Editing AxisymmetricConfigurations/PGE (section)

==Spherical Coordinate Base==
Here we choose to &hellip;

<ol>
<li>Express each of the multidimensional spatial operators in spherical coordinates (<math>r, \theta, \varphi</math>)  (see, for example, the [http://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates Wikipedia discussion of vector calculus formulae in spherical coordinates]) and set to zero all spatial derivatives that are taken with respect to the angular coordinate <math>\varphi</math>:

<table align="center" border="0" cellpadding="5">
<tr>
<td colspan="3" align="center">
<font color="#770000"><b>Spatial Operators in Spherical Coordinates</b></font>
</td>
</tr>

<tr>
<td align="right">
<math>
\nabla f
</math>
</td>
<td align="center"> 
=
</td>
<td align="left">
<math> 
{\hat{e}}_r \biggl[ \frac{\partial f}{\partial r} \biggr] 
+ {\hat{e}}_\theta \biggl[ \frac{1}{r} \frac{\partial f}{\partial\theta} \biggr] 
+  {\hat{e}}_\varphi \cancel{\biggl[\frac{1}{r\sin\theta}~ \frac{\partial f}{\partial \varphi} \biggr]} ;
</math>
</td>
</tr>
<tr>
  <td align="center" colspan="3">
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 649, Eq. (1B-38)
  </td>
</tr>

<tr>
<td align="right">
<math>
\nabla^2 f 
</math>
</td>
<td align="center"> 
=
</td>
<td align="left">
<math>
\frac{1}{r^2} \frac{\partial }{\partial r} \biggl[ r^2 \frac{\partial f}{\partial r} \biggr] 
+ \frac{1}{r^2 \sin\theta} \frac{\partial }{\partial \theta}\biggl(\sin\theta \frac{\partial f}{\partial\theta}\biggr) 
+ \cancel{ \biggl[\frac{1}{r^2 \sin^2\theta} \frac{\partial^2 f}{\partial \varphi^2} \biggr]} ;
</math>
</td>
</tr>
<tr>
  <td align="center" colspan="3">
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 650, Eq. (1B-51)
  </td>
</tr>

<tr>
<td align="right">
<math>
(\vec{v}\cdot\nabla)f
</math>
</td>
<td align="center"> 
=
</td>
<td align="left">
<math> 
\biggl[ v_r \frac{\partial f}{\partial r} \biggr] 
+ \biggl[ \frac{v_\theta}{r} \frac{\partial f}{\partial\theta} \biggr] 
+ \cancel{\biggl[\frac{v_\varphi}{r\sin\theta}~ \frac{\partial f}{\partial \varphi} \biggr]} ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\nabla \cdot \vec{F} 
</math>
</td>
<td align="center"> 
=
</td>
<td align="left">
<math>
\frac{1}{r^2} \frac{\partial (r^2 F_r)}{\partial r} 
+ \frac{1}{r\sin\theta} \frac{\partial }{\partial\theta} \biggl( F_\theta \sin\theta \biggr)
+ \cancel{ \biggl[ \frac{1}{r\sin\theta}~\frac{\partial F_\varphi}{\partial \varphi} \biggr]} ;
</math>
</td>
</tr>
<tr>
  <td align="center" colspan="3">
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 650, Eq. (1B-46)
  </td>
</tr>
</table>

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

<tr>
<td align="right">
<math>
(\vec{F} \cdot \nabla )\vec{B} 
</math>
</td>
<td align="center"> 
=
</td>
<td align="left">
<math>
\hat{e}_r \biggl[ F_r \frac{\partial B_r}{\partial r} + \frac{F_\theta}{r} \frac{\partial B_r}{\partial \theta} + \cancel{ \frac{F_\varphi}{r\sin\theta}  \frac{\partial B_r}{\partial \varphi} } - \frac{(F_\theta B_\theta + F_\varphi B_\varphi)}{r}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
&nbsp;
</td>
<td align="center"> 
&nbsp;
</td>
<td align="left">
<math>
+ \hat{e}_\theta \biggl[ F_r  \frac{\partial B_\theta}{\partial r} + \frac{F_\theta}{r}  \frac{\partial B_\theta}{\partial \theta } + \cancel{ \frac{F_\varphi}{r\sin\theta}  \frac{\partial B_\theta}{\partial \varphi} }
+ \frac{F_\theta B_r}{r} - \frac{F_\varphi B_\varphi \cot\theta}{r} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
&nbsp;
</td>
<td align="center"> 
&nbsp;
</td>
<td align="left">
<math>
+ \hat{e}_\varphi \biggl[ F_r  \frac{\partial B_\varphi}{\partial r} + \frac{F_\theta}{r}  \frac{\partial B_\varphi}{\partial \theta} + \cancel{ \frac{F_\varphi}{r\sin\theta}  \frac{\partial B_\varphi}{\partial \varphi} }
+ \frac{F_\varphi B_r}{r} + \frac{F_\varphi B_\theta \cot\theta}{r}  \biggr] \, .
</math>
</td>
</tr>
<tr>
  <td align="center" colspan="3">
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 651, Eq. (1B-55)
  </td>
</tr>
</table>
From this last expression &#8212; the so-called ''convective operator'' &#8212; we conclude as well that, for axisymmetric systems,

<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}_r \biggl[ v_r \frac{\partial v_r}{\partial r} + \frac{v_\theta}{r} \frac{\partial v_r}{\partial \theta}  - \frac{(v_\theta^2 + v_\varphi^2 )}{r}\biggr]
+ \hat{e}_\theta \biggl[ v_r  \frac{\partial v_\theta}{\partial r} + \frac{v_\theta}{r}  \frac{\partial v_\theta}{\partial \theta } 
+ \frac{v_\theta v_r}{r} - \frac{v_\varphi^2 \cot\theta}{r} \biggr]
+ \hat{e}_\varphi \biggl[ v_r  \frac{\partial v_\varphi}{\partial r} + \frac{v_\theta}{r}  \frac{\partial v_\varphi}{\partial \theta} 
+ \frac{v_\varphi v_r}{r} + \frac{v_\varphi v_\theta \cot\theta}{r}  \biggr] \, .
</math>
</td>
</tr>
</table>

<li>Express all vector time-derivatives in spherical coordinates:

<table align="center" border="0" cellpadding="5">
<tr>
<td colspan="3" align="center">
<font color="#770000"><b>Vector Time-Derivatives in Spherical Coordinates</b></font>
</td>
</tr>

<tr>
<td align="right">
<math>
\frac{d}{dt}\vec{F} 
</math>
</td>
<td align="center"> 
=
</td>
<td align="left">
<math>
{\hat{e}}_r \frac{dF_r}{dt} + F_r \frac{d{\hat{e}}_r}{dt} + {\hat{e}}_\theta \frac{dF_\theta}{dt} + F_\theta \frac{d{\hat{e}}_\theta}{dt} + {\hat{e}}_\varphi \frac{dF_\varphi}{dt} + F_\varphi \frac{d{\hat{e}}_\varphi}{dt}
</math>
</td>
</tr>

<tr>
<td align="right">
&nbsp;
</td>
<td align="center"> 
=
</td>
<td align="left">
<math>
{\hat{e}}_r \frac{dF_r}{dt} + F_r \biggl[ {\hat{e}}_\theta \dot\theta + {\hat{e}}_\varphi \dot\varphi \sin\theta \biggr] 
+ {\hat{e}}_\theta \frac{dF_\theta}{dt} + F_\theta \biggl[ - {\hat{e}}_r \dot\theta + {\hat{e}}_\varphi \dot\varphi \cos\theta \biggr] 
+ {\hat{e}}_\varphi \frac{dF_\varphi}{dt} + F_\varphi \biggl[ - {\hat{e}}_r \dot\varphi \sin\theta - {\hat{e}}_\theta \dot\varphi \cos\theta \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
&nbsp;
</td>
<td align="center"> 
=
</td>
<td align="left">
<math>
{\hat{e}}_r \biggl[ \frac{dF_r}{dt} -  F_\theta \dot\theta - F_\varphi \dot\varphi \sin\theta  \biggr] 
+ {\hat{e}}_\theta \biggl[ \frac{dF_\theta}{dt} + F_r \dot\theta - F_\varphi \dot\varphi \cos\theta \biggr]  
+ {\hat{e}}_\varphi \biggl[ \frac{dF_\varphi}{dt} + F_r \dot\varphi \sin\theta + F_\theta \dot\varphi \cos\theta \biggr] ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\vec{v} = \frac{d\vec{x}}{dt} 
</math>
</td>
<td align="center"> 
=
</td>
<td align="left">
<math>
\frac{d}{dt}\biggl[ \hat{e}_r r  \biggr]
= {\hat{e}}_r \dot{r} 
+ {\hat{e}}_\theta~ r \dot\theta  
+ {\hat{e}}_\varphi ~r \sin\theta ~ \dot\varphi  .
</math>
</td>
</tr>
<tr>
  <td align="center" colspan="3">
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 648, Eq. (1B-30)
  </td>
</tr>
</table>

</ol>
===Governing Equations (SPH.)===

Introducing the above expressions into the [[PGE|principal governing equations]] gives,

<div align="center">
<span id="Continuity"><font color="#770000">'''Equation of Continuity'''</font></span><br />

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

<tr>
  <td align="right">
<math>~\frac{d\rho}{dt} + \rho \biggl[ \frac{1}{r^2} \frac{\partial (r^2 \dot{r})}{\partial r} 
+ \frac{1}{r\sin\theta} \frac{\partial }{\partial\theta} \biggl( \dot\theta r \sin\theta \biggr)
 \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0</math>
  </td>
</tr>
</table>


<span id="PGE:Euler">
<font color="#770000">'''Euler Equation'''</font>
</span><br />

<table border="0" cellpadding="5" align="center">
<!--
<tr>
  <td align="right">
<math>~\frac{d\vec{v}}{dt}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{1}{\rho} \nabla P - \nabla\Phi</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~
{\hat{e}}_r \biggl[ \frac{dv_r}{dt} -  v_\theta \dot\theta - v_\varphi \dot\varphi \sin\theta  \biggr] 
+ {\hat{e}}_\theta \biggl[ \frac{dv_\theta}{dt} + v_r \dot\theta - v_\varphi \dot\varphi \cos\theta \biggr]  
+ {\hat{e}}_\varphi \biggl[ \frac{dv_\varphi}{dt} + v_r \dot\varphi \sin\theta + v_\theta \dot\varphi \cos\theta \biggr] 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- {\hat{e}}_r \biggl[ \frac{1}{\rho} \frac{\partial P}{\partial r}+  \frac{\partial \Phi }{\partial r} \biggr] 
- {\hat{e}}_\theta \biggl[ \frac{1}{\rho r}  \frac{\partial P}{\partial\theta} +  \frac{1}{r} \frac{\partial \Phi}{\partial\theta}  \biggr]  
</math>
  </td>
</tr>
-->

<tr>
  <td align="right">
<math>~
{\hat{e}}_r \biggl[ \frac{d\dot{r}}{dt} -  r {\dot\theta}^2 - r {\dot\varphi}^2 \sin^2\theta  \biggr] 
+ {\hat{e}}_\theta \biggl[ \frac{d(r\dot\theta)}{dt} + \dot{r} \dot\theta - r { \dot\varphi }^2 \sin\theta \cos\theta \biggr]  
+ {\hat{e}}_\varphi \biggl[ \frac{d(r \sin\theta \dot\varphi)}{dt} + \dot{r} \dot\varphi \sin\theta + r \dot\theta \dot\varphi \cos\theta \biggr] 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- {\hat{e}}_r \biggl[ \frac{1}{\rho} \frac{\partial P}{\partial r}+  \frac{\partial \Phi }{\partial r} \biggr] 
- {\hat{e}}_\theta \biggl[ \frac{1}{\rho r}  \frac{\partial P}{\partial\theta} +  \frac{1}{r} \frac{\partial \Phi}{\partial\theta}  \biggr]  
</math>
  </td>
</tr>
</table>


<span id="PGE:AdiabaticFirstLaw">Adiabatic Form of the<br />
<font color="#770000">'''First Law of Thermodynamics'''</font></span><br />

{{Math/EQ_FirstLaw02}}


<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />

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

<tr>
  <td align="right">
<math>~
\frac{1}{r^2} \frac{\partial }{\partial r} \biggl[ r^2 \frac{\partial \Phi }{\partial r} \biggr] 
+ \frac{1}{r^2 \sin\theta} \frac{\partial }{\partial \theta}\biggl(\sin\theta ~ \frac{\partial \Phi}{\partial\theta}\biggr) 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi G\rho</math>
  </td>
</tr>
</table>

</div>

===Conservation of Specific Angular Momentum (SPH.)===

The <math>\hat{e}_\varphi</math> component of the Euler equation leads to a statement of conservation of specific angular momentum, <math>~j</math>, as follows.  

<div align="center">

<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>~
\frac{d(r \sin\theta \dot\varphi)}{dt} + \dot{r} \dot\varphi \sin\theta + r \dot\theta \dot\varphi \cos\theta
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{r\sin\theta} \biggl[ r\sin\theta\frac{d(r \sin\theta \dot\varphi)}{dt} + r\sin\theta  \dot\varphi ( \dot{r}\sin\theta +  r\dot\theta  \cos\theta) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{r\sin\theta} \biggl[\frac{d(r^2 \sin^2\theta \dot\varphi )}{dt}  \biggr] \, .
</math>
  </td>
</tr>
</table>

<math>
\Rightarrow ~~~~~ j(r,\theta) \equiv (r\sin\theta)^2 \dot\varphi =  \mathrm{constant} ~(\mathrm{i.e.,}~\mathrm{independent~of~time})
</math><br />

</div>


<span id="RelevantSphericalComponents">So, for axisymmetric configurations, the <math>\hat{e}_r</math> and <math>\hat{e}_\theta</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}}_r</math>: &nbsp; &nbsp;</td>
  <td align="right">
<math>
 \frac{d\dot{r}}{dt} -  r {\dot\theta}^2 - \biggl[ \frac{j^2}{r^3 \sin^3\theta} \biggr]\sin\theta
</math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>
- \biggl[ \frac{1}{\rho} \frac{\partial P}{\partial r}+  \frac{\partial \Phi }{\partial r} \biggr]   \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>~{\hat{e}}_\theta</math>: &nbsp; &nbsp;</td>
  <td align="right">
<math>
\frac{d(r\dot\theta)}{dt} + \dot{r} \dot\theta -  \biggl[ \frac{j^2}{r^3 \sin^3\theta} \biggr] \cos\theta
</math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>
- \biggl[ \frac{1}{\rho r}  \frac{\partial P}{\partial\theta} +  \frac{1}{r} \frac{\partial \Phi}{\partial\theta}  \biggr] \, .
</math>
  </td>
</tr>
</table>
</td></tr></table>

===Eulerian Formulation (SPH.)===

Each of the above simplified governing equations has been written in terms of Lagrangian time derivatives.  An Eulerian formulation of each equation can be obtained by replacing each Lagrangian time derivative by its Eulerian counterpart.  Specifically, for any scalar function, <math>f</math>, 


<div align="center">
<math>
\frac{df}{dt} \rightarrow \frac{\partial f}{\partial t} + (\vec{v}\cdot \nabla)f =
\frac{\partial f}{\partial t} + 
\biggl[ v_r \frac{\partial f}{\partial r} \biggr] 
+ \biggl[ \frac{v_\theta}{r} \frac{\partial f}{\partial\theta} \biggr] =
\frac{\partial f}{\partial t} + 
\biggl[ \dot{r} \frac{\partial f}{\partial r} \biggr] 
+ \biggl[ \dot\theta \frac{\partial f}{\partial\theta} \biggr] 
\, .
</math>
</div>

Making this substitution throughout the set of governing relations gives: 

<div align="center">
<span id="Continuity"><font color="#770000">'''Equation of Continuity'''</font></span><br />

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

<tr>
  <td align="right">
<math>~
\frac{\partial \rho}{\partial t}   
+ \biggl[ \frac{1}{r^2} \frac{\partial (\rho r^2 \dot{r})}{\partial r} 
+ \frac{1}{r\sin\theta} \frac{\partial }{\partial\theta} \biggl( \rho \dot\theta r \sin\theta \biggr)
 \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0</math>
  </td>
</tr>
</table>


<span id="PGE:Euler">The Two Relevant Components of the<br />
<font color="#770000">'''Euler Equation'''</font>
</span><br />

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right"><math>~{\hat{e}}_r</math>: &nbsp; &nbsp;</td>
  <td align="right">
<math>
\biggl\{ \frac{\partial \dot{r}}{\partial t} + \biggl[ \dot{r} \frac{\partial \dot{r}}{\partial r} \biggr] + \biggl[ \dot\theta \frac{\partial \dot{r}}{\partial\theta} \biggr] \biggr\}
-  r {\dot\theta}^2
</math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>
- \biggl[ \frac{1}{\rho} \frac{\partial P}{\partial r}+  \frac{\partial \Phi }{\partial r} \biggr]  + \biggl[ \frac{j^2}{r^3 \sin^2\theta} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>~{\hat{e}}_\theta</math>: &nbsp; &nbsp;</td>
  <td align="right">
<math>
r \biggl\{ \frac{\partial \dot{\theta}}{\partial t} + \biggl[ \dot{r} \frac{\partial \dot{\theta}}{\partial r} \biggr] + \biggl[ \dot\theta \frac{\partial \dot{\theta}}{\partial\theta} \biggr] \biggr\} + 2\dot{r} \dot\theta 
</math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>
- \biggl[ \frac{1}{\rho r}  \frac{\partial P}{\partial\theta} +  \frac{1}{r} \frac{\partial \Phi}{\partial\theta}  \biggr] +  \biggl[ \frac{j^2}{r^3 \sin^3\theta} \biggr] \cos\theta
</math>
  </td>
</tr>
</table>

<span id="PGE:AdiabaticFirstLaw">Adiabatic Form of the<br />
<font color="#770000">'''First Law of Thermodynamics'''</font></span><br />

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

<tr>
  <td align="right">
<math>~
\biggl\{ \frac{\partial \epsilon}{\partial t} + \biggl[ \dot{r} \frac{\partial \epsilon}{\partial r} \biggr] + \biggl[ \dot\theta \frac{\partial \epsilon}{\partial\theta} \biggr]  \biggr\}
+ P\biggl\{ \frac{\partial }{\partial t} \biggl( \frac{1}{\rho}\biggr) 
+ \biggl[ \dot{r} \frac{\partial }{\partial r} \biggl( \frac{1}{\rho}\biggr) \biggr] 
+ \biggl[ \dot\theta \frac{\partial }{\partial\theta} \biggl( \frac{1}{\rho}\biggr) \biggr]  \biggr\}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0</math>
  </td>
</tr>
</table>


<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />

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

<tr>
  <td align="right">
<math>~
\frac{1}{r^2} \frac{\partial }{\partial r} \biggl[ r^2 \frac{\partial \Phi }{\partial r} \biggr] 
+ \frac{1}{r^2 \sin\theta} \frac{\partial }{\partial \theta}\biggl(\sin\theta ~ \frac{\partial \Phi}{\partial\theta}\biggr) 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi G\rho</math>
  </td>
</tr>
</table>

</div>