Editing ParabolicDensity/Axisymmetric/Structure/Try8thru10 (section)

===6<sup>th</sup> Try===

====Euler Equation====

From, for example, [[PGE/Euler#in_terms_of_velocity:_2|here]] we can write the,

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

{{Template:Math/EQ_Euler02}}
</div>
In  steady-state, we should set <math>\partial\vec{v}/\partial t = 0</math>.  There are various ways of expressing the nonlinear term on the LHS; from [[PGE/Euler#in_terms_of_the_vorticity:|here]], for example, we find,
<div align="center">
<math>
(\vec{v}\cdot\nabla)\vec{v} = \frac{1}{2}\nabla(\vec{v}\cdot\vec{v}) - \vec{v}\times(\nabla\times\vec{v})
= \frac{1}{2}\nabla(v^2) + \vec{\zeta}\times \vec{v} ,
</math>
</div>
where,
<div align="center">
<math>
\vec\zeta \equiv \nabla\times\vec{v}
</math>
</div>
is commonly referred to as the [https://en.wikipedia.org/wiki/Vorticity vorticity].

====Axisymmetric Configurations====

From, for example, [[AxisymmetricConfigurations/PGE#CYLconvectiveOperator|here]], we appreciate that, quite generally, for axisymmetric systems when written in cylindrical coordinates,

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

<tr>
<td align="right">
<math>
(\vec{v} \cdot \nabla )\vec{v} 
</math>
</td>
<td align="center"> 
=
</td>
<td align="left">
<math>
\hat{e}_\varpi \biggl[ v_\varpi \frac{\partial v_\varpi}{\partial\varpi} + v_z \frac{\partial v_\varpi}{\partial z} - \frac{v_\varphi v_\varphi}{\varpi}  \biggr]
+ \hat{e}_\varphi \biggl[ v_\varpi \frac{\partial v_\varphi}{\partial \varpi}  + v_z \frac{\partial v_\varphi}{\partial z} + \frac{v_\varphi v_\varpi}{\varpi}  \biggr]
+ \hat{e}_z \biggl[ v_\varpi \frac{\partial v_z}{\partial\varpi}  + v_z \frac{\partial v_z}{\partial z} \biggr] \, .
</math>
</td>
</tr>
</table>
We seek steady-state configurations for which <math>v_\varpi =0</math>  and <math>v_z = 0</math>, in which case this expression simplifies considerably to,

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

<tr>
  <td align="right">
<math>
(\vec{v} \cdot \nabla )\vec{v} 
</math>
  </td>
  <td align="center"> 
<math>=</math>
  </td>
  <td align="left">
<math>
\hat{e}_\varpi \biggl[ - \frac{v_\varphi v_\varphi}{\varpi}  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"> 
<math>=</math>
  </td>
  <td align="left">
<math>
\hat{e}_\varpi \biggl[ - \frac{j^2}{\varpi^3}  \biggr]
\, ,
</math>
  </td>
</tr>
</table>
where, in this last expression we have replaced <math>v_\varphi</math> with the specific angular momentum, <math>j \equiv \varpi v_\varphi = (\varpi^2 \dot\varphi)</math>, which is a [[AxisymmetricConfigurations/PGE#Conservation_of_Specific_Angular_Momentum_(CYL.)|conserved quantity in dynamically evolving systems]].  NOTE: &nbsp; Up to this point in our discussion, <math>j</math> can be a function of both coordinates, that is, <math>j = j(\varpi, z)</math>.

As has been highlighted [[AxisymmetricConfigurations/PGE#RelevantCylindricalComponents|here]] for example &#8212; for the axisymmetric configurations under consideration &#8212; the <math>\hat{e}_\varpi</math> and <math>\hat{e}_z</math> components of the Euler equation become, respectively,</span>
<table border="1" align="center" cellpadding="10"><tr><td align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right"><math>{\hat{e}}_\varpi</math>: &nbsp; &nbsp;</td>
  <td align="right">
<math>
- \frac{j^2}{\varpi^3}  
</math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>
- \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial\varpi} + \frac{\partial \Phi}{\partial\varpi}\biggr]  
</math>
  </td>
</tr>
<tr>
  <td align="right"><math>{\hat{e}}_z</math>: &nbsp; &nbsp;</td>
  <td align="right">
<math>
0 
</math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>
- \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial z} + \frac{\partial \Phi}{\partial z} \biggr] 
</math>
  </td>
</tr>
</table>
</td></tr></table>