Editing PGE/RotatingFrame (section)

==Overview==
<span id="InertialFrame">Among</span> the [[PGE#Principal_Governing_Equations|principal governing equations]] we have included the
&nbsp;<br />
&nbsp;<br />

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

{{Template:Math/EQ_Euler01}}

[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 671, Appendix Eq. (1E-6)<br />
[<b>[[Appendix/References#BLRY07|<font color="red">BLRY07</font>]]</b>], p. 13, Eq. (1.55)
</div>
Alternatively, a rewrite of the LHS gives what we refer to as the,
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="center" colspan="3">
<font color="#770000">'''Eulerian Representation'''</font><br />
of the intertial-frame Euler Equation
  </td>
</tr>
<tr>
  <td align="right">
<math>
\frac{\partial \vec{v}}{\partial t} + (\vec{v} \cdot \nabla)\vec{v}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-\frac{1}{\rho}\nabla P - \nabla \Phi \, . 
</math>
  </td>
</tr>
</table>
At times, it can be useful to view the motion of a fluid from a frame of reference that is rotating with an angular velocity
<div align="center"> 
<math>\vec{\Omega} = \hat\imath ~\Omega_1 + \hat\jmath ~\Omega_2 + \hat{k}~\Omega_3 \, .</math> 
</div>

----

<table border="0" align="center" width="90%" cellpadding="8"><tr><td align="left">Often it suffices to align <math>\vec\Omega</math> with the z-axis of the chosen coordinate system &#8212; in which case, <math>\Omega_1 =\Omega_2 =0</math> &#8212; and to set <math>d\vec\Omega/dt = 0</math>, in which case the nonzero component of the frame's angular velocity, <math>\Omega_3</math>, is independent of time.</td></tr></table>

----

In what follows we show that, when viewed from this rotating reference frame, we have what will be referred to as the
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="center" colspan="3"><font color="#770000">'''Lagrangian Representation'''</font><br />of the ''rotating-frame'' Euler Equation</td>
</tr>

<tr>
  <td align="right">
<math>
\frac{d\vec{u}}{dt}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
2\vec{u}\times \vec\Omega + \vec\Omega \times (\vec{x}\times \vec\Omega)
+ \vec{x}\times \frac{d\vec\Omega}{dt}
- ~\biggl[\frac{1}{\rho}\nabla P + \nabla\Phi \biggr]_\mathrm{rotating} \, , 
</math>
  </td>
</tr>

<tr>
  <td align="center" colspan="3">{{ Rossner67 }}, &sect;II, Eq. (1)<br />
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], Appendix 1.D, &sect;3, (p. 664) Eq. (1D-42)</td>
</tr>
</table>
where the difference between the rotating-frame velocity, <math>\vec{u}</math>, and the inertial-frame velocity, <math>\vec{v}</math>, is given by the expression,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>
\vec{v} - \vec{u}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\vec\Omega \times \vec{x} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\hat\imath (\Omega_2 z - \Omega_3 y) 
+ \hat\jmath (\Omega_3 x - \Omega_1 z) 
+ \hat{k} (\Omega_1 y - \Omega_2 x)) 
\, .
</math>
  </td>
</tr>
</table>

As above, a rewrite of the LHS gives what we will refer to as the, 
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="center" colspan="3"><font color="#770000">'''Eulerian Representation'''</font><br />of the ''rotating-frame'' Euler Equation</td>
</tr>

<tr>
  <td align="right">
<math>
\frac{\partial\vec{u}}{\partial t} + (\vec{u} \cdot \nabla)\vec{u}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
2\vec{u}\times \vec\Omega + \vec\Omega \times (\vec{x}\times \vec\Omega)
+ \vec{x}\times \frac{d\vec\Omega}{dt}
- ~\biggl[\frac{1}{\rho}\nabla P + \nabla\Phi \biggr]_\mathrm{rotating} \, . 
</math>
  </td>
</tr>
</table>

Along the way, and being guided by Chandrasekhar's presentation in Chapter 4, &sect;25 of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], we appreciate that it can be useful to highlight a ''hybrid'' representation of the Euler Equation that involves a mixture of the velocity variables, <math>\vec{u}</math> along with <math>\vec{v}</math>.  For example, beginning with this last expressions, we can write,

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

<tr>
  <td align="right">
<math>
2\vec{u}\times \vec\Omega + \vec\Omega \times (\vec{x}\times \vec\Omega)
+ \vec{x}\times \frac{d\vec\Omega}{dt}
- ~\biggl[\frac{1}{\rho}\nabla P + \nabla\Phi \biggr] 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\partial\vec{u}}{\partial t} + (\vec{u} \cdot \nabla)\vec{u}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\partial}{\partial t} \biggl[\vec{v} - \vec\Omega \times \vec{x} \biggr]
+ 
(\vec{u} \cdot \nabla)\biggl[\vec{v} - \vec\Omega \times \vec{x} \biggr]
</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} + (\vec{u}\cdot \nabla)\vec{v}\biggr]
+ \frac{\partial}{\partial t} \biggl[- \vec\Omega \times \vec{x} \biggr]
+ 
(\vec{u} \cdot \nabla)\biggl[- \vec\Omega \times \vec{x} \biggr]
</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} + (\vec{u}\cdot \nabla)\vec{v}\biggr]
- \frac{d}{dt}\biggl[\vec\Omega \times \vec{x} \biggr]
</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} + (\vec{u}\cdot \nabla)\vec{v}\biggr]
+
\vec{x}\times \frac{d\vec\Omega}{\partial t} - \vec\Omega \times \vec{v}
</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} + (\vec{u}\cdot \nabla)\vec{v}\biggr]
+
\vec{x}\times \frac{d\vec\Omega}{dt} - \vec\Omega \times \biggl[\vec{u} + \vec\Omega \times \vec{x}\biggr]
</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} + (\vec{u}\cdot \nabla)\vec{v}\biggr]
+
\vec{x}\times \frac{d\vec\Omega}{dt} 
+ \vec{u}\times \vec\Omega
+ \vec\Omega \times (\vec{x}\times \vec\Omega )
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~
\biggl[ \frac{\partial \vec{v}}{\partial t} + (\vec{u}\cdot \nabla)\vec{v}\biggr]
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\vec{u}\times \vec\Omega 
- ~\biggl[\frac{1}{\rho}\nabla P + \nabla\Phi \biggr] 
</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>-\frac{1}{\rho}\nabla P - \nabla \Phi</math>
  </td>
  <td align="center"><math>=</math>  </td>
  <td align="left">
<math>\frac{\partial\vec{v}}{\partial t} + (\vec{v}\cdot \nabla)[\vec{u} + \vec\Omega \times \vec{x}]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math>  </td>
  <td align="left">
<math>\frac{\partial\vec{v}}{\partial t} 
+ (\vec{v}\cdot \nabla)\vec{u}
+ (\vec{v}\cdot \nabla)[ \vec\Omega \times \vec{x}]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math>  </td>
  <td align="left">
<math>\frac{\partial\vec{v}}{\partial t} + (\vec{v}\cdot \nabla)\vec{u}
+ \biggl[ v_x \frac{\partial}{\partial x} + v_y \frac{\partial}{\partial y} + v_z \frac{\partial}{\partial z} \biggr]
\biggl[ 
\hat\imath (\Omega_2 z - \Omega_3 y) 
+ \hat\jmath (\Omega_3 x - \Omega_1 z) 
+ \hat{k} (\Omega_1 y - \Omega_2 x)) 
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math>  </td>
  <td align="left">
<math>\frac{\partial\vec{v}}{\partial t} + (\vec{v}\cdot \nabla)\vec{u}
+ v_x
\biggl[ 
\hat\jmath (\Omega_3 ) 
- \hat{k} (\Omega_2 ) 
\biggr]
+ v_y 
\biggl[ 
-\hat\imath (\Omega_3) 
+ \hat{k} (\Omega_1 )
\biggr]
+ v_z 
\biggl[ 
\hat\imath (\Omega_2 ) 
- \hat\jmath (\Omega_1) 
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math>  </td>
  <td align="left">
<math>\frac{\partial\vec{v}}{\partial t} + (\vec{v}\cdot \nabla)\vec{u}
+ \underbrace{\hat\imath (\Omega_2 v_z - \Omega_3 v_y)
+ \hat\jmath (\Omega_3 v_x - \Omega_1 v_z)
+\hat{k} (\Omega_1 v_y - \Omega_2 v_x)}_{\vec\Omega\times\vec{v}} \, .
</math>
  </td>
</tr>
</table>

That is, along the way we derive a,
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="center" colspan="3"><font color="#770000">'''Hybrid (Eulerian) Representation'''</font><br />of the rotating-frame Euler Equation</td>
</tr>

<tr>
  <td align="right">
<math>\frac{\partial\vec{v}}{\partial t} + (\vec{v}\cdot \nabla)\vec{u}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-~\vec\Omega \times \vec{v}
- ~\biggl[\frac{1}{\rho}\nabla P + \nabla\Phi \biggr] 
\, .
</math>
  </td>
</tr>
</table>

<table border="1"  align="center" width="80%" cellpadding="8"><tr><td align="left">

<div align="center"><font color="red"><b>CAUTION!</b></font></div>
If our interpretation of Chandrasekhar's discussion of "moving frames" is correct &#8212; see, Chap. 4, &sect;25 of  [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] &#8212; the RHS of his Eq. (18) should match the LHS of our "hybrid" equation, but it does not:  the pair of vector velocities in his advection term are swapped.  That is, based on our interpretation, the RHS of his Eq. (18) reads,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
RHS
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{\partial\vec{v}}{\partial t} + (\vec{u}\cdot \nabla)\vec{v} \, ;</math>
  </td>
</tr>
</table>
and this expression carries over to the LHS of his Eq. (19).  This is either a mistake in his presentation, or our interpretation of his presentation is incorrect.
</td></tr></table>