Editing Appendix/Ramblings/HybridSchemeImplications (section)

====Rotating Frame====

Drawing from an [[PGE/RotatingFrame#Rotating_Reference_Frame|accompanying discussion of rotating reference frames]], let's build our model in a cylindrical coordinate system that is spinning about its <math>~\bold{\hat{k}}</math>-axis with a time-independent angular velocity, <math>~\bold\Omega_f = \bold{\hat{k}} \Omega_f</math>.  Furthermore, let's use <math>~\bold{u}</math> &#8212; instead of <math>~\bold{v}</math> &#8212; to represent the velocity as viewed in the rotating frame.  We know that,

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

<tr>
  <td align="right">
<math>~\bold{v}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\bold{u} + \bold\Omega_f \times \bold{x}_\mathrm{rot} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\bold{u} + \bold{\hat{e}}_\varphi \varpi \Omega_f \, ,
</math>
  </td>
</tr>
</table>
and,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="left">
<math>~\bold{a} = \frac{d\bold{v}}{dt} =
\frac{d\bold{u}}{dt} - \overbrace{ \biggl[
\underbrace{(- 2\bold\Omega_f \times \bold{u}) }_\text{Coriolis}
~+~
\underbrace{(- \bold\Omega_f \times (\bold\Omega_f \times \bold{x}_\mathrm{rot} )) }_\text{Centrifugal} \biggr]
}^\text{Fictitious accelerations} \, .
</math>
  </td>
</tr>
<tr><td align="center" colspan="1">
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 664, Appendix &sect;1.D.3, Eq. (1D-43)
</td></tr>
</table>

<table border="1" align="center" width="80%" cellpadding="10"><tr><td align="left">
Note that, in the particular case being considered here,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\bold{a}_\mathrm{fict}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- 2\bold\Omega_f \times \bold{u} 
- \bold\Omega_f \times (\bold\Omega_f \times \bold{x}_\mathrm{rot} ) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- 2\bold\Omega_f \times [ \bold{v} - \bold\Omega_f \times \bold{x}_\mathrm{rot} ] 
- \bold\Omega_f \times (\bold\Omega_f \times \bold{x}_\mathrm{rot} ) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- 2\bold\Omega_f \times \bold{v}  
+ \bold\Omega_f \times (\bold\Omega_f \times \bold{x}_\mathrm{rot} ) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- 2\bold\Omega_f \times \bold{v}  
+ \bold{\hat{k}}\Omega_f \times (\bold{\hat{e}}_\varphi \varpi \Omega_f  ) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- 2\Omega_f [\bold{\hat{e}}_\varphi v_\varpi - \bold{\hat{e}}_\varpi v_\varphi]
- (\bold{\hat{e}}_\varpi \varpi \Omega_f^2  ) \, .
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
+ \bold{\hat{e}}_\varpi [2\Omega_f v_\varphi
- \varpi \Omega_f^2  ] 
- \bold{\hat{e}}_\varphi 2\Omega_f v_\varpi 
\, .
</math>
  </td>
</tr>
</table>

</td></tr></table>

We may therefore also write,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right" colspan="2">
<math>~\bold{a} + \bold{a}_\mathrm{fict} = \frac{d\bold{u}}{dt}</math>
  </td>
  <td align="left">
<math>~= \frac{\partial \bold{u}}{\partial t} + (\bold{u} \cdot \bold\nabla)\bold{u} </math>
  </td>
</tr>

<tr>
  <td align="right" colspan="2">
&nbsp;
  </td>
  <td align="left">
<math>~=~ \frac{\partial \bold{u}}{\partial t} + 
\bold{\hat{e}}_\varpi \biggl[ \mathcal{L}_u u_\varpi \biggr]
+ \bold{\hat{e}}_\varphi \biggl[ \mathcal{L}_u u_\varphi \biggr]
+ \bold{\hat{e}}_z \biggl[ \mathcal{L}_u u_z \biggr] 
~+ \underbrace{\bold{\hat{e}}_\varphi \biggl( \frac{ u_\varpi u_\varphi}{\varpi} \biggr)
-~
\bold{\hat{e}}_\varpi \biggl( \frac{u_\varphi^2}{\varpi} \biggr) }_\text{curvature terms} \, ,
</math>
  </td>
</tr>
</table>
where the operator,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{L}_u</math>
  </td>
  <td align="center">
<math>~\equiv</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] \, .
</math>
  </td>
</tr>
</table>

In numerical simulations that are carried out on a cylindrical grid and in a rotating reference frame, it is customary to group the "curvature terms" with the fictitious acceleration to obtain,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right" colspan="1">
<math>~\frac{\partial \bold{u}}{\partial t} + 
\bold{\hat{e}}_\varpi \biggl[ \mathcal{L}_u u_\varpi \biggr]
+ \bold{\hat{e}}_\varphi \biggl[ \mathcal{L}_u u_\varphi \biggr]
+ \bold{\hat{e}}_z \biggl[ \mathcal{L}_u u_z \biggr] 
</math>  
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \bold{a} + \bold{a}_\mathrm{fict}
~- \bold{\hat{e}}_\varphi \biggl( \frac{ u_\varpi u_\varphi}{\varpi} \biggr)
+~
\bold{\hat{e}}_\varpi \biggl( \frac{u_\varphi^2}{\varpi} \biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right" colspan="1">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \bold{a} 
+ \bold{\hat{e}}_\varpi \biggl[2\Omega_f v_\varphi
- \varpi \Omega_f^2  + \biggl( \frac{u_\varphi^2}{\varpi} \biggr) \biggr] 
- \bold{\hat{e}}_\varphi \biggl[ 2\Omega_f v_\varpi 
~+ \biggl( \frac{ u_\varpi u_\varphi}{\varpi} \biggr)\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right" colspan="1">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \bold{a} 
+ \bold{\hat{e}}_\varpi \biggl[2\Omega_f (u_\varphi + \varpi \Omega_f)
- \varpi \Omega_f^2  + \biggl( \frac{u_\varphi^2}{\varpi} \biggr) \biggr] 
- \bold{\hat{e}}_\varphi \biggl[ 
2\varpi \Omega_f  + u_\varphi 
\biggr] \frac{u_\varpi}{\varpi}
</math>
  </td>
</tr>

<tr>
  <td align="right" colspan="1">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \bold{a} 
+ \bold{\hat{e}}_\varpi \biggl[ \varpi\Omega_f + u_\varphi \biggr]^2 \frac{1}{\varpi}
- \bold{\hat{e}}_\varphi \biggl[ 2\varpi \Omega_f  + u_\varphi \biggr] \frac{u_\varpi}{\varpi} 
\, ,
</math>
  </td>
</tr>
</table>

treating the ensemble as an additional "source" of acceleration.

<table border="1" align="center" width="80%" cellpadding="10"><tr><td align="left">
<div align="center"><b>Example from the Literature</b><br />(see an accompanying [[Appendix/Ramblings/HybridSchemeOld#NW78|related discussion]])</div>

We begin with the version of the Euler equation that has just been derived, namely,
<table border="0" cellpadding="5" align="center">

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

----


In examining and rearranging terms in each of the three components of this Euler equation, we will recognize that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{L}_u u_i</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(\bold{u} \cdot \bold\nabla)u_i \, ;</math>
  </td>
</tr>
</table>
and that, after multiplying the [[PGE/ConservingMass#Various_Forms|standard Lagrangian representation of the continuity equation]] through by <math>~u_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>~
u_i \biggl[\frac{d \rho}{dt} + \rho \bold\nabla \cdot \bold{v} \biggr]
=
u_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>~u_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 u_i}{\partial t} - \rho \frac{\partial u_i}{\partial t} 
+ \bold\nabla \cdot (\rho u_i \bold{u})
- \rho (\bold{u}\cdot \bold\nabla) u_i 
</math>
  </td>
</tr>

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

----


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

<tr>
  <td align="right" colspan="1">
<math>~\rho \frac{\partial u_z}{\partial t} 
+ \rho \biggl[ \mathcal{L}_u u_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 u_z}{\partial t} + \bold\nabla \cdot (\rho u_z \bold{u})
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\rho a_z \, .</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[https://ui.adsabs.harvard.edu/abs/1978ApJ...224..497N/abstract Norman &amp; Wilson (1978)], ApJ, 224, pp. 497 - 511, &sect;III.b, Eq. (5)<br />
[https://ui.adsabs.harvard.edu/abs/1997ApJ...490..311N/abstract New &amp; Tohline (1997)], ApJ, 490, pp. 311 - 237, &sect;2, Eq. (3)
</td>
</tr>
</table>

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

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

<tr>
  <td align="right" colspan="1">
<math>~\Rightarrow ~~~
\frac{\partial \rho u_\varpi}{\partial t} 
+ \bold\nabla \cdot (\rho u_\varpi \bold{u})</math>  
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \rho a_\varpi 
+ \biggl[\frac{(\rho \varpi u_\varphi)^2}{\rho \varpi^3} + \rho\Omega_f^2 \varpi + \frac{ 2\Omega_f (\rho \varpi u_\varphi )}{\varpi} \biggr] \, .
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[https://ui.adsabs.harvard.edu/abs/1978ApJ...224..497N/abstract Norman &amp; Wilson (1978)], ApJ, 224, pp. 497 - 511, &sect;III.b, Eq. (6)<br />
[https://ui.adsabs.harvard.edu/abs/1997ApJ...490..311N/abstract New &amp; Tohline (1997)], ApJ, 490, pp. 311 - 237, &sect;2.2, Eq. (11)
</td>
</tr>
</table>

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

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

<tr>
  <td align="right" colspan="1">
<math>~\Rightarrow ~~~
\frac{\partial \rho u_\varphi}{\partial t} 
+ \bold\nabla \cdot (\rho u_\varphi \bold{u})</math>  
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \rho a_\varphi 
- \biggl[ 2\varpi \Omega_f  + u_\varphi \biggr] \frac{\rho u_\varpi}{\varpi}  \, .
</math>
  </td>
</tr>
</table>

Then, multiplying through by <math>~\varpi</math>, gives,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right" colspan="1">
<math>~
\varpi \frac{\partial \rho u_\varphi}{\partial t} 
+ \varpi \bold\nabla \cdot (\rho u_\varphi \bold{u})</math>  
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \rho \varpi a_\varphi 
- \biggl[ 2\varpi \Omega_f  + u_\varphi \biggr] \rho u_\varpi  
</math>
  </td>
</tr>

<tr>
  <td align="right" colspan="1">
<math>~\Rightarrow~~~
\frac{\partial \rho \varpi u_\varphi}{\partial t} 
+ \bold\nabla \cdot (\rho \varpi u_\varphi \bold{u})
- \rho u_\varphi \cancelto{u_\varpi}{\biggl[ \mathcal{L}_u \varpi \biggr]} 
</math>  
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \rho \varpi a_\varphi 
- 2\Omega_f \varpi \rho u_\varpi  - \rho u_\varphi u_\varpi   
</math>
  </td>
</tr>

<tr>
  <td align="right" colspan="1">
<math>~\Rightarrow~~~
\frac{\partial \rho \varpi u_\varphi}{\partial t} 
+ \bold\nabla \cdot (\rho \varpi u_\varphi \bold{u})
</math>  
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \rho \varpi a_\varphi 
- 2\Omega_f \varpi \rho u_\varpi  \, .  
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[https://ui.adsabs.harvard.edu/abs/1978ApJ...224..497N/abstract Norman &amp; Wilson (1978)], ApJ, 224, pp. 497 - 511, &sect;III.b, Eq. (7)<br />
[https://ui.adsabs.harvard.edu/abs/1997ApJ...490..311N/abstract New &amp; Tohline (1997)], ApJ, 490, pp. 311 - 237, &sect;2.2, Eq. (12)
</td>
</tr>
</table>

</td></tr></table>