<math>~\frac{\partial \bold{v} }{\partial t} + 
\bold{\hat{e}}_\varpi \biggl[ \mathcal{L}_u v_\varpi \biggr]
+ \bold{\hat{e}}_\varphi \biggl[ \mathcal{L}_u v_\varphi - v_\varpi \Omega_f \biggr]
+ \bold{\hat{e}}_z \biggl[ \mathcal{L}_u v_z \biggr] 
</math>  
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \bold{a} 
+ \bold{\hat{e}}_\varpi  \biggl[ \frac{v_\varphi^2}{\varpi} \biggr]
- \bold{\hat{e}}_\varphi \biggl[ \varpi \Omega_f  + v_\varphi \biggr] \frac{v_\varpi}{\varpi} 
</math>
  </td>
</tr>

<tr>
  <td align="right" colspan="1">
<math>~\Rightarrow ~~~
\frac{\partial \bold{v} }{\partial t} + 
\bold{\hat{e}}_\varpi \biggl[ \mathcal{L}_u v_\varpi \biggr]
+ \bold{\hat{e}}_\varphi \biggl[ \mathcal{L}_u v_\varphi \biggr]
+ \bold{\hat{e}}_z \biggl[ \mathcal{L}_u v_z \biggr] 
</math>  
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \bold{a} 
+ \bold{\hat{e}}_\varpi  \biggl[ \frac{v_\varphi^2}{\varpi} \biggr]
- \bold{\hat{e}}_\varphi \biggl[ \frac{v_\varpi v_\varphi}{\varpi} \biggr] \, .
</math>
  </td>
</tr>
</table>

Also, if we multiply the [[PGE/ConservingMass#Various_Forms|standard Lagrangian representation of the continuity equation]] through by <math>~v_i</math>, we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
v_i \biggl[\frac{d \rho}{dt} + \rho \bold\nabla \cdot \bold{v} \biggr]
=
v_i \biggl[\frac{d \rho}{dt} + \rho \bold\nabla \cdot \bold{u} + \rho \cancelto{0}{\bold\nabla \cdot (\bold{\hat{e}}_\varphi \varpi \Omega_f)} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~v_i \biggl[\frac{\partial \rho}{\partial t} + \bold\nabla \cdot (\rho \bold{u}) \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\partial \rho v_i}{\partial t} - \rho \frac{\partial v_i}{\partial t} 
+ \bold\nabla \cdot (\rho v_i \bold{u})
- \rho (\bold{u}\cdot \bold\nabla) v_i 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
\rho \frac{\partial v_i}{\partial t} + \rho \biggl[ \mathcal{L}_u  v_i \biggr]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\partial \rho v_i}{\partial t} 
+ \bold\nabla \cdot (\rho v_i \bold{u}) \, .
</math>
  </td>
</tr>
</table>

----


<font color="red">'''Vertical Component:'''</font> &nbsp;Multiplying the <math>~\bold{\hat{k}}</math> component of our modified Euler equation through by <math>~\rho</math>, then incorporating this version of the continuity equation, gives,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right" colspan="1">
<math>~\rho \frac{\partial v_z}{\partial t} 
+ \rho \biggl[ \mathcal{L}_u v_z \biggr] 
</math>  
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\rho  a_z 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
\frac{\partial \rho v_z}{\partial t} + \bold\nabla \cdot (\rho v_z \bold{u})
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\rho a_z \, .</math>
  </td>
</tr>
</table>

<font color="red">'''Radial Component:'''</font> &nbsp;Multiplying the <math>~\bold{\hat{e}}_\varpi</math> component of our modified Euler equation through by <math>~\rho</math>, then incorporating the continuity equation, gives,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right" colspan="1">
<math>~\rho \frac{\partial v_\varpi}{\partial t} 
+ \rho \biggl[ \mathcal{L}_u v_\varpi \biggr] 
</math>  
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\rho  a_\varpi + \frac{v_\varphi^2}{\varpi} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
\frac{\partial \rho v_\varpi}{\partial t} + \bold\nabla \cdot (\rho v_\varpi \bold{u})
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\rho  a_\varpi + \frac{v_\varphi^2}{\varpi} \, .</math>
  </td>
</tr>
</table>

<font color="red">'''Azimuthal Component:'''</font> &nbsp;Multiplying the <math>~\bold{\hat{e}}_\varphi</math> component of our modified Euler equation through by <math>~\rho</math>, then incorporating the continuity equation, gives,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right" colspan="1">
<math>~\rho \frac{\partial v_\varphi}{\partial t} 
+ \rho \biggl[ \mathcal{L}_u v_\varphi \biggr] 
</math>  
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\rho  a_\varphi - \frac{\rho v_\varphi v_\varpi}{\varpi} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
\frac{\partial \rho v_\varphi}{\partial t} + \bold\nabla \cdot (\rho v_\varphi \bold{u})
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\rho  a_\varphi - \frac{\rho v_\varphi v_\varpi}{\varpi} \, .</math>
  </td>
</tr>
</table>
Then, multiplying through by <math>~\varpi</math>, we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\rho \varpi  a_\varphi - \rho v_\varphi v_\varpi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\varpi \frac{\partial \rho v_\varphi}{\partial t} + \varpi \bold\nabla \cdot (\rho v_\varphi \bold{u})
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{\partial(\rho \varpi v_\varphi )}{\partial t} - (\rho v_\varphi )\cancelto{0}{\frac{\partial \varpi}{\partial t}}
+ \bold\nabla \cdot (\rho \varpi v_\varphi \bold{u})
- \rho v_\varphi (\bold{u}\cdot \bold\nabla) \varpi
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{\partial(\rho \varpi v_\varphi )}{\partial t} 
+ \bold\nabla \cdot (\rho \varpi v_\varphi \bold{u})
- \rho v_\varphi \biggl[ \cancelto{v_\varpi}{\mathcal{L}_u \varpi} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~ \Rightarrow~~~
\frac{\partial(\rho \varpi v_\varphi )}{\partial t} 
+ \bold\nabla \cdot (\rho \varpi v_\varphi \bold{u})
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\rho \varpi  a_\varphi \, .</math>
  </td>
</tr>
</table>