View source for Cylindrical3D/Linearization

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=Linearized Equations in Cylindrical Coordinates=

==Eulerian Formulation of Nonlinear Governing Equations==

From our more detailed, [[Cylindrical3D#Eulerian_Formulation|accompanying discussion]] we pull the Eulerian representation of the set of principal governing equations written in cylindrical coordinates.


<div align="center">
<span id="PGE:Euler:R">
<font color="#770000">'''<math>\varpi</math> Component of Euler Equation'''</font>
</span><br />

<math>
\frac{\partial \dot\varpi}{\partial t} + \biggl[ \dot\varpi \frac{\partial \dot\varpi}{\partial\varpi} \biggr] + 
\biggl[ \dot\varphi \frac{\partial \dot\varpi}{\partial\varphi} \biggr] +
\biggl[ \dot{z} \frac{\partial \dot\varpi}{\partial z} \biggr] -  \varpi {\dot\varphi}^2   = 
- \frac{1}{\rho}\frac{\partial P}{\partial\varpi} - \frac{\partial \Phi}{\partial\varpi}
</math><br />



<span id="PGE:Euler:Phi">
<font color="#770000">'''<math>\varphi</math> Component of Euler Equation'''</font>
</span><br />


<math>
\frac{\partial (\varpi\dot\varphi)}{\partial t} + \biggl[ \dot\varpi \frac{\partial (\varpi\dot\varphi)}{\partial\varpi} \biggr] + 
\biggl[ \dot\varphi \frac{\partial (\varpi\dot\varphi)}{\partial\varphi} \biggr] +
\biggl[ \dot{z} \frac{\partial (\varpi\dot\varphi)}{\partial z} \biggr] + \dot\varpi \dot\varphi = 
-  \frac{1}{\varpi} \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial \varphi} + \frac{\partial \Phi}{\partial \varphi} \biggr] 
</math><br />



<span id="PGE:Euler:Z">
<font color="#770000">'''<math>z</math> Component of Euler Equation'''</font>
</span><br />

<math>
\frac{\partial \dot{z}}{\partial t} + \biggl[ \dot\varpi \frac{\partial \dot{z}}{\partial\varpi} \biggr] 
+ \biggl[ \dot\varphi \frac{\partial \dot{z}}{\partial\varphi} \biggr] +\biggl[ \dot{z} \frac{\partial \dot{z}}{\partial z} \biggr] = 
- \frac{1}{\rho}\frac{\partial P}{\partial z} - \frac{\partial \Phi}{\partial z} 
</math><br />



</div>

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

<math>
\frac{\partial\rho}{\partial t} + \frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \rho \varpi \dot\varpi \biggr] 
+ \frac{1}{\varpi} \frac{\partial}{\partial \varphi} \biggl[ \rho \varpi \dot\varphi \biggr]
+ \frac{\partial}{\partial z} \biggl[ \rho \dot{z} \biggr] = 0 
</math><br />
</div>

These match, for example, equations (3.1) - (3.4)  of [http://adsabs.harvard.edu/abs/1984MNRAS.208..721P Papaloizou &amp; Pringle] (1984, MNRAS, 208, 721-750), hereafter, PPI.

==Linearization==

If we assume that the initial equilibrium configuration is axisymmetric with no radial or vertical velocity, the linearized equations become:

===Linearizing Radial Component of Euler Equation===
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial {\dot\varpi}^'}{\partial t} + 
\biggl[ {\dot\varphi}_0 \frac{\partial {\dot\varpi}^'}{\partial\varphi} \biggr] 
-  \varpi ( { {\dot\varphi}_0 + {\dot\varphi}^'})^2 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{1}{(\rho_0 + \rho^')}\frac{\partial (P_0 + P^')}{\partial\varpi} - \frac{\partial (\Phi_0+\Phi^')}{\partial\varpi}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~
\frac{\partial {\dot\varpi}^'}{\partial t} + 
\biggl[ {\dot\varphi}_0 \frac{\partial {\dot\varpi}^'}{\partial\varphi} \biggr] 
-  \varpi ( {\dot\varphi}_0)^2 -  2\varpi ( {\dot\varphi}_0 {\dot\varphi}^')</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{1}{\rho_0}\frac{\partial P^'}{\partial\varpi} 
- \biggl[\frac{1}{\rho_0}\frac{\partial P_0 }{\partial\varpi}\biggr]\biggl(1 - \frac{\rho^'}{\rho_0}  \biggr) 
- \frac{\partial (\Phi_0+\Phi^')}{\partial\varpi}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~
\frac{\partial {\dot\varpi}^'}{\partial t} + 
{\dot\varphi}_0 \frac{\partial {\dot\varpi}^'}{\partial\varphi} 
-  2\varpi ( {\dot\varphi}_0 {\dot\varphi}^')
+ \biggl[ \frac{1}{\rho_0}\frac{\partial P^'}{\partial\varpi}- \frac{\rho^'}{\rho_0^2}\frac{\partial P_0 }{\partial\varpi}\biggr] 
+ \frac{\partial \Phi^'}{\partial \varpi}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{  \varpi ( {\dot\varphi}_0)^2 
- \frac{1}{\rho_0}\frac{\partial P_0 }{\partial\varpi} 
- \frac{\partial \Phi_0}{\partial\varpi} \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~
\frac{\partial {\dot\varpi}^'}{\partial t} + 
{\dot\varphi}_0 \frac{\partial {\dot\varpi}^'}{\partial\varphi} 
-  2\varpi ( {\dot\varphi}_0 {\dot\varphi}^')
+ \biggl[ \frac{\partial}{\partial\varpi}\biggl( \frac{P^'}{\rho_0} \biggr) \biggr] + \frac{\partial \Phi^'}{\partial \varpi}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, .
</math>
  </td>
</tr>
</table>
</div>

This last expression has been obtained by recognizing that, in the next-to-last expression: (1) The terms inside the curly braces on the right-hand side collectively provide a statement of equilibrium (in the radial-coordinate direction) in the initial, unperturbed configuration and therefore the terms sum to zero; and (2) the terms inside square brackets on the left-hand side can be rewritten in a more compact form because we have adopted a polytropic equation of state to build the unperturbed initial equilibrium configuration and are examining only adiabatic perturbations with <math>~\gamma = (n+1)/n</math>, in which case,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\nabla P_0}{P_0} = \frac{(n+1)}{n} \cdot \frac{\nabla \rho_0}{\rho_0} \, ,</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\frac{P^'}{P_0} = \frac{\gamma \rho^'}{\rho_0} \, .</math>
  </td>
</tr>
</table>
</div>


===Linearizing Azimuthal Component of Euler Equation===
Keeping in mind that the initial equilibrium configuration is axisymmetric &#8212; that is, equilibrium parameters exhibit no variation in the azimuthal direction &#8212; and, in addition, <math>~\dot\varphi_0</math> exhibits no variation in the vertical direction, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial (\varpi {\dot\varphi}^')}{\partial t} + ( {\dot\varpi}^') \frac{\partial (\varpi\dot\varphi_0)}{\partial\varpi}  + 
( \dot\varphi_0)\frac{\partial (\varpi{\dot\varphi}^')}{\partial\varphi} +
( {\dot\varpi}^') {\dot\varphi_0} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-  \frac{1}{\varpi} \biggl[ \frac{1}{\rho_0}\frac{\partial P^'}{\partial \varphi} + \frac{\partial \Phi^'}{\partial \varphi} \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~\frac{\partial (\varpi {\dot\varphi}^')}{\partial t} + 
( \dot\varphi_0)\frac{\partial (\varpi{\dot\varphi}^')}{\partial\varphi} +
\frac{{\dot\varpi}^'}{\varpi}\biggl[ \frac{\partial (\varpi^2\dot\varphi_0)}{\partial\varpi} \biggr] 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-  \frac{1}{\varpi} \biggl[ \frac{\partial }{\partial \varphi} \biggl(\frac{P^'}{\rho_0}\biggr)+ \frac{\partial \Phi^'}{\partial \varphi} \biggr] 
\, .</math>
  </td>
</tr>

</table>
</div>


===Linearizing Vertical Component of Euler Equation===

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

<tr>
  <td align="right">
<math>~
\frac{\partial {\dot{z}}^'}{\partial t} 
+ (\dot\varphi_0) \frac{\partial {\dot{z}}^'}{\partial\varphi}  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{1}{(\rho_0 + \rho^')}\frac{\partial (P_0 + P^')}{\partial z} - \frac{\partial (\Phi_0+\Phi^')}{\partial z}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{1}{\rho_0}\frac{\partial P^'}{\partial z} 
- \biggl[\frac{1}{\rho_0}\frac{\partial P_0 }{\partial z}\biggr]\biggl(1 - \frac{\rho^'}{\rho_0}  \biggr) 
- \frac{\partial (\Phi_0+\Phi^')}{\partial z}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~
\frac{\partial {\dot{z}}^'}{\partial t} 
+ (\dot\varphi_0) \frac{\partial {\dot{z}}^'}{\partial\varphi}  
+ \biggl[ \frac{1}{\rho_0}\frac{\partial P^'}{\partial z}- \frac{\rho^'}{\rho_0^2}\frac{\partial P_0 }{\partial z}\biggr] 
+ \frac{\partial \Phi^'}{\partial z}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{ 
- \frac{1}{\rho_0}\frac{\partial P_0 }{\partial z} 
- \frac{\partial \Phi_0}{\partial z} \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~
\frac{\partial {\dot{z}}^'}{\partial t} 
+ (\dot\varphi_0) \frac{\partial {\dot{z}}^'}{\partial\varphi}  
+ \biggl[ \frac{\partial}{\partial z}\biggl( \frac{P^'}{\rho_0} \biggr) \biggr]  
+ \frac{\partial \Phi^'}{\partial z}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, ,
</math>
  </td>
</tr>
</table>
</div>
where the logic followed in deriving the last expression from the next-to-last one is directly analogous to [[#Linearizing_Radial_Component_of_Euler_Equation|the logic used, above]], in obtaining the final expression for the radial component of the linearized Euler equation.

===Linearizing Continuity Equation===

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

<tr>
  <td align="right">
<math>~\frac{\partial\rho^'}{\partial t} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \rho_0 \varpi {\dot\varpi}^' \biggr] 
- \frac{1}{\varpi} \frac{\partial}{\partial \varphi} \biggl[ \rho_0 \varpi {\dot\varphi}^' + \rho^' \varpi {\dot\varphi}_0 \biggr]
- \frac{\partial}{\partial z} \biggl[ \rho_0 {\dot{z}}^' \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~\frac{\partial\rho^'}{\partial t} + ( {\dot\varphi}_0 )\frac{\partial \rho^'}{\partial \varphi} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \rho_0 \varpi {\dot\varpi}^' \biggr] 
- \frac{1}{\varpi} \frac{\partial }{\partial \varphi} \biggl[ \rho_0 \varpi {\dot\varphi}^' \biggr]
- \frac{\partial}{\partial z} \biggl[ \rho_0 {\dot{z}}^' \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

===Summary===
<div align="center">
<table border="1" cellpadding="5" align="center">
<tr>
  <th align="center">
Set of Linearized Principal Governing Equations in Cylindrical Coordinates
  </th>
</tr>
<tr><td align="center">

<table border="0" cellpadding="8" align="center">

<tr><td align="center" colspan="3"><font color="#770000">'''Continuity Equation'''</font></td></tr>
<tr>
  <td align="right">
<math>~\frac{\partial\rho^'}{\partial t} + ( {\dot\varphi}_0 )\frac{\partial \rho^'}{\partial \varphi} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \rho_0 \varpi {\dot\varpi}^' \biggr] 
- \frac{1}{\varpi} \frac{\partial }{\partial \varphi} \biggl[ \rho_0 \varpi {\dot\varphi}^' \biggr]
- \frac{\partial}{\partial z} \biggl[ \rho_0 {\dot{z}}^' \biggr] \, .
</math>
  </td>
</tr>

<tr><td align="center" colspan="3"><font color="#770000">'''<math>\varpi</math> Component of Euler Equation'''</font></td></tr>
<tr>
  <td align="right">
<math>~
\frac{\partial {\dot\varpi}^'}{\partial t} + 
( {\dot\varphi}_0 ) \frac{\partial {\dot\varpi}^'}{\partial\varphi} 
-  2\varpi ( {\dot\varphi}_0 {\dot\varphi}^')
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{\partial}{\partial\varpi}\biggl( \frac{P^'}{\rho_0} \biggr)  - \frac{\partial \Phi^'}{\partial \varpi}
</math>
  </td>
</tr>

<tr><td align="center" colspan="3"><font color="#770000">'''<math>\varphi</math> Component of Euler Equation'''</font></td></tr>
<tr>
  <td align="right">
<math>~\frac{\partial (\varpi {\dot\varphi}^')}{\partial t} + 
( \dot\varphi_0)\frac{\partial (\varpi{\dot\varphi}^')}{\partial\varphi} +
\frac{{\dot\varpi}^'}{\varpi}\biggl[ \frac{\partial (\varpi^2\dot\varphi_0)}{\partial\varpi} \biggr] 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-  \frac{1}{\varpi} \biggl[ \frac{\partial }{\partial \varphi} \biggl(\frac{P^'}{\rho_0}\biggr)+ \frac{\partial \Phi^'}{\partial \varphi} \biggr] 
</math>
  </td>
</tr>

<tr><td align="center" colspan="3"><font color="#770000">'''<math>~z</math> Component of Euler Equation'''</font></td></tr>
<tr>
  <td align="right">
<math>~
\frac{\partial {\dot{z}}^'}{\partial t} 
+ (\dot\varphi_0) \frac{\partial {\dot{z}}^'}{\partial\varphi}  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{\partial}{\partial z}\biggl( \frac{P^'}{\rho_0} \biggr) 
- \frac{\partial \Phi^'}{\partial z}
</math>
  </td>
</tr>

<tr><td align="center" colspan="3"><font color="#770000">'''Adiabatic Form of the 1<sup>st</sup> Law of Thermodynamics'''</font></td></tr>
<tr>
  <td align="right">
<math>~\frac{P^'}{P_0}</math>
  </td>
  <td align="center">
<math>~=</math> 
  </td>
  <td align="left">
<math>~ \frac{\gamma \rho^'}{\rho_0} </math>
  </td>
</tr>

<tr><td align="center" colspan="3"><font color="#770000">'''Poisson Equation'''</font></td></tr>
<tr>
  <td align="right">
<math>~\nabla^2 \Phi^'
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
4\pi G\rho^'
</math>
  </td>
</tr>
</table>

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

=See Also=


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